Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

Simple arguments on consecutive power residues

By some extremely simple arguments, we point out the following:

(i)

If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth power residues mod m is smaller than m/n.

(ii)

Let OK be the ring of algebraic integers in a quadratic field with d∈{−1,−2,−3,−7,−11}. Then, for any irreducible π∈OK and positive integer k not relatively prime to , there exists a kth power non-residue ω∈OK modulo π such that .

     

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

Let H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b∈H, let ω(H,b) be the smallest  N∈N₀∪{∞} having the following property: if  n∈N and  a₁,…,a_n∈H are such that b divides  a₁⋅…⋅a_n, then b already divides a subproduct of a₁⋅…⋅a_n consisting of at most N factors. The monoid H is called tame if . This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M.     

A combinatorial identity with application to Catalan numbers

By a very simple argument, we prove that if l,m,n∈{0,1,2,…} then

     

Convergence preserving mappings on topological groups

It is well known that a mapping is convergence preserving, that is, whenever an infinite series ∑an converges, the series ∑φ(an) converges, if and only if there exists m∈R such that φ(x)=mx in some neighborhood of 0. We explore convergence preserving mappings on Hausdorff topological groups, showing in particular, that if G×G is a Fréchet group, and H has no small subgroups, then a mapping is convergence preserving if and only if there is a neighborhood of the identity in G on which φ is a sequentially continuous homomorphism.     

Maps of several variables of finite total variation. II. E. Helly-type pointwise selection principles

Given a=(a₁,…,an), b=(b₁,…,bn)∈Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), denote by the Hildebrandt-Leonov total variation of f on , which has been recently studied in [V.V. Chistyakov, Yu.V. Tretyachenko, Maps of several variables of finite total variation. I, J. Math. Anal. Appl. (2010), submitted for publication]. The following Helly-type pointwise selection principle is proved: If a sequence{fj}_j∈Nof maps frominto M is such that the closure in M of the set{fj(x)}_j∈Nis compact for eachandis finite, then there exists a subsequence of{fj}_j∈N, which converges pointwise onto a map f such that. A variant of this result is established concerning the weak pointwise convergence when values of maps lie in a reflexive Banach space (M,‖⋅‖) with separable dual M^∗.     

Strong convergence of the composite Halpern iteration

Let C be a closed convex subset of a uniformly smooth Banach space E and let T:C→C be a nonexpansive mapping with a nonempty fixed points set. Given a point u∈C, the initial guess x₀∈C is chosen arbitrarily and given sequences , and in (0,1), the following conditions are satisfied:

(i)

;

(ii)

αn→0, βn→0 and 0<a?γn, for some a∈(0,1);

(iii)

, and . Let be a composite iteration process defined by

     

A further three critical points theorem

If X is a real Banach space, we denote by W_X the class of all functionals possessing the following property: if {u_n} is a sequence in X converging weakly to u∈X and lim inf_n→∞Φ(u_n)≤Φ(u), then {u_n} has a subsequence converging strongly to u.In this paper, we prove the following result:Let X be a separable and reflexive real Banach space; an interval; a sequentially weakly lower semicontinuous C¹ functional, belonging to W_X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X^∗; a C¹ functional with compact derivative. Assume that, for each λ∈I, the functional Φ−λJ is coercive and has a strict local, not global minimum, say .Then, for each compact interval [a,b]⊆I for which , there exists r>0 with the following property: for every λ∈[a,b] and every C¹ functional with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation

Φ^′(x)=λJ^′(x)+μΨ^′(x)     

Definable sets of generators in maximal cofinitary groups

A group G?Sym(N) is cofinitary if g has finitely many fixed points for every g∈G except the identity element. In this paper, we discuss the definability of maximal cofinitary groups and some related structures. More precisely, we show the following two results:

(1)

Assuming V=L, there is a set of permutations on N which generates a maximal cofinitary group.

(2)

Assuming V=L, there is a mad permutation family in Sym(N).

     

Super-stationary set, subword problem and the complexity

Let Ω⊂{0,1}^N be a nonempty closed set with N={0,1,2,…}. For N={N₀<N₁<N₂<?}⊂N and ω∈{0,1}^N, define ω[N]∈{0,1}^N by and

     

Sequences and filters of characters characterizing subgroups of compact Abelian groups

Let H be a countable subgroup of the metrizable compact Abelian group G and a (not necessarily continuous) character of H. Then there exists a sequence of (continuous) characters of G such that limn→∞χn(α)=f(α) for all α∈H and does not converge whenever α∈G?H. If one drops the countability and metrizability requirement one can obtain similar results by using filters of characters instead of sequences. Furthermore the introduced methods allow to answer questions of Dikranjan et al.     

A note on the Jacobian Conjecture

In this note, we show that, if the Druzkowski mappings F(X)=X+(AX)^∗3, i.e. F(X)=(x₁+(a₁₁x₁+?+a_1nx_n)³,…,x_n+(a_n1x₁+?+a_nnx_n)³), satisfies TrJ((AX)^∗3)=0, then where δ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension ?9 in the case .     

Precompact abelian groups and topological annihilators

For a compact Hausdorff abelian group K and its subgroup H≤K, one defines the g-closureg_K(H) of H in K as the subgroup consisting of χ∈K such that χ(a_n)?0 in T=R/Z for every sequence {a_n} in (the Pontryagin dual of K) that converges to 0 in the topology that H induces on . We prove that every countable subgroup of a compact Hausdorff group is g-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every g-closed subgroup of a compact Hausdorff group is realcompact. The techniques developed in the paper are used to construct a close relative of the closure operator g that coincides with the G_δ-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.     

Values of coefficients of cyclotomic polynomials

Let a(k,n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that . In this paper, we improve this result and prove that for any prime p and any integer l≥1, we have

{a(k,p^ln)∣n,k∈N}=Z.     

On the quadratic character of quadratic units

Let be a prime. Let a,b∈Z with p?a(a²+b²). In the paper we mainly determine by assuming p=c²+d² or p=Ax²+2Bxy+Cy² with AC−B²=a²+b². As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1).     

Generalized Haar integral

The aim of the paper is to generalize the notion of the Haar integral. For a compact semigroup S acting continuously on a Hausdorff compact space Ω, the algebra A(S)⊂C(Ω,R) of S-invariant functions and the linear space M(S) of S-invariant (real-valued) finite signed measures are considered. It is shown that if S has a left and right invariant measure, then the dual space of A(S) is isometrically lattice-isomorphic to M(S) and that there exists a unique linear operator (called the Haar integral) such that for each f∈A(S) and for any f∈C(Ω,R) and s∈S, , where .