Volume 32, Issue 4

The paper is dedicated to formulate an open problem concerning the existence of solutions of an infinite system of nonlinear differential (or integral) equations which are obtained during the modelling of the so-called birth-and-death stochastic process. Namely, creating a model of the mentioned birth-and-death process and also taking into account nonlinear components which are usually deleted during standard considerations, we obtain an infinite system of nonlinear differential (or integral) equations on a bounded or unbounded interval. It turns out that the operator associated with that infinite system is unbounded in all known classical Banach or Fréchet spaces. This fact initiates an open problem which will be described and raised in the paper.

In this paper, we introduce a new norm, christened the t -operator norm, on the space of all bounded linear operators defined on a complex Hilbert space ℋ {\mathcal{H}} as ∥ T ∥ t := sup ∥ y ∥ = 1 ∥ x ∥ = 1 ⁡ ( t ⁢ | 〈 T ⁢ x , y 〉 | + ( 1 - t ) ⁢ | 〈 x , T ⁢ y 〉 | ) , \|T\|_{t}:=\sup_{\begin{subarray}{c}\|y\|=1\\ \|x\|=1\end{subarray}}(t|\langle Tx,y\rangle|+(1-t)|\langle x,Ty\rangle|), where x , y ∈ ℋ {x,y\in\mathcal{H}} and t ∈ [ 0 , 1 ] {t\in[0,1]} . This norm satisfies 1 2 ⁢ ∥ T ∥ t ≤ w ⁢ ( T ) ≤ ∥ T ∥ t {\frac{1}{2}\|T\|_{t}\leq w(T)\leq\|T\|_{t}} and we explore its properties. This norm characterizes those invertible operators that are also unitary. We obtain various inequalities involving the t -operator norm and the usual operator norm. We show that w ⁢ ( T ) ≤ min t ∈ [ 0 , 1 ] ⁢ ∥ T ∥ t {w(T)\leq\min_{t\in[0,1]}\|T\|_{t}} improves the existing bounds w ⁢ ( T ) ≤ 1 2 ⁢ ( ∥ T ∥ + ∥ T 2 ∥ ) w(T)\leq\frac{1}{2}\Bigl{(}\|T\|+\sqrt{\|T^{2}\|}\Bigr{)} (see [F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158 2003, 1, 11–17]) and w ⁢ ( T ) ≤ 1 2 ⁢ ∥ T * ⁢ T + T ⁢ T * ∥ w(T)\leq\sqrt{\frac{1}{2}\|T^{*}T+TT^{*}\|} (see [F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math. 168 2005, 1, 73–80]). We show that ∥ T ∥ - w ⁢ ( T ) ≤ min | λ | = 1 ⁡ ∥ T + λ ⁢ T * 2 ∥ {\|T\|-w(T)\leq\min_{|\lambda|=1}\|\frac{T+\lambda T^{*}}{2}\|} . Further, we study the t -operator norm of operator matrices.

In the present paper, we concern on discussing some results of the fractional Laplacian with respect to a given function μ. In addition, we also discuss the Nemytskii operators Ψ ↦ | Ψ | {\Psi\mapsto|\Psi|} and Ψ ↦ Ψ ± {\Psi\mapsto\Psi^{\pm}} in μ-fractional Sobolev–Slobodetskii space.

In this paper, quasi-statical boundary contact problems of couple-stress viscoelasticity for inhomogeneous anisotropic bodies with regard to friction are investigated. Non-coercive cases considered, when a friction force acts on the entire boundary of a viscoelastic body. The posed boundary-contact problem equivalently reduced to a spatial variational inequality. After factorization of the main spaces with respect to the vector space of the function of generalized rigid displacements, the necessary condition for the existence of solutions is obtained. This condition, under some additional assumptions, becomes a sufficient condition for the existence of solutions.

In the paper, some approximation properties of angular partial sums are established.

In the present paper, we consider isotype submodules of those QTAG \mathrm{QTAG} -modules that have countable length and give an in-depth study of their basic and specific properties either in parallel or in contrast to the direct sum of countably generated modules. In addition, we also examine the QTAG \mathrm{QTAG} -modules whose h -pure submodules are direct sums of countably generated modules.

The main purpose of this paper is to provide a simple and short way to obtain Leibniz rule for various types of derivations. As an application of this discussion, the generalized types of higher derivations are also studied.

For an uncountable commutative group ( G , + ) {(G,+)} and for an infinite cardinal κ, the notions of a ( G , κ ) {(G,\kappa)} -thick set in G and of a ( G , κ ) {(G,\kappa)} -thin set in G are introduced and examined. It is shown that some statements about these sets are closely connected with the existence of absolutely nonmeasurable subsets of ( G , + ) {(G,+)} and with additional set-theoretical axioms.

In this paper, we mainly study the necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators on weighted Bergman spaces over a tubular domains by using the Carlson measures on tubular domains. We also give some related results about Carlson measures.

There is a resent paper claiming that every hyponormal operator which is not a multiple of the identity (operator) has a nontrivial hyperinvariant subspace. If this claim is true, then every hyponormal operator has a nontrivial invariant subspace, and every subnormal operator which is not multiple of the identity has a nontrivial hyperinvariant subspace. But we find out that the proof of the above claim goes wrong. Thus, the invariant subspace problem of the hyponormal operator and the hyperinvariant problem of subnormal operator remain unresolved. Moreover, it is well known that these are two important research topics in operator theory that have not been solved for a long time, and that many researchers have been trying to solve these two problems. So, it is very meaningful to clarify whether these two problems have been solved.

The k -tuple total dominator chromatic number is a graph parameter that measures the minimum number of colors required for a proper coloring, where each vertex must be adjacent to every vertex of k distinct color classes. In this paper, we investigate the k -tuple total dominator chromatic number in the context of Mycielskian graphs. Mycielskian graphs are obtained through a graph transformation introduced by Mycielski in 1955, and they have been widely studied for their intriguing properties and applications in graph theory. The generalized Mycielskians are natural generalizations of Mycielskian graphs, denoted by μ m ⁢ ( G ) {\mu_{m}(G)} , where m ≥ 1 {m\geq 1} . Specifically, we determine both the upper and lower bounds for the k -tuple total dominator chromatic number of Mycielskian graphs, providing insights into the minimum number of colors needed to achieve a k -tuple total dominator coloring in these graphs. We also discuss the exact values of the k -tuple total dominator chromatic number for specific instances of Mycielskian graphs. Moreover, we discuss the χ k , d t {\chi_{k,d}^{t}} of the generalized Mycielskian graph μ m ⁢ ( G ) {\mu_{m}(G)} , where m ≥ 2 {m\geq 2} , by giving some bounds in terms of k , m {k,m} and that of G .

The Sombor index ( SO {\mathrm{SO}} ) is a recently introduced degree-based graph invariant, defined as the sum over all pairs of adjacent vertices u , v {u,v} of the term d u 2 + d v 2 {\sqrt{{d_{u}^{2}+d_{v}^{2}}}} , where d u {d_{u}} and d v {d_{v}} denote the degrees of vertices u and v , respectively. The matrix associated with SO {\mathrm{SO}} is the Sombor matrix, and its spectrum is the Sombor spectrum. In this paper, the connected graphs having exactly two and exactly three Sombor eigenvalues are characterized. Bounds are obtained for the spectral radius and energy of the Sombor matrix, and the corresponding extremal graphs are determined. In addition, the Sombor spectra of several families of graphs are calculated.

In this paper, we give some new characterizations of SEP elements. Mainly, we first discuss some properties of SEP elements. Then we characterize SEP elements by using left and right a -idempotents, projections and constructed regular vectors.

In this paper, the boundedness and compactness of Toeplitz operators T ϕ {T_{\phi}} with the symbol ϕ ∈ L 1 ⁢ ( 𝔻 ) {\phi\in L^{1}(\mathbb{D})} between different Bergman spaces with the log-Hölder continuous variable exponents are investigated.

In this paper, the linear model of Moore–Gibson–Thompson thermoporoelasticity is considered and the governing equations of motion and steady vibrations are given. The basic system of equations of steady vibrations with respect to the displacement vector, the changes of temperature and fluid pressure are proposed. Then the radiation conditions are established and Green’s first identity is obtained. Finally, on the basis of this identity, the uniqueness theorems for classical solutions of the boundary value problems of steady vibrations in the theory of MGT thermoporoelaticity are proved.

We introduce the stable perturbation of matrices over an arbitrary ∗ {\ast} -ring with respect to pseudo core inverses, which extends the stable perturbation of square complex matrices with respect to core-EP inverses. Let A and B be pseudo core invertible matrices over a unital ∗ {\ast} -ring. Then B is called a stable perturbation of A with respect to pseudo core inverses if ℛ ⁢ ( A k ) ∩ 𝒩 ⁢ ( ( B s ) ∗ ) = { 0 } {\mathcal{R}(A^{k})\cap\mathcal{N}((B^{s})^{\ast})=\{0\}} and ℛ ⁢ ( B s ) ∩ 𝒩 ⁢ ( ( A k ) ∗ ) = { 0 } {\mathcal{R}(B^{s})\cap\mathcal{N}((A^{k})^{\ast})=\{0\}} , where k and s are pseudo core indices of A and B , respectively. It is shown that if R is a right FP-injective ring, then B is a stable perturbation of A if and only if I - ( B π - A π ) 2 {I-(B^{\pi}-A^{\pi})^{2}} is invertible, where A π = I - A ⁢ A \⃝raisebox{0.569055pt}{$\scriptstyle{D}$} {A^{\pi}=I-AA^{\text{\textcircled{\raisebox{0.2mm}{$\scriptstyle{D}$}}}}} and B π = I - B ⁢ B \⃝raisebox{0.569055pt}{$\scriptstyle{D}$} {B^{\pi}=I-BB^{\text{\textcircled{\raisebox{0.2mm}{$\scriptstyle{D}$}}}}} are spectral projectors of A and B , respectively. Necessary and sufficient conditions are given for B to be a stable perturbation of A in terms of the invertibility of certain matrices involving pseudo core inverses of A and B .

Let 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} be abelian categories, and 𝐓 : 𝒜 → ℬ {\mathbf{T}:\mathcal{A}\rightarrow\mathcal{B}} be a left exact additive functor. Then one has a comma category ( ℬ ↓ 𝐓 ) {(\mathcal{B}{\downarrow}\mathbf{T})} . Under some natural assumptions, we prove that ( 𝒳 , 𝒴 ) {(\mathcal{X},\mathcal{Y})} and ( 𝒳 ′ , 𝒴 ′ ) {(\mathcal{X}^{\prime},\mathcal{Y}^{\prime})} are n -tilting pairs in 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} , respectively, if and only if ( 𝔇 𝒳 ′ 𝒳 , 𝔇 𝒴 ′ 𝒴 ) {(\mathfrak{D}^{\mathcal{X}}_{\mathcal{X}^{\prime}},\mathfrak{D}^{\mathcal{Y}}% _{\mathcal{Y}^{\prime}})} is an n -tilting pair in ( ℬ ↓ 𝐓 ) {(\mathcal{B}{\downarrow}\mathbf{T})} . Furthermore, we also show that 𝒳 {\mathcal{X}} and 𝒴 {\mathcal{Y}} are n -cotilting subcategories of 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} , respectively, if and only if 𝔇 𝒴 𝒳 {\mathfrak{D}^{\mathcal{X}}_{\mathcal{Y}}} is an n -cotilting subcategory of ( ℬ ↓ 𝐓 ) {(\mathcal{B}{\downarrow}\mathbf{T})} . Finally, we construct cotilting cotorsion pairs over ( ℬ ↓ 𝐓 ) {(\mathcal{B}{\downarrow}\mathbf{T})} using cotilting cotorsion pairs over 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} .