Matrix Power S-box Analysis

* Work supported by the Lithuanian State Science and Studies Foundation.

On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem

Pólya’s fundamental enumeration theorem and some results from Williamson’s generalized setup of it are proved in terms of Schur- Macdonald’s theory (S-MT) of “invariant matrices”. Given a permutation group W ≤ Sd and a one-dimensional character χ of W , the polynomial functor Fχ corresponding via S-MT to the induced monomial representation Uχ = ind|Sdv/W (χ) of Sd , is studied. It turns out that the characteristic ch(Fχ ) is the weighted inventory of some set J(χ) of W -orbits in the integer-valued hypercube [0, ∞)d . The elements of J(χ) can be distinguished among all W -orbits by a maximum property. The identity ch(Fχ ) = ch(Uχ ) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson. Pólya’s theorem can be obtained from the above identity by the specialization χ = 1W , where 1W is the unit character of W.

Cayley-Hamilton Theorem for Matrices over an Arbitrary Ring

2000 Mathematics Subject Classification: 15A15, 15A24, 15A33, 16S50.

Numerical Solution of Fractional Diffusion-Wave Equation with two Space Variables by Matrix Method

Mathematics Subject Classi¯cation 2010: 26A33, 65D25, 65M06, 65Z05.

The Variety of Leibniz Algebras Defined by the Identity x(y(zt)) ≡ 0

2000 Mathematics Subject Classification: Primary: 17A32; Secondary: 16R10, 16P99, 17B01, 17B30, 20C30