Natural number
The basic number system for counting and discrete order.
atlas-natural-number · canonical_mko
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
Integer
A number system extending the natural numbers to represent positive values, negative values and zero.
atlas-integer · canonical_mko
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Rational number
A number expressible as a ratio of two integers.
atlas-rational-number · canonical_mko
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Real number
A number system containing rational and irrational numbers and supporting continuous quantities and real analysis.
atlas-real-number · canonical_mko
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Arithmetic operations
Addition, subtraction, multiplication, division and their basic rules.
atlas-arithmetic-operations · canonical_mko
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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
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Ratio and 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
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Natural-number powers and principal square roots
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
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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
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Usual order on the real numbers
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
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Set
A basic mathematical structure treating objects as a collection.
atlas-set · canonical_mko
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Set membership
The relation describing whether an element belongs to a set.
atlas-set-membership · canonical_mko
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Subset
The relation that every element of one set also belongs to another set.
atlas-subset · canonical_mko
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Set operations
Operations on sets, including union, intersection, difference and relative complement.
atlas-set-operations · canonical_mko
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Proposition
A mathematical statement with a truth value.
atlas-proposition · canonical_mko
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Logical connectives
Ways to combine propositions, including negation, conjunction, disjunction, implication and equivalence.
atlas-logical-connectives · canonical_mko
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Quantifiers and explicit domains
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
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Mathematical proof
A process establishing a conclusion from explicit premises through valid reasoning.
atlas-proof · canonical_mko
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Counterexamples and domain eligibility
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
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Binary relations and set projections
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
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Variables and algebraic expressions
Symbols representing unknown quantities and the structure of operations.
atlas-variable-expression · canonical_mko
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Equation
A condition requiring two algebraic expressions to be equal.
atlas-equation · canonical_mko
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Linear equation in one variable
An equation over an explicit number system with one unknown and a nonzero linear coefficient.
atlas-linear-equation · canonical_mko
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Quadratic equations over the reals
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
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Univariate polynomials over the rationals
A polynomial is finitely supported rational-coefficient data; combine like powers before reading coefficients or degree.
atlas-polynomial · canonical_mko
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Polynomial factorization over the rationals
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
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Linear systems in two variables and degenerate constraints
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
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Plane vectors in a fixed basis
This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise.
atlas-vector · canonical_mko
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Finite matrices: dimensions, indexing, and transpose
Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances.
atlas-matrix · canonical_mko
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Matrix-represented linear maps on the real plane
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
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Determinants of real 2×2 matrices
This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices.
atlas-determinant · canonical_mko
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Real eigenvalues and nonzero eigenvectors of 2×2 matrices
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
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Points, straight lines and planes
Basic objects of position and extension in Euclidean geometry.
atlas-point-line-plane · canonical_mko
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Angle
A turning relation between two rays or directions.
atlas-angle · canonical_mko
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Euclidean length
The distance between two points in a plane or in space.
atlas-euclidean-length · canonical_mko
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Triangle
A basic polygon formed by three noncollinear points and the segments joining them.
atlas-triangle · canonical_mko
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Right triangle
A triangle with one right angle.
atlas-right-triangle · canonical_mko
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Congruence of nondegenerate triangles with explicit correspondence
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
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Labeled triangle similarity with a specified positive scale
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
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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
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Euclidean distance in an equal-unit orthonormal frame
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
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Pythagorean theorem
In a right triangle, the sum of the squared leg lengths equals the squared hypotenuse length.
atlas-pythagorean-theorem · canonical_mko
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Circle boundaries and zero-radius degeneracy
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
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Plane-coordinate isometries and uniform scaling
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
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Function mapping
A mapping assigning exactly one output to each input in its domain.
atlas-function-mapping · canonical_mko
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Domain, codomain, and 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
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Graphs of real-valued functions
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
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Affine functions and degree-one functions
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
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Quadratic functions in vertex form
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
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Injectivity and distinct inputs
Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain.
atlas-injective · canonical_mko
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Surjectivity and codomain coverage
Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages.
atlas-surjective · canonical_mko
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Bijectivity and unique preimages
A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage.
atlas-bijective · canonical_mko
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Function 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
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Inverse functions of bijections
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
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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
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Tends-to relation
A core relation describing a variable, sequence or function approaching a target state.
atlas-tends-to-relation · canonical_mko
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Sequence limits and exact tail cutoffs
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
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Function limits on punctured real domains
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
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Point values and continuity of real functions
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
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Average rate of change between distinct inputs
Average rate of change is an endpoint difference quotient with a nonzero input difference, not an instantaneous derivative.
atlas-rate-of-change · canonical_mko
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Real derivatives at a point and finite difference quotients
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
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Differentiating polynomial sums, products, and compositions
Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials.
atlas-derivative-rules · canonical_mko
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Polynomial antiderivatives and the choice of constant
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
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Exact area of half-open rectangles and unit-cell unions
This entry computes exact area for half-open integer rectangles and finite unit-cell unions, including empty and overlapping cases.
atlas-area · canonical_mko
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Signed polynomial definite integrals on finite intervals
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
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Fundamental theorem of calculus: continuity and oriented endpoints
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
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Sample space
The set of all possible outcomes of a random experiment.
atlas-sample-space · canonical_mko
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Event
A set of outcomes in a sample space.
atlas-event · canonical_mko
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Probability on finite sample spaces
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
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Conditional probability in finite models
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
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Real random variables on finite sample spaces
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
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Expectation in finite models
Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence.
atlas-expectation · canonical_mko
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Variance in finite probability models
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
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Descriptive summaries of nonempty observation lists
Mean, median, range, and the explicitly defined maximal-frequency set summarize a nonempty finite observation list while retaining multiplicities.
atlas-descriptive-statistics · canonical_mko
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Basic counting principles
Addition and multiplication principles for counting finite choices.
atlas-counting-principle · canonical_mko
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Factorial
The function taking a natural number n to the product of the first n positive integers.
atlas-factorial · canonical_mko
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Permutations and combinations
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
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Finite simple undirected graphs
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
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Nonempty finite trees: connectivity, cycles, and unique paths
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
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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
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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
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Open and closed sets
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
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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
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Continuous maps between topological spaces
A map is topologically continuous when every open set in its target has an open preimage in its source.
atlas-topological-continuity · canonical_mko
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Natural-number 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
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Greatest common divisor and coprimality
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
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Prime natural number
A prime natural number is at least 2 and has no positive divisors other than 1 and itself.
atlas-prime-number · canonical_mko
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Natural-number congruence modulo a positive modulus
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
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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
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Abstract 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
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Algorithm correctness
Algorithm correctness separates partial correctness from termination; total correctness requires both.
atlas-algorithm-correctness · canonical_mko
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Asymptotic resource bounds
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
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Formal statements in a proof assistant
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
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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
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Discrete-time 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
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Orbits, fixed points and periodic points
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
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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
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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
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Numerical approximation and 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
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Finite sign-bracket updates in the 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
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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
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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
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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
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Linear 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
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Discrete exponential growth model
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
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Discrete logistic increment model
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
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Compound-interest model
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
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Present value and 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
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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
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Diagnostic mathematical assessment records
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
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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
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Complex modulus and conjugation
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
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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
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Linear combinations and 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
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Linear independence and bases
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
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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
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Subgroups and group homomorphisms
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
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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
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Convergence in a metric space
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
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Sigma-algebras and measure spaces
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
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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
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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
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Discrete-time 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
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Finite-state 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
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Multivariable real-valued 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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Finite field
A finite commutative unital ring is a finite field when every nonzero element has a multiplicative inverse.
atlas-finite-field · canonical_mko
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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
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Quotient ring of a finite 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
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Polynomial ring over a 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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Linear map between real vector spaces
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
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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
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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
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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
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Independence of random variables
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
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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
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Finite 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
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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
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Finite 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
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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
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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
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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
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Independent identically distributed 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
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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
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Graph walks and simple paths
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
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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
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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
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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
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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
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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
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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
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Cardinality comparison of sets
An injection, surjection or bijection between explicit finite source and target sets supplies a corresponding finite cardinality-comparison certificate.
atlas-cardinality-comparison · canonical_mko
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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
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Real 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
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Kernel of a linear map
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
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Image of a linear map
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
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Dimension of a finite-dimensional vector space
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
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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
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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
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Direct sum of subspaces
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
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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
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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
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Topological 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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Weak law of 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
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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
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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
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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
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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
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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
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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
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Euler trails and circuits
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
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Spanning tree
A spanning tree is a connected acyclic subgraph containing every vertex of its parent graph.
atlas-spanning-tree · canonical_mko
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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
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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
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Cliques and independent sets
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
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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
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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
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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
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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
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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
Prerequisite nodes
Fermat's 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
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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
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Deterministic 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
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Regular language
A language is regular when a complete DFA recognizes all and exactly its words.
atlas-regular-language · canonical_mko
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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
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Orbit of a finite group action
The images of one point under all group elements form its orbit; all orbits partition the point carrier.
atlas-finite-group-orbit · canonical_mko
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Stabilizer of a finite group action
The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup.
atlas-finite-group-stabilizer · canonical_mko
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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
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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
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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
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Conjugacy class in a finite group
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
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Center of a finite group
The center consists of all elements commuting with every group element and is a normal subgroup.
atlas-finite-group-center · canonical_mko
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Lagrange theorem for 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
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Kernel of a finite group homomorphism
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
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First isomorphism theorem for 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
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Cauchy theorem for 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
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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
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Subsequence
A subsequence selects terms by a strictly increasing natural-index map and preserves their original order.
atlas-subsequence · canonical_mko
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Monotone bounded 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
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Bolzano-Weierstrass theorem for real sequences
Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map.
atlas-bolzano-weierstrass-theorem-real · canonical_mko
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Intermediate value theorem
A continuous real function on a closed interval attains each value between its endpoint values.
atlas-intermediate-value-theorem · canonical_mko
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Mean value theorem
If a<b, f is continuous on [a,b] and differentiable on (a,b), some interior c has derivative equal to the endpoint secant slope.
atlas-mean-value-theorem · canonical_mko
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Higher-order derivative
Define each existing finite derivative recursively by f^(0)=f and f^(n+1)=(f^(n))'.
atlas-higher-order-derivative · canonical_mko
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Taylor polynomial
At center a, derivative values of orders0 through n, scaled by their factorials, form the order-n Taylor polynomial.
atlas-taylor-polynomial · canonical_mko
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Taylor remainder theorem
Under sufficient differentiability, a Lagrange remainder and derivative bound control the Taylor approximation error.
atlas-taylor-remainder-theorem · canonical_mko
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Comparison tests for series
Eventual termwise bounds transfer convergence from a nonnegative upper series or divergence from a nonnegative lower series.
atlas-series-comparison-test · canonical_mko
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Ratio test for series
For eventually nonzero terms, an absolute successive-ratio limit below1 gives absolute convergence, above1 gives divergence, and equal to1 is inconclusive.
atlas-ratio-test-series · canonical_mko
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Radius of convergence of a power series
Around its center c, a power series has absolute convergence inside radius R and divergence outside; endpoints require separate decisions.
atlas-power-series-radius-of-convergence · canonical_mko
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A uniform limit of continuous functions is continuous
Continuous functions on one common domain have a continuous limit when convergence is uniform on that domain.
atlas-uniform-limit-continuity-theorem · canonical_mko
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Real exponential function
Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0.
atlas-real-exponential-function · canonical_mko
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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
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Topological basis
A topological basis generates a topology through carrier coverage and pointwise local intersection refinement.
atlas-topological-basis · canonical_mko
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Subspace topology
For Y⊆X, the relative topology consists of every intersection of an ambient open set with Y.
atlas-subspace-topology · canonical_mko
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Finite product topology
Open rectangles generate the topology of a finite Cartesian product of topological spaces.
atlas-finite-product-topology · canonical_mko
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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
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Hausdorff space
A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods.
atlas-hausdorff-space · canonical_mko
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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
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Continuous image of a compact set
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
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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
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Connected component of a topological space
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
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A compact subset of a Hausdorff space is closed
If K is compact in the Hausdorff space X, then K is closed in X.
atlas-compact-subset-closed-in-hausdorff · canonical_mko
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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
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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
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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
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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
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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
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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
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Central limit theorem for finite-variance IID sequences
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
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Strong law of large numbers for finite-valued IID sequences
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
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Maximum-likelihood estimator in a finite model
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
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Mean squared error of an estimator
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
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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
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Sufficient statistic in a finite model
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
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Confidence-interval coverage in a finite model
Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter.
atlas-finite-confidence-interval-coverage · canonical_mko
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Topological ordering of a directed acyclic graph
A finite directed graph has a topological ordering exactly when it has no directed cycle.
atlas-topological-ordering-dag · canonical_mko
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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
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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
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Euler's theorem in 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
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Multiplicative order modulo 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
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Unique prime factorization of integers
Every positive integer n>1 has a prime factorization whose prime multiset is unique up to factor order.
atlas-unique-prime-factorization · canonical_mko
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Wilson theorem
An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n.
atlas-wilson-theorem · canonical_mko
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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
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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
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Euler criterion for quadratic residues
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
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p-adic valuation of 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
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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
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Mobius inversion on finite divisor sets
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
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Regular expression
Regular expressions recursively build languages from the empty language, epsilon, alphabet symbols, union, concatenation and Kleene star.
atlas-regular-expression · canonical_mko
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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
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Subset construction from an NFA to a 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
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Pumping lemma for 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
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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
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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
Prerequisite nodes
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
Prerequisite nodes
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
Prerequisite nodes
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
Prerequisite nodes
Newton method for root finding
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
Prerequisite nodes
Composite 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
Prerequisite nodes
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
Prerequisite nodes
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
Prerequisite nodes
Necessary Lagrange multiplier condition for equality constraints
At a regular feasible constrained local extremum, the objective gradient is a linear combination of the constraint gradients.
atlas-lagrange-multiplier-condition · canonical_mko
Prerequisite nodes
Standard form of a linear program
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
Prerequisite nodes
Linear-programming 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