Abstract

Let $(X,d)$ be a metric space and ${Iso}(X,d)$ the associated isometry group. We study the subadditivity of the minimal displacement function $f:{Iso}(X,d)\to {R}$ for different metric spaces. When $(X,d)$ is ultrametric, we prove that the minimal displacement function is subadditive. We show, by a simple algebraic argument, that subadditivity does not hold for the direct isometry group of the hyperbolic plane. The same argument can be used for other metric spaces.

Keywords: Minimal displacement function; metric space; subadditivity.

MSC: 51F99, 51K05

Abstract

In this paper, we prove that if ${A}$ is a Banach $*$-subalgebra of $B(H)$, $T$ is a compact Hausdorff space equipped with a Radon measure $\mu$¦ and $\alpha:T\rightarrow [0,\infty)$ is a integrable function and $(A_t), (B_t)$ are appropriate integrable fields of operators in ${A}$ having the almost synchronous property for the Hadamard product, then $$ \int_T\!\alpha(s)d\mu(s)\int_T\!\alpha(t)\big(A_t\circ B_t\big) d\mu(t) \geq \int_T\!\alpha(t)A_td\mu(t)\circ\int_T\!\alpha(t)B_td\mu(t). $$ We also introduce a semi-inner product for square integrable fields of operators in a Hilbert space and using it, we prove the Schwarz and Chebyshev type inequalities dealing with the Hadamard product and the trace of operators.

Keywords: Grüss inequality; Chebyshev inequality; operator inequality.

MSC: 26D10, 26D15, 46C50, 46G12

Abstract

In this paper we study the property $(gaz)$ for a bounded linear operator $T\in L(X)$ on a Banach space $X$, introduced by Zariouh in [\emph{Property $(gz)$ for bounded linear operators}, Mat.\ Vesnik, {\bf 65(1)}(2013), 94--103], through the methods of local spectral theory. This property is a stronger variant of generalized $a$-Browder's theorem. In particular, we shall give several characterizations of property $(gaz)$, by using the localized SVEP.

Keywords: Property $(gaz)$; localized SVEP; Browder type theorems.

MSC: 47A10, 47A11, 47A53, 47A55

Abstract

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)$ and $D^{Q}(G)$ respectively be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix and the distance signless Laplacian matrix of a graph $G$. The convex linear combination $D_{\alpha}(G)$ of $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^{Q}(G)$, $D_{1}(G)=Tr(G)$, this matrix reduces to merging the distance spectral, signless distance Laplacian spectral theories. In this paper, we study the spectral radius of the generalized distance matrix $D_{\alpha}(G)$ of a graph $G$. We obtain bounds for the generalized distance spectral radius of a bipartite graph in terms of various parameters associated with the structure of the graph and characterize the extremal graphs. For $\alpha=0$, our results improve some previously known bounds.

Keywords: Distance matrix (spectrum); Distance signless Laplacian matrix (spectrum); generalized distance matrix; spectral radius.

MSC: 05C30, 05C50

Abstract

The paper is concerned with the interpretation of the fixed points of an involution as invariant solutions under certain Lie algebra of symmetries of a given equation. Our aim is to study the involutivity in terms of the symmetries of an equation. We prove that if $\pi:E\to M$ is a fiber bundle and $\nabla:T^*M\to J^1T^*M$ is a linear connection on the base space, then there exists a unique involutive linear automorphism, $\alpha_{_{\nabla}}$ in $J^1J^1E$, that commutes with the projections $\pi_{11}$ and $J^1\pi_{1,0}$. Moreover, we prove that the space $J^k(\pi)$ is the quotient space of the iterated sesqui-holonomics jets $\^{J}^1J^{k-1}(\pi)$ relative to the subgroup of symmetries determined by some involution $\alpha_{g}$.

Keywords: Geometric structures on manifolds; differential systems; contact theory; co-contact of higher order.

MSC: 53C15, 53B25

Abstract

A graph $\Gamma$ is called an $n$-Cayley graph over a group $G$ if its automorphism contains a semi-regular subgroup isomorphic to $G$ with $n$ orbits. Every $n$-Cayley graph over a group $G$ is completely determined by $n^2$ suitable subsets of $G$. If each of these subsets is a union of conjugacy classes of $G$, then it is called a quasi-abelian $n$-Cayley graph over $G$. In this paper, we determine the characteristic polynomial of quasi-abelian $n$-Cayley graphs. Then we exactly determine the eigenvalues and the number of closed walks of quasi-abelian semi-Cayley graphs. Furthermore, we construct some integral graphs.

Keywords: Semi-Cayley graph; $n$-Cayley graph; quasi-abelian; eigenvalue.

MSC: 05C50, 05C25, 05C31

Abstract

In this work, we are concerned with the existence of at least one non-trivial weak solution for a second-order impulsive Hamiltonian system. The proof of the main result is based on the critical point theory.

Keywords: Weak solution; impulsive Hamiltonian system; critical point theory; variational methods.

MSC: 34B15, 47J10

Abstract

In this paper, we investigate two fixed point theorems in the framework of $b$-metric spaces and rectangular $b$-metric spaces, using interpolative approach. One is Hardy-Rogers and the other is Reich-Rus-{Ć}iri{ć} type contraction. Examples are provided in support of the results.

Keywords: Fixed point; $b$-metric space; rectangular $b$-metric space; contraction map.

MSC: 47H10, 54H25, 54E50