# Mathematical Atlas This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt. [Catalogue presentation and source-binding data](https://ocme.evemisslab.com/data/i18n/catalogue.en.json) ## [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) The basic number system for counting and discrete order. atlas-natural-number · canonical_mko [Natural number](https://ocme.evemisslab.com/en/mko/mko-natural-number/) ## Prerequisite nodes ## [Integer](https://ocme.evemisslab.com/en/atlas/atlas-integer/) A number system extending the natural numbers to represent positive values, negative values and zero. atlas-integer · canonical_mko [Integer](https://ocme.evemisslab.com/en/mko/mko-integer/) ## Prerequisite nodes - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) ## [Rational number](https://ocme.evemisslab.com/en/atlas/atlas-rational-number/) A number expressible as a ratio of two integers. atlas-rational-number · canonical_mko [Rational number](https://ocme.evemisslab.com/en/mko/mko-rational-number/) ## Prerequisite nodes - [Integer](https://ocme.evemisslab.com/en/atlas/atlas-integer/) ## [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) A number system containing rational and irrational numbers and supporting continuous quantities and real analysis. atlas-real-number · canonical_mko [Real number](https://ocme.evemisslab.com/en/mko/mko-real-number/) ## Prerequisite nodes - [Rational number](https://ocme.evemisslab.com/en/atlas/atlas-rational-number/) ## [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) Addition, subtraction, multiplication, division and their basic rules. atlas-arithmetic-operations · canonical_mko [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) ## Prerequisite nodes - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) ## [Fraction](https://ocme.evemisslab.com/en/atlas/atlas-fraction/) A fraction p/q is a representation of a rational number whose numerator and denominator are integers and whose denominator is nonzero. atlas-fraction · canonical_mko [Fraction](https://ocme.evemisslab.com/en/mko/mko-fraction/) ## Prerequisite nodes - [Rational number](https://ocme.evemisslab.com/en/atlas/atlas-rational-number/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Ratio and proportion](https://ocme.evemisslab.com/en/atlas/atlas-ratio-proportion/) This entry interprets the numerical ratio a:b as a/b and defines a proportion as the equality of two valid ratios. atlas-ratio-proportion · canonical_mko [Ratio and proportion](https://ocme.evemisslab.com/en/mko/mko-ratio-proportion/) ## Prerequisite nodes - [Fraction](https://ocme.evemisslab.com/en/atlas/atlas-fraction/) ## [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) 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. atlas-power-root · canonical_mko [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/mko/mko-power-root/) ## Prerequisite nodes - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) ## [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) The nonnegative distance of a real number from zero, given by x or −x according to its sign. atlas-absolute-value · canonical_mko [Absolute value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/) ## Prerequisite nodes - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) The usual real order compares real values and supports finite nonempty extrema, closed intervals, and translation by a common addend. atlas-order-relation · canonical_mko [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) ## Prerequisite nodes - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Rational number](https://ocme.evemisslab.com/en/atlas/atlas-rational-number/) ## [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) A basic mathematical structure treating objects as a collection. atlas-set · canonical_mko [Set](https://ocme.evemisslab.com/en/mko/mko-set/) ## Prerequisite nodes ## [Set membership](https://ocme.evemisslab.com/en/atlas/atlas-set-membership/) The relation describing whether an element belongs to a set. atlas-set-membership · canonical_mko [Set membership](https://ocme.evemisslab.com/en/mko/mko-set-membership/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) ## [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) The relation that every element of one set also belongs to another set. atlas-subset · canonical_mko [Subset](https://ocme.evemisslab.com/en/mko/mko-subset/) ## Prerequisite nodes - [Set membership](https://ocme.evemisslab.com/en/atlas/atlas-set-membership/) ## [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) Operations on sets, including union, intersection, difference and relative complement. atlas-set-operations · canonical_mko [Set operations](https://ocme.evemisslab.com/en/mko/mko-set-operations/) ## Prerequisite nodes - [Set membership](https://ocme.evemisslab.com/en/atlas/atlas-set-membership/) ## [Proposition](https://ocme.evemisslab.com/en/atlas/atlas-proposition/) A mathematical statement with a truth value. atlas-proposition · canonical_mko [Proposition](https://ocme.evemisslab.com/en/mko/mko-proposition/) ## Prerequisite nodes ## [Logical connectives](https://ocme.evemisslab.com/en/atlas/atlas-logical-connectives/) Ways to combine propositions, including negation, conjunction, disjunction, implication and equivalence. atlas-logical-connectives · canonical_mko [Logical connectives](https://ocme.evemisslab.com/en/mko/mko-logical-connectives/) ## Prerequisite nodes - [Proposition](https://ocme.evemisslab.com/en/atlas/atlas-proposition/) ## [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) 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. atlas-quantifier · canonical_mko [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) ## Prerequisite nodes - [Proposition](https://ocme.evemisslab.com/en/atlas/atlas-proposition/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) ## [Mathematical proof](https://ocme.evemisslab.com/en/atlas/atlas-proof/) A process establishing a conclusion from explicit premises through valid reasoning. atlas-proof · canonical_mko [Mathematical proof](https://ocme.evemisslab.com/en/mko/mko-proof/) ## Prerequisite nodes - [Logical connectives](https://ocme.evemisslab.com/en/atlas/atlas-logical-connectives/) ## [Counterexamples and domain eligibility](https://ocme.evemisslab.com/en/atlas/atlas-counterexample/) 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. atlas-counterexample · canonical_mko [Counterexamples and domain eligibility](https://ocme.evemisslab.com/en/mko/mko-counterexample/) ## Prerequisite nodes - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) - [Mathematical proof](https://ocme.evemisslab.com/en/atlas/atlas-proof/) ## [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) 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. atlas-relation · canonical_mko [Binary relations and set projections](https://ocme.evemisslab.com/en/mko/mko-relation/) ## Prerequisite nodes - [Set membership](https://ocme.evemisslab.com/en/atlas/atlas-set-membership/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) ## [Variables and algebraic expressions](https://ocme.evemisslab.com/en/atlas/atlas-variable-expression/) Symbols representing unknown quantities and the structure of operations. atlas-variable-expression · canonical_mko [Variables and algebraic expressions](https://ocme.evemisslab.com/en/mko/mko-variable-expression/) ## Prerequisite nodes - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) A condition requiring two algebraic expressions to be equal. atlas-equation · canonical_mko [Equation](https://ocme.evemisslab.com/en/mko/mko-equation/) ## Prerequisite nodes - [Variables and algebraic expressions](https://ocme.evemisslab.com/en/atlas/atlas-variable-expression/) ## [Linear equation in one variable](https://ocme.evemisslab.com/en/atlas/atlas-linear-equation/) An equation over an explicit number system with one unknown and a nonzero linear coefficient. atlas-linear-equation · canonical_mko [Linear equation in one variable](https://ocme.evemisslab.com/en/mko/mko-linear-equation/) ## Prerequisite nodes - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) ## [Quadratic equations over the reals](https://ocme.evemisslab.com/en/atlas/atlas-quadratic-equation/) 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. atlas-quadratic-equation · canonical_mko [Quadratic equations over the reals](https://ocme.evemisslab.com/en/mko/mko-quadratic-equation/) ## Prerequisite nodes - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) - [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/atlas/atlas-polynomial/) ## [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/atlas/atlas-polynomial/) A polynomial is finitely supported rational-coefficient data; combine like powers before reading coefficients or degree. atlas-polynomial · canonical_mko [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/mko/mko-polynomial/) ## Prerequisite nodes - [Variables and algebraic expressions](https://ocme.evemisslab.com/en/atlas/atlas-variable-expression/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) - [Rational number](https://ocme.evemisslab.com/en/atlas/atlas-rational-number/) ## [Polynomial factorization over the rationals](https://ocme.evemisslab.com/en/atlas/atlas-factorization/) Factorization writes a polynomial as a product in a declared coefficient field. Validation compares all coefficients rather than a few evaluated values. atlas-factorization · canonical_mko [Polynomial factorization over the rationals](https://ocme.evemisslab.com/en/mko/mko-factorization/) ## Prerequisite nodes - [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/atlas/atlas-polynomial/) ## [Linear systems in two variables and degenerate constraints](https://ocme.evemisslab.com/en/atlas/atlas-system-linear-equations/) 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. atlas-system-linear-equations · canonical_mko [Linear systems in two variables and degenerate constraints](https://ocme.evemisslab.com/en/mko/mko-system-linear-equations/) ## Prerequisite nodes - [Linear equation in one variable](https://ocme.evemisslab.com/en/atlas/atlas-linear-equation/) - [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/atlas/atlas-vector/) ## [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/atlas/atlas-vector/) This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise. atlas-vector · canonical_mko [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/mko/mko-vector/) ## Prerequisite nodes - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) - [Coordinate plane](https://ocme.evemisslab.com/en/atlas/atlas-coordinate-plane/) ## [Finite matrices: dimensions, indexing, and transpose](https://ocme.evemisslab.com/en/atlas/atlas-matrix/) Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances. atlas-matrix · canonical_mko [Finite matrices: dimensions, indexing, and transpose](https://ocme.evemisslab.com/en/mko/mko-matrix/) ## Prerequisite nodes - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) ## [Matrix-represented linear maps on the real plane](https://ocme.evemisslab.com/en/atlas/atlas-linear-transformation/) Use the explicit R²→R² matrix model for additivity and scalar compatibility; an affine expression with nonzero translation is not linear. atlas-linear-transformation · canonical_mko [Matrix-represented linear maps on the real plane](https://ocme.evemisslab.com/en/mko/mko-linear-transformation/) ## Prerequisite nodes - [Finite matrices: dimensions, indexing, and transpose](https://ocme.evemisslab.com/en/atlas/atlas-matrix/) - [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/atlas/atlas-vector/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Determinants of real 2×2 matrices](https://ocme.evemisslab.com/en/atlas/atlas-determinant/) This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices. atlas-determinant · canonical_mko [Determinants of real 2×2 matrices](https://ocme.evemisslab.com/en/mko/mko-determinant/) ## Prerequisite nodes - [Finite matrices: dimensions, indexing, and transpose](https://ocme.evemisslab.com/en/atlas/atlas-matrix/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) ## [Real eigenvalues and nonzero eigenvectors of 2×2 matrices](https://ocme.evemisslab.com/en/atlas/atlas-eigenvalue/) An eigenvector must be nonzero, whereas a candidate-scalar kernel includes zero; a real 2×2 matrix may have no real eigenvalues. atlas-eigenvalue · canonical_mko [Real eigenvalues and nonzero eigenvectors of 2×2 matrices](https://ocme.evemisslab.com/en/mko/mko-eigenvalue/) ## Prerequisite nodes - [Matrix-represented linear maps on the real plane](https://ocme.evemisslab.com/en/atlas/atlas-linear-transformation/) - [Determinants of real 2×2 matrices](https://ocme.evemisslab.com/en/atlas/atlas-determinant/) - [Quadratic equations over the reals](https://ocme.evemisslab.com/en/atlas/atlas-quadratic-equation/) ## [Points, straight lines and planes](https://ocme.evemisslab.com/en/atlas/atlas-point-line-plane/) Basic objects of position and extension in Euclidean geometry. atlas-point-line-plane · canonical_mko [Points, lines, and planes](https://ocme.evemisslab.com/en/mko/mko-point-line-plane/) ## Prerequisite nodes ## [Angle](https://ocme.evemisslab.com/en/atlas/atlas-angle/) A turning relation between two rays or directions. atlas-angle · canonical_mko [Angle](https://ocme.evemisslab.com/en/mko/mko-angle/) ## Prerequisite nodes - [Points, straight lines and planes](https://ocme.evemisslab.com/en/atlas/atlas-point-line-plane/) ## [Euclidean length](https://ocme.evemisslab.com/en/atlas/atlas-euclidean-length/) The distance between two points in a plane or in space. atlas-euclidean-length · canonical_mko [Euclidean length](https://ocme.evemisslab.com/en/mko/mko-euclidean-length/) ## Prerequisite nodes - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Points, straight lines and planes](https://ocme.evemisslab.com/en/atlas/atlas-point-line-plane/) ## [Triangle](https://ocme.evemisslab.com/en/atlas/atlas-triangle/) A basic polygon formed by three noncollinear points and the segments joining them. atlas-triangle · canonical_mko [Triangle](https://ocme.evemisslab.com/en/mko/mko-triangle/) ## Prerequisite nodes - [Points, straight lines and planes](https://ocme.evemisslab.com/en/atlas/atlas-point-line-plane/) ## [Right triangle](https://ocme.evemisslab.com/en/atlas/atlas-right-triangle/) A triangle with one right angle. atlas-right-triangle · canonical_mko [Right triangle](https://ocme.evemisslab.com/en/mko/mko-right-triangle/) ## Prerequisite nodes - [Triangle](https://ocme.evemisslab.com/en/atlas/atlas-triangle/) - [Angle](https://ocme.evemisslab.com/en/atlas/atlas-angle/) ## [Congruence of nondegenerate triangles with explicit correspondence](https://ocme.evemisslab.com/en/atlas/atlas-congruence/) In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection. atlas-congruence · canonical_mko [Congruence of nondegenerate triangles with explicit correspondence](https://ocme.evemisslab.com/en/mko/mko-congruence/) ## Prerequisite nodes - [Triangle](https://ocme.evemisslab.com/en/atlas/atlas-triangle/) - [Euclidean distance in an equal-unit orthonormal frame](https://ocme.evemisslab.com/en/atlas/atlas-distance-formula/) - [Determinants of real 2×2 matrices](https://ocme.evemisslab.com/en/atlas/atlas-determinant/) ## [Labeled triangle similarity with a specified positive scale](https://ocme.evemisslab.com/en/atlas/atlas-similarity/) 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. atlas-similarity · canonical_mko [Labeled triangle similarity with a specified positive scale](https://ocme.evemisslab.com/en/mko/mko-similarity/) ## Prerequisite nodes - [Congruence of nondegenerate triangles with explicit correspondence](https://ocme.evemisslab.com/en/atlas/atlas-congruence/) - [Ratio and proportion](https://ocme.evemisslab.com/en/atlas/atlas-ratio-proportion/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) ## [Coordinate plane](https://ocme.evemisslab.com/en/atlas/atlas-coordinate-plane/) 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). atlas-coordinate-plane · canonical_mko [Coordinate plane](https://ocme.evemisslab.com/en/mko/mko-coordinate-plane/) ## Prerequisite nodes - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Points, straight lines and planes](https://ocme.evemisslab.com/en/atlas/atlas-point-line-plane/) ## [Euclidean distance in an equal-unit orthonormal frame](https://ocme.evemisslab.com/en/atlas/atlas-distance-formula/) The nonnegative L2 distance in one equal-unit orthonormal coordinate frame is the principal square root of the sum of squared coordinate differences. atlas-distance-formula · canonical_mko [Euclidean distance in an equal-unit orthonormal frame](https://ocme.evemisslab.com/en/mko/mko-distance-formula/) ## Prerequisite nodes - [Coordinate plane](https://ocme.evemisslab.com/en/atlas/atlas-coordinate-plane/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) - [Euclidean length](https://ocme.evemisslab.com/en/atlas/atlas-euclidean-length/) ## [Pythagorean theorem](https://ocme.evemisslab.com/en/atlas/atlas-pythagorean-theorem/) In a right triangle, the sum of the squared leg lengths equals the squared hypotenuse length. atlas-pythagorean-theorem · canonical_mko [Pythagorean theorem](https://ocme.evemisslab.com/en/mko/mko-euclid-pythagorean-theorem/) ## Prerequisite nodes - [Right triangle](https://ocme.evemisslab.com/en/atlas/atlas-right-triangle/) - [Euclidean length](https://ocme.evemisslab.com/en/atlas/atlas-euclidean-length/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) ## [Circle boundaries and zero-radius degeneracy](https://ocme.evemisslab.com/en/atlas/atlas-circle/) A circle here is the boundary point set at a nonnegative radius from its center, not the filled disk; radius zero yields a singleton. atlas-circle · canonical_mko [Circle boundaries and zero-radius degeneracy](https://ocme.evemisslab.com/en/mko/mko-circle/) ## Prerequisite nodes - [Euclidean length](https://ocme.evemisslab.com/en/atlas/atlas-euclidean-length/) - [Points, straight lines and planes](https://ocme.evemisslab.com/en/atlas/atlas-point-line-plane/) - [Euclidean distance in an equal-unit orthonormal frame](https://ocme.evemisslab.com/en/atlas/atlas-distance-formula/) ## [Plane-coordinate isometries and uniform scaling](https://ocme.evemisslab.com/en/atlas/atlas-geometric-transformations/) Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients. atlas-geometric-transformations · canonical_mko [Plane-coordinate isometries and uniform scaling](https://ocme.evemisslab.com/en/mko/mko-geometric-transformations/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Coordinate plane](https://ocme.evemisslab.com/en/atlas/atlas-coordinate-plane/) - [Euclidean distance in an equal-unit orthonormal frame](https://ocme.evemisslab.com/en/atlas/atlas-distance-formula/) - [Labeled triangle similarity with a specified positive scale](https://ocme.evemisslab.com/en/atlas/atlas-similarity/) ## [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) A mapping assigning exactly one output to each input in its domain. atlas-function-mapping · canonical_mko [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) ## [Domain, codomain, and range](https://ocme.evemisslab.com/en/atlas/atlas-domain-codomain-range/) 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. atlas-domain-codomain-range · canonical_mko [Domain, codomain, and range](https://ocme.evemisslab.com/en/mko/mko-domain-codomain-range/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) ## [Graphs of real-valued functions](https://ocme.evemisslab.com/en/atlas/atlas-function-graph/) 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. atlas-function-graph · canonical_mko [Graphs of real-valued functions](https://ocme.evemisslab.com/en/mko/mko-function-graph/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Coordinate plane](https://ocme.evemisslab.com/en/atlas/atlas-coordinate-plane/) ## [Affine functions and degree-one functions](https://ocme.evemisslab.com/en/atlas/atlas-linear-function/) A real function f(x)=mx+b is affine. If m≠0 it has degree one; if m=0 it is constant. atlas-linear-function · canonical_mko [Affine functions and degree-one functions](https://ocme.evemisslab.com/en/mko/mko-linear-function/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Linear equation in one variable](https://ocme.evemisslab.com/en/atlas/atlas-linear-equation/) ## [Quadratic functions in vertex form](https://ocme.evemisslab.com/en/atlas/atlas-quadratic-function/) 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. atlas-quadratic-function · canonical_mko [Quadratic functions in vertex form](https://ocme.evemisslab.com/en/mko/mko-quadratic-function/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Quadratic equations over the reals](https://ocme.evemisslab.com/en/atlas/atlas-quadratic-equation/) ## [Injectivity and distinct inputs](https://ocme.evemisslab.com/en/atlas/atlas-injective/) Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain. atlas-injective · canonical_mko [Injectivity and distinct inputs](https://ocme.evemisslab.com/en/mko/mko-injective/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Surjectivity and codomain coverage](https://ocme.evemisslab.com/en/atlas/atlas-surjective/) Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages. atlas-surjective · canonical_mko [Surjectivity and codomain coverage](https://ocme.evemisslab.com/en/mko/mko-surjective/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/atlas/atlas-bijective/) A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage. atlas-bijective · canonical_mko [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/mko/mko-bijective/) ## Prerequisite nodes - [Injectivity and distinct inputs](https://ocme.evemisslab.com/en/atlas/atlas-injective/) - [Surjectivity and codomain coverage](https://ocme.evemisslab.com/en/atlas/atlas-surjective/) ## [Function composition](https://ocme.evemisslab.com/en/atlas/atlas-composition/) The composition g∘f first evaluates f and then supplies its output to g; the intermediate value must be an allowed input of g. atlas-composition · canonical_mko [Function composition](https://ocme.evemisslab.com/en/mko/mko-composition/) ## Prerequisite nodes - [Domain, codomain, and range](https://ocme.evemisslab.com/en/atlas/atlas-domain-codomain-range/) ## [Inverse functions of bijections](https://ocme.evemisslab.com/en/atlas/atlas-inverse-function/) 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. atlas-inverse-function · canonical_mko [Inverse functions of bijections](https://ocme.evemisslab.com/en/mko/mko-inverse-function/) ## Prerequisite nodes - [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/atlas/atlas-bijective/) - [Function composition](https://ocme.evemisslab.com/en/atlas/atlas-composition/) - [Affine functions and degree-one functions](https://ocme.evemisslab.com/en/atlas/atlas-linear-function/) ## [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) 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. atlas-sequence · canonical_mko [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) ## [Tends-to relation](https://ocme.evemisslab.com/en/atlas/atlas-tends-to-relation/) A core relation describing a variable, sequence or function approaching a target state. atlas-tends-to-relation · canonical_mko [Tends-to relation](https://ocme.evemisslab.com/en/mko/mko-tends-to-relation/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) ## [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/atlas/atlas-sequence-limit/) 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. atlas-sequence-limit · canonical_mko [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) ## Prerequisite nodes - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Tends-to relation](https://ocme.evemisslab.com/en/atlas/atlas-tends-to-relation/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) - [Rational number](https://ocme.evemisslab.com/en/atlas/atlas-rational-number/) ## [Function limits on punctured real domains](https://ocme.evemisslab.com/en/atlas/atlas-function-limit/) An ε–δ limit controls values near a while excluding a itself; the center value is not part of the punctured-limit condition. atlas-function-limit · canonical_mko [Function limits on punctured real domains](https://ocme.evemisslab.com/en/mko/mko-function-limit/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Tends-to relation](https://ocme.evemisslab.com/en/atlas/atlas-tends-to-relation/) - [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) - [Polynomial factorization over the rationals](https://ocme.evemisslab.com/en/atlas/atlas-factorization/) ## [Point values and continuity of real functions](https://ocme.evemisslab.com/en/atlas/atlas-continuity/) Continuity on the whole real domain requires a defined center value agreeing with the two-sided limit; extension and original-function continuity are distinct. atlas-continuity · canonical_mko [Point values and continuity of real functions](https://ocme.evemisslab.com/en/mko/mko-continuity/) ## Prerequisite nodes - [Function limits on punctured real domains](https://ocme.evemisslab.com/en/atlas/atlas-function-limit/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Average rate of change between distinct inputs](https://ocme.evemisslab.com/en/atlas/atlas-rate-of-change/) Average rate of change is an endpoint difference quotient with a nonzero input difference, not an instantaneous derivative. atlas-rate-of-change · canonical_mko [Average rate of change between distinct inputs](https://ocme.evemisslab.com/en/mko/mko-rate-of-change/) ## Prerequisite nodes - [Ratio and proportion](https://ocme.evemisslab.com/en/atlas/atlas-ratio-proportion/) - [Graphs of real-valued functions](https://ocme.evemisslab.com/en/atlas/atlas-function-graph/) - [Affine functions and degree-one functions](https://ocme.evemisslab.com/en/atlas/atlas-linear-function/) - [Quadratic functions in vertex form](https://ocme.evemisslab.com/en/atlas/atlas-quadratic-function/) ## [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/atlas/atlas-derivative/) A derivative is the two-sided limit of nonzero-step quotients, not any one finite-step quotient; an absolute-value corner retains nonexistence. atlas-derivative · canonical_mko [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/mko/mko-derivative/) ## Prerequisite nodes - [Function limits on punctured real domains](https://ocme.evemisslab.com/en/atlas/atlas-function-limit/) - [Average rate of change between distinct inputs](https://ocme.evemisslab.com/en/atlas/atlas-rate-of-change/) - [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/atlas/atlas-polynomial/) - [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) ## [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/atlas/atlas-derivative-rules/) Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials. atlas-derivative-rules · canonical_mko [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/mko/mko-derivative-rules/) ## Prerequisite nodes - [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/atlas/atlas-derivative/) - [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/atlas/atlas-polynomial/) - [Function composition](https://ocme.evemisslab.com/en/atlas/atlas-composition/) ## [Polynomial antiderivatives and the choice of constant](https://ocme.evemisslab.com/en/atlas/atlas-antiderivative/) An antiderivative differentiates to f on its declared interval; this pulse selects a zero-constant rational-coefficient representative for full-real polynomials. atlas-antiderivative · canonical_mko [Polynomial antiderivatives and the choice of constant](https://ocme.evemisslab.com/en/mko/mko-antiderivative/) ## Prerequisite nodes - [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/atlas/atlas-derivative/) - [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/atlas/atlas-derivative-rules/) - [Rational number](https://ocme.evemisslab.com/en/atlas/atlas-rational-number/) ## [Exact area of half-open rectangles and unit-cell unions](https://ocme.evemisslab.com/en/atlas/atlas-area/) This entry computes exact area for half-open integer rectangles and finite unit-cell unions, including empty and overlapping cases. atlas-area · canonical_mko [Exact area of half-open rectangles and unit-cell unions](https://ocme.evemisslab.com/en/mko/mko-area/) ## Prerequisite nodes - [Euclidean length](https://ocme.evemisslab.com/en/atlas/atlas-euclidean-length/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) - [Coordinate plane](https://ocme.evemisslab.com/en/atlas/atlas-coordinate-plane/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) ## [Signed polynomial definite integrals on finite intervals](https://ocme.evemisslab.com/en/atlas/atlas-definite-integral/) 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. atlas-definite-integral · canonical_mko [Signed polynomial definite integrals on finite intervals](https://ocme.evemisslab.com/en/mko/mko-definite-integral/) ## Prerequisite nodes - [Function limits on punctured real domains](https://ocme.evemisslab.com/en/atlas/atlas-function-limit/) - [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/atlas/atlas-polynomial/) - [Polynomial antiderivatives and the choice of constant](https://ocme.evemisslab.com/en/atlas/atlas-antiderivative/) - [Exact area of half-open rectangles and unit-cell unions](https://ocme.evemisslab.com/en/atlas/atlas-area/) ## [Fundamental theorem of calculus: continuity and oriented endpoints](https://ocme.evemisslab.com/en/atlas/atlas-fundamental-theorem-calculus/) With continuity on all R, differentiation of the accumulating integral recovers the integrand; an eligible antiderivative gives an oriented endpoint difference. atlas-fundamental-theorem-calculus · canonical_mko [Fundamental theorem of calculus: continuity and oriented endpoints](https://ocme.evemisslab.com/en/mko/mko-fundamental-theorem-calculus/) ## Prerequisite nodes - [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/atlas/atlas-derivative/) - [Signed polynomial definite integrals on finite intervals](https://ocme.evemisslab.com/en/atlas/atlas-definite-integral/) - [Point values and continuity of real functions](https://ocme.evemisslab.com/en/atlas/atlas-continuity/) - [Polynomial antiderivatives and the choice of constant](https://ocme.evemisslab.com/en/atlas/atlas-antiderivative/) - [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/atlas/atlas-derivative-rules/) ## [Sample space](https://ocme.evemisslab.com/en/atlas/atlas-sample-space/) The set of all possible outcomes of a random experiment. atlas-sample-space · canonical_mko [Sample space](https://ocme.evemisslab.com/en/mko/mko-sample-space/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) ## [Event](https://ocme.evemisslab.com/en/atlas/atlas-event/) A set of outcomes in a sample space. atlas-event · canonical_mko [Event](https://ocme.evemisslab.com/en/mko/mko-event/) ## Prerequisite nodes - [Sample space](https://ocme.evemisslab.com/en/atlas/atlas-sample-space/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) ## [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) Assign nonnegative weights summing to one on a nonempty finite sample space. An event probability is the sum of the weights of its outcomes. atlas-probability · canonical_mko [Probability on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-probability/) ## Prerequisite nodes - [Event](https://ocme.evemisslab.com/en/atlas/atlas-event/) - [Ratio and proportion](https://ocme.evemisslab.com/en/atlas/atlas-ratio-proportion/) ## [Conditional probability in finite models](https://ocme.evemisslab.com/en/atlas/atlas-conditional-probability/) 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. atlas-conditional-probability · canonical_mko [Conditional probability in finite models](https://ocme.evemisslab.com/en/mko/mko-conditional-probability/) ## Prerequisite nodes - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) ## [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) 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. atlas-random-variable · canonical_mko [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-random-variable/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Sample space](https://ocme.evemisslab.com/en/atlas/atlas-sample-space/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) ## [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence. atlas-expectation · canonical_mko [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) ## Prerequisite nodes - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) ## [Variance in finite probability models](https://ocme.evemisslab.com/en/atlas/atlas-variance/) 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. atlas-variance · canonical_mko [Variance in finite probability models](https://ocme.evemisslab.com/en/mko/mko-variance/) ## Prerequisite nodes - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) ## [Descriptive summaries of nonempty observation lists](https://ocme.evemisslab.com/en/atlas/atlas-descriptive-statistics/) Mean, median, range, and the explicitly defined maximal-frequency set summarize a nonempty finite observation list while retaining multiplicities. atlas-descriptive-statistics · canonical_mko [Descriptive summaries of nonempty observation lists](https://ocme.evemisslab.com/en/mko/mko-descriptive-statistics/) ## Prerequisite nodes - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) - [Rational number](https://ocme.evemisslab.com/en/atlas/atlas-rational-number/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Basic counting principles](https://ocme.evemisslab.com/en/atlas/atlas-counting-principle/) Addition and multiplication principles for counting finite choices. atlas-counting-principle · canonical_mko [Basic counting principles](https://ocme.evemisslab.com/en/mko/mko-counting-principle/) ## Prerequisite nodes - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Factorial](https://ocme.evemisslab.com/en/atlas/atlas-factorial/) The function taking a natural number n to the product of the first n positive integers. atlas-factorial · canonical_mko [Factorial](https://ocme.evemisslab.com/en/mko/mko-factorial/) ## Prerequisite nodes - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Permutations and combinations](https://ocme.evemisslab.com/en/atlas/atlas-permutation-combination/) 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. atlas-permutation-combination · canonical_mko [Permutations and combinations](https://ocme.evemisslab.com/en/mko/mko-permutation-combination/) ## Prerequisite nodes - [Basic counting principles](https://ocme.evemisslab.com/en/atlas/atlas-counting-principle/) - [Factorial](https://ocme.evemisslab.com/en/atlas/atlas-factorial/) ## [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors. atlas-graph · canonical_mko [Finite simple undirected graphs](https://ocme.evemisslab.com/en/mko/mko-graph/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) ## [Nonempty finite trees: connectivity, cycles, and unique paths](https://ocme.evemisslab.com/en/atlas/atlas-tree/) A finite simple undirected graph is a tree only when nonempty, connected, and acyclic; retain singleton, degree-one leaf, and new-edge cycle conventions. atlas-tree · canonical_mko [Nonempty finite trees: connectivity, cycles, and unique paths](https://ocme.evemisslab.com/en/mko/mko-tree/) ## Prerequisite nodes - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) ## [Mathematical induction](https://ocme.evemisslab.com/en/atlas/atlas-mathematical-induction/) A method proving a natural-number-indexed family of propositions by a base case and an inductive step. atlas-mathematical-induction · canonical_mko [Mathematical induction](https://ocme.evemisslab.com/en/mko/mko-mathematical-induction/) ## Prerequisite nodes - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) - [Proposition](https://ocme.evemisslab.com/en/atlas/atlas-proposition/) ## [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) 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. atlas-topological-space · canonical_mko [Topological space](https://ocme.evemisslab.com/en/mko/mko-topological-space/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) ## [Open and closed sets](https://ocme.evemisslab.com/en/atlas/atlas-open-closed-set/) 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. atlas-open-closed-set · canonical_mko [Open and closed sets](https://ocme.evemisslab.com/en/mko/mko-open-closed-set/) ## Prerequisite nodes - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) ## [Neighborhood](https://ocme.evemisslab.com/en/atlas/atlas-neighborhood/) A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open. atlas-neighborhood · canonical_mko [Neighborhood](https://ocme.evemisslab.com/en/mko/mko-neighborhood/) ## Prerequisite nodes - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Open and closed sets](https://ocme.evemisslab.com/en/atlas/atlas-open-closed-set/) - [Set membership](https://ocme.evemisslab.com/en/atlas/atlas-set-membership/) ## [Continuous maps between topological spaces](https://ocme.evemisslab.com/en/atlas/atlas-topological-continuity/) A map is topologically continuous when every open set in its target has an open preimage in its source. atlas-topological-continuity · canonical_mko [Continuous maps between topological spaces](https://ocme.evemisslab.com/en/mko/mko-topological-continuity/) ## Prerequisite nodes - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Open and closed sets](https://ocme.evemisslab.com/en/atlas/atlas-open-closed-set/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Natural-number divisibility](https://ocme.evemisslab.com/en/atlas/atlas-natural-divisibility/) A natural number a divides b when a natural multiplier k satisfies b=a·k; the two roles of zero must be distinguished. atlas-natural-divisibility · canonical_mko [Natural-number divisibility](https://ocme.evemisslab.com/en/mko/mko-natural-divisibility/) ## Prerequisite nodes - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/atlas/atlas-gcd-coprime/) 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. atlas-gcd-coprime · canonical_mko [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/mko/mko-gcd-coprime/) ## Prerequisite nodes - [Natural-number divisibility](https://ocme.evemisslab.com/en/atlas/atlas-natural-divisibility/) ## [Prime natural number](https://ocme.evemisslab.com/en/atlas/atlas-prime-number/) A prime natural number is at least 2 and has no positive divisors other than 1 and itself. atlas-prime-number · canonical_mko [Prime natural number](https://ocme.evemisslab.com/en/mko/mko-prime-number/) ## Prerequisite nodes - [Natural-number divisibility](https://ocme.evemisslab.com/en/atlas/atlas-natural-divisibility/) ## [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/atlas/atlas-modular-congruence/) 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. atlas-modular-congruence · canonical_mko [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/mko/mko-modular-congruence/) ## Prerequisite nodes - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Decision problem](https://ocme.evemisslab.com/en/atlas/atlas-decision-problem/) 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. atlas-decision-problem · canonical_mko [Decision problem](https://ocme.evemisslab.com/en/mko/mko-decision-problem/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Proposition](https://ocme.evemisslab.com/en/atlas/atlas-proposition/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Abstract computational model](https://ocme.evemisslab.com/en/atlas/atlas-computational-model/) A finite-state computational model describes possible executions through its state set, initial-state set, step relation and halt-state set. atlas-computational-model · canonical_mko [Abstract computational model](https://ocme.evemisslab.com/en/mko/mko-computational-model/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) ## [Algorithm correctness](https://ocme.evemisslab.com/en/atlas/atlas-algorithm-correctness/) Algorithm correctness separates partial correctness from termination; total correctness requires both. atlas-algorithm-correctness · canonical_mko [Algorithm correctness](https://ocme.evemisslab.com/en/mko/mko-algorithm-correctness/) ## Prerequisite nodes - [Abstract computational model](https://ocme.evemisslab.com/en/atlas/atlas-computational-model/) - [Proposition](https://ocme.evemisslab.com/en/atlas/atlas-proposition/) - [Mathematical proof](https://ocme.evemisslab.com/en/atlas/atlas-proof/) ## [Asymptotic resource bounds](https://ocme.evemisslab.com/en/atlas/atlas-asymptotic-resource-bound/) Asymptotic upper, lower and tight bounds require an input-size convention, cost model and quantification over a full tail. atlas-asymptotic-resource-bound · canonical_mko [Asymptotic resource bounds](https://ocme.evemisslab.com/en/mko/mko-asymptotic-resource-bound/) ## Prerequisite nodes - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Formal statements in a proof assistant](https://ocme.evemisslab.com/en/atlas/atlas-formal-statement/) A formal statement is a core term of type Prop obtained by successful elaboration in a fixed version, import environment and local context. atlas-formal-statement · canonical_mko [Formal statements in a proof assistant](https://ocme.evemisslab.com/en/mko/mko-lean-formal-statement/) ## Prerequisite nodes - [Proposition](https://ocme.evemisslab.com/en/atlas/atlas-proposition/) ## [Proof term](https://ocme.evemisslab.com/en/atlas/atlas-proof-term/) A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel. atlas-proof-term · canonical_mko [Proof term](https://ocme.evemisslab.com/en/mko/mko-lean-proof-term/) ## Prerequisite nodes - [Formal statements in a proof assistant](https://ocme.evemisslab.com/en/atlas/atlas-formal-statement/) - [Mathematical proof](https://ocme.evemisslab.com/en/atlas/atlas-proof/) ## [Discrete-time dynamical system](https://ocme.evemisslab.com/en/atlas/atlas-discrete-dynamical-system/) 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. atlas-discrete-dynamical-system · canonical_mko [Discrete-time dynamical system](https://ocme.evemisslab.com/en/mko/mko-discrete-dynamical-system/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) ## [Orbits, fixed points and periodic points](https://ocme.evemisslab.com/en/atlas/atlas-orbit-fixed-periodic/) 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. atlas-orbit-fixed-periodic · canonical_mko [Orbits, fixed points and periodic points](https://ocme.evemisslab.com/en/mko/mko-orbit-fixed-periodic/) ## Prerequisite nodes - [Discrete-time dynamical system](https://ocme.evemisslab.com/en/atlas/atlas-discrete-dynamical-system/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) ## [Ordinary differential equation](https://ocme.evemisslab.com/en/atlas/atlas-ordinary-differential-equation/) An ordinary differential equation relates an unknown single-variable function and finitely many derivatives; candidate residual checks are not existence or uniqueness theorems. atlas-ordinary-differential-equation · canonical_mko [Ordinary differential equation](https://ocme.evemisslab.com/en/mko/mko-ordinary-differential-equation/) ## Prerequisite nodes - [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/atlas/atlas-derivative/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) ## [Initial-value problem](https://ocme.evemisslab.com/en/atlas/atlas-initial-value-problem/) 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. atlas-initial-value-problem · canonical_mko [Initial-value problem](https://ocme.evemisslab.com/en/mko/mko-initial-value-problem/) ## Prerequisite nodes - [Ordinary differential equation](https://ocme.evemisslab.com/en/atlas/atlas-ordinary-differential-equation/) - [Domain, codomain, and range](https://ocme.evemisslab.com/en/atlas/atlas-domain-codomain-range/) ## [Numerical approximation and error](https://ocme.evemisslab.com/en/atlas/atlas-numerical-approximation-error/) Absolute error is the absolute difference between approximation and reference; relative error additionally requires a nonzero reference and divides by its absolute value. atlas-numerical-approximation-error · canonical_mko [Numerical approximation and error](https://ocme.evemisslab.com/en/mko/mko-numerical-approximation-error/) ## Prerequisite nodes - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) - [Ratio and proportion](https://ocme.evemisslab.com/en/atlas/atlas-ratio-proportion/) ## [Finite sign-bracket updates in the bisection method](https://ocme.evemisslab.com/en/atlas/atlas-bisection-method/) 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. atlas-bisection-method · canonical_mko [Finite sign-bracket updates in the bisection method](https://ocme.evemisslab.com/en/mko/mko-bisection-method/) ## Prerequisite nodes - [Numerical approximation and error](https://ocme.evemisslab.com/en/atlas/atlas-numerical-approximation-error/) - [Point values and continuity of real functions](https://ocme.evemisslab.com/en/atlas/atlas-continuity/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) ## [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects. atlas-optimization-problem · canonical_mko [Optimization problem](https://ocme.evemisslab.com/en/mko/mko-optimization-problem/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Linear programming](https://ocme.evemisslab.com/en/atlas/atlas-linear-programming/) 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. atlas-linear-programming · canonical_mko [Linear programming](https://ocme.evemisslab.com/en/mko/mko-linear-programming/) ## Prerequisite nodes - [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) - [Linear equation in one variable](https://ocme.evemisslab.com/en/atlas/atlas-linear-equation/) - [Linear systems in two variables and degenerate constraints](https://ocme.evemisslab.com/en/atlas/atlas-system-linear-equations/) - [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/atlas/atlas-vector/) ## [Dimensional analysis](https://ocme.evemisslab.com/en/atlas/atlas-dimensional-analysis/) Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides. atlas-dimensional-analysis · canonical_mko [Dimensional analysis](https://ocme.evemisslab.com/en/mko/mko-dimensional-analysis/) ## Prerequisite nodes - [Ratio and proportion](https://ocme.evemisslab.com/en/atlas/atlas-ratio-proportion/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) ## [Linear harmonic-oscillator model](https://ocme.evemisslab.com/en/atlas/atlas-harmonic-oscillator-model/) 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. atlas-harmonic-oscillator-model · canonical_mko [Linear harmonic-oscillator model](https://ocme.evemisslab.com/en/mko/mko-harmonic-oscillator-model/) ## Prerequisite nodes - [Ordinary differential equation](https://ocme.evemisslab.com/en/atlas/atlas-ordinary-differential-equation/) - [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/atlas/atlas-derivative-rules/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) ## [Discrete exponential growth model](https://ocme.evemisslab.com/en/atlas/atlas-discrete-exponential-growth/) 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. atlas-discrete-exponential-growth · canonical_mko [Discrete exponential growth model](https://ocme.evemisslab.com/en/mko/mko-discrete-exponential-growth/) ## Prerequisite nodes - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Discrete logistic increment model](https://ocme.evemisslab.com/en/atlas/atlas-discrete-logistic-growth/) 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. atlas-discrete-logistic-growth · canonical_mko [Discrete logistic increment model](https://ocme.evemisslab.com/en/mko/mko-discrete-logistic-growth/) ## Prerequisite nodes - [Discrete exponential growth model](https://ocme.evemisslab.com/en/atlas/atlas-discrete-exponential-growth/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Compound-interest model](https://ocme.evemisslab.com/en/atlas/atlas-compound-interest/) 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. atlas-compound-interest · canonical_mko [Compound-interest model](https://ocme.evemisslab.com/en/mko/mko-compound-interest/) ## Prerequisite nodes - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) - [Ratio and proportion](https://ocme.evemisslab.com/en/atlas/atlas-ratio-proportion/) ## [Present value and discounting](https://ocme.evemisslab.com/en/atlas/atlas-present-value-discounting/) For a fixed positive accumulation factor and period count, divide a future amount by the factor power to obtain present value. atlas-present-value-discounting · canonical_mko [Present value and discounting](https://ocme.evemisslab.com/en/mko/mko-present-value-discounting/) ## Prerequisite nodes - [Compound-interest model](https://ocme.evemisslab.com/en/atlas/atlas-compound-interest/) - [Fraction](https://ocme.evemisslab.com/en/atlas/atlas-fraction/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) ## [Mathematical representation](https://ocme.evemisslab.com/en/atlas/atlas-mathematical-representation/) 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. atlas-mathematical-representation · canonical_mko [Mathematical representation](https://ocme.evemisslab.com/en/mko/mko-mathematical-representation/) ## Prerequisite nodes - [Variables and algebraic expressions](https://ocme.evemisslab.com/en/atlas/atlas-variable-expression/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Coordinate plane](https://ocme.evemisslab.com/en/atlas/atlas-coordinate-plane/) ## [Diagnostic mathematical assessment records](https://ocme.evemisslab.com/en/atlas/atlas-diagnostic-mathematical-assessment/) 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. atlas-diagnostic-mathematical-assessment · canonical_mko [Diagnostic mathematical assessment records](https://ocme.evemisslab.com/en/mko/mko-diagnostic-mathematical-assessment/) ## Prerequisite nodes - [Proposition](https://ocme.evemisslab.com/en/atlas/atlas-proposition/) - [Counterexamples and domain eligibility](https://ocme.evemisslab.com/en/atlas/atlas-counterexample/) - [Mathematical representation](https://ocme.evemisslab.com/en/atlas/atlas-mathematical-representation/) ## [Complex number](https://ocme.evemisslab.com/en/atlas/atlas-complex-number/) A complex number a+b*i is represented by the ordered real pair (a,b), with addition, subtraction and multiplication defined using i²=−1. atlas-complex-number · canonical_mko [Complex number](https://ocme.evemisslab.com/en/mko/mko-complex-number/) ## Prerequisite nodes - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Coordinate plane](https://ocme.evemisslab.com/en/atlas/atlas-coordinate-plane/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Complex modulus and conjugation](https://ocme.evemisslab.com/en/atlas/atlas-complex-modulus-conjugate/) Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares. atlas-complex-modulus-conjugate · canonical_mko [Complex modulus and conjugation](https://ocme.evemisslab.com/en/mko/mko-complex-modulus-conjugate/) ## Prerequisite nodes - [Complex number](https://ocme.evemisslab.com/en/atlas/atlas-complex-number/) - [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) ## [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) A real vector space has vector addition and real scalar multiplication, satisfying the additive abelian-group and compatible scalar-action axioms. atlas-real-vector-space · canonical_mko [Real vector space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) ## Prerequisite nodes - [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/atlas/atlas-vector/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Linear combinations and span](https://ocme.evemisslab.com/en/atlas/atlas-linear-combination-span/) A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination. atlas-linear-combination-span · canonical_mko [Linear combinations and span](https://ocme.evemisslab.com/en/mko/mko-linear-combination-span/) ## Prerequisite nodes - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) - [Linear systems in two variables and degenerate constraints](https://ocme.evemisslab.com/en/atlas/atlas-system-linear-equations/) ## [Linear independence and bases](https://ocme.evemisslab.com/en/atlas/atlas-linear-independence-basis/) Linear independence excludes nontrivial zero-sum coefficient relations; a basis is both independent and spanning for the declared space. atlas-linear-independence-basis · canonical_mko [Linear independence and bases](https://ocme.evemisslab.com/en/mko/mko-linear-independence-basis/) ## Prerequisite nodes - [Linear combinations and span](https://ocme.evemisslab.com/en/atlas/atlas-linear-combination-span/) - [Linear systems in two variables and degenerate constraints](https://ocme.evemisslab.com/en/atlas/atlas-system-linear-equations/) - [Determinants of real 2×2 matrices](https://ocme.evemisslab.com/en/atlas/atlas-determinant/) ## [Finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group/) A finite group has a finite carrier and complete closed Cayley table, satisfying associativity, a unique identity and inverses for every element. atlas-finite-group · canonical_mko [Finite group](https://ocme.evemisslab.com/en/mko/mko-finite-group/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) - [Mathematical proof](https://ocme.evemisslab.com/en/atlas/atlas-proof/) ## [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/atlas/atlas-subgroup-homomorphism/) 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. atlas-subgroup-homomorphism · canonical_mko [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/mko/mko-subgroup-homomorphism/) ## Prerequisite nodes - [Finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Metric space](https://ocme.evemisslab.com/en/atlas/atlas-metric-space/) 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. atlas-metric-space · canonical_mko [Metric space](https://ocme.evemisslab.com/en/mko/mko-metric-space/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Convergence in a metric space](https://ocme.evemisslab.com/en/atlas/atlas-metric-convergence/) 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. atlas-metric-convergence · canonical_mko [Convergence in a metric space](https://ocme.evemisslab.com/en/mko/mko-metric-convergence/) ## Prerequisite nodes - [Metric space](https://ocme.evemisslab.com/en/atlas/atlas-metric-space/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/atlas/atlas-sigma-algebra-measure-space/) 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. atlas-sigma-algebra-measure-space · canonical_mko [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/mko/mko-sigma-algebra-measure-space/) ## Prerequisite nodes - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) ## [Measurable function](https://ocme.evemisslab.com/en/atlas/atlas-measurable-function/) 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. atlas-measurable-function · canonical_mko [Measurable function](https://ocme.evemisslab.com/en/mko/mko-measurable-function/) ## Prerequisite nodes - [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/atlas/atlas-sigma-algebra-measure-space/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) ## [Finite discrete probability distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-discrete-probability-distribution/) A finite discrete distribution assigns nonnegative probability masses to finitely many outcome values, with exact total mass one. atlas-finite-discrete-probability-distribution · canonical_mko [Finite discrete probability distribution](https://ocme.evemisslab.com/en/mko/mko-finite-discrete-probability-distribution/) ## Prerequisite nodes - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Discrete-time stochastic process](https://ocme.evemisslab.com/en/atlas/atlas-discrete-stochastic-process/) 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. atlas-discrete-stochastic-process · canonical_mko [Discrete-time stochastic process](https://ocme.evemisslab.com/en/mko/mko-discrete-stochastic-process/) ## Prerequisite nodes - [Finite discrete probability distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-discrete-probability-distribution/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) ## [Finite-state Markov chain](https://ocme.evemisslab.com/en/atlas/atlas-finite-markov-chain/) A finite time-homogeneous Markov chain uses a row-stochastic transition kernel and, when needed, an initial law to specify finite path probabilities. atlas-finite-markov-chain · canonical_mko [Finite-state Markov chain](https://ocme.evemisslab.com/en/mko/mko-finite-markov-chain/) ## Prerequisite nodes - [Discrete-time stochastic process](https://ocme.evemisslab.com/en/atlas/atlas-discrete-stochastic-process/) - [Finite matrices: dimensions, indexing, and transpose](https://ocme.evemisslab.com/en/atlas/atlas-matrix/) - [Conditional probability in finite models](https://ocme.evemisslab.com/en/atlas/atlas-conditional-probability/) ## [Multivariable real-valued function](https://ocme.evemisslab.com/en/atlas/atlas-multivariable-real-function/) A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data. atlas-multivariable-real-function · canonical_mko [Multivariable real-valued function](https://ocme.evemisslab.com/en/mko/mko-multivariable-real-function/) ## Prerequisite nodes - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/atlas/atlas-vector/) - [Coordinate plane](https://ocme.evemisslab.com/en/atlas/atlas-coordinate-plane/) ## [Partial derivative](https://ocme.evemisslab.com/en/atlas/atlas-partial-derivative/) A partial derivative fixes every other coordinate and takes the one-variable derivative along one coordinate slice as h tends to zero. atlas-partial-derivative · canonical_mko [Partial derivative](https://ocme.evemisslab.com/en/mko/mko-partial-derivative/) ## Prerequisite nodes - [Multivariable real-valued function](https://ocme.evemisslab.com/en/atlas/atlas-multivariable-real-function/) - [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/atlas/atlas-derivative/) - [Average rate of change between distinct inputs](https://ocme.evemisslab.com/en/atlas/atlas-rate-of-change/) ## [Partial differential equation](https://ocme.evemisslab.com/en/atlas/atlas-partial-differential-equation/) 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. atlas-partial-differential-equation · canonical_mko [Partial differential equation](https://ocme.evemisslab.com/en/mko/mko-partial-differential-equation/) ## Prerequisite nodes - [Partial derivative](https://ocme.evemisslab.com/en/atlas/atlas-partial-derivative/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) - [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/atlas/atlas-derivative-rules/) ## [Boundary-value problem](https://ocme.evemisslab.com/en/atlas/atlas-boundary-value-problem/) 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. atlas-boundary-value-problem · canonical_mko [Boundary-value problem](https://ocme.evemisslab.com/en/mko/mko-boundary-value-problem/) ## Prerequisite nodes - [Partial differential equation](https://ocme.evemisslab.com/en/atlas/atlas-partial-differential-equation/) - [Domain, codomain, and range](https://ocme.evemisslab.com/en/atlas/atlas-domain-codomain-range/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) ## [Finite convex combination](https://ocme.evemisslab.com/en/atlas/atlas-finite-convex-combination/) A finite convex combination weights a finite nonempty ordered vector list by exact nonnegative weights whose total is exactly one. atlas-finite-convex-combination · canonical_mko [Finite convex combination](https://ocme.evemisslab.com/en/mko/mko-finite-convex-combination/) ## Prerequisite nodes - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) - [Ratio and proportion](https://ocme.evemisslab.com/en/atlas/atlas-ratio-proportion/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Convex set](https://ocme.evemisslab.com/en/atlas/atlas-convex-set/) 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. atlas-convex-set · canonical_mko [Convex set](https://ocme.evemisslab.com/en/mko/mko-convex-set/) ## Prerequisite nodes - [Finite convex combination](https://ocme.evemisslab.com/en/atlas/atlas-finite-convex-combination/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) ## [Convex function](https://ocme.evemisslab.com/en/atlas/atlas-convex-function/) 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. atlas-convex-function · canonical_mko [Convex function](https://ocme.evemisslab.com/en/mko/mko-convex-function/) ## Prerequisite nodes - [Convex set](https://ocme.evemisslab.com/en/atlas/atlas-convex-set/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Convex optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-convex-optimization-problem/) Convex optimization minimizes a convex objective on a convex feasible domain; problem qualification, feasibility, evaluation and global optimality are distinct obligations. atlas-convex-optimization-problem · canonical_mko [Convex optimization problem](https://ocme.evemisslab.com/en/mko/mko-convex-optimization-problem/) ## Prerequisite nodes - [Convex function](https://ocme.evemisslab.com/en/atlas/atlas-convex-function/) - [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) ## [Finite abelian group](https://ocme.evemisslab.com/en/atlas/atlas-finite-abelian-group/) A finite group is abelian when its complete Cayley table satisfies commutativity for every pair of elements. atlas-finite-abelian-group · canonical_mko [Finite abelian group](https://ocme.evemisslab.com/en/mko/mko-finite-abelian-group/) ## Prerequisite nodes - [Finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Finite unital ring](https://ocme.evemisslab.com/en/atlas/atlas-finite-unital-ring/) 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. atlas-finite-unital-ring · canonical_mko [Finite unital ring](https://ocme.evemisslab.com/en/mko/mko-finite-unital-ring/) ## Prerequisite nodes - [Finite abelian group](https://ocme.evemisslab.com/en/atlas/atlas-finite-abelian-group/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Finite commutative ring](https://ocme.evemisslab.com/en/atlas/atlas-finite-commutative-ring/) A finite unital ring is commutative when multiplication commutes for every pair of elements; zero divisors may still exist. atlas-finite-commutative-ring · canonical_mko [Finite commutative ring](https://ocme.evemisslab.com/en/mko/mko-finite-commutative-ring/) ## Prerequisite nodes - [Finite unital ring](https://ocme.evemisslab.com/en/atlas/atlas-finite-unital-ring/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Finite integral domain](https://ocme.evemisslab.com/en/atlas/atlas-finite-integral-domain/) A finite commutative unital ring with 1 distinct from 0 is an integral domain when it has no nonzero zero divisors. atlas-finite-integral-domain · canonical_mko [Finite integral domain](https://ocme.evemisslab.com/en/mko/mko-finite-integral-domain/) ## Prerequisite nodes - [Finite commutative ring](https://ocme.evemisslab.com/en/atlas/atlas-finite-commutative-ring/) - [Proposition](https://ocme.evemisslab.com/en/atlas/atlas-proposition/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Finite field](https://ocme.evemisslab.com/en/atlas/atlas-finite-field/) A finite commutative unital ring is a finite field when every nonzero element has a multiplicative inverse. atlas-finite-field · canonical_mko [Finite field](https://ocme.evemisslab.com/en/mko/mko-finite-field/) ## Prerequisite nodes - [Finite integral domain](https://ocme.evemisslab.com/en/atlas/atlas-finite-integral-domain/) - [Finite unital ring](https://ocme.evemisslab.com/en/atlas/atlas-finite-unital-ring/) - [Finite abelian group](https://ocme.evemisslab.com/en/atlas/atlas-finite-abelian-group/) ## [Finite two-sided ideal](https://ocme.evemisslab.com/en/atlas/atlas-finite-two-sided-ideal/) 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. atlas-finite-two-sided-ideal · canonical_mko [Finite two-sided ideal](https://ocme.evemisslab.com/en/mko/mko-finite-two-sided-ideal/) ## Prerequisite nodes - [Finite unital ring](https://ocme.evemisslab.com/en/atlas/atlas-finite-unital-ring/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) - [Finite abelian group](https://ocme.evemisslab.com/en/atlas/atlas-finite-abelian-group/) ## [Quotient ring of a finite ring](https://ocme.evemisslab.com/en/atlas/atlas-finite-quotient-ring/) Additive cosets of a proper two-sided ideal in a finite ring form a quotient ring when its operations are independent of representative choice. atlas-finite-quotient-ring · canonical_mko [Quotient ring of a finite ring](https://ocme.evemisslab.com/en/mko/mko-finite-quotient-ring/) ## Prerequisite nodes - [Finite unital ring](https://ocme.evemisslab.com/en/atlas/atlas-finite-unital-ring/) - [Finite two-sided ideal](https://ocme.evemisslab.com/en/atlas/atlas-finite-two-sided-ideal/) - [Equivalence relation](https://ocme.evemisslab.com/en/atlas/atlas-equivalence-relation/) - [Quotient set](https://ocme.evemisslab.com/en/atlas/atlas-quotient-set/) ## [Polynomial ring over a finite commutative ring](https://ocme.evemisslab.com/en/atlas/atlas-polynomial-ring-over-finite-commutative-ring/) Ordinary polynomials over a finite commutative coefficient ring have finite-support coefficient sequences, but the collection over all finite degrees is still infinite. atlas-polynomial-ring-over-finite-commutative-ring · canonical_mko [Polynomial ring over a finite commutative ring](https://ocme.evemisslab.com/en/mko/mko-polynomial-ring-over-finite-commutative-ring/) ## Prerequisite nodes - [Finite commutative ring](https://ocme.evemisslab.com/en/atlas/atlas-finite-commutative-ring/) - [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/atlas/atlas-polynomial/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) ## [Real normed vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-normed-vector-space/) A real vector space equipped with a positive-definite, absolutely homogeneous real-valued norm satisfying the triangle inequality. atlas-real-normed-vector-space · canonical_mko [Real normed vector space](https://ocme.evemisslab.com/en/mko/mko-real-normed-vector-space/) ## Prerequisite nodes - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) - [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Norm-induced metric](https://ocme.evemisslab.com/en/atlas/atlas-norm-induced-metric/) 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. atlas-norm-induced-metric · canonical_mko [Norm-induced metric](https://ocme.evemisslab.com/en/mko/mko-norm-induced-metric/) ## Prerequisite nodes - [Real normed vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-normed-vector-space/) - [Metric space](https://ocme.evemisslab.com/en/atlas/atlas-metric-space/) - [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/atlas/atlas-vector/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Cauchy sequence](https://ocme.evemisslab.com/en/atlas/atlas-cauchy-sequence/) A sequence in a metric space is Cauchy if, for every positive epsilon, all pairs of sufficiently late terms are less than epsilon apart. atlas-cauchy-sequence · canonical_mko [Cauchy sequence](https://ocme.evemisslab.com/en/mko/mko-cauchy-sequence/) ## Prerequisite nodes - [Metric space](https://ocme.evemisslab.com/en/atlas/atlas-metric-space/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Complete metric space](https://ocme.evemisslab.com/en/atlas/atlas-complete-metric-space/) A metric space is complete when every Cauchy sequence converges to a point still belonging to that space. atlas-complete-metric-space · canonical_mko [Complete metric space](https://ocme.evemisslab.com/en/mko/mko-complete-metric-space/) ## Prerequisite nodes - [Cauchy sequence](https://ocme.evemisslab.com/en/atlas/atlas-cauchy-sequence/) - [Convergence in a metric space](https://ocme.evemisslab.com/en/atlas/atlas-metric-convergence/) - [Metric space](https://ocme.evemisslab.com/en/atlas/atlas-metric-space/) ## [Real Banach space](https://ocme.evemisslab.com/en/atlas/atlas-real-banach-space/) A real normed vector space is a Banach space when it is complete in the metric induced by its own norm. atlas-real-banach-space · canonical_mko [Real Banach space](https://ocme.evemisslab.com/en/mko/mko-real-banach-space/) ## Prerequisite nodes - [Real normed vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-normed-vector-space/) - [Norm-induced metric](https://ocme.evemisslab.com/en/atlas/atlas-norm-induced-metric/) - [Complete metric space](https://ocme.evemisslab.com/en/atlas/atlas-complete-metric-space/) - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) ## [Real bilinear form](https://ocme.evemisslab.com/en/atlas/atlas-real-bilinear-form/) A bilinear form on a real vector space preserves addition and real scalar multiplication separately in each input. atlas-real-bilinear-form · canonical_mko [Real bilinear form](https://ocme.evemisslab.com/en/mko/mko-real-bilinear-form/) ## Prerequisite nodes - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) - [Linear combinations and span](https://ocme.evemisslab.com/en/atlas/atlas-linear-combination-span/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Real inner-product space](https://ocme.evemisslab.com/en/atlas/atlas-real-inner-product-space/) A symmetric, positive-definite real bilinear form is a real inner product; positive semidefiniteness or positive sampled values do not suffice. atlas-real-inner-product-space · canonical_mko [Real inner-product space](https://ocme.evemisslab.com/en/mko/mko-real-inner-product-space/) ## Prerequisite nodes - [Real bilinear form](https://ocme.evemisslab.com/en/atlas/atlas-real-bilinear-form/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) ## [Inner-product-induced norm](https://ocme.evemisslab.com/en/atlas/atlas-inner-product-induced-norm/) 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. atlas-inner-product-induced-norm · canonical_mko [Inner-product-induced norm](https://ocme.evemisslab.com/en/mko/mko-inner-product-induced-norm/) ## Prerequisite nodes - [Real inner-product space](https://ocme.evemisslab.com/en/atlas/atlas-real-inner-product-space/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Real Hilbert space](https://ocme.evemisslab.com/en/atlas/atlas-real-hilbert-space/) A real inner-product space is Hilbert when it is complete in the norm and metric induced by that same inner product. atlas-real-hilbert-space · canonical_mko [Real Hilbert space](https://ocme.evemisslab.com/en/mko/mko-real-hilbert-space/) ## Prerequisite nodes - [Real inner-product space](https://ocme.evemisslab.com/en/atlas/atlas-real-inner-product-space/) - [Inner-product-induced norm](https://ocme.evemisslab.com/en/atlas/atlas-inner-product-induced-norm/) - [Norm-induced metric](https://ocme.evemisslab.com/en/atlas/atlas-norm-induced-metric/) - [Complete metric space](https://ocme.evemisslab.com/en/atlas/atlas-complete-metric-space/) ## [Linear map between real vector spaces](https://ocme.evemisslab.com/en/atlas/atlas-general-linear-map/) A map between real vector spaces is linear when it preserves linear combinations for all vectors and real scalars. atlas-general-linear-map · canonical_mko [Linear map between real vector spaces](https://ocme.evemisslab.com/en/mko/mko-general-linear-map/) ## Prerequisite nodes - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Linear combinations and span](https://ocme.evemisslab.com/en/atlas/atlas-linear-combination-span/) ## [Bounded linear operator](https://ocme.evemisslab.com/en/atlas/atlas-bounded-linear-operator/) A linear map between normed spaces is bounded when one nonnegative constant uniformly controls all output norms relative to input norms. atlas-bounded-linear-operator · canonical_mko [Bounded linear operator](https://ocme.evemisslab.com/en/mko/mko-bounded-linear-operator/) ## Prerequisite nodes - [Linear map between real vector spaces](https://ocme.evemisslab.com/en/atlas/atlas-general-linear-map/) - [Real normed vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-normed-vector-space/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Event independence](https://ocme.evemisslab.com/en/atlas/atlas-event-independence/) 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. atlas-event-independence · canonical_mko [Event independence](https://ocme.evemisslab.com/en/mko/mko-event-independence/) ## Prerequisite nodes - [Event](https://ocme.evemisslab.com/en/atlas/atlas-event/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) - [Conditional probability in finite models](https://ocme.evemisslab.com/en/atlas/atlas-conditional-probability/) ## [Finite joint distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-joint-distribution/) Value tuples of random variables on one common finite probability space, together with exact masses, specify a finite joint distribution. atlas-finite-joint-distribution · canonical_mko [Finite joint distribution](https://ocme.evemisslab.com/en/mko/mko-finite-joint-distribution/) ## Prerequisite nodes - [Finite discrete probability distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-discrete-probability-distribution/) - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) - [Sample space](https://ocme.evemisslab.com/en/atlas/atlas-sample-space/) ## [Independence of random variables](https://ocme.evemisslab.com/en/atlas/atlas-random-variable-independence/) Under one finite joint law, independence requires every value-pair mass to factor as the product of its marginal masses. atlas-random-variable-independence · canonical_mko [Independence of random variables](https://ocme.evemisslab.com/en/mko/mko-random-variable-independence/) ## Prerequisite nodes - [Event independence](https://ocme.evemisslab.com/en/atlas/atlas-event-independence/) - [Finite joint distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-joint-distribution/) - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) - [Measurable function](https://ocme.evemisslab.com/en/atlas/atlas-measurable-function/) ## [Covariance](https://ocme.evemisslab.com/en/atlas/atlas-covariance/) Under one finite joint distribution, covariance is E[XY]−E[X]E[Y]; zero covariance does not in general imply independence. atlas-covariance · canonical_mko [Covariance](https://ocme.evemisslab.com/en/mko/mko-covariance/) ## Prerequisite nodes - [Finite joint distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-joint-distribution/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Variance in finite probability models](https://ocme.evemisslab.com/en/atlas/atlas-variance/) - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) ## [Finite conditional expectation](https://ocme.evemisslab.com/en/atlas/atlas-conditional-expectation/) 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. atlas-conditional-expectation · canonical_mko [Finite conditional expectation](https://ocme.evemisslab.com/en/mko/mko-conditional-expectation/) ## Prerequisite nodes - [Finite joint distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-joint-distribution/) - [Measurable function](https://ocme.evemisslab.com/en/atlas/atlas-measurable-function/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Conditional probability in finite models](https://ocme.evemisslab.com/en/atlas/atlas-conditional-probability/) - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) ## [Finite filtration](https://ocme.evemisslab.com/en/atlas/atlas-finite-filtration/) A finite-time filtration is a sequence of sigma-algebras on the same probability space whose information increases with time. atlas-finite-filtration · canonical_mko [Finite filtration](https://ocme.evemisslab.com/en/mko/mko-finite-filtration/) ## Prerequisite nodes - [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/atlas/atlas-sigma-algebra-measure-space/) - [Discrete-time stochastic process](https://ocme.evemisslab.com/en/atlas/atlas-discrete-stochastic-process/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Measurable function](https://ocme.evemisslab.com/en/atlas/atlas-measurable-function/) ## [Finite martingale](https://ocme.evemisslab.com/en/atlas/atlas-martingale/) 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. atlas-martingale · canonical_mko [Finite martingale](https://ocme.evemisslab.com/en/mko/mko-martingale/) ## Prerequisite nodes - [Finite filtration](https://ocme.evemisslab.com/en/atlas/atlas-finite-filtration/) - [Finite conditional expectation](https://ocme.evemisslab.com/en/atlas/atlas-conditional-expectation/) - [Discrete-time stochastic process](https://ocme.evemisslab.com/en/atlas/atlas-discrete-stochastic-process/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Measurable function](https://ocme.evemisslab.com/en/atlas/atlas-measurable-function/) ## [Convergence in probability](https://ocme.evemisslab.com/en/atlas/atlas-convergence-in-probability/) X_n converges in probability to X when, for every epsilon>0, the probability of deviation greater than epsilon tends to zero. atlas-convergence-in-probability · canonical_mko [Convergence in probability](https://ocme.evemisslab.com/en/mko/mko-convergence-in-probability/) ## Prerequisite nodes - [Finite joint distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-joint-distribution/) - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Convergence in a metric space](https://ocme.evemisslab.com/en/atlas/atlas-metric-convergence/) ## [Cumulative distribution function](https://ocme.evemisslab.com/en/atlas/atlas-cumulative-distribution-function/) 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. atlas-cumulative-distribution-function · canonical_mko [Cumulative distribution function](https://ocme.evemisslab.com/en/mko/mko-cumulative-distribution-function/) ## Prerequisite nodes - [Finite discrete probability distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-discrete-probability-distribution/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Convergence in distribution](https://ocme.evemisslab.com/en/atlas/atlas-convergence-in-distribution/) Convergence in distribution means that F_n tends pointwise to F at every continuity point of the limiting CDF. atlas-convergence-in-distribution · canonical_mko [Convergence in distribution](https://ocme.evemisslab.com/en/mko/mko-convergence-in-distribution/) ## Prerequisite nodes - [Cumulative distribution function](https://ocme.evemisslab.com/en/atlas/atlas-cumulative-distribution-function/) - [Function limits on punctured real domains](https://ocme.evemisslab.com/en/atlas/atlas-function-limit/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Finite joint distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-joint-distribution/) ## [Independent identically distributed sequence](https://ocme.evemisslab.com/en/atlas/atlas-iid-sequence/) An IID sequence requires every variable to have the same distribution and every finite subfamily to be mutually independent. atlas-iid-sequence · canonical_mko [Independent identically distributed sequence](https://ocme.evemisslab.com/en/mko/mko-iid-sequence/) ## Prerequisite nodes - [Independence of random variables](https://ocme.evemisslab.com/en/atlas/atlas-random-variable-independence/) - [Discrete-time stochastic process](https://ocme.evemisslab.com/en/atlas/atlas-discrete-stochastic-process/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Finite joint distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-joint-distribution/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) ## [Graph isomorphism](https://ocme.evemisslab.com/en/atlas/atlas-graph-isomorphism/) An isomorphism of finite simple graphs is a vertex bijection preserving both adjacency and nonadjacency for every pair. atlas-graph-isomorphism · canonical_mko [Graph isomorphism](https://ocme.evemisslab.com/en/mko/mko-graph-isomorphism/) ## Prerequisite nodes - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) - [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/atlas/atlas-bijective/) - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) ## [Graph walks and simple paths](https://ocme.evemisslab.com/en/atlas/atlas-graph-walk-simple-path/) 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. atlas-graph-walk-simple-path · canonical_mko [Graph walks and simple paths](https://ocme.evemisslab.com/en/mko/mko-graph-walk-simple-path/) ## Prerequisite nodes - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) ## [Connected component](https://ocme.evemisslab.com/en/atlas/atlas-connected-component/) In a finite undirected simple graph, connected components are exactly the vertex equivalence classes defined by the existence of a path. atlas-connected-component · canonical_mko [Connected component](https://ocme.evemisslab.com/en/mko/mko-connected-component/) ## Prerequisite nodes - [Graph walks and simple paths](https://ocme.evemisslab.com/en/atlas/atlas-graph-walk-simple-path/) - [Equivalence relation](https://ocme.evemisslab.com/en/atlas/atlas-equivalence-relation/) - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) ## [Bipartite graph](https://ocme.evemisslab.com/en/atlas/atlas-bipartite-graph/) A finite undirected simple graph is bipartite when all vertices split into two disjoint parts and every edge crosses between the parts. atlas-bipartite-graph · canonical_mko [Bipartite graph](https://ocme.evemisslab.com/en/mko/mko-bipartite-graph/) ## Prerequisite nodes - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Graph coloring](https://ocme.evemisslab.com/en/atlas/atlas-graph-coloring/) A proper k-coloring uses at most k colors and gives different colors to the endpoints of every edge. atlas-graph-coloring · canonical_mko [Graph coloring](https://ocme.evemisslab.com/en/mko/mko-graph-coloring/) ## Prerequisite nodes - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) ## [Equivalence relation](https://ocme.evemisslab.com/en/atlas/atlas-equivalence-relation/) A relation on a fixed carrier is an equivalence relation when it fully satisfies reflexivity, symmetry and transitivity. atlas-equivalence-relation · canonical_mko [Equivalence relation](https://ocme.evemisslab.com/en/mko/mko-equivalence-relation/) ## Prerequisite nodes - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) - [Logical connectives](https://ocme.evemisslab.com/en/atlas/atlas-logical-connectives/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Equivalence class](https://ocme.evemisslab.com/en/atlas/atlas-equivalence-class/) For a fixed complete equivalence relation, the class of an element a consists of every carrier element related to a. atlas-equivalence-class · canonical_mko [Equivalence class](https://ocme.evemisslab.com/en/mko/mko-equivalence-class/) ## Prerequisite nodes - [Equivalence relation](https://ocme.evemisslab.com/en/atlas/atlas-equivalence-relation/) - [Set membership](https://ocme.evemisslab.com/en/atlas/atlas-set-membership/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) ## [Quotient set](https://ocme.evemisslab.com/en/atlas/atlas-quotient-set/) The distinct equivalence classes of a complete equivalence relation form its quotient set and partition the original carrier. atlas-quotient-set · canonical_mko [Quotient set](https://ocme.evemisslab.com/en/mko/mko-quotient-set/) ## Prerequisite nodes - [Equivalence class](https://ocme.evemisslab.com/en/atlas/atlas-equivalence-class/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Equivalence relation](https://ocme.evemisslab.com/en/atlas/atlas-equivalence-relation/) ## [Cardinality comparison of sets](https://ocme.evemisslab.com/en/atlas/atlas-cardinality-comparison/) An injection, surjection or bijection between explicit finite source and target sets supplies a corresponding finite cardinality-comparison certificate. atlas-cardinality-comparison · canonical_mko [Cardinality comparison of sets](https://ocme.evemisslab.com/en/mko/mko-cardinality-comparison/) ## Prerequisite nodes - [Injectivity and distinct inputs](https://ocme.evemisslab.com/en/atlas/atlas-injective/) - [Surjectivity and codomain coverage](https://ocme.evemisslab.com/en/atlas/atlas-surjective/) - [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/atlas/atlas-bijective/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) ## [At-most-countable set](https://ocme.evemisslab.com/en/atlas/atlas-at-most-countable-set/) 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. atlas-at-most-countable-set · canonical_mko [At-most-countable set](https://ocme.evemisslab.com/en/mko/mko-at-most-countable-set/) ## Prerequisite nodes - [Cardinality comparison of sets](https://ocme.evemisslab.com/en/atlas/atlas-cardinality-comparison/) - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) - [Injectivity and distinct inputs](https://ocme.evemisslab.com/en/atlas/atlas-injective/) - [Surjectivity and codomain coverage](https://ocme.evemisslab.com/en/atlas/atlas-surjective/) ## [Real vector subspace](https://ocme.evemisslab.com/en/atlas/atlas-vector-subspace/) 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. atlas-vector-subspace · canonical_mko [Real vector subspace](https://ocme.evemisslab.com/en/mko/mko-vector-subspace/) ## Prerequisite nodes - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Kernel of a linear map](https://ocme.evemisslab.com/en/atlas/atlas-linear-map-kernel/) The kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source. atlas-linear-map-kernel · canonical_mko [Kernel of a linear map](https://ocme.evemisslab.com/en/mko/mko-linear-map-kernel/) ## Prerequisite nodes - [Linear map between real vector spaces](https://ocme.evemisslab.com/en/atlas/atlas-general-linear-map/) - [Real vector subspace](https://ocme.evemisslab.com/en/atlas/atlas-vector-subspace/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) ## [Image of a linear map](https://ocme.evemisslab.com/en/atlas/atlas-linear-map-image/) The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span. atlas-linear-map-image · canonical_mko [Image of a linear map](https://ocme.evemisslab.com/en/mko/mko-linear-map-image/) ## Prerequisite nodes - [Linear map between real vector spaces](https://ocme.evemisslab.com/en/atlas/atlas-general-linear-map/) - [Real vector subspace](https://ocme.evemisslab.com/en/atlas/atlas-vector-subspace/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Dimension of a finite-dimensional vector space](https://ocme.evemisslab.com/en/atlas/atlas-vector-space-dimension/) The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality. atlas-vector-space-dimension · canonical_mko [Dimension of a finite-dimensional vector space](https://ocme.evemisslab.com/en/mko/mko-vector-space-dimension/) ## Prerequisite nodes - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) - [Linear independence and bases](https://ocme.evemisslab.com/en/atlas/atlas-linear-independence-basis/) - [Cardinality comparison of sets](https://ocme.evemisslab.com/en/atlas/atlas-cardinality-comparison/) ## [Matrix rank](https://ocme.evemisslab.com/en/atlas/atlas-matrix-rank/) 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. atlas-matrix-rank · canonical_mko [Matrix rank](https://ocme.evemisslab.com/en/mko/mko-matrix-rank/) ## Prerequisite nodes - [Finite matrices: dimensions, indexing, and transpose](https://ocme.evemisslab.com/en/atlas/atlas-matrix/) - [Linear combinations and span](https://ocme.evemisslab.com/en/atlas/atlas-linear-combination-span/) - [Dimension of a finite-dimensional vector space](https://ocme.evemisslab.com/en/atlas/atlas-vector-space-dimension/) ## [Rank-nullity theorem](https://ocme.evemisslab.com/en/atlas/atlas-rank-nullity-theorem/) A linear map T:V→W with finite-dimensional source satisfies dim(V)=dim(ker T)+rank(T). atlas-rank-nullity-theorem · canonical_mko [Rank-nullity theorem](https://ocme.evemisslab.com/en/mko/mko-rank-nullity-theorem/) ## Prerequisite nodes - [Kernel of a linear map](https://ocme.evemisslab.com/en/atlas/atlas-linear-map-kernel/) - [Image of a linear map](https://ocme.evemisslab.com/en/atlas/atlas-linear-map-image/) - [Matrix rank](https://ocme.evemisslab.com/en/atlas/atlas-matrix-rank/) - [Dimension of a finite-dimensional vector space](https://ocme.evemisslab.com/en/atlas/atlas-vector-space-dimension/) ## [Direct sum of subspaces](https://ocme.evemisslab.com/en/atlas/atlas-direct-sum/) 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. atlas-direct-sum · canonical_mko [Direct sum of subspaces](https://ocme.evemisslab.com/en/mko/mko-direct-sum/) ## Prerequisite nodes - [Real vector subspace](https://ocme.evemisslab.com/en/atlas/atlas-vector-subspace/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) - [Linear independence and bases](https://ocme.evemisslab.com/en/atlas/atlas-linear-independence-basis/) ## [Quotient vector space](https://ocme.evemisslab.com/en/atlas/atlas-quotient-vector-space/) Cosets of a subspace W form the quotient vector space V/W, whose operations must be independent of representative choices. atlas-quotient-vector-space · canonical_mko [Quotient vector space](https://ocme.evemisslab.com/en/mko/mko-quotient-vector-space/) ## Prerequisite nodes - [Real vector subspace](https://ocme.evemisslab.com/en/atlas/atlas-vector-subspace/) - [Quotient set](https://ocme.evemisslab.com/en/atlas/atlas-quotient-set/) - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) - [Equivalence class](https://ocme.evemisslab.com/en/atlas/atlas-equivalence-class/) - [Dimension of a finite-dimensional vector space](https://ocme.evemisslab.com/en/atlas/atlas-vector-space-dimension/) ## [Dual vector space](https://ocme.evemisslab.com/en/atlas/atlas-dual-vector-space/) The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis. atlas-dual-vector-space · canonical_mko [Dual vector space](https://ocme.evemisslab.com/en/mko/mko-dual-vector-space/) ## Prerequisite nodes - [Linear map between real vector spaces](https://ocme.evemisslab.com/en/atlas/atlas-general-linear-map/) - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) ## [Topological closure and interior](https://ocme.evemisslab.com/en/atlas/atlas-closure-and-interior/) 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. atlas-closure-and-interior · canonical_mko [Topological closure and interior](https://ocme.evemisslab.com/en/mko/mko-closure-and-interior/) ## Prerequisite nodes - [Open and closed sets](https://ocme.evemisslab.com/en/atlas/atlas-open-closed-set/) - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) ## [Topological boundary](https://ocme.evemisslab.com/en/atlas/atlas-topological-boundary/) The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement. atlas-topological-boundary · canonical_mko [Topological boundary](https://ocme.evemisslab.com/en/mko/mko-topological-boundary/) ## Prerequisite nodes - [Topological closure and interior](https://ocme.evemisslab.com/en/atlas/atlas-closure-and-interior/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) ## [Open cover and subcover](https://ocme.evemisslab.com/en/atlas/atlas-open-cover-and-subcover/) 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. atlas-open-cover-and-subcover · canonical_mko [Open cover and subcover](https://ocme.evemisslab.com/en/mko/mko-open-cover-and-subcover/) ## Prerequisite nodes - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Open and closed sets](https://ocme.evemisslab.com/en/atlas/atlas-open-closed-set/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) ## [Compact topological space](https://ocme.evemisslab.com/en/atlas/atlas-compact-topological-space/) A space is compact when every open cover has a finite subcover; every finite topological space is compact. atlas-compact-topological-space · canonical_mko [Compact topological space](https://ocme.evemisslab.com/en/mko/mko-compact-topological-space/) ## Prerequisite nodes - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Open cover and subcover](https://ocme.evemisslab.com/en/atlas/atlas-open-cover-and-subcover/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Connected topological space](https://ocme.evemisslab.com/en/atlas/atlas-connected-topological-space/) A space is connected when it has no separation into two disjoint nonempty open sets, equivalently no nontrivial clopen subset. atlas-connected-topological-space · canonical_mko [Connected topological space](https://ocme.evemisslab.com/en/mko/mko-connected-topological-space/) ## Prerequisite nodes - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Open and closed sets](https://ocme.evemisslab.com/en/atlas/atlas-open-closed-set/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) ## [Homeomorphism](https://ocme.evemisslab.com/en/atlas/atlas-homeomorphism/) A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism. atlas-homeomorphism · canonical_mko [Homeomorphism](https://ocme.evemisslab.com/en/mko/mko-homeomorphism/) ## Prerequisite nodes - [Continuous maps between topological spaces](https://ocme.evemisslab.com/en/atlas/atlas-topological-continuity/) - [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/atlas/atlas-bijective/) - [Inverse functions of bijections](https://ocme.evemisslab.com/en/atlas/atlas-inverse-function/) - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) ## [Pointwise convergence of functions](https://ocme.evemisslab.com/en/atlas/atlas-pointwise-convergence-of-functions/) 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. atlas-pointwise-convergence-of-functions · canonical_mko [Pointwise convergence of functions](https://ocme.evemisslab.com/en/mko/mko-pointwise-convergence-of-functions/) ## Prerequisite nodes - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/atlas/atlas-sequence-limit/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Uniform convergence of functions](https://ocme.evemisslab.com/en/atlas/atlas-uniform-convergence-of-functions/) Uniform convergence requires one witness N(epsilon) controlling every point of the domain simultaneously. atlas-uniform-convergence-of-functions · canonical_mko [Uniform convergence of functions](https://ocme.evemisslab.com/en/mko/mko-uniform-convergence-of-functions/) ## Prerequisite nodes - [Pointwise convergence of functions](https://ocme.evemisslab.com/en/atlas/atlas-pointwise-convergence-of-functions/) - [Metric space](https://ocme.evemisslab.com/en/atlas/atlas-metric-space/) - [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Infinite series](https://ocme.evemisslab.com/en/atlas/atlas-infinite-series/) An infinite series is defined through the limit of its partial-sum sequence; terms tending to zero are necessary but not sufficient for convergence. atlas-infinite-series · canonical_mko [Infinite series](https://ocme.evemisslab.com/en/mko/mko-infinite-series/) ## Prerequisite nodes - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/atlas/atlas-sequence-limit/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Absolute convergence of series](https://ocme.evemisslab.com/en/atlas/atlas-absolute-convergence-of-series/) A series is absolutely convergent when the sum of absolute terms converges; absolute convergence implies convergence, but the converse does not hold in general. atlas-absolute-convergence-of-series · canonical_mko [Absolute convergence of series](https://ocme.evemisslab.com/en/mko/mko-absolute-convergence-of-series/) ## Prerequisite nodes - [Infinite series](https://ocme.evemisslab.com/en/atlas/atlas-infinite-series/) - [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) - [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/atlas/atlas-sequence-limit/) ## [Almost-sure convergence](https://ocme.evemisslab.com/en/atlas/atlas-almost-sure-convergence/) Random variables on the same probability space converge almost surely when they converge pointwise outside an exceptional set of probability zero. atlas-almost-sure-convergence · canonical_mko [Almost-sure convergence](https://ocme.evemisslab.com/en/mko/mko-almost-sure-convergence/) ## Prerequisite nodes - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) - [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/atlas/atlas-sigma-algebra-measure-space/) - [Convergence in probability](https://ocme.evemisslab.com/en/atlas/atlas-convergence-in-probability/) ## [Lp convergence](https://ocme.evemisslab.com/en/atlas/atlas-lp-convergence/) On a common probability space with fixed integer p≥1, Lp convergence means E|X_n−X|^p tends to zero. atlas-lp-convergence · canonical_mko [Lp convergence](https://ocme.evemisslab.com/en/mko/mko-lp-convergence/) ## Prerequisite nodes - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) - [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/atlas/atlas-sequence-limit/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) - [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/atlas/atlas-sigma-algebra-measure-space/) ## [Weak law of large numbers](https://ocme.evemisslab.com/en/atlas/atlas-weak-law-large-numbers/) For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean. atlas-weak-law-large-numbers · canonical_mko [Weak law of large numbers](https://ocme.evemisslab.com/en/mko/mko-weak-law-large-numbers/) ## Prerequisite nodes - [Independent identically distributed sequence](https://ocme.evemisslab.com/en/atlas/atlas-iid-sequence/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Variance in finite probability models](https://ocme.evemisslab.com/en/atlas/atlas-variance/) - [Convergence in probability](https://ocme.evemisslab.com/en/atlas/atlas-convergence-in-probability/) ## [Bayes theorem](https://ocme.evemisslab.com/en/atlas/atlas-bayes-theorem/) With positive conditioning denominators, Bayes theorem reverses conditional probabilities using the same intersection together with priors and likelihoods. atlas-bayes-theorem · canonical_mko [Bayes theorem](https://ocme.evemisslab.com/en/mko/mko-bayes-theorem/) ## Prerequisite nodes - [Conditional probability in finite models](https://ocme.evemisslab.com/en/atlas/atlas-conditional-probability/) - [Event](https://ocme.evemisslab.com/en/atlas/atlas-event/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) ## [Parametric statistical model](https://ocme.evemisslab.com/en/atlas/atlas-parametric-statistical-model/) A parametric statistical model assigns each parameter theta a complete probability law P_theta on the same finite measurable sample space. atlas-parametric-statistical-model · canonical_mko [Parametric statistical model](https://ocme.evemisslab.com/en/mko/mko-parametric-statistical-model/) ## Prerequisite nodes - [Finite joint distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-joint-distribution/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) ## [Statistical estimator](https://ocme.evemisslab.com/en/atlas/atlas-statistical-estimator/) An estimator is a fixed measurable total statistic independent of the unknown parameter; its observed numerical value is an estimate. atlas-statistical-estimator · canonical_mko [Statistical estimator](https://ocme.evemisslab.com/en/mko/mko-statistical-estimator/) ## Prerequisite nodes - [Parametric statistical model](https://ocme.evemisslab.com/en/atlas/atlas-parametric-statistical-model/) - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) - [Sample space](https://ocme.evemisslab.com/en/atlas/atlas-sample-space/) ## [Sampling distribution](https://ocme.evemisslab.com/en/atlas/atlas-sampling-distribution/) At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws. atlas-sampling-distribution · canonical_mko [Sampling distribution](https://ocme.evemisslab.com/en/mko/mko-sampling-distribution/) ## Prerequisite nodes - [Parametric statistical model](https://ocme.evemisslab.com/en/atlas/atlas-parametric-statistical-model/) - [Statistical estimator](https://ocme.evemisslab.com/en/atlas/atlas-statistical-estimator/) - [Finite discrete probability distribution](https://ocme.evemisslab.com/en/atlas/atlas-finite-discrete-probability-distribution/) ## [Estimator bias](https://ocme.evemisslab.com/en/atlas/atlas-estimator-bias/) Bias at theta is the estimator expectation under its theta sampling law minus the target parameter; unbiasedness requires zero bias at every theta. atlas-estimator-bias · canonical_mko [Estimator bias](https://ocme.evemisslab.com/en/mko/mko-estimator-bias/) ## Prerequisite nodes - [Sampling distribution](https://ocme.evemisslab.com/en/atlas/atlas-sampling-distribution/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Parametric statistical model](https://ocme.evemisslab.com/en/atlas/atlas-parametric-statistical-model/) ## [Estimator variance](https://ocme.evemisslab.com/en/atlas/atlas-estimator-variance/) 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. atlas-estimator-variance · canonical_mko [Estimator variance](https://ocme.evemisslab.com/en/mko/mko-estimator-variance/) ## Prerequisite nodes - [Sampling distribution](https://ocme.evemisslab.com/en/atlas/atlas-sampling-distribution/) - [Variance in finite probability models](https://ocme.evemisslab.com/en/atlas/atlas-variance/) ## [Euler trails and circuits](https://ocme.evemisslab.com/en/atlas/atlas-eulerian-trail/) 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. atlas-eulerian-trail · canonical_mko [Euler trails and circuits](https://ocme.evemisslab.com/en/mko/mko-eulerian-trail/) ## Prerequisite nodes - [Graph walks and simple paths](https://ocme.evemisslab.com/en/atlas/atlas-graph-walk-simple-path/) - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) - [Connected component](https://ocme.evemisslab.com/en/atlas/atlas-connected-component/) ## [Spanning tree](https://ocme.evemisslab.com/en/atlas/atlas-spanning-tree/) A spanning tree is a connected acyclic subgraph containing every vertex of its parent graph. atlas-spanning-tree · canonical_mko [Spanning tree](https://ocme.evemisslab.com/en/mko/mko-spanning-tree/) ## Prerequisite nodes - [Nonempty finite trees: connectivity, cycles, and unique paths](https://ocme.evemisslab.com/en/atlas/atlas-tree/) - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) - [Connected component](https://ocme.evemisslab.com/en/atlas/atlas-connected-component/) ## [Graph matching](https://ocme.evemisslab.com/en/atlas/atlas-graph-matching/) A matching in a finite graph is an edge subset whose edges share no endpoints; the empty matching is allowed. atlas-graph-matching · canonical_mko [Graph matching](https://ocme.evemisslab.com/en/mko/mko-graph-matching/) ## Prerequisite nodes - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) ## [Maximum matching](https://ocme.evemisslab.com/en/atlas/atlas-maximum-matching/) 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. atlas-maximum-matching · canonical_mko [Maximum matching](https://ocme.evemisslab.com/en/mko/mko-maximum-matching/) ## Prerequisite nodes - [Graph matching](https://ocme.evemisslab.com/en/atlas/atlas-graph-matching/) - [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) ## [Cliques and independent sets](https://ocme.evemisslab.com/en/atlas/atlas-clique-and-independent-set/) 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. atlas-clique-and-independent-set · canonical_mko [Cliques and independent sets](https://ocme.evemisslab.com/en/mko/mko-clique-and-independent-set/) ## Prerequisite nodes - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) ## [Chromatic number](https://ocme.evemisslab.com/en/atlas/atlas-chromatic-number/) 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. atlas-chromatic-number · canonical_mko [Chromatic number](https://ocme.evemisslab.com/en/mko/mko-chromatic-number/) ## Prerequisite nodes - [Graph coloring](https://ocme.evemisslab.com/en/atlas/atlas-graph-coloring/) - [Natural number](https://ocme.evemisslab.com/en/atlas/atlas-natural-number/) - [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) ## [Euler totient function](https://ocme.evemisslab.com/en/atlas/atlas-euler-totient-function/) Euler phi(n) counts the representatives coprime to n in a complete residue-representative system, retaining the convention phi(1)=1. atlas-euler-totient-function · canonical_mko [Euler totient function](https://ocme.evemisslab.com/en/mko/mko-euler-totient-function/) ## Prerequisite nodes - [Prime natural number](https://ocme.evemisslab.com/en/atlas/atlas-prime-number/) - [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/atlas/atlas-gcd-coprime/) - [Basic counting principles](https://ocme.evemisslab.com/en/atlas/atlas-counting-principle/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Chinese remainder theorem](https://ocme.evemisslab.com/en/atlas/atlas-chinese-remainder-theorem/) For positive pairwise-coprime moduli, an integer congruence system has a unique solution class modulo their product. atlas-chinese-remainder-theorem · canonical_mko [Chinese remainder theorem](https://ocme.evemisslab.com/en/mko/mko-chinese-remainder-theorem/) ## Prerequisite nodes - [Integer](https://ocme.evemisslab.com/en/atlas/atlas-integer/) - [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/atlas/atlas-modular-congruence/) - [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/atlas/atlas-gcd-coprime/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Modular inverse](https://ocme.evemisslab.com/en/atlas/atlas-modular-inverse/) 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. atlas-modular-inverse · canonical_mko [Modular inverse](https://ocme.evemisslab.com/en/mko/mko-modular-inverse/) ## Prerequisite nodes - [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/atlas/atlas-gcd-coprime/) - [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/atlas/atlas-modular-congruence/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) ## [Linear Diophantine equation](https://ocme.evemisslab.com/en/atlas/atlas-linear-diophantine-equation/) 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. atlas-linear-diophantine-equation · canonical_mko [Linear Diophantine equation](https://ocme.evemisslab.com/en/mko/mko-linear-diophantine-equation/) ## Prerequisite nodes - [Integer](https://ocme.evemisslab.com/en/atlas/atlas-integer/) - [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/atlas/atlas-gcd-coprime/) - [Linear equation in one variable](https://ocme.evemisslab.com/en/atlas/atlas-linear-equation/) - [Euclidean algorithm](https://ocme.evemisslab.com/en/atlas/atlas-euclidean-algorithm/) ## [Fermat's little theorem](https://ocme.evemisslab.com/en/atlas/atlas-fermat-little-theorem/) 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. atlas-fermat-little-theorem · canonical_mko [Fermat's little theorem](https://ocme.evemisslab.com/en/mko/mko-fermat-little-theorem/) ## Prerequisite nodes - [Prime natural number](https://ocme.evemisslab.com/en/atlas/atlas-prime-number/) - [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/atlas/atlas-modular-congruence/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) ## [Euclidean algorithm](https://ocme.evemisslab.com/en/atlas/atlas-euclidean-algorithm/) The Euclidean algorithm computes the gcd using exact quotient-remainder divisions, gcd invariance and strictly decreasing nonnegative remainders. atlas-euclidean-algorithm · canonical_mko [Euclidean algorithm](https://ocme.evemisslab.com/en/mko/mko-euclidean-algorithm/) ## Prerequisite nodes - [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/atlas/atlas-gcd-coprime/) - [Natural-number divisibility](https://ocme.evemisslab.com/en/atlas/atlas-natural-divisibility/) - [Algorithm correctness](https://ocme.evemisslab.com/en/atlas/atlas-algorithm-correctness/) ## [Deterministic finite automaton](https://ocme.evemisslab.com/en/atlas/atlas-finite-automaton/) A DFA consists of finite states, a finite alphabet, a total transition function, one initial state and a subset of accepting states. atlas-finite-automaton · canonical_mko [Deterministic finite automaton](https://ocme.evemisslab.com/en/mko/mko-finite-automaton/) ## Prerequisite nodes - [Abstract computational model](https://ocme.evemisslab.com/en/atlas/atlas-computational-model/) - [Discrete-time dynamical system](https://ocme.evemisslab.com/en/atlas/atlas-discrete-dynamical-system/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) ## [Regular language](https://ocme.evemisslab.com/en/atlas/atlas-regular-language/) A language is regular when a complete DFA recognizes all and exactly its words. atlas-regular-language · canonical_mko [Regular language](https://ocme.evemisslab.com/en/mko/mko-regular-language/) ## Prerequisite nodes - [Deterministic finite automaton](https://ocme.evemisslab.com/en/atlas/atlas-finite-automaton/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Decision problem](https://ocme.evemisslab.com/en/atlas/atlas-decision-problem/) ## [Finite group action](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-action/) An action of a finite group on a finite set is a total action table satisfying the identity and multiplication-compatibility laws. atlas-finite-group-action · canonical_mko [Finite group action](https://ocme.evemisslab.com/en/mko/mko-finite-group-action/) ## Prerequisite nodes - [Finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) ## [Orbit of a finite group action](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-orbit/) The images of one point under all group elements form its orbit; all orbits partition the point carrier. atlas-finite-group-orbit · canonical_mko [Orbit of a finite group action](https://ocme.evemisslab.com/en/mko/mko-finite-group-orbit/) ## Prerequisite nodes - [Finite group action](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-action/) - [Equivalence class](https://ocme.evemisslab.com/en/atlas/atlas-equivalence-class/) ## [Stabilizer of a finite group action](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-stabilizer/) The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup. atlas-finite-group-stabilizer · canonical_mko [Stabilizer of a finite group action](https://ocme.evemisslab.com/en/mko/mko-finite-group-stabilizer/) ## Prerequisite nodes - [Finite group action](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-action/) - [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/atlas/atlas-subgroup-homomorphism/) ## [Orbit-stabilizer theorem](https://ocme.evemisslab.com/en/atlas/atlas-orbit-stabilizer-theorem/) For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order. atlas-orbit-stabilizer-theorem · canonical_mko [Orbit-stabilizer theorem](https://ocme.evemisslab.com/en/mko/mko-orbit-stabilizer-theorem/) ## Prerequisite nodes - [Orbit of a finite group action](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-orbit/) - [Stabilizer of a finite group action](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-stabilizer/) - [Cardinality comparison of sets](https://ocme.evemisslab.com/en/atlas/atlas-cardinality-comparison/) ## [Finite normal subgroup](https://ocme.evemisslab.com/en/atlas/atlas-finite-normal-subgroup/) 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. atlas-finite-normal-subgroup · canonical_mko [Finite normal subgroup](https://ocme.evemisslab.com/en/mko/mko-finite-normal-subgroup/) ## Prerequisite nodes - [Finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group/) - [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/atlas/atlas-subgroup-homomorphism/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Finite quotient group](https://ocme.evemisslab.com/en/atlas/atlas-finite-quotient-group/) Cosets of a normal subgroup N form a finite quotient group, with (gN)(hN)=(g*h)N independent of representatives. atlas-finite-quotient-group · canonical_mko [Finite quotient group](https://ocme.evemisslab.com/en/mko/mko-finite-quotient-group/) ## Prerequisite nodes - [Finite normal subgroup](https://ocme.evemisslab.com/en/atlas/atlas-finite-normal-subgroup/) - [Quotient set](https://ocme.evemisslab.com/en/atlas/atlas-quotient-set/) - [Finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group/) ## [Conjugacy class in a finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-conjugacy-class/) A finite group acts on itself by g*x=g*x*g^(-1); the orbits are conjugacy classes partitioning the group. atlas-finite-group-conjugacy-class · canonical_mko [Conjugacy class in a finite group](https://ocme.evemisslab.com/en/mko/mko-finite-group-conjugacy-class/) ## Prerequisite nodes - [Finite group action](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-action/) - [Orbit of a finite group action](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-orbit/) - [Finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group/) ## [Center of a finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-center/) The center consists of all elements commuting with every group element and is a normal subgroup. atlas-finite-group-center · canonical_mko [Center of a finite group](https://ocme.evemisslab.com/en/mko/mko-finite-group-center/) ## Prerequisite nodes - [Finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group/) - [Conjugacy class in a finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-conjugacy-class/) - [Finite normal subgroup](https://ocme.evemisslab.com/en/atlas/atlas-finite-normal-subgroup/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Lagrange theorem for finite groups](https://ocme.evemisslab.com/en/atlas/atlas-lagrange-theorem-finite-groups/) Equal-size left cosets of a subgroup H partition the finite group G, so |G|=[G:H]*|H| and |H| divides |G|. atlas-lagrange-theorem-finite-groups · canonical_mko [Lagrange theorem for finite groups](https://ocme.evemisslab.com/en/mko/mko-lagrange-theorem-finite-groups/) ## Prerequisite nodes - [Finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group/) - [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/atlas/atlas-subgroup-homomorphism/) - [Natural-number divisibility](https://ocme.evemisslab.com/en/atlas/atlas-natural-divisibility/) - [Cardinality comparison of sets](https://ocme.evemisslab.com/en/atlas/atlas-cardinality-comparison/) - [Quotient set](https://ocme.evemisslab.com/en/atlas/atlas-quotient-set/) ## [Kernel of a finite group homomorphism](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-homomorphism-kernel/) 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. atlas-finite-group-homomorphism-kernel · canonical_mko [Kernel of a finite group homomorphism](https://ocme.evemisslab.com/en/mko/mko-finite-group-homomorphism-kernel/) ## Prerequisite nodes - [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/atlas/atlas-subgroup-homomorphism/) - [Finite normal subgroup](https://ocme.evemisslab.com/en/atlas/atlas-finite-normal-subgroup/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [First isomorphism theorem for finite groups](https://ocme.evemisslab.com/en/atlas/atlas-first-isomorphism-theorem-finite-groups/) A finite group homomorphism induces an isomorphism from its kernel quotient to its image, not automatically to the entire codomain. atlas-first-isomorphism-theorem-finite-groups · canonical_mko [First isomorphism theorem for finite groups](https://ocme.evemisslab.com/en/mko/mko-first-isomorphism-theorem-finite-groups/) ## Prerequisite nodes - [Kernel of a finite group homomorphism](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-homomorphism-kernel/) - [Finite quotient group](https://ocme.evemisslab.com/en/atlas/atlas-finite-quotient-group/) - [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/atlas/atlas-subgroup-homomorphism/) - [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/atlas/atlas-bijective/) ## [Cauchy theorem for finite groups](https://ocme.evemisslab.com/en/atlas/atlas-cauchy-theorem-finite-groups/) If a prime p divides the order of a finite group G, G contains an element of exact order p. atlas-cauchy-theorem-finite-groups · canonical_mko [Cauchy theorem for finite groups](https://ocme.evemisslab.com/en/mko/mko-cauchy-theorem-finite-groups/) ## Prerequisite nodes - [Lagrange theorem for finite groups](https://ocme.evemisslab.com/en/atlas/atlas-lagrange-theorem-finite-groups/) - [Prime natural number](https://ocme.evemisslab.com/en/atlas/atlas-prime-number/) - [Finite group](https://ocme.evemisslab.com/en/atlas/atlas-finite-group/) - [Finite group action](https://ocme.evemisslab.com/en/atlas/atlas-finite-group-action/) ## [Supremum and infimum](https://ocme.evemisslab.com/en/atlas/atlas-supremum-and-infimum/) 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. atlas-supremum-and-infimum · canonical_mko [Supremum and infimum](https://ocme.evemisslab.com/en/mko/mko-supremum-and-infimum/) ## Prerequisite nodes - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Subsequence](https://ocme.evemisslab.com/en/atlas/atlas-subsequence/) A subsequence selects terms by a strictly increasing natural-index map and preserves their original order. atlas-subsequence · canonical_mko [Subsequence](https://ocme.evemisslab.com/en/mko/mko-subsequence/) ## Prerequisite nodes - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Injectivity and distinct inputs](https://ocme.evemisslab.com/en/atlas/atlas-injective/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Monotone bounded sequence convergence theorem](https://ocme.evemisslab.com/en/atlas/atlas-monotone-sequence-convergence-theorem/) A nondecreasing real sequence bounded above converges to the supremum of its terms; the nonincreasing bounded-below version converges to the infimum. atlas-monotone-sequence-convergence-theorem · canonical_mko [Monotone bounded sequence convergence theorem](https://ocme.evemisslab.com/en/mko/mko-monotone-sequence-convergence-theorem/) ## Prerequisite nodes - [Supremum and infimum](https://ocme.evemisslab.com/en/atlas/atlas-supremum-and-infimum/) - [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/atlas/atlas-sequence-limit/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Bolzano-Weierstrass theorem for real sequences](https://ocme.evemisslab.com/en/atlas/atlas-bolzano-weierstrass-theorem-real/) Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map. atlas-bolzano-weierstrass-theorem-real · canonical_mko [Bolzano-Weierstrass theorem for real sequences](https://ocme.evemisslab.com/en/mko/mko-bolzano-weierstrass-theorem-real/) ## Prerequisite nodes - [Subsequence](https://ocme.evemisslab.com/en/atlas/atlas-subsequence/) - [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/atlas/atlas-sequence-limit/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Supremum and infimum](https://ocme.evemisslab.com/en/atlas/atlas-supremum-and-infimum/) ## [Intermediate value theorem](https://ocme.evemisslab.com/en/atlas/atlas-intermediate-value-theorem/) A continuous real function on a closed interval attains each value between its endpoint values. atlas-intermediate-value-theorem · canonical_mko [Intermediate value theorem](https://ocme.evemisslab.com/en/mko/mko-intermediate-value-theorem/) ## Prerequisite nodes - [Point values and continuity of real functions](https://ocme.evemisslab.com/en/atlas/atlas-continuity/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Supremum and infimum](https://ocme.evemisslab.com/en/atlas/atlas-supremum-and-infimum/) ## [Mean value theorem](https://ocme.evemisslab.com/en/atlas/atlas-mean-value-theorem/) If a0. atlas-real-exponential-function · canonical_mko [Real exponential function](https://ocme.evemisslab.com/en/mko/mko-real-exponential-function/) ## Prerequisite nodes - [Infinite series](https://ocme.evemisslab.com/en/atlas/atlas-infinite-series/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/atlas/atlas-sequence-limit/) - [Radius of convergence of a power series](https://ocme.evemisslab.com/en/atlas/atlas-power-series-radius-of-convergence/) - [Factorial](https://ocme.evemisslab.com/en/atlas/atlas-factorial/) - [Absolute convergence of series](https://ocme.evemisslab.com/en/atlas/atlas-absolute-convergence-of-series/) - [Binomial theorem](https://ocme.evemisslab.com/en/atlas/atlas-binomial-theorem/) ## [Improper integral](https://ocme.evemisslab.com/en/atlas/atlas-improper-integral/) Integrals over unbounded intervals or with singularities are defined by appropriate one-sided limits, with every required piece converging independently. atlas-improper-integral · canonical_mko [Improper integral](https://ocme.evemisslab.com/en/mko/mko-improper-integral/) ## Prerequisite nodes - [Signed polynomial definite integrals on finite intervals](https://ocme.evemisslab.com/en/atlas/atlas-definite-integral/) - [Function limits on punctured real domains](https://ocme.evemisslab.com/en/atlas/atlas-function-limit/) - [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/atlas/atlas-sequence-limit/) - [Absolute convergence of series](https://ocme.evemisslab.com/en/atlas/atlas-absolute-convergence-of-series/) ## [Topological basis](https://ocme.evemisslab.com/en/atlas/atlas-topological-basis/) A topological basis generates a topology through carrier coverage and pointwise local intersection refinement. atlas-topological-basis · canonical_mko [Topological basis](https://ocme.evemisslab.com/en/mko/mko-topological-basis/) ## Prerequisite nodes - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Open and closed sets](https://ocme.evemisslab.com/en/atlas/atlas-open-closed-set/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) ## [Subspace topology](https://ocme.evemisslab.com/en/atlas/atlas-subspace-topology/) For Y⊆X, the relative topology consists of every intersection of an ambient open set with Y. atlas-subspace-topology · canonical_mko [Subspace topology](https://ocme.evemisslab.com/en/mko/mko-subspace-topology/) ## Prerequisite nodes - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) - [Topological basis](https://ocme.evemisslab.com/en/atlas/atlas-topological-basis/) ## [Finite product topology](https://ocme.evemisslab.com/en/atlas/atlas-finite-product-topology/) Open rectangles generate the topology of a finite Cartesian product of topological spaces. atlas-finite-product-topology · canonical_mko [Finite product topology](https://ocme.evemisslab.com/en/mko/mko-finite-product-topology/) ## Prerequisite nodes - [Topological basis](https://ocme.evemisslab.com/en/atlas/atlas-topological-basis/) - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) - [Continuous maps between topological spaces](https://ocme.evemisslab.com/en/atlas/atlas-topological-continuity/) - [Function composition](https://ocme.evemisslab.com/en/atlas/atlas-composition/) ## [Quotient topology](https://ocme.evemisslab.com/en/atlas/atlas-quotient-topology/) For a surjection q:X→Y, the quotient topology declares U open exactly when its full preimage under q is open. atlas-quotient-topology · canonical_mko [Quotient topology](https://ocme.evemisslab.com/en/mko/mko-quotient-topology/) ## Prerequisite nodes - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Quotient set](https://ocme.evemisslab.com/en/atlas/atlas-quotient-set/) - [Surjectivity and codomain coverage](https://ocme.evemisslab.com/en/atlas/atlas-surjective/) - [Continuous maps between topological spaces](https://ocme.evemisslab.com/en/atlas/atlas-topological-continuity/) - [Function composition](https://ocme.evemisslab.com/en/atlas/atlas-composition/) ## [Hausdorff space](https://ocme.evemisslab.com/en/atlas/atlas-hausdorff-space/) A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods. atlas-hausdorff-space · canonical_mko [Hausdorff space](https://ocme.evemisslab.com/en/mko/mko-hausdorff-space/) ## Prerequisite nodes - [Topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-space/) - [Neighborhood](https://ocme.evemisslab.com/en/atlas/atlas-neighborhood/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Compact subset](https://ocme.evemisslab.com/en/atlas/atlas-compact-subset/) A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover. atlas-compact-subset · canonical_mko [Compact subset](https://ocme.evemisslab.com/en/mko/mko-compact-subset/) ## Prerequisite nodes - [Compact topological space](https://ocme.evemisslab.com/en/atlas/atlas-compact-topological-space/) - [Subspace topology](https://ocme.evemisslab.com/en/atlas/atlas-subspace-topology/) ## [Continuous image of a compact set](https://ocme.evemisslab.com/en/atlas/atlas-continuous-image-of-compact/) If K is compact in X and f:X→Y is continuous, then f(K) is compact in Y. atlas-continuous-image-of-compact · canonical_mko [Continuous image of a compact set](https://ocme.evemisslab.com/en/mko/mko-continuous-image-of-compact/) ## Prerequisite nodes - [Compact subset](https://ocme.evemisslab.com/en/atlas/atlas-compact-subset/) - [Continuous maps between topological spaces](https://ocme.evemisslab.com/en/atlas/atlas-topological-continuity/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Path-connected space](https://ocme.evemisslab.com/en/atlas/atlas-path-connected-space/) A space is path connected when every pair of its points is joined by a continuous map from the unit interval. atlas-path-connected-space · canonical_mko [Path-connected space](https://ocme.evemisslab.com/en/mko/mko-path-connected-space/) ## Prerequisite nodes - [Continuous maps between topological spaces](https://ocme.evemisslab.com/en/atlas/atlas-topological-continuity/) - [Connected topological space](https://ocme.evemisslab.com/en/atlas/atlas-connected-topological-space/) - [Real number](https://ocme.evemisslab.com/en/atlas/atlas-real-number/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) - [Finite product topology](https://ocme.evemisslab.com/en/atlas/atlas-finite-product-topology/) - [Intermediate value theorem](https://ocme.evemisslab.com/en/atlas/atlas-intermediate-value-theorem/) - [Function composition](https://ocme.evemisslab.com/en/atlas/atlas-composition/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) ## [Connected component of a topological space](https://ocme.evemisslab.com/en/atlas/atlas-topological-connected-component/) The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed. atlas-topological-connected-component · canonical_mko [Connected component of a topological space](https://ocme.evemisslab.com/en/mko/mko-topological-connected-component/) ## Prerequisite nodes - [Connected topological space](https://ocme.evemisslab.com/en/atlas/atlas-connected-topological-space/) - [Equivalence class](https://ocme.evemisslab.com/en/atlas/atlas-equivalence-class/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) - [Topological closure and interior](https://ocme.evemisslab.com/en/atlas/atlas-closure-and-interior/) - [Connected component](https://ocme.evemisslab.com/en/atlas/atlas-connected-component/) - [Subspace topology](https://ocme.evemisslab.com/en/atlas/atlas-subspace-topology/) ## [A compact subset of a Hausdorff space is closed](https://ocme.evemisslab.com/en/atlas/atlas-compact-subset-closed-in-hausdorff/) If K is compact in the Hausdorff space X, then K is closed in X. atlas-compact-subset-closed-in-hausdorff · canonical_mko [A compact subset of a Hausdorff space is closed](https://ocme.evemisslab.com/en/mko/mko-compact-subset-closed-in-hausdorff/) ## Prerequisite nodes - [Hausdorff space](https://ocme.evemisslab.com/en/atlas/atlas-hausdorff-space/) - [Compact subset](https://ocme.evemisslab.com/en/atlas/atlas-compact-subset/) - [Open and closed sets](https://ocme.evemisslab.com/en/atlas/atlas-open-closed-set/) ## [Markov inequality](https://ocme.evemisslab.com/en/atlas/atlas-markov-inequality/) For a nonnegative random variable with finite expectation, its tail probability at a positive threshold is at most E[X] divided by that threshold. atlas-markov-inequality · canonical_mko [Markov inequality](https://ocme.evemisslab.com/en/mko/mko-markov-inequality/) ## Prerequisite nodes - [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-random-variable/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Chebyshev inequality](https://ocme.evemisslab.com/en/atlas/atlas-chebyshev-inequality/) For a variable with finite mean and variance, the probability of absolute deviation at least epsilon is at most variance/epsilon^2. atlas-chebyshev-inequality · canonical_mko [Chebyshev inequality](https://ocme.evemisslab.com/en/mko/mko-chebyshev-inequality/) ## Prerequisite nodes - [Markov inequality](https://ocme.evemisslab.com/en/atlas/atlas-markov-inequality/) - [Variance in finite probability models](https://ocme.evemisslab.com/en/atlas/atlas-variance/) - [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) ## [Law of total expectation](https://ocme.evemisslab.com/en/atlas/atlas-law-of-total-expectation/) On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X]. atlas-law-of-total-expectation · canonical_mko [Law of total expectation](https://ocme.evemisslab.com/en/mko/mko-law-of-total-expectation/) ## Prerequisite nodes - [Finite conditional expectation](https://ocme.evemisslab.com/en/atlas/atlas-conditional-expectation/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Event](https://ocme.evemisslab.com/en/atlas/atlas-event/) ## [Law of total variance](https://ocme.evemisslab.com/en/atlas/atlas-law-of-total-variance/) For the same finite law and partition-generated G, total variance is the expected within-cell variance plus the variance of the cell means. atlas-law-of-total-variance · canonical_mko [Law of total variance](https://ocme.evemisslab.com/en/mko/mko-law-of-total-variance/) ## Prerequisite nodes - [Law of total expectation](https://ocme.evemisslab.com/en/atlas/atlas-law-of-total-expectation/) - [Variance in finite probability models](https://ocme.evemisslab.com/en/atlas/atlas-variance/) - [Finite conditional expectation](https://ocme.evemisslab.com/en/atlas/atlas-conditional-expectation/) ## [Continuous real distribution function](https://ocme.evemisslab.com/en/atlas/atlas-continuous-real-distribution-function/) Here a continuous distribution means that its CDF is continuous; the CDF must still be nondecreasing, right-continuous and have endpoint limits0 and1. atlas-continuous-real-distribution-function · canonical_mko [Continuous real distribution function](https://ocme.evemisslab.com/en/mko/mko-continuous-real-distribution-function/) ## Prerequisite nodes - [Cumulative distribution function](https://ocme.evemisslab.com/en/atlas/atlas-cumulative-distribution-function/) - [Point values and continuity of real functions](https://ocme.evemisslab.com/en/atlas/atlas-continuity/) - [Function limits on punctured real domains](https://ocme.evemisslab.com/en/atlas/atlas-function-limit/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) - [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/atlas/atlas-sigma-algebra-measure-space/) ## [Standard normal distribution](https://ocme.evemisslab.com/en/atlas/atlas-standard-normal-distribution/) The standard normal law has density phi(x)=(2*pi)^(-1/2)*exp(−x^2/2), continuous CDF Phi, mean0 and variance1. atlas-standard-normal-distribution · canonical_mko [Standard normal distribution](https://ocme.evemisslab.com/en/mko/mko-standard-normal-distribution/) ## Prerequisite nodes - [Continuous real distribution function](https://ocme.evemisslab.com/en/atlas/atlas-continuous-real-distribution-function/) - [Improper integral](https://ocme.evemisslab.com/en/atlas/atlas-improper-integral/) - [Real exponential function](https://ocme.evemisslab.com/en/atlas/atlas-real-exponential-function/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Variance in finite probability models](https://ocme.evemisslab.com/en/atlas/atlas-variance/) ## [Central limit theorem for finite-variance IID sequences](https://ocme.evemisslab.com/en/atlas/atlas-central-limit-theorem-finite-variance-iid/) 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. atlas-central-limit-theorem-finite-variance-iid · canonical_mko [Central limit theorem for finite-variance IID sequences](https://ocme.evemisslab.com/en/mko/mko-central-limit-theorem-finite-variance-iid/) ## Prerequisite nodes - [Independent identically distributed sequence](https://ocme.evemisslab.com/en/atlas/atlas-iid-sequence/) - [Convergence in distribution](https://ocme.evemisslab.com/en/atlas/atlas-convergence-in-distribution/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Variance in finite probability models](https://ocme.evemisslab.com/en/atlas/atlas-variance/) - [Standard normal distribution](https://ocme.evemisslab.com/en/atlas/atlas-standard-normal-distribution/) - [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/atlas/atlas-sigma-algebra-measure-space/) - [Binomial theorem](https://ocme.evemisslab.com/en/atlas/atlas-binomial-theorem/) ## [Strong law of large numbers for finite-valued IID sequences](https://ocme.evemisslab.com/en/atlas/atlas-strong-law-large-numbers-finite-valued-iid/) The sample means of a finite-valued infinite IID sequence converge almost surely to the common expectation. atlas-strong-law-large-numbers-finite-valued-iid · canonical_mko [Strong law of large numbers for finite-valued IID sequences](https://ocme.evemisslab.com/en/mko/mko-strong-law-large-numbers-finite-valued-iid/) ## Prerequisite nodes - [Independent identically distributed sequence](https://ocme.evemisslab.com/en/atlas/atlas-iid-sequence/) - [Almost-sure convergence](https://ocme.evemisslab.com/en/atlas/atlas-almost-sure-convergence/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/atlas/atlas-sigma-algebra-measure-space/) - [Variance in finite probability models](https://ocme.evemisslab.com/en/atlas/atlas-variance/) ## [Maximum-likelihood estimator in a finite model](https://ocme.evemisslab.com/en/atlas/atlas-finite-maximum-likelihood-estimator/) A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter. atlas-finite-maximum-likelihood-estimator · canonical_mko [Maximum-likelihood estimator in a finite model](https://ocme.evemisslab.com/en/mko/mko-finite-maximum-likelihood-estimator/) ## Prerequisite nodes - [Parametric statistical model](https://ocme.evemisslab.com/en/atlas/atlas-parametric-statistical-model/) - [Statistical estimator](https://ocme.evemisslab.com/en/atlas/atlas-statistical-estimator/) - [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) ## [Mean squared error of an estimator](https://ocme.evemisslab.com/en/atlas/atlas-estimator-mean-squared-error/) For scalar parameter theta, MSE_theta(T)=E_theta[(T−theta)^2] is defined under each complete sampling distribution separately. atlas-estimator-mean-squared-error · canonical_mko [Mean squared error of an estimator](https://ocme.evemisslab.com/en/mko/mko-estimator-mean-squared-error/) ## Prerequisite nodes - [Estimator bias](https://ocme.evemisslab.com/en/atlas/atlas-estimator-bias/) - [Estimator variance](https://ocme.evemisslab.com/en/atlas/atlas-estimator-variance/) - [Expectation in finite models](https://ocme.evemisslab.com/en/atlas/atlas-expectation/) - [Sampling distribution](https://ocme.evemisslab.com/en/atlas/atlas-sampling-distribution/) - [Parametric statistical model](https://ocme.evemisslab.com/en/atlas/atlas-parametric-statistical-model/) ## [Bias-variance decomposition](https://ocme.evemisslab.com/en/atlas/atlas-bias-variance-decomposition/) For each parameter value and its sampling law, scalar-estimator MSE equals variance plus squared bias. atlas-bias-variance-decomposition · canonical_mko [Bias-variance decomposition](https://ocme.evemisslab.com/en/mko/mko-bias-variance-decomposition/) ## Prerequisite nodes - [Mean squared error of an estimator](https://ocme.evemisslab.com/en/atlas/atlas-estimator-mean-squared-error/) - [Estimator bias](https://ocme.evemisslab.com/en/atlas/atlas-estimator-bias/) - [Estimator variance](https://ocme.evemisslab.com/en/atlas/atlas-estimator-variance/) ## [Sufficient statistic in a finite model](https://ocme.evemisslab.com/en/atlas/atlas-finite-sufficient-statistic/) A statistic is sufficient when the conditional sample law given its value does not depend on theta wherever that condition has positive mass. atlas-finite-sufficient-statistic · canonical_mko [Sufficient statistic in a finite model](https://ocme.evemisslab.com/en/mko/mko-finite-sufficient-statistic/) ## Prerequisite nodes - [Parametric statistical model](https://ocme.evemisslab.com/en/atlas/atlas-parametric-statistical-model/) - [Statistical estimator](https://ocme.evemisslab.com/en/atlas/atlas-statistical-estimator/) - [Sampling distribution](https://ocme.evemisslab.com/en/atlas/atlas-sampling-distribution/) - [Conditional probability in finite models](https://ocme.evemisslab.com/en/atlas/atlas-conditional-probability/) ## [Confidence-interval coverage in a finite model](https://ocme.evemisslab.com/en/atlas/atlas-finite-confidence-interval-coverage/) Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter. atlas-finite-confidence-interval-coverage · canonical_mko [Confidence-interval coverage in a finite model](https://ocme.evemisslab.com/en/mko/mko-finite-confidence-interval-coverage/) ## Prerequisite nodes - [Sampling distribution](https://ocme.evemisslab.com/en/atlas/atlas-sampling-distribution/) - [Statistical estimator](https://ocme.evemisslab.com/en/atlas/atlas-statistical-estimator/) - [Parametric statistical model](https://ocme.evemisslab.com/en/atlas/atlas-parametric-statistical-model/) - [Probability on finite sample spaces](https://ocme.evemisslab.com/en/atlas/atlas-probability/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Pigeonhole principle](https://ocme.evemisslab.com/en/atlas/atlas-pigeonhole-principle/) A total map from a larger finite domain into a smaller finite codomain sends two distinct source elements to the same value. atlas-pigeonhole-principle · canonical_mko [Pigeonhole principle](https://ocme.evemisslab.com/en/mko/mko-pigeonhole-principle/) ## Prerequisite nodes - [Basic counting principles](https://ocme.evemisslab.com/en/atlas/atlas-counting-principle/) - [Cardinality comparison of sets](https://ocme.evemisslab.com/en/atlas/atlas-cardinality-comparison/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Inclusion-exclusion principle](https://ocme.evemisslab.com/en/atlas/atlas-inclusion-exclusion-principle/) The union cardinality of a finite indexed set family is the alternating sum of intersection sizes over every nonempty index subset. atlas-inclusion-exclusion-principle · canonical_mko [Inclusion-exclusion principle](https://ocme.evemisslab.com/en/mko/mko-inclusion-exclusion-principle/) ## Prerequisite nodes - [Set operations](https://ocme.evemisslab.com/en/atlas/atlas-set-operations/) - [Basic counting principles](https://ocme.evemisslab.com/en/atlas/atlas-counting-principle/) - [Cardinality comparison of sets](https://ocme.evemisslab.com/en/atlas/atlas-cardinality-comparison/) - [Mathematical induction](https://ocme.evemisslab.com/en/atlas/atlas-mathematical-induction/) ## [Binomial theorem](https://ocme.evemisslab.com/en/atlas/atlas-binomial-theorem/) With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients. atlas-binomial-theorem · canonical_mko [Binomial theorem](https://ocme.evemisslab.com/en/mko/mko-binomial-theorem/) ## Prerequisite nodes - [Permutations and combinations](https://ocme.evemisslab.com/en/atlas/atlas-permutation-combination/) - [Mathematical induction](https://ocme.evemisslab.com/en/atlas/atlas-mathematical-induction/) - [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/atlas/atlas-polynomial/) ## [Recurrence relation](https://ocme.evemisslab.com/en/atlas/atlas-recurrence-relation/) An order-r recurrence uses explicit initial values and a total single-valued state update to determine successive sequence terms. atlas-recurrence-relation · canonical_mko [Recurrence relation](https://ocme.evemisslab.com/en/mko/mko-recurrence-relation/) ## Prerequisite nodes - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Analytic ordinary generating function](https://ocme.evemisslab.com/en/atlas/atlas-analytic-ordinary-generating-function/) 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. atlas-analytic-ordinary-generating-function · canonical_mko [Analytic ordinary generating function](https://ocme.evemisslab.com/en/mko/mko-analytic-ordinary-generating-function/) ## Prerequisite nodes - [Recurrence relation](https://ocme.evemisslab.com/en/atlas/atlas-recurrence-relation/) - [Radius of convergence of a power series](https://ocme.evemisslab.com/en/atlas/atlas-power-series-radius-of-convergence/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) ## [Catalan numbers](https://ocme.evemisslab.com/en/atlas/atlas-catalan-numbers/) With C_0=1, the convolution recurrence and two exact closed forms define the same zero-based Catalan sequence. atlas-catalan-numbers · canonical_mko [Catalan numbers](https://ocme.evemisslab.com/en/mko/mko-catalan-numbers/) ## Prerequisite nodes - [Recurrence relation](https://ocme.evemisslab.com/en/atlas/atlas-recurrence-relation/) - [Binomial theorem](https://ocme.evemisslab.com/en/atlas/atlas-binomial-theorem/) - [Basic counting principles](https://ocme.evemisslab.com/en/atlas/atlas-counting-principle/) - [Natural-number divisibility](https://ocme.evemisslab.com/en/atlas/atlas-natural-divisibility/) ## [Vertex cover](https://ocme.evemisslab.com/en/atlas/atlas-vertex-cover/) A vertex subset C covers a finite simple undirected graph if every edge has at least one endpoint in C. atlas-vertex-cover · canonical_mko [Vertex cover](https://ocme.evemisslab.com/en/mko/mko-vertex-cover/) ## Prerequisite nodes - [Finite simple undirected graphs](https://ocme.evemisslab.com/en/atlas/atlas-graph/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) ## [Minimum vertex cover](https://ocme.evemisslab.com/en/atlas/atlas-minimum-vertex-cover/) A minimum vertex cover has least cardinality among all vertex covers of the finite simple graph. atlas-minimum-vertex-cover · canonical_mko [Minimum vertex cover](https://ocme.evemisslab.com/en/mko/mko-minimum-vertex-cover/) ## Prerequisite nodes - [Vertex cover](https://ocme.evemisslab.com/en/atlas/atlas-vertex-cover/) - [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) - [Cardinality comparison of sets](https://ocme.evemisslab.com/en/atlas/atlas-cardinality-comparison/) ## [Finite directed graph](https://ocme.evemisslab.com/en/atlas/atlas-finite-directed-graph/) 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. atlas-finite-directed-graph · canonical_mko [Finite directed graph](https://ocme.evemisslab.com/en/mko/mko-finite-directed-graph/) ## Prerequisite nodes - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Cardinality comparison of sets](https://ocme.evemisslab.com/en/atlas/atlas-cardinality-comparison/) ## [Topological ordering of a directed acyclic graph](https://ocme.evemisslab.com/en/atlas/atlas-topological-ordering-dag/) A finite directed graph has a topological ordering exactly when it has no directed cycle. atlas-topological-ordering-dag · canonical_mko [Topological ordering of a directed acyclic graph](https://ocme.evemisslab.com/en/mko/mko-topological-ordering-dag/) ## Prerequisite nodes - [Finite directed graph](https://ocme.evemisslab.com/en/atlas/atlas-finite-directed-graph/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) - [Algorithm correctness](https://ocme.evemisslab.com/en/atlas/atlas-algorithm-correctness/) - [Mathematical induction](https://ocme.evemisslab.com/en/atlas/atlas-mathematical-induction/) ## [Finite flow network](https://ocme.evemisslab.com/en/atlas/atlas-finite-flow-network/) A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink. atlas-finite-flow-network · canonical_mko [Finite flow network](https://ocme.evemisslab.com/en/mko/mko-finite-flow-network/) ## Prerequisite nodes - [Finite directed graph](https://ocme.evemisslab.com/en/atlas/atlas-finite-directed-graph/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Linear programming](https://ocme.evemisslab.com/en/atlas/atlas-linear-programming/) ## [Max-flow min-cut theorem](https://ocme.evemisslab.com/en/atlas/atlas-max-flow-min-cut-theorem/) In a finite capacitated network the largest feasible flow value equals the least capacity among all source–sink cuts. atlas-max-flow-min-cut-theorem · canonical_mko [Max-flow min-cut theorem](https://ocme.evemisslab.com/en/mko/mko-max-flow-min-cut-theorem/) ## Prerequisite nodes - [Finite flow network](https://ocme.evemisslab.com/en/atlas/atlas-finite-flow-network/) - [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) - [Linear programming](https://ocme.evemisslab.com/en/atlas/atlas-linear-programming/) ## [Euler's theorem in number theory](https://ocme.evemisslab.com/en/atlas/atlas-euler-theorem-number-theory/) For n≥2 and gcd(a,n)=1, a^phi(n) is congruent to1 modulo n. atlas-euler-theorem-number-theory · canonical_mko [Euler's theorem in number theory](https://ocme.evemisslab.com/en/mko/mko-euler-theorem-number-theory/) ## Prerequisite nodes - [Euler totient function](https://ocme.evemisslab.com/en/atlas/atlas-euler-totient-function/) - [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/atlas/atlas-gcd-coprime/) - [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/atlas/atlas-modular-congruence/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) ## [Multiplicative order modulo n](https://ocme.evemisslab.com/en/atlas/atlas-multiplicative-order-mod-n/) For gcd(a,n)=1, the multiplicative order is the smallest positive k with a^k congruent to1 modulo n. atlas-multiplicative-order-mod-n · canonical_mko [Multiplicative order modulo n](https://ocme.evemisslab.com/en/mko/mko-multiplicative-order-mod-n/) ## Prerequisite nodes - [Modular inverse](https://ocme.evemisslab.com/en/atlas/atlas-modular-inverse/) - [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/atlas/atlas-gcd-coprime/) - [Natural-number divisibility](https://ocme.evemisslab.com/en/atlas/atlas-natural-divisibility/) - [Euler's theorem in number theory](https://ocme.evemisslab.com/en/atlas/atlas-euler-theorem-number-theory/) ## [Unique prime factorization of integers](https://ocme.evemisslab.com/en/atlas/atlas-unique-prime-factorization/) Every positive integer n>1 has a prime factorization whose prime multiset is unique up to factor order. atlas-unique-prime-factorization · canonical_mko [Unique prime factorization of integers](https://ocme.evemisslab.com/en/mko/mko-unique-prime-factorization/) ## Prerequisite nodes - [Prime natural number](https://ocme.evemisslab.com/en/atlas/atlas-prime-number/) - [Natural-number divisibility](https://ocme.evemisslab.com/en/atlas/atlas-natural-divisibility/) - [Mathematical induction](https://ocme.evemisslab.com/en/atlas/atlas-mathematical-induction/) - [Euclidean algorithm](https://ocme.evemisslab.com/en/atlas/atlas-euclidean-algorithm/) - [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/atlas/atlas-gcd-coprime/) ## [Wilson theorem](https://ocme.evemisslab.com/en/atlas/atlas-wilson-theorem/) An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n. atlas-wilson-theorem · canonical_mko [Wilson theorem](https://ocme.evemisslab.com/en/mko/mko-wilson-theorem/) ## Prerequisite nodes - [Prime natural number](https://ocme.evemisslab.com/en/atlas/atlas-prime-number/) - [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/atlas/atlas-modular-congruence/) - [Factorial](https://ocme.evemisslab.com/en/atlas/atlas-factorial/) - [Modular inverse](https://ocme.evemisslab.com/en/atlas/atlas-modular-inverse/) ## [Quadratic residue](https://ocme.evemisslab.com/en/atlas/atlas-quadratic-residue/) 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. atlas-quadratic-residue · canonical_mko [Quadratic residue](https://ocme.evemisslab.com/en/mko/mko-quadratic-residue/) ## Prerequisite nodes - [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/atlas/atlas-modular-congruence/) - [Prime natural number](https://ocme.evemisslab.com/en/atlas/atlas-prime-number/) - [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/atlas/atlas-power-root/) ## [Legendre symbol](https://ocme.evemisslab.com/en/atlas/atlas-legendre-symbol/) For odd prime p, the Legendre symbol uses0,1,−1 for the zero class, nonzero quadratic residues and nonresidues respectively. atlas-legendre-symbol · canonical_mko [Legendre symbol](https://ocme.evemisslab.com/en/mko/mko-legendre-symbol/) ## Prerequisite nodes - [Quadratic residue](https://ocme.evemisslab.com/en/atlas/atlas-quadratic-residue/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Euler criterion for quadratic residues](https://ocme.evemisslab.com/en/atlas/atlas-euler-criterion-quadratic-residue/) 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). atlas-euler-criterion-quadratic-residue · canonical_mko [Euler criterion for quadratic residues](https://ocme.evemisslab.com/en/mko/mko-euler-criterion-quadratic-residue/) ## Prerequisite nodes - [Euler's theorem in number theory](https://ocme.evemisslab.com/en/atlas/atlas-euler-theorem-number-theory/) - [Quadratic residue](https://ocme.evemisslab.com/en/atlas/atlas-quadratic-residue/) - [Legendre symbol](https://ocme.evemisslab.com/en/atlas/atlas-legendre-symbol/) ## [p-adic valuation of integers](https://ocme.evemisslab.com/en/atlas/atlas-p-adic-valuation-integers/) 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. atlas-p-adic-valuation-integers · canonical_mko [p-adic valuation of integers](https://ocme.evemisslab.com/en/mko/mko-p-adic-valuation-integers/) ## Prerequisite nodes - [Prime natural number](https://ocme.evemisslab.com/en/atlas/atlas-prime-number/) - [Natural-number divisibility](https://ocme.evemisslab.com/en/atlas/atlas-natural-divisibility/) - [Unique prime factorization of integers](https://ocme.evemisslab.com/en/atlas/atlas-unique-prime-factorization/) - [Integer](https://ocme.evemisslab.com/en/atlas/atlas-integer/) - [Absolute value](https://ocme.evemisslab.com/en/atlas/atlas-absolute-value/) ## [Mobius function](https://ocme.evemisslab.com/en/atlas/atlas-mobius-function/) 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. atlas-mobius-function · canonical_mko [Mobius function](https://ocme.evemisslab.com/en/mko/mko-mobius-function/) ## Prerequisite nodes - [Prime natural number](https://ocme.evemisslab.com/en/atlas/atlas-prime-number/) - [Unique prime factorization of integers](https://ocme.evemisslab.com/en/atlas/atlas-unique-prime-factorization/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) ## [Mobius inversion on finite divisor sets](https://ocme.evemisslab.com/en/atlas/atlas-mobius-inversion-finite-divisors/) If F(n) sums f(d) over every positive divisor of n, Mobius weights exactly recover f(n). atlas-mobius-inversion-finite-divisors · canonical_mko [Mobius inversion on finite divisor sets](https://ocme.evemisslab.com/en/mko/mko-mobius-inversion-finite-divisors/) ## Prerequisite nodes - [Mobius function](https://ocme.evemisslab.com/en/atlas/atlas-mobius-function/) - [Natural-number divisibility](https://ocme.evemisslab.com/en/atlas/atlas-natural-divisibility/) - [Basic counting principles](https://ocme.evemisslab.com/en/atlas/atlas-counting-principle/) ## [Regular expression](https://ocme.evemisslab.com/en/atlas/atlas-regular-expression/) Regular expressions recursively build languages from the empty language, epsilon, alphabet symbols, union, concatenation and Kleene star. atlas-regular-expression · canonical_mko [Regular expression](https://ocme.evemisslab.com/en/mko/mko-regular-expression/) ## Prerequisite nodes - [Regular language](https://ocme.evemisslab.com/en/atlas/atlas-regular-language/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Mathematical induction](https://ocme.evemisslab.com/en/atlas/atlas-mathematical-induction/) ## [Nondeterministic finite automaton](https://ocme.evemisslab.com/en/atlas/atlas-nondeterministic-finite-automaton/) An epsilon-NFA relates a state and input symbol or epsilon to possible successor states; one full-input final path suffices for acceptance. atlas-nondeterministic-finite-automaton · canonical_mko [Nondeterministic finite automaton](https://ocme.evemisslab.com/en/mko/mko-nondeterministic-finite-automaton/) ## Prerequisite nodes - [Deterministic finite automaton](https://ocme.evemisslab.com/en/atlas/atlas-finite-automaton/) - [Binary relations and set projections](https://ocme.evemisslab.com/en/atlas/atlas-relation/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) ## [Subset construction from an NFA to a DFA](https://ocme.evemisslab.com/en/atlas/atlas-subset-construction-nfa-to-dfa/) Use epsilon-NFA state subsets as DFA states, defining start, transitions and acceptance with epsilon closure to preserve the language. atlas-subset-construction-nfa-to-dfa · canonical_mko [Subset construction from an NFA to a DFA](https://ocme.evemisslab.com/en/mko/mko-subset-construction-nfa-to-dfa/) ## Prerequisite nodes - [Nondeterministic finite automaton](https://ocme.evemisslab.com/en/atlas/atlas-nondeterministic-finite-automaton/) - [Deterministic finite automaton](https://ocme.evemisslab.com/en/atlas/atlas-finite-automaton/) - [Subset](https://ocme.evemisslab.com/en/atlas/atlas-subset/) ## [Pumping lemma for regular languages](https://ocme.evemisslab.com/en/atlas/atlas-pumping-lemma-regular-languages/) 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. atlas-pumping-lemma-regular-languages · canonical_mko [Pumping lemma for regular languages](https://ocme.evemisslab.com/en/mko/mko-pumping-lemma-regular-languages/) ## Prerequisite nodes - [Regular language](https://ocme.evemisslab.com/en/atlas/atlas-regular-language/) - [Deterministic finite automaton](https://ocme.evemisslab.com/en/atlas/atlas-finite-automaton/) - [Mathematical proof](https://ocme.evemisslab.com/en/atlas/atlas-proof/) - [Pigeonhole principle](https://ocme.evemisslab.com/en/atlas/atlas-pigeonhole-principle/) - [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/atlas/atlas-quantifier/) - [Mathematical induction](https://ocme.evemisslab.com/en/atlas/atlas-mathematical-induction/) ## [Context-free grammar](https://ocme.evemisslab.com/en/atlas/atlas-context-free-grammar/) A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side. atlas-context-free-grammar · canonical_mko [Context-free grammar](https://ocme.evemisslab.com/en/mko/mko-context-free-grammar/) ## Prerequisite nodes - [Abstract computational model](https://ocme.evemisslab.com/en/atlas/atlas-computational-model/) - [Set](https://ocme.evemisslab.com/en/atlas/atlas-set/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) ## [Pushdown automaton](https://ocme.evemisslab.com/en/atlas/atlas-pushdown-automaton/) A nondeterministic PDA uses finite control, input and stack transitions; this slice accepts only after reading the whole input in a final state. atlas-pushdown-automaton · canonical_mko [Pushdown automaton](https://ocme.evemisslab.com/en/mko/mko-pushdown-automaton/) ## Prerequisite nodes - [Abstract computational model](https://ocme.evemisslab.com/en/atlas/atlas-computational-model/) - [Deterministic finite automaton](https://ocme.evemisslab.com/en/atlas/atlas-finite-automaton/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Context-free grammar](https://ocme.evemisslab.com/en/atlas/atlas-context-free-grammar/) ## [Turing-machine decision model](https://ocme.evemisslab.com/en/atlas/atlas-turing-machine-decision-model/) Under fixed deterministic single-tape semantics, a decider must halt on every legal input and return the correct accept/reject answer. atlas-turing-machine-decision-model · canonical_mko [Turing-machine decision model](https://ocme.evemisslab.com/en/mko/mko-turing-machine-decision-model/) ## Prerequisite nodes - [Abstract computational model](https://ocme.evemisslab.com/en/atlas/atlas-computational-model/) - [Decision problem](https://ocme.evemisslab.com/en/atlas/atlas-decision-problem/) - [Deterministic finite automaton](https://ocme.evemisslab.com/en/atlas/atlas-finite-automaton/) - [Algorithm correctness](https://ocme.evemisslab.com/en/atlas/atlas-algorithm-correctness/) ## [Many-one reduction](https://ocme.evemisslab.com/en/atlas/atlas-many-one-reduction/) A≤mB means a total computable map f satisfies x∈A exactly when f(x)∈B for every x. atlas-many-one-reduction · canonical_mko [Many-one reduction](https://ocme.evemisslab.com/en/mko/mko-many-one-reduction/) ## Prerequisite nodes - [Decision problem](https://ocme.evemisslab.com/en/atlas/atlas-decision-problem/) - [Function mapping](https://ocme.evemisslab.com/en/atlas/atlas-function-mapping/) - [Function composition](https://ocme.evemisslab.com/en/atlas/atlas-composition/) - [Turing-machine decision model](https://ocme.evemisslab.com/en/atlas/atlas-turing-machine-decision-model/) ## [Contraction mapping theorem](https://ocme.evemisslab.com/en/atlas/atlas-contraction-mapping-theorem/) 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. atlas-contraction-mapping-theorem · canonical_mko [Contraction mapping theorem](https://ocme.evemisslab.com/en/mko/mko-contraction-mapping-theorem/) ## Prerequisite nodes - [Orbits, fixed points and periodic points](https://ocme.evemisslab.com/en/atlas/atlas-orbit-fixed-periodic/) - [Complete metric space](https://ocme.evemisslab.com/en/atlas/atlas-complete-metric-space/) - [Convergence in a metric space](https://ocme.evemisslab.com/en/atlas/atlas-metric-convergence/) - [Infinite series](https://ocme.evemisslab.com/en/atlas/atlas-infinite-series/) ## [Newton method for root finding](https://ocme.evemisslab.com/en/atlas/atlas-newton-method/) 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. atlas-newton-method · canonical_mko [Newton method for root finding](https://ocme.evemisslab.com/en/mko/mko-newton-method/) ## Prerequisite nodes - [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/atlas/atlas-derivative/) - [Numerical approximation and error](https://ocme.evemisslab.com/en/atlas/atlas-numerical-approximation-error/) - [Algorithm correctness](https://ocme.evemisslab.com/en/atlas/atlas-algorithm-correctness/) - [Equation](https://ocme.evemisslab.com/en/atlas/atlas-equation/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/atlas/atlas-derivative-rules/) ## [Composite trapezoidal quadrature](https://ocme.evemisslab.com/en/atlas/atlas-trapezoidal-quadrature/) On a strictly increasing partition, add each subinterval width times its two-endpoint average to approximate the definite integral. atlas-trapezoidal-quadrature · canonical_mko [Composite trapezoidal quadrature](https://ocme.evemisslab.com/en/mko/mko-trapezoidal-quadrature/) ## Prerequisite nodes - [Signed polynomial definite integrals on finite intervals](https://ocme.evemisslab.com/en/atlas/atlas-definite-integral/) - [Numerical approximation and error](https://ocme.evemisslab.com/en/atlas/atlas-numerical-approximation-error/) - [Arithmetic operations](https://ocme.evemisslab.com/en/atlas/atlas-arithmetic-operations/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) ## [Gradient vector](https://ocme.evemisslab.com/en/atlas/atlas-gradient-vector/) In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector. atlas-gradient-vector · canonical_mko [Gradient vector](https://ocme.evemisslab.com/en/mko/mko-gradient-vector/) ## Prerequisite nodes - [Partial derivative](https://ocme.evemisslab.com/en/atlas/atlas-partial-derivative/) - [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/atlas/atlas-vector/) - [Multivariable real-valued function](https://ocme.evemisslab.com/en/atlas/atlas-multivariable-real-function/) ## [Gradient descent](https://ocme.evemisslab.com/en/atlas/atlas-gradient-descent/) In fixed Euclidean coordinates, a positive step moves opposite the gradient; descent and convergence still require objective and step-size hypotheses. atlas-gradient-descent · canonical_mko [Gradient descent](https://ocme.evemisslab.com/en/mko/mko-gradient-descent/) ## Prerequisite nodes - [Gradient vector](https://ocme.evemisslab.com/en/atlas/atlas-gradient-vector/) - [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) - [Sequence](https://ocme.evemisslab.com/en/atlas/atlas-sequence/) - [Real vector space](https://ocme.evemisslab.com/en/atlas/atlas-real-vector-space/) ## [Necessary Lagrange multiplier condition for equality constraints](https://ocme.evemisslab.com/en/atlas/atlas-lagrange-multiplier-condition/) At a regular feasible constrained local extremum, the objective gradient is a linear combination of the constraint gradients. atlas-lagrange-multiplier-condition · canonical_mko [Necessary Lagrange multiplier condition for equality constraints](https://ocme.evemisslab.com/en/mko/mko-lagrange-multiplier-condition/) ## Prerequisite nodes - [Gradient vector](https://ocme.evemisslab.com/en/atlas/atlas-gradient-vector/) - [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) - [Linear systems in two variables and degenerate constraints](https://ocme.evemisslab.com/en/atlas/atlas-system-linear-equations/) - [Matrix rank](https://ocme.evemisslab.com/en/atlas/atlas-matrix-rank/) - [Linear independence and bases](https://ocme.evemisslab.com/en/atlas/atlas-linear-independence-basis/) ## [Standard form of a linear program](https://ocme.evemisslab.com/en/atlas/atlas-linear-program-standard-form/) This node fixes one primal convention: maximize c^T*x subject to A*x≤b and x≥0. atlas-linear-program-standard-form · canonical_mko [Standard form of a linear program](https://ocme.evemisslab.com/en/mko/mko-linear-program-standard-form/) ## Prerequisite nodes - [Linear programming](https://ocme.evemisslab.com/en/atlas/atlas-linear-programming/) - [Finite matrices: dimensions, indexing, and transpose](https://ocme.evemisslab.com/en/atlas/atlas-matrix/) - [Linear systems in two variables and degenerate constraints](https://ocme.evemisslab.com/en/atlas/atlas-system-linear-equations/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/) ## [Linear-programming duality](https://ocme.evemisslab.com/en/atlas/atlas-linear-program-duality/) 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. atlas-linear-program-duality · canonical_mko [Linear-programming duality](https://ocme.evemisslab.com/en/mko/mko-linear-program-duality/) ## Prerequisite nodes - [Standard form of a linear program](https://ocme.evemisslab.com/en/atlas/atlas-linear-program-standard-form/) - [Optimization problem](https://ocme.evemisslab.com/en/atlas/atlas-optimization-problem/) - [Usual order on the real numbers](https://ocme.evemisslab.com/en/atlas/atlas-order-relation/)