Authors:S. Norvidas Pages: 378 - 388 Abstract: We consider the conditions under which a continuous function \({\varphi \colon {\mathbb{R}}^n \to \mathbb {R}}\) is the imaginary part \({\Im f}\) of the characteristic function f of a probability measure on \({{\mathbb{R}}^n}\) . A similar problem about such an \({\varphi}\) that it is the argument of the characteristic function was solved by Ilinskii [Theory Probab. Appl. 20 (1975), 410–415]. In this paper, a characterization of what \({\varphi}\) might serve as the imaginary part of the characteristic function f is given. As a consequence, we provide an answer to the following question posed by N. G. Ushakov [7]: Is it true that f is never determined by its imaginary part \({\Im f}\) ' In other words, is it true that for any characteristic function f there exists a characteristic function g such that \({\Im f\equiv \Im g}\) but \({ f\not\equiv g}\) ' We prove that the answer to this question is negative. In addition, several examples of characteristic functions which are uniquely determined by their imaginary parts are given. PubDate: 2018-04-01 DOI: 10.1007/s10474-017-0779-2 Issue No:Vol. 154, No. 2 (2018)
Authors:A. K. Bhuniya; T. K. Mondal Pages: 470 - 479 Abstract: We generalize the notion of k-radical of Green’s \({\overline{\mathcal{J}}}\) -relation, study the decompositions of semirings and investigate the semirings on which the powers and transitive closures of k-radicals are distributive lattice congruences. PubDate: 2018-04-01 DOI: 10.1007/s10474-018-0786-y Issue No:Vol. 154, No. 2 (2018)
Authors:W. Lin Pages: 480 - 500 Abstract: Let \(\mathcal{H}\) be an infinite dimensional complex Hilbert space and \(\mathcal{A}\) be a standard operator algebra on \(\mathcal{H}\) which is closed under the adjoint operation. It is shown that each nonlinear *-Lie-type derivation δ on \(\mathcal{A}\) is a linear *-derivation. Moreover, δ is an inner *-derivation as well. PubDate: 2018-04-01 DOI: 10.1007/s10474-017-0783-6 Issue No:Vol. 154, No. 2 (2018)
Authors:M. Balcerowski Abstract: We generalize a theorem proved by R. Ger and T. Kochanek [4]. Hence we obtain the solution of a problem posed by Z. Daróczy, J. Jarczyk and W. Jarczyk [2]. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0816-9
Authors:F. A. A. Almahdi; K. Louartiti; M. Tamekkante Abstract: Let R be a commutative ring and let \({n >1}\) be an integer. We introduce a simple graph, denoted by \({\Gamma_t(M_n(R))}\) , which we call the trace graph of the matrix ring \({M_n(R)}\) , such that its vertex set is \({M_n(R)^{\ast}}\) and such that two distinct vertices A and B are joined by an edge if and only if \({{\rm Tr} (AB)=0}\) where \({ {\rm Tr} (AB)}\) denotes the trace of the matrix AB. We prove that \({\Gamma_t(M_n(R))}\) is connected with \({{\rm diam}(\Gamma_{t}(M_{n}(R)))=2}\) and \({{\rm gr} (\Gamma_t(M_n(R)))=3}\) . We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of \({\Gamma_t(M_n(R))}\) . Hence, we use the notion of the irregularity index of a graph to characterize rings with exactly one nontrivial ideal. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0815-x
Authors:M. Henk; H. Pollehn Abstract: We study the log-Minkowski inequality for centered convex bodies when the cone-volume body is a simplex or a parallelepiped. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0822-y
Authors:B. Csikós; M. Horváth Abstract: We show that the perimeter of the convex hull of finitely many disks lying in the hyperbolic or Euclidean plane, or in a hemisphere does not increase when the disks are rearranged so that the distances between their centers do not increase. This generalizes the theorem on the monotonicity of the perimeter of the convex hull of a finite set under contractions, proved in the Euclidean plane by V. N. Sudakov [8], R. Alexander [1], V. Capoyleas and J. Pach [3]. We also prove that the area of the intersection of finitely many disks in the hyperbolic plane does not decrease after such a contractive rearrangement. The Euclidean analogue of the latter statement was proved by K. Bezdek and R. Connelly [2]. Both theorems are proved by a suitable adaptation of a recently published method of I. Gorbovickis [4]. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0820-0
Authors:M. Tărnăuceanu Abstract: Let G be a finite group. We give a criterion of nilpotency of G based on the existence of elements of certain order in each section of G. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0826-7
Authors:H. Baños; D. Oliveros Abstract: By extending the definition of boxicity, we extend a Hellytype result given by Danzer and Grünbaum on 2-piercings of families of boxes in d-dimensional Euclidean space by lowering the dimension of the boxes in the ambient space. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0817-8
Authors:M.-J. Huang Abstract: The Dirichlet eigenvalues \({\{\lambda_{n}\}_{n=1}^{\infty}}\) and Neumann eigenvalues \({\{\mu_{n}\}_{n=1}^{\infty}}\) of the string equation \({\varphi'' (x) +\lambda \rho (x) \varphi(x) =0}\) are considered. It is known that \({ \mu_{n} < \lambda_{n} < \mu_{n+2}}\) for all n. The purpose of this paper is to provide conditions on the mass density \({\rho(x)}\) under which \({\lambda_{n} < \mu_{n+1}}\) or \({\mu_{n+1} < \lambda_{n}.}\) PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0824-9
Authors:J. H. Schmerl Abstract: A theorem of Erdős asserts that every infinite \({X \subseteq \mathbb{R}^n}\) has a subset of the same cardinality having no repeated distances. This theorem is generalized here as follows: If \({(\mathbb{R}^n,E)}\) is an algebraic hypergraph that does not have an infinite, complete subset, then every infinite subset has an independent subset of the same cardinality. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0830-y
Authors:S. V. Medvedev Abstract: J. E. Jayne and C. A. Rogers [3] proved that a mapping \({f \colon {X \rightarrow Y}}\) of an absolute Souslin- \({\mathcal{F}}\) set X to a metric space Y is \({\mathbf{\Delta}^0_2}\) -measurable if and only if it is piecewise continuous. We give a similar result for a perfectly paracompact first-countable space X and a regular space Y. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0827-6
Authors:M. Djorić; M. Okumura Abstract: We prove that there do not exist CR submanifolds M n of maximal CR dimension of a complex projective space \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) with flat normal connection D of M, when the distinguished normal vector field is parallel with respect to D. If D is lift-flat, then there exists a totally geodesic complex projective subspace \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\) of \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) such that M is a real hypersurface of \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\) . PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0821-z
Authors:N. C. Bonciocat Abstract: We provide irreducibility criteria for compositions of multivariate polynomials over a field K, of the form \({f(X_{1},\ldots ,X_{r-1},g(X_{1},\ldots ,X_{r}))}\) , with \({f,g\in K[X_{1},\ldots ,X_{r}]}\) , for the case that \({f}\) as a polynomial in X r is irreducible over \({K(X_{1},\ldots ,X_{r-1})}\) and has leading coefficient divisible by a power of an irreducible polynomial \({p(X_{1},\ldots ,X_{r-1})}\) of sufficiently large degree with respect to \({X_{r-1}}\) . PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0818-7
Authors:S.-J. Lv Abstract: For strictly increasing concave functions \({\varphi}\) whose inverse functions are log-concave, the \({\varphi}\) -Brunn–Minkowski inequality for planar convex bodies is established. It is shown that for convex bodies in \({\mathbb{R}^n}\) the \({\varphi}\) -Brunn–Minkowski is equivalent to the \({\varphi}\) -Minkowski mixed volume inequalities. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0825-8
Authors:C. Cao; M. T. Hussain; L. Zhang Abstract: Let \({\sigma =\{\sigma_i i\in I\}}\) be some partition of the set of all primes \({\mathbb{P}}\) , G be a finite group and \({\sigma(G)=\{\sigma_i \sigma_i\cap \pi(G)\neq \emptyset\}}\) . A set \({\mathcal{H}}\) of subgroups of G is said to be a complete Hall \({\sigma}\) -set of G if every non-identity member of \({\mathcal{H}}\) is a Hall \({\sigma_i}\) -subgroup of \({G}\) and \({\mathcal{H}}\) contains exactly one Hall \({\sigma_i}\) -subgroup of G for every \({\sigma_i\in \sigma(G)}\) . A subgroup H of G is \({\sigma}\) -permutable in G if G possesses a complete Hall \({\sigma}\) -set \({\mathcal{H}}\) such that HA x = A x H for all \({A\in \mathcal{H}}\) and all \({x\in G}\) . We say that a subgroup H of G is n- \({\sigma}\) -embedded in G if there exists a normal subgroup T of G such that HT is \({\sigma}\) -permutable in G and \({H\cap T\leq H_{\sigma G}}\) , where \({H_{\sigma G}}\) is the subgroup of H generated by all those subgroups of H which are \({\sigma}\) -permutable in G. In this paper, we study the properties of the n- \({\sigma}\) -embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0819-6
Authors:Q. Du; L. Yuan; T. Zamfirescu Abstract: If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family \({\mathcal{F}}\) has property \({T-m}\) , if there exists a subfamily \({\mathcal{G} \subset \mathcal{F}}\) with \({ \mathcal{F} - \mathcal{G} \le m}\) admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if \({\mathcal{F}}\) is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which \({\mathcal{F}}\) has property \({T-m}\) . In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon \({(n \ge 5)}\) . We find out that, for \({5 \le n \le 34}\) , the family has property \({T - 3}\) ; for \({n \ge 35}\) , the family has property \({T - 2}\) . PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0823-x
Authors:O. R. Musin Abstract: This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in 4-dimensional Euclidean space. We consider two long-standing open problems: the uniqueness of maximum kissing arrangements in 4 dimensions and the 24-cell conjecture. Note that a proof of the 24-cell conjecture also proves that the lattice packing D4 is the densest sphere packing in 4 dimensions. PubDate: 2018-04-24 DOI: 10.1007/s10474-018-0828-5
Authors:F. Matúš Abstract: Given finitely many events in a probability space, conditional independences among the indicators of events are considered simultaneously with the signs of covariances. Resulting discrete structures are studied restricting attention mostly to all couples and triples of events. Necessary and sufficient conditions for such structures to be represented by events are found. Consequences of the results for the patterns of conjunctive forks are discussed. PubDate: 2018-02-22 DOI: 10.1007/s10474-018-0799-6
Authors:S. N. Begum; C. Nag; M. R. Talukder Abstract: We discuss congruences of p-algebras. We characterize kernel ideals of a p-algebra. Indeed, we show that an ideal of a p-algebra is a p-ideal if and only if it is a kernel ideal. We study cokernel filters of a p-algebra. We construct a class of p-algebras in which every cokernel filter is a p-filter. We also give some characterizations of Boolean congruences of a p-algebra. PubDate: 2018-02-22 DOI: 10.1007/s10474-018-0793-z
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