{ "schema_version": "ocme-catalogue-localization-v0.1", "locale": "en", "translation_state": "ai_candidate", "model_claim": "unresolved", "presentation_only": true, "source": { "commit": "518a2b1a20af9e9caee5f8847f6dec3dd80a02bb", "files": { "public/data/index.json": "e7c83469cba9ba79e50636f5d5904f0a6e7bc2ce86faccc67abd734c44277755", "public/data/atlas/core-atlas.json": "c017b9147625ed0af747001ed4e2a3c84e886dd2f51a3afda752d6d789d34a7c", "public/data/atlas/extended-atlas.json": "3a1a1d3bfed1bed3792ae701119a4d2e6d36b2adfd12df07f5822fe5dd6d04b2", "public/data/architecture/domains.json": "b7b0d3d78c41e1952baeaf527eedb1e53e62afb3d3b91f7d7c25b2fd80721868", "public/data/architecture/methods.json": "498589f2a1f5ac88890b7be66d7dd8b39c4c5d02537f98bd974878dacf3edfba" } }, "provenance": { "created_at": "2026-10-17", "authority": "presentation_only" }, "atlas": { "atlas-natural-number": { "source_snapshot": { "id": "atlas-natural-number", "title_zh": "自然數", "summary_zh": "用於計數與離散次序的基本數系。", "canonical_mko_id": "mko-natural-number" }, "fields": { "title": "Natural number", "summary": "The basic number system for counting and discrete order." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-natural-number", "edition_id": "edition-mko-natural-number-en-v1", "source_field": "title", "source_mko_sha256": "222b72502fb661f3372012f602fd2f1eeef754b9e2069498c6b24234778730c4", "edition_fields_sha256": "a1c985fdd2daac3b122433fdb7be7ddbdbe0a0b31622494467adad784eb31e3c" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-integer": { "source_snapshot": { "id": "atlas-integer", "title_zh": "整數", "summary_zh": "由自然數擴張得到，可表示正、負與零的數系。", "canonical_mko_id": "mko-integer" }, "fields": { "title": "Integer", "summary": "A number system extending the natural numbers to represent positive values, negative values and zero." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-integer", "edition_id": "edition-mko-integer-en-v1", "source_field": "title", "source_mko_sha256": "e1e3776eaf56e003f2eb07a66a39c7e221d26759df8fbc4db2652a2733318d66", "edition_fields_sha256": "66e022b5c423ab26b13456255068aea92ed7330a2236bc7a7843da2c91b384db" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-rational-number": { "source_snapshot": { "id": "atlas-rational-number", "title_zh": "有理數", "summary_zh": "可表示為兩個整數之比的數。", "canonical_mko_id": "mko-rational-number" }, "fields": { "title": "Rational number", "summary": "A number expressible as a ratio of two integers." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-rational-number", "edition_id": "edition-mko-rational-number-en-v1", "source_field": "title", "source_mko_sha256": "f1b52aad1867316fd94d6e07d6e94237a202d6769454898a0831d8d1bb89f23f", "edition_fields_sha256": "e460a2fdf5b23db2531164f06641fa5a2ff00441f3eaaa388fe811e84e2032d8" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-real-number": { "source_snapshot": { "id": "atlas-real-number", "title_zh": "實數", "summary_zh": "包含有理數與無理數，支撐連續量與實分析。", "canonical_mko_id": "mko-real-number" }, "fields": { "title": "Real number", "summary": "A number system containing rational and irrational numbers and supporting continuous quantities and real analysis." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-real-number", "edition_id": "edition-mko-real-number-en-v1", "source_field": "title", "source_mko_sha256": "2bc8c0a1defae2276ce50808a0761bf4d62f26892e443fcab79fc8090113d880", "edition_fields_sha256": "df3a8682a053a98e4c3516e4bd84f1961adcbf329f7c2fe79aab3f7c20770a12" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-arithmetic-operations": { "source_snapshot": { "id": "atlas-arithmetic-operations", "title_zh": "四則運算", "summary_zh": "加、減、乘、除及其基本規則。", "canonical_mko_id": "mko-arithmetic-operations" }, "fields": { "title": "Arithmetic operations", "summary": "Addition, subtraction, multiplication, division and their basic rules." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-arithmetic-operations", "edition_id": "edition-mko-arithmetic-operations-en-v1", "source_field": "title", "source_mko_sha256": "452d77028e2659b860c94200c1de159c4184dde86d6d73950e1998a0080e265a", "edition_fields_sha256": "4272d5bf3025f3b34368aedc404bba94dfe7d57262a0c55188e88f19cd03e80f" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-fraction": { "source_snapshot": { "id": "atlas-fraction", "title_zh": "分數", "summary_zh": "分數 p/q 是有理數的一種表示；分子與分母為整數，分母必須非零。", "canonical_mko_id": "mko-fraction" }, "fields": { "title": "Fraction", "summary": "A fraction p/q is a representation of a rational number whose numerator and denominator are integers and whose denominator is nonzero." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-fraction", "edition_id": "edition-mko-fraction-en-v1", "source_field": "title", "source_mko_sha256": "aa89902ad405685be331872579647cec33e7ab6792273fe60ad8fefe6129f919", "edition_fields_sha256": "8cc5dd33e60b6cdaad6202378abe183e0e5b0b6a9997718e03c63ee8a80f2976" }, "summary": { "kind": "mko_edition", "mko_id": "mko-fraction", "edition_id": "edition-mko-fraction-en-v1", "source_field": "summary", "source_mko_sha256": "aa89902ad405685be331872579647cec33e7ab6792273fe60ad8fefe6129f919", "edition_fields_sha256": "8cc5dd33e60b6cdaad6202378abe183e0e5b0b6a9997718e03c63ee8a80f2976" } } }, "atlas-ratio-proportion": { "source_snapshot": { "id": "atlas-ratio-proportion", "title_zh": "比與比例", "summary_zh": "本條目把數值比 a:b 解讀為 a/b，把比例定義為兩個合法比值相等。", "canonical_mko_id": "mko-ratio-proportion" }, "fields": { "title": "Ratio and proportion", "summary": "This entry interprets the numerical ratio a:b as a/b and defines a proportion as the equality of two valid ratios." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-ratio-proportion", "edition_id": "edition-mko-ratio-proportion-en-v1", "source_field": "title", "source_mko_sha256": "b0200b9dd1e1ceb4c2b34f37f0279a1abaf31c3021957de6cd9ea72919da47ec", "edition_fields_sha256": "986f33936b5bfeb144d108e6f9f1dc850abd7925095e68822c3a82887b9ed60c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-ratio-proportion", "edition_id": "edition-mko-ratio-proportion-en-v1", "source_field": "summary", "source_mko_sha256": "b0200b9dd1e1ceb4c2b34f37f0279a1abaf31c3021957de6cd9ea72919da47ec", "edition_fields_sha256": "986f33936b5bfeb144d108e6f9f1dc850abd7925095e68822c3a82887b9ed60c" } } }, "atlas-power-root": { "source_snapshot": { "id": "atlas-power-root", "title_zh": "自然數冪與主平方根", "summary_zh": "本輪限定實數底數的自然數冪，以及非負實數的主平方根，分開冪運算、根值與解集。", "canonical_mko_id": "mko-power-root" }, "fields": { "title": "Natural-number powers and principal square roots", "summary": "This entry covers natural-number powers of real bases and principal square roots of nonnegative real numbers, keeping powers, root values, and solution sets distinct." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-power-root", "edition_id": "edition-mko-power-root-en-v1", "source_field": "title", "source_mko_sha256": "29380d066f7612e50db0cbd16c05855d0464ed8f50858e7f348e62e899557c19", "edition_fields_sha256": "df3190c5ea5d96c4a7711513d2d437e98e240445df8dd1f0f4cab1355ee161e9" }, "summary": { "kind": "mko_edition", "mko_id": "mko-power-root", "edition_id": "edition-mko-power-root-en-v1", "source_field": "summary", "source_mko_sha256": "29380d066f7612e50db0cbd16c05855d0464ed8f50858e7f348e62e899557c19", "edition_fields_sha256": "df3190c5ea5d96c4a7711513d2d437e98e240445df8dd1f0f4cab1355ee161e9" } } }, "atlas-absolute-value": { "source_snapshot": { "id": "atlas-absolute-value", "title_zh": "絕對值", "summary_zh": "實數到零的非負距離，依正負號分段取 x 或 -x。", "canonical_mko_id": "mko-absolute-value" }, "fields": { "title": "Absolute value", "summary": "The nonnegative distance of a real number from zero, given by x or −x according to its sign." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-absolute-value", "edition_id": "edition-mko-absolute-value-en-v1", "source_field": "title", "source_mko_sha256": "7df0aed9ca41a0e67b84a464f450422348753d1171802220b41a103488fb5f95", "edition_fields_sha256": "4be05c0339ea8e55dfee05a88ea382c6cf13e8287571af093cea60431f2733fc" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-order-relation": { "source_snapshot": { "id": "atlas-order-relation", "title_zh": "實數大小與順序", "summary_zh": "在實數論域比較大小；精確有理數交叉乘積、有限極值及含端點區間是本輪切片。", "canonical_mko_id": "mko-order-relation" }, "fields": { "title": "Usual order on the real numbers", "summary": "The usual real order compares real values and supports finite nonempty extrema, closed intervals, and translation by a common addend." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-order-relation", "edition_id": "edition-mko-order-relation-en-v1", "source_field": "title", "source_mko_sha256": "95554189da3a451f8a69edd139e154fd9254524119837ea1643487a5646a2dc3", "edition_fields_sha256": "5949bf3dcffd92504cc94285b3b7d5c888a17beed66ccc9130f111386eb66866" }, "summary": { "kind": "mko_edition", "mko_id": "mko-order-relation", "edition_id": "edition-mko-order-relation-en-v1", "source_field": "summary", "source_mko_sha256": "95554189da3a451f8a69edd139e154fd9254524119837ea1643487a5646a2dc3", "edition_fields_sha256": "5949bf3dcffd92504cc94285b3b7d5c888a17beed66ccc9130f111386eb66866" } } }, "atlas-set": { "source_snapshot": { "id": "atlas-set", "title_zh": "集合", "summary_zh": "把對象視為一個整體的基本數學結構。", "canonical_mko_id": "mko-set" }, "fields": { "title": "Set", "summary": "A basic mathematical structure treating objects as a collection." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-set", "edition_id": "edition-mko-set-en-v1", "source_field": "title", "source_mko_sha256": "4f736510823c29158d78264f57fa9f96c5c2107ddc99999622656fdacc7a5728", "edition_fields_sha256": "5e96bc501c5c49cfecbfb514eea17c7da86b29b8aaf1f84e2880da66cd1f9a7e" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-set-membership": { "source_snapshot": { "id": "atlas-set-membership", "title_zh": "集合隸屬", "summary_zh": "描述元素是否屬於某個集合。", "canonical_mko_id": "mko-set-membership" }, "fields": { "title": "Set membership", "summary": "The relation describing whether an element belongs to a set." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-set-membership", "edition_id": "edition-mko-set-membership-en-v1", "source_field": "title", "source_mko_sha256": "2a438aa79c23e6d37869e1e63eb48420bcf2f114f44b58fe5bea386076b4f0db", "edition_fields_sha256": "23b5cd56068e4daf002405f26b7f536a2a57a4857bda84c9f40c7d59ec19bfd4" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-subset": { "source_snapshot": { "id": "atlas-subset", "title_zh": "子集合", "summary_zh": "描述一個集合的每個元素都屬於另一集合。", "canonical_mko_id": "mko-subset" }, "fields": { "title": "Subset", "summary": "The relation that every element of one set also belongs to another set." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-subset", "edition_id": "edition-mko-subset-en-v1", "source_field": "title", "source_mko_sha256": "984a77e67dc9322e36b869843513f1cb76f9527912202782848120c123ce99b3", "edition_fields_sha256": "a78cbf64f04cb5ad62ffec5df5217b99464d6735b84dc941efa108fc9c80a1ed" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-set-operations": { "source_snapshot": { "id": "atlas-set-operations", "title_zh": "集合運算", "summary_zh": "聯集、交集、差集與相對補集等集合操作。", "canonical_mko_id": "mko-set-operations" }, "fields": { "title": "Set operations", "summary": "Operations on sets, including union, intersection, difference and relative complement." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-set-operations", "edition_id": "edition-mko-set-operations-en-v1", "source_field": "title", "source_mko_sha256": "e857b27da045df1a33d58d7cd56b319d6c8ec23162a8cfc3c15f7c70d30f317b", "edition_fields_sha256": "b75890da49ea69660f918ba5695e78aabf815d2c0acde62db4b5f60066f16526" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-proposition": { "source_snapshot": { "id": "atlas-proposition", "title_zh": "命題", "summary_zh": "具有真值的數學陳述。", "canonical_mko_id": "mko-proposition" }, "fields": { "title": "Proposition", "summary": "A mathematical statement with a truth value." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-proposition", "edition_id": "edition-mko-proposition-en-v1", "source_field": "title", "source_mko_sha256": "77a5e0d8403fa47d1ee57431b11f6fc52bfdde80a575db9dac3589fb0b9e8510", "edition_fields_sha256": "2598f50588150ae792ada3fc83125c760f390209a77f452624a299de34dd4a7b" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-logical-connectives": { "source_snapshot": { "id": "atlas-logical-connectives", "title_zh": "邏輯連接詞", "summary_zh": "否定、且、或、蘊涵與等價等命題組合方式。", "canonical_mko_id": "mko-logical-connectives" }, "fields": { "title": "Logical connectives", "summary": "Ways to combine propositions, including negation, conjunction, disjunction, implication and equivalence." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-logical-connectives", "edition_id": "edition-mko-logical-connectives-en-v1", "source_field": "title", "source_mko_sha256": "d102be1d207477db8f00af868ed046578922ba9eeef220b0f6049bfca3964270", "edition_fields_sha256": "2a0d30eeb8920f0aed2c042e964645db638bb94fe6d21cedc8026cf21c133e3b" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-quantifier": { "source_snapshot": { "id": "atlas-quantifier", "title_zh": "量詞與明示論域", "summary_zh": "全稱與存在量詞必須綁定論域；古典真值語義下，空論域的全稱為真、存在為假。", "canonical_mko_id": "mko-quantifier" }, "fields": { "title": "Quantifiers and explicit domains", "summary": "Universal and existential quantifiers must bind a domain. Under classical truth semantics, a universal statement over the empty domain is true and an existential statement is false." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-quantifier", "edition_id": "edition-mko-quantifier-en-v1", "source_field": "title", "source_mko_sha256": "3b2f73bd57aa43b5441edef2b27d44470f8cbf04a229fb31e98243708e818341", "edition_fields_sha256": "f3b5c72ed5213216cf0cc4b81126ff09857384e0756eacc6903fc953c413d186" }, "summary": { "kind": "mko_edition", "mko_id": "mko-quantifier", "edition_id": "edition-mko-quantifier-en-v1", "source_field": "summary", "source_mko_sha256": "3b2f73bd57aa43b5441edef2b27d44470f8cbf04a229fb31e98243708e818341", "edition_fields_sha256": "f3b5c72ed5213216cf0cc4b81126ff09857384e0756eacc6903fc953c413d186" } } }, "atlas-proof": { "source_snapshot": { "id": "atlas-proof", "title_zh": "數學證明", "summary_zh": "由明示前提與合法推理建立結論的過程。", "canonical_mko_id": "mko-proof" }, "fields": { "title": "Mathematical proof", "summary": "A process establishing a conclusion from explicit premises through valid reasoning." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-proof", "edition_id": "edition-mko-proof-en-v1", "source_field": "title", "source_mko_sha256": "72f6009b912caa3c05a2948a97ca422367574a19ad5fc665964a540875417b91", "edition_fields_sha256": "bbbbb0652e419ed2f061a3092971b6a7e397c87d092510dd40cd02d7390b679e" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-counterexample": { "source_snapshot": { "id": "atlas-counterexample", "title_zh": "反例與論域資格", "summary_zh": "反例必須屬於被聲稱的論域且使全稱述詞失敗；域外值和找不到反例都不能自動證明原命題。", "canonical_mko_id": "mko-counterexample" }, "fields": { "title": "Counterexamples and domain eligibility", "summary": "A counterexample must belong to the claimed domain and falsify the universal predicate. An outside value or failure to find a counterexample does not by itself prove the original statement." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-counterexample", "edition_id": "edition-mko-counterexample-en-v1", "source_field": "title", "source_mko_sha256": "c3528c0ad339526d31db8324f3a6cbcdc28e756c638ba57e2792a24d8cc5beef", "edition_fields_sha256": "ff728389aaecf5f21286c46340a179d2115e000393729a7ec945647f89ed51b8" }, "summary": { "kind": "mko_edition", "mko_id": "mko-counterexample", "edition_id": "edition-mko-counterexample-en-v1", "source_field": "summary", "source_mko_sha256": "c3528c0ad339526d31db8324f3a6cbcdc28e756c638ba57e2792a24d8cc5beef", "edition_fields_sha256": "ff728389aaecf5f21286c46340a179d2115e000393729a7ec945647f89ed51b8" } } }, "atlas-relation": { "source_snapshot": { "id": "atlas-relation", "title_zh": "二元關係與集合投影", "summary_zh": "二元關係是指定 A×B 中有序對的集合，不要求每個輸入唯一或必有輸出。", "canonical_mko_id": "mko-relation" }, "fields": { "title": "Binary relations and set projections", "summary": "A binary relation is a set of ordered pairs in a specified A×B. An input need not have a unique output or any output." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-relation", "edition_id": "edition-mko-relation-en-v1", "source_field": "title", "source_mko_sha256": "3788215401bc9a5610691985883d04b506880cc9bd5e2ce0a98054cd5a5f0774", "edition_fields_sha256": "aa3b7e92c01181e2cd3b2ce510ed1591401c1bf99137db88489b2d35d1115a34" }, "summary": { "kind": "mko_edition", "mko_id": "mko-relation", "edition_id": "edition-mko-relation-en-v1", "source_field": "summary", "source_mko_sha256": "3788215401bc9a5610691985883d04b506880cc9bd5e2ce0a98054cd5a5f0774", "edition_fields_sha256": "aa3b7e92c01181e2cd3b2ce510ed1591401c1bf99137db88489b2d35d1115a34" } } }, "atlas-variable-expression": { "source_snapshot": { "id": "atlas-variable-expression", "title_zh": "變數與代數式", "summary_zh": "以符號表示未知量與運算結構。", "canonical_mko_id": "mko-variable-expression" }, "fields": { "title": "Variables and algebraic expressions", "summary": "Symbols representing unknown quantities and the structure of operations." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-variable-expression", "edition_id": "edition-mko-variable-expression-en-v1", "source_field": "title", "source_mko_sha256": "da4b6e4a4edfd379a38b6535fd3030cc5c552e2b549572c0d4733d68974a292c", "edition_fields_sha256": "106f062cb5072d8cd5bbdae7bcf17ec434154358a8f1c6fe464bf8ebaaee81a7" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-equation": { "source_snapshot": { "id": "atlas-equation", "title_zh": "方程式", "summary_zh": "要求兩個代數表達式相等的條件。", "canonical_mko_id": "mko-equation" }, "fields": { "title": "Equation", "summary": "A condition requiring two algebraic expressions to be equal." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-equation", "edition_id": "edition-mko-equation-en-v1", "source_field": "title", "source_mko_sha256": "6f8d6ca379da35727c14f320d99d0bc8c53c47ad478d0f462d30a4127aea0a78", "edition_fields_sha256": "8dc153a2e58c90d1fe542c903f69d9dd05a2342395ab94f1ecf414d6351aab19" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-linear-equation": { "source_snapshot": { "id": "atlas-linear-equation", "title_zh": "一元一次方程式", "summary_zh": "在明示數系中，具有一個未知數與非零一次係數的方程式。", "canonical_mko_id": "mko-linear-equation" }, "fields": { "title": "Linear equation in one variable", "summary": "An equation over an explicit number system with one unknown and a nonzero linear coefficient." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-equation", "edition_id": "edition-mko-linear-equation-en-v1", "source_field": "title", "source_mko_sha256": "70cc85ab6a989077a196caad95d1800fef7d3e5988f0a9cdc5c0c700600ef109", "edition_fields_sha256": "7e619a2ba9801d05b2baff1c4885e3468fe2ff2f006336a86807ce4ba1c5d444" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-quadratic-equation": { "source_snapshot": { "id": "atlas-quadratic-equation", "title_zh": "一元實二次方程式", "summary_zh": "對非零二次係數的實方程，用判別式區分實根數，將方程式、根集合與重根分開。", "canonical_mko_id": "mko-quadratic-equation" }, "fields": { "title": "Quadratic equations over the reals", "summary": "For a real equation with a nonzero quadratic coefficient, the discriminant classifies its distinct real roots. The equation, root set, and repeated roots are kept distinct." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-quadratic-equation", "edition_id": "edition-mko-quadratic-equation-en-v1", "source_field": "title", "source_mko_sha256": "d04aa30d9e57bfe7e7df1f57f60f8361ce982217aa4548374680659f56b38851", "edition_fields_sha256": "56bd95b511d37dc03232bccef4a361565dbee76f992d2e364cce95961e42b7c4" }, "summary": { "kind": "mko_edition", "mko_id": "mko-quadratic-equation", "edition_id": "edition-mko-quadratic-equation-en-v1", "source_field": "summary", "source_mko_sha256": "d04aa30d9e57bfe7e7df1f57f60f8361ce982217aa4548374680659f56b38851", "edition_fields_sha256": "56bd95b511d37dc03232bccef4a361565dbee76f992d2e364cce95961e42b7c4" } } }, "atlas-polynomial": { "source_snapshot": { "id": "atlas-polynomial", "title_zh": "一元有理係數多項式", "summary_zh": "把多項式視為有限支撐的有理係數資料；合併同次項後再讀取係數與次數。", "canonical_mko_id": "mko-polynomial" }, "fields": { "title": "Univariate polynomials over the rationals", "summary": "A polynomial is finitely supported rational-coefficient data; combine like powers before reading coefficients or degree." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-polynomial", "edition_id": "edition-mko-polynomial-en-v1", "source_field": "title", "source_mko_sha256": "0a0d76b12363d10d9ecbe8b45d7bebb1136637db1c41e6308481c675c8db9e6c", "edition_fields_sha256": "f7a3e99abef2ab48ec6b088245203dcfbacf804d5ae91cfb1de1a79a5db242dc" }, "summary": { "kind": "mko_edition", "mko_id": "mko-polynomial", "edition_id": "edition-mko-polynomial-en-v1", "source_field": "summary", "source_mko_sha256": "0a0d76b12363d10d9ecbe8b45d7bebb1136637db1c41e6308481c675c8db9e6c", "edition_fields_sha256": "f7a3e99abef2ab48ec6b088245203dcfbacf804d5ae91cfb1de1a79a5db242dc" } } }, "atlas-factorization": { "source_snapshot": { "id": "atlas-factorization", "title_zh": "有理係數多項式因式分解", "summary_zh": "因式分解是在指定係數域中把多項式寫成因子乘積；驗收比較全部係數而不是若干代入值。", "canonical_mko_id": "mko-factorization" }, "fields": { "title": "Polynomial factorization over the rationals", "summary": "Factorization writes a polynomial as a product in a declared coefficient field. Validation compares all coefficients rather than a few evaluated values." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-factorization", "edition_id": "edition-mko-factorization-en-v1", "source_field": "title", "source_mko_sha256": "375bd5e9c171dd33297e9e1954a6c9c2b48d97cebbbc4dd93b44c36108f04015", "edition_fields_sha256": "6c8b37982fcb0d3b68aa5c97d2108a73ed29b1834ae8cc911c2fc654e815f036" }, "summary": { "kind": "mko_edition", "mko_id": "mko-factorization", "edition_id": "edition-mko-factorization-en-v1", "source_field": "summary", "source_mko_sha256": "375bd5e9c171dd33297e9e1954a6c9c2b48d97cebbbc4dd93b44c36108f04015", "edition_fields_sha256": "6c8b37982fcb0d3b68aa5c97d2108a73ed29b1834ae8cc911c2fc654e815f036" } } }, "atlas-system-linear-equations": { "source_snapshot": { "id": "atlas-system-linear-equations", "title_zh": "二元聯立一次方程組與退化邊界", "summary_zh": "同時滿足兩個實數線性限制才是解；非零行列式給唯一解，奇異與零列須另判一致性。", "canonical_mko_id": "mko-system-linear-equations" }, "fields": { "title": "Linear systems in two variables and degenerate constraints", "summary": "A solution must satisfy two real linear constraints simultaneously. A nonzero determinant gives a unique solution; singular systems and zero rows require separate consistency checks." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-system-linear-equations", "edition_id": "edition-mko-system-linear-equations-en-v1", "source_field": "title", "source_mko_sha256": "62dc0d59fbb66d23a8ef47c7b2725a56713d925c7f2c8ef0a380c46552972403", "edition_fields_sha256": "0b4d6c90bdba716aae8c3ead2f2aea590308680dba7942e40059c0a271ba9452" }, "summary": { "kind": "mko_edition", "mko_id": "mko-system-linear-equations", "edition_id": "edition-mko-system-linear-equations-en-v1", "source_field": "summary", "source_mko_sha256": "62dc0d59fbb66d23a8ef47c7b2725a56713d925c7f2c8ef0a380c46552972403", "edition_fields_sha256": "0b4d6c90bdba716aae8c3ead2f2aea590308680dba7942e40059c0a271ba9452" } } }, "atlas-vector": { "source_snapshot": { "id": "atlas-vector", "title_zh": "固定基底下的平面向量", "summary_zh": "本輪把平面向量限制為固定基底下的 R² 有序對，逐分量定義加法與純量乘法。", "canonical_mko_id": "mko-vector" }, "fields": { "title": "Plane vectors in a fixed basis", "summary": "This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-vector", "edition_id": "edition-mko-vector-en-v1", "source_field": "title", "source_mko_sha256": "0d8db0f5e7511aaa8401ebe39e808af23991063892ca349e742b0551c91cba3d", "edition_fields_sha256": "602086f1fe1df71eab93b2ad3e7eebbd9293c3947c252e15052f75751508e71f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-vector", "edition_id": "edition-mko-vector-en-v1", "source_field": "summary", "source_mko_sha256": "0d8db0f5e7511aaa8401ebe39e808af23991063892ca349e742b0551c91cba3d", "edition_fields_sha256": "602086f1fe1df71eab93b2ad3e7eebbd9293c3947c252e15052f75751508e71f" } } }, "atlas-matrix": { "source_snapshot": { "id": "atlas-matrix", "title_zh": "有限矩陣的尺寸、索引與轉置", "summary_zh": "矩陣以行、列組織元素；本輪使用非空有限實矩陣及精確整數陣列切片。", "canonical_mko_id": "mko-matrix" }, "fields": { "title": "Finite matrices: dimensions, indexing, and transpose", "summary": "Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-matrix", "edition_id": "edition-mko-matrix-en-v1", "source_field": "title", "source_mko_sha256": "c0233ecc6ee07427af1589feb16108500eb2e48b0b2fb84b5b39f0f38b02f109", "edition_fields_sha256": "410465dbdfad3535a9632502caa5d72ff29df907c0173aef0ea5c53bc7773bc3" }, "summary": { "kind": "mko_edition", "mko_id": "mko-matrix", "edition_id": "edition-mko-matrix-en-v1", "source_field": "summary", "source_mko_sha256": "c0233ecc6ee07427af1589feb16108500eb2e48b0b2fb84b5b39f0f38b02f109", "edition_fields_sha256": "410465dbdfad3535a9632502caa5d72ff29df907c0173aef0ea5c53bc7773bc3" } } }, "atlas-linear-transformation": { "source_snapshot": { "id": "atlas-linear-transformation", "title_zh": "矩陣表示的實平面線性映射", "summary_zh": "先在 R²→R² 的矩陣切片驗證加法與純量相容，不把帶非零平移的仿射式叫線性映射。", "canonical_mko_id": "mko-linear-transformation" }, "fields": { "title": "Matrix-represented linear maps on the real plane", "summary": "Use the explicit R²→R² matrix model for additivity and scalar compatibility; an affine expression with nonzero translation is not linear." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-transformation", "edition_id": "edition-mko-linear-transformation-en-v1", "source_field": "title", "source_mko_sha256": "59ce4d8fac03871ea16eb62a76c9fae255470dc391ff7312ad4da120bafbf132", "edition_fields_sha256": "54f62659b55b60cd78deb2b080c3750ec4c7305cf48112d46ebf18a05e764029" }, "summary": { "kind": "mko_edition", "mko_id": "mko-linear-transformation", "edition_id": "edition-mko-linear-transformation-en-v1", "source_field": "summary", "source_mko_sha256": "59ce4d8fac03871ea16eb62a76c9fae255470dc391ff7312ad4da120bafbf132", "edition_fields_sha256": "54f62659b55b60cd78deb2b080c3750ec4c7305cf48112d46ebf18a05e764029" } } }, "atlas-determinant": { "source_snapshot": { "id": "atlas-determinant", "title_zh": "實二階方陣的行列式", "summary_zh": "2×2實矩陣的 ad−bc、非零行列式可逆性、乘積與純量縮放切片。", "canonical_mko_id": "mko-determinant" }, "fields": { "title": "Determinants of real 2×2 matrices", "summary": "This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-determinant", "edition_id": "edition-mko-determinant-en-v1", "source_field": "title", "source_mko_sha256": "d49050d70b58e0d9d2cab2f5ddc64133813f5a99418865b5941af4f1ec240093", "edition_fields_sha256": "8c5b00243e33a32549abc164fac8f7fe8e8fab5db42ce81db576cc5833334d59" }, "summary": { "kind": "mko_edition", "mko_id": "mko-determinant", "edition_id": "edition-mko-determinant-en-v1", "source_field": "summary", "source_mko_sha256": "d49050d70b58e0d9d2cab2f5ddc64133813f5a99418865b5941af4f1ec240093", "edition_fields_sha256": "8c5b00243e33a32549abc164fac8f7fe8e8fab5db42ce81db576cc5833334d59" } } }, "atlas-eigenvalue": { "source_snapshot": { "id": "atlas-eigenvalue", "title_zh": "實二階方陣的特徵值與非零特徵向量", "summary_zh": "特徵向量須非零；候選特徵空間核含零向量，實二階矩陣可沒有實特徵值。", "canonical_mko_id": "mko-eigenvalue" }, "fields": { "title": "Real eigenvalues and nonzero eigenvectors of 2×2 matrices", "summary": "An eigenvector must be nonzero, whereas a candidate-scalar kernel includes zero; a real 2×2 matrix may have no real eigenvalues." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-eigenvalue", "edition_id": "edition-mko-eigenvalue-en-v1", "source_field": "title", "source_mko_sha256": "5e1e1186120be48f1e86332bf1d679143e44074e2b3e9f59c5c219cb76a99e36", "edition_fields_sha256": "8a873da64236a6eadf02ecc3082bade7301ab2bd8561371519c293952d0fbbfb" }, "summary": { "kind": "mko_edition", "mko_id": "mko-eigenvalue", "edition_id": "edition-mko-eigenvalue-en-v1", "source_field": "summary", "source_mko_sha256": "5e1e1186120be48f1e86332bf1d679143e44074e2b3e9f59c5c219cb76a99e36", "edition_fields_sha256": "8a873da64236a6eadf02ecc3082bade7301ab2bd8561371519c293952d0fbbfb" } } }, "atlas-point-line-plane": { "source_snapshot": { "id": "atlas-point-line-plane", "title_zh": "點、直線與平面", "summary_zh": "歐幾里得幾何中的基本位置與延伸對象。", "canonical_mko_id": "mko-point-line-plane" }, "fields": { "title": "Points, straight lines and planes", "summary": "Basic objects of position and extension in Euclidean geometry." }, "field_sources": { "title": { "kind": "catalogue_literal" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-angle": { "source_snapshot": { "id": "atlas-angle", "title_zh": "角", "summary_zh": "兩條射線或方向之間的轉動關係。", "canonical_mko_id": "mko-angle" }, "fields": { "title": "Angle", "summary": "A turning relation between two rays or directions." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-angle", "edition_id": "edition-mko-angle-en-v1", "source_field": "title", "source_mko_sha256": "e77eca47694501f576639c69b49c366d6053b453055167a92e61e9a9b88626da", "edition_fields_sha256": "d5cf62bad4c911e2705ac61ff0fa344ba55b6bd1d85b3e20726baba99d56e0b9" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-euclidean-length": { "source_snapshot": { "id": "atlas-euclidean-length", "title_zh": "歐幾里得長度", "summary_zh": "平面或空間中兩點之間的距離量。", "canonical_mko_id": "mko-euclidean-length" }, "fields": { "title": "Euclidean length", "summary": "The distance between two points in a plane or in space." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-euclidean-length", "edition_id": "edition-mko-euclidean-length-en-v1", "source_field": "title", "source_mko_sha256": "485948172a6b39e7ec2cf3170ab03489fed98190d0c868161fbbd037f7151e91", "edition_fields_sha256": "58f429b7c08e832a70ea211e04213afa13d4de23947fc4fe02143039b2436576" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-triangle": { "source_snapshot": { "id": "atlas-triangle", "title_zh": "三角形", "summary_zh": "由三個不共線點及其連線形成的基本多邊形。", "canonical_mko_id": "mko-triangle" }, "fields": { "title": "Triangle", "summary": "A basic polygon formed by three noncollinear points and the segments joining them." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-triangle", "edition_id": "edition-mko-triangle-en-v1", "source_field": "title", "source_mko_sha256": "1556dabd42bf5d132f41296684eb2b037900a511e659c48627bee6d0c4dcd461", "edition_fields_sha256": "3dbf0429aed3aca96b459532ef553b51482846eadddff27fa4f81341f78d76c2" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-right-triangle": { "source_snapshot": { "id": "atlas-right-triangle", "title_zh": "直角三角形", "summary_zh": "具有一個直角的三角形。", "canonical_mko_id": "mko-right-triangle" }, "fields": { "title": "Right triangle", "summary": "A triangle with one right angle." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-right-triangle", "edition_id": "edition-mko-right-triangle-en-v1", "source_field": "title", "source_mko_sha256": "189cad28fbc4231852d6848d7f5209ff5d91a3b2dabd6380708fc678de0a758e", "edition_fields_sha256": "6c03173002dd9bb36aecbc526dda0e6036501c2d81416a970c07100765642afe" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-congruence": { "source_snapshot": { "id": "atlas-congruence", "title_zh": "明示對應的非退化三角形全等", "summary_zh": "限定正交等單位平面中的標記非退化三角形，以三組對應邊長判全等，允許鏡射。", "canonical_mko_id": "mko-congruence" }, "fields": { "title": "Congruence of nondegenerate triangles with explicit correspondence", "summary": "In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-congruence", "edition_id": "edition-mko-congruence-en-v1", "source_field": "title", "source_mko_sha256": "493629f3decdf708c53ca64d9ba2a4332bee59a979caa6d4767e0577fac7fbec", "edition_fields_sha256": "e29906e6bfe4b7efc4e7b6d14d2462b382cea4a9b9cb12beacc52719032ceeb9" }, "summary": { "kind": "mko_edition", "mko_id": "mko-congruence", "edition_id": "edition-mko-congruence-en-v1", "source_field": "summary", "source_mko_sha256": "493629f3decdf708c53ca64d9ba2a4332bee59a979caa6d4767e0577fac7fbec", "edition_fields_sha256": "e29906e6bfe4b7efc4e7b6d14d2462b382cea4a9b9cb12beacc52719032ceeb9" } } }, "atlas-similarity": { "source_snapshot": { "id": "atlas-similarity", "title_zh": "正倍率的標記三角形相似切片", "summary_zh": "指定由A到B的正長度倍率 k，平方長度與指定面積以 k² 縮放，不把負座標係數當負長度倍率。", "canonical_mko_id": "mko-similarity" }, "fields": { "title": "Labeled triangle similarity with a specified positive scale", "summary": "Specify the positive length scale k from A to B; squared lengths and a given area scale by k². A negative coordinate coefficient is not a negative length scale." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-similarity", "edition_id": "edition-mko-similarity-en-v1", "source_field": "title", "source_mko_sha256": "d60b59d396063a61d7b04c2039c88873846f8e3486fe0fa326adf55123c006f4", "edition_fields_sha256": "55f4182e49a273088088f2f44bfd57a93788cd912915b8f109773bf47e4706da" }, "summary": { "kind": "mko_edition", "mko_id": "mko-similarity", "edition_id": "edition-mko-similarity-en-v1", "source_field": "summary", "source_mko_sha256": "d60b59d396063a61d7b04c2039c88873846f8e3486fe0fa326adf55123c006f4", "edition_fields_sha256": "55f4182e49a273088088f2f44bfd57a93788cd912915b8f109773bf47e4706da" } } }, "atlas-coordinate-plane": { "source_snapshot": { "id": "atlas-coordinate-plane", "title_zh": "座標平面", "summary_zh": "固定原點、互相垂直的有向實數軸與单位後，平面點用有序實數對 (x,y) 表示。", "canonical_mko_id": "mko-coordinate-plane" }, "fields": { "title": "Coordinate plane", "summary": "After fixing an origin, perpendicular oriented real axes, and units, a point in the plane is represented by an ordered pair of real numbers (x,y)." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-coordinate-plane", "edition_id": "edition-mko-coordinate-plane-en-v1", "source_field": "title", "source_mko_sha256": "eb010001cb85a8fe8eee5a876ee1ec40caddc67fe621f671741caacf48ab3c78", "edition_fields_sha256": "64b5ee2077081e97c6e9b3107449935477bf89684384b974e10c1f6cb7d83f4d" }, "summary": { "kind": "mko_edition", "mko_id": "mko-coordinate-plane", "edition_id": "edition-mko-coordinate-plane-en-v1", "source_field": "summary", "source_mko_sha256": "eb010001cb85a8fe8eee5a876ee1ec40caddc67fe621f671741caacf48ab3c78", "edition_fields_sha256": "64b5ee2077081e97c6e9b3107449935477bf89684384b974e10c1f6cb7d83f4d" } } }, "atlas-distance-formula": { "source_snapshot": { "id": "atlas-distance-formula", "title_zh": "等單位正交座標的歐氏距離", "summary_zh": "在同一等尺度正交座標系，用非負主平方根讀取二維歐氏距離。", "canonical_mko_id": "mko-distance-formula" }, "fields": { "title": "Euclidean distance in an equal-unit orthonormal frame", "summary": "The nonnegative L2 distance in one equal-unit orthonormal coordinate frame is the principal square root of the sum of squared coordinate differences." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-distance-formula", "edition_id": "edition-mko-distance-formula-en-v1", "source_field": "title", "source_mko_sha256": "153d0d0feabadaf33e08081677be0f81709df28ffa5bffaf0176a0f7875e5904", "edition_fields_sha256": "9aa093bb09854885f18d0f53019e8ce1fdafaa608dfd1e893473e1f0b1f86401" }, "summary": { "kind": "mko_edition", "mko_id": "mko-distance-formula", "edition_id": "edition-mko-distance-formula-en-v1", "source_field": "summary", "source_mko_sha256": "153d0d0feabadaf33e08081677be0f81709df28ffa5bffaf0176a0f7875e5904", "edition_fields_sha256": "9aa093bb09854885f18d0f53019e8ce1fdafaa608dfd1e893473e1f0b1f86401" } } }, "atlas-pythagorean-theorem": { "source_snapshot": { "id": "atlas-pythagorean-theorem", "title_zh": "畢達哥拉斯定理", "summary_zh": "直角三角形兩股平方和等於斜邊平方。", "canonical_mko_id": "mko-euclid-pythagorean-theorem" }, "fields": { "title": "Pythagorean theorem", "summary": "In a right triangle, the sum of the squared leg lengths equals the squared hypotenuse length." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-euclid-pythagorean-theorem", "edition_id": "edition-mko-euclid-pythagorean-theorem-en-v1", "source_field": "title", "source_mko_sha256": "f91873f82805e9a6d9ee8358fa64afa78e7594fc6568999814e34c1205ad51c6", "edition_fields_sha256": "f0798d65b2e63fd952d3244cecd5ab72bb4d7af460ea6456ac92e9b8ccd3dad7" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-circle": { "source_snapshot": { "id": "atlas-circle", "title_zh": "圓的邊界與座標判定", "summary_zh": "圓是到固定中心距離恰等於半徑的點集，不是含內部的圓盤。", "canonical_mko_id": "mko-circle" }, "fields": { "title": "Circle boundaries and zero-radius degeneracy", "summary": "A circle here is the boundary point set at a nonnegative radius from its center, not the filled disk; radius zero yields a singleton." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-circle", "edition_id": "edition-mko-circle-en-v1", "source_field": "title", "source_mko_sha256": "f5666d01288a4c7140be59abad5dbf093293c9136d7d60dc752a79a497d99d2e", "edition_fields_sha256": "08a76069a0913021dac35f93c2e19cea979eebff44a391fa0eb18e95bc39bdea" }, "summary": { "kind": "mko_edition", "mko_id": "mko-circle", "edition_id": "edition-mko-circle-en-v1", "source_field": "summary", "source_mko_sha256": "f5666d01288a4c7140be59abad5dbf093293c9136d7d60dc752a79a497d99d2e", "edition_fields_sha256": "08a76069a0913021dac35f93c2e19cea979eebff44a391fa0eb18e95bc39bdea" } } }, "atlas-geometric-transformations": { "source_snapshot": { "id": "atlas-geometric-transformations", "title_zh": "平面座標的基本等距與均勻縮放", "summary_zh": "平移、原點逆時針四分之一圈的整數倍旋轉、指定軸鏡射是全平面等距；均勻縮放先分清零、±1與其他係數。", "canonical_mko_id": "mko-geometric-transformations" }, "fields": { "title": "Plane-coordinate isometries and uniform scaling", "summary": "Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-geometric-transformations", "edition_id": "edition-mko-geometric-transformations-en-v1", "source_field": "title", "source_mko_sha256": "13176ddfbd967892818086ef2859a68ada71d09c120c932f688634fbec64e489", "edition_fields_sha256": "0dded57378b675466ebe6e0504231686d8431cadd28c7001ad0159ce515a4d82" }, "summary": { "kind": "mko_edition", "mko_id": "mko-geometric-transformations", "edition_id": "edition-mko-geometric-transformations-en-v1", "source_field": "summary", "source_mko_sha256": "13176ddfbd967892818086ef2859a68ada71d09c120c932f688634fbec64e489", "edition_fields_sha256": "0dded57378b675466ebe6e0504231686d8431cadd28c7001ad0159ce515a4d82" } } }, "atlas-function-mapping": { "source_snapshot": { "id": "atlas-function-mapping", "title_zh": "函數映射", "summary_zh": "把定義域中每個輸入指定到唯一輸出的映射。", "canonical_mko_id": "mko-function-mapping" }, "fields": { "title": "Function mapping", "summary": "A mapping assigning exactly one output to each input in its domain." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-function-mapping", "edition_id": "edition-mko-function-mapping-en-v1", "source_field": "title", "source_mko_sha256": "f103408037c3680d770316d998ec3b92d19797f2865d8c94fa74efaf38bbc2b0", "edition_fields_sha256": "4b0337d7be21c22c34f68195a25dc7a36c740e92d6002a673095a448d2a812a9" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-domain-codomain-range": { "source_snapshot": { "id": "atlas-domain-codomain-range", "title_zh": "定義域、陪域與值域", "summary_zh": "總函數 f:A→B 的定義域是 A，陪域是 B，值域是實際取得的輸出集合 f[A]。", "canonical_mko_id": "mko-domain-codomain-range" }, "fields": { "title": "Domain, codomain, and range", "summary": "For a total function f:A→B, A is the domain, B is the codomain, and f[A] is the set of outputs actually attained." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-domain-codomain-range", "edition_id": "edition-mko-domain-codomain-range-en-v1", "source_field": "title", "source_mko_sha256": "94e55c31535caafd763f4e1f805a7da15848c153c844351ec087944dea68d451", "edition_fields_sha256": "4e694f4f6def29b458a5da030371ee48d074cd471c7f427a325c7411a18cc37b" }, "summary": { "kind": "mko_edition", "mko_id": "mko-domain-codomain-range", "edition_id": "edition-mko-domain-codomain-range-en-v1", "source_field": "summary", "source_mko_sha256": "94e55c31535caafd763f4e1f805a7da15848c153c844351ec087944dea68d451", "edition_fields_sha256": "4e694f4f6def29b458a5da030371ee48d074cd471c7f427a325c7411a18cc37b" } } }, "atlas-function-graph": { "source_snapshot": { "id": "atlas-function-graph", "title_zh": "實值函數圖形", "summary_zh": "固定直角座標系後，實值函數的圖形是全部合法輸入與其唯一輸出組成的有序對集合。", "canonical_mko_id": "mko-function-graph" }, "fields": { "title": "Graphs of real-valued functions", "summary": "In a fixed Cartesian coordinate system, the graph of a real-valued function is the set of ordered pairs of every permitted input with its unique output." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-function-graph", "edition_id": "edition-mko-function-graph-en-v1", "source_field": "title", "source_mko_sha256": "6a0892c9773f3366d44da22a063aa3c99163828135681c4e15719eda4e3638be", "edition_fields_sha256": "866cd2b7893a22f5d2c97e4e7e043e25a7dce080c7e25436af6c81d38afb72de" }, "summary": { "kind": "mko_edition", "mko_id": "mko-function-graph", "edition_id": "edition-mko-function-graph-en-v1", "source_field": "summary", "source_mko_sha256": "6a0892c9773f3366d44da22a063aa3c99163828135681c4e15719eda4e3638be", "edition_fields_sha256": "866cd2b7893a22f5d2c97e4e7e043e25a7dce080c7e25436af6c81d38afb72de" } } }, "atlas-linear-function": { "source_snapshot": { "id": "atlas-linear-function", "title_zh": "仿射函數與一次函數", "summary_zh": "實函數 f(x)=mx+b 稱仿射函數；m≠0 時為一次函數，m=0 時為常數函數。", "canonical_mko_id": "mko-linear-function" }, "fields": { "title": "Affine functions and degree-one functions", "summary": "A real function f(x)=mx+b is affine. If m≠0 it has degree one; if m=0 it is constant." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-function", "edition_id": "edition-mko-linear-function-en-v1", "source_field": "title", "source_mko_sha256": "718236b5b7dfecd9f4d276590f1ddc012cfd5df96864540a3840a2b4e6650abe", "edition_fields_sha256": "122d0d8b57c2717b960017cda1f85ea069c1e1de947f6dcb5992632018eea73f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-linear-function", "edition_id": "edition-mko-linear-function-en-v1", "source_field": "summary", "source_mko_sha256": "718236b5b7dfecd9f4d276590f1ddc012cfd5df96864540a3840a2b4e6650abe", "edition_fields_sha256": "122d0d8b57c2717b960017cda1f85ea069c1e1de947f6dcb5992632018eea73f" } } }, "atlas-quadratic-function": { "source_snapshot": { "id": "atlas-quadratic-function", "title_zh": "頂點式二次函數", "summary_zh": "實數全論域的 q(x)=a(x−h)²+k、a≠0；頂點、全域極值與對稱點的明示切片。", "canonical_mko_id": "mko-quadratic-function" }, "fields": { "title": "Quadratic functions in vertex form", "summary": "A nonzero-coefficient quadratic on all of R has a vertex, an axis of symmetry, and a global minimum or maximum determined by the coefficient sign." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-quadratic-function", "edition_id": "edition-mko-quadratic-function-en-v1", "source_field": "title", "source_mko_sha256": "439528a9247668c73092d919fb031abee74709f0633c254061b2256ed99f6441", "edition_fields_sha256": "732038404b1c581a741500ff347c03c870b4ebbd78c58327e68e6931833f3abc" }, "summary": { "kind": "mko_edition", "mko_id": "mko-quadratic-function", "edition_id": "edition-mko-quadratic-function-en-v1", "source_field": "summary", "source_mko_sha256": "439528a9247668c73092d919fb031abee74709f0633c254061b2256ed99f6441", "edition_fields_sha256": "732038404b1c581a741500ff347c03c870b4ebbd78c58327e68e6931833f3abc" } } }, "atlas-injective": { "source_snapshot": { "id": "atlas-injective", "title_zh": "單射與不同輸入的辨識", "summary_zh": "單射要求不同合法輸入不能有同一輸出，不要求覆蓋整個陪域。", "canonical_mko_id": "mko-injective" }, "fields": { "title": "Injectivity and distinct inputs", "summary": "Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-injective", "edition_id": "edition-mko-injective-en-v1", "source_field": "title", "source_mko_sha256": "3075724e6925d827d24d227d1a1ae7ef1d5fb63e41ca8cbf87adcb12c8c76941", "edition_fields_sha256": "ace77af56963b89fe5cd202a5a5ad04a1bbbaafca6b5819931eccc3d5ecee203" }, "summary": { "kind": "mko_edition", "mko_id": "mko-injective", "edition_id": "edition-mko-injective-en-v1", "source_field": "summary", "source_mko_sha256": "3075724e6925d827d24d227d1a1ae7ef1d5fb63e41ca8cbf87adcb12c8c76941", "edition_fields_sha256": "ace77af56963b89fe5cd202a5a5ad04a1bbbaafca6b5819931eccc3d5ecee203" } } }, "atlas-surjective": { "source_snapshot": { "id": "atlas-surjective", "title_zh": "滿射與陪域覆蓋", "summary_zh": "滿射要求陪域每個元素都有合法原像；相同輸出可以有多個原像。", "canonical_mko_id": "mko-surjective" }, "fields": { "title": "Surjectivity and codomain coverage", "summary": "Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-surjective", "edition_id": "edition-mko-surjective-en-v1", "source_field": "title", "source_mko_sha256": "9e23f524f4cd5d7a32f8fed322c2a1717b66148676cc519dc1288a1346303feb", "edition_fields_sha256": "b1a20763ddf4539295200424d7fe0e51f3b40227b4cc518601a32b8c7ce974b4" }, "summary": { "kind": "mko_edition", "mko_id": "mko-surjective", "edition_id": "edition-mko-surjective-en-v1", "source_field": "summary", "source_mko_sha256": "9e23f524f4cd5d7a32f8fed322c2a1717b66148676cc519dc1288a1346303feb", "edition_fields_sha256": "b1a20763ddf4539295200424d7fe0e51f3b40227b4cc518601a32b8c7ce974b4" } } }, "atlas-bijective": { "source_snapshot": { "id": "atlas-bijective", "title_zh": "雙射與唯一原像", "summary_zh": "雙射同時單射與滿射，等價於陪域每個元素恰有一個合法原像。", "canonical_mko_id": "mko-bijective" }, "fields": { "title": "Bijectivity and unique preimages", "summary": "A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-bijective", "edition_id": "edition-mko-bijective-en-v1", "source_field": "title", "source_mko_sha256": "3f8aacffa7cd2d06121165a4baccd488e1af92185dc745b6ee2cd5733c59c73c", "edition_fields_sha256": "41b29dfe8778bb263f9c2e881279cd394f9c32e61e393d857821937f8e2f79a7" }, "summary": { "kind": "mko_edition", "mko_id": "mko-bijective", "edition_id": "edition-mko-bijective-en-v1", "source_field": "summary", "source_mko_sha256": "3f8aacffa7cd2d06121165a4baccd488e1af92185dc745b6ee2cd5733c59c73c", "edition_fields_sha256": "41b29dfe8778bb263f9c2e881279cd394f9c32e61e393d857821937f8e2f79a7" } } }, "atlas-composition": { "source_snapshot": { "id": "atlas-composition", "title_zh": "函數合成", "summary_zh": "合成 g∘f 先求 f 的輸出，再把它送入 g；合成前須確認中間值可作為 g 的輸入。", "canonical_mko_id": "mko-composition" }, "fields": { "title": "Function composition", "summary": "The composition g∘f first evaluates f and then supplies its output to g; the intermediate value must be an allowed input of g." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-composition", "edition_id": "edition-mko-composition-en-v1", "source_field": "title", "source_mko_sha256": "a915b980c52ed5cfa4fd9ef59e2a23ef237a0eebbe262cd6f1a67b15f052bfa9", "edition_fields_sha256": "2c5421fa7ec1a20320b748adefa5108cc5155e4a57c3f136ee51c94ec3d435bb" }, "summary": { "kind": "mko_edition", "mko_id": "mko-composition", "edition_id": "edition-mko-composition-en-v1", "source_field": "summary", "source_mko_sha256": "a915b980c52ed5cfa4fd9ef59e2a23ef237a0eebbe262cd6f1a67b15f052bfa9", "edition_fields_sha256": "2c5421fa7ec1a20320b748adefa5108cc5155e4a57c3f136ee51c94ec3d435bb" } } }, "atlas-inverse-function": { "source_snapshot": { "id": "atlas-inverse-function", "title_zh": "雙射的反函數", "summary_zh": "雙射 f:A→B 的反函數 g:B→A 須驗證兩個方向的還原，不能把交換資料當成無條件反函數。", "canonical_mko_id": "mko-inverse-function" }, "fields": { "title": "Inverse functions of bijections", "summary": "For a bijection f:A→B, an inverse g:B→A restores values in both directions; exchanging a table is not an unconditional inverse construction." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-inverse-function", "edition_id": "edition-mko-inverse-function-en-v1", "source_field": "title", "source_mko_sha256": "bad163b9a218429518937fb4e86abe0e8707ebda0fc06c7e27f625184bc05c4f", "edition_fields_sha256": "6407be46587424baa5962c35e7920ca708e88b58464c50532400c17ec73108db" }, "summary": { "kind": "mko_edition", "mko_id": "mko-inverse-function", "edition_id": "edition-mko-inverse-function-en-v1", "source_field": "summary", "source_mko_sha256": "bad163b9a218429518937fb4e86abe0e8707ebda0fc06c7e27f625184bc05c4f", "edition_fields_sha256": "6407be46587424baa5962c35e7920ca708e88b58464c50532400c17ec73108db" } } }, "atlas-sequence": { "source_snapshot": { "id": "atlas-sequence", "title_zh": "數列", "summary_zh": "本條目採從 0 起始的實數數列 a:N→R；數列項依索引排列，不能任意重排成集合。", "canonical_mko_id": "mko-sequence" }, "fields": { "title": "Sequence", "summary": "This entry uses real sequences a:N→R indexed from 0. Sequence terms are ordered by their indices and cannot be rearranged into an unordered set." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-sequence", "edition_id": "edition-mko-sequence-en-v1", "source_field": "title", "source_mko_sha256": "61e652778c86fb6151fba32a0e61c914e4f99a1c496dd1c3d933c1f9297782f7", "edition_fields_sha256": "c09af713b8c918d5bfc260a10aac873b0c3ec1084c16830366a59c8f6523e855" }, "summary": { "kind": "mko_edition", "mko_id": "mko-sequence", "edition_id": "edition-mko-sequence-en-v1", "source_field": "summary", "source_mko_sha256": "61e652778c86fb6151fba32a0e61c914e4f99a1c496dd1c3d933c1f9297782f7", "edition_fields_sha256": "c09af713b8c918d5bfc260a10aac873b0c3ec1084c16830366a59c8f6523e855" } } }, "atlas-tends-to-relation": { "source_snapshot": { "id": "atlas-tends-to-relation", "title_zh": "趨近關係", "summary_zh": "描述變數、數列或函數朝目標狀態接近的核心關係。", "canonical_mko_id": "mko-tends-to-relation" }, "fields": { "title": "Tends-to relation", "summary": "A core relation describing a variable, sequence or function approaching a target state." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-tends-to-relation", "edition_id": "edition-mko-tends-to-relation-en-v1", "source_field": "title", "source_mko_sha256": "2fbb3c700378e80e68833bcada18866ed6439803d8b639e304fe76d31903f70e", "edition_fields_sha256": "42bf1105dc02c8a79941b5b3966ae5a0a823ef4367b8b405e53eebef617ca655" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-sequence-limit": { "source_snapshot": { "id": "atlas-sequence-limit", "title_zh": "數列極限與精確尾界", "summary_zh": "從0起算的實數列以任意精度、某一自然尾界及整個尾段定義極限；本輪精確處理倒數尾項模型。", "canonical_mko_id": "mko-sequence-limit" }, "fields": { "title": "Sequence limits and exact tail cutoffs", "summary": "A zero-based real sequence converges when every precision admits a natural cutoff controlling the entire tail; this pulse treats an exact reciprocal-tail model." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-sequence-limit", "edition_id": "edition-mko-sequence-limit-en-v1", "source_field": "title", "source_mko_sha256": "3018ee44ad1a41db719f4c5275fd215d436e1dc4e73fdc44993604a211486fff", "edition_fields_sha256": "479b0ea89a0a5610866237f8b3267d0f3279b5f1d7ccf2bc58843deb7f698f95" }, "summary": { "kind": "mko_edition", "mko_id": "mko-sequence-limit", "edition_id": "edition-mko-sequence-limit-en-v1", "source_field": "summary", "source_mko_sha256": "3018ee44ad1a41db719f4c5275fd215d436e1dc4e73fdc44993604a211486fff", "edition_fields_sha256": "479b0ea89a0a5610866237f8b3267d0f3279b5f1d7ccf2bc58843deb7f698f95" } } }, "atlas-function-limit": { "source_snapshot": { "id": "atlas-function-limit", "title_zh": "去心實數論域的函數極限", "summary_zh": "在a附近排除a本身，以ε-δ控制函數值；點值不屬於去心極限條件。", "canonical_mko_id": "mko-function-limit" }, "fields": { "title": "Function limits on punctured real domains", "summary": "An ε–δ limit controls values near a while excluding a itself; the center value is not part of the punctured-limit condition." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-function-limit", "edition_id": "edition-mko-function-limit-en-v1", "source_field": "title", "source_mko_sha256": "2dce49b3fcec9f9cf28bc85107ed2884a57b36ccc297c60e4ef5836b4a3c5ca3", "edition_fields_sha256": "ed6e34cedaf7c2521a75ac9f7e4bf88090f5b7c68fcce56ce47e89f7b1846db9" }, "summary": { "kind": "mko_edition", "mko_id": "mko-function-limit", "edition_id": "edition-mko-function-limit-en-v1", "source_field": "summary", "source_mko_sha256": "2dce49b3fcec9f9cf28bc85107ed2884a57b36ccc297c60e4ef5836b4a3c5ca3", "edition_fields_sha256": "ed6e34cedaf7c2521a75ac9f7e4bf88090f5b7c68fcce56ce47e89f7b1846db9" } } }, "atlas-continuity": { "source_snapshot": { "id": "atlas-continuity", "title_zh": "實函數的點值與連續", "summary_zh": "在全實數論域連續需要點值存在且與兩側極限一致；補點與原函數連續不可混用。", "canonical_mko_id": "mko-continuity" }, "fields": { "title": "Point values and continuity of real functions", "summary": "Continuity on the whole real domain requires a defined center value agreeing with the two-sided limit; extension and original-function continuity are distinct." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-continuity", "edition_id": "edition-mko-continuity-en-v1", "source_field": "title", "source_mko_sha256": "500bf8bf13b82d10311338ec1e1ec96d1b7b1d3fcff6113697be491180bdb9e6", "edition_fields_sha256": "ccab45420026ac3f114e5ecfbb3db7d6e36d652c04cd4afa0dc0f26e9988d24b" }, "summary": { "kind": "mko_edition", "mko_id": "mko-continuity", "edition_id": "edition-mko-continuity-en-v1", "source_field": "summary", "source_mko_sha256": "500bf8bf13b82d10311338ec1e1ec96d1b7b1d3fcff6113697be491180bdb9e6", "edition_fields_sha256": "ccab45420026ac3f114e5ecfbb3db7d6e36d652c04cd4afa0dc0f26e9988d24b" } } }, "atlas-rate-of-change": { "source_snapshot": { "id": "atlas-rate-of-change", "title_zh": "兩點間的平均變化率", "summary_zh": "平均變化率是輸出差除以非零輸入差，不是瞬時導數。", "canonical_mko_id": "mko-rate-of-change" }, "fields": { "title": "Average rate of change between distinct inputs", "summary": "Average rate of change is an endpoint difference quotient with a nonzero input difference, not an instantaneous derivative." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-rate-of-change", "edition_id": "edition-mko-rate-of-change-en-v1", "source_field": "title", "source_mko_sha256": "8c12a896998e745bbd3743c80ae005d1ffbc28d69ea33072b8e08d03c3837416", "edition_fields_sha256": "e2a896ecd6cc13555a7012770d5470bab0fda7dd63e4aa67abd00da4bfd0085b" }, "summary": { "kind": "mko_edition", "mko_id": "mko-rate-of-change", "edition_id": "edition-mko-rate-of-change-en-v1", "source_field": "summary", "source_mko_sha256": "8c12a896998e745bbd3743c80ae005d1ffbc28d69ea33072b8e08d03c3837416", "edition_fields_sha256": "e2a896ecd6cc13555a7012770d5470bab0fda7dd63e4aa67abd00da4bfd0085b" } } }, "atlas-derivative": { "source_snapshot": { "id": "atlas-derivative", "title_zh": "實函數在一點的導數與差商", "summary_zh": "導數是非零步差商的兩側極限，不是任何一個有限步差商；絕對值尖點另保留不存在。", "canonical_mko_id": "mko-derivative" }, "fields": { "title": "Real derivatives at a point and finite difference quotients", "summary": "A derivative is the two-sided limit of nonzero-step quotients, not any one finite-step quotient; an absolute-value corner retains nonexistence." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-derivative", "edition_id": "edition-mko-derivative-en-v1", "source_field": "title", "source_mko_sha256": "b523a35b9e5c20d66a7346f84ac5bece2c4167b17d46208fdb9b40115deeaf64", "edition_fields_sha256": "cc2fec17e901a5214a90804582bf9f388945c90671c6b65f82b6e76428edd3bf" }, "summary": { "kind": "mko_edition", "mko_id": "mko-derivative", "edition_id": "edition-mko-derivative-en-v1", "source_field": "summary", "source_mko_sha256": "b523a35b9e5c20d66a7346f84ac5bece2c4167b17d46208fdb9b40115deeaf64", "edition_fields_sha256": "cc2fec17e901a5214a90804582bf9f388945c90671c6b65f82b6e76428edd3bf" } } }, "atlas-derivative-rules": { "source_snapshot": { "id": "atlas-derivative-rules", "title_zh": "多項式的和、乘積與合成微分", "summary_zh": "四條微分規則先保留可微前提；本輪只在有限整數係數多項式上精確計算。", "canonical_mko_id": "mko-derivative-rules" }, "fields": { "title": "Differentiating polynomial sums, products, and compositions", "summary": "Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-derivative-rules", "edition_id": "edition-mko-derivative-rules-en-v1", "source_field": "title", "source_mko_sha256": "4756c1f6226c8cecc330975377ab5d42ed0e461b0fe039c05959fc60ab7c628f", "edition_fields_sha256": "36fdb07898d05dc2a9d7bd34619b17e26bdd31ad5dd0b98f03ac5c8ba64a7919" }, "summary": { "kind": "mko_edition", "mko_id": "mko-derivative-rules", "edition_id": "edition-mko-derivative-rules-en-v1", "source_field": "summary", "source_mko_sha256": "4756c1f6226c8cecc330975377ab5d42ed0e461b0fe039c05959fc60ab7c628f", "edition_fields_sha256": "36fdb07898d05dc2a9d7bd34619b17e26bdd31ad5dd0b98f03ac5c8ba64a7919" } } }, "atlas-antiderivative": { "source_snapshot": { "id": "atlas-antiderivative", "title_zh": "多項式原函數與常數選擇", "summary_zh": "原函數須在宣告區間導回f；本輪全R多項式以常數項固定0的有理係數原函數作可重播代表。", "canonical_mko_id": "mko-antiderivative" }, "fields": { "title": "Polynomial antiderivatives and the choice of constant", "summary": "An antiderivative differentiates to f on its declared interval; this pulse selects a zero-constant rational-coefficient representative for full-real polynomials." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-antiderivative", "edition_id": "edition-mko-antiderivative-en-v1", "source_field": "title", "source_mko_sha256": "5f2ecff6866e471d1f8da7fea1fa5e121364ae2d24b2fea405d2300fe9ff1cae", "edition_fields_sha256": "93fa3b39a3fa7cb3c5e301c68e355856d8b6947ffff2834fa7c34b247c1bda12" }, "summary": { "kind": "mko_edition", "mko_id": "mko-antiderivative", "edition_id": "edition-mko-antiderivative-en-v1", "source_field": "summary", "source_mko_sha256": "5f2ecff6866e471d1f8da7fea1fa5e121364ae2d24b2fea405d2300fe9ff1cae", "edition_fields_sha256": "93fa3b39a3fa7cb3c5e301c68e355856d8b6947ffff2834fa7c34b247c1bda12" } } }, "atlas-area": { "source_snapshot": { "id": "atlas-area", "title_zh": "座標矩形與單位格的面積", "summary_zh": "先精確處理半開矩形及有限單位格聯集，含重疊、觸邊與空矩形；不冒充一般面積理論。", "canonical_mko_id": "mko-area" }, "fields": { "title": "Exact area of half-open rectangles and unit-cell unions", "summary": "This entry computes exact area for half-open integer rectangles and finite unit-cell unions, including empty and overlapping cases." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-area", "edition_id": "edition-mko-area-en-v1", "source_field": "title", "source_mko_sha256": "dd727d645461b67b46876e9fedc6a33242ba8cb9ef182cf6bd74290b45ac1b58", "edition_fields_sha256": "09e9453992a5cab26201c55821f73fdb068065973d19d3180e2d3fba9617625f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-area", "edition_id": "edition-mko-area-en-v1", "source_field": "summary", "source_mko_sha256": "dd727d645461b67b46876e9fedc6a33242ba8cb9ef182cf6bd74290b45ac1b58", "edition_fields_sha256": "09e9453992a5cab26201c55821f73fdb068065973d19d3180e2d3fba9617625f" } } }, "atlas-definite-integral": { "source_snapshot": { "id": "atlas-definite-integral", "title_zh": "有限區間上的多項式定積分", "summary_zh": "連續多項式的有向定積分與精確矩公式；有限左端黎曼和只是有限和，不冒充一般積分或微積分基本定理完成。", "canonical_mko_id": "mko-definite-integral" }, "fields": { "title": "Signed polynomial definite integrals on finite intervals", "summary": "Continuous polynomials have oriented definite integrals with exact moment formulas; a finite left Riemann sum is not a general integral or a completed general FTC bridge." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-definite-integral", "edition_id": "edition-mko-definite-integral-en-v1", "source_field": "title", "source_mko_sha256": "f1fc56fc2ac48870dcaa53e6269bb2e365c1529c5540683ba6b6cd103ddcc5f7", "edition_fields_sha256": "94fea8c4e266737cd0cd572bd270aee5d03bf38e9002cacb8ae596f7c05fe67f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-definite-integral", "edition_id": "edition-mko-definite-integral-en-v1", "source_field": "summary", "source_mko_sha256": "f1fc56fc2ac48870dcaa53e6269bb2e365c1529c5540683ba6b6cd103ddcc5f7", "edition_fields_sha256": "94fea8c4e266737cd0cd572bd270aee5d03bf38e9002cacb8ae596f7c05fe67f" } } }, "atlas-fundamental-theorem-calculus": { "source_snapshot": { "id": "atlas-fundamental-theorem-calculus", "title_zh": "微積分基本定理：連續前提與有向端點", "summary_zh": "在全實數連續前提下，累積定積分的導數還原被積函數；合法原函數給有向積分端點差。", "canonical_mko_id": "mko-fundamental-theorem-calculus" }, "fields": { "title": "Fundamental theorem of calculus: continuity and oriented endpoints", "summary": "With continuity on all R, differentiation of the accumulating integral recovers the integrand; an eligible antiderivative gives an oriented endpoint difference." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-fundamental-theorem-calculus", "edition_id": "edition-mko-fundamental-theorem-calculus-en-v1", "source_field": "title", "source_mko_sha256": "6b22aa2d4edd9541d84e10a77e3f555318aaef766fcb9ac34f8008c0e2b43b73", "edition_fields_sha256": "42d1398fcc3b1220b48de0284014138022b93da02d617c0b0a129368bb1c322b" }, "summary": { "kind": "mko_edition", "mko_id": "mko-fundamental-theorem-calculus", "edition_id": "edition-mko-fundamental-theorem-calculus-en-v1", "source_field": "summary", "source_mko_sha256": "6b22aa2d4edd9541d84e10a77e3f555318aaef766fcb9ac34f8008c0e2b43b73", "edition_fields_sha256": "42d1398fcc3b1220b48de0284014138022b93da02d617c0b0a129368bb1c322b" } } }, "atlas-sample-space": { "source_snapshot": { "id": "atlas-sample-space", "title_zh": "樣本空間", "summary_zh": "隨機試驗所有可能結果的集合。", "canonical_mko_id": "mko-sample-space" }, "fields": { "title": "Sample space", "summary": "The set of all possible outcomes of a random experiment." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-sample-space", "edition_id": "edition-mko-sample-space-en-v1", "source_field": "title", "source_mko_sha256": "c58b0d07252ece257a4143fd10f0f25b249023f4ed50bc6d393f4752a91267f7", "edition_fields_sha256": "59d4efb33c612f020f37d08844a33b4911d8d1310f03ce00801ea1014f317114" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-event": { "source_snapshot": { "id": "atlas-event", "title_zh": "事件", "summary_zh": "樣本空間中的結果集合。", "canonical_mko_id": "mko-event" }, "fields": { "title": "Event", "summary": "A set of outcomes in a sample space." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-event", "edition_id": "edition-mko-event-en-v1", "source_field": "title", "source_mko_sha256": "5962ae6672cb79c34888bdda4ac66de557a4f9b21243f2445414d7dd326b9c74", "edition_fields_sha256": "21ec691b371a81feb066a9c7bf9ecc2b916f98457cd22c577498bb880c139c6a" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-probability": { "source_snapshot": { "id": "atlas-probability", "title_zh": "有限樣本空間上的機率", "summary_zh": "在非空有限樣本空間上指定非負且總和為一的權重，事件機率為其成員權重之和。", "canonical_mko_id": "mko-probability" }, "fields": { "title": "Probability on finite sample spaces", "summary": "Assign nonnegative weights summing to one on a nonempty finite sample space. An event probability is the sum of the weights of its outcomes." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-probability", "edition_id": "edition-mko-probability-en-v1", "source_field": "title", "source_mko_sha256": "07f2b780895b31a69725da388129f9624510427b4a2937b15a61418f94e267ca", "edition_fields_sha256": "6d9def9ee5255c4b2d8273d5cc8db804096e51dd386a0dddbee22f58d2c093a7" }, "summary": { "kind": "mko_edition", "mko_id": "mko-probability", "edition_id": "edition-mko-probability-en-v1", "source_field": "summary", "source_mko_sha256": "07f2b780895b31a69725da388129f9624510427b4a2937b15a61418f94e267ca", "edition_fields_sha256": "6d9def9ee5255c4b2d8273d5cc8db804096e51dd386a0dddbee22f58d2c093a7" } } }, "atlas-conditional-probability": { "source_snapshot": { "id": "atlas-conditional-probability", "title_zh": "有限模型的條件機率", "summary_zh": "條件事件必須有正機率；有限模型中 P(A|B)=P(A∩B)/P(B)，方向不能任意交換。", "canonical_mko_id": "mko-conditional-probability" }, "fields": { "title": "Conditional probability in finite models", "summary": "The conditioning event must have positive probability. In a finite model, P(A|B)=P(A∩B)/P(B), and the direction cannot be freely exchanged." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-conditional-probability", "edition_id": "edition-mko-conditional-probability-en-v1", "source_field": "title", "source_mko_sha256": "7b0f4824aa1f3e4c64ea306c5bed04843566b8e4bbdf96d94dbc1367dbb68419", "edition_fields_sha256": "4f55a0241121df11e909abd53ac87ca1639a38bbd43eb0faab393dd1eba21fc8" }, "summary": { "kind": "mko_edition", "mko_id": "mko-conditional-probability", "edition_id": "edition-mko-conditional-probability-en-v1", "source_field": "summary", "source_mko_sha256": "7b0f4824aa1f3e4c64ea306c5bed04843566b8e4bbdf96d94dbc1367dbb68419", "edition_fields_sha256": "4f55a0241121df11e909abd53ac87ca1639a38bbd43eb0faab393dd1eba21fc8" } } }, "atlas-random-variable": { "source_snapshot": { "id": "atlas-random-variable", "title_zh": "有限樣本空間的實值隨機變數", "summary_zh": "有限模型中的隨機變數是基本結果到實數的總函數；它不是基本結果，也不是權重本身。", "canonical_mko_id": "mko-random-variable" }, "fields": { "title": "Real random variables on finite sample spaces", "summary": "A random variable in a finite model is a total function from elementary outcomes to real numbers, distinct from an outcome or its probability weight." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-random-variable", "edition_id": "edition-mko-random-variable-en-v1", "source_field": "title", "source_mko_sha256": "bba00300a619b37481007904b211059750f93a4a2c4ba2b8209eac472ad33e06", "edition_fields_sha256": "9b58acc17d709a8d15e15b834f3315492db4b18eac4781ef5eb065bff4c21aa6" }, "summary": { "kind": "mko_edition", "mko_id": "mko-random-variable", "edition_id": "edition-mko-random-variable-en-v1", "source_field": "summary", "source_mko_sha256": "bba00300a619b37481007904b211059750f93a4a2c4ba2b8209eac472ad33e06", "edition_fields_sha256": "9b58acc17d709a8d15e15b834f3315492db4b18eac4781ef5eb065bff4c21aa6" } } }, "atlas-expectation": { "source_snapshot": { "id": "atlas-expectation", "title_zh": "有限模型的期望值", "summary_zh": "有限期望是合法機率權重與隨機變數值的加權總和，線性性不要求獨立。", "canonical_mko_id": "mko-expectation" }, "fields": { "title": "Expectation in finite models", "summary": "Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-expectation", "edition_id": "edition-mko-expectation-en-v1", "source_field": "title", "source_mko_sha256": "71bc68faf69ca0df9cdd11d093055262aa277d07abd77982d0aa1afd849daf3c", "edition_fields_sha256": "a8b7c106a7a94c8b0258666ce4586413b17da07d344459741558299519b993e3" }, "summary": { "kind": "mko_edition", "mko_id": "mko-expectation", "edition_id": "edition-mko-expectation-en-v1", "source_field": "summary", "source_mko_sha256": "71bc68faf69ca0df9cdd11d093055262aa277d07abd77982d0aa1afd849daf3c", "edition_fields_sha256": "a8b7c106a7a94c8b0258666ce4586413b17da07d344459741558299519b993e3" } } }, "atlas-variance": { "source_snapshot": { "id": "atlas-variance", "title_zh": "有限機率模型的變異數", "summary_zh": "本輪變異數為 E[(X−E[X])²]，不是 n−1 分母的樣本變異數；零權重離群值不影響它。", "canonical_mko_id": "mko-variance" }, "fields": { "title": "Variance in finite probability models", "summary": "This entry defines variance as E[(X−E[X])²], distinct from sample variance with an n−1 denominator. Zero-weight outliers do not affect it." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-variance", "edition_id": "edition-mko-variance-en-v1", "source_field": "title", "source_mko_sha256": "282681c1ed2d55f1c853c9759022470c991dd46c2b150626fa4bf654642032bc", "edition_fields_sha256": "71909eaa10506a4006c97aad9a292cbfab087d581f9949b42ed66a2a57ce415d" }, "summary": { "kind": "mko_edition", "mko_id": "mko-variance", "edition_id": "edition-mko-variance-en-v1", "source_field": "summary", "source_mko_sha256": "282681c1ed2d55f1c853c9759022470c991dd46c2b150626fa4bf654642032bc", "edition_fields_sha256": "71909eaa10506a4006c97aad9a292cbfab087d581f9949b42ed66a2a57ce415d" } } }, "atlas-descriptive-statistics": { "source_snapshot": { "id": "atlas-descriptive-statistics", "title_zh": "有限觀測列的描述統計", "summary_zh": "保留重數的非空觀測列，精確計算均值、中位數、全距與所有最大頻率值。", "canonical_mko_id": "mko-descriptive-statistics" }, "fields": { "title": "Descriptive summaries of nonempty observation lists", "summary": "Mean, median, range, and the explicitly defined maximal-frequency set summarize a nonempty finite observation list while retaining multiplicities." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-descriptive-statistics", "edition_id": "edition-mko-descriptive-statistics-en-v1", "source_field": "title", "source_mko_sha256": "bb188b4a4f67bc4e07da170bd9ea1b92aa0a51719cac24e1570cb156d8f76204", "edition_fields_sha256": "9a9e2981a1f55b36ddb878632b63675645a41292a540a2ab97abc85625bf18bc" }, "summary": { "kind": "mko_edition", "mko_id": "mko-descriptive-statistics", "edition_id": "edition-mko-descriptive-statistics-en-v1", "source_field": "summary", "source_mko_sha256": "bb188b4a4f67bc4e07da170bd9ea1b92aa0a51719cac24e1570cb156d8f76204", "edition_fields_sha256": "9a9e2981a1f55b36ddb878632b63675645a41292a540a2ab97abc85625bf18bc" } } }, "atlas-counting-principle": { "source_snapshot": { "id": "atlas-counting-principle", "title_zh": "基本計數原理", "summary_zh": "以加法與乘法原理計算有限選擇數量。", "canonical_mko_id": "mko-counting-principle" }, "fields": { "title": "Basic counting principles", "summary": "Addition and multiplication principles for counting finite choices." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-counting-principle", "edition_id": "edition-mko-counting-principle-en-v1", "source_field": "title", "source_mko_sha256": "0a0619ebbab98b96543b42abc4e66d927aeea705c65ee85b3a9badb6d3b3cf57", "edition_fields_sha256": "8afea4044d1226c965b440c3abfa33fa5318725ef98b7cdb2ec605395e7585df" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-factorial": { "source_snapshot": { "id": "atlas-factorial", "title_zh": "階乘", "summary_zh": "自然數到前 n 個正整數乘積的函數。", "canonical_mko_id": "mko-factorial" }, "fields": { "title": "Factorial", "summary": "The function taking a natural number n to the product of the first n positive integers." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-factorial", "edition_id": "edition-mko-factorial-en-v1", "source_field": "title", "source_mko_sha256": "fc2d013658e47191e660f97c927ceeec97905dd4fc520329d3df326b343a4b8f", "edition_fields_sha256": "af77b4e1f999fa7dbf24d29aef5f86ebd423945dd1f4a92840dd17096176175c" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-permutation-combination": { "source_snapshot": { "id": "atlas-permutation-combination", "title_zh": "排列與組合", "summary_zh": "由 n 個相異物件不重複取 k 個，有序選取是排列，無序選取是組合。", "canonical_mko_id": "mko-permutation-combination" }, "fields": { "title": "Permutations and combinations", "summary": "When k distinct objects are selected without replacement from n distinct objects, an ordered selection is a permutation and an unordered selection is a combination." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-permutation-combination", "edition_id": "edition-mko-permutation-combination-en-v1", "source_field": "title", "source_mko_sha256": "bd1ee61b3eaee90e08b3bb7a753128635ac36421438a78c41bba6dc768077c96", "edition_fields_sha256": "d76f351b31f7ba6188520a58923ae50bd4fdbd20c851e9e13fbe89c5e150207a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-permutation-combination", "edition_id": "edition-mko-permutation-combination-en-v1", "source_field": "summary", "source_mko_sha256": "bd1ee61b3eaee90e08b3bb7a753128635ac36421438a78c41bba6dc768077c96", "edition_fields_sha256": "d76f351b31f7ba6188520a58923ae50bd4fdbd20c851e9e13fbe89c5e150207a" } } }, "atlas-graph": { "source_snapshot": { "id": "atlas-graph", "title_zh": "有限簡單無向圖", "summary_zh": "本輪限定有限簡單無向圖，邊是兩個相異頂點的無序集合，度數只計相鄰頂點。", "canonical_mko_id": "mko-graph" }, "fields": { "title": "Finite simple undirected graphs", "summary": "This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-graph", "edition_id": "edition-mko-graph-en-v1", "source_field": "title", "source_mko_sha256": "47d40107cd3836bb878ae46e5c4766288abb1b91e480c38ed14f61f749d27d55", "edition_fields_sha256": "8c966f33e718cbfc14cf6dfcaba59c7f985bb5d6120ad401f81cd60bd7ee8bc9" }, "summary": { "kind": "mko_edition", "mko_id": "mko-graph", "edition_id": "edition-mko-graph-en-v1", "source_field": "summary", "source_mko_sha256": "47d40107cd3836bb878ae46e5c4766288abb1b91e480c38ed14f61f749d27d55", "edition_fields_sha256": "8c966f33e718cbfc14cf6dfcaba59c7f985bb5d6120ad401f81cd60bd7ee8bc9" } } }, "atlas-tree": { "source_snapshot": { "id": "atlas-tree", "title_zh": "非空有限樹：連通、無環與唯一道路", "summary_zh": "非空有限簡單無向圖同時連通且無環才是樹；保留單點、度1葉及加一條新邊的唯一環。", "canonical_mko_id": "mko-tree" }, "fields": { "title": "Nonempty finite trees: connectivity, cycles, and unique paths", "summary": "A finite simple undirected graph is a tree only when nonempty, connected, and acyclic; retain singleton, degree-one leaf, and new-edge cycle conventions." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-tree", "edition_id": "edition-mko-tree-en-v1", "source_field": "title", "source_mko_sha256": "48c7310136a8fd9749e9a9af278b0ade15fdecb8f4cc5657f4f0895eef89aebf", "edition_fields_sha256": "c554401f8fb51fa786623e7d0cfee1cc03e516576211905aaa9e7f35c30cf9d9" }, "summary": { "kind": "mko_edition", "mko_id": "mko-tree", "edition_id": "edition-mko-tree-en-v1", "source_field": "summary", "source_mko_sha256": "48c7310136a8fd9749e9a9af278b0ade15fdecb8f4cc5657f4f0895eef89aebf", "edition_fields_sha256": "c554401f8fb51fa786623e7d0cfee1cc03e516576211905aaa9e7f35c30cf9d9" } } }, "atlas-mathematical-induction": { "source_snapshot": { "id": "atlas-mathematical-induction", "title_zh": "數學歸納法", "summary_zh": "以基礎情況與歸納步驟證明自然數族命題。", "canonical_mko_id": "mko-mathematical-induction" }, "fields": { "title": "Mathematical induction", "summary": "A method proving a natural-number-indexed family of propositions by a base case and an inductive step." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-mathematical-induction", "edition_id": "edition-mko-mathematical-induction-en-v1", "source_field": "title", "source_mko_sha256": "53fe3a341f5fa25780e7aa742a25087665b257b3c91a3e4f718b50f368a8dd67", "edition_fields_sha256": "4bc3882331b446b62db091be9866432810fe9317a57ddc390b0a17a2a9662bc0" }, "summary": { "kind": "catalogue_literal" } } }, "atlas-topological-space": { "source_snapshot": { "id": "atlas-topological-space", "title_zh": "拓撲空間", "summary_zh": "拓撲空間由底集合與一族開集組成；開集族包含空集與全空間，並對任意聯集與有限交封閉。", "canonical_mko_id": "mko-topological-space" }, "fields": { "title": "Topological space", "summary": "A topological space consists of an underlying set and a family of open sets containing the empty set and whole space and closed under arbitrary unions and finite intersections." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-topological-space", "edition_id": "edition-mko-topological-space-en-v1", "source_field": "title", "source_mko_sha256": "95f78b4f323b3139dd05068842f7209ac33b39b3d86feb5b73e0eb8849cdb43d", "edition_fields_sha256": "08ea016b8dcaec7d0d0de55568355100c92d597a9e629c236dbb7094768275f6" }, "summary": { "kind": "mko_edition", "mko_id": "mko-topological-space", "edition_id": "edition-mko-topological-space-en-v1", "source_field": "summary", "source_mko_sha256": "95f78b4f323b3139dd05068842f7209ac33b39b3d86feb5b73e0eb8849cdb43d", "edition_fields_sha256": "08ea016b8dcaec7d0d0de55568355100c92d597a9e629c236dbb7094768275f6" } } }, "atlas-open-closed-set": { "source_snapshot": { "id": "atlas-open-closed-set", "title_zh": "開集與閉集", "summary_zh": "在固定拓撲空間中，開集屬於拓撲；閉集的補集為開集，同一集合可以同時開且閉。", "canonical_mko_id": "mko-open-closed-set" }, "fields": { "title": "Open and closed sets", "summary": "In a fixed topological space, open sets belong to the topology and closed sets have open complements; one set can be both open and closed." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-open-closed-set", "edition_id": "edition-mko-open-closed-set-en-v1", "source_field": "title", "source_mko_sha256": "78fdc40df7a4161259566db404d47fa805aa334d6f661420675bbc8c139c6508", "edition_fields_sha256": "cfef0068dc5200e92d88402adcb67b18201422034cf830a14011ec0f39f2866e" }, "summary": { "kind": "mko_edition", "mko_id": "mko-open-closed-set", "edition_id": "edition-mko-open-closed-set-en-v1", "source_field": "summary", "source_mko_sha256": "78fdc40df7a4161259566db404d47fa805aa334d6f661420675bbc8c139c6508", "edition_fields_sha256": "cfef0068dc5200e92d88402adcb67b18201422034cf830a14011ec0f39f2866e" } } }, "atlas-neighborhood": { "source_snapshot": { "id": "atlas-neighborhood", "title_zh": "鄰域", "summary_zh": "集合 N 是點 x 的鄰域，當且僅當 N 包含某個含 x 的開集；N 本身不必開。", "canonical_mko_id": "mko-neighborhood" }, "fields": { "title": "Neighborhood", "summary": "A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-neighborhood", "edition_id": "edition-mko-neighborhood-en-v1", "source_field": "title", "source_mko_sha256": "175c50e3f4cbb3dfa266c387a13492faeffdacb3df5d699b2ad94f7ea5f91ef1", "edition_fields_sha256": "85f93b0a78aba177f0eb298dc78822f1637b65b3346589126b02e5950b20f4a8" }, "summary": { "kind": "mko_edition", "mko_id": "mko-neighborhood", "edition_id": "edition-mko-neighborhood-en-v1", "source_field": "summary", "source_mko_sha256": "175c50e3f4cbb3dfa266c387a13492faeffdacb3df5d699b2ad94f7ea5f91ef1", "edition_fields_sha256": "85f93b0a78aba177f0eb298dc78822f1637b65b3346589126b02e5950b20f4a8" } } }, "atlas-topological-continuity": { "source_snapshot": { "id": "atlas-topological-continuity", "title_zh": "拓撲連續映射", "summary_zh": "拓撲映射連續，表示目標空間每個開集的逆像在來源空間仍是開集。", "canonical_mko_id": "mko-topological-continuity" }, "fields": { "title": "Continuous maps between topological spaces", "summary": "A map is topologically continuous when every open set in its target has an open preimage in its source." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-topological-continuity", "edition_id": "edition-mko-topological-continuity-en-v1", "source_field": "title", "source_mko_sha256": "559fd18172a255b51e3254df6aeeb3ac6e1121e20bded6913c4cd34f3f1484b4", "edition_fields_sha256": "98f01b8ccbdf22e8dc05bc8f8848a8e5fdbc11d1a3859fd993eac77074b60128" }, "summary": { "kind": "mko_edition", "mko_id": "mko-topological-continuity", "edition_id": "edition-mko-topological-continuity-en-v1", "source_field": "summary", "source_mko_sha256": "559fd18172a255b51e3254df6aeeb3ac6e1121e20bded6913c4cd34f3f1484b4", "edition_fields_sha256": "98f01b8ccbdf22e8dc05bc8f8848a8e5fdbc11d1a3859fd993eac77074b60128" } } }, "atlas-natural-divisibility": { "source_snapshot": { "id": "atlas-natural-divisibility", "title_zh": "自然數整除", "summary_zh": "自然數 a 整除 b，表示存在自然數乘數 k 使 b=a·k；零的兩側行為必須分開。", "canonical_mko_id": "mko-natural-divisibility" }, "fields": { "title": "Natural-number divisibility", "summary": "A natural number a divides b when a natural multiplier k satisfies b=a·k; the two roles of zero must be distinguished." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-natural-divisibility", "edition_id": "edition-mko-natural-divisibility-en-v1", "source_field": "title", "source_mko_sha256": "2e5d00faa3ae3031b2cb2a650b12db0ab8e8aa1b59a7ea6b2350f49a792758d7", "edition_fields_sha256": "5192d24e4e2d8fb91aaefb2523bb623bc2d8bd6d379059d8fd36d217557471bc" }, "summary": { "kind": "mko_edition", "mko_id": "mko-natural-divisibility", "edition_id": "edition-mko-natural-divisibility-en-v1", "source_field": "summary", "source_mko_sha256": "2e5d00faa3ae3031b2cb2a650b12db0ab8e8aa1b59a7ea6b2350f49a792758d7", "edition_fields_sha256": "5192d24e4e2d8fb91aaefb2523bb623bc2d8bd6d379059d8fd36d217557471bc" } } }, "atlas-gcd-coprime": { "source_snapshot": { "id": "atlas-gcd-coprime", "title_zh": "最大公因數與互質", "summary_zh": "gcd(a,b) 是同時整除 a、b 且被每個公因數整除的自然數；gcd=1 時稱互質。", "canonical_mko_id": "mko-gcd-coprime" }, "fields": { "title": "Greatest common divisor and coprimality", "summary": "The natural number gcd(a,b) divides both a and b and is divisible by every common divisor; the inputs are coprime when their gcd is 1." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-gcd-coprime", "edition_id": "edition-mko-gcd-coprime-en-v1", "source_field": "title", "source_mko_sha256": "a8f5934d6354ea07a83057c61f045034e2d964850bed4759aedd8721848b3b9b", "edition_fields_sha256": "0cb57814d31a1dee0eb123f96c10fa7719e5741967d9a6827297eb9a1d86d164" }, "summary": { "kind": "mko_edition", "mko_id": "mko-gcd-coprime", "edition_id": "edition-mko-gcd-coprime-en-v1", "source_field": "summary", "source_mko_sha256": "a8f5934d6354ea07a83057c61f045034e2d964850bed4759aedd8721848b3b9b", "edition_fields_sha256": "0cb57814d31a1dee0eb123f96c10fa7719e5741967d9a6827297eb9a1d86d164" } } }, "atlas-prime-number": { "source_snapshot": { "id": "atlas-prime-number", "title_zh": "自然數質數", "summary_zh": "自然數質數是大於或等於 2，且正因數只有 1 與自身的自然數。", "canonical_mko_id": "mko-prime-number" }, "fields": { "title": "Prime natural number", "summary": "A prime natural number is at least 2 and has no positive divisors other than 1 and itself." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-prime-number", "edition_id": "edition-mko-prime-number-en-v1", "source_field": "title", "source_mko_sha256": "5ebeb1330ba52c9685111730d0c0c2255e653887f8869f119ebbbb767ed139b5", "edition_fields_sha256": "38f3648f1a5f554a4198d5ea1b5de73344c110e823aec3452296e617498594ce" }, "summary": { "kind": "mko_edition", "mko_id": "mko-prime-number", "edition_id": "edition-mko-prime-number-en-v1", "source_field": "summary", "source_mko_sha256": "5ebeb1330ba52c9685111730d0c0c2255e653887f8869f119ebbbb767ed139b5", "edition_fields_sha256": "38f3648f1a5f554a4198d5ea1b5de73344c110e823aec3452296e617498594ce" } } }, "atlas-modular-congruence": { "source_snapshot": { "id": "atlas-modular-congruence", "title_zh": "自然數模同餘", "summary_zh": "固定正模數 n 時，a 與 b 模 n 同餘表示兩者除以 n 的餘數相同。", "canonical_mko_id": "mko-modular-congruence" }, "fields": { "title": "Natural-number congruence modulo a positive modulus", "summary": "For a fixed positive modulus n, two natural numbers a and b are congruent modulo n when they have the same remainder on division by n." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-modular-congruence", "edition_id": "edition-mko-modular-congruence-en-v1", "source_field": "title", "source_mko_sha256": "f2701dae3fc0037b15acf2146bea327e803c9f727d640db0a85d02de29f3f635", "edition_fields_sha256": "03179c5223863f014515f0f5a2c2b74c622639ed78392fe074d39efd59d9643f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-modular-congruence", "edition_id": "edition-mko-modular-congruence-en-v1", "source_field": "summary", "source_mko_sha256": "f2701dae3fc0037b15acf2146bea327e803c9f727d640db0a85d02de29f3f635", "edition_fields_sha256": "03179c5223863f014515f0f5a2c2b74c622639ed78392fe074d39efd59d9643f" } } }, "atlas-decision-problem": { "source_snapshot": { "id": "atlas-decision-problem", "title_zh": "決定問題", "summary_zh": "決定問題由輸入集合 X 與 yes-instance 子集合 Y⊆X 指定；decider 必須對每個合法輸入終止並回傳正確布林答案。", "canonical_mko_id": "mko-decision-problem" }, "fields": { "title": "Decision problem", "summary": "A decision problem specifies an input set X and a yes-instance subset Y⊆X; a decider must terminate on every legal input and return the correct Boolean answer." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-decision-problem", "edition_id": "edition-mko-decision-problem-en-v1", "source_field": "title", "source_mko_sha256": "b63ff13299c4ce36349133c27ce4fdcc5554615d69e54610ae5c15fe80b3005c", "edition_fields_sha256": "36af4729927a408eededb9dca87a9308bc0aaf89dc0ea8ae8d92d55c144fba8c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-decision-problem", "edition_id": "edition-mko-decision-problem-en-v1", "source_field": "summary", "source_mko_sha256": "b63ff13299c4ce36349133c27ce4fdcc5554615d69e54610ae5c15fe80b3005c", "edition_fields_sha256": "36af4729927a408eededb9dca87a9308bc0aaf89dc0ea8ae8d92d55c144fba8c" } } }, "atlas-computational-model": { "source_snapshot": { "id": "atlas-computational-model", "title_zh": "抽象計算模型", "summary_zh": "有限狀態計算模型以狀態集、初始狀態集、步驟關係與停止狀態集描述可能執行。", "canonical_mko_id": "mko-computational-model" }, "fields": { "title": "Abstract computational model", "summary": "A finite-state computational model describes possible executions through its state set, initial-state set, step relation and halt-state set." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-computational-model", "edition_id": "edition-mko-computational-model-en-v1", "source_field": "title", "source_mko_sha256": "ceedb5463ff015fc7cd0f74870bfb9033cfb9a8ff893490b0335fe4bf9112ace", "edition_fields_sha256": "537cc5b50c418541fa94c0ea0e8e376c886733e1bc266fc83228c9de7e3205c7" }, "summary": { "kind": "mko_edition", "mko_id": "mko-computational-model", "edition_id": "edition-mko-computational-model-en-v1", "source_field": "summary", "source_mko_sha256": "ceedb5463ff015fc7cd0f74870bfb9033cfb9a8ff893490b0335fe4bf9112ace", "edition_fields_sha256": "537cc5b50c418541fa94c0ea0e8e376c886733e1bc266fc83228c9de7e3205c7" } } }, "atlas-algorithm-correctness": { "source_snapshot": { "id": "atlas-algorithm-correctness", "title_zh": "算法正確性", "summary_zh": "算法正確性分開部分正確性與終止性；總正確性需要兩者同時成立。", "canonical_mko_id": "mko-algorithm-correctness" }, "fields": { "title": "Algorithm correctness", "summary": "Algorithm correctness separates partial correctness from termination; total correctness requires both." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-algorithm-correctness", "edition_id": "edition-mko-algorithm-correctness-en-v1", "source_field": "title", "source_mko_sha256": "d9789607b7391a897a2a291245a0699d81ef6dbbdfac86b9b86c04cf7dc0ad27", "edition_fields_sha256": "b6f4b8c346b3ba54ad773c63be8400883d88a27c5b32a54e84a0475160499047" }, "summary": { "kind": "mko_edition", "mko_id": "mko-algorithm-correctness", "edition_id": "edition-mko-algorithm-correctness-en-v1", "source_field": "summary", "source_mko_sha256": "d9789607b7391a897a2a291245a0699d81ef6dbbdfac86b9b86c04cf7dc0ad27", "edition_fields_sha256": "b6f4b8c346b3ba54ad773c63be8400883d88a27c5b32a54e84a0475160499047" } } }, "atlas-asymptotic-resource-bound": { "source_snapshot": { "id": "atlas-asymptotic-resource-bound", "title_zh": "漸近資源界", "summary_zh": "漸近上界、下界與緊界必須相對輸入大小、成本模型及全尾段量化定義。", "canonical_mko_id": "mko-asymptotic-resource-bound" }, "fields": { "title": "Asymptotic resource bounds", "summary": "Asymptotic upper, lower and tight bounds require an input-size convention, cost model and quantification over a full tail." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-asymptotic-resource-bound", "edition_id": "edition-mko-asymptotic-resource-bound-en-v1", "source_field": "title", "source_mko_sha256": "2b5dc69aebd1ca2d1e4c3c4bb10e9c0c889c653d311aebba7e59d53def35d42e", "edition_fields_sha256": "25a19ae252d626a12e3ee95cfa54a2778c64a15a278824e415a3cb9424314fa0" }, "summary": { "kind": "mko_edition", "mko_id": "mko-asymptotic-resource-bound", "edition_id": "edition-mko-asymptotic-resource-bound-en-v1", "source_field": "summary", "source_mko_sha256": "2b5dc69aebd1ca2d1e4c3c4bb10e9c0c889c653d311aebba7e59d53def35d42e", "edition_fields_sha256": "25a19ae252d626a12e3ee95cfa54a2778c64a15a278824e415a3cb9424314fa0" } } }, "atlas-formal-statement": { "source_snapshot": { "id": "atlas-formal-statement", "title_zh": "證明助理中的形式陳述", "summary_zh": "形式陳述是在固定版本、imports、環境與 local context 中成功 elaboration，且型別為 Prop 的核心項。", "canonical_mko_id": "mko-lean-formal-statement" }, "fields": { "title": "Formal statements in a proof assistant", "summary": "A formal statement is a core term of type Prop obtained by successful elaboration in a fixed version, import environment and local context." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-lean-formal-statement", "edition_id": "edition-mko-lean-formal-statement-en-v1", "source_field": "title", "source_mko_sha256": "b60e51a622407c4c4ee426df9c55830652f67ff5e0a8344c040c14f26dd49128", "edition_fields_sha256": "4e7526014d64a778cd337edf17965de584a8ee15c3ebbe85383b0685652da1d1" }, "summary": { "kind": "mko_edition", "mko_id": "mko-lean-formal-statement", "edition_id": "edition-mko-lean-formal-statement-en-v1", "source_field": "summary", "source_mko_sha256": "b60e51a622407c4c4ee426df9c55830652f67ff5e0a8344c040c14f26dd49128", "edition_fields_sha256": "4e7526014d64a778cd337edf17965de584a8ee15c3ebbe85383b0685652da1d1" } } }, "atlas-proof-term": { "source_snapshot": { "id": "atlas-proof-term", "title_zh": "證明項", "summary_zh": "證明項是在固定形式環境與 context 中具有命題 P 型別，並可由 kernel 檢查的項 t。", "canonical_mko_id": "mko-lean-proof-term" }, "fields": { "title": "Proof term", "summary": "A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-lean-proof-term", "edition_id": "edition-mko-lean-proof-term-en-v1", "source_field": "title", "source_mko_sha256": "a11c7813a596785a48d1af6603780643970769d714b6a167fec2a29245c6e1b2", "edition_fields_sha256": "245599832aebdb2393a2df7ce3509dfa7b1f29b8ae57cd676de6599d74305469" }, "summary": { "kind": "mko_edition", "mko_id": "mko-lean-proof-term", "edition_id": "edition-mko-lean-proof-term-en-v1", "source_field": "summary", "source_mko_sha256": "a11c7813a596785a48d1af6603780643970769d714b6a167fec2a29245c6e1b2", "edition_fields_sha256": "245599832aebdb2393a2df7ce3509dfa7b1f29b8ae57cd676de6599d74305469" } } }, "atlas-discrete-dynamical-system": { "source_snapshot": { "id": "atlas-discrete-dynamical-system", "title_zh": "離散時間動力系統", "summary_zh": "離散時間動力系統以狀態空間、自映射與自然數時間描述逐步演化；有限軌道核對不自動給出一般穩定性。", "canonical_mko_id": "mko-discrete-dynamical-system" }, "fields": { "title": "Discrete-time dynamical system", "summary": "A discrete-time dynamical system uses a state space, a self-map and natural-number time to describe stepwise evolution; a finite orbit check does not establish general stability." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-discrete-dynamical-system", "edition_id": "edition-mko-discrete-dynamical-system-en-v1", "source_field": "title", "source_mko_sha256": "83db6a4ce40522862935c10fd58632bc890dbcf7e2d2277368dad51ee0b72837", "edition_fields_sha256": "d3c32a79cb9a896eed9238ce2419daf5283e70d52d27972799cfe2dd4599ba69" }, "summary": { "kind": "mko_edition", "mko_id": "mko-discrete-dynamical-system", "edition_id": "edition-mko-discrete-dynamical-system-en-v1", "source_field": "summary", "source_mko_sha256": "83db6a4ce40522862935c10fd58632bc890dbcf7e2d2277368dad51ee0b72837", "edition_fields_sha256": "d3c32a79cb9a896eed9238ce2419daf5283e70d52d27972799cfe2dd4599ba69" } } }, "atlas-orbit-fixed-periodic": { "source_snapshot": { "id": "atlas-orbit-fixed-periodic", "title_zh": "軌道、固定點與週期點", "summary_zh": "反覆迭代自映射形成軌道；固定點週期為1，週期點的最小正週期須和任意返回時間分開。", "canonical_mko_id": "mko-orbit-fixed-periodic" }, "fields": { "title": "Orbits, fixed points and periodic points", "summary": "Repeated iteration of a self-map forms an orbit. Fixed points have period 1; the least positive period of a periodic point must be distinguished from an arbitrary return time." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-orbit-fixed-periodic", "edition_id": "edition-mko-orbit-fixed-periodic-en-v1", "source_field": "title", "source_mko_sha256": "702889d6acab1a9c1c6c8d26447c0b2a0201751460d118874f55c0486060083f", "edition_fields_sha256": "3434ccc1fc18d245aae4552143bd5ccb6bb48ad98f13bf7df53a4134a3257bbd" }, "summary": { "kind": "mko_edition", "mko_id": "mko-orbit-fixed-periodic", "edition_id": "edition-mko-orbit-fixed-periodic-en-v1", "source_field": "summary", "source_mko_sha256": "702889d6acab1a9c1c6c8d26447c0b2a0201751460d118874f55c0486060083f", "edition_fields_sha256": "3434ccc1fc18d245aae4552143bd5ccb6bb48ad98f13bf7df53a4134a3257bbd" } } }, "atlas-ordinary-differential-equation": { "source_snapshot": { "id": "atlas-ordinary-differential-equation", "title_zh": "常微分方程", "summary_zh": "常微分方程關聯未知單變數函數及其有限階導數；候選函數的殘差核對不等於存在唯一性定理。", "canonical_mko_id": "mko-ordinary-differential-equation" }, "fields": { "title": "Ordinary differential equation", "summary": "An ordinary differential equation relates an unknown single-variable function and finitely many derivatives; candidate residual checks are not existence or uniqueness theorems." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-ordinary-differential-equation", "edition_id": "edition-mko-ordinary-differential-equation-en-v1", "source_field": "title", "source_mko_sha256": "78a220ac6f5f781865fa9d3cdc17fd9d5f931ff83cfdd5890bfaae2b91641a13", "edition_fields_sha256": "3d9865f51e7b8fa2bcc02b33e3ee8768feae1e0d824d07e869a2fbb5cabfdc3f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-ordinary-differential-equation", "edition_id": "edition-mko-ordinary-differential-equation-en-v1", "source_field": "summary", "source_mko_sha256": "78a220ac6f5f781865fa9d3cdc17fd9d5f931ff83cfdd5890bfaae2b91641a13", "edition_fields_sha256": "3d9865f51e7b8fa2bcc02b33e3ee8768feae1e0d824d07e869a2fbb5cabfdc3f" } } }, "atlas-initial-value-problem": { "source_snapshot": { "id": "atlas-initial-value-problem", "title_zh": "初值問題", "summary_zh": "初值問題在微分方程外指定某點的函數值或導數值；候選核對與存在唯一性是不同責任。", "canonical_mko_id": "mko-initial-value-problem" }, "fields": { "title": "Initial-value problem", "summary": "An initial-value problem adds function or derivative values at a specified point to a differential equation; candidate checks are distinct from existence and uniqueness." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-initial-value-problem", "edition_id": "edition-mko-initial-value-problem-en-v1", "source_field": "title", "source_mko_sha256": "84ab0a91c84b74df8cfa9a3a8033b7ad83825d55e89f1d0ca847ed81f85e9925", "edition_fields_sha256": "74474b90cd96281cb3b8244ad70c59ea617ab6f11445db6e9411bf49d3cb7cf0" }, "summary": { "kind": "mko_edition", "mko_id": "mko-initial-value-problem", "edition_id": "edition-mko-initial-value-problem-en-v1", "source_field": "summary", "source_mko_sha256": "84ab0a91c84b74df8cfa9a3a8033b7ad83825d55e89f1d0ca847ed81f85e9925", "edition_fields_sha256": "74474b90cd96281cb3b8244ad70c59ea617ab6f11445db6e9411bf49d3cb7cf0" } } }, "atlas-numerical-approximation-error": { "source_snapshot": { "id": "atlas-numerical-approximation-error", "title_zh": "數值近似與誤差", "summary_zh": "絕對誤差是近似值與參照值之差的絕對值；相對誤差另要求參照值非零並除以其絕對值。", "canonical_mko_id": "mko-numerical-approximation-error" }, "fields": { "title": "Numerical approximation and error", "summary": "Absolute error is the absolute difference between approximation and reference; relative error additionally requires a nonzero reference and divides by its absolute value." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-numerical-approximation-error", "edition_id": "edition-mko-numerical-approximation-error-en-v1", "source_field": "title", "source_mko_sha256": "ced94d430d6417b9e607be1f09be7234591c3138507fecd536d3d7dc024ecbe0", "edition_fields_sha256": "a1edb0723817030df811184ca53172796a0c40ea0934411c03eaecb4b8c025a3" }, "summary": { "kind": "mko_edition", "mko_id": "mko-numerical-approximation-error", "edition_id": "edition-mko-numerical-approximation-error-en-v1", "source_field": "summary", "source_mko_sha256": "ced94d430d6417b9e607be1f09be7234591c3138507fecd536d3d7dc024ecbe0", "edition_fields_sha256": "a1edb0723817030df811184ca53172796a0c40ea0934411c03eaecb4b8c025a3" } } }, "atlas-bisection-method": { "source_snapshot": { "id": "atlas-bisection-method", "title_zh": "二分法的有限 bracket 更新", "summary_zh": "二分法在已提供的端點與中點符號資料上取中點並保留符號相反或零值的半區，有限步寬度減半。", "canonical_mko_id": "mko-bisection-method" }, "fields": { "title": "Finite sign-bracket updates in the bisection method", "summary": "Using supplied endpoint and midpoint signs, bisection takes a midpoint and retains the opposite-sign half or a detected zero; finite nonterminal bisections halve the interval width." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-bisection-method", "edition_id": "edition-mko-bisection-method-en-v1", "source_field": "title", "source_mko_sha256": "ab834bbfb6637765ed0a0282df9021cdc9509c95f67eb41e02c6d5db1774d835", "edition_fields_sha256": "9a653032a0f8265ee9a9a15077ff895a01c49e0b0bc2b6e70aa0f19ab3317d7f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-bisection-method", "edition_id": "edition-mko-bisection-method-en-v1", "source_field": "summary", "source_mko_sha256": "ab834bbfb6637765ed0a0282df9021cdc9509c95f67eb41e02c6d5db1774d835", "edition_fields_sha256": "9a653032a0f8265ee9a9a15077ff895a01c49e0b0bc2b6e70aa0f19ab3317d7f" } } }, "atlas-optimization-problem": { "source_snapshot": { "id": "atlas-optimization-problem", "title_zh": "最佳化問題", "summary_zh": "最佳化問題在明示可行集合上尋找使目標函數最小或最大的點；最優值與最優解集合需分開。", "canonical_mko_id": "mko-optimization-problem" }, "fields": { "title": "Optimization problem", "summary": "An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-optimization-problem", "edition_id": "edition-mko-optimization-problem-en-v1", "source_field": "title", "source_mko_sha256": "55c8accf83a0b3a16cb86f438bdc6575f8dcec45e670f324e863e79a1a233514", "edition_fields_sha256": "46c6ef69fe65cee9de285fc66e90456a8ffbbbaec52436fcb869b9a7dd70b482" }, "summary": { "kind": "mko_edition", "mko_id": "mko-optimization-problem", "edition_id": "edition-mko-optimization-problem-en-v1", "source_field": "summary", "source_mko_sha256": "55c8accf83a0b3a16cb86f438bdc6575f8dcec45e670f324e863e79a1a233514", "edition_fields_sha256": "46c6ef69fe65cee9de285fc66e90456a8ffbbbaec52436fcb869b9a7dd70b482" } } }, "atlas-linear-programming": { "source_snapshot": { "id": "atlas-linear-programming", "title_zh": "線性規劃", "summary_zh": "線性規劃在線性等式或不等式限制的可行域上最佳化線性目標函數；本輪題庫只驗完整有限候選網格。", "canonical_mko_id": "mko-linear-programming" }, "fields": { "title": "Linear programming", "summary": "Linear programming optimizes a linear objective over a feasible domain defined by linear equalities or inequalities; this exercise slice checks only a complete explicit finite candidate grid." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-programming", "edition_id": "edition-mko-linear-programming-en-v1", "source_field": "title", "source_mko_sha256": "e954454f13d345f635417e689ccae76261e754c182baf343173c0279585e8ba4", "edition_fields_sha256": "814f16737ef2a1edb00d5b0b02cae3b2027187d39e085fc7cf22edd24233323a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-linear-programming", "edition_id": "edition-mko-linear-programming-en-v1", "source_field": "summary", "source_mko_sha256": "e954454f13d345f635417e689ccae76261e754c182baf343173c0279585e8ba4", "edition_fields_sha256": "814f16737ef2a1edb00d5b0b02cae3b2027187d39e085fc7cf22edd24233323a" } } }, "atlas-dimensional-analysis": { "source_snapshot": { "id": "atlas-dimensional-analysis", "title_zh": "量綱分析", "summary_zh": "量綱分析用基本量綱指數向量追蹤乘、除與冪，並核對方程兩側量綱是否一致。", "canonical_mko_id": "mko-dimensional-analysis" }, "fields": { "title": "Dimensional analysis", "summary": "Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-dimensional-analysis", "edition_id": "edition-mko-dimensional-analysis-en-v1", "source_field": "title", "source_mko_sha256": "05ee29db7e8f4b787e36b2e536eb249e1b92597007252752dd6347b2dffab5eb", "edition_fields_sha256": "c47527a371dcae8cd2ad54ab2e3017dfc932a1b857e1f8ebdc918cd44fa47e2f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-dimensional-analysis", "edition_id": "edition-mko-dimensional-analysis-en-v1", "source_field": "summary", "source_mko_sha256": "05ee29db7e8f4b787e36b2e536eb249e1b92597007252752dd6347b2dffab5eb", "edition_fields_sha256": "c47527a371dcae8cd2ad54ab2e3017dfc932a1b857e1f8ebdc918cd44fa47e2f" } } }, "atlas-harmonic-oscillator-model": { "source_snapshot": { "id": "atlas-harmonic-oscillator-model", "title_zh": "線性簡諧振子模型", "summary_zh": "無阻尼線性簡諧振子以m x''+k x=0及m,k>0定義；本輪不宣稱三角函數閉式解。", "canonical_mko_id": "mko-harmonic-oscillator-model" }, "fields": { "title": "Linear harmonic-oscillator model", "summary": "The undamped linear harmonic oscillator is defined by m*x″+k*x=0 with m,k>0; this slice asserts no trigonometric closed-form solution." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-harmonic-oscillator-model", "edition_id": "edition-mko-harmonic-oscillator-model-en-v1", "source_field": "title", "source_mko_sha256": "920969c8aeda483138db1c073acdd7c8acdf5a854c7cef34fb51509539f7ebb4", "edition_fields_sha256": "b103cf6e22e581a195e08e87a89b23372000fefc96203a0a3ed488441f4958f2" }, "summary": { "kind": "mko_edition", "mko_id": "mko-harmonic-oscillator-model", "edition_id": "edition-mko-harmonic-oscillator-model-en-v1", "source_field": "summary", "source_mko_sha256": "920969c8aeda483138db1c073acdd7c8acdf5a854c7cef34fb51509539f7ebb4", "edition_fields_sha256": "b103cf6e22e581a195e08e87a89b23372000fefc96203a0a3ed488441f4958f2" } } }, "atlas-discrete-exponential-growth": { "source_snapshot": { "id": "atlas-discrete-exponential-growth", "title_zh": "離散指數成長模型", "summary_zh": "固定時間步與固定非負因子 r 的離散模型滿足 x_(n+1)=r x_n，因此 x_n=x_0 r^n。", "canonical_mko_id": "mko-discrete-exponential-growth" }, "fields": { "title": "Discrete exponential growth model", "summary": "With a fixed time step and fixed nonnegative factor r, the discrete recurrence x_(n+1)=r*x_n gives x_n=x_0*r^n." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-discrete-exponential-growth", "edition_id": "edition-mko-discrete-exponential-growth-en-v1", "source_field": "title", "source_mko_sha256": "81d7d953b56dc3b4f26d1051ae74ae30b298a2c6085c59c5f06439b68ea8e780", "edition_fields_sha256": "09b01662fb38e799dac43f2d5311657aa8533f84291a6cec28c585913f477dc9" }, "summary": { "kind": "mko_edition", "mko_id": "mko-discrete-exponential-growth", "edition_id": "edition-mko-discrete-exponential-growth-en-v1", "source_field": "summary", "source_mko_sha256": "81d7d953b56dc3b4f26d1051ae74ae30b298a2c6085c59c5f06439b68ea8e780", "edition_fields_sha256": "09b01662fb38e799dac43f2d5311657aa8533f84291a6cec28c585913f477dc9" } } }, "atlas-discrete-logistic-growth": { "source_snapshot": { "id": "atlas-discrete-logistic-growth", "title_zh": "離散 logistic 增量模型", "summary_zh": "固定容量K與0≤r≤1時，本條目以x_(n+1)=x_n+r x_n(1-x_n/K)描述每步受限增量。", "canonical_mko_id": "mko-discrete-logistic-growth" }, "fields": { "title": "Discrete logistic increment model", "summary": "For fixed capacity K and 0≤r≤1, this entry uses x_(n+1)=x_n+r*x_n*(1−x_n/K) to describe each capacity-limited discrete increment." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-discrete-logistic-growth", "edition_id": "edition-mko-discrete-logistic-growth-en-v1", "source_field": "title", "source_mko_sha256": "463931fbaa17a793076cfbf7af76f4490ba8c752e31fe52a9dc774d188ac3e4c", "edition_fields_sha256": "f21b5a14f7933fc7a3d09619682c20a0c49852a1f00d0f549b25fd3f0b7c1c64" }, "summary": { "kind": "mko_edition", "mko_id": "mko-discrete-logistic-growth", "edition_id": "edition-mko-discrete-logistic-growth-en-v1", "source_field": "summary", "source_mko_sha256": "463931fbaa17a793076cfbf7af76f4490ba8c752e31fe52a9dc774d188ac3e4c", "edition_fields_sha256": "f21b5a14f7933fc7a3d09619682c20a0c49852a1f00d0f549b25fd3f0b7c1c64" } } }, "atlas-compound-interest": { "source_snapshot": { "id": "atlas-compound-interest", "title_zh": "複利模型", "summary_zh": "固定每期利率 i 且無額外現金流時，餘額滿足 B_(n+1)=(1+i)B_n，故 B_n=B_0(1+i)^n。", "canonical_mko_id": "mko-compound-interest" }, "fields": { "title": "Compound-interest model", "summary": "With a fixed rate i per period and no additional cash flows, balances satisfy B_(n+1)=(1+i)B_n and hence B_n=B_0(1+i)^n." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-compound-interest", "edition_id": "edition-mko-compound-interest-en-v1", "source_field": "title", "source_mko_sha256": "5e3f823637101eaaf3ba73ed1b448b858ef9d83163c30769859bfcd7bbb22bb8", "edition_fields_sha256": "a1458a3a5249fc14134bff7f8035cfbdeeabfa877c07ed889ff2edba6962acea" }, "summary": { "kind": "mko_edition", "mko_id": "mko-compound-interest", "edition_id": "edition-mko-compound-interest-en-v1", "source_field": "summary", "source_mko_sha256": "5e3f823637101eaaf3ba73ed1b448b858ef9d83163c30769859bfcd7bbb22bb8", "edition_fields_sha256": "a1458a3a5249fc14134bff7f8035cfbdeeabfa877c07ed889ff2edba6962acea" } } }, "atlas-present-value-discounting": { "source_snapshot": { "id": "atlas-present-value-discounting", "title_zh": "現值與折現", "summary_zh": "固定正累積因子與期數時，未來金額除以累積因子冪得到現值。", "canonical_mko_id": "mko-present-value-discounting" }, "fields": { "title": "Present value and discounting", "summary": "For a fixed positive accumulation factor and period count, divide a future amount by the factor power to obtain present value." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-present-value-discounting", "edition_id": "edition-mko-present-value-discounting-en-v1", "source_field": "title", "source_mko_sha256": "2e47f9d121504d577795a46f4235f98114b5701015039cc83bef029cead220b0", "edition_fields_sha256": "4a54dc23212c3795ba5750750500406d2dcd33a380bad63a62b750d727c6c0c9" }, "summary": { "kind": "mko_edition", "mko_id": "mko-present-value-discounting", "edition_id": "edition-mko-present-value-discounting-en-v1", "source_field": "summary", "source_mko_sha256": "2e47f9d121504d577795a46f4235f98114b5701015039cc83bef029cead220b0", "edition_fields_sha256": "4a54dc23212c3795ba5750750500406d2dcd33a380bad63a62b750d727c6c0c9" } } }, "atlas-mathematical-representation": { "source_snapshot": { "id": "atlas-mathematical-representation", "title_zh": "數學表徵", "summary_zh": "數學表徵把同一對象的文字規則、符號式、有限表格、序對與座標資料分開記錄，並以明示語義橋核對一致性。", "canonical_mko_id": "mko-mathematical-representation" }, "fields": { "title": "Mathematical representation", "summary": "Mathematical representation separately records verbal rules, symbolic expressions, finite tables, ordered pairs and coordinate data for the same object, using explicit semantic bridges to check agreement." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-mathematical-representation", "edition_id": "edition-mko-mathematical-representation-en-v1", "source_field": "title", "source_mko_sha256": "4f70691fec7da4288ceee8ebb2ce291dfee4c28ac4a2e8aa851fa23fe5757fd5", "edition_fields_sha256": "a78f70b831bdb64221301bc549f2bffa168edfed04c6f05cd8c999d6d08ac167" }, "summary": { "kind": "mko_edition", "mko_id": "mko-mathematical-representation", "edition_id": "edition-mko-mathematical-representation-en-v1", "source_field": "summary", "source_mko_sha256": "4f70691fec7da4288ceee8ebb2ce291dfee4c28ac4a2e8aa851fa23fe5757fd5", "edition_fields_sha256": "a78f70b831bdb64221301bc549f2bffa168edfed04c6f05cd8c999d6d08ac167" } } }, "atlas-diagnostic-mathematical-assessment": { "source_snapshot": { "id": "atlas-diagnostic-mathematical-assessment", "title_zh": "診斷性數學評量記錄", "summary_zh": "診斷性數學評量記錄把題目、期望、作答與明示判定規則綁在一起，只產生可追溯的規則符合狀態。", "canonical_mko_id": "mko-diagnostic-mathematical-assessment" }, "fields": { "title": "Diagnostic mathematical assessment records", "summary": "A diagnostic mathematical assessment record binds an item, its expected result, a response and an explicit judgment rule, producing only a traceable rule-agreement status." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-diagnostic-mathematical-assessment", "edition_id": "edition-mko-diagnostic-mathematical-assessment-en-v1", "source_field": "title", "source_mko_sha256": "a28efe40ed4cdaae147307d664cab323409357f5deb8a840a56e4cb0ce9a40ce", "edition_fields_sha256": "28ff4a342c1ffc34fc66cbab71aec1c2e5eb65df8aacd7cec401f12b6b8b07d4" }, "summary": { "kind": "mko_edition", "mko_id": "mko-diagnostic-mathematical-assessment", "edition_id": "edition-mko-diagnostic-mathematical-assessment-en-v1", "source_field": "summary", "source_mko_sha256": "a28efe40ed4cdaae147307d664cab323409357f5deb8a840a56e4cb0ce9a40ce", "edition_fields_sha256": "28ff4a342c1ffc34fc66cbab71aec1c2e5eb65df8aacd7cec401f12b6b8b07d4" } } }, "atlas-complex-number": { "source_snapshot": { "id": "atlas-complex-number", "title_zh": "複數", "summary_zh": "複數以實數有序對(a,b)表示a+bi，並以i²=-1定義加減乘法。", "canonical_mko_id": "mko-complex-number" }, "fields": { "title": "Complex number", "summary": "A complex number a+b*i is represented by the ordered real pair (a,b), with addition, subtraction and multiplication defined using i²=−1." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-complex-number", "edition_id": "edition-mko-complex-number-en-v1", "source_field": "title", "source_mko_sha256": "a0d0b2223ccef6be06ad99f3ba6f05b76e1570d35ae43f7caac2509712c0431a", "edition_fields_sha256": "a83f15fc94e06982d937e9f62e4c47f9547569be7e0220dd2abb710b46b3d5dd" }, "summary": { "kind": "mko_edition", "mko_id": "mko-complex-number", "edition_id": "edition-mko-complex-number-en-v1", "source_field": "summary", "source_mko_sha256": "a0d0b2223ccef6be06ad99f3ba6f05b76e1570d35ae43f7caac2509712c0431a", "edition_fields_sha256": "a83f15fc94e06982d937e9f62e4c47f9547569be7e0220dd2abb710b46b3d5dd" } } }, "atlas-complex-modulus-conjugate": { "source_snapshot": { "id": "atlas-complex-modulus-conjugate", "title_zh": "複數模與共軛", "summary_zh": "複數共軛保留實部並反轉虛部；模平方為實部與虛部平方和。", "canonical_mko_id": "mko-complex-modulus-conjugate" }, "fields": { "title": "Complex modulus and conjugation", "summary": "Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-complex-modulus-conjugate", "edition_id": "edition-mko-complex-modulus-conjugate-en-v1", "source_field": "title", "source_mko_sha256": "c4c1f275b4ae1708bd308a1ea5ce57ee90a10921251139a1eb880ffd2101842d", "edition_fields_sha256": "853c5a7d9931d78ed4f3fd1c5a3100c80b220ea3c4277a5d2ccaf42ff047c53c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-complex-modulus-conjugate", "edition_id": "edition-mko-complex-modulus-conjugate-en-v1", "source_field": "summary", "source_mko_sha256": "c4c1f275b4ae1708bd308a1ea5ce57ee90a10921251139a1eb880ffd2101842d", "edition_fields_sha256": "853c5a7d9931d78ed4f3fd1c5a3100c80b220ea3c4277a5d2ccaf42ff047c53c" } } }, "atlas-real-vector-space": { "source_snapshot": { "id": "atlas-real-vector-space", "title_zh": "實向量空間", "summary_zh": "實向量空間配備向量加法與實數純量乘法，滿足加法交換群及相容的純量作用公理。", "canonical_mko_id": "mko-real-vector-space" }, "fields": { "title": "Real vector space", "summary": "A real vector space has vector addition and real scalar multiplication, satisfying the additive abelian-group and compatible scalar-action axioms." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-real-vector-space", "edition_id": "edition-mko-real-vector-space-en-v1", "source_field": "title", "source_mko_sha256": "d57c98eb1c006f761ecaaf2870e2a09b30e5a29627351005b7e3bb7b30b33356", "edition_fields_sha256": "db4a1a3f3f9603da03118ae5f9b56713840edcf11ed9e5aee02a609b7aedd196" }, "summary": { "kind": "mko_edition", "mko_id": "mko-real-vector-space", "edition_id": "edition-mko-real-vector-space-en-v1", "source_field": "summary", "source_mko_sha256": "d57c98eb1c006f761ecaaf2870e2a09b30e5a29627351005b7e3bb7b30b33356", "edition_fields_sha256": "db4a1a3f3f9603da03118ae5f9b56713840edcf11ed9e5aee02a609b7aedd196" } } }, "atlas-linear-combination-span": { "source_snapshot": { "id": "atlas-linear-combination-span", "title_zh": "線性組合與生成", "summary_zh": "線性組合是有限個向量的純量加權和；生成空間收集全部有限線性組合。", "canonical_mko_id": "mko-linear-combination-span" }, "fields": { "title": "Linear combinations and span", "summary": "A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-combination-span", "edition_id": "edition-mko-linear-combination-span-en-v1", "source_field": "title", "source_mko_sha256": "6f4f8ff504708ab9492740d70255cb2d85c406d8eaa891eef387935b3d46dad3", "edition_fields_sha256": "535d6e3a8e4fb1851318c735037ef908e527b7db1faf6845f38995e7d80986ba" }, "summary": { "kind": "mko_edition", "mko_id": "mko-linear-combination-span", "edition_id": "edition-mko-linear-combination-span-en-v1", "source_field": "summary", "source_mko_sha256": "6f4f8ff504708ab9492740d70255cb2d85c406d8eaa891eef387935b3d46dad3", "edition_fields_sha256": "535d6e3a8e4fb1851318c735037ef908e527b7db1faf6845f38995e7d80986ba" } } }, "atlas-linear-independence-basis": { "source_snapshot": { "id": "atlas-linear-independence-basis", "title_zh": "線性獨立與基底", "summary_zh": "線性獨立排除非平凡零和關係；基底同時線性獨立並生成已宣告空間。", "canonical_mko_id": "mko-linear-independence-basis" }, "fields": { "title": "Linear independence and bases", "summary": "Linear independence excludes nontrivial zero-sum coefficient relations; a basis is both independent and spanning for the declared space." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-independence-basis", "edition_id": "edition-mko-linear-independence-basis-en-v1", "source_field": "title", "source_mko_sha256": "78c45988a718f6a7ca4acf2aba00ea4df794c8f141cefe822bf72206fa2b2f62", "edition_fields_sha256": "7789a1ce3f3c74109c2ff9f86568326eb2281717a4994f68a5674a4d60c87b83" }, "summary": { "kind": "mko_edition", "mko_id": "mko-linear-independence-basis", "edition_id": "edition-mko-linear-independence-basis-en-v1", "source_field": "summary", "source_mko_sha256": "78c45988a718f6a7ca4acf2aba00ea4df794c8f141cefe822bf72206fa2b2f62", "edition_fields_sha256": "7789a1ce3f3c74109c2ff9f86568326eb2281717a4994f68a5674a4d60c87b83" } } }, "atlas-finite-group": { "source_snapshot": { "id": "atlas-finite-group", "title_zh": "有限群", "summary_zh": "有限群由有限集合與完整封閉Cayley表構成，並滿足結合律、唯一單位元及每元素逆元。", "canonical_mko_id": "mko-finite-group" }, "fields": { "title": "Finite group", "summary": "A finite group has a finite carrier and complete closed Cayley table, satisfying associativity, a unique identity and inverses for every element." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-group", "edition_id": "edition-mko-finite-group-en-v1", "source_field": "title", "source_mko_sha256": "491d28b1773ac105187d1f17664e8e55b1ab4831613c354693901e40fae75f01", "edition_fields_sha256": "efb57cafe7e26d4d0a7775479a2a3a729a49ca8fa12ddb5960435b246083b3ca" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-group", "edition_id": "edition-mko-finite-group-en-v1", "source_field": "summary", "source_mko_sha256": "491d28b1773ac105187d1f17664e8e55b1ab4831613c354693901e40fae75f01", "edition_fields_sha256": "efb57cafe7e26d4d0a7775479a2a3a729a49ca8fa12ddb5960435b246083b3ca" } } }, "atlas-subgroup-homomorphism": { "source_snapshot": { "id": "atlas-subgroup-homomorphism", "title_zh": "子群與群同態", "summary_zh": "子群在繼承運算下自身成群；群同態保持乘法，kernel與image由完整映射表導出。", "canonical_mko_id": "mko-subgroup-homomorphism" }, "fields": { "title": "Subgroups and group homomorphisms", "summary": "A subgroup is a group under the inherited operation; a group homomorphism preserves multiplication, and its kernel and image follow from the complete map table." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-subgroup-homomorphism", "edition_id": "edition-mko-subgroup-homomorphism-en-v1", "source_field": "title", "source_mko_sha256": "2121cf9c71fab6aa9bfa10d881cdffbab5367fdfebbee9381b35fefec60837c0", "edition_fields_sha256": "4567e1e6b312d53df4f69ffeb8c6013d123535b6c81471959bffa5ea1b1a27e7" }, "summary": { "kind": "mko_edition", "mko_id": "mko-subgroup-homomorphism", "edition_id": "edition-mko-subgroup-homomorphism-en-v1", "source_field": "summary", "source_mko_sha256": "2121cf9c71fab6aa9bfa10d881cdffbab5367fdfebbee9381b35fefec60837c0", "edition_fields_sha256": "4567e1e6b312d53df4f69ffeb8c6013d123535b6c81471959bffa5ea1b1a27e7" } } }, "atlas-metric-space": { "source_snapshot": { "id": "atlas-metric-space", "title_zh": "度量空間", "summary_zh": "度量空間由集合與距離函數組成，距離非負、只在相同點為零、對稱且滿足三角不等式。", "canonical_mko_id": "mko-metric-space" }, "fields": { "title": "Metric space", "summary": "A metric space has a carrier set and a distance function that is nonnegative, zero exactly at equal points, symmetric and satisfies the triangle inequality." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-metric-space", "edition_id": "edition-mko-metric-space-en-v1", "source_field": "title", "source_mko_sha256": "50cc56a6caf28683227e2793c67e48b7091d583a514abf7fc895eb8fa92898a3", "edition_fields_sha256": "1943ec397a9f5b8d0ad437eb26c6fe951b1d27b95db5fe2727c183b09396bc88" }, "summary": { "kind": "mko_edition", "mko_id": "mko-metric-space", "edition_id": "edition-mko-metric-space-en-v1", "source_field": "summary", "source_mko_sha256": "50cc56a6caf28683227e2793c67e48b7091d583a514abf7fc895eb8fa92898a3", "edition_fields_sha256": "1943ec397a9f5b8d0ad437eb26c6fe951b1d27b95db5fe2727c183b09396bc88" } } }, "atlas-metric-convergence": { "source_snapshot": { "id": "atlas-metric-convergence", "title_zh": "度量空間中的收斂", "summary_zh": "度量空間序列收斂到x，表示每個正epsilon終有所有n≥N落入x的epsilon球。", "canonical_mko_id": "mko-metric-convergence" }, "fields": { "title": "Convergence in a metric space", "summary": "A sequence in a metric space converges to x when, for every positive epsilon, eventually all n≥N lie in the epsilon-ball around x." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-metric-convergence", "edition_id": "edition-mko-metric-convergence-en-v1", "source_field": "title", "source_mko_sha256": "185db7b6b678014d6791d0448e2eccc1efc643b288d9b95c2b7b91f3ea710969", "edition_fields_sha256": "94e1cebef1216cd8e88dd2a85b7b7d99d32cb24315ece2d04a6775dd5b645646" }, "summary": { "kind": "mko_edition", "mko_id": "mko-metric-convergence", "edition_id": "edition-mko-metric-convergence-en-v1", "source_field": "summary", "source_mko_sha256": "185db7b6b678014d6791d0448e2eccc1efc643b288d9b95c2b7b91f3ea710969", "edition_fields_sha256": "94e1cebef1216cd8e88dd2a85b7b7d99d32cb24315ece2d04a6775dd5b645646" } } }, "atlas-sigma-algebra-measure-space": { "source_snapshot": { "id": "atlas-sigma-algebra-measure-space", "title_zh": "σ-代數與測度空間", "summary_zh": "σ-代數含空集並對補集與可數聯集封閉；測度非負、空集為零且可數可加。", "canonical_mko_id": "mko-sigma-algebra-measure-space" }, "fields": { "title": "Sigma-algebras and measure spaces", "summary": "A sigma-algebra contains the empty set and is closed under complements and countable unions; a measure is nonnegative, zero on the empty set and countably additive." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-sigma-algebra-measure-space", "edition_id": "edition-mko-sigma-algebra-measure-space-en-v1", "source_field": "title", "source_mko_sha256": "36f29056738113842fd5fc437c89ae6f234815fa3abaea4cc68e968a358d6650", "edition_fields_sha256": "cd8dd03cbb0da57d66c01ad08071e0fe618e4a647e45a16084e368b2262a96b2" }, "summary": { "kind": "mko_edition", "mko_id": "mko-sigma-algebra-measure-space", "edition_id": "edition-mko-sigma-algebra-measure-space-en-v1", "source_field": "summary", "source_mko_sha256": "36f29056738113842fd5fc437c89ae6f234815fa3abaea4cc68e968a358d6650", "edition_fields_sha256": "cd8dd03cbb0da57d66c01ad08071e0fe618e4a647e45a16084e368b2262a96b2" } } }, "atlas-measurable-function": { "source_snapshot": { "id": "atlas-measurable-function", "title_zh": "可測函數", "summary_zh": "來源與目標各固定sigma代數時，函數可測表示每個目標可測集的逆像皆在來源sigma代數。", "canonical_mko_id": "mko-measurable-function" }, "fields": { "title": "Measurable function", "summary": "With fixed source and target sigma-algebras, a function is measurable when the preimage of every target measurable set belongs to the source sigma-algebra." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-measurable-function", "edition_id": "edition-mko-measurable-function-en-v1", "source_field": "title", "source_mko_sha256": "0c21c3c35f2f6b0451adf573f920b3dd8e797c631bcae8c14bf6bd959dc1571b", "edition_fields_sha256": "1f5576cd6a6eb68ede8a196ac418655e612e2ea2e10215213f85fc7d0f7eeb88" }, "summary": { "kind": "mko_edition", "mko_id": "mko-measurable-function", "edition_id": "edition-mko-measurable-function-en-v1", "source_field": "summary", "source_mko_sha256": "0c21c3c35f2f6b0451adf573f920b3dd8e797c631bcae8c14bf6bd959dc1571b", "edition_fields_sha256": "1f5576cd6a6eb68ede8a196ac418655e612e2ea2e10215213f85fc7d0f7eeb88" } } }, "atlas-finite-discrete-probability-distribution": { "source_snapshot": { "id": "atlas-finite-discrete-probability-distribution", "title_zh": "有限離散機率分布", "summary_zh": "有限離散分布是在有限結果值上指定非負機率質量，全部質量精確和為一。", "canonical_mko_id": "mko-finite-discrete-probability-distribution" }, "fields": { "title": "Finite discrete probability distribution", "summary": "A finite discrete distribution assigns nonnegative probability masses to finitely many outcome values, with exact total mass one." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-discrete-probability-distribution", "edition_id": "edition-mko-finite-discrete-probability-distribution-en-v1", "source_field": "title", "source_mko_sha256": "a77aa248d456c6fff168a9df6f8b4ad85be410912a036130257ff1d3e0aa1b2a", "edition_fields_sha256": "a0d94aaf8a80a463ca472650a2b3051e53b0934b8ec6270d4a4afccf3ea9e9ae" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-discrete-probability-distribution", "edition_id": "edition-mko-finite-discrete-probability-distribution-en-v1", "source_field": "summary", "source_mko_sha256": "a77aa248d456c6fff168a9df6f8b4ad85be410912a036130257ff1d3e0aa1b2a", "edition_fields_sha256": "a0d94aaf8a80a463ca472650a2b3051e53b0934b8ec6270d4a4afccf3ea9e9ae" } } }, "atlas-discrete-stochastic-process": { "source_snapshot": { "id": "atlas-discrete-stochastic-process", "title_zh": "離散隨機過程", "summary_zh": "離散隨機過程是共同機率空間上的離散時間隨機變數族；邊際分布不決定聯合律。", "canonical_mko_id": "mko-discrete-stochastic-process" }, "fields": { "title": "Discrete-time stochastic process", "summary": "A discrete-time stochastic process is a time-indexed family of random variables on one probability space; marginal distributions do not determine its joint law." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-discrete-stochastic-process", "edition_id": "edition-mko-discrete-stochastic-process-en-v1", "source_field": "title", "source_mko_sha256": "1b73d458f056e14bbaaf318ebfecf7e93214b9895dae967bfcffc48910854832", "edition_fields_sha256": "351b6a80457e846aeb52110ab5efcaa47fb0be12fb0f02e0ce8161f2e87c7d50" }, "summary": { "kind": "mko_edition", "mko_id": "mko-discrete-stochastic-process", "edition_id": "edition-mko-discrete-stochastic-process-en-v1", "source_field": "summary", "source_mko_sha256": "1b73d458f056e14bbaaf318ebfecf7e93214b9895dae967bfcffc48910854832", "edition_fields_sha256": "351b6a80457e846aeb52110ab5efcaa47fb0be12fb0f02e0ce8161f2e87c7d50" } } }, "atlas-finite-markov-chain": { "source_snapshot": { "id": "atlas-finite-markov-chain", "title_zh": "有限狀態 Markov 鏈", "summary_zh": "有限時間齊次Markov鏈以row-stochastic轉移核與需要時的初始律指定有限路徑機率。", "canonical_mko_id": "mko-finite-markov-chain" }, "fields": { "title": "Finite-state Markov chain", "summary": "A finite time-homogeneous Markov chain uses a row-stochastic transition kernel and, when needed, an initial law to specify finite path probabilities." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-markov-chain", "edition_id": "edition-mko-finite-markov-chain-en-v1", "source_field": "title", "source_mko_sha256": "713cef8fb3c1efc7627452592ab30fb69b9ebb7992c47aad78b9ae322ee7fd55", "edition_fields_sha256": "aec970bc3fab9d11a510a8e8b7d14c1efc8959128a3e7604753df1f10eedbcfd" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-markov-chain", "edition_id": "edition-mko-finite-markov-chain-en-v1", "source_field": "summary", "source_mko_sha256": "713cef8fb3c1efc7627452592ab30fb69b9ebb7992c47aad78b9ae322ee7fd55", "edition_fields_sha256": "aec970bc3fab9d11a510a8e8b7d14c1efc8959128a3e7604753df1f10eedbcfd" } } }, "atlas-multivariable-real-function": { "source_snapshot": { "id": "atlas-multivariable-real-function", "title_zh": "多變數實函數", "summary_zh": "多變數實函數把明示實向量域中的輸入映到實數；座標角色與定義域是函數資料。", "canonical_mko_id": "mko-multivariable-real-function" }, "fields": { "title": "Multivariable real-valued function", "summary": "A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-multivariable-real-function", "edition_id": "edition-mko-multivariable-real-function-en-v1", "source_field": "title", "source_mko_sha256": "f96ebe8babf93fa8de6b32cb4156ad2b710b938550218a6aff16be79ba4ef928", "edition_fields_sha256": "b20bbacf20775d913062ba633583c71693ee1018bae1502a4625158df0ac40c7" }, "summary": { "kind": "mko_edition", "mko_id": "mko-multivariable-real-function", "edition_id": "edition-mko-multivariable-real-function-en-v1", "source_field": "summary", "source_mko_sha256": "f96ebe8babf93fa8de6b32cb4156ad2b710b938550218a6aff16be79ba4ef928", "edition_fields_sha256": "b20bbacf20775d913062ba633583c71693ee1018bae1502a4625158df0ac40c7" } } }, "atlas-partial-derivative": { "source_snapshot": { "id": "atlas-partial-derivative", "title_zh": "偏導數", "summary_zh": "偏導數固定其他座標，只沿一個座標切片取h趨近零的單變數導數。", "canonical_mko_id": "mko-partial-derivative" }, "fields": { "title": "Partial derivative", "summary": "A partial derivative fixes every other coordinate and takes the one-variable derivative along one coordinate slice as h tends to zero." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-partial-derivative", "edition_id": "edition-mko-partial-derivative-en-v1", "source_field": "title", "source_mko_sha256": "7159526c8e30cc8cb9086f9690d279fd3852328abe371206b3441ecf6ef69ce8", "edition_fields_sha256": "0efcc125f7938357bff2763461ecd48125092bc04835163a20293dee8292886c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-partial-derivative", "edition_id": "edition-mko-partial-derivative-en-v1", "source_field": "summary", "source_mko_sha256": "7159526c8e30cc8cb9086f9690d279fd3852328abe371206b3441ecf6ef69ce8", "edition_fields_sha256": "0efcc125f7938357bff2763461ecd48125092bc04835163a20293dee8292886c" } } }, "atlas-partial-differential-equation": { "source_snapshot": { "id": "atlas-partial-differential-equation", "title_zh": "偏微分方程", "summary_zh": "偏微分方程關聯多變數未知函數與一個或多個偏導數；殘差與主部分類需固定算子convention。", "canonical_mko_id": "mko-partial-differential-equation" }, "fields": { "title": "Partial differential equation", "summary": "A partial differential equation relates a multivariable unknown function and one or more partial derivatives; residuals and principal-part classification require a fixed operator convention." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-partial-differential-equation", "edition_id": "edition-mko-partial-differential-equation-en-v1", "source_field": "title", "source_mko_sha256": "b78ef1b1a613523b784d17e0507361cf263e495850e2b7aa880558f1294758b8", "edition_fields_sha256": "39301c2597005a00404eb4c47751b76e05c72d39246deb5f2636f17c165ad1b0" }, "summary": { "kind": "mko_edition", "mko_id": "mko-partial-differential-equation", "edition_id": "edition-mko-partial-differential-equation-en-v1", "source_field": "summary", "source_mko_sha256": "b78ef1b1a613523b784d17e0507361cf263e495850e2b7aa880558f1294758b8", "edition_fields_sha256": "39301c2597005a00404eb4c47751b76e05c72d39246deb5f2636f17c165ad1b0" } } }, "atlas-boundary-value-problem": { "source_snapshot": { "id": "atlas-boundary-value-problem", "title_zh": "邊界值問題", "summary_zh": "邊界值問題把定義域內的微分方程責任與邊界上的值或導數資料同時指定；候選通過不等於適定性。", "canonical_mko_id": "mko-boundary-value-problem" }, "fields": { "title": "Boundary-value problem", "summary": "A boundary-value problem specifies both an interior differential equation and boundary values or derivative data; successful candidate checks do not establish general well-posedness." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-boundary-value-problem", "edition_id": "edition-mko-boundary-value-problem-en-v1", "source_field": "title", "source_mko_sha256": "4ffaeecac13219cb387fd43a669823ff3c2b402bfadf2e3d04f3938d83bf3f0d", "edition_fields_sha256": "fc5e1e16f513c2be6a04ffb01caae37015c5324cf946e6a046486cbc7aa7992c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-boundary-value-problem", "edition_id": "edition-mko-boundary-value-problem-en-v1", "source_field": "summary", "source_mko_sha256": "4ffaeecac13219cb387fd43a669823ff3c2b402bfadf2e3d04f3938d83bf3f0d", "edition_fields_sha256": "fc5e1e16f513c2be6a04ffb01caae37015c5324cf946e6a046486cbc7aa7992c" } } }, "atlas-finite-convex-combination": { "source_snapshot": { "id": "atlas-finite-convex-combination", "title_zh": "有限凸組合", "summary_zh": "有限凸組合用非負精確權重加權有限非空向量列，且權重總和恰為一。", "canonical_mko_id": "mko-finite-convex-combination" }, "fields": { "title": "Finite convex combination", "summary": "A finite convex combination weights a finite nonempty ordered vector list by exact nonnegative weights whose total is exactly one." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-convex-combination", "edition_id": "edition-mko-finite-convex-combination-en-v1", "source_field": "title", "source_mko_sha256": "84206c94ef6c80e08ea032783fce4ef3d1e4b15b55088556463d360b07c78d80", "edition_fields_sha256": "51e217ea4eabcd65eaf3752d9526e95c690f868741a81ecc9ebd412d0766636b" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-convex-combination", "edition_id": "edition-mko-finite-convex-combination-en-v1", "source_field": "summary", "source_mko_sha256": "84206c94ef6c80e08ea032783fce4ef3d1e4b15b55088556463d360b07c78d80", "edition_fields_sha256": "51e217ea4eabcd65eaf3752d9526e95c690f868741a81ecc9ebd412d0766636b" } } }, "atlas-convex-set": { "source_snapshot": { "id": "atlas-convex-set", "title_zh": "凸集合", "summary_zh": "集合凸表示任兩點及每個lambda∈[0,1]形成的整條線段仍在集合內。", "canonical_mko_id": "mko-convex-set" }, "fields": { "title": "Convex set", "summary": "A set is convex when the entire segment formed by every pair of its points and every lambda in [0,1] remains in the set." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-convex-set", "edition_id": "edition-mko-convex-set-en-v1", "source_field": "title", "source_mko_sha256": "0d52be71dca290f856018d176bbb26dcd877c887031f690bd1d656db017ce02d", "edition_fields_sha256": "c96ec6dc78bd9b7650261251dbddb55845d3c1f35229283fcd6f4f8d0277fde9" }, "summary": { "kind": "mko_edition", "mko_id": "mko-convex-set", "edition_id": "edition-mko-convex-set-en-v1", "source_field": "summary", "source_mko_sha256": "0d52be71dca290f856018d176bbb26dcd877c887031f690bd1d656db017ce02d", "edition_fields_sha256": "c96ec6dc78bd9b7650261251dbddb55845d3c1f35229283fcd6f4f8d0277fde9" } } }, "atlas-convex-function": { "source_snapshot": { "id": "atlas-convex-function", "title_zh": "凸函數", "summary_zh": "凸域上的函數為凸，若每個兩點凸組合處的函數值不超過相同權重的端點函數值。", "canonical_mko_id": "mko-convex-function" }, "fields": { "title": "Convex function", "summary": "A function on a convex domain is convex when its value at every two-point convex combination is at most the same weighted combination of endpoint values." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-convex-function", "edition_id": "edition-mko-convex-function-en-v1", "source_field": "title", "source_mko_sha256": "d5af460bbe41c8e967949f9398bde368d76471cd2b4a24e4f49f14ad90a05fed", "edition_fields_sha256": "31033502108883a0d8cc426de539cbb610e918bd6860145f91df6e035365761b" }, "summary": { "kind": "mko_edition", "mko_id": "mko-convex-function", "edition_id": "edition-mko-convex-function-en-v1", "source_field": "summary", "source_mko_sha256": "d5af460bbe41c8e967949f9398bde368d76471cd2b4a24e4f49f14ad90a05fed", "edition_fields_sha256": "31033502108883a0d8cc426de539cbb610e918bd6860145f91df6e035365761b" } } }, "atlas-convex-optimization-problem": { "source_snapshot": { "id": "atlas-convex-optimization-problem", "title_zh": "凸最佳化問題", "summary_zh": "凸最佳化問題在凸可行域上最小化凸目標；問題資格、可行性、求值與全域最優是不同責任。", "canonical_mko_id": "mko-convex-optimization-problem" }, "fields": { "title": "Convex optimization problem", "summary": "Convex optimization minimizes a convex objective on a convex feasible domain; problem qualification, feasibility, evaluation and global optimality are distinct obligations." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-convex-optimization-problem", "edition_id": "edition-mko-convex-optimization-problem-en-v1", "source_field": "title", "source_mko_sha256": "0484bac345126653c2ee68f9057b0c7bf124c996ee7e7cfd461d32ad4e78747b", "edition_fields_sha256": "bddb7c5f729c0979e12eb6cfb348ea40b021804a9ffe175bc22e6d8e87bff57c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-convex-optimization-problem", "edition_id": "edition-mko-convex-optimization-problem-en-v1", "source_field": "summary", "source_mko_sha256": "0484bac345126653c2ee68f9057b0c7bf124c996ee7e7cfd461d32ad4e78747b", "edition_fields_sha256": "bddb7c5f729c0979e12eb6cfb348ea40b021804a9ffe175bc22e6d8e87bff57c" } } }, "atlas-finite-abelian-group": { "source_snapshot": { "id": "atlas-finite-abelian-group", "title_zh": "有限阿貝爾群", "summary_zh": "有限群若完整Cayley表對所有元素對都滿足交換律，則為有限阿貝爾群。", "canonical_mko_id": "mko-finite-abelian-group" }, "fields": { "title": "Finite abelian group", "summary": "A finite group is abelian when its complete Cayley table satisfies commutativity for every pair of elements." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-abelian-group", "edition_id": "edition-mko-finite-abelian-group-en-v1", "source_field": "title", "source_mko_sha256": "1486476569952aee082941d6f91d2921526de39dd732a11bacce98c252b95415", "edition_fields_sha256": "a03e1a04bdeb36b3d0ec8e812daf7dc1fcb8e15635bc61292bf96d7a5f9ad3c8" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-abelian-group", "edition_id": "edition-mko-finite-abelian-group-en-v1", "source_field": "summary", "source_mko_sha256": "1486476569952aee082941d6f91d2921526de39dd732a11bacce98c252b95415", "edition_fields_sha256": "a03e1a04bdeb36b3d0ec8e812daf7dc1fcb8e15635bc61292bf96d7a5f9ad3c8" } } }, "atlas-finite-unital-ring": { "source_snapshot": { "id": "atlas-finite-unital-ring", "title_zh": "有限含一環", "summary_zh": "有限含一環在同一完整有限載體上具加法阿貝爾群、乘法結合律、左右分配律及與零不同的乘法單位元。", "canonical_mko_id": "mko-finite-unital-ring" }, "fields": { "title": "Finite unital ring", "summary": "On one complete finite carrier, this unital-ring protocol requires an additive abelian group, associative multiplication, both distributive laws and a multiplicative identity different from zero." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-unital-ring", "edition_id": "edition-mko-finite-unital-ring-en-v1", "source_field": "title", "source_mko_sha256": "e930e72ae14b44b56feb441e48526b614a1bfb2f281a1518ec6c7b19b1de48bf", "edition_fields_sha256": "92abfc8ab707950262bcd31689bfc952dea7bf9693ea9da2c7d6e2cd0d00fc2d" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-unital-ring", "edition_id": "edition-mko-finite-unital-ring-en-v1", "source_field": "summary", "source_mko_sha256": "e930e72ae14b44b56feb441e48526b614a1bfb2f281a1518ec6c7b19b1de48bf", "edition_fields_sha256": "92abfc8ab707950262bcd31689bfc952dea7bf9693ea9da2c7d6e2cd0d00fc2d" } } }, "atlas-finite-commutative-ring": { "source_snapshot": { "id": "atlas-finite-commutative-ring", "title_zh": "有限交換環", "summary_zh": "有限含一環若乘法對全部元素對交換，則為有限交換環；零因子仍可能存在。", "canonical_mko_id": "mko-finite-commutative-ring" }, "fields": { "title": "Finite commutative ring", "summary": "A finite unital ring is commutative when multiplication commutes for every pair of elements; zero divisors may still exist." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-commutative-ring", "edition_id": "edition-mko-finite-commutative-ring-en-v1", "source_field": "title", "source_mko_sha256": "5a488ffcb874034a7b2144c7585ccef8981c9d6bc0d2959107aa85650113234a", "edition_fields_sha256": "fa030a14a5533d55e93a416337b50a90c40799e3c69525905db7348988c43926" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-commutative-ring", "edition_id": "edition-mko-finite-commutative-ring-en-v1", "source_field": "summary", "source_mko_sha256": "5a488ffcb874034a7b2144c7585ccef8981c9d6bc0d2959107aa85650113234a", "edition_fields_sha256": "fa030a14a5533d55e93a416337b50a90c40799e3c69525905db7348988c43926" } } }, "atlas-finite-integral-domain": { "source_snapshot": { "id": "atlas-finite-integral-domain", "title_zh": "有限整環", "summary_zh": "有限交換含一環若1不等於0且沒有非零零因子，則為有限整環。", "canonical_mko_id": "mko-finite-integral-domain" }, "fields": { "title": "Finite integral domain", "summary": "A finite commutative unital ring with 1 distinct from 0 is an integral domain when it has no nonzero zero divisors." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-integral-domain", "edition_id": "edition-mko-finite-integral-domain-en-v1", "source_field": "title", "source_mko_sha256": "5aefd54faece27112b7f0cc473cb7465bf2c77336f1d8ebfa180261027ba56fc", "edition_fields_sha256": "66b127cfc7cf87a333416e96ed636215a8c8cbc6c85eaa66a944ec73d3a4094c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-integral-domain", "edition_id": "edition-mko-finite-integral-domain-en-v1", "source_field": "summary", "source_mko_sha256": "5aefd54faece27112b7f0cc473cb7465bf2c77336f1d8ebfa180261027ba56fc", "edition_fields_sha256": "66b127cfc7cf87a333416e96ed636215a8c8cbc6c85eaa66a944ec73d3a4094c" } } }, "atlas-finite-field": { "source_snapshot": { "id": "atlas-finite-field", "title_zh": "有限域", "summary_zh": "有限交換含一環中每個非零元素皆有乘法逆元時構成有限域。", "canonical_mko_id": "mko-finite-field" }, "fields": { "title": "Finite field", "summary": "A finite commutative unital ring is a finite field when every nonzero element has a multiplicative inverse." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-field", "edition_id": "edition-mko-finite-field-en-v1", "source_field": "title", "source_mko_sha256": "a0f82af13b19a69bc7a3bab914cede8b8bae9db78543343f63b0b4086a684c5e", "edition_fields_sha256": "97c87cc49977f461ed0bef167ae77dc820567236c581652061463d0511aeca8a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-field", "edition_id": "edition-mko-finite-field-en-v1", "source_field": "summary", "source_mko_sha256": "a0f82af13b19a69bc7a3bab914cede8b8bae9db78543343f63b0b4086a684c5e", "edition_fields_sha256": "97c87cc49977f461ed0bef167ae77dc820567236c581652061463d0511aeca8a" } } }, "atlas-finite-two-sided-ideal": { "source_snapshot": { "id": "atlas-finite-two-sided-ideal", "title_zh": "有限雙邊理想", "summary_zh": "有限環子集若為加法子群且吸收父環的左右乘法，則為雙邊理想。", "canonical_mko_id": "mko-finite-two-sided-ideal" }, "fields": { "title": "Finite two-sided ideal", "summary": "A subset of a finite ring is a two-sided ideal when it is an additive subgroup and absorbs multiplication by parent-ring elements on both sides." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-two-sided-ideal", "edition_id": "edition-mko-finite-two-sided-ideal-en-v1", "source_field": "title", "source_mko_sha256": "56473b0d9ce61008676f55869388fbd0ce017b0555d5a29cc9a088626d1ef36c", "edition_fields_sha256": "b31eb9681d07bf4f363cc61a3d4fe0f91cfdc0e2e366bfbb3974ad4c4ee001c7" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-two-sided-ideal", "edition_id": "edition-mko-finite-two-sided-ideal-en-v1", "source_field": "summary", "source_mko_sha256": "56473b0d9ce61008676f55869388fbd0ce017b0555d5a29cc9a088626d1ef36c", "edition_fields_sha256": "b31eb9681d07bf4f363cc61a3d4fe0f91cfdc0e2e366bfbb3974ad4c4ee001c7" } } }, "atlas-finite-quotient-ring": { "source_snapshot": { "id": "atlas-finite-quotient-ring", "title_zh": "有限環的商環", "summary_zh": "有限環依proper雙邊理想的加法cosets形成商環，運算須與代表元選擇無關。", "canonical_mko_id": "mko-finite-quotient-ring" }, "fields": { "title": "Quotient ring of a finite ring", "summary": "Additive cosets of a proper two-sided ideal in a finite ring form a quotient ring when its operations are independent of representative choice." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-quotient-ring", "edition_id": "edition-mko-finite-quotient-ring-en-v1", "source_field": "title", "source_mko_sha256": "8c2b45b38428b422d01b869c06eaca3a0db6e14e6fafc691ef583d4614d39216", "edition_fields_sha256": "0daa3fadea4a1bc81119d62ffb4c8df8cd3b53e110fd8ad60593b63ce64401ec" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-quotient-ring", "edition_id": "edition-mko-finite-quotient-ring-en-v1", "source_field": "summary", "source_mko_sha256": "8c2b45b38428b422d01b869c06eaca3a0db6e14e6fafc691ef583d4614d39216", "edition_fields_sha256": "0daa3fadea4a1bc81119d62ffb4c8df8cd3b53e110fd8ad60593b63ce64401ec" } } }, "atlas-polynomial-ring-over-finite-commutative-ring": { "source_snapshot": { "id": "atlas-polynomial-ring-over-finite-commutative-ring", "title_zh": "有限交換環上的多項式環", "summary_zh": "係數取自有限交換環的普通多項式以有限支撐係數序列表示，但所有任意有限次多項式合成的載體仍是無限。", "canonical_mko_id": "mko-polynomial-ring-over-finite-commutative-ring" }, "fields": { "title": "Polynomial ring over a finite commutative ring", "summary": "Ordinary polynomials over a finite commutative coefficient ring have finite-support coefficient sequences, but the collection over all finite degrees is still infinite." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-polynomial-ring-over-finite-commutative-ring", "edition_id": "edition-mko-polynomial-ring-over-finite-commutative-ring-en-v1", "source_field": "title", "source_mko_sha256": "738ac2a85874f0a45327552d44abd7bfe8f789812c2f6c29c8f80d592ebb8a67", "edition_fields_sha256": "c214c57f727b33d23afe1d642a1bcf332086fda81a810752b8103e1aaf3da26f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-polynomial-ring-over-finite-commutative-ring", "edition_id": "edition-mko-polynomial-ring-over-finite-commutative-ring-en-v1", "source_field": "summary", "source_mko_sha256": "738ac2a85874f0a45327552d44abd7bfe8f789812c2f6c29c8f80d592ebb8a67", "edition_fields_sha256": "c214c57f727b33d23afe1d642a1bcf332086fda81a810752b8103e1aaf3da26f" } } }, "atlas-real-normed-vector-space": { "source_snapshot": { "id": "atlas-real-normed-vector-space", "title_zh": "實賦範向量空間", "summary_zh": "實向量空間配備正定、絕對齊次且滿足三角不等式的實值範數。", "canonical_mko_id": "mko-real-normed-vector-space" }, "fields": { "title": "Real normed vector space", "summary": "A real vector space equipped with a positive-definite, absolutely homogeneous real-valued norm satisfying the triangle inequality." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-real-normed-vector-space", "edition_id": "edition-mko-real-normed-vector-space-en-v1", "source_field": "title", "source_mko_sha256": "8f2547c8a3312254b34134e96a3083f313e6e7334ab0a87a1ba73cc147931964", "edition_fields_sha256": "8bf3d8f29dee3a2e66de383294d3b91649e07c451a5aeb4de3d288d94b230d43" }, "summary": { "kind": "mko_edition", "mko_id": "mko-real-normed-vector-space", "edition_id": "edition-mko-real-normed-vector-space-en-v1", "source_field": "summary", "source_mko_sha256": "8f2547c8a3312254b34134e96a3083f313e6e7334ab0a87a1ba73cc147931964", "edition_fields_sha256": "8bf3d8f29dee3a2e66de383294d3b91649e07c451a5aeb4de3d288d94b230d43" } } }, "atlas-norm-induced-metric": { "source_snapshot": { "id": "atlas-norm-induced-metric", "title_zh": "範數誘導度量", "summary_zh": "同一實賦範向量空間上的範數以向量差的範數定義度量，度量公理由範數公理全稱推出。", "canonical_mko_id": "mko-norm-induced-metric" }, "fields": { "title": "Norm-induced metric", "summary": "On the same real normed vector space, taking the norm of a vector difference defines a metric whose axioms follow universally from the norm axioms." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-norm-induced-metric", "edition_id": "edition-mko-norm-induced-metric-en-v1", "source_field": "title", "source_mko_sha256": "579d0833bcf98eaea0df0559e17c92ecaaa60c6760d51154ff23dcc92d83da4f", "edition_fields_sha256": "1afba8895040f0e342c5c6dc0a573b6ac3461f113fba04abe167ae5febc37181" }, "summary": { "kind": "mko_edition", "mko_id": "mko-norm-induced-metric", "edition_id": "edition-mko-norm-induced-metric-en-v1", "source_field": "summary", "source_mko_sha256": "579d0833bcf98eaea0df0559e17c92ecaaa60c6760d51154ff23dcc92d83da4f", "edition_fields_sha256": "1afba8895040f0e342c5c6dc0a573b6ac3461f113fba04abe167ae5febc37181" } } }, "atlas-cauchy-sequence": { "source_snapshot": { "id": "atlas-cauchy-sequence", "title_zh": "Cauchy序列", "summary_zh": "度量空間序列為Cauchy，若每個正epsilon最終所有兩項彼此距離皆小於epsilon。", "canonical_mko_id": "mko-cauchy-sequence" }, "fields": { "title": "Cauchy sequence", "summary": "A sequence in a metric space is Cauchy if, for every positive epsilon, all pairs of sufficiently late terms are less than epsilon apart." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-cauchy-sequence", "edition_id": "edition-mko-cauchy-sequence-en-v1", "source_field": "title", "source_mko_sha256": "f833bc771b9e0a259993b3560086625a99bfce9069edf8b08acd37a09b08c004", "edition_fields_sha256": "13f299ccb93cd689e0506634cfdb1a327a481eb238b70b951e0974d3e016aa8d" }, "summary": { "kind": "mko_edition", "mko_id": "mko-cauchy-sequence", "edition_id": "edition-mko-cauchy-sequence-en-v1", "source_field": "summary", "source_mko_sha256": "f833bc771b9e0a259993b3560086625a99bfce9069edf8b08acd37a09b08c004", "edition_fields_sha256": "13f299ccb93cd689e0506634cfdb1a327a481eb238b70b951e0974d3e016aa8d" } } }, "atlas-complete-metric-space": { "source_snapshot": { "id": "atlas-complete-metric-space", "title_zh": "完備度量空間", "summary_zh": "每個Cauchy序列都收斂到空間內某點的度量空間稱為完備。", "canonical_mko_id": "mko-complete-metric-space" }, "fields": { "title": "Complete metric space", "summary": "A metric space is complete when every Cauchy sequence converges to a point still belonging to that space." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-complete-metric-space", "edition_id": "edition-mko-complete-metric-space-en-v1", "source_field": "title", "source_mko_sha256": "783e45783b1dc07ccecb89f0c76b128f6c4a9a3da036e1678e908f2250530c73", "edition_fields_sha256": "ff0481a26452c6e2f6e26f374fa73a4b4076fa194dd263f05b5318e6e3dc2391" }, "summary": { "kind": "mko_edition", "mko_id": "mko-complete-metric-space", "edition_id": "edition-mko-complete-metric-space-en-v1", "source_field": "summary", "source_mko_sha256": "783e45783b1dc07ccecb89f0c76b128f6c4a9a3da036e1678e908f2250530c73", "edition_fields_sha256": "ff0481a26452c6e2f6e26f374fa73a4b4076fa194dd263f05b5318e6e3dc2391" } } }, "atlas-real-banach-space": { "source_snapshot": { "id": "atlas-real-banach-space", "title_zh": "實Banach空間", "summary_zh": "在自身範數誘導度量下完備的實賦範向量空間稱為實Banach空間。", "canonical_mko_id": "mko-real-banach-space" }, "fields": { "title": "Real Banach space", "summary": "A real normed vector space is a Banach space when it is complete in the metric induced by its own norm." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-real-banach-space", "edition_id": "edition-mko-real-banach-space-en-v1", "source_field": "title", "source_mko_sha256": "1c66f08e81b1ae1e47f887e3890cc179fd0aa8071970f275641c4c720de626ca", "edition_fields_sha256": "fdb1504012d4b32240ec10696bc85deef8513b07855042263d61da8336404d4e" }, "summary": { "kind": "mko_edition", "mko_id": "mko-real-banach-space", "edition_id": "edition-mko-real-banach-space-en-v1", "source_field": "summary", "source_mko_sha256": "1c66f08e81b1ae1e47f887e3890cc179fd0aa8071970f275641c4c720de626ca", "edition_fields_sha256": "fdb1504012d4b32240ec10696bc85deef8513b07855042263d61da8336404d4e" } } }, "atlas-real-bilinear-form": { "source_snapshot": { "id": "atlas-real-bilinear-form", "title_zh": "實雙線性形式", "summary_zh": "實向量空間上的雙線性形式分別對兩個輸入保持向量加法與實純量乘法。", "canonical_mko_id": "mko-real-bilinear-form" }, "fields": { "title": "Real bilinear form", "summary": "A bilinear form on a real vector space preserves addition and real scalar multiplication separately in each input." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-real-bilinear-form", "edition_id": "edition-mko-real-bilinear-form-en-v1", "source_field": "title", "source_mko_sha256": "fa2ccad4a26e9cf94b590f02594c9e1c0db0ba86fc50c60ee410e2b934d4c7d5", "edition_fields_sha256": "f4e35ffa42d582e2559d4d552306d21c7d186006c842edc954f7b1694e622a2e" }, "summary": { "kind": "mko_edition", "mko_id": "mko-real-bilinear-form", "edition_id": "edition-mko-real-bilinear-form-en-v1", "source_field": "summary", "source_mko_sha256": "fa2ccad4a26e9cf94b590f02594c9e1c0db0ba86fc50c60ee410e2b934d4c7d5", "edition_fields_sha256": "f4e35ffa42d582e2559d4d552306d21c7d186006c842edc954f7b1694e622a2e" } } }, "atlas-real-inner-product-space": { "source_snapshot": { "id": "atlas-real-inner-product-space", "title_zh": "實內積空間", "summary_zh": "對稱且正定的實雙線性形式構成實內積；半正定或抽樣為正不足。", "canonical_mko_id": "mko-real-inner-product-space" }, "fields": { "title": "Real inner-product space", "summary": "A symmetric, positive-definite real bilinear form is a real inner product; positive semidefiniteness or positive sampled values do not suffice." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-real-inner-product-space", "edition_id": "edition-mko-real-inner-product-space-en-v1", "source_field": "title", "source_mko_sha256": "70d1a304624117e0e814b5b9ca816a5b851798e54cc213f1f2b3460e6450f963", "edition_fields_sha256": "2fe635f9ef3662f46bac1b251eb804d701d7af6b3572549dc1189a5f4776907b" }, "summary": { "kind": "mko_edition", "mko_id": "mko-real-inner-product-space", "edition_id": "edition-mko-real-inner-product-space-en-v1", "source_field": "summary", "source_mko_sha256": "70d1a304624117e0e814b5b9ca816a5b851798e54cc213f1f2b3460e6450f963", "edition_fields_sha256": "2fe635f9ef3662f46bac1b251eb804d701d7af6b3572549dc1189a5f4776907b" } } }, "atlas-inner-product-induced-norm": { "source_snapshot": { "id": "atlas-inner-product-induced-norm", "title_zh": "內積誘導範數", "summary_zh": "同一實內積以向量自內積的非負平方根定義範數；三角不等式依Cauchy–Schwarz全稱證書。", "canonical_mko_id": "mko-inner-product-induced-norm" }, "fields": { "title": "Inner-product-induced norm", "summary": "On the same real inner-product space, the norm is the nonnegative square root of the self-inner-product; the triangle inequality uses the universal Cauchy–Schwarz certificate." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-inner-product-induced-norm", "edition_id": "edition-mko-inner-product-induced-norm-en-v1", "source_field": "title", "source_mko_sha256": "9b9498dd5a3aa838a1854d97c526a01a657f0f88bab64b00a78ef20fb1f80c97", "edition_fields_sha256": "c15cb973b4b769e7f4eba8519f0b48141be5d92aaf09a88663880764aba8f3a5" }, "summary": { "kind": "mko_edition", "mko_id": "mko-inner-product-induced-norm", "edition_id": "edition-mko-inner-product-induced-norm-en-v1", "source_field": "summary", "source_mko_sha256": "9b9498dd5a3aa838a1854d97c526a01a657f0f88bab64b00a78ef20fb1f80c97", "edition_fields_sha256": "c15cb973b4b769e7f4eba8519f0b48141be5d92aaf09a88663880764aba8f3a5" } } }, "atlas-real-hilbert-space": { "source_snapshot": { "id": "atlas-real-hilbert-space", "title_zh": "實Hilbert空間", "summary_zh": "在同一內積誘導範數與度量下完備的實內積空間為實Hilbert空間。", "canonical_mko_id": "mko-real-hilbert-space" }, "fields": { "title": "Real Hilbert space", "summary": "A real inner-product space is Hilbert when it is complete in the norm and metric induced by that same inner product." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-real-hilbert-space", "edition_id": "edition-mko-real-hilbert-space-en-v1", "source_field": "title", "source_mko_sha256": "89b1eab58b853963348761bf140ffff91d2903053edbdf9baf157b32a5ac731e", "edition_fields_sha256": "17aad4f00c69e188b16d67455637206fd1c8ab06fc3538de46b496e385e11255" }, "summary": { "kind": "mko_edition", "mko_id": "mko-real-hilbert-space", "edition_id": "edition-mko-real-hilbert-space-en-v1", "source_field": "summary", "source_mko_sha256": "89b1eab58b853963348761bf140ffff91d2903053edbdf9baf157b32a5ac731e", "edition_fields_sha256": "17aad4f00c69e188b16d67455637206fd1c8ab06fc3538de46b496e385e11255" } } }, "atlas-general-linear-map": { "source_snapshot": { "id": "atlas-general-linear-map", "title_zh": "實向量空間間線性映射", "summary_zh": "一般實向量空間間映射若對所有向量與實純量保持線性組合，則為線性映射。", "canonical_mko_id": "mko-general-linear-map" }, "fields": { "title": "Linear map between real vector spaces", "summary": "A map between real vector spaces is linear when it preserves linear combinations for all vectors and real scalars." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-general-linear-map", "edition_id": "edition-mko-general-linear-map-en-v1", "source_field": "title", "source_mko_sha256": "184a885b199aff8dc05ec273c0fe5b2dcdd38a9ef98c8ec790ce2668da9a07c7", "edition_fields_sha256": "3cb4710604d4347f348f8f762d4d8f36d98461ecaa909b3359ed3b01b3d4f3af" }, "summary": { "kind": "mko_edition", "mko_id": "mko-general-linear-map", "edition_id": "edition-mko-general-linear-map-en-v1", "source_field": "summary", "source_mko_sha256": "184a885b199aff8dc05ec273c0fe5b2dcdd38a9ef98c8ec790ce2668da9a07c7", "edition_fields_sha256": "3cb4710604d4347f348f8f762d4d8f36d98461ecaa909b3359ed3b01b3d4f3af" } } }, "atlas-bounded-linear-operator": { "source_snapshot": { "id": "atlas-bounded-linear-operator", "title_zh": "有界線性算子", "summary_zh": "賦範空間間線性映射若有單一非負常數統一控制所有輸出範數，則為有界線性算子。", "canonical_mko_id": "mko-bounded-linear-operator" }, "fields": { "title": "Bounded linear operator", "summary": "A linear map between normed spaces is bounded when one nonnegative constant uniformly controls all output norms relative to input norms." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-bounded-linear-operator", "edition_id": "edition-mko-bounded-linear-operator-en-v1", "source_field": "title", "source_mko_sha256": "c673fba2cd8ab7e707c55dc85252e9d84fa9bed89d6cfd51ac95bcc4cee707f9", "edition_fields_sha256": "8d9bf2ab9bfbeb4e19b74ab29e47c947b32f8f2bedb52edddec8c6ab6ec456e5" }, "summary": { "kind": "mko_edition", "mko_id": "mko-bounded-linear-operator", "edition_id": "edition-mko-bounded-linear-operator-en-v1", "source_field": "summary", "source_mko_sha256": "c673fba2cd8ab7e707c55dc85252e9d84fa9bed89d6cfd51ac95bcc4cee707f9", "edition_fields_sha256": "8d9bf2ab9bfbeb4e19b74ab29e47c947b32f8f2bedb52edddec8c6ab6ec456e5" } } }, "atlas-event-independence": { "source_snapshot": { "id": "atlas-event-independence", "title_zh": "事件獨立", "summary_zh": "兩事件獨立以交集機率等於邊際機率乘積定義；此定義不依賴零機率條件式。", "canonical_mko_id": "mko-event-independence" }, "fields": { "title": "Event independence", "summary": "Two events are independent when their intersection probability equals the product of their marginal probabilities; this definition does not rely on conditioning on a zero-probability event." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-event-independence", "edition_id": "edition-mko-event-independence-en-v1", "source_field": "title", "source_mko_sha256": "9b6f08c178cbc612effac540bd933bfde0d1b9ebc7708cc7ab928ac4505ad563", "edition_fields_sha256": "6e0c250d9060e4bbb1a4fb37d2117d5b256667cf99928a276c794b5c5ca19d15" }, "summary": { "kind": "mko_edition", "mko_id": "mko-event-independence", "edition_id": "edition-mko-event-independence-en-v1", "source_field": "summary", "source_mko_sha256": "9b6f08c178cbc612effac540bd933bfde0d1b9ebc7708cc7ab928ac4505ad563", "edition_fields_sha256": "6e0c250d9060e4bbb1a4fb37d2117d5b256667cf99928a276c794b5c5ca19d15" } } }, "atlas-finite-joint-distribution": { "source_snapshot": { "id": "atlas-finite-joint-distribution", "title_zh": "有限聯合分布", "summary_zh": "共同有限機率空間上多個隨機變數的值組與精確質量共同指定有限聯合分布。", "canonical_mko_id": "mko-finite-joint-distribution" }, "fields": { "title": "Finite joint distribution", "summary": "Value tuples of random variables on one common finite probability space, together with exact masses, specify a finite joint distribution." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-joint-distribution", "edition_id": "edition-mko-finite-joint-distribution-en-v1", "source_field": "title", "source_mko_sha256": "d04aa2266aeda810e9e7e400220d366a61da23cdc33192e7286eac1d0fed0340", "edition_fields_sha256": "7bdf23d56999afd96c952f3a6c0eab7c07abbc122d2096e6f1b37db360b30fca" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-joint-distribution", "edition_id": "edition-mko-finite-joint-distribution-en-v1", "source_field": "summary", "source_mko_sha256": "d04aa2266aeda810e9e7e400220d366a61da23cdc33192e7286eac1d0fed0340", "edition_fields_sha256": "7bdf23d56999afd96c952f3a6c0eab7c07abbc122d2096e6f1b37db360b30fca" } } }, "atlas-random-variable-independence": { "source_snapshot": { "id": "atlas-random-variable-independence", "title_zh": "隨機變數獨立", "summary_zh": "同一有限joint law下，隨機變數獨立要求全部值對的joint mass分解為邊際質量乘積。", "canonical_mko_id": "mko-random-variable-independence" }, "fields": { "title": "Independence of random variables", "summary": "Under one finite joint law, independence requires every value-pair mass to factor as the product of its marginal masses." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-random-variable-independence", "edition_id": "edition-mko-random-variable-independence-en-v1", "source_field": "title", "source_mko_sha256": "e8f6e97a13cd93c4cf37384ba6ab187d3ad4ca22c8f56260ed3707df0b8cc94e", "edition_fields_sha256": "af91ddad37c51ac0f7a0029861810bc50ea7610d82a9c9323a2d5e2025e5261e" }, "summary": { "kind": "mko_edition", "mko_id": "mko-random-variable-independence", "edition_id": "edition-mko-random-variable-independence-en-v1", "source_field": "summary", "source_mko_sha256": "e8f6e97a13cd93c4cf37384ba6ab187d3ad4ca22c8f56260ed3707df0b8cc94e", "edition_fields_sha256": "af91ddad37c51ac0f7a0029861810bc50ea7610d82a9c9323a2d5e2025e5261e" } } }, "atlas-covariance": { "source_snapshot": { "id": "atlas-covariance", "title_zh": "協方差", "summary_zh": "同一有限聯合分布下協方差為E[XY]-E[X]E[Y]；零協方差通常不推出獨立。", "canonical_mko_id": "mko-covariance" }, "fields": { "title": "Covariance", "summary": "Under one finite joint distribution, covariance is E[XY]−E[X]E[Y]; zero covariance does not in general imply independence." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-covariance", "edition_id": "edition-mko-covariance-en-v1", "source_field": "title", "source_mko_sha256": "3b0480b894efd088fcbcb2c9c856fb6e46bc8b62a223285723b060bf0070143d", "edition_fields_sha256": "a6bf925e8660bf5b27bef91f8cd39fc3d5b83b5cadb88f72ff0274c1db2ed11c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-covariance", "edition_id": "edition-mko-covariance-en-v1", "source_field": "summary", "source_mko_sha256": "3b0480b894efd088fcbcb2c9c856fb6e46bc8b62a223285723b060bf0070143d", "edition_fields_sha256": "a6bf925e8660bf5b27bef91f8cd39fc3d5b83b5cadb88f72ff0274c1db2ed11c" } } }, "atlas-conditional-expectation": { "source_snapshot": { "id": "atlas-conditional-expectation", "title_zh": "有限條件期望", "summary_zh": "有限機率空間中，相對分割生成sigma代數的條件期望在每個正質量區塊取加權平均並保持區塊積分。", "canonical_mko_id": "mko-conditional-expectation" }, "fields": { "title": "Finite conditional expectation", "summary": "Relative to a partition-generated sigma-algebra on a finite probability space, conditional expectation takes the weighted mean on each positive-mass block and preserves block integrals." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-conditional-expectation", "edition_id": "edition-mko-conditional-expectation-en-v1", "source_field": "title", "source_mko_sha256": "35d7f871dac286e45d4661dcffbafe25a02cca5c55a4c58ef60bd41d2408bd95", "edition_fields_sha256": "540007c839aa138746f52f4733c46639dd8a2597bbb091d5e5fe184ebcacb308" }, "summary": { "kind": "mko_edition", "mko_id": "mko-conditional-expectation", "edition_id": "edition-mko-conditional-expectation-en-v1", "source_field": "summary", "source_mko_sha256": "35d7f871dac286e45d4661dcffbafe25a02cca5c55a4c58ef60bd41d2408bd95", "edition_fields_sha256": "540007c839aa138746f52f4733c46639dd8a2597bbb091d5e5fe184ebcacb308" } } }, "atlas-finite-filtration": { "source_snapshot": { "id": "atlas-finite-filtration", "title_zh": "有限filtration", "summary_zh": "有限時間filtration是在同一機率空間上隨時間資訊遞增的sigma代數序列。", "canonical_mko_id": "mko-finite-filtration" }, "fields": { "title": "Finite filtration", "summary": "A finite-time filtration is a sequence of sigma-algebras on the same probability space whose information increases with time." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-filtration", "edition_id": "edition-mko-finite-filtration-en-v1", "source_field": "title", "source_mko_sha256": "1b51e644ad6518d67dc1c6b78e1ff5043eb9a2fe5576f71a9f732b6b1c168a62", "edition_fields_sha256": "7b6d4f19fa98f9ec4ec30c071a193542659a13b0f1607392c5e363b509ba7e0c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-filtration", "edition_id": "edition-mko-finite-filtration-en-v1", "source_field": "summary", "source_mko_sha256": "1b51e644ad6518d67dc1c6b78e1ff5043eb9a2fe5576f71a9f732b6b1c168a62", "edition_fields_sha256": "7b6d4f19fa98f9ec4ec30c071a193542659a13b0f1607392c5e363b509ba7e0c" } } }, "atlas-martingale": { "source_snapshot": { "id": "atlas-martingale", "title_zh": "有限Martingale", "summary_zh": "相對同一filtration適應且可積的有限過程，若每一步下一值條件期望等於現在值a.s.，則為Martingale。", "canonical_mko_id": "mko-martingale" }, "fields": { "title": "Finite martingale", "summary": "An integrable finite process adapted to the same filtration is a martingale when the next value's conditional expectation equals the current value almost surely at every step." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-martingale", "edition_id": "edition-mko-martingale-en-v1", "source_field": "title", "source_mko_sha256": "25744aeb70a10daa6dcabb09f5b71bc0ee3e9f2c266412b727873f5e2abddf7b", "edition_fields_sha256": "ab892243a5a4e324f6272ec7b55f26b37141a35b3cf0635d5ca9f575afe63655" }, "summary": { "kind": "mko_edition", "mko_id": "mko-martingale", "edition_id": "edition-mko-martingale-en-v1", "source_field": "summary", "source_mko_sha256": "25744aeb70a10daa6dcabb09f5b71bc0ee3e9f2c266412b727873f5e2abddf7b", "edition_fields_sha256": "ab892243a5a4e324f6272ec7b55f26b37141a35b3cf0635d5ca9f575afe63655" } } }, "atlas-convergence-in-probability": { "source_snapshot": { "id": "atlas-convergence-in-probability", "title_zh": "依機率收斂", "summary_zh": "X_n依機率收斂到X，若對每個epsilon>0，偏差超過epsilon的機率趨近0。", "canonical_mko_id": "mko-convergence-in-probability" }, "fields": { "title": "Convergence in probability", "summary": "X_n converges in probability to X when, for every epsilon>0, the probability of deviation greater than epsilon tends to zero." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-convergence-in-probability", "edition_id": "edition-mko-convergence-in-probability-en-v1", "source_field": "title", "source_mko_sha256": "f6a36c4c48518f81f1515a2208543968a39ea38fccfb6c38d7f3613607756354", "edition_fields_sha256": "92954b658aafd798f756c34e723a96e99da681780052627c0e3ee1f051fa7704" }, "summary": { "kind": "mko_edition", "mko_id": "mko-convergence-in-probability", "edition_id": "edition-mko-convergence-in-probability-en-v1", "source_field": "summary", "source_mko_sha256": "f6a36c4c48518f81f1515a2208543968a39ea38fccfb6c38d7f3613607756354", "edition_fields_sha256": "92954b658aafd798f756c34e723a96e99da681780052627c0e3ee1f051fa7704" } } }, "atlas-cumulative-distribution-function": { "source_snapshot": { "id": "atlas-cumulative-distribution-function", "title_zh": "累積分布函數", "summary_zh": "實值隨機變數的CDF以F(t)=P(X小於等於t)定義，為單調右連續且有限離散跳躍等於點質量。", "canonical_mko_id": "mko-cumulative-distribution-function" }, "fields": { "title": "Cumulative distribution function", "summary": "For a real-valued random variable, F(t)=P(X≤t) is nondecreasing and right-continuous; in a finite discrete law, its jump is the point mass." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-cumulative-distribution-function", "edition_id": "edition-mko-cumulative-distribution-function-en-v1", "source_field": "title", "source_mko_sha256": "4111d0774535349038ca5452c94102adeb111e4fa99147bfbaf9d380f06da4fc", "edition_fields_sha256": "164c76b8e59f9f86d67faf38afb513bb0f01dd8dc9dcf3c43e22c27fea32293e" }, "summary": { "kind": "mko_edition", "mko_id": "mko-cumulative-distribution-function", "edition_id": "edition-mko-cumulative-distribution-function-en-v1", "source_field": "summary", "source_mko_sha256": "4111d0774535349038ca5452c94102adeb111e4fa99147bfbaf9d380f06da4fc", "edition_fields_sha256": "164c76b8e59f9f86d67faf38afb513bb0f01dd8dc9dcf3c43e22c27fea32293e" } } }, "atlas-convergence-in-distribution": { "source_snapshot": { "id": "atlas-convergence-in-distribution", "title_zh": "依分布收斂", "summary_zh": "極限CDF每個連續點上F_n逐點收斂到F時，稱依分布收斂。", "canonical_mko_id": "mko-convergence-in-distribution" }, "fields": { "title": "Convergence in distribution", "summary": "Convergence in distribution means that F_n tends pointwise to F at every continuity point of the limiting CDF." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-convergence-in-distribution", "edition_id": "edition-mko-convergence-in-distribution-en-v1", "source_field": "title", "source_mko_sha256": "082a2a414602a248555b80f0fa923b8b69256bdacb1fe1fe32b38bb0924de21a", "edition_fields_sha256": "2b30f1bcc0411c8836057e9af4742ca0e97e9f3c8fa1f372f0b5511c9a15ff94" }, "summary": { "kind": "mko_edition", "mko_id": "mko-convergence-in-distribution", "edition_id": "edition-mko-convergence-in-distribution-en-v1", "source_field": "summary", "source_mko_sha256": "082a2a414602a248555b80f0fa923b8b69256bdacb1fe1fe32b38bb0924de21a", "edition_fields_sha256": "2b30f1bcc0411c8836057e9af4742ca0e97e9f3c8fa1f372f0b5511c9a15ff94" } } }, "atlas-iid-sequence": { "source_snapshot": { "id": "atlas-iid-sequence", "title_zh": "獨立同分布序列", "summary_zh": "IID序列同時要求每個變數有相同分布，且每個有限子族滿足mutual independence。", "canonical_mko_id": "mko-iid-sequence" }, "fields": { "title": "Independent identically distributed sequence", "summary": "An IID sequence requires every variable to have the same distribution and every finite subfamily to be mutually independent." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-iid-sequence", "edition_id": "edition-mko-iid-sequence-en-v1", "source_field": "title", "source_mko_sha256": "bac79da4634be201a10f08ec4489946a636db5ebc2e628c0b1830b743f4ddbd0", "edition_fields_sha256": "7d82c4dede9155fde654ed008f92e79b39d7d09ec50318620230a4abd0d67d40" }, "summary": { "kind": "mko_edition", "mko_id": "mko-iid-sequence", "edition_id": "edition-mko-iid-sequence-en-v1", "source_field": "summary", "source_mko_sha256": "bac79da4634be201a10f08ec4489946a636db5ebc2e628c0b1830b743f4ddbd0", "edition_fields_sha256": "7d82c4dede9155fde654ed008f92e79b39d7d09ec50318620230a4abd0d67d40" } } }, "atlas-graph-isomorphism": { "source_snapshot": { "id": "atlas-graph-isomorphism", "title_zh": "圖同構", "summary_zh": "有限簡單圖同構是頂點雙射，對所有頂點對完整保持鄰接與非鄰接。", "canonical_mko_id": "mko-graph-isomorphism" }, "fields": { "title": "Graph isomorphism", "summary": "An isomorphism of finite simple graphs is a vertex bijection preserving both adjacency and nonadjacency for every pair." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-graph-isomorphism", "edition_id": "edition-mko-graph-isomorphism-en-v1", "source_field": "title", "source_mko_sha256": "36b491efcb2f7e6297c71cc86792ccbd2cc61cace4f61f1b9324257b32a5d09f", "edition_fields_sha256": "14c9376c477fc7e747299f83564b8b78214d085e063c4a8947bee6a723458094" }, "summary": { "kind": "mko_edition", "mko_id": "mko-graph-isomorphism", "edition_id": "edition-mko-graph-isomorphism-en-v1", "source_field": "summary", "source_mko_sha256": "36b491efcb2f7e6297c71cc86792ccbd2cc61cace4f61f1b9324257b32a5d09f", "edition_fields_sha256": "14c9376c477fc7e747299f83564b8b78214d085e063c4a8947bee6a723458094" } } }, "atlas-graph-walk-simple-path": { "source_snapshot": { "id": "atlas-graph-walk-simple-path", "title_zh": "圖walk與simple path", "summary_zh": "有限無向簡單圖的walk要求每對連續頂點相鄰；simple path另要求所有頂點互異。", "canonical_mko_id": "mko-graph-walk-simple-path" }, "fields": { "title": "Graph walks and simple paths", "summary": "In a finite undirected simple graph, a walk requires each consecutive pair of vertices to be adjacent; a simple path additionally requires all vertices to be distinct." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-graph-walk-simple-path", "edition_id": "edition-mko-graph-walk-simple-path-en-v1", "source_field": "title", "source_mko_sha256": "e48ce83c081f9aa69ead3069dd9ed3c1f58b44d320e7dc35d7794f060c83b554", "edition_fields_sha256": "25b81707b2687429eed6132f981d7b4b6f10944cd99caa3b83672abd30a3523a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-graph-walk-simple-path", "edition_id": "edition-mko-graph-walk-simple-path-en-v1", "source_field": "summary", "source_mko_sha256": "e48ce83c081f9aa69ead3069dd9ed3c1f58b44d320e7dc35d7794f060c83b554", "edition_fields_sha256": "25b81707b2687429eed6132f981d7b4b6f10944cd99caa3b83672abd30a3523a" } } }, "atlas-connected-component": { "source_snapshot": { "id": "atlas-connected-component", "title_zh": "連通分量", "summary_zh": "有限無向簡單圖中，由存在路徑所定義的頂點等價類恰為連通分量。", "canonical_mko_id": "mko-connected-component" }, "fields": { "title": "Connected component", "summary": "In a finite undirected simple graph, connected components are exactly the vertex equivalence classes defined by the existence of a path." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-connected-component", "edition_id": "edition-mko-connected-component-en-v1", "source_field": "title", "source_mko_sha256": "f921926c2c42fafc8d64364eb7bd06488901e081b0117ba17fc4265610e54eab", "edition_fields_sha256": "97b71a94a5cb8d1f898a9679f85d44dcc602f0bba56f3efefef15df6f0569953" }, "summary": { "kind": "mko_edition", "mko_id": "mko-connected-component", "edition_id": "edition-mko-connected-component-en-v1", "source_field": "summary", "source_mko_sha256": "f921926c2c42fafc8d64364eb7bd06488901e081b0117ba17fc4265610e54eab", "edition_fields_sha256": "97b71a94a5cb8d1f898a9679f85d44dcc602f0bba56f3efefef15df6f0569953" } } }, "atlas-bipartite-graph": { "source_snapshot": { "id": "atlas-bipartite-graph", "title_zh": "二分圖", "summary_zh": "有限無向簡單圖若全體頂點可分為兩個互斥部分且每條邊跨部，則為二分圖。", "canonical_mko_id": "mko-bipartite-graph" }, "fields": { "title": "Bipartite graph", "summary": "A finite undirected simple graph is bipartite when all vertices split into two disjoint parts and every edge crosses between the parts." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-bipartite-graph", "edition_id": "edition-mko-bipartite-graph-en-v1", "source_field": "title", "source_mko_sha256": "c54116e1bcf2bbb83b39260399271673ba3ad43ec2e0b27732fd148d843f7aa1", "edition_fields_sha256": "582e19307476435706bccab23c47121da0952befac4490e3de5ae016fcfd96ac" }, "summary": { "kind": "mko_edition", "mko_id": "mko-bipartite-graph", "edition_id": "edition-mko-bipartite-graph-en-v1", "source_field": "summary", "source_mko_sha256": "c54116e1bcf2bbb83b39260399271673ba3ad43ec2e0b27732fd148d843f7aa1", "edition_fields_sha256": "582e19307476435706bccab23c47121da0952befac4490e3de5ae016fcfd96ac" } } }, "atlas-graph-coloring": { "source_snapshot": { "id": "atlas-graph-coloring", "title_zh": "圖著色", "summary_zh": "proper k-coloring使用至多k種顏色並使每條邊兩端顏色不同。", "canonical_mko_id": "mko-graph-coloring" }, "fields": { "title": "Graph coloring", "summary": "A proper k-coloring uses at most k colors and gives different colors to the endpoints of every edge." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-graph-coloring", "edition_id": "edition-mko-graph-coloring-en-v1", "source_field": "title", "source_mko_sha256": "6c2084e53867b83ec1c3c9f3a21f1023e1d3b0910c9b23c6f774aa05a08ff54d", "edition_fields_sha256": "4ad05189f699543d089320bfe0a5e163720dc4889599d8414ee90e470ccf9517" }, "summary": { "kind": "mko_edition", "mko_id": "mko-graph-coloring", "edition_id": "edition-mko-graph-coloring-en-v1", "source_field": "summary", "source_mko_sha256": "6c2084e53867b83ec1c3c9f3a21f1023e1d3b0910c9b23c6f774aa05a08ff54d", "edition_fields_sha256": "4ad05189f699543d089320bfe0a5e163720dc4889599d8414ee90e470ccf9517" } } }, "atlas-equivalence-relation": { "source_snapshot": { "id": "atlas-equivalence-relation", "title_zh": "等價關係", "summary_zh": "固定載體上的關係若完整滿足反身、對稱與傳遞，則為等價關係。", "canonical_mko_id": "mko-equivalence-relation" }, "fields": { "title": "Equivalence relation", "summary": "A relation on a fixed carrier is an equivalence relation when it fully satisfies reflexivity, symmetry and transitivity." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-equivalence-relation", "edition_id": "edition-mko-equivalence-relation-en-v1", "source_field": "title", "source_mko_sha256": "ceccd69c4881e4b2eba57eb1a7c6ab6c6616ab09fa42eb8591999522a5948fd8", "edition_fields_sha256": "ef02bff35fc649a9be697c07bee8877e7fe20c6ea76f93b83d976c6d26d2bdeb" }, "summary": { "kind": "mko_edition", "mko_id": "mko-equivalence-relation", "edition_id": "edition-mko-equivalence-relation-en-v1", "source_field": "summary", "source_mko_sha256": "ceccd69c4881e4b2eba57eb1a7c6ab6c6616ab09fa42eb8591999522a5948fd8", "edition_fields_sha256": "ef02bff35fc649a9be697c07bee8877e7fe20c6ea76f93b83d976c6d26d2bdeb" } } }, "atlas-equivalence-class": { "source_snapshot": { "id": "atlas-equivalence-class", "title_zh": "等價類", "summary_zh": "固定完整等價關係下，元素a的等價類收集全部與a相關的載體元素。", "canonical_mko_id": "mko-equivalence-class" }, "fields": { "title": "Equivalence class", "summary": "For a fixed complete equivalence relation, the class of an element a consists of every carrier element related to a." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-equivalence-class", "edition_id": "edition-mko-equivalence-class-en-v1", "source_field": "title", "source_mko_sha256": "6ead5a95cd13a163d636e78db505bb8f7aad6744f9e25afee8148c838a3a44d9", "edition_fields_sha256": "caf0655fb32ff03074f0b44fda74f0d7c24b28e92fed0acf5c033106fb547569" }, "summary": { "kind": "mko_edition", "mko_id": "mko-equivalence-class", "edition_id": "edition-mko-equivalence-class-en-v1", "source_field": "summary", "source_mko_sha256": "6ead5a95cd13a163d636e78db505bb8f7aad6744f9e25afee8148c838a3a44d9", "edition_fields_sha256": "caf0655fb32ff03074f0b44fda74f0d7c24b28e92fed0acf5c033106fb547569" } } }, "atlas-quotient-set": { "source_snapshot": { "id": "atlas-quotient-set", "title_zh": "商集", "summary_zh": "完整等價關係的全部互異等價類組成商集，並分割原載體。", "canonical_mko_id": "mko-quotient-set" }, "fields": { "title": "Quotient set", "summary": "The distinct equivalence classes of a complete equivalence relation form its quotient set and partition the original carrier." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-quotient-set", "edition_id": "edition-mko-quotient-set-en-v1", "source_field": "title", "source_mko_sha256": "3741dd3a04bde75f8d82e8fd31849761617fb2410c548387ba4434f502159236", "edition_fields_sha256": "a5c521de48b5a4684dc9a13a495ac0d1471d0d72ccc52f9877ec161b96898de8" }, "summary": { "kind": "mko_edition", "mko_id": "mko-quotient-set", "edition_id": "edition-mko-quotient-set-en-v1", "source_field": "summary", "source_mko_sha256": "3741dd3a04bde75f8d82e8fd31849761617fb2410c548387ba4434f502159236", "edition_fields_sha256": "a5c521de48b5a4684dc9a13a495ac0d1471d0d72ccc52f9877ec161b96898de8" } } }, "atlas-cardinality-comparison": { "source_snapshot": { "id": "atlas-cardinality-comparison", "title_zh": "集合基數比較", "summary_zh": "明示source與target間的單射、滿射、雙射分別提供有限集合基數比較證書。", "canonical_mko_id": "mko-cardinality-comparison" }, "fields": { "title": "Cardinality comparison of sets", "summary": "An injection, surjection or bijection between explicit finite source and target sets supplies a corresponding finite cardinality-comparison certificate." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-cardinality-comparison", "edition_id": "edition-mko-cardinality-comparison-en-v1", "source_field": "title", "source_mko_sha256": "50efb74b57a576c8cec0b8230fc37bf7cd9c7ecd6303efe09810085d9a570227", "edition_fields_sha256": "98569dbaa5d77381395f3f22f184348aa0dd7bcd52ea9c0b7d04838202479379" }, "summary": { "kind": "mko_edition", "mko_id": "mko-cardinality-comparison", "edition_id": "edition-mko-cardinality-comparison-en-v1", "source_field": "summary", "source_mko_sha256": "50efb74b57a576c8cec0b8230fc37bf7cd9c7ecd6303efe09810085d9a570227", "edition_fields_sha256": "98569dbaa5d77381395f3f22f184348aa0dd7bcd52ea9c0b7d04838202479379" } } }, "atlas-at-most-countable-set": { "source_snapshot": { "id": "atlas-at-most-countable-set", "title_zh": "至多可數集", "summary_zh": "集合若有限或存在到自然數的明示單射，則為至多可數；不等同可數無限。", "canonical_mko_id": "mko-at-most-countable-set" }, "fields": { "title": "At-most-countable set", "summary": "A set is at most countable if it is finite or has an explicit injection into the natural numbers; this is not the same as countably infinite." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-at-most-countable-set", "edition_id": "edition-mko-at-most-countable-set-en-v1", "source_field": "title", "source_mko_sha256": "9c7748686b7974341920a59d07628f20017ef6d65476d751b2feb7424f7b3b57", "edition_fields_sha256": "4b83db0655be76ff5bbd8a540dcf36ad4dade5a7458daef3ce127b9c9f5235cd" }, "summary": { "kind": "mko_edition", "mko_id": "mko-at-most-countable-set", "edition_id": "edition-mko-at-most-countable-set-en-v1", "source_field": "summary", "source_mko_sha256": "9c7748686b7974341920a59d07628f20017ef6d65476d751b2feb7424f7b3b57", "edition_fields_sha256": "4b83db0655be76ff5bbd8a540dcf36ad4dade5a7458daef3ce127b9c9f5235cd" } } }, "atlas-vector-subspace": { "source_snapshot": { "id": "atlas-vector-subspace", "title_zh": "實向量子空間", "summary_zh": "實向量空間的非空子集若含零向量並對向量加法與每個實純量乘法封閉，則為向量子空間。", "canonical_mko_id": "mko-vector-subspace" }, "fields": { "title": "Real vector subspace", "summary": "A nonempty subset of a real vector space is a subspace when it contains zero and is closed under vector addition and multiplication by every real scalar." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-vector-subspace", "edition_id": "edition-mko-vector-subspace-en-v1", "source_field": "title", "source_mko_sha256": "a92727827e6ab175bea8b4da7021fe6b063af40e68f09beb9ad7437cd72c1abc", "edition_fields_sha256": "1bc40a2ff66fab1db9c3025eba6e134abdd00880b1b83025f40cb4bdd53d9ecf" }, "summary": { "kind": "mko_edition", "mko_id": "mko-vector-subspace", "edition_id": "edition-mko-vector-subspace-en-v1", "source_field": "summary", "source_mko_sha256": "a92727827e6ab175bea8b4da7021fe6b063af40e68f09beb9ad7437cd72c1abc", "edition_fields_sha256": "1bc40a2ff66fab1db9c3025eba6e134abdd00880b1b83025f40cb4bdd53d9ecf" } } }, "atlas-linear-map-kernel": { "source_snapshot": { "id": "atlas-linear-map-kernel", "title_zh": "線性映射的核", "summary_zh": "線性映射T的核是全部滿足T(x)=0的來源向量，並形成來源子空間。", "canonical_mko_id": "mko-linear-map-kernel" }, "fields": { "title": "Kernel of a linear map", "summary": "The kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-map-kernel", "edition_id": "edition-mko-linear-map-kernel-en-v1", "source_field": "title", "source_mko_sha256": "01a060bc8fa8fe1e302b8dcb559e8b1a9d5f6607cde6d5973ab75c1e4d0d3abc", "edition_fields_sha256": "d3ce524f367b83e5c1a2d86849107621308d29b5bcd27a50b10dfa3510423947" }, "summary": { "kind": "mko_edition", "mko_id": "mko-linear-map-kernel", "edition_id": "edition-mko-linear-map-kernel-en-v1", "source_field": "summary", "source_mko_sha256": "01a060bc8fa8fe1e302b8dcb559e8b1a9d5f6607cde6d5973ab75c1e4d0d3abc", "edition_fields_sha256": "d3ce524f367b83e5c1a2d86849107621308d29b5bcd27a50b10dfa3510423947" } } }, "atlas-linear-map-image": { "source_snapshot": { "id": "atlas-linear-map-image", "title_zh": "線性映射的像", "summary_zh": "線性映射的像是所有輸出T(x)的集合；矩陣映射的像等於column span。", "canonical_mko_id": "mko-linear-map-image" }, "fields": { "title": "Image of a linear map", "summary": "The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-map-image", "edition_id": "edition-mko-linear-map-image-en-v1", "source_field": "title", "source_mko_sha256": "9901554d389b316f90187d99a94cc8c17a03fc69cde338962d359b8d5e95eb8d", "edition_fields_sha256": "b66b978543b18ecb9254a9d2701b08c72333b8674a5aa86906dac3d95c3dc5c1" }, "summary": { "kind": "mko_edition", "mko_id": "mko-linear-map-image", "edition_id": "edition-mko-linear-map-image-en-v1", "source_field": "summary", "source_mko_sha256": "9901554d389b316f90187d99a94cc8c17a03fc69cde338962d359b8d5e95eb8d", "edition_fields_sha256": "b66b978543b18ecb9254a9d2701b08c72333b8674a5aa86906dac3d95c3dc5c1" } } }, "atlas-vector-space-dimension": { "source_snapshot": { "id": "atlas-vector-space-dimension", "title_zh": "有限維向量空間的維度", "summary_zh": "有限維向量空間的維度是任一有限基底的元素數；基底不同但基數不變。", "canonical_mko_id": "mko-vector-space-dimension" }, "fields": { "title": "Dimension of a finite-dimensional vector space", "summary": "The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-vector-space-dimension", "edition_id": "edition-mko-vector-space-dimension-en-v1", "source_field": "title", "source_mko_sha256": "0d547787130d8aa91b60c2231053891922efe324656000b1e67b8cd7f2658dd8", "edition_fields_sha256": "e68e1ea3e19469c372de665d7389d83cf4229e3c9039473287c85e7350b69fe3" }, "summary": { "kind": "mko_edition", "mko_id": "mko-vector-space-dimension", "edition_id": "edition-mko-vector-space-dimension-en-v1", "source_field": "summary", "source_mko_sha256": "0d547787130d8aa91b60c2231053891922efe324656000b1e67b8cd7f2658dd8", "edition_fields_sha256": "e68e1ea3e19469c372de665d7389d83cf4229e3c9039473287c85e7350b69fe3" } } }, "atlas-matrix-rank": { "source_snapshot": { "id": "atlas-matrix-rank", "title_zh": "矩陣秩", "summary_zh": "矩陣秩是row/column space維度，可由精確rank-preserving消去的pivot數計算。", "canonical_mko_id": "mko-matrix-rank" }, "fields": { "title": "Matrix rank", "summary": "Matrix rank is the dimension of the row or column space and can be computed as the number of pivots after exact rank-preserving elimination." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-matrix-rank", "edition_id": "edition-mko-matrix-rank-en-v1", "source_field": "title", "source_mko_sha256": "fdddbb93e9e8a00070f3e18fbc7fb5a9406d8ddab249ec9f7798986636ffd45e", "edition_fields_sha256": "4669d7b179c42901b2f7000b3d7ddc1a0ce028a1457cd4f14cc854fafe52d0d0" }, "summary": { "kind": "mko_edition", "mko_id": "mko-matrix-rank", "edition_id": "edition-mko-matrix-rank-en-v1", "source_field": "summary", "source_mko_sha256": "fdddbb93e9e8a00070f3e18fbc7fb5a9406d8ddab249ec9f7798986636ffd45e", "edition_fields_sha256": "4669d7b179c42901b2f7000b3d7ddc1a0ce028a1457cd4f14cc854fafe52d0d0" } } }, "atlas-rank-nullity-theorem": { "source_snapshot": { "id": "atlas-rank-nullity-theorem", "title_zh": "秩－零度定理", "summary_zh": "有限維線性映射T:V→W滿足dim V=dim ker T+rank T。", "canonical_mko_id": "mko-rank-nullity-theorem" }, "fields": { "title": "Rank-nullity theorem", "summary": "A linear map T:V→W with finite-dimensional source satisfies dim(V)=dim(ker T)+rank(T)." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-rank-nullity-theorem", "edition_id": "edition-mko-rank-nullity-theorem-en-v1", "source_field": "title", "source_mko_sha256": "e567ac132f3923949776cbd2866741a3564ba2c4d3e79d654abe99ba42b5ee0a", "edition_fields_sha256": "3b134c80229b4dce5738abbe0332748bef79c5e5f673ec844b3e0123233d6c3c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-rank-nullity-theorem", "edition_id": "edition-mko-rank-nullity-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "e567ac132f3923949776cbd2866741a3564ba2c4d3e79d654abe99ba42b5ee0a", "edition_fields_sha256": "3b134c80229b4dce5738abbe0332748bef79c5e5f673ec844b3e0123233d6c3c" } } }, "atlas-direct-sum": { "source_snapshot": { "id": "atlas-direct-sum", "title_zh": "子空間直和", "summary_zh": "二個子空間之和若每個向量有唯一分解便是內直和；要宣稱覆蓋母空間還須另證生成。", "canonical_mko_id": "mko-direct-sum" }, "fields": { "title": "Direct sum of subspaces", "summary": "The sum of two subspaces is an internal direct sum when every vector in that sum has a unique decomposition; covering an ambient space is a separate spanning obligation." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-direct-sum", "edition_id": "edition-mko-direct-sum-en-v1", "source_field": "title", "source_mko_sha256": "3eedbed2a646d34aa3bf6b0c5eb4e3482fcd8c816d40ca7c5652a6bf7224b9e8", "edition_fields_sha256": "23923687e23a223c502fb1b8ca19984f9b5c55cc2f5d531a6f54636f2114fc74" }, "summary": { "kind": "mko_edition", "mko_id": "mko-direct-sum", "edition_id": "edition-mko-direct-sum-en-v1", "source_field": "summary", "source_mko_sha256": "3eedbed2a646d34aa3bf6b0c5eb4e3482fcd8c816d40ca7c5652a6bf7224b9e8", "edition_fields_sha256": "23923687e23a223c502fb1b8ca19984f9b5c55cc2f5d531a6f54636f2114fc74" } } }, "atlas-quotient-vector-space": { "source_snapshot": { "id": "atlas-quotient-vector-space", "title_zh": "商向量空間", "summary_zh": "V依子空間W的cosets形成商向量空間，運算須與代表元選擇無關。", "canonical_mko_id": "mko-quotient-vector-space" }, "fields": { "title": "Quotient vector space", "summary": "Cosets of a subspace W form the quotient vector space V/W, whose operations must be independent of representative choices." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-quotient-vector-space", "edition_id": "edition-mko-quotient-vector-space-en-v1", "source_field": "title", "source_mko_sha256": "0f381f20543e1db106973cf749847f066bf4cb237412005cfb871209df548ed0", "edition_fields_sha256": "d13c8960af106c91530e9bdf378453d5319f425de9497baab8de950689774fd3" }, "summary": { "kind": "mko_edition", "mko_id": "mko-quotient-vector-space", "edition_id": "edition-mko-quotient-vector-space-en-v1", "source_field": "summary", "source_mko_sha256": "0f381f20543e1db106973cf749847f066bf4cb237412005cfb871209df548ed0", "edition_fields_sha256": "d13c8960af106c91530e9bdf378453d5319f425de9497baab8de950689774fd3" } } }, "atlas-dual-vector-space": { "source_snapshot": { "id": "atlas-dual-vector-space", "title_zh": "對偶向量空間", "summary_zh": "實向量空間的對偶空間由所有實值線性泛函組成；座標covector表示依賴基底。", "canonical_mko_id": "mko-dual-vector-space" }, "fields": { "title": "Dual vector space", "summary": "The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-dual-vector-space", "edition_id": "edition-mko-dual-vector-space-en-v1", "source_field": "title", "source_mko_sha256": "1994bbbcc73dff239e5c7cfdaba60f5dc5aadb1a0a4603f613e4b1371d59234a", "edition_fields_sha256": "10451f948dc275ef2249e02c8297073cde72dbbf7ecacc1d24db99d41aab739e" }, "summary": { "kind": "mko_edition", "mko_id": "mko-dual-vector-space", "edition_id": "edition-mko-dual-vector-space-en-v1", "source_field": "summary", "source_mko_sha256": "1994bbbcc73dff239e5c7cfdaba60f5dc5aadb1a0a4603f613e4b1371d59234a", "edition_fields_sha256": "10451f948dc275ef2249e02c8297073cde72dbbf7ecacc1d24db99d41aab739e" } } }, "atlas-closure-and-interior": { "source_snapshot": { "id": "atlas-closure-and-interior", "title_zh": "拓撲閉包與內部", "summary_zh": "集合閉包是所有閉超集的交，內部是所有開子集的聯；兩者相對於指定拓撲。", "canonical_mko_id": "mko-closure-and-interior" }, "fields": { "title": "Topological closure and interior", "summary": "The closure of a set is the intersection of all closed supersets, and its interior is the union of all open subsets, relative to the specified topology." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-closure-and-interior", "edition_id": "edition-mko-closure-and-interior-en-v1", "source_field": "title", "source_mko_sha256": "72aef7da5935f151015d5eecaa4d9cdde8226564638885f222596bf4ef2433e4", "edition_fields_sha256": "6fd68d2f742cc7977a0059b2a02f6d0441b75d54e9b2f0b119d4a3cbdc1e5bfa" }, "summary": { "kind": "mko_edition", "mko_id": "mko-closure-and-interior", "edition_id": "edition-mko-closure-and-interior-en-v1", "source_field": "summary", "source_mko_sha256": "72aef7da5935f151015d5eecaa4d9cdde8226564638885f222596bf4ef2433e4", "edition_fields_sha256": "6fd68d2f742cc7977a0059b2a02f6d0441b75d54e9b2f0b119d4a3cbdc1e5bfa" } } }, "atlas-topological-boundary": { "source_snapshot": { "id": "atlas-topological-boundary", "title_zh": "拓撲邊界", "summary_zh": "集合A的拓撲邊界為closure(A)減去interior(A)，亦等於closure(A)交closure(complement A)。", "canonical_mko_id": "mko-topological-boundary" }, "fields": { "title": "Topological boundary", "summary": "The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-topological-boundary", "edition_id": "edition-mko-topological-boundary-en-v1", "source_field": "title", "source_mko_sha256": "31f941e14ac65e7366835ceef8264cd169c8739bdaa6e04182c4f7b160ec8a13", "edition_fields_sha256": "4d64572504094b6648466e74f7325952ace25e4fbc64989b97e2aeba436bd6fe" }, "summary": { "kind": "mko_edition", "mko_id": "mko-topological-boundary", "edition_id": "edition-mko-topological-boundary-en-v1", "source_field": "summary", "source_mko_sha256": "31f941e14ac65e7366835ceef8264cd169c8739bdaa6e04182c4f7b160ec8a13", "edition_fields_sha256": "4d64572504094b6648466e74f7325952ace25e4fbc64989b97e2aeba436bd6fe" } } }, "atlas-open-cover-and-subcover": { "source_snapshot": { "id": "atlas-open-cover-and-subcover", "title_zh": "開覆蓋與子覆蓋", "summary_zh": "開覆蓋由指定拓撲的開集組成且聯集包含目標集合；子覆蓋須取自原覆蓋並仍覆蓋同一目標。", "canonical_mko_id": "mko-open-cover-and-subcover" }, "fields": { "title": "Open cover and subcover", "summary": "An open cover consists of open sets in the specified topology whose union contains the target; a subcover selects from that original cover and still covers the same target." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-open-cover-and-subcover", "edition_id": "edition-mko-open-cover-and-subcover-en-v1", "source_field": "title", "source_mko_sha256": "bd5a852d24868cb59747e9030f8da58e604c78a87a2442558dc2643a01d8ecb1", "edition_fields_sha256": "4ca8e019224a0fa388f8b3cca38d2fd55db504f6d607a5c4a4436b7e25fa325c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-open-cover-and-subcover", "edition_id": "edition-mko-open-cover-and-subcover-en-v1", "source_field": "summary", "source_mko_sha256": "bd5a852d24868cb59747e9030f8da58e604c78a87a2442558dc2643a01d8ecb1", "edition_fields_sha256": "4ca8e019224a0fa388f8b3cca38d2fd55db504f6d607a5c4a4436b7e25fa325c" } } }, "atlas-compact-topological-space": { "source_snapshot": { "id": "atlas-compact-topological-space", "title_zh": "緊緻拓撲空間", "summary_zh": "拓撲空間緊緻，若每個開覆蓋都有有限子覆蓋；每個有限拓撲空間皆緊緻。", "canonical_mko_id": "mko-compact-topological-space" }, "fields": { "title": "Compact topological space", "summary": "A space is compact when every open cover has a finite subcover; every finite topological space is compact." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-compact-topological-space", "edition_id": "edition-mko-compact-topological-space-en-v1", "source_field": "title", "source_mko_sha256": "13f302a171c38a633a5a145da984410cab9bef79a09c4aef91bdaf9e016226e6", "edition_fields_sha256": "76041f6b967c8723bb091273ea689f8d9ab70793faa0c2ef8c2be4ff4bc7ac86" }, "summary": { "kind": "mko_edition", "mko_id": "mko-compact-topological-space", "edition_id": "edition-mko-compact-topological-space-en-v1", "source_field": "summary", "source_mko_sha256": "13f302a171c38a633a5a145da984410cab9bef79a09c4aef91bdaf9e016226e6", "edition_fields_sha256": "76041f6b967c8723bb091273ea689f8d9ab70793faa0c2ef8c2be4ff4bc7ac86" } } }, "atlas-connected-topological-space": { "source_snapshot": { "id": "atlas-connected-topological-space", "title_zh": "連通拓撲空間", "summary_zh": "拓撲空間連通，若不存在由兩個不交非空開集組成的separation；等價於無非平凡clopen subset。", "canonical_mko_id": "mko-connected-topological-space" }, "fields": { "title": "Connected topological space", "summary": "A space is connected when it has no separation into two disjoint nonempty open sets, equivalently no nontrivial clopen subset." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-connected-topological-space", "edition_id": "edition-mko-connected-topological-space-en-v1", "source_field": "title", "source_mko_sha256": "a3956f0120a382824010422c3639f93914810983c5f1dcf47b08e4f3b2d3d96f", "edition_fields_sha256": "9fd4aebeb589e9a82acc25dd8ac537e3a58fa8135270274c6d2216aa9bdfd7fa" }, "summary": { "kind": "mko_edition", "mko_id": "mko-connected-topological-space", "edition_id": "edition-mko-connected-topological-space-en-v1", "source_field": "summary", "source_mko_sha256": "a3956f0120a382824010422c3639f93914810983c5f1dcf47b08e4f3b2d3d96f", "edition_fields_sha256": "9fd4aebeb589e9a82acc25dd8ac537e3a58fa8135270274c6d2216aa9bdfd7fa" } } }, "atlas-homeomorphism": { "source_snapshot": { "id": "atlas-homeomorphism", "title_zh": "同胚", "summary_zh": "同胚是雙射、連續且逆映射亦連續的映射；一般continuous bijection仍可能不是同胚。", "canonical_mko_id": "mko-homeomorphism" }, "fields": { "title": "Homeomorphism", "summary": "A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-homeomorphism", "edition_id": "edition-mko-homeomorphism-en-v1", "source_field": "title", "source_mko_sha256": "2869b47d1ae828f6ec6ed61a0f7aa9d72ef812e946cb7b04184b3592d2f384de", "edition_fields_sha256": "60aee4c33428cf5b94d34ac985b0aece4cd40e422fe234860b02c1c2858c9021" }, "summary": { "kind": "mko_edition", "mko_id": "mko-homeomorphism", "edition_id": "edition-mko-homeomorphism-en-v1", "source_field": "summary", "source_mko_sha256": "2869b47d1ae828f6ec6ed61a0f7aa9d72ef812e946cb7b04184b3592d2f384de", "edition_fields_sha256": "60aee4c33428cf5b94d34ac985b0aece4cd40e422fe234860b02c1c2858c9021" } } }, "atlas-pointwise-convergence-of-functions": { "source_snapshot": { "id": "atlas-pointwise-convergence-of-functions", "title_zh": "函數列逐點收斂", "summary_zh": "同一domain上的函數列逐點收斂，若每個固定x的值序列收斂至f(x)，且N可依x與epsilon改變。", "canonical_mko_id": "mko-pointwise-convergence-of-functions" }, "fields": { "title": "Pointwise convergence of functions", "summary": "A function sequence on one domain converges pointwise when its value sequence converges at each fixed x; the witness N may depend on x and epsilon." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-pointwise-convergence-of-functions", "edition_id": "edition-mko-pointwise-convergence-of-functions-en-v1", "source_field": "title", "source_mko_sha256": "dde2c47154f28c5cd5dac55ff9f8a148e7761d085014f6ee6d2f72acbfb6012c", "edition_fields_sha256": "4e0276d57419d18a0dc49343c547c0aff4eac7c5272650571483b9f22f266113" }, "summary": { "kind": "mko_edition", "mko_id": "mko-pointwise-convergence-of-functions", "edition_id": "edition-mko-pointwise-convergence-of-functions-en-v1", "source_field": "summary", "source_mko_sha256": "dde2c47154f28c5cd5dac55ff9f8a148e7761d085014f6ee6d2f72acbfb6012c", "edition_fields_sha256": "4e0276d57419d18a0dc49343c547c0aff4eac7c5272650571483b9f22f266113" } } }, "atlas-uniform-convergence-of-functions": { "source_snapshot": { "id": "atlas-uniform-convergence-of-functions", "title_zh": "函數列一致收斂", "summary_zh": "一致收斂要求同一N(epsilon)同時控制domain全部x。", "canonical_mko_id": "mko-uniform-convergence-of-functions" }, "fields": { "title": "Uniform convergence of functions", "summary": "Uniform convergence requires one witness N(epsilon) controlling every point of the domain simultaneously." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-uniform-convergence-of-functions", "edition_id": "edition-mko-uniform-convergence-of-functions-en-v1", "source_field": "title", "source_mko_sha256": "c28f03cdaf36d8b06ca1718e4253a1d705745496ed7c655d09f55c2a82148ac0", "edition_fields_sha256": "245560a9bcee4e3f12273dd55d2e205739e53de24f947105b7ecac130e253587" }, "summary": { "kind": "mko_edition", "mko_id": "mko-uniform-convergence-of-functions", "edition_id": "edition-mko-uniform-convergence-of-functions-en-v1", "source_field": "summary", "source_mko_sha256": "c28f03cdaf36d8b06ca1718e4253a1d705745496ed7c655d09f55c2a82148ac0", "edition_fields_sha256": "245560a9bcee4e3f12273dd55d2e205739e53de24f947105b7ecac130e253587" } } }, "atlas-infinite-series": { "source_snapshot": { "id": "atlas-infinite-series", "title_zh": "無窮級數", "summary_zh": "無窮級數以partial-sum sequence的極限定義；項趨零必要但不充分。", "canonical_mko_id": "mko-infinite-series" }, "fields": { "title": "Infinite series", "summary": "An infinite series is defined through the limit of its partial-sum sequence; terms tending to zero are necessary but not sufficient for convergence." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-infinite-series", "edition_id": "edition-mko-infinite-series-en-v1", "source_field": "title", "source_mko_sha256": "f2c97043e047957807c3f303d9bf37bbe3d188f1721ae0dcfbab1537a45b8d0b", "edition_fields_sha256": "a3aa90ccbd9d921d5ee3ad807eb5fc5f74ac9328fb10a9c6366affdcc0da5b92" }, "summary": { "kind": "mko_edition", "mko_id": "mko-infinite-series", "edition_id": "edition-mko-infinite-series-en-v1", "source_field": "summary", "source_mko_sha256": "f2c97043e047957807c3f303d9bf37bbe3d188f1721ae0dcfbab1537a45b8d0b", "edition_fields_sha256": "a3aa90ccbd9d921d5ee3ad807eb5fc5f74ac9328fb10a9c6366affdcc0da5b92" } } }, "atlas-absolute-convergence-of-series": { "source_snapshot": { "id": "atlas-absolute-convergence-of-series", "title_zh": "級數絕對收斂", "summary_zh": "sum |a_n|收斂時sum a_n絕對收斂；絕對收斂推出收斂，反向不一般成立。", "canonical_mko_id": "mko-absolute-convergence-of-series" }, "fields": { "title": "Absolute convergence of series", "summary": "A series is absolutely convergent when the sum of absolute terms converges; absolute convergence implies convergence, but the converse does not hold in general." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-absolute-convergence-of-series", "edition_id": "edition-mko-absolute-convergence-of-series-en-v1", "source_field": "title", "source_mko_sha256": "4419bd0abbb51b07728caad18022eaace6aec3c8243880972704e8b1413cc1f8", "edition_fields_sha256": "6a3db3d982b0ce3467c6d7e330407a2da154e541c4b4b0c8643cecd95499ab85" }, "summary": { "kind": "mko_edition", "mko_id": "mko-absolute-convergence-of-series", "edition_id": "edition-mko-absolute-convergence-of-series-en-v1", "source_field": "summary", "source_mko_sha256": "4419bd0abbb51b07728caad18022eaace6aec3c8243880972704e8b1413cc1f8", "edition_fields_sha256": "6a3db3d982b0ce3467c6d7e330407a2da154e541c4b4b0c8643cecd95499ab85" } } }, "atlas-almost-sure-convergence": { "source_snapshot": { "id": "atlas-almost-sure-convergence", "title_zh": "幾乎必然收斂", "summary_zh": "同一機率空間上的隨機變數序列若在零機率例外集外逐點收斂，則幾乎必然收斂。", "canonical_mko_id": "mko-almost-sure-convergence" }, "fields": { "title": "Almost-sure convergence", "summary": "Random variables on the same probability space converge almost surely when they converge pointwise outside an exceptional set of probability zero." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-almost-sure-convergence", "edition_id": "edition-mko-almost-sure-convergence-en-v1", "source_field": "title", "source_mko_sha256": "a848e75d5e3643fc87b620fc61c2385c8ed080ba4eac3a8573d9cb5e575f0b70", "edition_fields_sha256": "be6abef02d5eb8455e09f71db44f3c9a09cad58dcb1ed52f3be0bab40db995fc" }, "summary": { "kind": "mko_edition", "mko_id": "mko-almost-sure-convergence", "edition_id": "edition-mko-almost-sure-convergence-en-v1", "source_field": "summary", "source_mko_sha256": "a848e75d5e3643fc87b620fc61c2385c8ed080ba4eac3a8573d9cb5e575f0b70", "edition_fields_sha256": "be6abef02d5eb8455e09f71db44f3c9a09cad58dcb1ed52f3be0bab40db995fc" } } }, "atlas-lp-convergence": { "source_snapshot": { "id": "atlas-lp-convergence", "title_zh": "Lp收斂", "summary_zh": "共同機率空間上固定整數p≥1時，E|X_n−X|^p→0稱Lp收斂。", "canonical_mko_id": "mko-lp-convergence" }, "fields": { "title": "Lp convergence", "summary": "On a common probability space with fixed integer p≥1, Lp convergence means E|X_n−X|^p tends to zero." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-lp-convergence", "edition_id": "edition-mko-lp-convergence-en-v1", "source_field": "title", "source_mko_sha256": "aec69290c870938d15b40e7ffb6849ffa5fa53265e0fa29d59021aa1f01ce47f", "edition_fields_sha256": "7e4f30576485016bdc1f27143822211e588254b190646c43c050d98b86a3e85f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-lp-convergence", "edition_id": "edition-mko-lp-convergence-en-v1", "source_field": "summary", "source_mko_sha256": "aec69290c870938d15b40e7ffb6849ffa5fa53265e0fa29d59021aa1f01ce47f", "edition_fields_sha256": "7e4f30576485016bdc1f27143822211e588254b190646c43c050d98b86a3e85f" } } }, "atlas-weak-law-large-numbers": { "source_snapshot": { "id": "atlas-weak-law-large-numbers", "title_zh": "弱大數律", "summary_zh": "IID且具有有限variance時，Chebyshev界給樣本平均依機率收斂到共同mean。", "canonical_mko_id": "mko-weak-law-large-numbers" }, "fields": { "title": "Weak law of large numbers", "summary": "For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-weak-law-large-numbers", "edition_id": "edition-mko-weak-law-large-numbers-en-v1", "source_field": "title", "source_mko_sha256": "7eb5ea7326a95ec1926f661b129d1f6ed74f95c53574aea5f3878111f80bb55c", "edition_fields_sha256": "c35691d039daaba143480380fd48d11b68119a8aa1876754b10609102325175b" }, "summary": { "kind": "mko_edition", "mko_id": "mko-weak-law-large-numbers", "edition_id": "edition-mko-weak-law-large-numbers-en-v1", "source_field": "summary", "source_mko_sha256": "7eb5ea7326a95ec1926f661b129d1f6ed74f95c53574aea5f3878111f80bb55c", "edition_fields_sha256": "c35691d039daaba143480380fd48d11b68119a8aa1876754b10609102325175b" } } }, "atlas-bayes-theorem": { "source_snapshot": { "id": "atlas-bayes-theorem", "title_zh": "貝氏定理", "summary_zh": "正分母條件下，Bayes定理以共同intersection與priors/likelihoods精確反轉條件機率。", "canonical_mko_id": "mko-bayes-theorem" }, "fields": { "title": "Bayes theorem", "summary": "With positive conditioning denominators, Bayes theorem reverses conditional probabilities using the same intersection together with priors and likelihoods." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-bayes-theorem", "edition_id": "edition-mko-bayes-theorem-en-v1", "source_field": "title", "source_mko_sha256": "29451b12d334baf07d1c6b72cf87989c5c58d98ccd89384b8b2a6591f52b6179", "edition_fields_sha256": "23ce6f1b7137511703a9ee7008a6a81288fec8c5df542f7ed320c7c69b612eda" }, "summary": { "kind": "mko_edition", "mko_id": "mko-bayes-theorem", "edition_id": "edition-mko-bayes-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "29451b12d334baf07d1c6b72cf87989c5c58d98ccd89384b8b2a6591f52b6179", "edition_fields_sha256": "23ce6f1b7137511703a9ee7008a6a81288fec8c5df542f7ed320c7c69b612eda" } } }, "atlas-parametric-statistical-model": { "source_snapshot": { "id": "atlas-parametric-statistical-model", "title_zh": "參數統計模型", "summary_zh": "參數統計模型為每個theta指定同一有限可測sample space上的完整機率law P_theta。", "canonical_mko_id": "mko-parametric-statistical-model" }, "fields": { "title": "Parametric statistical model", "summary": "A parametric statistical model assigns each parameter theta a complete probability law P_theta on the same finite measurable sample space." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-parametric-statistical-model", "edition_id": "edition-mko-parametric-statistical-model-en-v1", "source_field": "title", "source_mko_sha256": "ef431eac244935b2a2629f424d522b6d9b36a17dede2b668b4402aed44ac715d", "edition_fields_sha256": "101d8827f1d6af1d8e8870e5518c03fb7fc53e3ae82808bbe6f169a05850b747" }, "summary": { "kind": "mko_edition", "mko_id": "mko-parametric-statistical-model", "edition_id": "edition-mko-parametric-statistical-model-en-v1", "source_field": "summary", "source_mko_sha256": "ef431eac244935b2a2629f424d522b6d9b36a17dede2b668b4402aed44ac715d", "edition_fields_sha256": "101d8827f1d6af1d8e8870e5518c03fb7fc53e3ae82808bbe6f169a05850b747" } } }, "atlas-statistical-estimator": { "source_snapshot": { "id": "atlas-statistical-estimator", "title_zh": "統計估計量", "summary_zh": "估計量是固定且不依未知theta的可測total statistic；觀察後數值是estimate。", "canonical_mko_id": "mko-statistical-estimator" }, "fields": { "title": "Statistical estimator", "summary": "An estimator is a fixed measurable total statistic independent of the unknown parameter; its observed numerical value is an estimate." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-statistical-estimator", "edition_id": "edition-mko-statistical-estimator-en-v1", "source_field": "title", "source_mko_sha256": "ba7fbdff9e10ebc9977253d4edccfcba77e5e47c0404aae6d00ad578d8bf6d2f", "edition_fields_sha256": "f918bb670d5c38f709aae12a717dda3a8b87a8758c67f0442341a88196b15393" }, "summary": { "kind": "mko_edition", "mko_id": "mko-statistical-estimator", "edition_id": "edition-mko-statistical-estimator-en-v1", "source_field": "summary", "source_mko_sha256": "ba7fbdff9e10ebc9977253d4edccfcba77e5e47c0404aae6d00ad578d8bf6d2f", "edition_fields_sha256": "f918bb670d5c38f709aae12a717dda3a8b87a8758c67f0442341a88196b15393" } } }, "atlas-sampling-distribution": { "source_snapshot": { "id": "atlas-sampling-distribution", "title_zh": "抽樣分布", "summary_zh": "固定theta時，估計量把樣本分布pushforward成統計量的抽樣分布；全部theta形成一族laws。", "canonical_mko_id": "mko-sampling-distribution" }, "fields": { "title": "Sampling distribution", "summary": "At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-sampling-distribution", "edition_id": "edition-mko-sampling-distribution-en-v1", "source_field": "title", "source_mko_sha256": "0878462afb0517679eb337ad1d5f12bd6025d7039d27a541544de8493ab9cdc9", "edition_fields_sha256": "e622ad72d88d6d2a0d60de76db2d25a225b9e9728d5e6bc24db66bad2513460f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-sampling-distribution", "edition_id": "edition-mko-sampling-distribution-en-v1", "source_field": "summary", "source_mko_sha256": "0878462afb0517679eb337ad1d5f12bd6025d7039d27a541544de8493ab9cdc9", "edition_fields_sha256": "e622ad72d88d6d2a0d60de76db2d25a225b9e9728d5e6bc24db66bad2513460f" } } }, "atlas-estimator-bias": { "source_snapshot": { "id": "atlas-estimator-bias", "title_zh": "估計量偏誤", "summary_zh": "估計量在theta的偏誤是其theta抽樣分布期望減去目標參數；無偏要求對所有theta成立。", "canonical_mko_id": "mko-estimator-bias" }, "fields": { "title": "Estimator bias", "summary": "Bias at theta is the estimator expectation under its theta sampling law minus the target parameter; unbiasedness requires zero bias at every theta." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-estimator-bias", "edition_id": "edition-mko-estimator-bias-en-v1", "source_field": "title", "source_mko_sha256": "be64de19751560ba6bd5142b5efa74407be5c35def263eb936120c6b63925da6", "edition_fields_sha256": "da796195dbe3857c131ddbd84c124a1539375f4fc2f6078c0dabc3ddafe01c6d" }, "summary": { "kind": "mko_edition", "mko_id": "mko-estimator-bias", "edition_id": "edition-mko-estimator-bias-en-v1", "source_field": "summary", "source_mko_sha256": "be64de19751560ba6bd5142b5efa74407be5c35def263eb936120c6b63925da6", "edition_fields_sha256": "da796195dbe3857c131ddbd84c124a1539375f4fc2f6078c0dabc3ddafe01c6d" } } }, "atlas-estimator-variance": { "source_snapshot": { "id": "atlas-estimator-variance", "title_zh": "估計量變異數", "summary_zh": "估計量變異數逐theta衡量sampling law相對自身期望的二次離差，不等同偏誤或MSE。", "canonical_mko_id": "mko-estimator-variance" }, "fields": { "title": "Estimator variance", "summary": "At each theta, estimator variance is the second central moment of its sampling law around its own mean; it is not bias or MSE by itself." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-estimator-variance", "edition_id": "edition-mko-estimator-variance-en-v1", "source_field": "title", "source_mko_sha256": "8e31f372ae9c9485e8d279c1850f034c30f4a1d657c84cd603be31da9a55e53f", "edition_fields_sha256": "786ba3be651557be1e70401b3a5a60bfaf47a7540f87c5a05f667e71dcc399a9" }, "summary": { "kind": "mko_edition", "mko_id": "mko-estimator-variance", "edition_id": "edition-mko-estimator-variance-en-v1", "source_field": "summary", "source_mko_sha256": "8e31f372ae9c9485e8d279c1850f034c30f4a1d657c84cd603be31da9a55e53f", "edition_fields_sha256": "786ba3be651557be1e70401b3a5a60bfaf47a7540f87c5a05f667e71dcc399a9" } } }, "atlas-eulerian-trail": { "source_snapshot": { "id": "atlas-eulerian-trail", "title_zh": "Euler trail與circuit", "summary_zh": "Euler trail恰用每條邊一次；非零邊支撐連通且奇度數為0或2時分別有circuit或open trail。", "canonical_mko_id": "mko-eulerian-trail" }, "fields": { "title": "Euler trails and circuits", "summary": "An Euler trail uses each edge once; on connected nonzero-edge support, zero odd-degree vertices give a circuit and two give an open Euler trail." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-eulerian-trail", "edition_id": "edition-mko-eulerian-trail-en-v1", "source_field": "title", "source_mko_sha256": "bd4c7accbd4091e00d7a37abe72e796dc663a93d6cdb230cfaa24a0ec125ad41", "edition_fields_sha256": "2b1f0a7f0aa56ba4b7a0f0fc28ca70455305dcb35af4068aa95dc23285bf4e06" }, "summary": { "kind": "mko_edition", "mko_id": "mko-eulerian-trail", "edition_id": "edition-mko-eulerian-trail-en-v1", "source_field": "summary", "source_mko_sha256": "bd4c7accbd4091e00d7a37abe72e796dc663a93d6cdb230cfaa24a0ec125ad41", "edition_fields_sha256": "2b1f0a7f0aa56ba4b7a0f0fc28ca70455305dcb35af4068aa95dc23285bf4e06" } } }, "atlas-spanning-tree": { "source_snapshot": { "id": "atlas-spanning-tree", "title_zh": "生成樹", "summary_zh": "生成樹是包含父圖全部頂點的連通無圈子圖。", "canonical_mko_id": "mko-spanning-tree" }, "fields": { "title": "Spanning tree", "summary": "A spanning tree is a connected acyclic subgraph containing every vertex of its parent graph." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-spanning-tree", "edition_id": "edition-mko-spanning-tree-en-v1", "source_field": "title", "source_mko_sha256": "f93b1e5d271c80d9a571d19d5ddba8bceb178dce9bc9c8e6f3b1b0279a0d3833", "edition_fields_sha256": "f9724b97f3ab4eb58d02ce4ee1929e399c4918bde084adb05d4df2abb4b732bf" }, "summary": { "kind": "mko_edition", "mko_id": "mko-spanning-tree", "edition_id": "edition-mko-spanning-tree-en-v1", "source_field": "summary", "source_mko_sha256": "f93b1e5d271c80d9a571d19d5ddba8bceb178dce9bc9c8e6f3b1b0279a0d3833", "edition_fields_sha256": "f9724b97f3ab4eb58d02ce4ee1929e399c4918bde084adb05d4df2abb4b732bf" } } }, "atlas-graph-matching": { "source_snapshot": { "id": "atlas-graph-matching", "title_zh": "圖匹配", "summary_zh": "有限圖的matching是彼此不共享端點的邊子集；空matching合法。", "canonical_mko_id": "mko-graph-matching" }, "fields": { "title": "Graph matching", "summary": "A matching in a finite graph is an edge subset whose edges share no endpoints; the empty matching is allowed." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-graph-matching", "edition_id": "edition-mko-graph-matching-en-v1", "source_field": "title", "source_mko_sha256": "a8718b439d32013b3b1b480ad7e54789abe034e6599f18fdc9261be210f5f8ed", "edition_fields_sha256": "12d95a6009b212bb5763742c852e36f906bc56e7ecb5e493ed7aad1bb556796b" }, "summary": { "kind": "mko_edition", "mko_id": "mko-graph-matching", "edition_id": "edition-mko-graph-matching-en-v1", "source_field": "summary", "source_mko_sha256": "a8718b439d32013b3b1b480ad7e54789abe034e6599f18fdc9261be210f5f8ed", "edition_fields_sha256": "12d95a6009b212bb5763742c852e36f906bc56e7ecb5e493ed7aad1bb556796b" } } }, "atlas-maximum-matching": { "source_snapshot": { "id": "atlas-maximum-matching", "title_zh": "最大匹配", "summary_zh": "最大匹配是在完整有限圖全部合法matching中邊數最大者；可有多個且不必perfect。", "canonical_mko_id": "mko-maximum-matching" }, "fields": { "title": "Maximum matching", "summary": "A maximum matching has the greatest edge count among all valid matchings of the complete finite graph; it may be nonunique and need not be perfect." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-maximum-matching", "edition_id": "edition-mko-maximum-matching-en-v1", "source_field": "title", "source_mko_sha256": "bb97ee67fcffa420b79baaaf56c7ebabe2977fb5d14f5826d1f1c0452f4ed7f9", "edition_fields_sha256": "6d7e90f7c4b5b67b081f503055e2bd2b8eb987235617f80e6d30bf889d4362fc" }, "summary": { "kind": "mko_edition", "mko_id": "mko-maximum-matching", "edition_id": "edition-mko-maximum-matching-en-v1", "source_field": "summary", "source_mko_sha256": "bb97ee67fcffa420b79baaaf56c7ebabe2977fb5d14f5826d1f1c0452f4ed7f9", "edition_fields_sha256": "6d7e90f7c4b5b67b081f503055e2bd2b8eb987235617f80e6d30bf889d4362fc" } } }, "atlas-clique-and-independent-set": { "source_snapshot": { "id": "atlas-clique-and-independent-set", "title_zh": "Clique與獨立集", "summary_zh": "Clique內每對不同頂點相鄰，獨立集內每對不同頂點不相鄰；最大與極大不可混同。", "canonical_mko_id": "mko-clique-and-independent-set" }, "fields": { "title": "Cliques and independent sets", "summary": "Every pair of distinct vertices of a clique is adjacent; every pair of an independent set is nonadjacent. Inclusion-maximal and maximum-cardinality are different." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-clique-and-independent-set", "edition_id": "edition-mko-clique-and-independent-set-en-v1", "source_field": "title", "source_mko_sha256": "9477f7b7e5b3af0c1c270f6f42b508ec6142c458868b507c850d56c9a5b946fe", "edition_fields_sha256": "dd187ac04382fd27fcf54e4c2939f73e3ff70bb612d6bb079230eeb71c83fd3d" }, "summary": { "kind": "mko_edition", "mko_id": "mko-clique-and-independent-set", "edition_id": "edition-mko-clique-and-independent-set-en-v1", "source_field": "summary", "source_mko_sha256": "9477f7b7e5b3af0c1c270f6f42b508ec6142c458868b507c850d56c9a5b946fe", "edition_fields_sha256": "dd187ac04382fd27fcf54e4c2939f73e3ff70bb612d6bb079230eeb71c83fd3d" } } }, "atlas-chromatic-number": { "source_snapshot": { "id": "atlas-chromatic-number", "title_zh": "色數", "summary_zh": "有限簡單圖的色數是proper頂點著色所需顏色數的最小值，需同時具備上界與下界證書。", "canonical_mko_id": "mko-chromatic-number" }, "fields": { "title": "Chromatic number", "summary": "The chromatic number of a finite simple graph is the minimum number of colors in a proper vertex coloring; equality needs both upper and lower certificates." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-chromatic-number", "edition_id": "edition-mko-chromatic-number-en-v1", "source_field": "title", "source_mko_sha256": "c6461c8fe8f1d88f27aaea185ce8ec6d8d3e4318855946ae3b5ef9daf6f5f109", "edition_fields_sha256": "d6f46e0c4e6dbc8a1dd72180811051cdaad32f9e52b8b67ade8fedc7fd945646" }, "summary": { "kind": "mko_edition", "mko_id": "mko-chromatic-number", "edition_id": "edition-mko-chromatic-number-en-v1", "source_field": "summary", "source_mko_sha256": "c6461c8fe8f1d88f27aaea185ce8ec6d8d3e4318855946ae3b5ef9daf6f5f109", "edition_fields_sha256": "d6f46e0c4e6dbc8a1dd72180811051cdaad32f9e52b8b67ade8fedc7fd945646" } } }, "atlas-euler-totient-function": { "source_snapshot": { "id": "atlas-euler-totient-function", "title_zh": "Euler φ函數", "summary_zh": "Euler φ(n)計數一個完整剩餘代表系中與n互質的代表；約定φ(1)=1。", "canonical_mko_id": "mko-euler-totient-function" }, "fields": { "title": "Euler totient function", "summary": "Euler phi(n) counts the representatives coprime to n in a complete residue-representative system, retaining the convention phi(1)=1." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-euler-totient-function", "edition_id": "edition-mko-euler-totient-function-en-v1", "source_field": "title", "source_mko_sha256": "2f96ece219d918a70b350ee38d51917541627e34c40240cf2b2cfa6d4ab48c31", "edition_fields_sha256": "4f20e6d94f371490b3efb2e719f29bba88b1ab946011f5450dcd7651a2790c42" }, "summary": { "kind": "mko_edition", "mko_id": "mko-euler-totient-function", "edition_id": "edition-mko-euler-totient-function-en-v1", "source_field": "summary", "source_mko_sha256": "2f96ece219d918a70b350ee38d51917541627e34c40240cf2b2cfa6d4ab48c31", "edition_fields_sha256": "4f20e6d94f371490b3efb2e719f29bba88b1ab946011f5450dcd7651a2790c42" } } }, "atlas-chinese-remainder-theorem": { "source_snapshot": { "id": "atlas-chinese-remainder-theorem", "title_zh": "中國剩餘定理", "summary_zh": "正兩兩互質模數的同餘系統在模乘積意義下存在唯一解。", "canonical_mko_id": "mko-chinese-remainder-theorem" }, "fields": { "title": "Chinese remainder theorem", "summary": "For positive pairwise-coprime moduli, an integer congruence system has a unique solution class modulo their product." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-chinese-remainder-theorem", "edition_id": "edition-mko-chinese-remainder-theorem-en-v1", "source_field": "title", "source_mko_sha256": "97d98262a4c1d1774983902029b0e964992d7472332d99d3c22e80f77a1a44cc", "edition_fields_sha256": "35934e4ca42168fa7dc82a5f25f26eeec1c240b03a4f12a675fcec6caf48cb81" }, "summary": { "kind": "mko_edition", "mko_id": "mko-chinese-remainder-theorem", "edition_id": "edition-mko-chinese-remainder-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "97d98262a4c1d1774983902029b0e964992d7472332d99d3c22e80f77a1a44cc", "edition_fields_sha256": "35934e4ca42168fa7dc82a5f25f26eeec1c240b03a4f12a675fcec6caf48cb81" } } }, "atlas-modular-inverse": { "source_snapshot": { "id": "atlas-modular-inverse", "title_zh": "模反元素", "summary_zh": "n≥2時，a模n有乘法反元素恰當gcd(a,n)=1；反元素在模n意義唯一。", "canonical_mko_id": "mko-modular-inverse" }, "fields": { "title": "Modular inverse", "summary": "For n≥2, a has a multiplicative inverse modulo n exactly when gcd(a,n)=1; the inverse is unique as a residue class modulo n." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-modular-inverse", "edition_id": "edition-mko-modular-inverse-en-v1", "source_field": "title", "source_mko_sha256": "d16e0773143022c00ff50dad2cd84ef53e2d8a9078f64c2d785413c676b002ab", "edition_fields_sha256": "fe0dacf2d2a88d67763944c9a900a84afab6c6adc40a973f1ec79c72cfdda40f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-modular-inverse", "edition_id": "edition-mko-modular-inverse-en-v1", "source_field": "summary", "source_mko_sha256": "d16e0773143022c00ff50dad2cd84ef53e2d8a9078f64c2d785413c676b002ab", "edition_fields_sha256": "fe0dacf2d2a88d67763944c9a900a84afab6c6adc40a973f1ec79c72cfdda40f" } } }, "atlas-linear-diophantine-equation": { "source_snapshot": { "id": "atlas-linear-diophantine-equation", "title_zh": "線性Diophantine方程", "summary_zh": "非退化整數方程ax+by=c有解恰當gcd(a,b)整除c，全部解由一個整數參數給出。", "canonical_mko_id": "mko-linear-diophantine-equation" }, "fields": { "title": "Linear Diophantine equation", "summary": "A nondegenerate integer equation a*x+b*y=c has a solution exactly when gcd(a,b) divides c; all solutions have one integer parameter." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-diophantine-equation", "edition_id": "edition-mko-linear-diophantine-equation-en-v1", "source_field": "title", "source_mko_sha256": "f0c63a1636cb208e054adfa59fdc5cb29078971741ce31fd2c59710199df6a79", "edition_fields_sha256": "104525843b8be78a96631e59a9adfa509c7513d1dd3b2f0c9c493413dc394cf7" }, "summary": { "kind": "mko_edition", "mko_id": "mko-linear-diophantine-equation", "edition_id": "edition-mko-linear-diophantine-equation-en-v1", "source_field": "summary", "source_mko_sha256": "f0c63a1636cb208e054adfa59fdc5cb29078971741ce31fd2c59710199df6a79", "edition_fields_sha256": "104525843b8be78a96631e59a9adfa509c7513d1dd3b2f0c9c493413dc394cf7" } } }, "atlas-fermat-little-theorem": { "source_snapshot": { "id": "atlas-fermat-little-theorem", "title_zh": "Fermat小定理", "summary_zh": "p為質數時a^p同餘a；若p不整除a則可化為a^(p-1)同餘1，逆命題不成立。", "canonical_mko_id": "mko-fermat-little-theorem" }, "fields": { "title": "Fermat's little theorem", "summary": "For a prime p, a^p is congruent to a modulo p; if p does not divide a, a^(p−1) is congruent to1 modulo p. The converse fails." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-fermat-little-theorem", "edition_id": "edition-mko-fermat-little-theorem-en-v1", "source_field": "title", "source_mko_sha256": "8ba0aab812152c67f3df53e149427ab6d311dab16d274314d1ad46b4ee3114d7", "edition_fields_sha256": "820ebb2e9e60fe2e7392c7ef15d78ad2b5aaeea445cca1abaa5f23261aeb9abb" }, "summary": { "kind": "mko_edition", "mko_id": "mko-fermat-little-theorem", "edition_id": "edition-mko-fermat-little-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "8ba0aab812152c67f3df53e149427ab6d311dab16d274314d1ad46b4ee3114d7", "edition_fields_sha256": "820ebb2e9e60fe2e7392c7ef15d78ad2b5aaeea445cca1abaa5f23261aeb9abb" } } }, "atlas-euclidean-algorithm": { "source_snapshot": { "id": "atlas-euclidean-algorithm", "title_zh": "Euclidean演算法", "summary_zh": "Euclidean演算法以精確帶餘除法、gcd invariant與嚴格餘數下降求最大公因數。", "canonical_mko_id": "mko-euclidean-algorithm" }, "fields": { "title": "Euclidean algorithm", "summary": "The Euclidean algorithm computes the gcd using exact quotient-remainder divisions, gcd invariance and strictly decreasing nonnegative remainders." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-euclidean-algorithm", "edition_id": "edition-mko-euclidean-algorithm-en-v1", "source_field": "title", "source_mko_sha256": "eadc65605f926f582e90694e2b3cd2fad1b79c8f39d0f5ecca8962c1ef2e95bf", "edition_fields_sha256": "a835b678c1ecac1f98cdf0a40d603fd38f318478006c1054aa6945b579caa2f3" }, "summary": { "kind": "mko_edition", "mko_id": "mko-euclidean-algorithm", "edition_id": "edition-mko-euclidean-algorithm-en-v1", "source_field": "summary", "source_mko_sha256": "eadc65605f926f582e90694e2b3cd2fad1b79c8f39d0f5ecca8962c1ef2e95bf", "edition_fields_sha256": "a835b678c1ecac1f98cdf0a40d603fd38f318478006c1054aa6945b579caa2f3" } } }, "atlas-finite-automaton": { "source_snapshot": { "id": "atlas-finite-automaton", "title_zh": "確定有限自動機", "summary_zh": "DFA由有限狀態集、有限字母表、總轉移函數、唯一初始狀態與接受狀態子集組成。", "canonical_mko_id": "mko-finite-automaton" }, "fields": { "title": "Deterministic finite automaton", "summary": "A DFA consists of finite states, a finite alphabet, a total transition function, one initial state and a subset of accepting states." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-automaton", "edition_id": "edition-mko-finite-automaton-en-v1", "source_field": "title", "source_mko_sha256": "6bae5f7a3af28d2877efade8771bee1a1adcddd1d88f055671859e37e3d0c0c8", "edition_fields_sha256": "7e1963ca2724bc904e8f29f2dec2a36c98bc8fc5bcb96fda032761f23eac1d21" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-automaton", "edition_id": "edition-mko-finite-automaton-en-v1", "source_field": "summary", "source_mko_sha256": "6bae5f7a3af28d2877efade8771bee1a1adcddd1d88f055671859e37e3d0c0c8", "edition_fields_sha256": "7e1963ca2724bc904e8f29f2dec2a36c98bc8fc5bcb96fda032761f23eac1d21" } } }, "atlas-regular-language": { "source_snapshot": { "id": "atlas-regular-language", "title_zh": "正規語言", "summary_zh": "正規語言是存在一個完整DFA可辨識其全部字串的語言。", "canonical_mko_id": "mko-regular-language" }, "fields": { "title": "Regular language", "summary": "A language is regular when a complete DFA recognizes all and exactly its words." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-regular-language", "edition_id": "edition-mko-regular-language-en-v1", "source_field": "title", "source_mko_sha256": "5b76e4da673cf35d7a7014ce76f103fa504f5f9986498711ed877c15dc35e57a", "edition_fields_sha256": "80a847850be166ef15235fa676301bc1c6effd72578b799240322289fa8d2bd4" }, "summary": { "kind": "mko_edition", "mko_id": "mko-regular-language", "edition_id": "edition-mko-regular-language-en-v1", "source_field": "summary", "source_mko_sha256": "5b76e4da673cf35d7a7014ce76f103fa504f5f9986498711ed877c15dc35e57a", "edition_fields_sha256": "80a847850be166ef15235fa676301bc1c6effd72578b799240322289fa8d2bd4" } } }, "atlas-finite-group-action": { "source_snapshot": { "id": "atlas-finite-group-action", "title_zh": "有限群作用", "summary_zh": "有限群在有限集合上的作用是滿足單位與乘法相容律的total action table。", "canonical_mko_id": "mko-finite-group-action" }, "fields": { "title": "Finite group action", "summary": "An action of a finite group on a finite set is a total action table satisfying the identity and multiplication-compatibility laws." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-group-action", "edition_id": "edition-mko-finite-group-action-en-v1", "source_field": "title", "source_mko_sha256": "2e31b8a871f6fd32317885aa68f63e2380a6c3a7d97c031b152b653c6a879de0", "edition_fields_sha256": "307906e1a6d35cddbfab64c211d37ae10e318f8e416ee89df8ec9744c55f8b88" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-group-action", "edition_id": "edition-mko-finite-group-action-en-v1", "source_field": "summary", "source_mko_sha256": "2e31b8a871f6fd32317885aa68f63e2380a6c3a7d97c031b152b653c6a879de0", "edition_fields_sha256": "307906e1a6d35cddbfab64c211d37ae10e318f8e416ee89df8ec9744c55f8b88" } } }, "atlas-finite-group-orbit": { "source_snapshot": { "id": "atlas-finite-group-orbit", "title_zh": "有限群作用的軌道", "summary_zh": "有限群作用下，一點被全部群元素送到的像集合形成軌道，全部軌道分割carrier。", "canonical_mko_id": "mko-finite-group-orbit" }, "fields": { "title": "Orbit of a finite group action", "summary": "The images of one point under all group elements form its orbit; all orbits partition the point carrier." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-group-orbit", "edition_id": "edition-mko-finite-group-orbit-en-v1", "source_field": "title", "source_mko_sha256": "090d257b39b4d7071dedd320490e40cf67e01ea92431b0b6dc042f6127849022", "edition_fields_sha256": "06da101b3b16f734e72a238c97fb41a0e1610eaae1938594e16b648600823bf0" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-group-orbit", "edition_id": "edition-mko-finite-group-orbit-en-v1", "source_field": "summary", "source_mko_sha256": "090d257b39b4d7071dedd320490e40cf67e01ea92431b0b6dc042f6127849022", "edition_fields_sha256": "06da101b3b16f734e72a238c97fb41a0e1610eaae1938594e16b648600823bf0" } } }, "atlas-finite-group-stabilizer": { "source_snapshot": { "id": "atlas-finite-group-stabilizer", "title_zh": "有限群作用的穩定子群", "summary_zh": "指定點的stabilizer是有限群作用中固定該點的全部群元素，並形成子群。", "canonical_mko_id": "mko-finite-group-stabilizer" }, "fields": { "title": "Stabilizer of a finite group action", "summary": "The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-group-stabilizer", "edition_id": "edition-mko-finite-group-stabilizer-en-v1", "source_field": "title", "source_mko_sha256": "49304c0f16f563ddb9f6ef0acf3ad8e4e8d4361d20320142e367f75e49497c65", "edition_fields_sha256": "d8174c464f118150aaa35546ab24d8ffaa3d33b88c79380e1e2f86a4c0611879" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-group-stabilizer", "edition_id": "edition-mko-finite-group-stabilizer-en-v1", "source_field": "summary", "source_mko_sha256": "49304c0f16f563ddb9f6ef0acf3ad8e4e8d4361d20320142e367f75e49497c65", "edition_fields_sha256": "d8174c464f118150aaa35546ab24d8ffaa3d33b88c79380e1e2f86a4c0611879" } } }, "atlas-orbit-stabilizer-theorem": { "source_snapshot": { "id": "atlas-orbit-stabilizer-theorem", "title_zh": "軌道－穩定子定理", "summary_zh": "完整有限群作用中，同一點的orbit大小乘以stabilizer大小等於群階。", "canonical_mko_id": "mko-orbit-stabilizer-theorem" }, "fields": { "title": "Orbit-stabilizer theorem", "summary": "For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-orbit-stabilizer-theorem", "edition_id": "edition-mko-orbit-stabilizer-theorem-en-v1", "source_field": "title", "source_mko_sha256": "0ae3563c72f6bb6e1702aa5a64f490a1f2b0ca0741f19d17ac3790775e66c38f", "edition_fields_sha256": "dceb87ce25ac99bb16c228eabc5002df4b1bdbb48b68004ae8abaa07a8a9e403" }, "summary": { "kind": "mko_edition", "mko_id": "mko-orbit-stabilizer-theorem", "edition_id": "edition-mko-orbit-stabilizer-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "0ae3563c72f6bb6e1702aa5a64f490a1f2b0ca0741f19d17ac3790775e66c38f", "edition_fields_sha256": "dceb87ce25ac99bb16c228eabc5002df4b1bdbb48b68004ae8abaa07a8a9e403" } } }, "atlas-finite-normal-subgroup": { "source_snapshot": { "id": "atlas-finite-normal-subgroup", "title_zh": "有限正規子群", "summary_zh": "有限群G的子群H若對每個g皆滿足gHg^{-1}=H，則H為正規子群；等價地每個左陪集等於相應右陪集。", "canonical_mko_id": "mko-finite-normal-subgroup" }, "fields": { "title": "Finite normal subgroup", "summary": "A subgroup H of a finite group G is normal when gHg^(-1)=H for every g, equivalently when each left coset equals the corresponding right coset." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-normal-subgroup", "edition_id": "edition-mko-finite-normal-subgroup-en-v1", "source_field": "title", "source_mko_sha256": "65f7894921104c1d8e907fd0f2a4dfca9c6a56325d6cff31cbafd83e26dc5727", "edition_fields_sha256": "5bf6d5a791308a375121818271ef8f760ecb473180e01c77146a2af7d27d3326" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-normal-subgroup", "edition_id": "edition-mko-finite-normal-subgroup-en-v1", "source_field": "summary", "source_mko_sha256": "65f7894921104c1d8e907fd0f2a4dfca9c6a56325d6cff31cbafd83e26dc5727", "edition_fields_sha256": "5bf6d5a791308a375121818271ef8f760ecb473180e01c77146a2af7d27d3326" } } }, "atlas-finite-quotient-group": { "source_snapshot": { "id": "atlas-finite-quotient-group", "title_zh": "有限商群", "summary_zh": "有限群G依正規子群N的陪集形成商群，元素是cosets且乘法(gN)(hN)=(gh)N必須與代表元無關。", "canonical_mko_id": "mko-finite-quotient-group" }, "fields": { "title": "Finite quotient group", "summary": "Cosets of a normal subgroup N form a finite quotient group, with (gN)(hN)=(g*h)N independent of representatives." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-quotient-group", "edition_id": "edition-mko-finite-quotient-group-en-v1", "source_field": "title", "source_mko_sha256": "f3e42a5b51685f805af8727447418b3b158bf4c512da0e78fd2804f7a04bba24", "edition_fields_sha256": "81c0bd5a5c5f14ebb4931851d1b860938af276e291e41486e9cab60c4366cd26" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-quotient-group", "edition_id": "edition-mko-finite-quotient-group-en-v1", "source_field": "summary", "source_mko_sha256": "f3e42a5b51685f805af8727447418b3b158bf4c512da0e78fd2804f7a04bba24", "edition_fields_sha256": "81c0bd5a5c5f14ebb4931851d1b860938af276e291e41486e9cab60c4366cd26" } } }, "atlas-finite-group-conjugacy-class": { "source_snapshot": { "id": "atlas-finite-group-conjugacy-class", "title_zh": "有限群共軛類", "summary_zh": "有限群G以conjugation g·x=gxg^{-1}作用於自身，其orbits形成共軛類並分割G。", "canonical_mko_id": "mko-finite-group-conjugacy-class" }, "fields": { "title": "Conjugacy class in a finite group", "summary": "A finite group acts on itself by g*x=g*x*g^(-1); the orbits are conjugacy classes partitioning the group." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-group-conjugacy-class", "edition_id": "edition-mko-finite-group-conjugacy-class-en-v1", "source_field": "title", "source_mko_sha256": "64bbc8cfc6df1fef690cba391584bc0ae2770f3af76115c24634669363f3bb26", "edition_fields_sha256": "b588ce41f9b63e812c9d2ba8b69b12e92d1ee1c5d6c570db3325322328754993" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-group-conjugacy-class", "edition_id": "edition-mko-finite-group-conjugacy-class-en-v1", "source_field": "summary", "source_mko_sha256": "64bbc8cfc6df1fef690cba391584bc0ae2770f3af76115c24634669363f3bb26", "edition_fields_sha256": "b588ce41f9b63e812c9d2ba8b69b12e92d1ee1c5d6c570db3325322328754993" } } }, "atlas-finite-group-center": { "source_snapshot": { "id": "atlas-finite-group-center", "title_zh": "有限群中心", "summary_zh": "有限群G的中心Z(G)由與每個群元素交換的全部元素構成，且是G的正規子群。", "canonical_mko_id": "mko-finite-group-center" }, "fields": { "title": "Center of a finite group", "summary": "The center consists of all elements commuting with every group element and is a normal subgroup." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-group-center", "edition_id": "edition-mko-finite-group-center-en-v1", "source_field": "title", "source_mko_sha256": "dd633886e48a7fb3d2d5a657d45487b7ff6cbff901ae305a94b8b1f8a966b58a", "edition_fields_sha256": "ffa4dfc02b0a280ee9e902cd84dcb64b89a74d691da3fa107b32f4794e8b87fb" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-group-center", "edition_id": "edition-mko-finite-group-center-en-v1", "source_field": "summary", "source_mko_sha256": "dd633886e48a7fb3d2d5a657d45487b7ff6cbff901ae305a94b8b1f8a966b58a", "edition_fields_sha256": "ffa4dfc02b0a280ee9e902cd84dcb64b89a74d691da3fa107b32f4794e8b87fb" } } }, "atlas-lagrange-theorem-finite-groups": { "source_snapshot": { "id": "atlas-lagrange-theorem-finite-groups", "title_zh": "有限群的Lagrange定理", "summary_zh": "有限群G的子群H以等大左陪集分割G，因此|G|=[G:H]|H|且|H|整除|G|。", "canonical_mko_id": "mko-lagrange-theorem-finite-groups" }, "fields": { "title": "Lagrange theorem for finite groups", "summary": "Equal-size left cosets of a subgroup H partition the finite group G, so |G|=[G:H]*|H| and |H| divides |G|." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-lagrange-theorem-finite-groups", "edition_id": "edition-mko-lagrange-theorem-finite-groups-en-v1", "source_field": "title", "source_mko_sha256": "c283a1941e05986f16993e7482210024c2ac94218c77a43426996a572e70d274", "edition_fields_sha256": "a6e23e27bd2a83195f63b0c50044dc284b06005417aea984d3057111684e2830" }, "summary": { "kind": "mko_edition", "mko_id": "mko-lagrange-theorem-finite-groups", "edition_id": "edition-mko-lagrange-theorem-finite-groups-en-v1", "source_field": "summary", "source_mko_sha256": "c283a1941e05986f16993e7482210024c2ac94218c77a43426996a572e70d274", "edition_fields_sha256": "a6e23e27bd2a83195f63b0c50044dc284b06005417aea984d3057111684e2830" } } }, "atlas-finite-group-homomorphism-kernel": { "source_snapshot": { "id": "atlas-finite-group-homomorphism-kernel", "title_zh": "有限群同態的核", "summary_zh": "有限群同態f:G→Q映到目標identity的完整原像是G的正規子群，且同態fibers恰為kernel的陪集。", "canonical_mko_id": "mko-finite-group-homomorphism-kernel" }, "fields": { "title": "Kernel of a finite group homomorphism", "summary": "For a finite group homomorphism f:G→Q, the full preimage of the target identity is a normal subgroup of G, and nonempty fibers are its cosets." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-group-homomorphism-kernel", "edition_id": "edition-mko-finite-group-homomorphism-kernel-en-v1", "source_field": "title", "source_mko_sha256": "2d8504a53dc49db696cced0fde35020459fee128373b3d2efc52f01168374d11", "edition_fields_sha256": "04e0d7b6e875cdd69b8f5f36dc8b42e684223262dd0265c45982ab2f5b88594e" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-group-homomorphism-kernel", "edition_id": "edition-mko-finite-group-homomorphism-kernel-en-v1", "source_field": "summary", "source_mko_sha256": "2d8504a53dc49db696cced0fde35020459fee128373b3d2efc52f01168374d11", "edition_fields_sha256": "04e0d7b6e875cdd69b8f5f36dc8b42e684223262dd0265c45982ab2f5b88594e" } } }, "atlas-first-isomorphism-theorem-finite-groups": { "source_snapshot": { "id": "atlas-first-isomorphism-theorem-finite-groups", "title_zh": "有限群第一同構定理", "summary_zh": "有限群同態f:G→Q由kernel cosets誘導出G/ker f到im f的群同構，而非自動同構到整個codomain。", "canonical_mko_id": "mko-first-isomorphism-theorem-finite-groups" }, "fields": { "title": "First isomorphism theorem for finite groups", "summary": "A finite group homomorphism induces an isomorphism from its kernel quotient to its image, not automatically to the entire codomain." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-first-isomorphism-theorem-finite-groups", "edition_id": "edition-mko-first-isomorphism-theorem-finite-groups-en-v1", "source_field": "title", "source_mko_sha256": "5586ce5ed3d47467ffd6c485dfcb495844143a17783d770ce2572420bfa28065", "edition_fields_sha256": "bf5cd1449b0dccf3ed4ac0e81e5fb8d80c0042b66e0d4b50219044e31596438e" }, "summary": { "kind": "mko_edition", "mko_id": "mko-first-isomorphism-theorem-finite-groups", "edition_id": "edition-mko-first-isomorphism-theorem-finite-groups-en-v1", "source_field": "summary", "source_mko_sha256": "5586ce5ed3d47467ffd6c485dfcb495844143a17783d770ce2572420bfa28065", "edition_fields_sha256": "bf5cd1449b0dccf3ed4ac0e81e5fb8d80c0042b66e0d4b50219044e31596438e" } } }, "atlas-cauchy-theorem-finite-groups": { "source_snapshot": { "id": "atlas-cauchy-theorem-finite-groups", "title_zh": "有限群的Cauchy定理", "summary_zh": "若質數p整除有限群G的階，則G含有一個exact order為p的元素。", "canonical_mko_id": "mko-cauchy-theorem-finite-groups" }, "fields": { "title": "Cauchy theorem for finite groups", "summary": "If a prime p divides the order of a finite group G, G contains an element of exact order p." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-cauchy-theorem-finite-groups", "edition_id": "edition-mko-cauchy-theorem-finite-groups-en-v1", "source_field": "title", "source_mko_sha256": "779e8ef831907c123fc1e7039941b328b38b54af646d6a171eab7e28b476927d", "edition_fields_sha256": "b979a8d21cd2ef3c53a791a4b7769c94899d36944bbfeaa8fd097c18a3935ed5" }, "summary": { "kind": "mko_edition", "mko_id": "mko-cauchy-theorem-finite-groups", "edition_id": "edition-mko-cauchy-theorem-finite-groups-en-v1", "source_field": "summary", "source_mko_sha256": "779e8ef831907c123fc1e7039941b328b38b54af646d6a171eab7e28b476927d", "edition_fields_sha256": "b979a8d21cd2ef3c53a791a4b7769c94899d36944bbfeaa8fd097c18a3935ed5" } } }, "atlas-supremum-and-infimum": { "source_snapshot": { "id": "atlas-supremum-and-infimum", "title_zh": "上確界與下確界", "summary_zh": "非空且有界實數集合的supremum是最小上界，infimum是最大下界，兩者未必屬於集合。", "canonical_mko_id": "mko-supremum-and-infimum" }, "fields": { "title": "Supremum and infimum", "summary": "For a nonempty appropriately bounded real set, the supremum is its least upper bound and the infimum its greatest lower bound; either need not belong to the set." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-supremum-and-infimum", "edition_id": "edition-mko-supremum-and-infimum-en-v1", "source_field": "title", "source_mko_sha256": "bb254a6cba1cbd62c80ad2e7af827c64072b05dda798122c533bdf47277f2434", "edition_fields_sha256": "bc8b9467da7f5b8cbbc918ea68b0225ea3dcc9bf1cf8d1347697f79f5d4dd897" }, "summary": { "kind": "mko_edition", "mko_id": "mko-supremum-and-infimum", "edition_id": "edition-mko-supremum-and-infimum-en-v1", "source_field": "summary", "source_mko_sha256": "bb254a6cba1cbd62c80ad2e7af827c64072b05dda798122c533bdf47277f2434", "edition_fields_sha256": "bc8b9467da7f5b8cbbc918ea68b0225ea3dcc9bf1cf8d1347697f79f5d4dd897" } } }, "atlas-subsequence": { "source_snapshot": { "id": "atlas-subsequence", "title_zh": "子數列", "summary_zh": "子數列由嚴格遞增的自然數索引映射選取原數列項並保留次序。", "canonical_mko_id": "mko-subsequence" }, "fields": { "title": "Subsequence", "summary": "A subsequence selects terms by a strictly increasing natural-index map and preserves their original order." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-subsequence", "edition_id": "edition-mko-subsequence-en-v1", "source_field": "title", "source_mko_sha256": "f4791c70590c6e099961027b14a591894bd691565f6e1bc8cd191792d0e2a44f", "edition_fields_sha256": "42c148616650877a59436362e02eeb159dbb376d7fa7f3ff5ab30a062713887f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-subsequence", "edition_id": "edition-mko-subsequence-en-v1", "source_field": "summary", "source_mko_sha256": "f4791c70590c6e099961027b14a591894bd691565f6e1bc8cd191792d0e2a44f", "edition_fields_sha256": "42c148616650877a59436362e02eeb159dbb376d7fa7f3ff5ab30a062713887f" } } }, "atlas-monotone-sequence-convergence-theorem": { "source_snapshot": { "id": "atlas-monotone-sequence-convergence-theorem", "title_zh": "單調有界數列收斂定理", "summary_zh": "非遞減且有上界的實數列收斂至其項集合supremum；非遞增版本對偶收斂至infimum。", "canonical_mko_id": "mko-monotone-sequence-convergence-theorem" }, "fields": { "title": "Monotone bounded sequence convergence theorem", "summary": "A nondecreasing real sequence bounded above converges to the supremum of its terms; the nonincreasing bounded-below version converges to the infimum." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-monotone-sequence-convergence-theorem", "edition_id": "edition-mko-monotone-sequence-convergence-theorem-en-v1", "source_field": "title", "source_mko_sha256": "7347d54ee6a5bad0a10d31d2740a6f5af3d2539f9dd91cfc6e85d51ea7c966dc", "edition_fields_sha256": "ce9417b4fa6665954f53d06481494ae0790dc9ef6f2cd4cbf4a3df59417fbc5a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-monotone-sequence-convergence-theorem", "edition_id": "edition-mko-monotone-sequence-convergence-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "7347d54ee6a5bad0a10d31d2740a6f5af3d2539f9dd91cfc6e85d51ea7c966dc", "edition_fields_sha256": "ce9417b4fa6665954f53d06481494ae0790dc9ef6f2cd4cbf4a3df59417fbc5a" } } }, "atlas-bolzano-weierstrass-theorem-real": { "source_snapshot": { "id": "atlas-bolzano-weierstrass-theorem-real", "title_zh": "實數列Bolzano－Weierstrass定理", "summary_zh": "每個整體有界的實數列都有一個以嚴格遞增無限索引取得的收斂子數列。", "canonical_mko_id": "mko-bolzano-weierstrass-theorem-real" }, "fields": { "title": "Bolzano-Weierstrass theorem for real sequences", "summary": "Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-bolzano-weierstrass-theorem-real", "edition_id": "edition-mko-bolzano-weierstrass-theorem-real-en-v1", "source_field": "title", "source_mko_sha256": "68509f22469c3dc3548c8128faec469688cb7a72f7b9fca9c4539568db8e44ea", "edition_fields_sha256": "1552a03d05d06601cfd459cac1cc21d0009073419431ae6691997a733048cb05" }, "summary": { "kind": "mko_edition", "mko_id": "mko-bolzano-weierstrass-theorem-real", "edition_id": "edition-mko-bolzano-weierstrass-theorem-real-en-v1", "source_field": "summary", "source_mko_sha256": "68509f22469c3dc3548c8128faec469688cb7a72f7b9fca9c4539568db8e44ea", "edition_fields_sha256": "1552a03d05d06601cfd459cac1cc21d0009073419431ae6691997a733048cb05" } } }, "atlas-intermediate-value-theorem": { "source_snapshot": { "id": "atlas-intermediate-value-theorem", "title_zh": "介值定理", "summary_zh": "連續實函數在閉區間端點值之間取到每個中間值。", "canonical_mko_id": "mko-intermediate-value-theorem" }, "fields": { "title": "Intermediate value theorem", "summary": "A continuous real function on a closed interval attains each value between its endpoint values." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-intermediate-value-theorem", "edition_id": "edition-mko-intermediate-value-theorem-en-v1", "source_field": "title", "source_mko_sha256": "52c0e9f2fe41f93f083f280990d519515da4d4698c3a483bd5f1a57ff44a71e7", "edition_fields_sha256": "f028a8706ca9e38261053388b3ae14be49b2883f768cfa0d66b868fa9775f5d4" }, "summary": { "kind": "mko_edition", "mko_id": "mko-intermediate-value-theorem", "edition_id": "edition-mko-intermediate-value-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "52c0e9f2fe41f93f083f280990d519515da4d4698c3a483bd5f1a57ff44a71e7", "edition_fields_sha256": "f028a8706ca9e38261053388b3ae14be49b2883f768cfa0d66b868fa9775f5d4" } } }, "atlas-mean-value-theorem": { "source_snapshot": { "id": "atlas-mean-value-theorem", "title_zh": "微分中值定理", "summary_zh": "若a0。", "canonical_mko_id": "mko-real-exponential-function" }, "fields": { "title": "Real exponential function", "summary": "Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-real-exponential-function", "edition_id": "edition-mko-real-exponential-function-en-v1", "source_field": "title", "source_mko_sha256": "7b0736c512a2b1e27562551d8a9de463446e4b54c5998868cfe7573aee0b4f17", "edition_fields_sha256": "74573172e5e603ddb2c55d1534a8ecc43486efac09ddff33de42090af87b5724" }, "summary": { "kind": "mko_edition", "mko_id": "mko-real-exponential-function", "edition_id": "edition-mko-real-exponential-function-en-v1", "source_field": "summary", "source_mko_sha256": "7b0736c512a2b1e27562551d8a9de463446e4b54c5998868cfe7573aee0b4f17", "edition_fields_sha256": "74573172e5e603ddb2c55d1534a8ecc43486efac09ddff33de42090af87b5724" } } }, "atlas-improper-integral": { "source_snapshot": { "id": "atlas-improper-integral", "title_zh": "瑕積分", "summary_zh": "無界區間或含奇點的積分以適當有限區間積分的one-sided limits定義，且每個必要分段須獨立收斂。", "canonical_mko_id": "mko-improper-integral" }, "fields": { "title": "Improper integral", "summary": "Integrals over unbounded intervals or with singularities are defined by appropriate one-sided limits, with every required piece converging independently." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-improper-integral", "edition_id": "edition-mko-improper-integral-en-v1", "source_field": "title", "source_mko_sha256": "db7f78b411add242ee46cc38c5fede57caf33f99eeb955e7bbe6271da2d44dec", "edition_fields_sha256": "8360f9b3463e6ce46a0d40362e8b210e90429279d8a48ffac210402a563b80d2" }, "summary": { "kind": "mko_edition", "mko_id": "mko-improper-integral", "edition_id": "edition-mko-improper-integral-en-v1", "source_field": "summary", "source_mko_sha256": "db7f78b411add242ee46cc38c5fede57caf33f99eeb955e7bbe6271da2d44dec", "edition_fields_sha256": "8360f9b3463e6ce46a0d40362e8b210e90429279d8a48ffac210402a563b80d2" } } }, "atlas-topological-basis": { "source_snapshot": { "id": "atlas-topological-basis", "title_zh": "拓撲基底", "summary_zh": "拓撲基底以覆蓋carrier與逐點局部交細化兩條公理生成拓撲。", "canonical_mko_id": "mko-topological-basis" }, "fields": { "title": "Topological basis", "summary": "A topological basis generates a topology through carrier coverage and pointwise local intersection refinement." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-topological-basis", "edition_id": "edition-mko-topological-basis-en-v1", "source_field": "title", "source_mko_sha256": "3aecebae96f5302a84e437770eaaec38b7992e61082720fa491302177ad43804", "edition_fields_sha256": "38ee346e2024dcf9b9884690336e5eff7074a9e37a697f2764a22ad8fc1b0266" }, "summary": { "kind": "mko_edition", "mko_id": "mko-topological-basis", "edition_id": "edition-mko-topological-basis-en-v1", "source_field": "summary", "source_mko_sha256": "3aecebae96f5302a84e437770eaaec38b7992e61082720fa491302177ad43804", "edition_fields_sha256": "38ee346e2024dcf9b9884690336e5eff7074a9e37a697f2764a22ad8fc1b0266" } } }, "atlas-subspace-topology": { "source_snapshot": { "id": "atlas-subspace-topology", "title_zh": "子空間拓撲", "summary_zh": "Y⊆X的相對拓撲由全部母空間open sets與Y的交集組成。", "canonical_mko_id": "mko-subspace-topology" }, "fields": { "title": "Subspace topology", "summary": "For Y⊆X, the relative topology consists of every intersection of an ambient open set with Y." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-subspace-topology", "edition_id": "edition-mko-subspace-topology-en-v1", "source_field": "title", "source_mko_sha256": "84259c8587e1acd913189fcf6b17200828b88d36622591d8bc5a52b3455544e7", "edition_fields_sha256": "9ebffb5e7d970e514016ef3179d24314fff247d86fd519b3158a3b4820bcb6d1" }, "summary": { "kind": "mko_edition", "mko_id": "mko-subspace-topology", "edition_id": "edition-mko-subspace-topology-en-v1", "source_field": "summary", "source_mko_sha256": "84259c8587e1acd913189fcf6b17200828b88d36622591d8bc5a52b3455544e7", "edition_fields_sha256": "9ebffb5e7d970e514016ef3179d24314fff247d86fd519b3158a3b4820bcb6d1" } } }, "atlas-finite-product-topology": { "source_snapshot": { "id": "atlas-finite-product-topology", "title_zh": "有限乘積拓撲", "summary_zh": "有限多個拓撲空間的Cartesian product由open rectangles生成乘積拓撲。", "canonical_mko_id": "mko-finite-product-topology" }, "fields": { "title": "Finite product topology", "summary": "Open rectangles generate the topology of a finite Cartesian product of topological spaces." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-product-topology", "edition_id": "edition-mko-finite-product-topology-en-v1", "source_field": "title", "source_mko_sha256": "9650dd3cf37e29ce4d8b2c6600ead8cb5c0ac577af1f7b69a0aa987341b02d5a", "edition_fields_sha256": "5469cdf3e7b27f6f40b145c7595fe3f2a44a1cd8d6253605ce03e99aa7ffa006" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-product-topology", "edition_id": "edition-mko-finite-product-topology-en-v1", "source_field": "summary", "source_mko_sha256": "9650dd3cf37e29ce4d8b2c6600ead8cb5c0ac577af1f7b69a0aa987341b02d5a", "edition_fields_sha256": "5469cdf3e7b27f6f40b145c7595fe3f2a44a1cd8d6253605ce03e99aa7ffa006" } } }, "atlas-quotient-topology": { "source_snapshot": { "id": "atlas-quotient-topology", "title_zh": "商拓撲", "summary_zh": "滿射q:X→Y使U⊆Y恰在q逆像開時為開，從而賦予Y商拓撲。", "canonical_mko_id": "mko-quotient-topology" }, "fields": { "title": "Quotient topology", "summary": "For a surjection q:X→Y, the quotient topology declares U open exactly when its full preimage under q is open." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-quotient-topology", "edition_id": "edition-mko-quotient-topology-en-v1", "source_field": "title", "source_mko_sha256": "6d39ac6c71e17262644cf9f224ea59cf647428b87cbb6eaf46dc2b3cdd563097", "edition_fields_sha256": "b7d86e0661f722a8fa1d80cdbc612b4310f1fa36fe8b8858cb35fec4c405d15a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-quotient-topology", "edition_id": "edition-mko-quotient-topology-en-v1", "source_field": "summary", "source_mko_sha256": "6d39ac6c71e17262644cf9f224ea59cf647428b87cbb6eaf46dc2b3cdd563097", "edition_fields_sha256": "b7d86e0661f722a8fa1d80cdbc612b4310f1fa36fe8b8858cb35fec4c405d15a" } } }, "atlas-hausdorff-space": { "source_snapshot": { "id": "atlas-hausdorff-space", "title_zh": "Hausdorff空間", "summary_zh": "拓撲空間X若每對不同點都有不交open neighborhoods即為Hausdorff。", "canonical_mko_id": "mko-hausdorff-space" }, "fields": { "title": "Hausdorff space", "summary": "A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-hausdorff-space", "edition_id": "edition-mko-hausdorff-space-en-v1", "source_field": "title", "source_mko_sha256": "084615a021bcdce41165bfa9dd88865f68a94405aa7108b97c7353e8f40ac5b3", "edition_fields_sha256": "c6c06492737b31ad495ef2842d4f613d9576e97eed7a3d236d5ee1c5367e8a40" }, "summary": { "kind": "mko_edition", "mko_id": "mko-hausdorff-space", "edition_id": "edition-mko-hausdorff-space-en-v1", "source_field": "summary", "source_mko_sha256": "084615a021bcdce41165bfa9dd88865f68a94405aa7108b97c7353e8f40ac5b3", "edition_fields_sha256": "c6c06492737b31ad495ef2842d4f613d9576e97eed7a3d236d5ee1c5367e8a40" } } }, "atlas-compact-subset": { "source_snapshot": { "id": "atlas-compact-subset", "title_zh": "緊緻子集", "summary_zh": "K⊆X在subspace topology下compact，等價地每個relative-open cover有有限subcover。", "canonical_mko_id": "mko-compact-subset" }, "fields": { "title": "Compact subset", "summary": "A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-compact-subset", "edition_id": "edition-mko-compact-subset-en-v1", "source_field": "title", "source_mko_sha256": "b4f23af78bd1197e0c8a2e10786c521b3e087c5ec8033bdec3c84135b9f380fd", "edition_fields_sha256": "dafd888e6af9dc27b8f1703f43b3d556b6a10222466928fdfcbfbeb3c2c7b31b" }, "summary": { "kind": "mko_edition", "mko_id": "mko-compact-subset", "edition_id": "edition-mko-compact-subset-en-v1", "source_field": "summary", "source_mko_sha256": "b4f23af78bd1197e0c8a2e10786c521b3e087c5ec8033bdec3c84135b9f380fd", "edition_fields_sha256": "dafd888e6af9dc27b8f1703f43b3d556b6a10222466928fdfcbfbeb3c2c7b31b" } } }, "atlas-continuous-image-of-compact": { "source_snapshot": { "id": "atlas-continuous-image-of-compact", "title_zh": "緊緻集的連續像", "summary_zh": "若K在X中compact且f:X→Y連續，則image f(K)在Y中compact。", "canonical_mko_id": "mko-continuous-image-of-compact" }, "fields": { "title": "Continuous image of a compact set", "summary": "If K is compact in X and f:X→Y is continuous, then f(K) is compact in Y." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-continuous-image-of-compact", "edition_id": "edition-mko-continuous-image-of-compact-en-v1", "source_field": "title", "source_mko_sha256": "033a72b2e9f25c030d4348c2fe05cda62d3477182efb9d2fb0e32b8c1178138c", "edition_fields_sha256": "67fcb0f45df4a93e5a5436436ca8af5c1692e6a32a28549713bdce9224639f10" }, "summary": { "kind": "mko_edition", "mko_id": "mko-continuous-image-of-compact", "edition_id": "edition-mko-continuous-image-of-compact-en-v1", "source_field": "summary", "source_mko_sha256": "033a72b2e9f25c030d4348c2fe05cda62d3477182efb9d2fb0e32b8c1178138c", "edition_fields_sha256": "67fcb0f45df4a93e5a5436436ca8af5c1692e6a32a28549713bdce9224639f10" } } }, "atlas-path-connected-space": { "source_snapshot": { "id": "atlas-path-connected-space", "title_zh": "道路連通空間", "summary_zh": "空間中任意兩點若可由單位區間的連續映射連接，則空間道路連通。", "canonical_mko_id": "mko-path-connected-space" }, "fields": { "title": "Path-connected space", "summary": "A space is path connected when every pair of its points is joined by a continuous map from the unit interval." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-path-connected-space", "edition_id": "edition-mko-path-connected-space-en-v1", "source_field": "title", "source_mko_sha256": "6df47a43396e20219abd3fe5ab3d088f1aaf4a1022bc9628ef0fe164b6a45e17", "edition_fields_sha256": "d1ff537e86867be886738961b3caa28d4168aaa38866751aee638ca48a60de8f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-path-connected-space", "edition_id": "edition-mko-path-connected-space-en-v1", "source_field": "summary", "source_mko_sha256": "6df47a43396e20219abd3fe5ab3d088f1aaf4a1022bc9628ef0fe164b6a45e17", "edition_fields_sha256": "d1ff537e86867be886738961b3caa28d4168aaa38866751aee638ca48a60de8f" } } }, "atlas-topological-connected-component": { "source_snapshot": { "id": "atlas-topological-connected-component", "title_zh": "拓撲連通分支", "summary_zh": "指定點的連通分支是包含該點的極大連通子集；所有分支分割空間且各自閉。", "canonical_mko_id": "mko-topological-connected-component" }, "fields": { "title": "Connected component of a topological space", "summary": "The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-topological-connected-component", "edition_id": "edition-mko-topological-connected-component-en-v1", "source_field": "title", "source_mko_sha256": "a7a9f0548153129b46182a41ce190097c4d35b3765ef3987fbc9f4789b930b35", "edition_fields_sha256": "26168969ce777257509d27db30643bd67beb794a20d44e32f4e11925900dd323" }, "summary": { "kind": "mko_edition", "mko_id": "mko-topological-connected-component", "edition_id": "edition-mko-topological-connected-component-en-v1", "source_field": "summary", "source_mko_sha256": "a7a9f0548153129b46182a41ce190097c4d35b3765ef3987fbc9f4789b930b35", "edition_fields_sha256": "26168969ce777257509d27db30643bd67beb794a20d44e32f4e11925900dd323" } } }, "atlas-compact-subset-closed-in-hausdorff": { "source_snapshot": { "id": "atlas-compact-subset-closed-in-hausdorff", "title_zh": "Hausdorff空間中的緊緻集為閉集", "summary_zh": "若K是Hausdorff空間X的緊緻子集，則K在X中為閉集。", "canonical_mko_id": "mko-compact-subset-closed-in-hausdorff" }, "fields": { "title": "A compact subset of a Hausdorff space is closed", "summary": "If K is compact in the Hausdorff space X, then K is closed in X." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-compact-subset-closed-in-hausdorff", "edition_id": "edition-mko-compact-subset-closed-in-hausdorff-en-v1", "source_field": "title", "source_mko_sha256": "13e0a8f9917f4a49a9cbe148c344b3f37d319fb16c750de2bb640ec93553d07d", "edition_fields_sha256": "2653d733088f3d93fa020093f67eaa7ae7391fbfe6c50ab0173547066bea0ed2" }, "summary": { "kind": "mko_edition", "mko_id": "mko-compact-subset-closed-in-hausdorff", "edition_id": "edition-mko-compact-subset-closed-in-hausdorff-en-v1", "source_field": "summary", "source_mko_sha256": "13e0a8f9917f4a49a9cbe148c344b3f37d319fb16c750de2bb640ec93553d07d", "edition_fields_sha256": "2653d733088f3d93fa020093f67eaa7ae7391fbfe6c50ab0173547066bea0ed2" } } }, "atlas-markov-inequality": { "source_snapshot": { "id": "atlas-markov-inequality", "title_zh": "Markov不等式", "summary_zh": "非負且有限期望的隨機變數在正閾值上的尾機率不超過E[X]/a。", "canonical_mko_id": "mko-markov-inequality" }, "fields": { "title": "Markov inequality", "summary": "For a nonnegative random variable with finite expectation, its tail probability at a positive threshold is at most E[X] divided by that threshold." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-markov-inequality", "edition_id": "edition-mko-markov-inequality-en-v1", "source_field": "title", "source_mko_sha256": "1363d255fe303067c7fcc800bd82bcecddd8331a0e2262159df06012d4c8bdbd", "edition_fields_sha256": "5aff2b9468c618ba9cdb1bc58706f008cc7b88724c33a10bac23f60eda1d2a06" }, "summary": { "kind": "mko_edition", "mko_id": "mko-markov-inequality", "edition_id": "edition-mko-markov-inequality-en-v1", "source_field": "summary", "source_mko_sha256": "1363d255fe303067c7fcc800bd82bcecddd8331a0e2262159df06012d4c8bdbd", "edition_fields_sha256": "5aff2b9468c618ba9cdb1bc58706f008cc7b88724c33a10bac23f60eda1d2a06" } } }, "atlas-chebyshev-inequality": { "source_snapshot": { "id": "atlas-chebyshev-inequality", "title_zh": "Chebyshev不等式", "summary_zh": "有限均值與變異數的隨機變數偏離均值至少epsilon的機率不超過variance/epsilon²。", "canonical_mko_id": "mko-chebyshev-inequality" }, "fields": { "title": "Chebyshev inequality", "summary": "For a variable with finite mean and variance, the probability of absolute deviation at least epsilon is at most variance/epsilon^2." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-chebyshev-inequality", "edition_id": "edition-mko-chebyshev-inequality-en-v1", "source_field": "title", "source_mko_sha256": "53f883dd0d24e24dd2c178a77c932cdf6caf9b65e2b443bee96a237be01b73e3", "edition_fields_sha256": "06e0f3b1fbe759e4e62c9f87ef587804331beb1dd38e4670204bdd1c0b3082ab" }, "summary": { "kind": "mko_edition", "mko_id": "mko-chebyshev-inequality", "edition_id": "edition-mko-chebyshev-inequality-en-v1", "source_field": "summary", "source_mko_sha256": "53f883dd0d24e24dd2c178a77c932cdf6caf9b65e2b443bee96a237be01b73e3", "edition_fields_sha256": "06e0f3b1fbe759e4e62c9f87ef587804331beb1dd38e4670204bdd1c0b3082ab" } } }, "atlas-law-of-total-expectation": { "source_snapshot": { "id": "atlas-law-of-total-expectation", "title_zh": "全期望公式", "summary_zh": "同一有限機率空間上，完整partition各cell的conditional mean按cell probability加權後等於E[X]。", "canonical_mko_id": "mko-law-of-total-expectation" }, "fields": { "title": "Law of total expectation", "summary": "On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X]." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-law-of-total-expectation", "edition_id": "edition-mko-law-of-total-expectation-en-v1", "source_field": "title", "source_mko_sha256": "135b6dedfa00c0fd52813c51ffe3c6af6b5eca3527d0e536110c62f0e27392ac", "edition_fields_sha256": "8f6c3852747373105a65856a117748c9df4cb21b01875fd182eb81a3fe174cf1" }, "summary": { "kind": "mko_edition", "mko_id": "mko-law-of-total-expectation", "edition_id": "edition-mko-law-of-total-expectation-en-v1", "source_field": "summary", "source_mko_sha256": "135b6dedfa00c0fd52813c51ffe3c6af6b5eca3527d0e536110c62f0e27392ac", "edition_fields_sha256": "8f6c3852747373105a65856a117748c9df4cb21b01875fd182eb81a3fe174cf1" } } }, "atlas-law-of-total-variance": { "source_snapshot": { "id": "atlas-law-of-total-variance", "title_zh": "全變異數公式", "summary_zh": "同一有限law與完整partition生成的G下，總變異數等於cell內變異數期望加cell均值變異數。", "canonical_mko_id": "mko-law-of-total-variance" }, "fields": { "title": "Law of total variance", "summary": "For the same finite law and partition-generated G, total variance is the expected within-cell variance plus the variance of the cell means." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-law-of-total-variance", "edition_id": "edition-mko-law-of-total-variance-en-v1", "source_field": "title", "source_mko_sha256": "603199e10801a37db817e6bb93a0c0e3caac715f1667385f7837bbca5fd374f3", "edition_fields_sha256": "a0388d4e434afbf1393bf3659858bfdfe42307be58993b3e2faef39f5c074d7c" }, "summary": { "kind": "mko_edition", "mko_id": "mko-law-of-total-variance", "edition_id": "edition-mko-law-of-total-variance-en-v1", "source_field": "summary", "source_mko_sha256": "603199e10801a37db817e6bb93a0c0e3caac715f1667385f7837bbca5fd374f3", "edition_fields_sha256": "a0388d4e434afbf1393bf3659858bfdfe42307be58993b3e2faef39f5c074d7c" } } }, "atlas-continuous-real-distribution-function": { "source_snapshot": { "id": "atlas-continuous-real-distribution-function", "title_zh": "連續實值分布函數", "summary_zh": "本節點以CDF本身連續定義continuous distribution；CDF仍須nondecreasing、right-continuous且兩端極限0與1。", "canonical_mko_id": "mko-continuous-real-distribution-function" }, "fields": { "title": "Continuous real distribution function", "summary": "Here a continuous distribution means that its CDF is continuous; the CDF must still be nondecreasing, right-continuous and have endpoint limits0 and1." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-continuous-real-distribution-function", "edition_id": "edition-mko-continuous-real-distribution-function-en-v1", "source_field": "title", "source_mko_sha256": "1dede2c575b28e28d08e5c509fe02847a319df1bd2e2069c0c6163ebfa3ebfac", "edition_fields_sha256": "d16bbe9813307fc1a038a6b33db2c32deb1ed718dd34a1924cbc79580d08f2dc" }, "summary": { "kind": "mko_edition", "mko_id": "mko-continuous-real-distribution-function", "edition_id": "edition-mko-continuous-real-distribution-function-en-v1", "source_field": "summary", "source_mko_sha256": "1dede2c575b28e28d08e5c509fe02847a319df1bd2e2069c0c6163ebfa3ebfac", "edition_fields_sha256": "d16bbe9813307fc1a038a6b33db2c32deb1ed718dd34a1924cbc79580d08f2dc" } } }, "atlas-standard-normal-distribution": { "source_snapshot": { "id": "atlas-standard-normal-distribution", "title_zh": "標準常態分布", "summary_zh": "標準常態Z以Gaussian density phi(x)=(2pi)^(-1/2)exp(-x^2/2)與其連續CDF Phi定義，並具有mean 0、variance 1。", "canonical_mko_id": "mko-standard-normal-distribution" }, "fields": { "title": "Standard normal distribution", "summary": "The standard normal law has density phi(x)=(2*pi)^(-1/2)*exp(−x^2/2), continuous CDF Phi, mean0 and variance1." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-standard-normal-distribution", "edition_id": "edition-mko-standard-normal-distribution-en-v1", "source_field": "title", "source_mko_sha256": "5dac4a8d8eb1b38c2dbe86853a836e63d55563a544cfe67d5e6b4095fdaa9ada", "edition_fields_sha256": "c603c16eff4f40318bde12ae9c149c5099e9d135d643592dac433bf46b005036" }, "summary": { "kind": "mko_edition", "mko_id": "mko-standard-normal-distribution", "edition_id": "edition-mko-standard-normal-distribution-en-v1", "source_field": "summary", "source_mko_sha256": "5dac4a8d8eb1b38c2dbe86853a836e63d55563a544cfe67d5e6b4095fdaa9ada", "edition_fields_sha256": "c603c16eff4f40318bde12ae9c149c5099e9d135d643592dac433bf46b005036" } } }, "atlas-central-limit-theorem-finite-variance-iid": { "source_snapshot": { "id": "atlas-central-limit-theorem-finite-variance-iid", "title_zh": "有限變異數IID中央極限定理", "summary_zh": "共同law且0<σ²<∞的無限IID序列，其正確中心化與sqrt(n)標準化和依分布收斂至標準常態。", "canonical_mko_id": "mko-central-limit-theorem-finite-variance-iid" }, "fields": { "title": "Central limit theorem for finite-variance IID sequences", "summary": "An infinite IID sequence with common finite positive variance has centered sums scaled by sigma*sqrt(n) converging in distribution to the standard normal law." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-central-limit-theorem-finite-variance-iid", "edition_id": "edition-mko-central-limit-theorem-finite-variance-iid-en-v1", "source_field": "title", "source_mko_sha256": "c37700292553db11f9702a86ba165e9287d7830d81ee8351eff28de1c07ea9fe", "edition_fields_sha256": "719cffd858ad1ef068501e8b615bc5d7e5d343f1b1d8121ede697f858885ab2a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-central-limit-theorem-finite-variance-iid", "edition_id": "edition-mko-central-limit-theorem-finite-variance-iid-en-v1", "source_field": "summary", "source_mko_sha256": "c37700292553db11f9702a86ba165e9287d7830d81ee8351eff28de1c07ea9fe", "edition_fields_sha256": "719cffd858ad1ef068501e8b615bc5d7e5d343f1b1d8121ede697f858885ab2a" } } }, "atlas-strong-law-large-numbers-finite-valued-iid": { "source_snapshot": { "id": "atlas-strong-law-large-numbers-finite-valued-iid", "title_zh": "有限值IID強大數律", "summary_zh": "有限值無限IID序列的樣本平均幾乎必然收斂至共同期望。", "canonical_mko_id": "mko-strong-law-large-numbers-finite-valued-iid" }, "fields": { "title": "Strong law of large numbers for finite-valued IID sequences", "summary": "The sample means of a finite-valued infinite IID sequence converge almost surely to the common expectation." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-strong-law-large-numbers-finite-valued-iid", "edition_id": "edition-mko-strong-law-large-numbers-finite-valued-iid-en-v1", "source_field": "title", "source_mko_sha256": "14ad2e3637b7d7efdbf46bd6d640bf2637a5ab65b1f9ceafa6ff1a08565e9037", "edition_fields_sha256": "a9631056d0f72999a17b7759ac6d651a20e9ca94215f634caba804e947f0e0a5" }, "summary": { "kind": "mko_edition", "mko_id": "mko-strong-law-large-numbers-finite-valued-iid", "edition_id": "edition-mko-strong-law-large-numbers-finite-valued-iid-en-v1", "source_field": "summary", "source_mko_sha256": "14ad2e3637b7d7efdbf46bd6d640bf2637a5ab65b1f9ceafa6ff1a08565e9037", "edition_fields_sha256": "a9631056d0f72999a17b7759ac6d651a20e9ca94215f634caba804e947f0e0a5" } } }, "atlas-finite-maximum-likelihood-estimator": { "source_snapshot": { "id": "atlas-finite-maximum-likelihood-estimator", "title_zh": "有限模型最大似然估計量", "summary_zh": "有限參數模型中，MLE選取使觀察likelihood最大的參數，tie需固定且與theta無關。", "canonical_mko_id": "mko-finite-maximum-likelihood-estimator" }, "fields": { "title": "Maximum-likelihood estimator in a finite model", "summary": "A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-maximum-likelihood-estimator", "edition_id": "edition-mko-finite-maximum-likelihood-estimator-en-v1", "source_field": "title", "source_mko_sha256": "44f2514c3f39a056585e86bd48bcfd2e135dc3ee54fdd4074dcdae1dc0a763f3", "edition_fields_sha256": "82b600cd12b32422f4964d9bfb63c5ffd835ef2a2e7c259bb5179daacc814b8a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-maximum-likelihood-estimator", "edition_id": "edition-mko-finite-maximum-likelihood-estimator-en-v1", "source_field": "summary", "source_mko_sha256": "44f2514c3f39a056585e86bd48bcfd2e135dc3ee54fdd4074dcdae1dc0a763f3", "edition_fields_sha256": "82b600cd12b32422f4964d9bfb63c5ffd835ef2a2e7c259bb5179daacc814b8a" } } }, "atlas-estimator-mean-squared-error": { "source_snapshot": { "id": "atlas-estimator-mean-squared-error", "title_zh": "估計量均方誤差", "summary_zh": "scalar parameter theta下，估計量T的MSE_theta(T)=E_theta[(T-theta)^2]由完整sampling distribution逐theta定義。", "canonical_mko_id": "mko-estimator-mean-squared-error" }, "fields": { "title": "Mean squared error of an estimator", "summary": "For scalar parameter theta, MSE_theta(T)=E_theta[(T−theta)^2] is defined under each complete sampling distribution separately." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-estimator-mean-squared-error", "edition_id": "edition-mko-estimator-mean-squared-error-en-v1", "source_field": "title", "source_mko_sha256": "b86734a34e0811ecbcceb6973118f1feff5e437465869c8706bc6a00c4cce3cb", "edition_fields_sha256": "b2a22ac141d347df8ab6742dd45eeaa152fdf35f4b9d8634cd727cd4ea933468" }, "summary": { "kind": "mko_edition", "mko_id": "mko-estimator-mean-squared-error", "edition_id": "edition-mko-estimator-mean-squared-error-en-v1", "source_field": "summary", "source_mko_sha256": "b86734a34e0811ecbcceb6973118f1feff5e437465869c8706bc6a00c4cce3cb", "edition_fields_sha256": "b2a22ac141d347df8ab6742dd45eeaa152fdf35f4b9d8634cd727cd4ea933468" } } }, "atlas-bias-variance-decomposition": { "source_snapshot": { "id": "atlas-bias-variance-decomposition", "title_zh": "偏誤－變異數分解", "summary_zh": "逐參數theta，scalar estimator的MSE等於其variance加上bias平方。", "canonical_mko_id": "mko-bias-variance-decomposition" }, "fields": { "title": "Bias-variance decomposition", "summary": "For each parameter value and its sampling law, scalar-estimator MSE equals variance plus squared bias." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-bias-variance-decomposition", "edition_id": "edition-mko-bias-variance-decomposition-en-v1", "source_field": "title", "source_mko_sha256": "03c45bb05c65b40d9f88d66e9009cd0182117446dd67211eee759fa9e0b2b377", "edition_fields_sha256": "34cb0fd69c3bad4e0043ecff580ebcb9924523879d202aa840e4f513c1d0e03d" }, "summary": { "kind": "mko_edition", "mko_id": "mko-bias-variance-decomposition", "edition_id": "edition-mko-bias-variance-decomposition-en-v1", "source_field": "summary", "source_mko_sha256": "03c45bb05c65b40d9f88d66e9009cd0182117446dd67211eee759fa9e0b2b377", "edition_fields_sha256": "34cb0fd69c3bad4e0043ecff580ebcb9924523879d202aa840e4f513c1d0e03d" } } }, "atlas-finite-sufficient-statistic": { "source_snapshot": { "id": "atlas-finite-sufficient-statistic", "title_zh": "有限模型充分統計量", "summary_zh": "finite model中，給定statistic value後sample conditional law不依theta時，T對theta充分。", "canonical_mko_id": "mko-finite-sufficient-statistic" }, "fields": { "title": "Sufficient statistic in a finite model", "summary": "A statistic is sufficient when the conditional sample law given its value does not depend on theta wherever that condition has positive mass." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-sufficient-statistic", "edition_id": "edition-mko-finite-sufficient-statistic-en-v1", "source_field": "title", "source_mko_sha256": "5b80cc27ea20e17864cb5068dedde0919001d6163cabe652473150401c131231", "edition_fields_sha256": "2f856db23e8ec4df60a83605d0026839f1e664812bacd5ed2075202745949722" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-sufficient-statistic", "edition_id": "edition-mko-finite-sufficient-statistic-en-v1", "source_field": "summary", "source_mko_sha256": "5b80cc27ea20e17864cb5068dedde0919001d6163cabe652473150401c131231", "edition_fields_sha256": "2f856db23e8ec4df60a83605d0026839f1e664812bacd5ed2075202745949722" } } }, "atlas-finite-confidence-interval-coverage": { "source_snapshot": { "id": "atlas-finite-confidence-interval-coverage", "title_zh": "有限模型信賴區間覆蓋率", "summary_zh": "有限模型中預先固定的interval-valued procedure逐參數覆蓋真值的機率，定義其frequentist coverage。", "canonical_mko_id": "mko-finite-confidence-interval-coverage" }, "fields": { "title": "Confidence-interval coverage in a finite model", "summary": "Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-confidence-interval-coverage", "edition_id": "edition-mko-finite-confidence-interval-coverage-en-v1", "source_field": "title", "source_mko_sha256": "5ac0ccb03c88f011ec1edd7c258fe04dc72e982b042a8dcea53ac59deb386363", "edition_fields_sha256": "c6e95f95480d7a4241d2ca4fed89888dcc7e4267b792a03a714ee4188d92a15f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-confidence-interval-coverage", "edition_id": "edition-mko-finite-confidence-interval-coverage-en-v1", "source_field": "summary", "source_mko_sha256": "5ac0ccb03c88f011ec1edd7c258fe04dc72e982b042a8dcea53ac59deb386363", "edition_fields_sha256": "c6e95f95480d7a4241d2ca4fed89888dcc7e4267b792a03a714ee4188d92a15f" } } }, "atlas-pigeonhole-principle": { "source_snapshot": { "id": "atlas-pigeonhole-principle", "title_zh": "鴿籠原理", "summary_zh": "有限total map的domain比codomain大時必有兩個不同元素共享像。", "canonical_mko_id": "mko-pigeonhole-principle" }, "fields": { "title": "Pigeonhole principle", "summary": "A total map from a larger finite domain into a smaller finite codomain sends two distinct source elements to the same value." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-pigeonhole-principle", "edition_id": "edition-mko-pigeonhole-principle-en-v1", "source_field": "title", "source_mko_sha256": "3a161ec4318e1b1c3993a14395e4e5710b5529f42a9311aa9b041380c9b9eaba", "edition_fields_sha256": "91ea00a5762fb5414948b72a883eb467ee4829a4ce16adf8a95c48449504ab1a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-pigeonhole-principle", "edition_id": "edition-mko-pigeonhole-principle-en-v1", "source_field": "summary", "source_mko_sha256": "3a161ec4318e1b1c3993a14395e4e5710b5529f42a9311aa9b041380c9b9eaba", "edition_fields_sha256": "91ea00a5762fb5414948b72a883eb467ee4829a4ce16adf8a95c48449504ab1a" } } }, "atlas-inclusion-exclusion-principle": { "source_snapshot": { "id": "atlas-inclusion-exclusion-principle", "title_zh": "容斥原理", "summary_zh": "有限indexed set family的union cardinality是全部非空index subsets交集大小的交替和。", "canonical_mko_id": "mko-inclusion-exclusion-principle" }, "fields": { "title": "Inclusion-exclusion principle", "summary": "The union cardinality of a finite indexed set family is the alternating sum of intersection sizes over every nonempty index subset." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-inclusion-exclusion-principle", "edition_id": "edition-mko-inclusion-exclusion-principle-en-v1", "source_field": "title", "source_mko_sha256": "2eb0f4de0d9b0d93d979ced97fcaadb56ea9045d427c0918e35505bde5f2a649", "edition_fields_sha256": "bbeb419390099ec81798bbc54d0832480bd1f777360034f52080c91e33d4acda" }, "summary": { "kind": "mko_edition", "mko_id": "mko-inclusion-exclusion-principle", "edition_id": "edition-mko-inclusion-exclusion-principle-en-v1", "source_field": "summary", "source_mko_sha256": "2eb0f4de0d9b0d93d979ced97fcaadb56ea9045d427c0918e35505bde5f2a649", "edition_fields_sha256": "bbeb419390099ec81798bbc54d0832480bd1f777360034f52080c91e33d4acda" } } }, "atlas-binomial-theorem": { "source_snapshot": { "id": "atlas-binomial-theorem", "title_zh": "二項式定理", "summary_zh": "交換乘法環境中，natural exponent n的二項式冪以binomial coefficients完整展開。", "canonical_mko_id": "mko-binomial-theorem" }, "fields": { "title": "Binomial theorem", "summary": "With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-binomial-theorem", "edition_id": "edition-mko-binomial-theorem-en-v1", "source_field": "title", "source_mko_sha256": "c8fefe3783fc4f7633b4b22b8cf149d96a683900f181e3b42c690409ce90e5b7", "edition_fields_sha256": "3207de2dac51c6b332ba8b8d74e2ff90d6e7945a60c3f0b205d9a729a1c5bbcb" }, "summary": { "kind": "mko_edition", "mko_id": "mko-binomial-theorem", "edition_id": "edition-mko-binomial-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "c8fefe3783fc4f7633b4b22b8cf149d96a683900f181e3b42c690409ce90e5b7", "edition_fields_sha256": "3207de2dac51c6b332ba8b8d74e2ff90d6e7945a60c3f0b205d9a729a1c5bbcb" } } }, "atlas-recurrence-relation": { "source_snapshot": { "id": "atlas-recurrence-relation", "title_zh": "遞迴關係", "summary_zh": "order-r recurrence以明示initial values與total single-valued state update逐項決定sequence。", "canonical_mko_id": "mko-recurrence-relation" }, "fields": { "title": "Recurrence relation", "summary": "An order-r recurrence uses explicit initial values and a total single-valued state update to determine successive sequence terms." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-recurrence-relation", "edition_id": "edition-mko-recurrence-relation-en-v1", "source_field": "title", "source_mko_sha256": "972d0f75cb7afb4eadac89d2faf5c5f9f37f9f1c30ba6672dcf403faf3747056", "edition_fields_sha256": "1adcbd36b6009171614b2665799f4859008379c36c7166babd1745e8c1723557" }, "summary": { "kind": "mko_edition", "mko_id": "mko-recurrence-relation", "edition_id": "edition-mko-recurrence-relation-en-v1", "source_field": "summary", "source_mko_sha256": "972d0f75cb7afb4eadac89d2faf5c5f9f37f9f1c30ba6672dcf403faf3747056", "edition_fields_sha256": "1adcbd36b6009171614b2665799f4859008379c36c7166babd1745e8c1723557" } } }, "atlas-analytic-ordinary-generating-function": { "source_snapshot": { "id": "atlas-analytic-ordinary-generating-function", "title_zh": "解析普通生成函數", "summary_zh": "數列(a_n)在明示收斂域內以ordinary power series A(x)=sum_{n≥0}a_nx^n解析編碼。", "canonical_mko_id": "mko-analytic-ordinary-generating-function" }, "fields": { "title": "Analytic ordinary generating function", "summary": "An ordinary power series A(x)=sum over n≥0 of a_n*x^n analytically encodes a sequence on an explicitly declared convergence domain." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-analytic-ordinary-generating-function", "edition_id": "edition-mko-analytic-ordinary-generating-function-en-v1", "source_field": "title", "source_mko_sha256": "bb78af651663a5b612aa987ab6c1ce997dd023e539de44fa7304c0b7363cb94c", "edition_fields_sha256": "9be94a3efc508ca9fa1507ae89de81e4829957446ad88f0e51339129e26f4169" }, "summary": { "kind": "mko_edition", "mko_id": "mko-analytic-ordinary-generating-function", "edition_id": "edition-mko-analytic-ordinary-generating-function-en-v1", "source_field": "summary", "source_mko_sha256": "bb78af651663a5b612aa987ab6c1ce997dd023e539de44fa7304c0b7363cb94c", "edition_fields_sha256": "9be94a3efc508ca9fa1507ae89de81e4829957446ad88f0e51339129e26f4169" } } }, "atlas-catalan-numbers": { "source_snapshot": { "id": "atlas-catalan-numbers", "title_zh": "Catalan數", "summary_zh": "Catalan數採C_0=1，並以convolution recurrence或兩個exact closed forms定義同一zero-based sequence。", "canonical_mko_id": "mko-catalan-numbers" }, "fields": { "title": "Catalan numbers", "summary": "With C_0=1, the convolution recurrence and two exact closed forms define the same zero-based Catalan sequence." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-catalan-numbers", "edition_id": "edition-mko-catalan-numbers-en-v1", "source_field": "title", "source_mko_sha256": "2d847df82bf2c534d59aedb78304a0f59d94a87bf4454e0eee1886df489e21b5", "edition_fields_sha256": "b243fd2fe596ae9bf23b552202357e1c33e851c12b42e9f3550a86ea609ab9f6" }, "summary": { "kind": "mko_edition", "mko_id": "mko-catalan-numbers", "edition_id": "edition-mko-catalan-numbers-en-v1", "source_field": "summary", "source_mko_sha256": "2d847df82bf2c534d59aedb78304a0f59d94a87bf4454e0eee1886df489e21b5", "edition_fields_sha256": "b243fd2fe596ae9bf23b552202357e1c33e851c12b42e9f3550a86ea609ab9f6" } } }, "atlas-vertex-cover": { "source_snapshot": { "id": "atlas-vertex-cover", "title_zh": "頂點覆蓋", "summary_zh": "有限simple undirected graph的頂點子集C若與每條edge至少一端相交，則C為vertex cover。", "canonical_mko_id": "mko-vertex-cover" }, "fields": { "title": "Vertex cover", "summary": "A vertex subset C covers a finite simple undirected graph if every edge has at least one endpoint in C." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-vertex-cover", "edition_id": "edition-mko-vertex-cover-en-v1", "source_field": "title", "source_mko_sha256": "4e6ca5a6921189d4aac44783f8135dd0f613679ec348ac53ddf8b3fd648ff22b", "edition_fields_sha256": "c94b90bed0210ef33471402cb108a865855f9458fa36b87ac8b7ffd26a5087d5" }, "summary": { "kind": "mko_edition", "mko_id": "mko-vertex-cover", "edition_id": "edition-mko-vertex-cover-en-v1", "source_field": "summary", "source_mko_sha256": "4e6ca5a6921189d4aac44783f8135dd0f613679ec348ac53ddf8b3fd648ff22b", "edition_fields_sha256": "c94b90bed0210ef33471402cb108a865855f9458fa36b87ac8b7ffd26a5087d5" } } }, "atlas-minimum-vertex-cover": { "source_snapshot": { "id": "atlas-minimum-vertex-cover", "title_zh": "最小頂點覆蓋", "summary_zh": "有限simple graph中，cardinality在全部vertex covers裡最小的候選是minimum vertex cover。", "canonical_mko_id": "mko-minimum-vertex-cover" }, "fields": { "title": "Minimum vertex cover", "summary": "A minimum vertex cover has least cardinality among all vertex covers of the finite simple graph." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-minimum-vertex-cover", "edition_id": "edition-mko-minimum-vertex-cover-en-v1", "source_field": "title", "source_mko_sha256": "4f8d510264312335a0f27c32a69f66f31350aede08398538e94a015583d8250b", "edition_fields_sha256": "0c63b790f10218991e65aa0e4c37baef33c541c4ce33c6182e4b0f008d416f97" }, "summary": { "kind": "mko_edition", "mko_id": "mko-minimum-vertex-cover", "edition_id": "edition-mko-minimum-vertex-cover-en-v1", "source_field": "summary", "source_mko_sha256": "4f8d510264312335a0f27c32a69f66f31350aede08398538e94a015583d8250b", "edition_fields_sha256": "0c63b790f10218991e65aa0e4c37baef33c541c4ce33c6182e4b0f008d416f97" } } }, "atlas-finite-directed-graph": { "source_snapshot": { "id": "atlas-finite-directed-graph", "title_zh": "有限有向圖", "summary_zh": "有限loop-free simple digraph由有限vertices與不含loop、無平行同向重邊的ordered-pair edge set組成。", "canonical_mko_id": "mko-finite-directed-graph" }, "fields": { "title": "Finite directed graph", "summary": "Under the loop-free simple convention, a finite digraph has a finite vertex carrier and an ordered-pair edge set without loops or duplicate same-direction edges." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-directed-graph", "edition_id": "edition-mko-finite-directed-graph-en-v1", "source_field": "title", "source_mko_sha256": "09311b0ec711e3f02a16b1c332e69c9f718fe2a288f2005683d393695a41161b", "edition_fields_sha256": "75a3f0380c9cca329f8332b4b12c1754478f204331f768ee90c1abebb8946a47" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-directed-graph", "edition_id": "edition-mko-finite-directed-graph-en-v1", "source_field": "summary", "source_mko_sha256": "09311b0ec711e3f02a16b1c332e69c9f718fe2a288f2005683d393695a41161b", "edition_fields_sha256": "75a3f0380c9cca329f8332b4b12c1754478f204331f768ee90c1abebb8946a47" } } }, "atlas-topological-ordering-dag": { "source_snapshot": { "id": "atlas-topological-ordering-dag", "title_zh": "有向無環圖的拓撲排序", "summary_zh": "有限有向圖存在topological ordering恰當其無directed cycle。", "canonical_mko_id": "mko-topological-ordering-dag" }, "fields": { "title": "Topological ordering of a directed acyclic graph", "summary": "A finite directed graph has a topological ordering exactly when it has no directed cycle." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-topological-ordering-dag", "edition_id": "edition-mko-topological-ordering-dag-en-v1", "source_field": "title", "source_mko_sha256": "826499909e9a38a23996ab91062cdf86e73a350078b0b11c2a5fb8828d12e5f9", "edition_fields_sha256": "4ef6f65bc89ff141457e6b811a07912ac523af355cdcfcd0fd215bc5a7235111" }, "summary": { "kind": "mko_edition", "mko_id": "mko-topological-ordering-dag", "edition_id": "edition-mko-topological-ordering-dag-en-v1", "source_field": "summary", "source_mko_sha256": "826499909e9a38a23996ab91062cdf86e73a350078b0b11c2a5fb8828d12e5f9", "edition_fields_sha256": "4ef6f65bc89ff141457e6b811a07912ac523af355cdcfcd0fd215bc5a7235111" } } }, "atlas-finite-flow-network": { "source_snapshot": { "id": "atlas-finite-flow-network", "title_zh": "有限流網路", "summary_zh": "有限有向網路中的非負流須逐邊不超過capacity，並在source與sink以外每個頂點滿足流量守恆。", "canonical_mko_id": "mko-finite-flow-network" }, "fields": { "title": "Finite flow network", "summary": "A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-finite-flow-network", "edition_id": "edition-mko-finite-flow-network-en-v1", "source_field": "title", "source_mko_sha256": "c6cc774f53f5114e2b238ff77e24ed61a2adc5f2d6d7ffa5a969b25f287ad5d0", "edition_fields_sha256": "f722ae01b9f5274379403cbef4ec8fa692e7bd9768f200ed26650ce8bd5fd945" }, "summary": { "kind": "mko_edition", "mko_id": "mko-finite-flow-network", "edition_id": "edition-mko-finite-flow-network-en-v1", "source_field": "summary", "source_mko_sha256": "c6cc774f53f5114e2b238ff77e24ed61a2adc5f2d6d7ffa5a969b25f287ad5d0", "edition_fields_sha256": "f722ae01b9f5274379403cbef4ec8fa692e7bd9768f200ed26650ce8bd5fd945" } } }, "atlas-max-flow-min-cut-theorem": { "source_snapshot": { "id": "atlas-max-flow-min-cut-theorem", "title_zh": "最大流最小割定理", "summary_zh": "有限容量網路的最大feasible flow value等於全部s–t cuts中的最小capacity。", "canonical_mko_id": "mko-max-flow-min-cut-theorem" }, "fields": { "title": "Max-flow min-cut theorem", "summary": "In a finite capacitated network the largest feasible flow value equals the least capacity among all source–sink cuts." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-max-flow-min-cut-theorem", "edition_id": "edition-mko-max-flow-min-cut-theorem-en-v1", "source_field": "title", "source_mko_sha256": "3757bdd0ce86cc1644a36f998a60ce8dbf6971de426f7cf31249a035f64db37c", "edition_fields_sha256": "398768e848779a6cfea53b2991f49da120b01b07dbf2537b848ea008305a793a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-max-flow-min-cut-theorem", "edition_id": "edition-mko-max-flow-min-cut-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "3757bdd0ce86cc1644a36f998a60ce8dbf6971de426f7cf31249a035f64db37c", "edition_fields_sha256": "398768e848779a6cfea53b2991f49da120b01b07dbf2537b848ea008305a793a" } } }, "atlas-euler-theorem-number-theory": { "source_snapshot": { "id": "atlas-euler-theorem-number-theory", "title_zh": "Euler定理", "summary_zh": "n>=2且gcd(a,n)=1時，a的phi(n)次方模n同餘於1。", "canonical_mko_id": "mko-euler-theorem-number-theory" }, "fields": { "title": "Euler's theorem in number theory", "summary": "For n≥2 and gcd(a,n)=1, a^phi(n) is congruent to1 modulo n." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-euler-theorem-number-theory", "edition_id": "edition-mko-euler-theorem-number-theory-en-v1", "source_field": "title", "source_mko_sha256": "d14018a9950716ffce9bc6bf1d38d0e4ef8a78193dee493ebaa0d0dda1a573b6", "edition_fields_sha256": "064c53ef05597389df8050c6859ea9a620d38782c84529f6cda0ffdc24cb6337" }, "summary": { "kind": "mko_edition", "mko_id": "mko-euler-theorem-number-theory", "edition_id": "edition-mko-euler-theorem-number-theory-en-v1", "source_field": "summary", "source_mko_sha256": "d14018a9950716ffce9bc6bf1d38d0e4ef8a78193dee493ebaa0d0dda1a573b6", "edition_fields_sha256": "064c53ef05597389df8050c6859ea9a620d38782c84529f6cda0ffdc24cb6337" } } }, "atlas-multiplicative-order-mod-n": { "source_snapshot": { "id": "atlas-multiplicative-order-mod-n", "title_zh": "模n乘法階", "summary_zh": "gcd(a,n)=1時，ord_n(a)是a^k≡1 mod n的最小正整數k。", "canonical_mko_id": "mko-multiplicative-order-mod-n" }, "fields": { "title": "Multiplicative order modulo n", "summary": "For gcd(a,n)=1, the multiplicative order is the smallest positive k with a^k congruent to1 modulo n." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-multiplicative-order-mod-n", "edition_id": "edition-mko-multiplicative-order-mod-n-en-v1", "source_field": "title", "source_mko_sha256": "70fb02dd0cfb34ce7cc4a848496bc4599fe6b0d22028bb31d567c1c29e501ec3", "edition_fields_sha256": "aeaffbefd29fccdf97947dd7a36ac280825975fc7b63b8c57911070736f73841" }, "summary": { "kind": "mko_edition", "mko_id": "mko-multiplicative-order-mod-n", "edition_id": "edition-mko-multiplicative-order-mod-n-en-v1", "source_field": "summary", "source_mko_sha256": "70fb02dd0cfb34ce7cc4a848496bc4599fe6b0d22028bb31d567c1c29e501ec3", "edition_fields_sha256": "aeaffbefd29fccdf97947dd7a36ac280825975fc7b63b8c57911070736f73841" } } }, "atlas-unique-prime-factorization": { "source_snapshot": { "id": "atlas-unique-prime-factorization", "title_zh": "整數唯一質因數分解", "summary_zh": "每個n>1有prime factorization且prime multiset唯一，只允許因數次序差異。", "canonical_mko_id": "mko-unique-prime-factorization" }, "fields": { "title": "Unique prime factorization of integers", "summary": "Every positive integer n>1 has a prime factorization whose prime multiset is unique up to factor order." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-unique-prime-factorization", "edition_id": "edition-mko-unique-prime-factorization-en-v1", "source_field": "title", "source_mko_sha256": "0f01853edccbfb8cdf298587be277da27032f1f31ab87712ee5745644aa66cff", "edition_fields_sha256": "86a55f19686809e98c0d1c64345c23b277945b411389c9e4287b3c9a62ecf6a8" }, "summary": { "kind": "mko_edition", "mko_id": "mko-unique-prime-factorization", "edition_id": "edition-mko-unique-prime-factorization-en-v1", "source_field": "summary", "source_mko_sha256": "0f01853edccbfb8cdf298587be277da27032f1f31ab87712ee5745644aa66cff", "edition_fields_sha256": "86a55f19686809e98c0d1c64345c23b277945b411389c9e4287b3c9a62ecf6a8" } } }, "atlas-wilson-theorem": { "source_snapshot": { "id": "atlas-wilson-theorem", "title_zh": "Wilson定理", "summary_zh": "整數n>1為質數，恰當(n-1)!模n同餘於-1。", "canonical_mko_id": "mko-wilson-theorem" }, "fields": { "title": "Wilson theorem", "summary": "An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-wilson-theorem", "edition_id": "edition-mko-wilson-theorem-en-v1", "source_field": "title", "source_mko_sha256": "32b4a0da9e832eb6c7c0c33d723d8f66608b7013dcd990ad19fa2596126d9eb8", "edition_fields_sha256": "bf7c94a67763000ff1ba48d0b6b3c78987bfe99ada20f99c6d72751d9748f527" }, "summary": { "kind": "mko_edition", "mko_id": "mko-wilson-theorem", "edition_id": "edition-mko-wilson-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "32b4a0da9e832eb6c7c0c33d723d8f66608b7013dcd990ad19fa2596126d9eb8", "edition_fields_sha256": "bf7c94a67763000ff1ba48d0b6b3c78987bfe99ada20f99c6d72751d9748f527" } } }, "atlas-quadratic-residue": { "source_snapshot": { "id": "atlas-quadratic-residue", "title_zh": "二次剩餘", "summary_zh": "本節點採odd prime p上的nonzero convention：a mod p非零且等於某個nonzero square時為quadratic residue。", "canonical_mko_id": "mko-quadratic-residue" }, "fields": { "title": "Quadratic residue", "summary": "Under this odd-prime nonzero convention, a class is a quadratic residue when its normalized nonzero value is a square of a nonzero class modulo p." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-quadratic-residue", "edition_id": "edition-mko-quadratic-residue-en-v1", "source_field": "title", "source_mko_sha256": "c93521b58b1876da85beb3308b403a1cd27209001660dfc6c04815276f035e21", "edition_fields_sha256": "50ce630833ed7e2625b7f64e0eade7bdddb0b0036d27ae876f5076c8a14d044d" }, "summary": { "kind": "mko_edition", "mko_id": "mko-quadratic-residue", "edition_id": "edition-mko-quadratic-residue-en-v1", "source_field": "summary", "source_mko_sha256": "c93521b58b1876da85beb3308b403a1cd27209001660dfc6c04815276f035e21", "edition_fields_sha256": "50ce630833ed7e2625b7f64e0eade7bdddb0b0036d27ae876f5076c8a14d044d" } } }, "atlas-legendre-symbol": { "source_snapshot": { "id": "atlas-legendre-symbol", "title_zh": "Legendre符號", "summary_zh": "對奇質數p，Legendre符號以0、1、-1分別編碼zero class、nonzero quadratic residue與nonresidue。", "canonical_mko_id": "mko-legendre-symbol" }, "fields": { "title": "Legendre symbol", "summary": "For odd prime p, the Legendre symbol uses0,1,−1 for the zero class, nonzero quadratic residues and nonresidues respectively." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-legendre-symbol", "edition_id": "edition-mko-legendre-symbol-en-v1", "source_field": "title", "source_mko_sha256": "56ce955775c56864a29a398a2adae655ad1f8d53a82d5d9970bc7d0e1a4cebde", "edition_fields_sha256": "3582408e711c0f584e471cf00b5ba322112db791f62bb6d68288e3529fec9200" }, "summary": { "kind": "mko_edition", "mko_id": "mko-legendre-symbol", "edition_id": "edition-mko-legendre-symbol-en-v1", "source_field": "summary", "source_mko_sha256": "56ce955775c56864a29a398a2adae655ad1f8d53a82d5d9970bc7d0e1a4cebde", "edition_fields_sha256": "3582408e711c0f584e471cf00b5ba322112db791f62bb6d68288e3529fec9200" } } }, "atlas-euler-criterion-quadratic-residue": { "source_snapshot": { "id": "atlas-euler-criterion-quadratic-residue", "title_zh": "Euler二次剩餘判準", "summary_zh": "若p為奇質數且p不整除a，則a^((p-1)/2)模p等於Legendre符號(a/p)的模p代表。", "canonical_mko_id": "mko-euler-criterion-quadratic-residue" }, "fields": { "title": "Euler criterion for quadratic residues", "summary": "For odd prime p and a not divisible by p, a^((p−1)/2) modulo p is the modulo-p representative of the Legendre symbol (a/p)." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-euler-criterion-quadratic-residue", "edition_id": "edition-mko-euler-criterion-quadratic-residue-en-v1", "source_field": "title", "source_mko_sha256": "da968b0cd6fc238e595782d4d9588d4c8089b176b4d05fc3d1c01eccdb99ce68", "edition_fields_sha256": "9a10a4f69cee1314c9192b3425efd1a28c4e8632352212c1d4c9abbf7963d0b3" }, "summary": { "kind": "mko_edition", "mko_id": "mko-euler-criterion-quadratic-residue", "edition_id": "edition-mko-euler-criterion-quadratic-residue-en-v1", "source_field": "summary", "source_mko_sha256": "da968b0cd6fc238e595782d4d9588d4c8089b176b4d05fc3d1c01eccdb99ce68", "edition_fields_sha256": "9a10a4f69cee1314c9192b3425efd1a28c4e8632352212c1d4c9abbf7963d0b3" } } }, "atlas-p-adic-valuation-integers": { "source_snapshot": { "id": "atlas-p-adic-valuation-integers", "title_zh": "整數p進賦值", "summary_zh": "對質數p與非零整數a，v_p(a)是p整除|a|的最高次數；本節固定v_p(0)=infinity。", "canonical_mko_id": "mko-p-adic-valuation-integers" }, "fields": { "title": "p-adic valuation of integers", "summary": "For prime p and nonzero integer a, v_p(a) is the largest exponent of p dividing |a|; this node fixes v_p(0)=infinity." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-p-adic-valuation-integers", "edition_id": "edition-mko-p-adic-valuation-integers-en-v1", "source_field": "title", "source_mko_sha256": "9805aa95014a92e3ee82b127895171c719f8deb0548a932c13fb85c07d782a40", "edition_fields_sha256": "2cd4ce4eac16916ac3f5ea76e49d5325c639f3a7376262f845df7a5988472c05" }, "summary": { "kind": "mko_edition", "mko_id": "mko-p-adic-valuation-integers", "edition_id": "edition-mko-p-adic-valuation-integers-en-v1", "source_field": "summary", "source_mko_sha256": "9805aa95014a92e3ee82b127895171c719f8deb0548a932c13fb85c07d782a40", "edition_fields_sha256": "2cd4ce4eac16916ac3f5ea76e49d5325c639f3a7376262f845df7a5988472c05" } } }, "atlas-mobius-function": { "source_snapshot": { "id": "atlas-mobius-function", "title_zh": "Möbius函數", "summary_zh": "Möbius函數在positive integers上由完整prime multiplicities決定：mu(1)=1、含平方質因數為0、squarefree取parity sign。", "canonical_mko_id": "mko-mobius-function" }, "fields": { "title": "Mobius function", "summary": "On positive integers, complete prime multiplicities determine the Mobius function: mu(1)=1, a repeated prime gives0, and a squarefree integer has the parity sign." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-mobius-function", "edition_id": "edition-mko-mobius-function-en-v1", "source_field": "title", "source_mko_sha256": "8991dece49cf8a59e05519f80e840deb92afafba5fef912edc1a03ad30438f84", "edition_fields_sha256": "333ce0d1f71f2a04ce219a1de69c97b737682175b062691f871bdf501d7d15b3" }, "summary": { "kind": "mko_edition", "mko_id": "mko-mobius-function", "edition_id": "edition-mko-mobius-function-en-v1", "source_field": "summary", "source_mko_sha256": "8991dece49cf8a59e05519f80e840deb92afafba5fef912edc1a03ad30438f84", "edition_fields_sha256": "333ce0d1f71f2a04ce219a1de69c97b737682175b062691f871bdf501d7d15b3" } } }, "atlas-mobius-inversion-finite-divisors": { "source_snapshot": { "id": "atlas-mobius-inversion-finite-divisors", "title_zh": "有限因數格上的Möbius反演", "summary_zh": "若F(n)是所有正因數d上的f(d)總和，則以Möbius權重可精確反演f(n)。", "canonical_mko_id": "mko-mobius-inversion-finite-divisors" }, "fields": { "title": "Mobius inversion on finite divisor sets", "summary": "If F(n) sums f(d) over every positive divisor of n, Mobius weights exactly recover f(n)." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-mobius-inversion-finite-divisors", "edition_id": "edition-mko-mobius-inversion-finite-divisors-en-v1", "source_field": "title", "source_mko_sha256": "903f02acdd799627c36103659ee206c7907a68eedbfb36d6a35a0e9dc6a2ef98", "edition_fields_sha256": "949ffbd2cbcae6fb78509ad96b95282572dd64be3f598686eafcfb294ed5637a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-mobius-inversion-finite-divisors", "edition_id": "edition-mko-mobius-inversion-finite-divisors-en-v1", "source_field": "summary", "source_mko_sha256": "903f02acdd799627c36103659ee206c7907a68eedbfb36d6a35a0e9dc6a2ef98", "edition_fields_sha256": "949ffbd2cbcae6fb78509ad96b95282572dd64be3f598686eafcfb294ed5637a" } } }, "atlas-regular-expression": { "source_snapshot": { "id": "atlas-regular-expression", "title_zh": "正規表示式", "summary_zh": "正規表示式由empty language、epsilon、alphabet symbols、union、concatenation與Kleene star遞迴建構語言。", "canonical_mko_id": "mko-regular-expression" }, "fields": { "title": "Regular expression", "summary": "Regular expressions recursively build languages from the empty language, epsilon, alphabet symbols, union, concatenation and Kleene star." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-regular-expression", "edition_id": "edition-mko-regular-expression-en-v1", "source_field": "title", "source_mko_sha256": "0d4eaa52f48c71240a6e4166e4f1a4ddb3b607f21c102c49d944833b87583695", "edition_fields_sha256": "a23da16fbd11e0f9b86b8c672ca94552be48d08fc679e5abda7795da7e34f8a2" }, "summary": { "kind": "mko_edition", "mko_id": "mko-regular-expression", "edition_id": "edition-mko-regular-expression-en-v1", "source_field": "summary", "source_mko_sha256": "0d4eaa52f48c71240a6e4166e4f1a4ddb3b607f21c102c49d944833b87583695", "edition_fields_sha256": "a23da16fbd11e0f9b86b8c672ca94552be48d08fc679e5abda7795da7e34f8a2" } } }, "atlas-nondeterministic-finite-automaton": { "source_snapshot": { "id": "atlas-nondeterministic-finite-automaton", "title_zh": "非確定有限自動機", "summary_zh": "epsilon-NFA以狀態、input symbol或epsilon對應可能後繼集合，存在一條full-input final path即接受。", "canonical_mko_id": "mko-nondeterministic-finite-automaton" }, "fields": { "title": "Nondeterministic finite automaton", "summary": "An epsilon-NFA relates a state and input symbol or epsilon to possible successor states; one full-input final path suffices for acceptance." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-nondeterministic-finite-automaton", "edition_id": "edition-mko-nondeterministic-finite-automaton-en-v1", "source_field": "title", "source_mko_sha256": "4ac739d18d7d5b29bc3d146c9707a5cbf0e98a1bd3b66ce6a5b064ec83f1380b", "edition_fields_sha256": "3bf99b0771f1cfb799f5ecfb22c7758a2cd04c0665e437d7b305c8150d76b3b1" }, "summary": { "kind": "mko_edition", "mko_id": "mko-nondeterministic-finite-automaton", "edition_id": "edition-mko-nondeterministic-finite-automaton-en-v1", "source_field": "summary", "source_mko_sha256": "4ac739d18d7d5b29bc3d146c9707a5cbf0e98a1bd3b66ce6a5b064ec83f1380b", "edition_fields_sha256": "3bf99b0771f1cfb799f5ecfb22c7758a2cd04c0665e437d7b305c8150d76b3b1" } } }, "atlas-subset-construction-nfa-to-dfa": { "source_snapshot": { "id": "atlas-subset-construction-nfa-to-dfa", "title_zh": "NFA到DFA的子集合構造", "summary_zh": "以epsilon-NFA狀態子集合為DFA狀態，start、transition與acceptance按epsilon closure定義即可保持語言。", "canonical_mko_id": "mko-subset-construction-nfa-to-dfa" }, "fields": { "title": "Subset construction from an NFA to a DFA", "summary": "Use epsilon-NFA state subsets as DFA states, defining start, transitions and acceptance with epsilon closure to preserve the language." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-subset-construction-nfa-to-dfa", "edition_id": "edition-mko-subset-construction-nfa-to-dfa-en-v1", "source_field": "title", "source_mko_sha256": "813266740a8269286c26a644c634a9940342e8c8909e1d7fd8233daa628c68b3", "edition_fields_sha256": "d97f91b35659ff98640f08bdd2b56f4632de1469dd6a057b6d94f13e8b92d810" }, "summary": { "kind": "mko_edition", "mko_id": "mko-subset-construction-nfa-to-dfa", "edition_id": "edition-mko-subset-construction-nfa-to-dfa-en-v1", "source_field": "summary", "source_mko_sha256": "813266740a8269286c26a644c634a9940342e8c8909e1d7fd8233daa628c68b3", "edition_fields_sha256": "d97f91b35659ff98640f08bdd2b56f4632de1469dd6a057b6d94f13e8b92d810" } } }, "atlas-pumping-lemma-regular-languages": { "source_snapshot": { "id": "atlas-pumping-lemma-regular-languages", "title_zh": "正規語言Pumping引理", "summary_zh": "每個正規語言都有p，使每個長度至少p的成員word存在可反覆pumping且仍留在語言的分解。", "canonical_mko_id": "mko-pumping-lemma-regular-languages" }, "fields": { "title": "Pumping lemma for regular languages", "summary": "Every regular language has a positive pumping length such that each sufficiently long member word has a split that remains in the language for every pump count." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-pumping-lemma-regular-languages", "edition_id": "edition-mko-pumping-lemma-regular-languages-en-v1", "source_field": "title", "source_mko_sha256": "7466612ae549a8cda880e2921d8c9163ab7681371f543eb905240583b66e6ed4", "edition_fields_sha256": "386cc13b168a4251b89123e0a0ce09a275ead407152afcb2d10717fc9d749337" }, "summary": { "kind": "mko_edition", "mko_id": "mko-pumping-lemma-regular-languages", "edition_id": "edition-mko-pumping-lemma-regular-languages-en-v1", "source_field": "summary", "source_mko_sha256": "7466612ae549a8cda880e2921d8c9163ab7681371f543eb905240583b66e6ed4", "edition_fields_sha256": "386cc13b168a4251b89123e0a0ce09a275ead407152afcb2d10717fc9d749337" } } }, "atlas-context-free-grammar": { "source_snapshot": { "id": "atlas-context-free-grammar", "title_zh": "上下文無關文法", "summary_zh": "CFG由有限且不交的nonterminals、terminals、start與單一nonterminal左側的finite productions生成語言。", "canonical_mko_id": "mko-context-free-grammar" }, "fields": { "title": "Context-free grammar", "summary": "A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-context-free-grammar", "edition_id": "edition-mko-context-free-grammar-en-v1", "source_field": "title", "source_mko_sha256": "0cffcd9eae42c9318c21e6233a518469b8747f149bd785b0f4c94ecbe19ea685", "edition_fields_sha256": "e4a7acd54d250171ab01c624404cc1e4db8624d94faad029dd32625500d1843f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-context-free-grammar", "edition_id": "edition-mko-context-free-grammar-en-v1", "source_field": "summary", "source_mko_sha256": "0cffcd9eae42c9318c21e6233a518469b8747f149bd785b0f4c94ecbe19ea685", "edition_fields_sha256": "e4a7acd54d250171ab01c624404cc1e4db8624d94faad029dd32625500d1843f" } } }, "atlas-pushdown-automaton": { "source_snapshot": { "id": "atlas-pushdown-automaton", "title_zh": "下推自動機", "summary_zh": "nondeterministic PDA以有限控制、input與stack transitions運算，本輪固定讀完input且到final state接受。", "canonical_mko_id": "mko-pushdown-automaton" }, "fields": { "title": "Pushdown automaton", "summary": "A nondeterministic PDA uses finite control, input and stack transitions; this slice accepts only after reading the whole input in a final state." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-pushdown-automaton", "edition_id": "edition-mko-pushdown-automaton-en-v1", "source_field": "title", "source_mko_sha256": "95321a6ccc573a74997a3ea1c00f5b63623fbe7f09a34e305b5357dfd4ce1ba7", "edition_fields_sha256": "b97a68c0a69b5e235a36b6e5ed3808844a386ea0624e4cb47200466b36500672" }, "summary": { "kind": "mko_edition", "mko_id": "mko-pushdown-automaton", "edition_id": "edition-mko-pushdown-automaton-en-v1", "source_field": "summary", "source_mko_sha256": "95321a6ccc573a74997a3ea1c00f5b63623fbe7f09a34e305b5357dfd4ce1ba7", "edition_fields_sha256": "b97a68c0a69b5e235a36b6e5ed3808844a386ea0624e4cb47200466b36500672" } } }, "atlas-turing-machine-decision-model": { "source_snapshot": { "id": "atlas-turing-machine-decision-model", "title_zh": "Turing機判定模型", "summary_zh": "固定deterministic single-tape Turing機語義；decider必須對每個合法輸入停機並給正確accept/reject答案。", "canonical_mko_id": "mko-turing-machine-decision-model" }, "fields": { "title": "Turing-machine decision model", "summary": "Under fixed deterministic single-tape semantics, a decider must halt on every legal input and return the correct accept/reject answer." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-turing-machine-decision-model", "edition_id": "edition-mko-turing-machine-decision-model-en-v1", "source_field": "title", "source_mko_sha256": "9fc9700d03f68194f76ce7dec138e0b7fcac9e14deae2adf44f6dba00627308c", "edition_fields_sha256": "1a1168e2c6cf504e3a3dc701bf218560f42d10e0c29714a98e9c8ec060029c11" }, "summary": { "kind": "mko_edition", "mko_id": "mko-turing-machine-decision-model", "edition_id": "edition-mko-turing-machine-decision-model-en-v1", "source_field": "summary", "source_mko_sha256": "9fc9700d03f68194f76ce7dec138e0b7fcac9e14deae2adf44f6dba00627308c", "edition_fields_sha256": "1a1168e2c6cf504e3a3dc701bf218560f42d10e0c29714a98e9c8ec060029c11" } } }, "atlas-many-one-reduction": { "source_snapshot": { "id": "atlas-many-one-reduction", "title_zh": "多對一歸約", "summary_zh": "A≤mB表示存在total computable f，使每個x皆滿足x∈A恰當f(x)∈B。", "canonical_mko_id": "mko-many-one-reduction" }, "fields": { "title": "Many-one reduction", "summary": "A≤mB means a total computable map f satisfies x∈A exactly when f(x)∈B for every x." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-many-one-reduction", "edition_id": "edition-mko-many-one-reduction-en-v1", "source_field": "title", "source_mko_sha256": "c1a3c40ef9e301b3c5bce75380905c6f44c8cfc3bdb5126ad469d6587f9af2c6", "edition_fields_sha256": "3d6e37d9491532c34da3c2d3307802360184a831c59a55ef90235f74d3fe1b73" }, "summary": { "kind": "mko_edition", "mko_id": "mko-many-one-reduction", "edition_id": "edition-mko-many-one-reduction-en-v1", "source_field": "summary", "source_mko_sha256": "c1a3c40ef9e301b3c5bce75380905c6f44c8cfc3bdb5126ad469d6587f9af2c6", "edition_fields_sha256": "3d6e37d9491532c34da3c2d3307802360184a831c59a55ef90235f74d3fe1b73" } } }, "atlas-contraction-mapping-theorem": { "source_snapshot": { "id": "atlas-contraction-mapping-theorem", "title_zh": "壓縮映射定理", "summary_zh": "非空完備度量空間上的self-map若以固定q<1壓縮所有距離，則有唯一fixed point且Picard orbit收斂。", "canonical_mko_id": "mko-contraction-mapping-theorem" }, "fields": { "title": "Contraction mapping theorem", "summary": "A self-map on a nonempty complete metric space with one global contraction constant q<1 has a unique fixed point and convergent Picard iterates." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-contraction-mapping-theorem", "edition_id": "edition-mko-contraction-mapping-theorem-en-v1", "source_field": "title", "source_mko_sha256": "ab6943f2789918c758734f9dd379c1ca82efe2111e3e7b2f6cd5c3be1eb4db5d", "edition_fields_sha256": "3ebde79ba32b0213e61309bea20eb1d03312c85f9958fac17891a6adeb9a3a5f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-contraction-mapping-theorem", "edition_id": "edition-mko-contraction-mapping-theorem-en-v1", "source_field": "summary", "source_mko_sha256": "ab6943f2789918c758734f9dd379c1ca82efe2111e3e7b2f6cd5c3be1eb4db5d", "edition_fields_sha256": "3ebde79ba32b0213e61309bea20eb1d03312c85f9958fac17891a6adeb9a3a5f" } } }, "atlas-newton-method": { "source_snapshot": { "id": "atlas-newton-method", "title_zh": "Newton根求解方法", "summary_zh": "在導數非零的目前點，以切線零點更新x至x-f(x)/f'(x)；收斂是另需局部假設的責任。", "canonical_mko_id": "mko-newton-method" }, "fields": { "title": "Newton method for root finding", "summary": "At a qualified point with nonzero derivative, Newton method moves to the tangent-line zero x−f(x)/f_prime(x); convergence requires separate local hypotheses." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-newton-method", "edition_id": "edition-mko-newton-method-en-v1", "source_field": "title", "source_mko_sha256": "893923a37c1598a5dbd5ccc0d20d475225dd3942fcbef402eb50c9fa3b35f945", "edition_fields_sha256": "87052eb985a445ba3896e4055c4f85a709dd700f967c5f38608084a527886177" }, "summary": { "kind": "mko_edition", "mko_id": "mko-newton-method", "edition_id": "edition-mko-newton-method-en-v1", "source_field": "summary", "source_mko_sha256": "893923a37c1598a5dbd5ccc0d20d475225dd3942fcbef402eb50c9fa3b35f945", "edition_fields_sha256": "87052eb985a445ba3896e4055c4f85a709dd700f967c5f38608084a527886177" } } }, "atlas-trapezoidal-quadrature": { "source_snapshot": { "id": "atlas-trapezoidal-quadrature", "title_zh": "複合梯形數值積分", "summary_zh": "在嚴格遞增partition上逐段以端點平均乘各自寬度，得到定積分的複合梯形近似。", "canonical_mko_id": "mko-trapezoidal-quadrature" }, "fields": { "title": "Composite trapezoidal quadrature", "summary": "On a strictly increasing partition, add each subinterval width times its two-endpoint average to approximate the definite integral." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-trapezoidal-quadrature", "edition_id": "edition-mko-trapezoidal-quadrature-en-v1", "source_field": "title", "source_mko_sha256": "bd70a77af9c5b1fc3dbca7b29f06de4dbb579be5a6c48bbb16df6141f4f90001", "edition_fields_sha256": "94155bee78cd0c1637def814bd89814b8ea24d581886206114b444cdda71c697" }, "summary": { "kind": "mko_edition", "mko_id": "mko-trapezoidal-quadrature", "edition_id": "edition-mko-trapezoidal-quadrature-en-v1", "source_field": "summary", "source_mko_sha256": "bd70a77af9c5b1fc3dbca7b29f06de4dbb579be5a6c48bbb16df6141f4f90001", "edition_fields_sha256": "94155bee78cd0c1637def814bd89814b8ea24d581886206114b444cdda71c697" } } }, "atlas-gradient-vector": { "source_snapshot": { "id": "atlas-gradient-vector", "title_zh": "梯度向量", "summary_zh": "在固定standard Euclidean coordinates中，可微實值函數的一階偏導依座標排列成梯度向量。", "canonical_mko_id": "mko-gradient-vector" }, "fields": { "title": "Gradient vector", "summary": "In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-gradient-vector", "edition_id": "edition-mko-gradient-vector-en-v1", "source_field": "title", "source_mko_sha256": "953e582b90145e3769b9b22891783891e5dbfe6a99446edc1b06a277683029fa", "edition_fields_sha256": "c3eecf0e76b46e480822b9a3b1325926e14566de0ca715ab48b94ee11bb56bf8" }, "summary": { "kind": "mko_edition", "mko_id": "mko-gradient-vector", "edition_id": "edition-mko-gradient-vector-en-v1", "source_field": "summary", "source_mko_sha256": "953e582b90145e3769b9b22891783891e5dbfe6a99446edc1b06a277683029fa", "edition_fields_sha256": "c3eecf0e76b46e480822b9a3b1325926e14566de0ca715ab48b94ee11bb56bf8" } } }, "atlas-gradient-descent": { "source_snapshot": { "id": "atlas-gradient-descent", "title_zh": "梯度下降法", "summary_zh": "在固定Euclidean coordinates中以正步長沿負梯度更新；下降與收斂仍需目標及步長假設。", "canonical_mko_id": "mko-gradient-descent" }, "fields": { "title": "Gradient descent", "summary": "In fixed Euclidean coordinates, a positive step moves opposite the gradient; descent and convergence still require objective and step-size hypotheses." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-gradient-descent", "edition_id": "edition-mko-gradient-descent-en-v1", "source_field": "title", "source_mko_sha256": "b886e6f39050667744f8b88cc6e5d065b39d33d9bbcaf87612a4129ae035973b", "edition_fields_sha256": "d4d06b3147f44c0524cbd8289ab0728d8af84d518a8441c98dedc20d584fce09" }, "summary": { "kind": "mko_edition", "mko_id": "mko-gradient-descent", "edition_id": "edition-mko-gradient-descent-en-v1", "source_field": "summary", "source_mko_sha256": "b886e6f39050667744f8b88cc6e5d065b39d33d9bbcaf87612a4129ae035973b", "edition_fields_sha256": "d4d06b3147f44c0524cbd8289ab0728d8af84d518a8441c98dedc20d584fce09" } } }, "atlas-lagrange-multiplier-condition": { "source_snapshot": { "id": "atlas-lagrange-multiplier-condition", "title_zh": "等式約束Lagrange乘數必要條件", "summary_zh": "正則等式約束局部極值點的目標梯度，必為constraint gradients的線性組合。", "canonical_mko_id": "mko-lagrange-multiplier-condition" }, "fields": { "title": "Necessary Lagrange multiplier condition for equality constraints", "summary": "At a regular feasible constrained local extremum, the objective gradient is a linear combination of the constraint gradients." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-lagrange-multiplier-condition", "edition_id": "edition-mko-lagrange-multiplier-condition-en-v1", "source_field": "title", "source_mko_sha256": "cfaf4ce60e4c881cdfb0ca770d8574ec63898460c86c1049720cf5c9546bfdd7", "edition_fields_sha256": "85eab8d79d9e1bf1866b87be9363599d2227163defb5395a83b46afbd1b0f038" }, "summary": { "kind": "mko_edition", "mko_id": "mko-lagrange-multiplier-condition", "edition_id": "edition-mko-lagrange-multiplier-condition-en-v1", "source_field": "summary", "source_mko_sha256": "cfaf4ce60e4c881cdfb0ca770d8574ec63898460c86c1049720cf5c9546bfdd7", "edition_fields_sha256": "85eab8d79d9e1bf1866b87be9363599d2227163defb5395a83b46afbd1b0f038" } } }, "atlas-linear-program-standard-form": { "source_snapshot": { "id": "atlas-linear-program-standard-form", "title_zh": "線性規劃標準形", "summary_zh": "本節點固定唯一primal convention：maximize c^T x，subject to Ax≤b與x≥0。", "canonical_mko_id": "mko-linear-program-standard-form" }, "fields": { "title": "Standard form of a linear program", "summary": "This node fixes one primal convention: maximize c^T*x subject to A*x≤b and x≥0." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-program-standard-form", "edition_id": "edition-mko-linear-program-standard-form-en-v1", "source_field": "title", "source_mko_sha256": "cd8b89ab19acb18f321ca46da48146c104e7dad057ad394dffacf6e741ae1314", "edition_fields_sha256": "ba65e5d9dedf401a8d586b2a391487a4020e376fc2469adc4fdc45f2f26be08f" }, "summary": { "kind": "mko_edition", "mko_id": "mko-linear-program-standard-form", "edition_id": "edition-mko-linear-program-standard-form-en-v1", "source_field": "summary", "source_mko_sha256": "cd8b89ab19acb18f321ca46da48146c104e7dad057ad394dffacf6e741ae1314", "edition_fields_sha256": "ba65e5d9dedf401a8d586b2a391487a4020e376fc2469adc4fdc45f2f26be08f" } } }, "atlas-linear-program-duality": { "source_snapshot": { "id": "atlas-linear-program-duality", "title_zh": "線性規劃對偶", "summary_zh": "對固定primal max c^Tx、Ax≤b、x≥0，其dual為min b^Ty、A^Ty≥c、y≥0；feasible pair滿足弱對偶。", "canonical_mko_id": "mko-linear-program-duality" }, "fields": { "title": "Linear-programming duality", "summary": "For primal maximize c^T*x with A*x≤b,x≥0, the dual minimizes b^T*y with A^T*y≥c,y≥0; feasible pairs satisfy weak duality." }, "field_sources": { "title": { "kind": "mko_edition", "mko_id": "mko-linear-program-duality", "edition_id": "edition-mko-linear-program-duality-en-v1", "source_field": "title", "source_mko_sha256": "89eec94cb92d6b32f73da15c2cad46fdeedeb59b91505437cded6bef94a6a6d5", "edition_fields_sha256": "5d8ad322c3690e66d39099a580f3c4a1ef1deb23eed6ae8abc2a61662b60ed5a" }, "summary": { "kind": "mko_edition", "mko_id": "mko-linear-program-duality", "edition_id": "edition-mko-linear-program-duality-en-v1", "source_field": "summary", "source_mko_sha256": "89eec94cb92d6b32f73da15c2cad46fdeedeb59b91505437cded6bef94a6a6d5", "edition_fields_sha256": "5d8ad322c3690e66d39099a580f3c4a1ef1deb23eed6ae8abc2a61662b60ed5a" } } } }, "groups": { "numbers": { "source_snapshot": { "id": "numbers", "title_zh": "數與運算" }, "fields": { "title": "Numbers and operations" } }, "sets_logic": { "source_snapshot": { "id": "sets_logic", "title_zh": "集合與邏輯" }, "fields": { "title": "Sets and logic" } }, "algebra": { "source_snapshot": { "id": "algebra", "title_zh": "代數與方程" }, "fields": { "title": "Algebra and equations" } }, "geometry": { "source_snapshot": { "id": "geometry", "title_zh": "幾何與向量" }, "fields": { "title": "Geometry and vectors" } }, "functions": { "source_snapshot": { "id": "functions", "title_zh": "函數與圖形" }, "fields": { "title": "Functions and graphs" } }, "calculus": { "source_snapshot": { "id": "calculus", "title_zh": "微積分基礎" }, "fields": { "title": "Foundations of calculus" } }, "probability": { "source_snapshot": { "id": "probability", "title_zh": "機率與統計" }, "fields": { "title": "Probability and statistics" } }, "discrete": { "source_snapshot": { "id": "discrete", "title_zh": "離散數學" }, "fields": { "title": "Discrete mathematics" } }, "topology": { "source_snapshot": { "id": "topology", "title_zh": "拓撲" }, "fields": { "title": "Topology" } }, "number_theory": { "source_snapshot": { "id": "number_theory", "title_zh": "數論" }, "fields": { "title": "Number theory" } }, "computer_science_mathematics": { "source_snapshot": { "id": "computer_science_mathematics", "title_zh": "計算機科學數學" }, "fields": { "title": "Mathematics of computer science" } }, "formalized_mathematics": { "source_snapshot": { "id": "formalized_mathematics", "title_zh": "形式化數學" }, "fields": { "title": "Formalized mathematics" } }, "dynamical_systems": { "source_snapshot": { "id": "dynamical_systems", "title_zh": "動力系統" }, "fields": { "title": "Dynamical systems" } }, "differential_equations": { "source_snapshot": { "id": "differential_equations", "title_zh": "微分方程" }, "fields": { "title": "Differential equations" } }, "numerical_computational": { "source_snapshot": { "id": "numerical_computational", "title_zh": "數值與計算數學" }, "fields": { "title": "Numerical and computational mathematics" } }, "optimization_operations_research": { "source_snapshot": { "id": "optimization_operations_research", "title_zh": "最佳化與作業研究" }, "fields": { "title": "Optimization and operations research" } }, "mathematical_physics": { "source_snapshot": { "id": "mathematical_physics", "title_zh": "數理物理" }, "fields": { "title": "Mathematical physics" } }, "mathematical_biology": { "source_snapshot": { "id": "mathematical_biology", "title_zh": "數學生物" }, "fields": { "title": "Mathematical biology" } }, "economics_finance": { "source_snapshot": { "id": "economics_finance", "title_zh": "經濟與金融數學" }, "fields": { "title": "Mathematics of economics and finance" } }, "mathematics_education": { "source_snapshot": { "id": "mathematics_education", "title_zh": "數學教育" }, "fields": { "title": "Mathematics education" } }, "complex_numbers": { "source_snapshot": { "id": "complex_numbers", "title_zh": "複數與複平面" }, "fields": { "title": "Complex numbers and the complex plane" } }, "abstract_algebra": { "source_snapshot": { "id": "abstract_algebra", "title_zh": "抽象代數" }, "fields": { "title": "Abstract algebra" } }, "metric_measure_analysis": { "source_snapshot": { "id": "metric_measure_analysis", "title_zh": "度量與測度分析" }, "fields": { "title": "Metric and measure analysis" } }, "stochastic_processes": { "source_snapshot": { "id": "stochastic_processes", "title_zh": "分布與隨機過程" }, "fields": { "title": "Distributions and stochastic processes" } }, "partial_differential_equations": { "source_snapshot": { "id": "partial_differential_equations", "title_zh": "偏微分方程" }, "fields": { "title": "Partial differential equations" } }, "convex_optimization": { "source_snapshot": { "id": "convex_optimization", "title_zh": "凸性與凸最佳化" }, "fields": { "title": "Convexity and convex optimization" } }, "advanced_algebra": { "source_snapshot": { "id": "advanced_algebra", "title_zh": "進階代數結構" }, "fields": { "title": "Advanced algebraic structures" } }, "functional_analysis": { "source_snapshot": { "id": "functional_analysis", "title_zh": "泛函分析基礎" }, "fields": { "title": "Foundations of functional analysis" } }, "probability_limits": { "source_snapshot": { "id": "probability_limits", "title_zh": "機率結構與收斂基礎" }, "fields": { "title": "Probability structures and convergence" } }, "graph_structures": { "source_snapshot": { "id": "graph_structures", "title_zh": "圖論結構" }, "fields": { "title": "Graph structures" } }, "relations_cardinality": { "source_snapshot": { "id": "relations_cardinality", "title_zh": "關係、商集與基數" }, "fields": { "title": "Relations, quotient sets and cardinality" } }, "wave5_linear_algebra": { "source_snapshot": { "id": "wave5_linear_algebra", "title_zh": "線性代數結構" }, "fields": { "title": "Linear-algebraic structures" } }, "wave5_topology_analysis": { "source_snapshot": { "id": "wave5_topology_analysis", "title_zh": "拓撲與分析核心" }, "fields": { "title": "Core topology and analysis" } }, "wave5_probability_statistics": { "source_snapshot": { "id": "wave5_probability_statistics", "title_zh": "機率極限與統計推論" }, "fields": { "title": "Probability limits and statistical inference" } }, "wave5_graph_combinatorics": { "source_snapshot": { "id": "wave5_graph_combinatorics", "title_zh": "圖論與組合結構" }, "fields": { "title": "Graph and combinatorial structures" } }, "wave5_number_computation": { "source_snapshot": { "id": "wave5_number_computation", "title_zh": "數論與計算理論" }, "fields": { "title": "Number theory and computation theory" } }, "wave6_algebra_group_theory": { "source_snapshot": { "id": "wave6_algebra_group_theory", "title_zh": "有限群作用與結構定理" }, "fields": { "title": "Finite group actions and structure theorems" } }, "wave6_real_analysis": { "source_snapshot": { "id": "wave6_real_analysis", "title_zh": "實分析核心定理與級數工具" }, "fields": { "title": "Core real-analysis theorems and series tools" } }, "wave6_topology": { "source_snapshot": { "id": "wave6_topology", "title_zh": "拓撲構造與分離性" }, "fields": { "title": "Topological constructions and separation" } }, "wave6_probability_statistics": { "source_snapshot": { "id": "wave6_probability_statistics", "title_zh": "機率不等式、極限定理與統計推論" }, "fields": { "title": "Probability inequalities, limit theorems and statistical inference" } }, "wave6_combinatorics_graphs": { "source_snapshot": { "id": "wave6_combinatorics_graphs", "title_zh": "組合原理、有向圖與網路" }, "fields": { "title": "Combinatorial principles, directed graphs and networks" } }, "wave6_number_theory": { "source_snapshot": { "id": "wave6_number_theory", "title_zh": "乘法數論與二次剩餘" }, "fields": { "title": "Multiplicative number theory and quadratic residues" } }, "wave6_computation_languages": { "source_snapshot": { "id": "wave6_computation_languages", "title_zh": "形式語言、自動機與可計算歸約" }, "fields": { "title": "Formal languages, automata and computable reductions" } }, "wave6_applied_continuous": { "source_snapshot": { "id": "wave6_applied_continuous", "title_zh": "數值迭代、最佳化與連續方法" }, "fields": { "title": "Numerical iteration, optimization and continuous methods" } } }, "domains": { "foundations_logic": { "source_snapshot": { "id": "foundations_logic", "title_zh": "基礎與邏輯", "description_zh": "研究數學語言、公理、集合、型別、可證明性與形式推理的基礎結構。" }, "fields": { "title": "Foundations and logic", "description": "The foundational structures of mathematical language, axioms, sets, types, provability and formal reasoning." } }, "arithmetic_number_systems": { "source_snapshot": { "id": "arithmetic_number_systems", "title_zh": "算術與數系", "description_zh": "研究數、運算、數系擴張與基本數量結構。" }, "fields": { "title": "Arithmetic and number systems", "description": "Numbers, operations, extensions of number systems and basic quantitative structures." } }, "algebra": { "source_snapshot": { "id": "algebra", "title_zh": "代數", "description_zh": "研究運算結構、方程、對稱與代數系統。" }, "fields": { "title": "Algebra", "description": "Operational structures, equations, symmetry and algebraic systems." } }, "geometry": { "source_snapshot": { "id": "geometry", "title_zh": "幾何", "description_zh": "研究形狀、空間、距離、角度與幾何結構。" }, "fields": { "title": "Geometry", "description": "Shape, space, distance, angle and geometric structures." } }, "topology": { "source_snapshot": { "id": "topology", "title_zh": "拓撲", "description_zh": "研究連續性、鄰域、收斂與在連續變形下保持的性質。" }, "fields": { "title": "Topology", "description": "Continuity, neighborhoods, convergence and properties preserved under continuous deformation." } }, "analysis": { "source_snapshot": { "id": "analysis", "title_zh": "分析", "description_zh": "研究極限、連續、微分、積分與無窮過程。" }, "fields": { "title": "Analysis", "description": "Limits, continuity, differentiation, integration and infinite processes." } }, "number_theory": { "source_snapshot": { "id": "number_theory", "title_zh": "數論", "description_zh": "研究整數及其算術結構。" }, "fields": { "title": "Number theory", "description": "Integers and their arithmetic structures." } }, "combinatorics_discrete": { "source_snapshot": { "id": "combinatorics_discrete", "title_zh": "組合與離散數學", "description_zh": "研究有限或可數結構、計數、圖與離散關係。" }, "fields": { "title": "Combinatorics and discrete mathematics", "description": "Finite or countable structures, counting, graphs and discrete relations." } }, "probability_statistics": { "source_snapshot": { "id": "probability_statistics", "title_zh": "機率與統計", "description_zh": "研究隨機性、分布、推論與資料不確定性。" }, "fields": { "title": "Probability and statistics", "description": "Randomness, distributions, inference and uncertainty in data." } }, "dynamical_systems": { "source_snapshot": { "id": "dynamical_systems", "title_zh": "動力系統", "description_zh": "研究狀態隨時間或迭代演化的系統。" }, "fields": { "title": "Dynamical systems", "description": "Systems whose states evolve through time or iteration." } }, "differential_equations": { "source_snapshot": { "id": "differential_equations", "title_zh": "微分方程", "description_zh": "研究以導數描述的關係與演化。" }, "fields": { "title": "Differential equations", "description": "Relations and evolution described by derivatives." } }, "numerical_computational": { "source_snapshot": { "id": "numerical_computational", "title_zh": "數值與計算數學", "description_zh": "研究近似算法、誤差、穩定性與可計算實作。" }, "fields": { "title": "Numerical and computational mathematics", "description": "Approximation algorithms, error, stability and computational implementations." } }, "optimization_operations_research": { "source_snapshot": { "id": "optimization_operations_research", "title_zh": "最佳化與作業研究", "description_zh": "研究在限制條件下尋找最佳決策。" }, "fields": { "title": "Optimization and operations research", "description": "Finding the best decisions under constraints." } }, "mathematical_physics": { "source_snapshot": { "id": "mathematical_physics", "title_zh": "數理物理", "description_zh": "研究物理理論中的數學結構與方法。" }, "fields": { "title": "Mathematical physics", "description": "Mathematical structures and methods in physical theories." } }, "mathematical_biology": { "source_snapshot": { "id": "mathematical_biology", "title_zh": "數學生物", "description_zh": "研究生命系統的數學模型。" }, "fields": { "title": "Mathematical biology", "description": "Mathematical models of living systems." } }, "economics_finance": { "source_snapshot": { "id": "economics_finance", "title_zh": "經濟與金融數學", "description_zh": "研究經濟、風險與金融系統的數學模型。" }, "fields": { "title": "Mathematics of economics and finance", "description": "Mathematical models of economic, risk and financial systems." } }, "computer_science_mathematics": { "source_snapshot": { "id": "computer_science_mathematics", "title_zh": "計算機科學數學", "description_zh": "研究演算法、計算、資訊與形式語言的數學基礎。" }, "fields": { "title": "Mathematics of computer science", "description": "Mathematical foundations of algorithms, computation, information and formal languages." } }, "mathematics_education": { "source_snapshot": { "id": "mathematics_education", "title_zh": "數學教育", "description_zh": "研究數學學習、教學、評量與課程。" }, "fields": { "title": "Mathematics education", "description": "Mathematical learning, teaching, assessment and curricula." } }, "history_philosophy_methodology": { "source_snapshot": { "id": "history_philosophy_methodology", "title_zh": "歷史、哲學與方法論", "description_zh": "研究數學思想的歷史、合法性標準與問題解決方法。" }, "fields": { "title": "History, philosophy and methodology", "description": "The history of mathematical ideas, standards of validity and problem-solving methods." } }, "formalized_mathematics": { "source_snapshot": { "id": "formalized_mathematics", "title_zh": "形式化數學", "description_zh": "研究以證明助理與形式系統表達、檢查及重播數學。" }, "fields": { "title": "Formalized mathematics", "description": "Expressing, checking and replaying mathematics with proof assistants and formal systems." } } }, "methods": { "method-direct-proof": { "source_snapshot": { "id": "method-direct-proof", "title_zh": "直接證明", "description_zh": "由已知條件沿合法推理直接推出結論。", "signals": [ "結論可由定義或已知定理逐步導出" ], "steps": [ "展開條件", "選取相關定義或定理", "逐步推出結論" ], "failure_modes": [ "隱藏使用未證明命題" ] }, "fields": { "title": "Direct proof", "description": "Derive the conclusion directly from known conditions through valid reasoning.", "signals": [ "The conclusion follows step by step from definitions or known theorems." ], "steps": [ "Expand the conditions.", "Select relevant definitions or theorems.", "Derive the conclusion step by step." ], "failure_modes": [ "Silently using an unproved proposition." ] } }, "method-contradiction": { "source_snapshot": { "id": "method-contradiction", "title_zh": "反證法", "description_zh": "假設結論不成立並推出矛盾。", "signals": [ "否定結論後更容易與已知條件衝突" ], "steps": [ "否定目標", "推導後果", "得到矛盾" ], "failure_modes": [ "矛盾與原假設無關" ] }, "fields": { "title": "Proof by contradiction", "description": "Assume the conclusion fails and derive a contradiction.", "signals": [ "Negating the conclusion makes a conflict with known conditions easier to obtain." ], "steps": [ "Negate the goal.", "Derive consequences.", "Obtain a contradiction." ], "failure_modes": [ "The contradiction is unrelated to the original assumption." ] } }, "method-induction": { "source_snapshot": { "id": "method-induction", "title_zh": "數學歸納", "description_zh": "由基礎情況與遞推步驟建立無限族命題。", "signals": [ "命題按自然數或遞迴結構分層" ], "steps": [ "證明基礎情況", "假設歸納命題", "證明下一步" ], "failure_modes": [ "歸納假設使用範圍錯誤" ] }, "fields": { "title": "Mathematical induction", "description": "Establish an infinite family of propositions through base cases and an inductive step.", "signals": [ "Propositions are organized by natural numbers or a recursive structure." ], "steps": [ "Prove the base case.", "State the inductive hypothesis.", "Prove the next step." ], "failure_modes": [ "Using the inductive hypothesis outside its permitted scope." ] } }, "method-construction": { "source_snapshot": { "id": "method-construction", "title_zh": "構造法", "description_zh": "明確建立滿足條件的數學對象。", "signals": [ "存在性問題要求具體對象" ], "steps": [ "分析限制", "提出候選構造", "驗證全部條件" ], "failure_modes": [ "只展示例子而未覆蓋一般情況" ] }, "fields": { "title": "Construction", "description": "Explicitly build a mathematical object satisfying the conditions.", "signals": [ "An existence problem asks for a concrete object." ], "steps": [ "Analyze the constraints.", "Propose a candidate construction.", "Verify every condition." ], "failure_modes": [ "Showing an example without covering the general case." ] } }, "method-invariant": { "source_snapshot": { "id": "method-invariant", "title_zh": "不變量", "description_zh": "尋找在操作或演化下保持不變的量。", "signals": [ "狀態反覆改變但可能保存某些性質" ], "steps": [ "定義狀態", "提出候選量", "證明保存", "比較初末狀態" ], "failure_modes": [ "候選量未真正保持" ] }, "fields": { "title": "Invariant", "description": "Find a quantity preserved by an operation or evolution.", "signals": [ "States change repeatedly while some properties may be preserved." ], "steps": [ "Define the states.", "Propose a candidate quantity.", "Prove preservation.", "Compare the initial and final states." ], "failure_modes": [ "The proposed quantity is not actually preserved." ] } }, "method-symmetry": { "source_snapshot": { "id": "method-symmetry", "title_zh": "對稱性", "description_zh": "利用變換下的等價性縮減問題情況。", "signals": [ "多個情況在變換下等價" ], "steps": [ "識別變換", "證明問題保持", "縮減代表情況" ], "failure_modes": [ "誤把近似對稱當成精確對稱" ] }, "fields": { "title": "Symmetry", "description": "Use equivalence under transformations to reduce the cases of a problem.", "signals": [ "Several cases are equivalent under a transformation." ], "steps": [ "Identify the transformation.", "Prove the problem is preserved.", "Reduce to representative cases." ], "failure_modes": [ "Mistaking approximate symmetry for exact symmetry." ] } }, "method-extremal-principle": { "source_snapshot": { "id": "method-extremal-principle", "title_zh": "極值原理", "description_zh": "選擇最大、最小或最早對象來逼出結構。", "signals": [ "有限集合或良序結構中存在極端元素" ], "steps": [ "選取極端對象", "分析其限制", "推出結論" ], "failure_modes": [ "極值不存在或未證明可取" ] }, "fields": { "title": "Extremal principle", "description": "Choose a largest, smallest or earliest object to force a structural conclusion.", "signals": [ "An extreme element exists in a finite set or a well-ordered structure." ], "steps": [ "Choose an extreme object.", "Analyze its constraints.", "Derive the conclusion." ], "failure_modes": [ "The extremum does not exist or its attainment has not been established." ] } }, "method-pigeonhole": { "source_snapshot": { "id": "method-pigeonhole", "title_zh": "抽屜原理", "description_zh": "由對象數量超過容器數量推出重合。", "signals": [ "問題具有有限分類與數量不平衡" ], "steps": [ "定義對象與容器", "比較數量", "推出重合" ], "failure_modes": [ "容器分類不互斥" ] }, "fields": { "title": "Pigeonhole principle", "description": "Infer a collision when there are more objects than containers.", "signals": [ "The problem has finitely many categories and an imbalance in counts." ], "steps": [ "Define the objects and containers.", "Compare their counts.", "Infer a collision." ], "failure_modes": [ "The container categories are not mutually exclusive." ] } }, "method-diagonalization": { "source_snapshot": { "id": "method-diagonalization", "title_zh": "對角化", "description_zh": "構造與列舉中每一項至少一處不同的對象。", "signals": [ "需要證明不可列舉或逃離給定族" ], "steps": [ "假設列舉", "逐項選取差異", "構造新對象" ], "failure_modes": [ "新對象不在允許空間" ] }, "fields": { "title": "Diagonalization", "description": "Construct an object differing from every listed item in at least one position.", "signals": [ "The goal is non-enumerability or escaping a given family." ], "steps": [ "Assume an enumeration.", "Choose a difference for each item.", "Construct the new object." ], "failure_modes": [ "The new object is outside the permitted space." ] } }, "method-compactness": { "source_snapshot": { "id": "method-compactness", "title_zh": "緊緻性", "description_zh": "把無限或全域問題轉化為有限或收斂資料。", "signals": [ "局部資訊需要推出全域結論" ], "steps": [ "選擇緊緻結構", "提取有限或收斂資料", "傳遞性質" ], "failure_modes": [ "未滿足緊緻條件" ] }, "fields": { "title": "Compactness", "description": "Reduce an infinite or global problem to finite or convergent data.", "signals": [ "Local information is needed to obtain a global conclusion." ], "steps": [ "Choose a compact structure.", "Extract finite or convergent data.", "Transfer the property." ], "failure_modes": [ "The compactness hypotheses are not satisfied." ] } }, "method-localization": { "source_snapshot": { "id": "method-localization", "title_zh": "局部化", "description_zh": "先研究局部資料，再考察如何拼接為全域結果。", "signals": [ "全域問題可由局部行為控制" ], "steps": [ "選擇局部尺度", "證明局部性質", "檢查拼接" ], "failure_modes": [ "局部解不能全域拼接" ] }, "fields": { "title": "Localization", "description": "Study local data first, then examine how it can be glued into a global result.", "signals": [ "Local behavior can control the global problem." ], "steps": [ "Choose a local scale.", "Prove the local property.", "Check the gluing." ], "failure_modes": [ "Local solutions cannot be glued globally." ] } }, "method-linearization": { "source_snapshot": { "id": "method-linearization", "title_zh": "線性化", "description_zh": "在局部或近似層把非線性問題轉成線性問題。", "signals": [ "局部一階行為占主導" ], "steps": [ "選擇展開點", "取線性主部", "估計餘項" ], "failure_modes": [ "忽略餘項適用範圍" ] }, "fields": { "title": "Linearization", "description": "Replace a nonlinear problem by a linear one at a local or approximation level.", "signals": [ "First-order local behavior dominates." ], "steps": [ "Choose an expansion point.", "Take the linear principal part.", "Estimate the remainder." ], "failure_modes": [ "Ignoring the range in which the remainder estimate applies." ] } }, "method-coordinate": { "source_snapshot": { "id": "method-coordinate", "title_zh": "座標化", "description_zh": "把幾何或抽象關係轉成座標與代數式。", "signals": [ "幾何關係可選取便利座標" ], "steps": [ "選擇座標系", "翻譯對象", "代數處理", "回譯意義" ], "failure_modes": [ "座標選擇引入退化" ] }, "fields": { "title": "Coordinates", "description": "Translate geometric or abstract relations into coordinates and algebraic expressions.", "signals": [ "Convenient coordinates can be chosen for the geometric relations." ], "steps": [ "Choose a coordinate system.", "Translate the objects.", "Work algebraically.", "Translate the meaning back." ], "failure_modes": [ "The coordinate choice introduces a degeneracy." ] } }, "method-generating-function": { "source_snapshot": { "id": "method-generating-function", "title_zh": "生成函數", "description_zh": "把數列或組合資料編碼成形式冪級數。", "signals": [ "遞迴、計數或卷積結構" ], "steps": [ "建立生成函數", "轉換關係", "求解關係", "提取係數" ], "failure_modes": [ "收斂與形式操作混淆" ] }, "fields": { "title": "Generating function", "description": "Encode sequence or combinatorial data as a formal power series.", "signals": [ "A recurrence, counting problem or convolution structure is present." ], "steps": [ "Construct the generating function.", "Transform the relations.", "Solve the relations.", "Extract coefficients." ], "failure_modes": [ "Confusing convergence with formal operations." ] } }, "method-fourier": { "source_snapshot": { "id": "method-fourier", "title_zh": "傅立葉方法", "description_zh": "把函數分解為頻率成分以分析運算與結構。", "signals": [ "平移、週期或卷積結構" ], "steps": [ "選擇轉換", "進入頻域", "處理頻率成分", "反轉換" ], "failure_modes": [ "忽略函數空間與收斂條件" ] }, "fields": { "title": "Fourier methods", "description": "Decompose functions into frequency components to analyze operations and structure.", "signals": [ "Translation, periodicity or convolution structure is present." ], "steps": [ "Choose a transform.", "Move to the frequency domain.", "Handle frequency components.", "Apply the inverse transform." ], "failure_modes": [ "Ignoring function-space and convergence hypotheses." ] } }, "method-probabilistic": { "source_snapshot": { "id": "method-probabilistic", "title_zh": "機率方法", "description_zh": "透過隨機選取證明結構存在或估計性質。", "signals": [ "直接構造困難但平均行為可控" ], "steps": [ "定義隨機模型", "估計機率或期望", "推出存在" ], "failure_modes": [ "隨機模型不覆蓋目標對象" ] }, "fields": { "title": "Probabilistic method", "description": "Use random selection to prove existence of a structure or estimate its properties.", "signals": [ "Direct construction is difficult but average behavior can be controlled." ], "steps": [ "Define a random model.", "Estimate probabilities or expectations.", "Infer existence." ], "failure_modes": [ "The random model does not cover the target objects." ] } }, "method-numerical-approximation": { "source_snapshot": { "id": "method-numerical-approximation", "title_zh": "數值近似", "description_zh": "以有限精度算法近似連續或無限問題。", "signals": [ "解析解不可得或成本過高" ], "steps": [ "選擇離散化", "分析誤差", "執行計算", "檢查穩定性" ], "failure_modes": [ "把近似結果當成精確證明" ] }, "fields": { "title": "Numerical approximation", "description": "Use finite-precision algorithms to approximate a continuous or infinite problem.", "signals": [ "An analytic solution is unavailable or too costly." ], "steps": [ "Choose a discretization.", "Analyze the error.", "Perform the computation.", "Check stability." ], "failure_modes": [ "Treating an approximation as an exact proof." ] } }, "method-computer-experiment": { "source_snapshot": { "id": "method-computer-experiment", "title_zh": "計算實驗", "description_zh": "以程式生成案例、探索模式與反例候選。", "signals": [ "需要大量案例或難以手工探索" ], "steps": [ "形式化可計算問題", "固定環境", "執行與記錄", "分析限制" ], "failure_modes": [ "有限測試冒充普遍證明" ] }, "fields": { "title": "Computational experiment", "description": "Use programs to generate cases and explore patterns or candidate counterexamples.", "signals": [ "Many cases are needed or manual exploration is difficult." ], "steps": [ "Formalize the computable problem.", "Fix the environment.", "Execute and record.", "Analyze the limitations." ], "failure_modes": [ "Presenting finite testing as a universal proof." ] } }, "method-counterexample-search": { "source_snapshot": { "id": "method-counterexample-search", "title_zh": "反例搜尋", "description_zh": "在宣告域內系統搜尋使命題失敗的輸入。", "signals": [ "命題可在有限或抽樣域內檢查" ], "steps": [ "定義宣告域", "建立判定器", "執行搜尋", "保存限制" ], "failure_modes": [ "未找到反例被誤解為證明" ] }, "fields": { "title": "Counterexample search", "description": "Systematically search an explicit domain for an input that falsifies a proposition.", "signals": [ "The proposition can be checked on a finite or sampled domain." ], "steps": [ "Define the declared domain.", "Build the decision procedure.", "Execute the search.", "Preserve the limitations." ], "failure_modes": [ "Treating not finding a counterexample as a proof." ] } }, "method-formal-verification": { "source_snapshot": { "id": "method-formal-verification", "title_zh": "形式驗證", "description_zh": "在明確形式系統中由核心檢查器驗證精確聲明。", "signals": [ "需要機器可重播的全稱聲明" ], "steps": [ "定義形式模型", "建立語義橋", "編譯證明", "拒絕佔位符" ], "failure_modes": [ "形式聲明與百科語義未完全對齊" ] }, "fields": { "title": "Formal verification", "description": "Use a kernel checker in an explicit formal system to verify an exact statement.", "signals": [ "A machine-replayable universally quantified statement is needed." ], "steps": [ "Define the formal model.", "Build the semantic bridge.", "Compile the proof.", "Reject placeholders." ], "failure_modes": [ "The formal statement and encyclopedia semantics are not fully aligned." ] } } } }