Abstract. We present an involution for a classical identity on the alternate sum of the Gauss coefficients in terms of the traditional Ferrers diagram. It turns out that the refinement of our involution with restrictions on the height of Ferrers diagram implies a generalization of the Gauss identity, which is a terminating form of the q-Kummer identity. Furthermore, we extend the Gauss identity...

Let A be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system An−1. Let B = B(q) be the Varchenko matrix for this arrangement with all hyperplane parameters equal to q. We show that B is the matrix with rows and columns indexed by permutations with σ, τ entry equal to qi(στ −1) where i(π) is the number of inversions of π. Equivalently B is the matrix for l...

In this lecture we work exclusively with first-order logic with equality. Fix a signature σ. A theory T is a set of sentences (closed formulas) that is closed under semantic entailment, i.e., if T |= F then F ∈ T . Given a σ-structure A it is clear that the set of sentences that hold in A is a theory. We denote this theory by Th(A) and call it the theory of A. We say that a theory is complete i...

Let F denote an algebraically closed field and assume that q ? is a primitive d th root of unity with ? 1 , 2 4 . The universal Askey–Wilson algebra ? unital associative -algebra defined by generators relations. are A B C the relations assert each + ? commutes We show every finite-dimensional irreducible -module dimension less than or equal to { if odd ; / even Moreover we provide example bound...

Let G be a simple connected graph with order n. $$\mathcal{L}(G)$$ and $$\mathcal{Q}(G)$$ the normalized Laplacian signless matrices of G, respectively. λk(G) k-th smallest eigenvalue G. Denote by ρ(A) spectral radius matrix A. In this paper, we study behaviors λ2(G) $$\rho(\mathcal{L}(G))$$ when is perturbed three operations. We also properties X for bipartite graphs, where unit eigenvector co...

We determine explicit quantum seeds for classes of quantized matrix algebras. Furthermore, we obtain results on centers and block diagonal forms of these algebras. In the case where q is an arbitrary root of unity, this further determines the degrees.

We show that computing e-th roots modulo n is easier than factoring n with currently known methods, given subexponential access to an oracle outputting the roots of numbers of the form xi + c. Here c is fixed and xi denotes small integers of the attacker’s choosing. The attack comes in two flavors: – A first version is illustrated here by producing selective roots of the form xi + c in Ln( 1 3 ...

We submitted a public-key encryption scheme, EPOC, and digital signature scheme, TSH-ESIGN, to IEEE P1363a. The security of EPOC and TSH-ESIGN is based on the intractability of factoring n = pq, where p and q are primes. TSH-ESIGN is also based on the intractability of the approximate e-th root (AERP) assumption, which is the approximate version of the RSA assumption. This draft describes the l...

1 Q I − − ) ( y , where the j th entry of the row vector y is the probability that the system state seen by the first arrival during a busy period is j and 1 Q I − − ) ( is the fundamental matrix associated with the standard GI/M/1 queue. In this paper, we present the entries of 1 Q I − − ) ( explicitly. Also, we illustrate how to find y by examples such as the N -policy GI/M/1 queue with or wi...