In this paper, for any two bipartite ordered sets P and Q; we de ne the convolution P Q of P and Q: For dim(P ) = s and dim(Q) = t; we prove that s+ t (U + V ) 2 dim(P Q) s+t (U+V )+2; where U+V is the max-min integer of the certain realizers. In particular, we also prove that dim(Pn;k) = n+k b n+k 3 c for 2 k n < 2k and dim(Pn;k) = n for n 2k; where Pn;k = Sn Sk is the convolution of two stand...

Let G be a group which has a finite number hn(G) of irreducible linear representations in GLn(C) for all n ≥ 1. Let ζ(G, s) = P∞ n=1 hn(G)n −s be its representation zeta function. First, in case G = H oXQ is a permutational wreath product with respect to a permutation group Q on a finite set X, we establish a formula for ζ(G, s) in terms of the zeta functions of H and of subgroups of Q, and of ...

Let q be a prime power, Fq be the field of q elements, and k, m be positive integers. A bipartite graph G = Gq(g, h, k,m) is defined as follows. The vertex set of G is a union of two copies P and L of two-dimensional vector spaces over Fq, with two vertices (p1, p2) ∈ P and [ l1, l2 ] ∈ L being adjacent if and only if p2 + l2 = p1l 1 . We prove that graphs Gq(k, m) and Gq′(k,m) are isomorphic i...

Let E/Q be an elliptic curve defined over Q of conductor N and let Gal(Q/Q) be the absolute Galois group of an algebraic closure Q of Q. For an automorphism σ ∈ Gal(Q/Q), we let Q be the fixed subfield of Q under σ. We prove that for every σ ∈ Gal(Q/Q), the Mordell-Weil group of E over the maximal Galois extension of Q contained in Q σ has infinite rank, so the rank of E(Q σ ) is infinite. Our ...

Let E/K be an abelian extension of number fields, with E 6= Q. Let ∆ and n denote the absolute discriminant and degree of E. Let σ denote an element of the Galois group of E/K. We prove the following theorems, assuming the Extended Riemann Hypothesis: (1) There is a degree-1 prime p of K such that ( p E/K ) = σ, satisfying Np ≤ (1 + o(1))(log ∆ + 2n)2. (2) There is a degree-1 prime p of K such ...

The paper studies the Lie symmetries of nonlinear Fokker-Planck equation in one dimension, which are associated to weighted Kaniadakis entropy. In particular, diffusive equation, entropy, found. MaxEnt problem entropy is given a complete solution, together with thermodynamic relations extend known ones from non-weighted case. Several different, but related, arguments point out subtle dichotomou...

Let V ∼= Fq be a v-dimensional vector space over the finite field Fq with q elements. By [ V k ] we denote the set of all k-dimensional subspaces in V . Its size is given by the q-binomial coefficient [ v k ]q = ∏k−1 i=0 qv−qi qk−qi for 0 ≤ k ≤ v and 0 otherwise. The set of all subspaces of V forms a metric space associated with the so-called subspace distance dS(U,W ) = dim(U + W ) − dim(U ∩ W...

A generalized Filbert matrix is introduced, sharing properties of the Hilbert matrix and Fibonacci numbers. Explicit formulæ are derived for the LU-decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q-analysis and to leave the justi…cation of the necessary identities to the q-version of Zeilberger’s celebrated algorithm. 1. Introduction The Filbert (...

Let X c pn be an irreducible projective variety defined over the finite field F q. We will say that X satisfies the property F F N (k, q) (Finite Field Nullstellensatz of degree k over F q) if every homogeneous polynomial over F q of degree k on pn vanishing on X (F q) vanishes on X, that is, it vanishes at all points of X defined over the algebraic closure of F q . We study when FFN(k,q) holds...

Let m > 2, ζm an m-th primitive root of 1, q ≡ 1 mod 2m a prime number, s = sq a primitive root modulo q and f = fq = (q − 1)/m. We study the Jacobi sums Ja,b = − ∑q−1 k=2 ζ a inds(k)+b inds(1−k) m , 0 ≤ a, b ≤ m−1, where inds(k) is the least nonnegative integer such that s inds(k) ≡ k mod q. We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE...