In this paper, we give the distance spectral radii of several classes of circulant graphs. We also list the elements in the first rows of their corresponding distance matrices, with which all other distance eigenvalues can be obtained. In addition, we get the relationships between the distance spectral radii and forwarding indices of circulant graphs. Finally, the exact values of the vertex-forwarding indices and some bounds of the edge-forwarding indices for these kinds of graphs are presented.

Using the combinatorial descriptions of stuffle product, we obtain recursive formulas for the stuffle product of multiple zeta values and of multiple zeta-star values. Then we apply the formulas to prove several stuffle product formulas with one or two strings of $z_p$'s. We also describe how to use our formulas in general cases.

In this paper, primitive submodules are defined and various properties of them are investigated. Some characterizations of co-semisimple modules are given and several conditions under which co-semisimple and regular modules coincide are discussed.

We set up the new models of product manifolds, namely a generalized circular cylinder and a generalized hyperbolic cylinder as cylindrical types of ruled submanifold in Minkowski space. We also establish some characterizations of generalized circular cylinders and hyperbolic cylinders in Minkowski space with the Gauss map. We also show that there do not exist non-cylindrical marginally trapped ruled submanifolds with the pointwise $1$-type Gauss map of the first kind, which gives a characterization of non-cylindrical minimal ruled submanifolds in Minkowski space.

Using the theory of cohomology support locus, we give a necessary condition for the Albanese map of a smooth projective surface to be a submersion. More precisely, assuming that the cohomology support locus of any finite abelian cover of a smooth projective surface consists of finitely many points, we prove that the surface has trivial first Betti number, or is a ruled surface of genus one, or is an abelian surface.

Theorem 4.1 of [1] is not correctly stated. In this erratum we make a correction on Theorem 4.1. As a consequence, we also make the corresponding correction on Theorem 5.2 of [1].

In this paper, an equivalent formulation of extendable local reflexivity (ELR) introduced by Oikhberg and Rosenthal is given. We introduced the positive version (PELR) of the ELR in Banach lattices to solve the lifting problem for the bounded positive approximation property (BPAP). It is proved that a Banach lattice $X$ has the BPAP and is PELR if and only if the dual space $X^{*}$ of $X$ has the BPAP. Finally, we give isometric factorizations of positive weakly compact operators and establish some new characterizations of positive approximation properties.

This paper is concerned with a nonlinear elliptic inclusion driven by a multivalued subdifferential of nonsmooth potential and a nonlinear inhomogeneous differential operator. We obtain two multiplicity theorems in the Orlicz-Sobolev space. In the first multiplicity theorem, we produce three nontrivial smooth solutions. Two of these solutions have constant sign (one is positive, the other is negative). In the second multiplicity theorem, we derive an unbounded sequence of critical points for the problem. Our approach is variational, based on the nonsmooth critical point theory. We also show that $C^1$-local minimizers are also local minimizers in the Orlicz-Sobolev space for a large class of locally Lipschitz functions.

In this paper, we establish a Liouville type theorem for the following integral system with negative exponents \[ \begin{cases} u(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} f(u,v)(y) \, dy, & x \in \mathbb{R}^n, \\ v(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} g(u,v)(y) \, dy, & x \in \mathbb{R}^n, \end{cases} \] where $n \geq 1$, $\nu \gt 0$, and $f$, $g$ are continuous functions defined on $\mathbb{R}_{+} \times \mathbb{R}_{+}$. Under nature structure conditions on $f$ and $g$, we classify each pair of positive solutions for above integral system by using the method of moving sphere in integral forms. Moreover, some other Liouville theorems are established for similar integral systems.

The present paper concerns the study of distributional travelling waves for the model problem $u_{t} + (u^{2}-v)_{x} = 0$, $v_{t} + (u^{3}/3-u)_{x} = 0$, also called the Keyfitz-Kranzer system. In the setting of a product of distributions, which is not defined by approximation processes, we are able to define a rigourous concept of a solution which extends the classical solution concept. As a consequence, we will establish necessary and sufficient conditions for the propagation of distributional profiles and explicit examples are given. A survey of the main ideas and formulas for multiplying distributions is also provided.

The reaction-diffusion equations provide a predictable mechanism for pattern formation. These equations have a limited applicability. Refining the reaction-diffusion equations must be a good way for supplying the gap between the mathematical simplicity of the model and the complexity of the real world. In this manuscript, we introduce a modified version of reaction-diffusion equation, which we have named ‘‘nonlocal energy-weighted reaction-diffusion equation’’. For any bounded smooth domain $\Omega \subset \mathbb{R}^n$, we establish the global existence of weak solutions $u \in L^2(0,T;H^1_0(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$ to the initial boundary value problem of the nonlocal energy-weighted reaction-diffusion equation for any initial data $u_0 \in H^1_0(\Omega)$.

A critical point result is exploited in order to prove that a class of discrete anisotropic boundary value problems possesses at least one solution under an asymptotical behaviour of the potential of the nonlinear term at zero. Some recent results are extended and improved. Some examples are presented to demonstrate the applications of our main results.

We consider the existence of a weak solution for a class of coupling system containing stationary electromagnetic coupling system associated with the Maxwell equations in a multi-connected domain. Mathematically we are concerned with the coupled system containing a $p$-curl equation and a $q$-Laplacian equation.