Showing papers in "Czechoslovak Mathematical Journal in 2018"

Some Berezin Number Inequalities for Operator Matrices

Abstract: The Berezin symbol A of an operator A acting on the reproducing kernel Hilbert space H = H(Ω) over some (nonempty) set is defined by \(\tilde A(\lambda ) = \left\langle {A\hat k_\lambda ,\hat k_\lambda } \right\rangle \), λ ∈ Ω, where \(\hat k_\lambda = k_\lambda /\left\| {k_\lambda } \right\|\) is the normalized reproducing kernel of H. The Berezin number of the operator A is defined by \(ber(A) = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\tilde A(\lambda )} \right| = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\left\langle {A\hat k_\lambda ,\hat k_\lambda } \right\rangle } \right|\). Moreover, ber(A) ⩽ w(A) (numerical radius). We present some Berezin number inequalities. Among other inequalities, it is shown that if \(T = \left[ {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right] \in \mathbb{B}(\mathcal{H}(\Omega _1 ) \oplus \mathcal{H}(\Omega _2 ))\), then $$ber(T) \leqslant \frac{1} {2}(ber(A) + ber(D)) + \frac{1} {2}\sqrt {(ber(A) - ber(D))^2 + \left( {\left\| B \right\| + \left\| C \right\|} \right)^2 } .$$

On realizability of sign patterns by real polynomials

Abstract: The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers (p, n), chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree 8 polynomials.

On Weak Supercyclicity II

Abstract: This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbertspace operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly l-sequentially supercyclic, and (iii) weak l-sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators: (iv) the point spectrum of the normed-space adjoint of a power bounded supercyclic operator is either empty or is a singleton in the open unit disk, (v) weak l-sequential supercyclicity coincides with supercyclicity for compact operators, and (vi) every compact weakly l-sequentially supercyclic operator is quasinilpotent.

The spectral determinations of the connected multicone graphs $ K_w\bigtriangledown mP_{17} $ and $ K_w\bigtriangledown mS $

Abstract: Finding and discovering any class of graphs which are determined by their spectra is always an important and interesting problem in the spectral graph theory. The main aim of this study is to characterize two classes of multicone graphs which are determined by both their adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let Kw denote a complete graph on w vertices, and let m be a positive integer number. In A.Z.Abdian (2016) it has been shown that multicone graphs Kw ▽ P17 and Kw ▽ S are determined by both their adjacency and Laplacian spectra, where P17 and S denote the Paley graph of order 17 and the Schlafli graph, respectively. In this paper, we generalize these results and we prove that multicone graphs Kw ▽mP17 and Kw▽mS are determined by their adjacency spectra as well as their Laplacian spectra.

Note on a conjecture for the sum of signless Laplacian eigenvalues

Abstract: For a simple graph G on n vertices and an integer k with 1 ⩽ k ⩽ n, denote by $$\mathcal{S}^+_k$$ (G) the sum of k largest signless Laplacian eigenvalues of G. It was conjectured that $$\mathcal{S}^+_k(G)\leqslant{e}(G)+(^{k+1}_{\;\;2})$$ (G) ⩽ e(G) + (k+1 2), where e(G) is the number of edges of G. This conjecture has been proved to be true for all graphs when k ∈ {1, 2, n − 1, n}, and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all k). In this note, this conjecture is proved to be true for all graphs when k = n − 2, and for some new classes of graphs.

Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

Abstract: We present a variation-of-constants formula for functional differential equations of the form $$\dot y = \mathcal{L}\left( t \right)y_t + f\left( {y_t,t} \right),\;y_{t_0}= \varphi $$ , where $$\mathcal{L}$$ is a bounded linear operator and φ is a regulated function Unlike the result by G Shanholt (1972), where the functions involved are continuous, the novelty here is that the application t $$t \mapsto f\left( {y_t,t} \right)$$ is Kurzweil integrable with t in an interval of ℝ, for each regulated function y This means that t $$t \mapsto f\left( {y_t,t} \right)$$ may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula Our main goal is achieved via theory of generalized ordinary differential equations introduced by JKurzweil (1957) As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type $$\frac{{dx}}{{d\tau }} = D\left[ {A\left( t \right)x} \right],x\left( {{t_0}} \right) = \tilde x$$ and the solutions of the perturbed Cauchy problem $$\frac{{dx}}{{d\tau }} = D\left[ {A\left( t \right)x + F\left( {x,t} \right)} \right],x\left( {{t_0}} \right) = \tilde x$$ Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form $$\dot y = \mathcal{L}\left( t \right)y_t,\;y_{t_0} = \varphi$$ , where $$\mathcal{L}$$ is a bounded linear operator and φ is a regulated function The main result comes as a consequence of such results Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs

Uniform Convexity and Associate Spaces

Abstract: We prove that the associate space of a generalized Orlicz space Lϕ(·) is given by the conjugate modular ϕ* even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling Φ-function is equivalent to a doubling Φ-function. As a consequence, we conclude that Lϕ(·) is uniformly convex if ϕ and ϕ* are weakly doubling.

Graphs with small diameter determined by their D -spectra

Abstract: Let G be a connected graph with vertex set V(G) = {v1, v2,..., v n }. The distance matrix D(G) = (d ij )n×n is the matrix indexed by the vertices of G, where d ij denotes the distance between the vertices v i and v j . Suppose that λ1(D) ≥ λ2(D) ≥... ≥ λ n (D) are the distance spectrum of G. The graph G is said to be determined by its D-spectrum if with respect to the distance matrix D(G), any graph having the same spectrum as G is isomorphic to G. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their D-spectra.

Zeros of a Certain Class of Gauss Hypergeometric Polynomials

Abstract: We prove that as n → ∞, the zeros of the polynomial $$_2 F_1 \left[ {\begin{array}{*{20}c} { - n,\alpha n + 1} \\ {\alpha n + 2} \\ \end{array} ;z} \right]$$ cluster on (a part of) a level curve of an explicit harmonic function. This generalizes previous results of Boggs, Driver, Duren et al. (1999–2001) to the case of a complex parameter α and partially proves a conjecture made by the authors in an earlier work.

Weighted generalization of the Ramadanov’S theorem and further considerations

Abstract: We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space ℂN, and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given by M. Skwarczynski (1980), highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover, we show that convergence of weighted Bergman kernels implies this property, which will give a characterization of the domains, for which the inverse of the Ramadanov’s theorem holds.

Generalized outerplanar index of a graph

Abstract: We define the generalized outerplanar index of a graph and give a full characterization of graphs with respect to this index.

Valency seven symmetric graphs of order 2pq

Abstract: A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, all connected valency seven symmetric graphs of order 2pq are classified, where p, q are distinct primes. It follows from the classification that there is a unique connected valency seven symmetric graph of order 4p, and that for odd primes p and q, there is an infinite family of connected valency seven one-regular graphs of order 2pq with solvable automorphism groups, and there are four sporadic ones with nonsolvable automorphism groups, which is 1, 2, 3-arc transitive, respectively. In particular, one of the four sporadic ones is primitive, and the other two of the four sporadic ones are bi-primitive.

Automorphism Group of Representation Ring of the Weak Hopf Algebra $$\widetilde{H_8 }$$

Abstract: Let H8 be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra $$\widetilde{H_8 }$$ based on H8, then we investigate the structure of the representation ring of $$\widetilde{H_8 }$$ . Finally, we prove that the automorphism group of $$r\left( {\widetilde{H_8 }} \right)$$ is just isomorphic to D6, where D6 is the dihedral group with order 12.

On the geometry of some solvable extensions of the Heisenberg group

Abstract: In this paper we first classify left-invariant generalized Ricci solitons on some solvable extensions of the Heisenberg group in both Riemannian and Lorentzian cases. Then we obtain the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and obtain all vector fields which are critical points for the energy functional restricted to vector fields of the same length. Furthermore, we determine all homogeneous Lorentzian structures and their types on these spaces and give a complete and explicit description of all parallel and totally geodesic hypersurfaces of these spaces. The nonexistence of harmonic maps in the non-abelian case is proved and it is shown that the existence of Einstein, Einstein-like metrics and some equations in the Riemannian case can not be extended to their Lorentzian analogues.

Strict Mittag-Leffler conditions and locally split morphisms

Abstract: In this paper, we prove that any pure submodule of a strict Mittag-Leffler module is a locally split submodule. As applications, we discuss some relations between locally split monomorphisms and locally split epimorphisms and give a partial answer to the open problem whether Gorenstein projective modules are Ding projective.

On the pointwise limits of sequences of Świątkowski functions

Abstract: The characterization of the pointwise limits of the sequences of Świątkowski functions is given. Modifications of Świątkowski property with respect to different topologies finer than the Euclidean topology are discussed.

Harmonic metrics on four dimensional non-reductive homogeneous manifolds

Abstract: We study harmonic metrics with respect to the class of invariant metrics on non-reductive homogeneous four dimensional manifolds. In particular, we consider harmonic lifted metrics with respect to the Sasaki lifts, horizontal lifts and complete lifts of the metrics under study.

Fundamental groupoids of digraphs and graphs

Abstract: We introduce the notion of fundamental groupoid of a digraph and prove its basic properties. In particular, we obtain a product theorem and an analogue of the Van Kampen theorem. Considering the category of (undirected) graphs as the full subcategory of digraphs, we transfer the results to the category of graphs. As a corollary we obtain the corresponding results for the fundamental groups of digraphs and graphs. We give an application to graph coloring.

Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces

Abstract: We show that if α > 1, then the logarithmically weighted Bergman space $$A_{{{\log }^\alpha }}^2$$ is mapped by the Libera operator L into the space $$A_{{{\log }^{\alpha - 1}}}^2$$ , while if α > 2 and 0 < e ≤ α−2, then the Hilbert matrix operator H maps $$A_{{{\log }^\alpha }}^2$$ into $$A_{{{\log }^{\alpha - 2 - \varepsilon }}}^2$$ . We show that the Libera operator L maps the logarithmically weighted Bloch space $${B_{{{\log }^\alpha }}}$$ , α ∈ R, into itself, while H maps $${B_{{{\log }^\alpha }}}$$ into $${B_{{{\log }^{\alpha + 1}}}}$$ . In Pavlovic’s paper (2016) it is shown that L maps the logarithmically weighted Hardy-Bloch space $$B_{{{\log }^\alpha }}^1$$ , α > 0, into $$B_{{{\log }^{\alpha - 1}}}^1$$ . We show that this result is sharp. We also show that H maps $$B_{{{\log }^\alpha }}^1$$ , α > 0, into $$B_{{{\log }^{\alpha - 1}}}^1$$ and that this result is sharp also.

Coherence relative to a weak torsion class

Abstract: Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules $$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$ such that F0,..., Fn−1 are finitely generated free and Kn−1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext n R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor R n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.

Generalized Morrey spaces associated to Schrödinger operators and applications

Abstract: We first introduce new weighted Morrey spaces related to certain non-negative potentials satisfying the reverse Holder inequality. Then we establish the weighted strong-type and weak-type estimates for the Riesz transforms and fractional integrals associated to Schrodinger operators. As an application, we prove the Calderon-Zygmund estimates for solutions to Schrodinger equation on these new spaces. Our results cover a number of known results.

Character Connes amenability of dual Banach algebras

Abstract: We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.

The exceptional set for Diophantine inequality with unlike powers of prime variables

Abstract: Suppose that λ1, λ2, λ3, λ4 are nonzero real numbers, not all negative, δ > 0, V is a well-spaced set, and the ratio λ1/λ2 is algebraic and irrational. Denote by E(V,N, δ) the number of v ∈ V with v ≤ N such that the inequality $$\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^3 + {\lambda _3}p_3^4 + {\lambda _4}p_4^5 - \upsilon } \right| < {\upsilon ^{ - \delta }}$$ has no solution in primes p1, p2, p3, p4. We show that $$E\left( {\upsilon ,N,\delta } \right) \ll {N^{1 + 2\delta - 1/72 + \varepsilon }}$$ for any ɛ > 0.

L p harmonic 1-form on submanifold with weighted Poincaré inequality

Abstract: We deal with complete submanifolds with weighted Poincare inequality. By assuming the submanifold is δ-stable or has sufficiently small total curvature, we establish two vanishing theorems for L p harmonic 1-forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitorio.

More on betweenness-uniform graphs

Abstract: We study graphs whose vertices possess the same value of betweenness centrality (which is defined as the sum of relative numbers of shortest paths passing through a given vertex). Extending previously known results of S. Gago, J. Hurajova, T. Madaras (2013), we show that, apart of cycles, such graphs cannot contain 2-valent vertices and, moreover, are 3-connected if their diameter is 2. In addition, we prove that the betweenness uniformity is satisfied in a wide graph family of semi-symmetric graphs, which enables us to construct a variety of nontrivial cubic betweenness-uniform graphs.

The holomorphic automorphism groups of twisted Fock-Bargmann-Hartogs domains

Abstract: We consider a certain class of unbounded nonhyperbolic Reinhardt domains which is called the twisted Fock-Bargmann-Hartogs domains. By showing Cartan’s linearity theorem for our unbounded nonhyperbolic domains, we give a complete description of the automorphism groups of twisted Fock-Bargmann-Hartogs domains.

When a line graph associated to annihilating-ideal graph of a lattice is planar or projective

Abstract: Let (L,∧, ∨) be a finite lattice with a least element 0. AG(L) is an annihilating-ideal graph of L in which the vertex set is the set of all nontrivial ideals of L, and two distinct vertices I and J are adjacent if and only if I ∧ J = 0. We completely characterize all finite lattices L whose line graph associated to an annihilating-ideal graph, denoted by L(AG(L)), is a planar or projective graph.

Cohen-Macaulay modifications of the vertex cover ideal of a graph

Abstract: We study when the modifications of the Cohen-Macaulay vertex cover ideal of a graph are Cohen-Macaulay.

Groups Satisfying the Two-Prime Hypothesis with a Composition Factor Isomorphic to PSL 2 ( q ) for q ⩾ 7

Abstract: Let G be a finite group and write cd(G) for the degree set of the complex irreducible characters of G. The group G is said to satisfy the two-prime hypothesis if for any distinct degrees a, b 2 cd(G), the total number of (not necessarily different) primes of the greatest common divisor gcd(a, b) is at most 2. We prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL2(q) for q ⩾ 7.

Recognition of characteristically simple group $A_5\times A_5$ by character degree graph and order

Abstract: The character degree graph of a finite group G is the graph whose vertices are the prime divisors of the irreducible character degrees of G and two vertices p and q are joined by an edge if pq divides some irreducible character degree of G. It is proved that some simple groups are uniquely determined by their orders and their character degree graphs. But since the character degree graphs of the characteristically simple groups are complete, there are very narrow class of characteristically simple groups which are characterizable by this method. We prove that the characteristically simple group A5 × A5 is uniquely determined by its order and its character degree graph. We note that this is the first example of a non simple group which is determined by order and character degree graph. As a consequence of our result we conclude that A5 × A5 is uniquely determined by its complex group algebra.