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Abstract: Abstract In the paper, we prove that the maximal commutator of conditional expectations \(\sup _{n\ge 0} [{\mathbb {E}}_n,g] \) is bounded from \(H_1^S\) to \(L_{1,\infty }\) provided \(g\in \textrm{BMO}_2\) . If \(g\in \textrm{bmo}_2\) , we also show that a modified maximal operator \(\sup _{n\ge 0}{\mathbb {E}}_n( [{\mathbb {E}}_n,g] )\) is bounded from \(\textrm{h}_1\) to \(L_{1,\infty }\) . PubDate: 2022-11-27

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Abstract: Abstract Several games that arise from graph coloring have been introduced and studied. Let \(\varphi \) denote a graph invariant that arises from such a game. If G is a graph and \(\varphi (G-x)\ne \varphi (G)=k\) , \(k \ge 1\) , holds true for every vertex \(x \in V(G)\) , then G is called a k- \(\varphi \) -game-vertex-critical graph. We study the concept of \(\varphi \) -game-vertex-criticality for \(\varphi \in \{\chi _g, \chi _i, \chi _\mathrm{{ig}}^\mathrm{{A}}, \chi _\mathrm{{ig}}^\mathrm{{AB}}\}\) , where \(\chi _g\) denotes the standard game chromatic number, \(\chi _i\) denotes the indicated game chromatic number and \(\chi _\mathrm{{ig}}^\mathrm{{A}}\) , \(\chi _\mathrm{{ig}}^\mathrm{{AB}}\) denote two versions of the independence game chromatic number. Since the game chromatic number \(\varphi (G-x)\) can either decrease or increase with respect to \(\varphi (G)\) , we distinguish between lower, upper and mixed vertex-criticality. We show that for \(\varphi \in \{\chi _g, \chi _\mathrm{{ig}}^\mathrm{{A}}, \chi _\mathrm{{ig}}^\mathrm{{AB}}\}\) the difference \(\varphi (G)-\varphi (G-x)\) , \(x \in V(G)\) , can be arbitrarily large. A characterization of 2- \(\varphi \) -game-vertex-critical and (connected) 3- \(\varphi \) -lower-game-vertex-critical graphs for all \(\varphi \in \{\chi _g, \chi _i, \chi _\mathrm{{ig}}^\mathrm{{A}}, \chi _\mathrm{{ig}}^\mathrm{{AB}}\}\) is given. It is shown that \(\chi _g\) -game-vertex-critical, \(\chi _\mathrm{{ig}}^\mathrm{{A}}\) -game-vertex-critical and \(\chi _\mathrm{{ig}}^\mathrm{{AB}}\) -game-vertex-critical graphs are not necessarily connected. However, it is also shown that \(\chi _i\) -lower-game-vertex-critical graphs are always connected. PubDate: 2022-11-25

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Abstract: Abstract The study on vanishing coefficients with arithmetic progressions in quotients of theta functions has its origin in the work by Richmond and Szekeres. Further investigations were subsequently considered by Andrews and Bressoud, Alladi and Gordon, and Mc Laughlin. Quite recently, Du and the author found that certain arithmetic progressions on vanishing coefficients could be enjoyed by a family of quotients of theta functions. In this paper, we prove that there are more such instances in two families of quotients of theta functions. Finally, we present several related conjectures that merit further investigation. PubDate: 2022-11-25

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Abstract: Abstract In this paper, we study chain mixing and various multi-transitivities in iterated function systems. We show that an iterated function system IFS( \({\mathscr {F}}\) ) is chain mixing if and only if IFS( \({\mathscr {F}}^r\) ) \(\times \) IFS( \({\mathscr {F}}^s\) ) is chain transitive for some positive integers r and s. We introduce and study the notions of multi-transitive, \(\Delta \) -transitive and \(\Delta \) -mixing for iterated function systems. Relations are established between multi-transitivity and shadowing properties. For example, D-multi-transitivity, \(\Delta \) -mixing and chain mixing are equivalent mutually for an IFS( \({\mathscr {F}}\) ) which has the shadowing property. Moreover, we point out that multi-transitivity (resp. \(\Delta \) -transitivity, \(\Delta \) -mixing) is different from Z-multi-transitivity (resp. Z- \(\Delta \) -transitivity, Z- \(\Delta \) -mixing) which is defined by Tiaoying Zeng. PubDate: 2022-11-24

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Abstract: Abstract We consider an initial-boundary-value problem of three-dimensional nonhomogeneous micropolar fluids with density-dependent viscosity. Based on the energy method, we establish the global existence and uniqueness of strong solutions when the initial energy is suitably small. Moreover, we show that the velocity and the micro-rotational velocity converge exponentially to zero in \(H^2\) as time goes to infinity. In particular, there is no need to impose some compatibility condition on the initial data despite the presence of vacuum. PubDate: 2022-11-24

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Abstract: Abstract In this paper, we establish several explicit formulas involving Arakawa–Kaneko zeta values and Kaneko–Tsumura \(\eta \) -values. In particular, we prove a more general duality theorem involving Kaneko–Tsumura \(\eta \) -values. PubDate: 2022-11-24

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Abstract: Abstract Two vertices x and y in a connected graph G are doubly resolved by two vertices \(u,v\in V(G)\) if \(d_{G}(x,u)-d_{G}(y,u)\ne d_{G}(x,v)-d_{G}(y,v)\) . A set S is a doubly resolving set of G if each pair of vertices of G is doubly resolved by some pair of vertices in S. The minimum cardinality of doubly resolving sets of a graph G is the doubly metric dimension of G. In this paper, we provide both a lower and an upper bound on the doubly metric dimension of Cartesian products \(G\Box H\) and generalize the known result on that of \(P_2\Box C_n\) . Moreover, we determine the doubly metric dimensions of grid graphs and torus graphs. PubDate: 2022-11-24

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Abstract: Abstract In this paper, we study various graphs, namely Colon power graph, cyclic graph, enhanced power graph, and commuting graph on a semigroup S. The purpose of this paper is twofold. First, we study the interconnection between the diameters of these graphs on semigroups having one idempotent. Consequently, the results on the connectedness and the diameter of the proper enhanced power graphs (or cyclic graph) of finite groups, viz., symmetric group and alternating group, are obtained. In the other part of this paper, for an arbitrary pair of these four graphs, we classify finite semigroups such that the graphs in this pair are equal. Our results generalize some of the corresponding results of these graphs on groups to semigroups. PubDate: 2022-11-24

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Abstract: Abstract A linear forest is a graph consisting of vertex disjoint paths. Let l(G) denote the maximum size of linear forests in G. Denote by \(\delta (G)\) the minimum degree of G. Recently, Duan, Wang and Yang gave an upper bound on the number of 3-cliques in n-vertex graphs with \(l(G)=k-1\) and \(\delta (G)=\delta .\) Duan et al. gave an upper bound \(h_s(n,\alpha ',\delta )\) on the number of s-cliques in n-vertex graphs with prescribed matching number \(\alpha '\) and minimum degree \(\delta .\) But in some cases, these two upper bounds are not obtained by the graph with minimum degree \(\delta .\) For example, \(h_2(15,7,3)=77\) is attained by a unique graph of minimum degree 7, not 3. Motivated by these works, we give sharp results about this problem. We determine the maximum number of s-cliques in n-vertex graphs with \(l(G)=k-1\) and \(\delta (G)=\delta .\) As a corollary of our main results, we determine the maximum number of s-cliques in n-vertex graphs with given matching number and minimum degree. Moreover, we also determine the maximum number of copies of \(K_{r_1,r_2},\) the complete bipartite graph with class sizes \(r_1\) and \(r_2,\) in n-vertex graphs with \(l(G)=k-1\) and \(\delta (G)=\delta .\) PubDate: 2022-11-24

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Abstract: Abstract Using Mujica’s linearization theorem, we extend to the holomorphic setting some classical characterizations of compact (weakly compact, Rosenthal, Asplund) linear operators between Banach spaces such as the Schauder, Gantmacher and Gantmacher–Nakamura theorems and the Davis–Figiel–Johnson–Pełczynski, Rosenthal and Asplund factorization theorems. PubDate: 2022-11-23

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Abstract: Abstract We investigate the norm maps of algebraic even K-groups of finite extensions of number fields. Namely, we show that they are surjective in most situations. In the event that they are not surjective, we give a criterion in determining when an element in the even K-group of the base field comes from a norm of an element from the even K-groups of the extension field. This latter criterion is only reliant on the real primes of the base field. PubDate: 2022-11-23

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Abstract: Abstract This paper aims to provide new sufficient conditions for the existence of solutions to a vector quasi-equilibrium problem with set-valued mappings. Using a very recent Browder-type fixed-point theorem, which allows us to relax the common lower semicontinuity assumptions, the results improve some theorems from the literature and they can be applied where others fail. PubDate: 2022-11-23

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Abstract: Abstract Let A and B be two M-matrices, \(A^{-1}\) be the inverse of A, and \(\tau (B\circ A^{-1})\) be the minimum eigenvalue of the Hadamard product of B and \(A^{-1}\) . Firstly, by using the theories of Schur complements, a lower bound of the main diagonal entries of \(A^{-1}\) is derived and used to present two types of lower bounds of \(\tau (B\circ A^{-1})\) . Secondly, in order to obtain bigger lower bounds of \(\tau (B\circ A^{-1})\) , two types of lower bounds of \(\tau (B\circ A^{-1})\) with non-negative parameters are constructed. Thirdly, by finding the optimal values of parameters, two preferable lower bounds of \(\tau (B\circ A^{-1})\) are yielded. Finally, numerical examples show the effectiveness of the new methods. PubDate: 2022-11-23

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Abstract: Abstract A new Brauer-type Z-eigenvalue inclusion set for an even-order real tensor is presented. It is proved that it is tighter than the existing inclusion sets. As an application, a sufficient condition for the positive definiteness of an even-order real symmetric tensor (also a homogeneous polynomial form) and asymptotically stability of time-invariant polynomial systems is given. PubDate: 2022-11-23

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Abstract: Abstract Let \(L=-\Delta _{{\mathbb {H}}^{n}}+V\) be a Schrödinger operator on Heisenberg groups \({\mathbb {H}}^{n}\) , where \(\Delta _{{\mathbb {H}}^{n}}\) is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class \(B_{q}\) , \(q\ge {\mathcal {Q}}/2\) and \({\mathcal {Q}}=2n+2\) is the homogeneous dimension of \({\mathbb {H}}^{n}\) . We establish a T1 criterion for the boundedness of \(\gamma \) -Schrödinger–Calderón–Zygmund operators on Campanato type spaces \(\textrm{BMO}^{\alpha }_{L}({\mathbb {H}}^{n})\) . As an application, by the aid of regularity estimate for fractional heat semigroup \(\{e^{-tL^{\beta }}\}_{t>0}\) , we prove the \(\textrm{BMO}^{\alpha }_{L}\) -boundedness of operators generated by fractional heat semigroups including the maximal operators, the square functions, the Laplace transform type multipliers and the fractional integral associated with Schrödinger operator L via T1 theorem, respectively. PubDate: 2022-11-23

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Abstract: Abstract The goal of this paper is to discuss the asymptotic behavior of weak solutions to a class of parabolic equations involving fractional Laplacian in cylindrical domains becoming unbounded in one direction. The results presented in this paper are new and extend some main results in the literature for local and nonlocal elliptic problems with Dirichlet boundary condition. PubDate: 2022-11-23

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Abstract: Abstract For a recollement \((\mathcal {A},\mathcal {B},\mathcal {C})\) of abelian categories, we show that n-tilting (resp. n-cotilting) subcategories in \(\mathcal {A}\) and \(\mathcal {C}\) can be glued to get n-tilting (resp. n-cotilting) subcategories in \(\mathcal {B}\) under certain conditions. PubDate: 2022-11-23

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Abstract: Abstract We establish the Lyapunov stability of the equilibrium (trivial) solution for the discrete diamond–alpha difference operator, using the imaginary diamond–alpha ellipse. This unifies and extends equilibrium analysis for first-order forward (Delta) and backward (nabla) difference equations with a constant complex coefficient. We prove that for coefficients with negative elliptical real part the equation is asymptotically stable, with zero elliptical real part the equation is stable, and with positive elliptical real part the equation is unstable, except in the critical case of alpha equals one half. We also provide an asymptotic stability result for the origin in the case of the corresponding inhomogeneous equation. PubDate: 2022-11-23

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Abstract: Abstract In this paper, the marginal distributions of concomitants of k-record values based on Sarmanov family of bivariate distributions are obtained, as an extension of several recent papers. Besides, we derive the joint distribution of concomitants of k-record values for this family. Furthermore, some new and useful properties of information measures, namely, the extropy, Shannon entropy, inaccuracy measure, cumulative entropy, cumulative residual entropy, and cumulative residual Fisher information are studied. Finally, we offered various examples accompanied by numerical investigations that backed up the theoretical findings. PubDate: 2022-11-22

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Abstract: Abstract This paper is concerned with a class of nonlocal dispersal problem with Dirichlet boundary conditions. We analyze the limit of solutions when the dispersal kernel is rescaled. Our main results reveal that the solutions of Dirichlet heat equation can be approximated by the nonlocal dispersal equation. The investigation also shows that the nonlocal dispersal equation is similar to the convection–diffusion equation by taking another special kernel function. PubDate: 2022-11-22