Acta Mathematica Hungarica最新文献

We determine the solution of the Drygas functional equation that satisfies the additional condition ((y^2+y)f(x)= (x^2+x)f(y)) on a restricted domain. Also, some other properties of Drygas functions are given as well.

Let (K_{3}) be a non-normal cubic extension over (mathbb{Q}), and let (a_{K_{3}}(n)) be the (n)-th coefficient of the Dedekind zeta function (zeta_{K_{3}}(s)). In this paper, we investigate the asymptotic behaviour of the type

where (ellgeq 2) is any fixed integer. As an application, we also establish the asymptotic formulae of the variance of (a_{K_{3}}^{2}(n^{ell})). Furthermore, we also consider the asymptotic relations for shifted convolution sums associated to (a_{K_{3}}(n)) with classical divisor function.

Let (Gamma) be a finite connected pentavalent graph admitting a nonabelian simple arc-transitive automorphism group (T) and soluble vertex stabilizers. Let (p>|T|_{2}) be an odd prime and ((p,|T|)=1), where (|T|_{2}) is the largest power of 2 dividing the order (|T|) of (|T|). Then we prove that there exists a (p)-radical cover (widetilde{Gamma}) of (Gamma) such that the full automorphism group (text{Aut}(widetilde{Gamma})) of (widetilde{Gamma}) is equal to (O_{p}(text{Aut}(widetilde{Gamma})).T) and the covering transformation group is (O_{p}(text{Aut}(widetilde{Gamma}))), where (O_{p}(text{Aut}(widetilde{Gamma}))) is the (p)-radical of (text{Aut}(widetilde{Gamma})).

A bounded linear operator (T) on a Hilbert space (H) is said to be absolutely norm attaining ((T in mathcal{AN}(H))) if the restriction of (T) to any non-zero closed subspace attains its norm and absolutely minimum attaining ((T in mathcal{AM}(H))) if every restriction to a non-zero closed subspace attains its minimum modulus.

In this article, we characterize normal operators in (overline{mathcal{AN}(H)}), the operator norm closure of (mathcal{AN}(H)), in terms of the essential spectrum. Later, we study representations of quasinormal and hyponormal operators in (overline{mathcal{AN}(H)}). Explicitly, we prove that any hyponormal operator in (overline{mathcal{AN}(H)}) is a direct sum of a normal (mathcal{AN})-operator and a (2times2) upper triangular (mathcal{AM})-operator matrix. Finally, we deduce some sufficient conditions implying the normality of them.

We construct the lower radical class and the semisimple closurefor a given class using class operators and detail some of the properties of theseoperators and their interplay with the operators already used in radical theory.The setting is the class of algebras introduced by Puczy lowski which ensures theresults hold in groups, multi-operator groups such as rings, as well as loops andhoops.

Let (sigma ={sigma_{i} mid iin I}) be some partition of the set of all primes and (G) a finite group. Then (G) is said to be (sigma)-full if (G) has a Hall (sigma _{i})-subgroup for all (i); (sigma)-primary if (G) is a (sigma _{i})-group for some (i); (sigma)-soluble if every chief factor of (G) is (sigma)-primary; (sigma)-nilpotent if (G) is the direct product of (sigma)-primary groups; (G^{mathfrak{N}_{sigma}}) denotes the (sigma)-nilpotent residual of (G), that is, the intersection of all normal subgroups (N) of (G) with (sigma)-nilpotent quotient (G/N).

A subgroup (A) of (G) is said to be: (sigma)-permutable in (G) provided (G) is (sigma)-full and (A) permutes with all Hall (sigma _{i})-subgroups (H) of (G) (that is, (AH=HA)) for all (i); (sigma)-subnormal in (G) if there is a subgroup chain (A=A_{0} leq A_{1} leq cdots leq A_{n}=G) such that either (A_{i-1} trianglelefteq A_{i}) or (A_{i}/(A_{i-1})_{A_{i}}) is (sigma)-primary for all (i=1, ldots , n).

Let (A_{sigma G}) be the subgroup of (A) generated by all (sigma)-permutable subgroups of (G) contained in (A) and (A^{sigma G}) be the intersection of all (sigma)-permutable subgroups of (G) containing (A).

We prove that if (G) is a finite (sigma)-soluble group, then the (sigma)-permutability is a transitive relation in (G) if and only if (G^{mathfrak{N}_{sigma}}) avoids the pair ((A^{sigma G}, A_{sigma G})), that is, (G^{mathfrak{N}_{sigma}}cap A^{sigma G}= G^{mathfrak{N}_{sigma}}cap A_{sigma G}) for every (sigma)-subnormal subgroup (A) of (G).

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.