Acta Mathematica Hungarica最新文献

Let (n) be a positive integer different from (8) and (n+1 neq 2^u) for any integer (ugeq 2). Let (phi(x)) belonging to (Z[x]) be a monic polynomial which is irreducible modulo all primes less than or equal to (n+1). Let (a_j(x)) with (0leq jleq n-1) belonging to (Z[x]) be polynomials having degree less than (degphi(x)). Assume that the content of (a_na_0(x)) is not divisible by any prime less than or equal to (n+1). We prove that the polynomial

is irreducible over the field (Q) of rational numbers. This generalises a well-known result of Schur which states that the polynomial ( sum _{j=0}^{n}a_jfrac{x^{j}}{(j+1)!}) with (a_j in Z) and (|a_0| = |a_n| = 1) is irreducible over (Q). For proving our results, we use the notion of (phi)-Newton polygons and a few results on primes from number theory. We illustrate our result through examples.

Geodesic loops on tetrahedra were studied only for the Euclidean space and it was known that there are no simple geodesic loops on regular tetrahedra. Here we prove that: 1) In the spherical space, there are no simple geodesic loops on tetrahedra with internal angles (pi/3 < a_i<pi/2)or regular tetrahedra with (a_i=pi/2), and there are three simple geodesic loops for each vertex of a tetrahedra with (a_i > pi/2)and the lengths of the edges (a_i>pi/2). 2) We obtain also a new theorem on simple closed geodesics: If the angles (a_i)of the faces of a tetraedron satisfy (pi/3 < a_i<pi/2)and all faces of the tetrahedron are congruent, then there exist at least (3) simple closed geodesics.3) In the hyperbolic space, for every regular tetrahedron (T)and every pair of coprime numbers ((p,q)), there is one simple geodesic loop of type ((p,q)) through every vertex of (T).The geodesic loops that we have found on the tetrahedra in the hyperbolic space are also quasi-geodesics.

We consider the ideal of nowhere dense sets in the common division topology (Szyszkowska’s ideal), and examine some of its basic properties. We also explore the possible inclusions between the studied ideal and Furstenberg’s and Rizza’s ideals, thus answering open questions posed in a recent article by A. Nowik and P. Szyszkowska [17]. Moreover, we discuss the relationships of the Szyszkowska’s ideal with selected well-known ideals playing an important role in number theory and combinatorics.

It is conjectured since long that for any convex body (Psubset mathbb{R}^n) there exists a point in its interior which belongs to at least (2n) normals from different points on the boundary of P. The conjecture is known to be true for (n=2,3,4).

We treat the same problem for convex polytopes in (mathbb{R}^3). It turns out that the PL concurrent normals problem differs a lot from the smooth one. One almost immediately proves that a convex polytope in (mathbb{R}^3) has 8 normals to its boundary emanating from some point in its interior. Moreover, we conjecture that each simple polytope in (mathbb{R}^3) has a point in its interior with 10 normals to the boundary. We confirm the conjecture for all tetrahedra and triangular prisms and give a sufficient condition for a simple polytope to have a point with 10 normals. Other related topics (average number of normals, minimal number of normals from an interior point, other dimensions) are discussed.

This work aims to investigate various properties of some special Z-symmetric manifolds and their applications on space-times. Having an important place of the study, classifications of second-order symmetric tensor fields on space-times and holonomy theory are considered. Z-symmetric manifolds in the holonomy structure are investigated and some results are obtained. Various special vector fields are examined on Z-recurrent and weakly Z-symmetric manifolds and some relations associated with the eigenvector structure of the Z-tensor are found. In addition, several examples related to the outcomes of the study are given. Finally, some links between the Z-tensor and Ricci solitons on space-times are determined.

We show that pseudocharacter turns out to be discretely reflexivein Lindelöf (Sigma)-groups but countable tightness is notdiscretely reflexive in hereditarily Lindelöf spaces. We alsoestablish that it is independent of ZFC whether countablecharacter, countable weight or countable network weight isdiscretely reflexive in spaces (C_p(X)). Furthermore, we provethat any hereditary topological property is discretely reflexivein spaces (C_p(X)) with the Lindelöf (Sigma)-property. If(C_p(X)) is a Lindelöf (Sigma)-space and (L D) is a(k)-space for any discrete subspace ( { D C_p(X) } ), then it isconsistent with ZFC that (C_p(X)) has the Fréchet–Urysohnproperty. Our results solve two published open questions.

Let (mathbb{H}^{n}) be the Heisenberg group. For (0 leq alpha < Q=2n+2) and (N in mathbb{N}) we consider exponent functions (p (cdot) colon mathbb{H}^{n} to (0, +infty)), which satisfy log-Hölder conditions, such that (frac{Q}{Q+N} < p_{-} leq p (cdot) leq p_{+} < frac{Q}{alpha}). In this article we prove the (H^{p (cdot)}(mathbb{H}^{n}) to L^{q (cdot)}(mathbb{H}^{n})) and (H^{p (cdot)}(mathbb{H}^{n}) to H^{q (cdot)}(mathbb{H}^{n})) boundedness of convolution operators with kernels of type ((alpha, N)) on (mathbb{H}^{n}), where (frac{1}{q (cdot)} = frac{1}{p (cdot)} - frac{alpha}{Q}). In particular, the Riesz potential on (mathbb{H}^{n}) satisfies such estimates.

We introduce the martingale local Morrey spaces. We establish the Doob's inequality, the Burkholder–Gundy inequality and the boundedness of the martingale transforms to martingale local Morrey spaces defined on complete probability spaces.

Let G be an additive finite abelian group. Denote by disc(G) the smallest positive integer t such that every sequence S over G of length (|S|geq t) has two nonempty zero-sum subsequences of distinct lengths. In this paper, we focus on the direct and inverse problems associated with disc(G) for certain groups of rank three. Explicitly, we first determine the exact value of disc(G) for (Gcong C_2oplus C_{n_1}oplus C_{n_2}) with (2mid n_1mid n_2) and (Gcong C_3oplus C_{6n_3}oplus C_{6n_3}) with (n_3geq 1). Then we investigate the inverse problem. Let (mathcal {L}_1(G)) denote the set of all positive integers t satisfying that there is a sequence S over G of length (|S|=operatorname{disc}(G)-1) such that every nonempty zero-sum subsequence of S has the same length t. We determine (mathcal {L}_1(G)) completely for certain groups of rank three.

Given a real interval (I), a group of homeomorphisms (mathcal{G} left(M,Iright)) is associated to every continuous mean defined (i)n (I). Twomeans (M), (N) defined in (I) will belong to the same class when (mathcal{G} (M, I) = mathcal{G} (N,I)). The equivalence relationdefined in this way in (mathcal{CM}(I)), the family ofcontinuous means defined in (I), gives a principle of classification basedon the algebrai object (mathcal{G}(M, I)). Two major questionsare raised by this classification: 1) the problem of computing (mathcal{G} (M, I)) for a given mean (M in mathcal{CM} (I)), and 2) the determination of general properties of the means belonging to asame class. Some instances of these questions will find suitable responsesin the present paper.