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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).
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
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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.
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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|.
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Higher-order derivative
Define each existing finite derivative recursively by f^(0)=f and f^(n+1)=(f^(n))'.
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Hausdorff space
A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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.
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Gradient vector
In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector.
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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.
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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.