Let $R$ be a commutative ring with identity. The co-annihilating-ideal graph of $R$ , denoted by ${{\mathcal{A}}_{R}}$ , is a graph whose vertex set is the set of all non-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent whenever $\text{Ann}\left( I \right)\,\cap \,\text{Ann}\left( J \right)\,=\,\left\{ 0 \right\}$ . In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties.

Let $H$ be a group. The co-maximal graph of subgroups of $H$ , denoted by $\Gamma \left( H \right)$ , is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma \left( H \right)$ if and only if $H\,=\,LK$ . In this paper, we study the connectivity, diameter, clique number, and vertex chromatic number of $\Gamma \left( H \right)$ . For instance, we show that if $\Gamma \left( H \right)$ has no isolated vertex, then $\Gamma \left( H \right)$ is connected with diameter at most 3. Also, we characterize all finitely groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma \left( H \right)$ is connected, and moreover, the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite.

All finite simple self 2-distance graphs with no square, diamond, or triangles with a common vertex as subgraph are determined. Utilizing these results, it is shown that there is no cubic self 2-distance graph.

Let $P$ be a finite $\text{N}$ -free poset. We consider the hypergraph $H\left( P \right)$ whose vertices are the elements of $P$ and whose edges are the maximal intervals of $P$ . We study the dual König property of $H\left( P \right)$ in two subclasses of $\text{N}$ -free class.

We identify when a tubular group (the fundamental group of a finite graph of groups with ${{\mathbb{Z}}^{2}}$ vertex and $\mathbb{Z}$ edge groups) is free by cyclic and show, using Wise’s equitable sets criterion, that every tubular free by cyclic group acts freely on a $\text{CAT}\left( 0 \right)$ cube complex.

Let $D$ be an integral domain, ${{X}^{1}}\left( D \right)$ be the set of height-one prime ideals of $D$ , $\left\{ {{X}_{\beta }} \right\}$ and $\left\{ {{X}_{\alpha }} \right\}$ be two disjoint nonempty sets of indeterminates over $D$ , $D\left[ \left\{ {{X}_{\beta }} \right\} \right]$ be the polynomial ring over $D$ , and $D\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1}}$ be the first type power series ring over $D\left[ \left\{ {{X}_{\beta }} \right\} \right]$ . Assume that $D$ is a Prüfer $v$ -multiplication domain $\left( \text{P}v\text{MD} \right)$ in which each proper integral $t$ -ideal has only finitely many minimal prime ideals (e.g., $t$ - $\text{SFT}$ $\text{P}v\text{MDs}$ , valuation domains, rings of Krull type). Among other things, we show that if ${{X}^{1}}\left( D \right)\,=\,\phi$ or ${{D}_{p}}$ is a $\text{DVR}$ for all $P\,\in \,{{X}^{1}}\left( D \right)$ , then $D\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1D-\left\{ 0 \right\}}}$ is a Krull domain. We also prove that if $D$ is a $t$ - $\text{SFT}\text{P}v\text{MD}$ , then the complete integral closure of $D$ is a Krull domain and $\text{ht}\left( M\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1}} \right)\,=\,1$ for every height-one maximal $t$ -ideal $M$ of $D$ .

Let $\mathbb{L}$ be a length function on a group $G$ , and let ${{M}_{\mathbb{L}}}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on ${{\ell }^{2}}\left( G \right)$ . Following Connes, ${{M}_{\mathbb{L}}}$ can be used as a “Dirac” operator for the reduced group ${{C}^{*}}$ -algebra $C_{r}^{*}\left( G \right)$ . It defines a Lipschitz seminorm on $C_{r}^{*}\left( G \right)$ , which defines a metric on the state space of $C_{r}^{*}\left( G \right)$ . We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak- $*$ topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Let $X$ be a real normed space, $Y$ a Banach space, and $f\,:\,X\,\to \,Y$ . We prove theUlam–Hyers stability theorem for the cubic functional equation

$$f\left( 2x\,+\,y \right)\,+\,f\left( 2x\,-\,y \right)\,-\,2f\left( x\,+\,y \right)\,-\,2f\left( x\,-\,y \right)\,-\,12f\left( x \right)\,=\,0$$

in restricted domains. As an application we consider a measure zero stability problem of the inequality

$$\left\| f\left( 2x\,+\,y \right)\,+\,f\left( 2x\,-\,y \right)\,-\,2f\left( x\,+\,y \right)\,-\,2f\left( x\,-\,y \right)\,-\,12f\left( x \right) \right\|\,\le \,\varepsilon$$

for all $\left( x,\,y \right)$ in $\Gamma \,\subset \,{{\mathbb{R}}^{2}}$ of Lebesgue measure 0.

A Nikishin–Maurey characterization is given for bounded subsets of weak-type Lebesgue spaces. New factorizations for linear and multilinear operators are shown to follow.

This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and let $H$ be a closed subgroup of $G$ . Let ${G}/{H}\;$ be the left coset space of $H$ in $G$ and let $\mu$ be the normalized $G$ -invariant measure on ${G}/{H}\;$ associated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space ${{L}^{2}}\left( {G}/{H,\,\mu }\; \right)$ .

We characterize two important notions of amenability and compactness of a locally compact quantum group $\mathbb{G}$ in terms of certain homological properties. For this, we show that $\mathbb{G}$ is character amenable if and only if it is both amenable and co-amenable. We finally apply our results to Arens regularity problems of the quantum group algebra ${{L}^{1}}\left( \mathbb{G} \right)$ . In particular, we improve an interesting result by Hu, Neufang, and Ruan.

In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.

It is shown that the Dirichlet problem for the slab $\left( a,\,b \right)\,\times \,{{\mathbb{R}}^{d}}$ with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function $g$ , the inhomogeneous difference equation $h\left( t\,+\,1,\,y \right)\,-\,h\left( t,\,y \right)\,=\,g\left( t,\,y \right)$ has an entire harmonic solution $h$ .

The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two finitely graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. The key ingredient in the proof is the use of stable partitions of graphs. The latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated with simple graphs using stable partitions. Our first result is the determination of the group of natural automorphisms of the chromatic functor, which is, in general, a larger group than the automorphism group of the graph. The second result is that the composition of the chromatic functor associated with a finitely graph restricted to the category FI of finitely sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups that is representation stable in the sense of Church–Farb.

Let $G$ be a finite group and let $A\left( G \right)$ denote the Burnside ring of $G$ . Then an inverse limit $L\left( G \right)$ of the groups $A\left( H \right)$ for proper subgroups $H$ of $G$ and a homomorphism res from $A\left( G \right)$ to $L\left( G \right)$ are obtained in a natural way. Let $Q\left( G \right)$ denote the cokernel of res. For a prime $p$ , let $N\left( p \right)$ be the minimal normal subgroup of $G$ such that the order of ${G}/{N}\;\left( p \right)$ is a power of $p$ , possibly 1. In this paper we prove that $Q\left( G \right)$ is isomorphic to the cartesian product of the groups $Q\left( {G}/{N\left( p \right)}\; \right)$ , where $p$ ranges over the primes dividing the order of $G$ .

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 actuallyHyers–Ulam stable. We thus extend and generalize Baak et al.’s result. Other questions concerningthe $^{*}$ -homomorphisms and the multipliers between ${{C}^{*}}$ -algebras are also considered.

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 $q$ .

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 $k\,\ge \,1$ .