Acta Mathematica Hungarica最新文献

Given a set (Xsubseteqmathbb{R}^2) of (n) points and a distance (d>0), the multiplicity of (d) is thenumber of times the distance (d) appears between points in (X). Let (a_1(X) geq a_2(X) geq cdots geq a_m(X)) denote the multiplicities of the (m) distances determined by (X) and let(a(X)=(a_1(X),dots,a_m(X))). In this paper, we study severalquestions from Erdős’s time regarding distance multiplicities. Among other results, we show that:

• (1) If (X) is convex or “not too convex”, then there exists a distance other than the diameter that has multiplicity at most (n).

• (2) There exists a set (Xsubseteqmathbb{R}^2) of (n) points, such that many distances occur with high multiplicity. In particular, at least (n^{Omega(1/loglog{n})}) distances have superlinear multiplicity in (n).

• (3) For any (not necessarily fixed) integer (1leq kleqlog{n}), there exists ( {Xsubseteqmathbb{R}^2 } ) of (n) points, such that the difference between the (k^{text{th}}) and ((k+1)^{text{th}}) largest multiplicities is at least (Omega(frac{nlog{n}}{k})). Moreover, the distances in (X) with the largest (k) multiplicities can be prescribed.

• (4) For every (nin N), there exists (Xsubseteqmathbb{R}^2) of (n) points, not all collinear or cocircular, such that (a(X)= (n-1,n-2,ldots,1)). There also exists (Xsubseteqmathbb{R}^2) of (n) points with pairwise distinct distance multiplicities and (a(Y) neq (n-1,n-2,ldots,1)).

We study Frobenius structures in higher dimensional p-adic analytic geometry and the corresponding p-adic functional analysis. This will build up foundations for further study on some generalized cohomology of Frobenius modules and the corresponding generalized Iwasawa theory and generalized noncommutative Tamagawa number conjectures in the spirit of Burns--Flach--Fukaya--Kato and Nakamura (as well as certainly the original noncommutative Tamagawa number conjectures as observed by Pal--Zábrádi). We will work in the program proposed by Carter-Kedlaya--Zábrádi and after Pal--Zábrádi, and we will follow closely the approach from Kedlaya--Pottharst--Xiao to investigate the corresponding deformation of the generalized p-adic Hodge structures.

Inspired by a classical result of Rényi, we prove that every even integer (Ngeq 4) can be written as the sum of a prime and a number with at most 395 prime factors. We also show, under assumption of the generalised Riemann hypothesis, that this result can be improved to 31 prime factors.

As the harmonic analysis associated with the modified Whittaker transform has been extensively investigated and has witnessed a remarkable development, it is natural to study several aspects of the time-frequency analysis associated with the Whittaker convolution operators (WCO). Knowing the fact that the study of the time-frequency analysis is both theoretically interesting and practically useful, this article aims to explore two other aspects of time-frequency analysis associated with WCO, namely spectral analysis associated with concentration operators and spectrogram analysis. As an application, we obtain some results concerning the Whittaker heat operator and the Whittaker Poisson integral.

Let (G) be a finite abelian group and (S) a sequence with elements of (G). Let (|S|) denote the length of (S) and (mathrm{supp}(S)) the set of all the distinct terms in (S). Let (Sigma(S)) denote the set of group elements which can be expressed as a sum of a nonempty subsequence of (S). It is known that if ({0notinSigma(S)}), then (|Sigma(S)| geq |S|+|mathrm{supp}(S)|-1). The sequence (S) satisfying (0notinSigma(S)) and (|Sigma(S)| = |S| + |mathrm{supp}(S)| - 1) has been described. In this paper, we determine the sequence (S) with (0notinSigma(S)) such that (|Sigma(S)|=|S|+|mathrm{supp}(S)|). As a consequence, we obtain some new results on the sequence (S) with (0notinSigma_{|G|}(S)) and (|Sigma_{|G|}(S)|=|S|-|G|+|mathrm{supp}(S)|-1), where (Sigma_{|G|}(S)) denotes the set of group elements which can be expressed as a sum of a subsequence of (S) of length (|G|).

We consider the set of partitions ( ped (n)) which counts the number of partitions of (n) wherein the even parts are distinct (and the odd parts are unrestricted). Using an algorithm developed by Radu, we prove congruences modulo 192 which were conjectured by Nath [10]. Further, we prove a few infinite families of congruences modulo 24 by using a result of Newman. Also, we prove that (ped (9n+7)) is lacunary modulo (2^{k+2}cdot 3) and (3^{k+1}cdot 4) for all positive integers (kgeq0). We further prove an infinite family of congruences for (ped (n)) modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators.

Suppose (X) and (Y) are topological spaces, (|X| = Delta(X)) and (|Y| = Delta(Y)). We investigate resolvability of the product (X times Y). We prove that:

• I. If (|X| = |Y| = omega) and (X,Y) are Hausdorff, then ( { X times Y } ) is maximally resolvable.

• II. If (2^kappa = kappa^+), ({|X|, mathrm{cf}|X|} cap {kappa, kappa^+} ne emptyset) and (mathrm{cf}|Y| = kappa^+), then the space ({ X times Y}) is (kappa^+)-resolvable. In particular, under GCH the space (X^2) is (mathrm{cf}|X|)-resolvable whenever (mathrm{cf}|X|) is an isolated cardinal.

• III. ((mathfrak{r} = mathfrak{c})) If (mathrm{cf}|X| = omega) and ({rm cf}|Y| = {rm cf}(mathfrak{c})), then the space ( { X times Y } ) is (omega)-resolvable. If, moreover, ({rm cf}(mathfrak{c}) = omega_1), then the space ( { X times Y } ) is (omega_1)-resolvable.

Let S = {1, 10, c} be an integer set. In this paper, we show that if the product of any two elements of S, decreased by one, forms a perfect square, and if there exists a positive integer d such that (d+1), (10d+1) and (cd+1) are all perfect squares, then (d=d^{pm}) where

Additionally, under certain conditions, we investigate the integer points on the parametrized family of elliptic curves:

We obtain the estimates from below and above for the average decay of Fourier transform of ((alpha_1,alpha_2))-Ahlfors regular measure. As an application, we obtain the upper bound of Beurling dimension of frame spectra for a class of Moran measures.

The exploration of graphs linked to algebraic structures has emerged as a dynamic field, bridging graph theory and module theory to reveal deeper insights into module properties. In this study, we introduce and analyze the semisimple intersection graph, denoted as (GSS_R(M)) associated with a right(R )-module (M). The vertices of (GSS_R(M)) are the nonzero submodules of (M), with two vertices (N) and (K) connected if their intersection forms a nonzero semisimple module. We examine fundamental properties of (GSS_R(M)) including its connectivity, diameter, and domination number, in relation to the module-theoretic characteristics of (M). Additionally, the girth of (GSS_R(M)) is determined.