Remarks on a generalization of the Davenport constant

A generalization of the Davenport constant is investigated. For a finite abelian group G and a positive integer k, let denote the smallest ? such that each sequence over G of length at least ? has k disjoint non-empty zero-sum subsequences. For general G, expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence is eventually an arithmetic progression with difference exp(G), and several questions arising from this fact are investigated. For elementary 2-groups, is investigated in detail; in particular, the exact values are determined for groups of rank four and five (for rank at most three they were already known).     

Davenport constant for semigroups

Let G be a finite commutative semigroup. The Davenport constant of G is the smallest integer d such that, every sequence S of d elements in G contains a subsequence T (≠S) with the same product of S. Let . Among other results, we determine D(R ^×)−D(U(R)), where R ^× is the multiplicative semigroup of R and U(R) is the group of units of R.     

Weighted Davenport’s constant and the weighted EGZ Theorem

Let G^∗,G be finite abelian groups with nontrivial homomorphism group . Let Ψ be a non-empty subset of . Let D_Ψ(G) denote the minimal integer, such that any sequence over G^∗ of length D_Ψ(G) must contain a nontrivial subsequence s₁,…,s_r, such that for some ψ_i∈Ψ. Let E_Ψ(G) denote the minimal integer such that any sequence over G^∗ of length E_Ψ(G) must contain a nontrivial subsequence of length |G|,s₁,…,s_|G|, such that for some ψ_i∈Ψ. In this paper, we show that E_Ψ(G)=|G|+D_Ψ(G)−1.     

Extremal product-one free sequences in Dihedral and Dicyclic Groups

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $D\left(G\right)$ such that every sequence of $G$ with $D\left(G\right)$ elements has a non-empty subsequence with product $1$. Let $D_{2n}$ be the Dihedral Group of order $2n$ and $Q_{4n}$ be the Dicyclic Group of order $4n$. Zhuang and Gao (2005) showed that $D\left(D_{2n}\right)=n+1$ and Bass (2007) showed that $D\left(Q_{4n}\right)=2n+1$. In this paper, we give explicit characterizations of all sequences $S$ of $G$ such that $\left|S\right|=D\left(G\right)?1$ and $S$ is free of subsequences whose product is 1, where $G$ is equal to $D_{2n}$ or $Q_{4n}$ for some $n$.     

The Erd?s-Ginzberg-Ziv theorem with units

Let x₁,…,x_r be a sequence of elements of Z_n, the integers modulo n. How large must r be to guarantee the existence of a subsequence x_i1,…,x_in and units α₁,…,α_n with α₁x_i1+?+α_nx_in=0? Our main aim in this paper is to show that r=n+a is large enough, where a is the sum of the exponents of primes in the prime factorisation of n. This result, which is best possible, could be viewed as a unit version of the Erd?s-Ginzberg-Ziv theorem. This proves a conjecture of Adhikari, Chen, Friedlander, Konyagin and Pappalardi.We also discuss a number of related questions, and make conjectures which would greatly extend a theorem of Gao.     

Zero-sum invariants on finite abelian groups with large exponent

Let $G$ be an additive finite abelian group with exponent $n$. Let $\mathsf{D}\left(G\right)$ be the Davenport constant of $G$, ${\mathsf{s}}_{kn}\left(G\right)$ the $k$th Erd?s–Ginzburg–Ziv constant of $G$, where $k$ is a positive integer. Recently, Gao, Han, Peng and Sun conjectured that ${\mathsf{s}}_{kn}\left(G\right)=kn+\mathsf{D}\left(G\right)?1$ holds if $k\ge ?\frac{\mathsf{D}\left(G\right)}{n}?$. Let $m,n$ be positive integers and $H$ an abelian $p$-group with $\mathsf{D}\left(H\right)\le p^n$. Let $G=H\oplus C_{m{p}^n}$. For any integer $k\ge 2$, we prove that ${\mathsf{s}}_{km{p}^n}\left(G\right)=(k+1)m{p}^n+\mathsf{D}\left(H\right)?2=km{p}^n+\mathsf{D}\left(G\right)?1.$ This verifies the above conjecture in this case. We also provide asymptotically tight bounds for zero-sum invariants $\mathsf{D}\left(G\right)$, ${\mathsf{s}}_{kn}\left(G\right)$ and $\eta \left(G\right)$ for a class of abelian groups with large exponent.     

Some zero-sum constants with weights

For an abelian group G, the Davenport constant D(G) is defined to be the smallest natural number k such that any sequence of k elements in G has a nonempty subsequence whose sum is zero (the identity element). Motivated by some recent developments around the notion of Davenport constant with weights, we study them in some basic cases. We also define a new combinatorial invariant related to (ℤ/nℤ) ^d , more in the spirit of some constants considered by Harborth and others and obtain its exact value in the case of (ℤ/nℤ)² where n is an odd integer.     

Existence and multiplicity results for some quasilinear elliptic equation with weights

Using variational methods, we show the existence and multiplicity of solutions of singular boundary value problems of the type

     

Periodic positive solutions for adelay nonlinear differential equation with piecewise constant arguments

Using a fixed-point theorem in cone, existence criteria are proved for the periodic positive solutions of a delay differential equation with piecewise constant arguments. Several examples concerning biological models for the survival of red blood cells are given.     

On some questions about KC and related spaces

Answering questions raised by O.T. Alas and R.G. Wilson, or by these two authors together with M.G. Tkachenko and V.V. Tkachuk, we show that every minimal SC space must be sequentially compact, and we produce the following examples:

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a KC space which cannot be embedded in any compact KC space;

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a countable KC space which does not admit any coarser compact KC topology;

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a minimal Hausdorff space which is not a k-space.

We also give an example of a compact KC space such that every nonempty open subset of it is dense, even if, as pointed out to us by the referee, a completely different construction carried out by E.K. van Douwen in 1993 leads to a space with the same properties.     

Integral representation with weights II, division and interpolation

Let f be a r×m-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic ψ such that ϕ=f ψ, provided that ϕ is holomorphic and annihilates a certain residue current with support on the set where f is not surjective. We also consider formulas for interpolation. As applications we obtain generalizations of various results previously known for the case r=1. The author was partially supported by the Swedish Research Council     

Mixed modulus of smoothness with Muckenhoupt weights and approximation by angle

Mixed modulus of smoothness in weighted Lebesgue spaces with Muckenhoupt weights are investigated. Using mixed modulus of smoothness we obtain Potapov type direct and inverse estimates of angular trigonometric approximation of functions in these spaces. Also we obtain equivalences between mixed modulus of smoothness and K-functional and realization functional. Fractional order modulus of smoothness is considered as well.     

The Dunkl-Williams constant, convexity, smoothness and normal structure

In this paper we exhibit some connections between the Dunkl-Williams constant and some other well-known constants and notions. We establish bounds for the Dunkl-Williams constant that explain and quantify a characterization of uniformly nonsquare Banach spaces in terms of the Dunkl-Williams constant given by M. Baronti and P.L. Papini. We also study the relationship between Dunkl-Williams constant, the fixed point property for nonexpansive mappings and normal structure.     

Green's function for second-order periodic boundary value problems with piecewise constant arguments

We obtain, under suitable conditions, the Green's function to express the unique solution for a second-order functional differential equation with periodic boundary conditions and functional dependence given by a piecewise constant function. This expression is given in terms of the solutions for certain associated problems. The sign of the solution is determined taking into account the sign of that Green's function.     

The James constant, the Jordan-von Neumann constant, weak orthogonality, and fixed points for multivalued mappings

We give some sufficient conditions for the Domínguez-Lorenzo condition in terms of the James constant, the Jordan-von Neumann constant, and the coefficient of weak orthogonality. As a consequence, we obtain fixed point theorems for multivalued nonexpansive mappings.     

Green's function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature

We extend the method of harmonic transplantation from Euclidean domains to spaces of constant positive or negative curvature. To this end the structure of the Green's function of the corresponding Laplace-Beltrami operator is investigated. By means of isoperimetric inequalities we derive complementary estimates for its distribution function. We apply the method of harmonic transplantation to the question of whether the best Sobolev constant for the critical exponent is attained, i.e. whether there is an extremal function for the best Sobolev constant in spaces of constant curvature. A fairly complete answer is given, based on a concentration-compactness argument and a Pohozaev identity. The result depends on the curvature.

     

The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces

We give some sufficient conditions for normal structure in terms of the von Neumann-Jordan constant, the James constant and the weak orthogonality coefficient introduced by B. Sims. In the rest of the paper, the von Neumann-Jordan constant and the James constant for the Bynum space are computed, and are used to show that our results are sharp.