On irreducible p,q-representations of gl(2)

The theory of irreducible p,q-representations of the complex Lie algebra gl(2) is developed. We construct a one variable model of irreducible p,q-representations of gl(2) in terms of p,q-derivative operator, and derive a generating function based on it.     

Applying incomplete statistics to nonextensive systems with different q indices

The nonextensive statistics based on the q-entropy has been so far applied to systems in which the q value is uniformly distributed. For the systems containing different q’s, the applicability of the theory is still a matter of investigation. The difficulty is that the class of systems to which the theory can be applied is actually limited by the usual nonadditivity rule of entropy which is no more valid when the systems contain non uniform distribution of q values. In this paper, within the framework of the so called incomplete information theory, we propose a more general nonadditivity rule of entropy prescribed by the zeroth law of thermodynamics. This new nonadditivity generalizes in a simple way the usual one and can be proved to lead uniquely to the q-entropy.     

A nonterminating 8φ7 summation for the root system Cr

Using multiple q-integrals and a determinant evaluation, we establish a nonterminating ₈φ₇ summation for the root system C_r. We also give some important specializations explicitly.     

Cubic theta functions

Some new identities for the four cubic theta functions a′(q,z), a(q,z), b(q,z) and c(q,z) are given. For example, we show that

a′(q,z)³=b(q,z)³+c(q)²c(q,z).

This is a counterpart of the identity

a(q,z)³=b(q)²b(q,z³)+c(q,z)³,

which was found by Hirschhorn et al.

The Laurent series expansions of the four cubic theta functions are given. Their transformation properties are established using an elementary approach due to K. Venkatachaliengar. By applying the modular transformation to the identities given by Hirschhorn et al., several new identities in which a′(q,z) plays the role of a(q,z) are obtained.     

Maximal partial spreads and the modular n-queen problem

We prove that for any integer n in the interval there is a maximal partial spread of size n in PG (3, q) where q is odd and q7. We also prove that there are maximal partial spreads of size (q²+3)/2 when gcd(q+1,24)=2 or 4 and of size (q²+5)/2 when gcd(q+1,24)=4.     

Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preserving formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq"(t)+ Bq(t)=f(q(t)), where A ∈ R^m×m is a symmetric positive definite matrix, B ∈ R^m×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f(q)=-▽_qV(q) for a real-valued function V(q). The energy-preserving formulae can exactly preserve the Hamiltonian H(q', q)=(1)/2q'^τ Aq' + (1)/2q^τ Bq + V(q). We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.     

Duality for finite multiple harmonic q-series

We define two finite q-analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q-analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial coefficients. Special cases of these identities—for example, with all parameters equal to 1—have occurred in the literature. The special case with only one parameter reduces to an identity for the divisor generating function, which has received some attention in connection with problems in sorting theory. The general case can be viewed as a duality result, reminiscent of the duality relation for the ordinary multiple zeta function.     

A shortcut to asymptotics for orthogonal polynomials

We consider the asymptotic behavior of the ratios q_n+1(z)/q_n(z) of polynomials orthonormal with respect to some positive measure μ. Let the recurrence coefficients _n and β_n converge to 0 and , respectively. Then, q_n+1(z)/q_n(z) Φ(z),for n→∞ locally uniformly for , where Φ maps conformally onto the exterior of the unit disc (Nevai (1979)). We provide a new and direct proof for this and some related results due to Nevai, and apply it to convergence acceleration of diagonal Padé approximants.     

Near vector spaces over GF(q) and (v,q+1,1)-BIBD''s

The usual construction of (v,q+1,1)−BIBD's from vector spaces over GF(q) is generalized to the class of near vector spaces over GF(q). It is shown that every (v,q+1,1)−BIBD can be constructed from a near vector space over GF(q). Some corollaries are: Given a (v₁,q+1,1)−BIBD P₁,B₁ and a (v₂,q+1,1)−BIBD P₂,B₂, there is a ((q−1)v₁v₂+v₁+v₂,q+1,1)−BIBD P₃,B₃ containing P₁,B₁ and P₂,B₂ as disjoint subdesigns. If there is a (v,q+1,1)−BIBD then there is a ((q−1)v+1,q,1)−BIBD. Every finite partial (v,q,1)−BIBD can be embedded in a finite (v′,q+1,1)−BIBD.     

Approximating geometric bottleneck shortest paths

In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: given two points p and q of S and a real number L, compute (or approximate) a shortest path between p and q in the subgraph of the complete graph on S consisting of all edges whose lengths are less than or equal to L. We present efficient algorithms for answering several query problems of this type. Our solutions are based on Euclidean minimum spanning trees, spanners, and the Delaunay triangulation. A result of independent interest is the following. For any two points p and q of S, there is a path between p and q in the Delaunay triangulation, whose length is less than or equal to 2π/(3cos(π/6)) times the Euclidean distance |pq| between p and q, and all of whose edges have length at most |pq|.     

On a hypergeometric identity of Gelfand, Graev and Retakh

A hypergeometric identity equating a triple sum to a single sum, originally found by Gelfand, et al. (Russian Math. Surveys 47 (1992)) by using systems of differential equations, is given hypergeometric proofs. As a bonus, several q-analogues can be derived.     

Constructions of (q, K, λ, t, Q) almost difference families

The concept of a (q, k, λ, t) almost difference family (ADF) has been introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, (q, K, λ, t, Q)-ADFs, where K = {k₁, k₂, ..., k_r} is a set of positive integers and Q = (q₁, q₂,..., q_r) is a given block-size distribution sequence. A necessary condition for the existence of a (q, K, λ, t, Q)-ADF is given, and several infinite classes of (q, K, λ, t, Q)-ADFs are constructed.     

Davis–Wielandt shell and q-numerical range

We describe the boundary of the q-numerical range of a square matrix using its Davis–Wielandt shell. The result is used to generate an algorithm for plotting the q-numerical range of the square matrix. Computations of the q-numerical ranges of a special class of matrices are explicitly given.     

Cohomology of the Universal Enveloping Algebras of Certain Bigraded Lie Algebras

Let p be an odd prime and q = 2(p-1).Up to total degree t-s max{(5p~3+ 6p~2+ 6 p +4)q-10,p~4q},the generators of H~(s,t)(U(L)),the cohomology of the universal enveloping algebra of a bigraded Lie algebra L,are determined and their convergence is also verified.Furthermore our results reveal that this cohomology satisfies an analogous Poinare duality property.This largely generalizes an earlier classical results due to J.P.May.     

5-regular 3-polytopal graphs with edges of only two types and shortness exponents less than one

It is shown that, if q 29 and q 0 (mod 3), the infinite class of 5-regular 3-polytopal graphs whose edges are incident with either two triangles or a triangle and a q-gon contains nonhamiltonian members and even has shortness exponent less than one.     

On the generalized Erdös-Szekeres Conjecture — a new upper bound

We prove the following result: For every two natural numbers n and q, n q + 2, there is a natural number E(n, q) satisfying the following:

1. (1) Let S be any set of points in the plane, no three on a line. If |S| E(n, q), then there exists a convex n-gon whose points belong to S, for which the number of points of S in its interior is 0 (mod q).

2. (2) For fixed q, E(n,q) 2^c(q)·n, c(q) is a constant depends on q only.

Part (1) was proved by Bialostocki et al. [2] and our proof is aimed to simplify the original proof. The proof of Part (2) is completely new and reduces the huge upper bound of [2] (a super-exponential bound) to an exponential upper bound.     

ContributionsSign-balanced covering matrices

A q × n array with entries from 0, 1,…,q − 1 is said to form a difference matrix if the vector difference (modulo q) of each pair of columns consists of a permutation of [0, 1,… q − 1]; this definition is inverted from the more standard one to be found, e.g., in Colbourn and de Launey (1996). The following idea generalizes this notion: Given an appropriate δ (-[−1, 1]^t, a λq × n array will be said to form a (t, q, λ, Δ) sign-balanced matrix if for each choice C₁, C₂,…, C_t of t columns and for each choice = (₁,…,_t) Δ of signs, the linear combination ∑_j=1^t _jC_j contains (mod q) each entry of [0, 1,…, q − 1] exactly λ times. We consider the following extremal problem in this paper: How large does the number k = k(n, t, q, λ, δ) of rows have to be so that for each choice of t columns and for each choice (₁, …, _t) of signs in δ, the linear combination ∑_j=1^t _jC_j contains each entry of [0, 1,…, q t- 1] at least λ times? We use probabilistic methods, in particular the Lovász local lemma and the Stein-Chen method of Poisson approximation to obtain general (logarithmic) upper bounds on the numbers k(n, t, q, λ, δ), and to provide Poisson approximations for the probability distribution of the number W of deficient sets of t columns, given a random array. It is proved, in addition, that arithmetic modulo q yields the smallest array - in a sense to be described.     

Cryptanalysis of Rijndael S-box and improvement

By the discovered correlation between linear functions over GF(qⁿ) and matrices over GF(q), a new scheme is presented to resolve the algebraic expression of Rijndael S-box in this paper. This new scheme has the advantage of predetermining in the case of a given random basis over GF(qⁿ). The reason why only nine terms are involved in the algebraic expression of Rijndael S-box is presented, which corrects the available inaccurate illustration. An improved AES S-box is presented to improve the complexity of AES S-box algebraic expression with terms increasing from 9 to 255 and algebraic degree invariable. The improved AES S-box also has good properties of Boolean functions in SAC and balance, and is capable of attacking against differential cryptanalysis with high reliable security. We finally summarize all the available methods to determine the algebraic expression of Rijndael S-box.     

Möbius Regular Maps of Order pq

M?bius regular maps are surface embeddings of graphs with doubled edges such that(i)the automorphism group of the embedding acts regularly on flags and(ii) each doubled edge is a center of a M?bius band on the surface. In this paper, we classify M?bius regular maps of order pq for any two primes p and q, where p≠q.     

Explicit construction of graphs with an arbitrary large girth and of large size

Let k 3 be a positive odd integer and 1 be a power of a prime. In this paper we give an explicit construction of a q-regular bipartite graph on v = 2q^k vertices with girth g k + 5. The constructed graph is the incidence graph of a flag-transitive semiplane. For any positive integer t we also give an example of a q = 2^t-regular bipartite graph on v = 2q^{k + 1} vertices with girth g k + 5 which is both vertex-transitive and edge-transitive.