Let $X$ and $Y$ be Banach spaces and let $f\,:\,X\,\to \,Y$ be an odd mapping. For any rational number $r\,\ne \,2$, C. Baak, D. H. Boo, and Th. M. Rassias proved the Hyers–Ulam stability of the functional equation

$$rf\left( \frac{\sum\nolimits_{j=1}^{d}{{{x}_{j}}}}{r} \right)\,+\,\sum\limits_{\begin{smallmatrix} i\left( j \right)\,\in \left\{ 0,\,1 \right\} \\ \sum\nolimits_{j=1}^{d}{i\left( j \right)=\ell } \end{smallmatrix}}{rf\left( \frac{\sum\nolimits_{j=1}^{d}{{{\left( -1 \right)}^{i\left( j \right)}}{{x}_{j}}}}{r} \right)}\,=\,\left( C_{d-1}^{\ell }\,-\,C_{d-1}^{\ell -1}\,+\,1 \right)\,\sum\limits_{j=1}^{d}{f\left( {{x}_{j}} \right),}$$

where $d$ and $\ell$ are positive integers so that $1\,<\,\ell \,<\,\frac{d}{2}$, and $C_{q}^{p}\,:=\frac{q!}{\left( q-p \right)!p!},\,p,\,q\,\in \,\mathbb{N}$ with $p\,\le \,q$.

In this note we solve this equation for arbitrary nonzero scalar $r$ and show that it is actually Hyers–Ulam stable. We thus extend and generalize Baak et al.’s result. Other questions concerning the $^{*}$-homomorphisms and the multipliers between ${{C}^{*}}$-algebras are also considered.

We prove a comprehensive version of the Ruelle–Perron–Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previously known estimates.

Let $R$ be a ring and $b,c\in R$. In this paper, we give some characterizations of the $(b,c)$-inverse in terms of the direct sum decomposition, the annihilator, and the invertible elements. Moreover, elements with equal $(b,c)$-idempotents related to their $(b,c)$-inverses are characterized, and the reverse order rule for the $(b,c)$-inverse is considered.

Let $\Omega \subset {{\mathbb{R}}^{n}}$ be a bounded Lipschitz domain and consider the energy functional

1

$$F[u,\Omega ]\,:=\,\,{{\int }_{\Omega }}\text{F}(\triangledown u(x))\,d\text{x,}$$

over the space of ${{W}^{1,2}}(\Omega ,{{\mathbb{R}}^{m}})$ where the integrand $\text{F}:{{\mathbb{M}}_{m\times n}}\to \mathbb{R}$ is a smooth uniformly convex function with bounded second derivatives. In this paper we address the question of regularity for solutions of the corresponding system of Euler–Lagrange equations. In particular, we introduce a class of singularmaps referred to as traceless and examine themas a new counterexample to the regularity of minimizers of the energy functional $F[\cdot ,\Omega ]$ using a method based on null Lagrangians.

In an attempt to resolve a folklore conjecture of Erdös regarding the non-vanishing at $s\,=\,1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\left\{ -1,\,1 \right\}$, Livingston conjectured the $\overline{\mathbb{Q}}$-linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston’s conjecture for composite $q\,\ge \,4$, highlighting that a new approach is required to settle Erdös conjecture. We also prove that the conjecture is true for prime $q\,\ge \,3$, and indicate that more ingredients will be needed to settle Erdös conjecture for prime $q$.

We study the semilinear Schrödinger equation

$$\left\{ _{u\,\,\in \,\,{{H}^{1}}({{\mathbf{R}}^{N}}),}^{-\Delta \,u+V(x)u=f(x,u),\,\,\,\,\,x\in \,\,{{\mathbf{R}}^{N}},} \right.$$

where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)={{V}_{0}}(x)+{{V}_{1}}(x)$, ${{V}_{0}}\in C({{\mathbf{R}}^{N}}),\,{{V}_{0}}(x)$ is 1-periodic in each of ${{x}_{1}},{{x}_{2}},...,{{x}_{N}}$, $\sup [\sigma (-\Delta +{{V}_{0}})\,\cap \,(-\infty ,0)]<0<$$\inf [\sigma (-\Delta +{{V}_{0}})\cap (0,\infty )],\,{{V}_{1}}\in C({{\mathbf{R}}^{N}})$, and ${{\lim }_{|x|\to \infty }}\,{{V}_{1}}(x)=0$. A new super-quadratic condition is obtained that is weaker than some well-known results.

In this paper, we construct two classes of rational function operators using the Poisson integrals of the function on the whole real axis. The convergence rates of the uniform and mean approximation of such rational function operators on the whole real axis are studied.

In this paper, the concept of the classical $f$-divergence for a pair of measures is extended to the mixed $f$-divergence formultiple pairs ofmeasures. The mixed $f$-divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed $f$-divergence are established, such as permutation invariance and symmetry in distributions. An Alexandrov–Fenchel type inequality and an isoperimetric inequality for the mixed $f$-divergence are proved.

We study a population model with nonlocal diffusion, which is a delayed integro-differential equation with double nonlinearity and two integrable kernels. By comparison method and analytical technique, we obtain globally asymptotic stability of the zero solution and the positive equilibrium. The results obtained reveal that the globally asymptotic stability only depends on the property of nonlinearity. As an application, we discuss an example for a population model with age structure.

For any ring $R$, we show that, in the bounded derived category ${{D}^{b}}(\text{Mod}\,R)$ of left $R$-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category $\underline{\text{GP}}(\text{Mod}\,R)\,(resp.\overline{GI}(Mod\,R))$ of Gorenstein projective (resp. injective) modules. As a consequence, we get that if $R$ is a left and right noetherian ring admitting a dualizing complex, then $\underline{\text{GP}}(\text{Mod}\,R)$ and $\overline{\text{GI}}(\text{Mod}\,R)$ are equivalent.

Let $\ell \in \mathbb{N}$ and $\alpha \in (0,2\ell )$. In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaces via the sequence ${{\{f-{{B}_{\ell ,{{2}^{-k}}}}f\}}_{k}}$ consisting of the difference between $f$ and the ball average ${{B}_{\ell ,{{2}^{-k}}}}f$. These results lead to the introduction of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaceswith any positive smoothness order onmetricmeasure spaces. As special cases, the authors obtain a new characterization of Morrey–Sobolev spaces and ${{\text{Q}}_{\alpha }}$ spaces with $\alpha \in (0,1)$, which are of independent interest.

Let $G$ be a connected graph with vertex set $V\left( G \right)$.The degree Kirchhoff index of $G$ is defined as ${{S}^{\prime }}\left( G \right)\,=\,\sum{_{\left\{ u,v \right\}\,\subseteq \,V\left( G \right)}d\left( u \right)d\left( v \right)R\left( u,\,v \right)}$, where $d\left( u \right)$ is the degree of vertex $u$, and $R\left( u,\,v \right)$ denotes the resistance distance between vertices $u$ and $v$. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoff index among all $n$-vertex bicyclic graphs with exactly two cycles.

Abstract. A subset $W$ of the vertex set of a graph $G$ is called a resolving set of $G$ if for every pair of distinct vertices $u,\,v$, of $G$, there is $w\,\in \,W$ such that the distance of $w$ and $u$ is different from the distance of $w$ and $v$. The cardinality of a smallest resolving set is called the metric dimension of $G$, denoted by $\dim\left( G \right)$. The circulant graph ${{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right)$ consists of the vertices ${{v}_{0}},\,{{v}_{1\,}},\,.\,.\,.\,,{{v}_{n\,-\,1}}$ and the edges ${{v}_{i}}{{v}_{i\,+\,j}}$, where $0\,\le \,i\,\le \,n\,-\,1,1\,\le \,j\,\le \,t\,\left( 2\,\le \,t\,\le \,\left\lfloor \frac{n}{2} \right\rfloor \right)$, the indices are taken modulo $n$. Grigorious, Manuel, Miller, Rajan, and Stephen proved that $\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right) \right)\,\ge \,t\,+\,1$ for $t\,<\,\left\lfloor \frac{n}{2} \right\rfloor ,\,n\,\ge \,3$, and they presented a conjecture saying that $\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right) \right)\,=\,t\,+\,p\,-\,1$ for $n\,=\,2tk\,+\,t\,+\,p$, where $3\,\le \,p\,\le \,t\,+\,1$. We disprove both statements. We show that if $t\,\ge \,4$ is even, there exists an infinite set of values of $n$ such that $\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,t \right) \right)\,=\,t$. We also prove that $\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right) \right)\,\le \,t\,+\,\frac{p}{2}$ for $n\,=\,2tk\,+\,t\,+\,p$, where $t$ and $p$ are even, $t\,\ge \,4,\,2\,\le \,p\,\le \,t$, and $k\,\ge \,1$.

In this paper, we study condition $C_{\hat{\ }}^{'}$, which is a projective tensor product analogue of condition $C'$. We show that the finite-dimensional $\text{OLLP}$ operator spaces have condition $C_{\hat{\ }}^{'}$, and ${{M}_{n}}\left( n\,>\,2 \right)$ does not have that property.