Acta Mathematica Hungarica最新文献

Finite groups are pervasive in mathematics.In studies into the structure of a finite group, the action of a finite group A on a finite group G such that the orders ((|A|, {rm and}, |G|)) of A and G are coprime is a basic topic. Here we present a Theorem in this topic which extends basic results “from a single prime to a finite set of primes” and implies that for any finite group G, if A acts trivially on (F^*(G)) (the generalized Fitting subgroup of G), then A acts trivially on G. Also our Theorem implies and extends a useful result of Z. Arad and G. Glauberman and implies and extends Thompson’s celebrated (P times Q) Lemma and an important Lemma of D. M. Goldschmidt.

This paper investigates the approximation of higher-order derivatives using integral operators constructed from Gegenbauer polynomials. We establish key equivalence results connecting the approximation error in the L²-norm to the decay of high-frequency components in the Fourier domain. A notable contribution is the derivation of Titchmarsh-type theorems that relate the solid average and Lanczos operators to the smoothness of the function being approximated. Additionally, we derive conditions under which the Fourier transform of functions belongs to (L^beta(mathbb{R})), depending on the smoothness of the function and the parameters of the approximation.

Let ({u_n}_{n=1}^{infty}) be Sylvester’s sequence (sequence A000058 in the OEIS), and let (a_1 < a_2 < cdots) be any other sequence of positive integers satisfying (sum_{i=1}^infty frac{1}{a_i} = 1). Erdős and Graham conjectured that

This conjecture has recently been resolved constructively by Kamio and independently by the present authors. In this paper, we focus on a generalization of this conjecture using a non-constructive approach. Specifically, assuming the unproven claim of Erdős and Graham that every rational number admits an eventually greedy best Egyptian underapproximation, we establish a more general asymptotic inequality involving best Egyptian underapproximations.

Let X and Y be two simply connected rational CW-complexes, and (n in mathbb{N}). We study the homotopy set ([P^nX, P^nY]), where PⁿX and PⁿY are the (n)-th Postnikov sections of X and Y, respectively. An equivalence relation is defined on this set, revealing connections with the cohomology groups. This approach uses rational homotopy theory to uncover new structural insights.

In the 1960s Moser asked how dense a subset of (mathbb{R}^d) can be if no pairs of points in the subset are exactly distance 1 apart.There has been a long line of work showing upper bounds on this density. One curious feature of dense unit distance avoiding sets is that they appear to be ''clumpy,'' i.e. forbidding unit distances comes hand in hand with having more than the expected number distance (approx 2) pairs.

In this work we rigorously establish this phenomenon in (mathbb{R}^2). We show that dense unit distance avoiding sets have over-represented distance (approx 2) pairs, and that this clustering extends to typical unit distance avoiding sets. To do so, we build off of the linear programming approach used previously to prove upper bounds on the density of unit distance avoiding sets.

We consider measure theoretic equicontinuity and sensitivity via Furstenberg family. We introduce the notion of (mathcal {F})-(mu)-equicontinuity which is the refined version of (mu )-equicontinuity using Furstenberg family (mathcal {F}) and prove that when (mathcal {F}) is a filter, a given dynamical system ((X,T)) is (mathcal {F})-(mu)-equicontinuous if and only if it is (mathcal {F})-(mu)-(f)-equicontinuous with respect to every continuous function (f colon {X to mathbb {C}} ). In addition, under certain conditinos, we prove that an ergodic measure theoretic dynamical system is either (kmathcal {F})-(mu )-sensitive or (mathcal {F})-(mu)-equicontinuous.

In the study of sums of finite sets of integers, most attention has been paid to sets with small sumsets (Freiman's theorem and related work) and to sets with large sumsets (Sidon sets and (B_h)-sets). This paper focuses on the full range of sizes of (h)-fold sums of a set of (k) integers. Many new results and open problems are presented.

We consider the space of functions almost in (L_p) and endow it with the topology of asymptotic (L_p)-convergence. This yields a completely metrizable topological vector space which, on finite measure spaces, coincides with the space of measurable functions equipped with the topology of (local) convergence in measure. We investigate analogs of classical results such as dominated convergence and Vitali convergence theorems. For (mathbb{R}^d) as the underlying measure space, we establish results on approximation by smooth functions and separability. Further aspects, including local boundedness, local convexity, and duality are examined in the (mathbb{R}^d) setting, revealing fundamental differences from standard (L_p) spaces.

We introduce the notion of an open-separated sequence, related to a free sequence, and define the cardinal function(text{os} ( X)). Nonregular and regular examples (X) are given such that (text{os} ( X)<F(X)). Motivated by a question of Angelo Bella, we show that if(X) is Hausdorff then (|X|leq hL(X)^{text{os} ( X)psi_c(X)}). As(text{os} ( X)psi_c(X)leq hL(X)) if (X) is Hausdorff, this gives astrengthening of the De Groot-Smirnov bound (2^{hL(X)}) for thecardinality of a Hausdorff space. Additionally we show ( text{nw}( X)leq text{hL}(X)^{text {os} ( X)}) if (X) is regular. A consequence is that if (X) is regular and either almost radial or hereditarily weakly Whyburn then ( { |X|leq hL(X)^{text{os} ( X)} } ).

We consider rotational beta expansions in dimensions 1, 2 and 4 and view them as expansions on real numbers, complex numbers, and quaternions, respectively.We give sufficient conditions on the parameters (alpha, beta in (0,1)) so that particular cylinder sets arising from the expansions are winning or losing Schmidt ((alpha,beta))-game.