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A stochastic matrix, also called a probability matrix, probability transition matrix, transition matrix, substitution matrix, or Markov matrix, is matrix used to characterize ...

By analogy with the tanc function, define the tanhc function by tanhc(z)={(tanhz)/z for z!=0; 1 for z=0. (1) It has derivative (dtanhc(z))/(dz)=(sech^2z)/z-(tanhz)/(z^2). (2) ...

A divisor, also called a factor, of a number n is a number d which divides n (written d|n). For integers, only positive divisors are usually considered, though obviously the ...

Apéry's constant is defined by zeta(3)=1.2020569..., (1) (OEIS A002117) where zeta(z) is the Riemann zeta function. Apéry (1979) proved that zeta(3) is irrational, although ...

Every finite-dimensional Lie algebra of characteristic p=0 has a faithful finite-dimensional representation.

Let G=SL(n,C). If lambda in Z^n is the highest weight of an irreducible holomorphic representation V of G, (i.e., lambda is a dominant integral weight), then the G-map ...

An operator Gamma=sum_(i=1)^me_i^Ru^(iR) on a representation R of a Lie algebra.

Let A be any algebra over a field F, and define a derivation of A as a linear operator D on A satisfying (xy)D=(xD)y+x(yD) for all x,y in A. Then the set D(A) of all ...

A finite-dimensional Lie algebra all of whose elements are ad-nilpotent is itself a nilpotent Lie algebra.

Every finite-dimensional Lie algebra of characteristic p!=0 has a faithful finite-dimensional representation.