Generating functions for real character degree sums of finite general linear and unitary groups

We compute generating functions for the sum of the real-valued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has Frobenius–Schur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of Hall–Littlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with Frobenius–Schur indicator 1 or ?1.     

Linear differential equations and multiple zeta values. II. A generalization of the WKB method

For linear differential equations x(n)+a₁x(n−1)+?+anx=0 (and corresponding linear differential systems) with large complex parameter λ and meromorphic coefficients aj=aj(t;λ) we prove existence of analogues of Stokes matrices for the asymptotic WKB solutions. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We also discuss a generalization of the Hukuhara-Levelt-Turritin theorem about formal reduction of a linear differential system near an irregular singular point t=0 to a normal form with ramified change of time to the case of systems with large parameter. These results are applied to some hypergeometric equations related with generating functions for multiple zeta values.     

Equality of P-Partition Generating Functions

To every labeled poset (P, ω), one can associate a quasisymmetric generating function for its (P, ω)-partitions. We ask: when do two labeled posets have the same generating function? Since the special case corresponding to skew Schur function equality is still open, a complete classification of equality among (P, ω) generating functions is likely too much to expect. Instead, we determine necessary conditions and separate sufficient conditions for two labeled posets to have equal generating functions. We conclude with a classification of all equalities for labeled posets with small numbers of linear extensions.     

Functional equations of the dilogarithm in motivic cohomology

We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH²(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generating the integral motivic cohomology groups of some number fields.     

An algorithm for rescaling a matrix positive definite

For a given square real matrix M, we present a general algorithm which decides the existence of a positive diagonal matrix D such that DM is positive definite and which constructs the D if it exists. It is shown that solving this matrix rescaling problem is equivalent to finding a solution of an infinite system of linear inequalities. The algorithm solves this infinite system of linear inequalities by generating and solving a sequence of linear programs.     

The perturbed compound Poisson risk model with linear dividend barrier

In this paper, we consider a diffusion perturbed classical compound Poisson risk model in the presence of a linear dividend barrier. Partial integro-differential equations for the moment generating function and the nth moment of the present value of all dividends until ruin are derived. Moreover, explicit solutions for the nth moment of the present value of dividend payments are obtained when the individual claim size distribution is exponential. We also provided some numerical examples to illustrate the applications of the explicit solutions. Finally we derive partial integro-differential equations with boundary conditions for the Gerber-Shiu function.     

Practical Band Toeplitz Preconditioning and Boundary Layer Effects

In the last decade many efficient iterative solvers for n×n Hermitian positive definite Toeplitz systems have been devised. Many of them are based on band Toeplitz preconditioners: they are optimal but require the knowledge of the zeros of the underlying generating function. In some cases this information is available and in some cases is not. In [27] an economic numerical procedure for finding these zeros within a given precision has been devised. Here we provide conditions on the approximation error of these zeros in order to maintain the optimality that is a convergence rate independent of the dimension n of the considered linear systems.     

Simple proofs of open problems about the structure of involutions in the Riordan group

We prove that if D=(g(x),f(x)) is an element of order 2 in the Riordan group then g(x)=±exp[Φ(x,xf(x)] for some antisymmetric function Φ(x,z). Also we prove that every element of order 2 in the Riordan group can be written as BMB^-1 for some element B and M=(1,-1) in the Riordan group. These proofs provide solutions to two open problems presented by L. Shapiro [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585-596].     

q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q-analogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C_p. Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.     

Linear operators that preserve Boolean rank of Boolean matrices

The Boolean rank of a nonzero m × n Boolean matrix A is the minimum number k such that there exist an m× k Boolean matrix B and a k × n Boolean matrix C such that A = BC. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and 2. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and k for some 1 < k ? m.     

Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients

We first propose a way for generating Lie algebras from which we get a few kinds of reduced 6 6 Lie algebras, denoted by R6, R8 and R1,R6/2, respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R6 whose compatibility gives rise to an integrable hierarchy with 4- potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R6 to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R62, we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

Enumeration aspects of maximal cliques and bicliques

We present a general framework to study enumeration algorithms for maximal cliques and maximal bicliques of a graph. Given a graph G, we introduce the notion of the transition graph T(G) whose vertices are maximal cliques of G and arcs are transitions between cliques. We show that T(G) is a strongly connected graph and characterize a rooted cover tree of T(G) which appears implicitly in [D.S. Johnson, M. Yannakakis, C.H. Papadimitriou, On generating all maximal independent sets, Information Processing Letters 27 (1988) 119-123; S. Tsukiyama, M. Ide, M. Aiyoshi, I. Shirawaka, A new algorithm for generating all the independent sets, SIAM Journal on Computing 6 (1977) 505-517]. When G is a bipartite graph, we show that the Galois lattice of G is a partial graph of T(G) and we deduce that algorithms based on the Galois lattice are a particular search of T(G). Moreover, we show that algorithms in [G. Alexe, S. Alexe, Y. Crama, S. Foldes, P.L. Hammer, B. Simeone, Consensus algorithms for the generation of all maximal bicliques, Discrete Applied Mathematics 145 (1) (2004) 11-21; L. Nourine, O. Raynaud, A fast algorithm for building lattices, Information Processing Letters 71 (1999) 199-204] generate maximal bicliques of a bipartite graph in O(n²) per maximal biclique, where n is the number of vertices in G. Finally, we show that under some specific numbering, the transition graph T(G) has a hamiltonian path for chordal and comparability graphs.     

Horse paths, restricted 132-avoiding permutations, continued fractions, and Chebyshev polynomials

Several authors have examined connections among 132-avoiding permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we find analogues for some of these results for permutations π avoiding 132 and 1□23 (there is no occurrence π_i<π_j<π_j+1 such that 1?i?j-2) and provide a combinatorial interpretation for such permutations in terms of lattice paths. Using tools developed to prove these analogues, we give enumerations and generating functions for permutations which avoid both 132 and 1□23, and certain additional patterns. We also give generating functions for permutations avoiding 132 and 1□23 and containing certain additional patterns exactly once. In all cases we express these generating functions in terms of Chebyshev polynomials of the second kind.     

Frobenius Problem and dead ends in integers

Let a and b be positive and relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In addition, for given integers a and b, we explicitly describe all such elements in Z. Finally, we show that Z has only finitely many dead ends with respect to any finite symmetric generating set. In Appendix A we show that every finitely generated group has a generating set with respect to which dead ends exist.     

On the topological linear spaces whose subspace lattices are modular

LetX be a topological linear space and letL(X) be a lattice of all closed subspaces ofX. We show that in many cases the modularity ofL(X) implies that every bounded subset ofX is finite-dimensional. We derive some topological consequences of the latter result. Due to the significance of the modularity condition forL(X) in quantum axiomatics and elswhere (see [1,14,15]) results also might find application outside the realm of topological linear spaces.     

How small can a lattice of order-dimension n be?

The aim of this paper is to study the order-dimension of partition lattices and linear lattices. Our investigations were motivated by a question due to Bill Sands: For a lattice L, does dim L=n always imply |L|≥2 ⁿ ? We will answer this question in the negative since both classes of lattices mentioned above form counterexamples. In the case of the partition lattices, we will determine the dimension up to an absolute constant. For the linear lattice over GF(2), L _n , we determine the dimension up to a factor C/n for an absolute constant C.     

Weyl Type Theorems and Restrictions

In this paper we show that for a bounded linear operator \({T\in L(X)}\) acting on a Banach space X, the study of generalized Weyl’s theorem for T can be reduced to the study of generalized Weyl’s theorem for some restrictions of T.