Acta Mathematica Hungarica最新文献

For a positive real number (gamma), let (A_{gamma})be the sequence ({lfloor gammarfloor, lfloor 2gammarfloor, lfloor 2^2gammarfloor, ldots }), where (lfloor xrfloor) denotes the greatest integer not greater than (x). For positive real numbers (alpha) and (beta), write(A_{alpha,beta}=A_{alpha}cup A_{beta}). Erdős and Graham [2] posed the following problem: suppose that (alpha) and (beta) are positive real numbers with (alpha/beta) irrational. Can all sufficiently large integers be represented as the sum of distinct terms of (A_{alpha,beta})? Afterwards, Hegyvári [3] proved that, for (alphage 2) and (beta=2^nalpha) for some positive integer (n), there exist infinitely many positive integers which cannot be represented as the sum of distinct terms of (A_{alpha,beta}). Recently, Jiang and Ma [5] further consider the case (1<alpha<2). For a sequence (A) of nonnegative integers, let (P(A)) be the set of all integers which can be represented as the sum of distinct terms of (A). In this paper, for a class of positive real numbers (alpha) and (beta(=2^lalpha)), we determine all positive integers (x) such that (x+sum_{i=0}^ua_{l+i}notin P(A_{alpha,beta})) for every nonnegative integer (u). That is, (x+sum_{i=0}^ua_{l+i}notin P(A_{alpha,beta})) for every nonnegative integer (u) if and only if (1le x<a_l) and (xnotin P({a_0, ldots ,a_{l-1}})). Other related results are also obtained.

We investigate a moving recurrent problem for the nonautonomous dynamical system induced by the Cantor series expansion.To be precise, let (Q={q_{k}}_{kgeq1}) be a sequence of positive integers with (q_{k}geq2) for all (kgeq1). Put(T_{Q}^{n}(x)=q_{1}cdots q_{n}x-lfloor q_{1}cdots q_{n}xrfloor) for each (ngeq1), which gives the (Q)-Cantor series expansion.We focus on the following ({n_{k},r_{k}})-moving recurrent points proposed by Boshernitzan and Glasner:

where ({n_{k}}_{kgeq1}) and ({r_{k}}_{kgeq1}) are two given sequences of integers. It is proved that when ({n_{k}}_{kgeq1})and ({r_{k}}_{kgeq1}) tend to infinity, the set of ({n_{k},r_{k}})-moving recurrent points is of full Lebesgue measure. In addition,we study the size of the following quantitative version of ({n_{k},r_{k}})-moving recurrent set:

where (varphi colon mathbb{N}rightarrowmathbb{R}^{+}) is a positive function and ``i.m.'' stands for ``infinitely many''. It is proved that when ({n_{k}}_{kgeq1}) and ({r_{k}}_{kgeq1}) tend to infinity, theLebesgue measure and Hausdorff measure of (R({n_{k},r_{k}})) respectively fulfill a dichotomy law according to the convergence or divergence of certain series.

We provide a method for reconstructing the underlying locallycompact Hausdorff space of an algebra of continuous functions vanishing at infinity, using the norm attainment sets of continuous functions. As an application,it is demonstrated that an order isomorphism of norm attainment sets betweenspaces of continuous functions induces a homeomorphism between the underlying topological spaces, even without linearity. Moreover, it turns out that a linearmap between algebras of continuous functions is an order isomorphism of normattainment sets if and only if it is a scalar multiple of an isometric isomorphismprovided that the underlying topological spaces are not two-point sets. We alsopresent a counterexample to the above statement in the two-point setting.

Let (G_1), ..., (G_m) be independentBernoulli random subgraphs of the complete graph (mathcal{K}_n) havingrandom sizes (X_1,dots, X_min {0,1,2,dots}) and edge densities (Q_1), ..., (Q_min [0,1]). Letting (n,mto+infty) we establish the connectivity threshold for the union ( bigcup_{i=1}^mG_i) defined on the vertex set of (mathcal{K}_n). We show that

where (lambda^{*}_{n,m}= ln n - frac{1}{n} sumnolimits_{i=1}^{m} textbf{E} X_{i}(1-(1-Q_i)^{|X_i-1|})).

Answering a question raised by V.V. Tkachuk in [10], we present several examples of (sigma)-compact spaces, some only consistent and some in ZFC, that are not countably tight but in which the closure of any discrete subset is countably tight. In fact, in some of our examples the closures of all discrete subsets are even first countable.

Let ( k geq 2 ) be an integer. One of the generalization of the classical Fibonacci sequence is defined by the recurrence relation( F_{n}^{(k)}=F_{n-1}^{(k)} + cdots + F_{n-k}^{(k)}) for all ( n geq 2) with the initial values ( F_{i}^{(k)}=0 ) for ( i=2-k, ldots, 0 ) and ( F_{1}^{(k)}=1) (. F_{n}^{(k)} ) is an order ( k ) generalization of the Fibonacci sequence and it is called ( k)-generalizedFibonacci sequence or shortly ( k)-Fibonacci sequence. Banks and Luca [7], among other things, determined all Fibonacci numbers which are concatenations of two Fibonacci numbers. In this paper, we consider the analogue of this problem in more general manner by taking into account the concatenations of two terms of the same sequence in base (b geq 2). First, we show that there exists only finitely many such concatenations for each ( k geq 2 ) and ( b geq 2 ). Next, we completely determine all these concatenations for all ( k geq 2) and ( 2 leq b leq 10 ).

The question of the relation between contractive conditions maybe extended to the problem of their iterates. It was shown by B. Fisher that fora Banach contraction there exists an iterate that fulfills Chatterjea contractivecondition, but the reverse relation holds under some restriction imposed on themetric space which assures that there exists an iterate of a Chatterjea contractionthat is a Banach contraction. However, the proposed restriction holds only for theidentity mapping which is not a Chatterjea contraction except for the singletondomain. We offer a possible adjustment of this approach with several examplesanswering some open questions on this topic.

This paper extends the Cohen-Ramanujan sum originally defined by Cohen to arbitrary residually finite Dedekind domains. We derive further properties that can be viewed as generalizations of those provided by Zheng [16] and Zheng-Chen-Hong [27]. In particular, we illustrate that the set of the Cohen-Ramanujan sums can be used as a basis for Fourier expansions just as the classical Ramanujan sums can.

We consider when a subset (Xsubsetmathbb{R}^{d}) has a Radon partition (X=X_{1}sqcup X_{2}) such that

showing that such a partition always exists when (X) has at least (lfloorfrac{3d}{2}rfloor+2) points in general position. The latter bound is sharp.

Let (mathcal L_{mathfrak F}) be the class of groups having a local system ({X_i : iin I}) of finite subgroups such that (X_i) is subnormal in (X_j) whenever (X_ileq X_j). It has been shown by Rae in [19] that the class of soluble (mathcal L_{mathfrak F})-groups is closer to the class of soluble periodic FC-groups than might be expected. The aim of this paper is to prove that, under some additional finite-rank assumptions, one can extend Rae's results to local systems of Černikov subgroups, showing for example that the locally nilpotent residual is always covered by normal Černikov subgroups of the group, and that the factor group by the Hirsch–Plotkin radical has Černikov conjugacy classes of elements (see Theorem 5.9).

In [2], Reinhold Baer introduced a characteristic subgroup of a group which coincides with the hypercentre in the finite case (we call this subgroup the Baer centre of the group); actually, as shown in [4], this subgroup coincides with the hypercentre even in periodic FC-groups. Extending these results, we prove that this equivalence holds in many relevant universes of locally finite groups (see Theorem 6.2) and in particular in certain classes of locally finite groups having local systems of the above-mentioned type (see Theorem 6.9).

Finally, in order to better understand the behaviour of the Baer centre in our context, we introduce and study a new class of groups that is strictly contained between the classes of periodic FC-groups and periodic BFC-groups, and that could be very useful from a computational point of view (see Section 7).