Irrationality proof of certain Lambert series using little q-Jacobi polynomials

We apply the Padé technique to find rational approximations to

     

The Andrews-Stanley partition function and Al-Salam-Chihara polynomials

For any partition λ let ω(λ) denote the four parameter weight

ω(λ)=a^{∑_i≥1⌈λ_2i−1/2⌉}b^{∑_i≥1⌊λ_2i−1/2⌋}c^{∑_i≥1⌈λ_2i/2⌉}d^{∑_i≥1⌊λ_2i/2⌋},     

Super-simple, pan-orientable and pan-decomposable GDDs with block size 4

In this paper we study (4,2μ)-GDDs of type gⁿ possessing both the pan-decomposable property introduced by Granville, Moisiadis, Rees, On complementary decompositions of the complete graph, Graphs and Combinatorics 5 (1989) 57-61 and the pan-orientable property introduced by Grüttmüller, Hartmann, Pan-orientable block designs, Australas. J. Combin. 40 (2008) 57-68. We show that the necessary condition for a (4,2μ)-GDD satisfying both of these properties, namely (1) n≥4, μg(n−1)≡0 (mod 3), and (2) g−1,n are not both even if μ is odd are sufficient. When λ=2, our designs are super-simple.We also determine the spectrum of (4,2)-GDDs which are super-simple and possess some of the decomposable/orientable conditions, but are not pan-decomposable or pan-orientable. In particular, we show that the necessary conditions for a super-simple directable (4,2)-GDD of type gⁿ are sufficient.     

Constructions of optimal quaternary constant weight codes via group divisible designs

Generalized Steiner systems were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g+1 with minimum Hamming distance 2k−3, in which each codeword has length v and weight k. As to the existence of a , a lot of work has been done for k=3, while not so much is known for k=4. The notion k-^∗GDD was first introduced by Chen et al. and used to construct . The necessary condition for the existence of a is v≥14. In this paper, it is proved that there exists a for any prime power and v≥19. By using this result, the known results on the existence of optimal quaternary constant weight codes are then extended.     

Binary relations and reduced hypergroups

Different partial hypergroupoids are associated with binary relations defined on a set H. In this paper we find sufficient and necessary conditions for these hypergroupoids in order to be reduced hypergroups. Given two binary relations ρ and σ on H we investigate when the hypergroups associated with the relations ρ∩σ, ρ∪σ and ρσ are reduced. We also determine when the cartesian product of two hypergroupoids associated with a binary relation is a reduced hypergroup.     

Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces

We extend the notion of L₂-B-discrepancy introduced in [E. Novak, H. Wo?niakowski, L₂ discrepancy and multivariate integration, in: W.W.L. Chen, W.T. Gowers, H. Halberstam, W.M. Schmidt, and R.C. Vaughan (Eds.), Analytic Number Theory. Essays in Honour of Klaus Roth, Cambridge University Press, Cambridge, 2009, pp. 359-388] to what we shall call weighted geometric L₂-discrepancy. This extension enables us to consider weights in order to moderate the importance of different groups of variables, as well as to consider volume measures different from the Lebesgue measure and classes of test sets different from measurable subsets of Euclidean spaces.We relate the weighted geometric L₂-discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wo?niakowski.Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite-dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by the algorithms.     

Independence polynomials of k-tree related graphs

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let α(G) denote the cardinality of a maximum independent set and f_s(G) for 0≤s≤α(G) denote the number of independent sets of s vertices. The independence polynomial defined first by Gutman and Harary has been the focus of considerable research recently. Wingard bounded the coefficients f_s(T) for trees T with n vertices: for s≥2. We generalize this result to bounds for a very large class of graphs, maximal k-degenerate graphs, a class which includes all k-trees. Additionally, we characterize all instances where our bounds are achieved, and determine exactly the independence polynomials of several classes of k-tree related graphs. Our main theorems generalize several related results known before.     

Multilinear extrapolation and applications to the bilinear Hilbert transform

We present two extrapolation methods for multi-sublinear operators that allow us to derive estimates for general functions from the corresponding estimates on characteristic functions. Of these methods, the first is applicable to general multi-sublinear operators while the second requires working with the so-called (ε,δ)-atomic operators. Among the applications, we discuss some new endpoint estimates for the bilinear Hilbert transform.     

Bilateral inversions and terminating basic hypergeometric series identities

The q-analogue of Legendre inversions is established and generalized to bilateral sequences. They are employed to investigate the dual relations of three basic formulae due to Jackson and Bailey, on balanced ₃?₂-series, well-poised ₈?₇-series and bilateral ₆ψ₆-series. Several terminating well-poised series identities are consequently derived, including the q-Dixon formulae on terminating ₃ψ₃-series and two terminating well-poised ₅ψ₅-series identities due to [F.H. Jackson, Certain q-identities, Quart. J. Math. (Oxford) 12 (1941) 167-172; W.N. Bailey, On the analogue of Dixon’s theorem for bilateral basic hypergeometric series, Quart. J. Math. (Oxford) 1 (1950) 318-320].     

Triangle path transit functions, betweenness and pseudo-modular graphs

The geodesic and induced path transit functions are the two well-studied interval functions in graphs. Two important transit functions related to the geodesic and induced path functions are the triangle path transit functions which consist of all vertices on all u,v-shortest (induced) paths or all vertices adjacent to two adjacent vertices on all u,v-shortest (induced) paths, for any two vertices u and v in a connected graph G. In this paper we study the two triangle path transit functions, namely the I^Δ and J^Δ on G. We discuss the betweenness axioms, for both triangle path transit functions. Also we present a characterization of pseudo-modular graphs using the transit function I^Δ by forbidden subgraphs.     

Absolutely convergent Fourier series and function classes

We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs either to one of the Lipschitz classes Lip(α) and lip(α) for some 0<α?1, or to one of the Zygmund classes Λ_∗(1) and λ_∗(1). Our theorems generalize some of those by Boas [R.P. Boas Jr., Fourier series with positive coefficients, J. Math. Anal. Appl. 17 (1967) 463-483] and one by Németh [J. Németh, Fourier series with positive coefficients and generalized Lipschitz classes, Acta Sci. Math. (Szeged) 54 (1990) 291-304]. We also prove a localized version of a theorem by Paley [R.E.A.C. Paley, On Fourier series with positive coefficients, J. London Math. Soc. 7 (1932) 205-208] on the existence and continuity of the derivative of f.     

Degree conditions for group connectivity

Let G be a 2-edge-connected simple graph on n≥13 vertices and A an (additive) abelian group with |A|≥4. In this paper, we prove that if for every uv∉E(G), max{d(u),d(v)}≥n/4, then either G is A-connected or G can be reduced to one of K_2,3,C₄ and C₅ by repeatedly contracting proper A-connected subgraphs, where C_k is a cycle of length k. We also show that the bound n≥13 is the best possible.     

Limit-point criteria for semi-degenerate singular Hamiltonian differential systems with perturbation terms

Some limit-point criteria are obtained for higher-dimensional semi-degenerate singular Hamiltonian differential systems with perturbation potential terms by using M(λ)-theory. Results in this paper cover many previous results of Hartman, Levinson, Titchmarsh and Read.     

Endomorphisms of the shift dynamical system, discrete derivatives, and applications

All continuous endomorphisms f_∞ of the shift dynamical system S on the 2-adic integers Z₂ are induced by some , where n is a positive integer, B_n is the set of n-blocks over {0, 1}, and f_∞(x)=y₀y₁y₂… where for all i∈N, y_i=f(x_ix_i+1…x_i+n−1). Define D:Z₂→Z₂ to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z₂→Z₂ by V(x)=−1−x. We prove that D, V°D, S, and V°S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x+1 function T:Z₂→Z₂ to a parity-neutral dynamical system. We also construct a conjugacy R from D to T. We apply these results to establish that, in order to prove the 3x+1 conjecture, it suffices to show that for any m∈Z⁺, there exists some n∈N such that R⁻¹(m) has binary representation of the form or .     

Matching polynomials for chains of cycles

Došli? and Måløy (2010) [2] obtained the extremal 6-cactus chains with respect to the number of matchings and of independent sets. Motivated by the prior paper, in this paper we give recurrences for matching polynomials of ortho-chains and meta-chains, and show that they are the h-cactus chains with the most matchings.     

Zeros of orthogonal Laurent polynomials and solutions of strong Stieltjes moment problems

The strong Stieltjes moment problem for a bisequence consists of finding positive measures μ with support in [0,∞) such that

     

Quantum Brownian motion and generalizations of Hermite polynomials

We introduce new families of orthogonal polynomials H_D,n, motivated by the non-equilibrium evolution of a quantum Brownian particle (qBp). The H_D,n’s generalize non-trivially the standard Hermite polynomials, employed for classical Brownian motion. We treat several models (labelled by D) for a non-equilibrium qBp, by means of the Wigner function W, in the presence of a “heat bath” at thermal equilibrium, with and without ab initio friction. For long times (for a suitable class of initial conditions), the non-equilibrium Wigner function W should approach, in some sense, the (time-independent) equilibrium Wigner function W_eq,D, which describes the thermal equilibrium of the qBp with the “heat bath” and plays a central role. W_eq,D is chosen to be the weight function which orthogonalizes the H_D,n’s. New results on W_eq,D and on the H_D,n’s are reported. We justify the key role of the H_D,n’s as follows. Using the H_D,n’s, moments W_eq,D,n and W_n are introduced for W_eq,D and W, respectively. At equilibrium, all moments W_eq,D,n except the lowest one (W_eq,D,0) vanish identically. Off-equilibrium, one expects that, for long times (for suitable initial conditions): (i) all non-equilibrium moments W_n (except the lowest moment W₀), will approach zero, while (ii) the lowest non-equilibrium moment W₀ will tend to W_eq,D,0(≠0). To complete the justification, we outline how the approximate long-time non-equilibrium theories determined by W₀ for the different models (D) yield Smoluchowski equations and irreversible evolutions of the qBp towards thermal equilibrium.     

Edge-connectivity and edge-disjoint spanning trees

Given a graph G, for an integer c∈{2,…,|V(G)|}, define λ_c(G)=min{|X|:X⊆E(G),ω(G−X)≥c}. For a graph G and for an integer c=1,2,…,|V(G)|−1, define,

     

Graphic sequences with a realization containing a complete multipartite subgraph

A nonincreasing sequence of nonnegative integers π=(d₁,d₂,…,d_n) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realizeπ. For a given graph H, a graphic sequence π is potentiallyH-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the terms of π. For a graph H and n∈Z⁺, σ(H,n) is defined as the smallest even integer m so that every n-term graphic sequence π with σ(π)≥m is potentially H-graphic. Let denote the complete t partite graph such that each partite set has exactly s vertices. We show that and obtain the exact value of σ(K_j+K_s,s,n) for n sufficiently large. Consequently, we obtain the exact value of for n sufficiently large.     

Pruning processes and a new characterization of convex geometries

We provide a new characterization of convex geometries via a multivariate version of an identity that was originally proved, in a special case arising from the k-SAT problem, by Maneva, Mossel and Wainwright. We thus highlight the connection between various characterizations of convex geometries and a family of removal processes studied in the literature on random structures.