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Abstract: Abstract We show it is consistent with \(ZFC\) that there is an everywhere Kurepa line which is order isomorphic to all of its dense \(\aleph_2\) -dense suborders. Moreover, this Kurepa line does not contain any Aronszajn suborder. We also show it is consistent with \(ZFC\) that there is a minimal Kurepa line which does not contain any Aronszajn suborder. PubDate: 2025-02-06 DOI: 10.1007/s10474-025-01509-3
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Abstract: Abstract For the Grassmann manifold \(\widetilde G_{n,4}\) of oriented 4-planes in \(\mathbb{R}^{n}\) no full description of its cohomology ring with coefficients in the two element field \(\mathbb {Z}_{2}\) is available. It is known however that it contains a subring that can be identified with a quotient of a polynomial ring by a certain ideal. Examining this quotient ring by means of Gröbner bases we are able to determine the \(\mathbb {Z}_{2}\) -cup-length of \(\widetilde G_{n,4}\) for \(n=2^t,2^t-1,2^t-2\) for all \(t \geq 4\) . PubDate: 2025-01-25 DOI: 10.1007/s10474-024-01502-2
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Abstract: Abstract Consider a nonstandard multidimensional risk model in which the claim sizes from all lines of businesses, sharing a common claim-arrival renewal process, constitute a sequence of independent and identically distributed nonnegative random vectors, the common inter-arrival times are assumed to be arbitrarily dependent and the dependence between claim size vectors and their waiting times are also allowed to be arbitrary. Moreover, the claim sizes from different lines of businesses are supposed to be extended negatively dependent. Under some mild conditions, this paper achieves some vector-type precise large deviation formulae for aggregate claims of such multidimensional risk model in the presence of dominatedly-varying claim sizes. The obtained results extend some existing ones in the literature. PubDate: 2025-01-20 DOI: 10.1007/s10474-024-01501-3
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Abstract: Abstract The study aims to investigate the weak convergence of nearest neighbor random walks in one-dimensional space, with the assumption that the transition probabilities tend towards a constant within the range \([ 0, 1/2 ]\) . The paper will demonstrate limit theorems based on the bias or balance of the random walk, utilizing the method of moments. PubDate: 2025-01-18 DOI: 10.1007/s10474-024-01497-w
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Abstract: Abstract Based on work of P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba [5], we show that if a covering system has distinct squarefree moduli, then the minimum modulus is at most 118. We also show that in general the \(k\) -th smallest modulus in a covering system with distinct moduli (provided it is required for the covering) is bounded by an absolute constant. PubDate: 2025-01-18 DOI: 10.1007/s10474-024-01496-x
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Abstract: Abstract 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. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01488-x
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Abstract: Abstract A Schmidt group is a non-nilpotent group whose every proper subgroup is nilpotent. A subgroup A of a group G is called OS-propermutablein G if there is a subgroup B such that \(G = NG(A)B\) , where AB is a subgroup of G and A permutes with all Schmidt subgroups of B. We proved \(p\) -solubility of a group in which a Sylow \(p\) -subgroup is OS-propermutable, where \(p\geq 7\) 7. For \(p < 7\) all non-Abelian composition factors of such group are listed. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01495-y
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Abstract: Abstract 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 \(\text{Aut}(\widetilde{\Gamma})\) . PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01491-2
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Abstract: Abstract We construct the lower radical class and the semisimple closure for a given class using class operators and detail some of the properties of these operators and their interplay with the operators already used in radical theory. The setting is the class of algebras introduced by Puczy lowski which ensures the results hold in groups, multi-operator groups such as rings, as well as loops and hoops. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01492-1
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Abstract: Abstract Let \(n\) be a positive integer different from \(8\) and \(n+1 \neq 2^u\) for any integer \(u\geq 2\) . 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 \(0\leq j\leq n-1\) belonging to \(Z[x]\) be polynomials having degree less than \(\deg\phi(x)\) . 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 $$ f(x) = a_n\frac{\phi(x)^n}{(n+1)!}+ \sum _{j=0}^{n-1}a_j(x)\frac{\phi(x)^{j}}{(j+1)!} $$ is irreducible over the field \(Q\) of rational numbers. This generalises a well-known result of Schur which states that the polynomial \( \sum _{j=0}^{n}a_j\frac{x^{j}}{(j+1)!}\) 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. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01478-z
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Abstract: Abstract 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. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01480-5
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Abstract: Abstract Let \(\sigma =\{\sigma_{i} \mid i\in I\}\) 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=H... PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01476-1
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Abstract: Abstract It is conjectured since long that for any convex body \(P\subset \mathbb{R}^n\) 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. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01483-2
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Abstract: Abstract Many remarkable results have been obtained on important problems combining arithmetic properties of the integers and some restricted conditions of their digits in a given base. Maynard considered the number of the polynomial values with missing digits and gave an asymptotic formula. In this paper we study truncated polynomials with restricted digits by using the estimates for character sums and exponential sums modulo prime powers. In the case where the polynomials are monomial we further give exact identities. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01490-3
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Abstract: Abstract 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 \(2\times2\) upper triangular \(\mathcal{AM}\) -operator matrix. Finally, we deduce some sufficient conditions implying the normality of them. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01493-0
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Abstract: Abstract We show that pseudocharacter turns out to be discretely reflexive in Lindelöf \(\Sigma\) -groups but countable tightness is not discretely reflexive in hereditarily Lindelöf spaces. We also establish that it is independent of ZFC whether countable character, countable weight or countable network weight is discretely reflexive in spaces \(C_p(X)\) . Furthermore, we prove that any hereditary topological property is discretely reflexive in 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 is consistent with ZFC that \(C_p(X)\) has the Fréchet–Urysohn property. Our results solve two published open questions. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01479-y
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Abstract: Abstract An almost cover of a finite set in the affine space is a collection of hyperplanes that together cover all points of the set except one. Using the polynomial method, we determine the minimum size of an almost cover of the vertex set of the permutohedron and address a few related questions. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01462-7
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Abstract: Abstract 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. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01481-4
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Abstract: Abstract 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. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01484-1
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Abstract: Abstract 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 $$ \notag \sum_{n\leq x}a_{K_{3}}^{2}(n^{\ell}),$$ where \(\ell\geq 2\) 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. PubDate: 2024-12-01 DOI: 10.1007/s10474-024-01489-w
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