This paper discusses homological properties of a finite valuated $p$-group $A$. A category equivalence between full subcategories of the category of valuated $p$-groups and the category of right modules over the endomorphism ring of $A$ is developed to study $A$-presented and $A$-valuated valuated $p$-groups. In particular, we show that these classes do not coincide if $|A/pA| \gt p$. Examples are given throughout the paper.

Evans, et al. (J. Graph Theory 18 (1994), 801--815), defined an equivalence relation $\sim $ on the set of vertices of a simple graph $G$ by taking $a\sim b$ if and only if their open neighborhoods are the same. They introduced a new graph $G_{red }={G}/{\sim } $, reduction graph of $G$, as follows. The vertices are $V (G_{red })=\{[a]: a \in V (G) \}$, and two equivalence classes $[a]$ and $[b]$ are adjacent if and only if $a$ and $b$ are adjacent in $G$. Recently, Anderson and LaGrange defined some equivalence relations on the set of vertices of the zero-divisor graph of a commutative ring, one of which yields the reduction graph of the zero-divisor graph. In this paper, we state some basic graph theoretic properties of $G_{red }$ and study the relations between some properties of graph $G$ and its subgraph, $G_{red }$, such as the chromatic number, clique number, girth and diameter. Moreover, we study the reduction graph of some algebraic graphs, such as the comaximal graph, zero-divisor graph and Cayley graph of a commutative ring. Among other results, we show that, for every commutative ring $R$, $\Gamma _2(R)_{red } \simeq \Gamma _1(\mathbb {Z}_2^n)$, where $\Gamma _1(\mathbb {Z}_2^n)$ is the zero-divisor graph of the Boolean ring $\mathbb {Z}_2^n$, $\Gamma _2(R)$ is the comaximal graph of $R$ and $n=|Max (R)|$.

In this paper, we investigate a kind of second-order nonlinear differential equation with attractive-repulsive singularities. By applications of Green's function and Schauder's fixed point theorem, we establish the existence of a positive periodic solution for this equation.

In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial $3$-polytopes. A $(d-1)$-dimensional simplicial complex is said to be balanced if its graph is $d$-colorable. If a simplicial complex is balanced, then its Stanley-Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stanley-Reisner ring of a balanced simplicial $3$-polytope with respect to this special system of parameters has the strong Lefschetz property if the characteristic of the base field is not two or three. Moreover, we characterize $(2,1)$-balanced simplicial polytopes, i.e., polytopes with exactly one red vertex and two blue vertices in each facet, such that an analogous property holds. In fact, we show that this is the case if and only if the induced graph on the blue vertices satisfies a Laman-type combinatorial condition.

A trace formulation of the Maclaurin spectral coefficients of the Schwartzian kernel of functions of the spherical Laplacian is given. A class of polynomials $\mathscr {P}^\nu _l(X)$ $(l \ge 0$, $\nu \gt -1/2)$ linking to the classical Gegenbauer polynomials through a differential-spectral identity is introduced, and its connection to the above spectral coefficients and their asymptotics analyzed. The paper discusses some applications of these ideas combined with the Funk-Hecke identity and semigroup techniques to geometric and variational-energy inequalities on the sphere and presents some examples.

In this paper, we look at properties of roots which can be written as sums of roots in crystallographic root systems. We derive properties of the poset associated to such a sum consisting of the subsums which are themselves roots.

We study a nonlinear fractional boundary value problem with nonlocal boundary conditions. An associated Green's function is constructed as a series of functions by the perturbation approach. Criteria for the existence of solutions are obtained based upon it.

In this paper, we obtain the general form of the periodic solutions of some higher order difference equations system \[x_{n+1}=\frac {\pm x_{n-k}y_{n-(2k+1)}}{y_{n-(2k+1)}\mp y_{n-k}},\] \[y_{n+1}=\frac {\pm y_{n-k}x_{n-(2k+1)}}{x_{n-(2k+1)}\mp x_{n-k}},\] $n,k\in \mathbb {N}_{0}$, where the initial values are arbitrary real numbers such that the denominator is always nonzero. Moreover, some numerical examples are presented to verify our theoretical results.

Let $\alpha _0\in \mathbb {C} \setminus \{0\}$, $A$ and $B$ be Banach function algebras. Also, let $\rho _1:\Omega _1 \rightarrow A$, $\rho _2:\Omega _2 \rightarrow A$, $\tau _1: \Omega _1 \rightarrow B$ and $\tau _2:\Omega _2 \rightarrow B$ be surjections such that $\|\rho _1(\omega _1)\rho _2(\omega _2)+\alpha _0\|_\infty =\|\tau _1(\omega _1)\tau _2(\omega _2)+\alpha _0\|_\infty $ for all $\omega _1\in \Omega _1, \omega _2\in \Omega _2$, where $\Omega _1$, $\Omega _2$ are two non-empty sets. Motivated by recent investigations on such maps between unital Banach function algebras, in this paper we characterize these maps for certain non-unital Banach function algebras including pointed Lipschitz algebras and abstract Segal algebras of the Talamanca-Herz algebras when the underlying groups are first countable. Moreover, sufficient conditions are given to guarantee such maps induce weighted composition operators.

The mixed van der Waerden number ${w}(k_1, \ldots ,k_r;r)$ is the least positive integer $n$ such that every $r$-coloring of $[1,n]$ admits a monochromatic arithmetic progression of length $k_i$, for at least one $i$. We denote by ${w}_2(k;r)$ the case in which $k_1=\cdots =k_{r-1}=2$ and $k_r=k$. For $k \le r$, we give upper and lower bounds for ${w}_2(k;r)$, also indicating cases when these bounds are achieved. We determine exact values in the cases where $(k,r) \in \{(p,p),(p,p+1),(p+1,p+1)\}$ and give bounds in the cases where $(k,r) \in \{(p,p+2),(p+2,p+2)\}$, for primes $p$. We provide a table of values for the cases $k \le r$ with $3 \le k \le 10$ and for several values of $r$, correcting some known values.

Let $\Gamma $ be a discrete group that acts on a semi-finite measure space $(\Omega , \mu )$ such that there is no $\Gamma $-invariant function in $L^1(\Omega , \mu )$. We show that the absence of the $\Gamma $-invariant mean on $L^\infty (\Omega ,\mu )$ is equivalent to the property $T$ of the reduced $C^*$-crossed product of $L^\infty (\Omega ,\mu )$ by $\Gamma $. In particular, if $\Lambda $ is a countable group acting ergodically on an infinite $\sigma $-finite measure space $(\Omega , \mu )$, then there exists a $\Lambda $-invariant mean on $L^\infty (\Omega , \mu )$ if and only if the corresponding crossed product does not have property $T$. Moreover, if $\Gamma $ is an ICC group, then $\Gamma $ is inner amenable if and only if $\ell ^\infty (\Gamma \setminus \{e\})\rtimes _{\mathbf {i},r} \Gamma $ does not have property $T$, where $\mathbf {i}$ is the conjugate action. On the other hand, a non-compact locally compact group $G$ is amenable if and only if $L^\infty (G)\rtimes _{\mathbf {lt}, r} G_\mathrm {d}$ does not have property $T$, where $G_\mathrm {d}$ is the group $G$ equipped with the discrete topology and $\mathbf {lt}$ is the left translation.

We classify all finite groups with planar cyclic graphs. Also, we compute the genus and crosscap number of some families of groups (by knowing that of the cyclic graph of particular proper subgroups in some cases).

The purpose of this paper is to study the singular Neumann problem involving the p$(x)$-Laplace operator: \begin{equation} (P_\lambda)\qquad\begin{cases} - \Delta_{p(x)} u +|u|^{p(x)-2}u = \frac{\lambda a(x)}{u^{\delta(x)}} &\mbox{in }\Omega, \\ u\gt0 &\mbox{in } \Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial\nu} = b(x) u^{q(x)-2}u &\mbox{on } \partial\Omega, \end{cases} \end{equation} where $\Omega\subset\mathbb{R}^N$, $N\geq 2$, is a bounded domain with $C^2$ boundary, $\lambda$ is a positive parameter, $a, b\in C(\overline{\Omega})$ are non-negative weight functions with compact support in $\Omega­$ and $\delta(x),$ $p(x),$ $q(x) \in C(\overline{\Omega})$ are assumed to satisfy the assumptions (A0)--(A1) in Section 1. We employ the Nehari manifold approach and some variational techniques in order to show the multiplicity of positive solutions for the $p(x)$-Laplacian singular problems.

Motivated by work of Hochster and Huneke, we investigate several constructions related to the $S_2$-ification $T$ of a complete equidimensional local ring $R$: the canonical module, the top local cohomology module, topological spaces of the form $Spec (R)-V(J)$, and the (finite simple) graph $\Gamma _R$ with vertex set $Min (R)$ defined by Hochster and Huneke. We generalize one of their results by showing, e.g., that the number of connected components of $\Gamma _R$ is equal to the maximum number of connected components of $Spec (R)-V(J)$ for all $J$ of height $2$. We further investigate this graph by exhibiting a technique for showing that certain graphs $G$ can be realized in the form $\Gamma _R$.

We prove that the crossing changes, Delta moves and sharp moves are unknotting operations on welded knots.

We consider support points of the class $S^0(\mathbb{D} ^n)$ of normalized univalent mappings on the polydisc $\mathbb{D} ^n$ with parametric representation, and we prove sharp estimates for coefficients of degree 2.

The aim of this paper is to prove several explicit formulas associated with the Frobenius-type Eulerian polynomials in terms of the weighted Stirling numbers of the second kind. As a consequence, we derive an explicit formula for the tangent numbers of higher order. We also give a recursive method for the calculation of the Frobenius-type Eulerian numbers and polynomials.

We combine harmonic analysis and operator algebraic techniques to give a concise argument that the left regular representation of a locally compact group is cyclic if and only if the group is first countable, a result first proved by Greenleaf and Moskowitz.

Let $\{\alpha _k\}_{k=1}^\infty$ be an arbitrary sequence of complex numbers in the upper half plane. We generalize the orthogonal rational functions $\phi _n$ based upon those points and obtain the Nevanlinna measure, together with the Riesz and Poisson kernels, for Caratheodory functions $F(z)$ on the upper half plane. Then, we study the relation between ORFs and their functions of the second kind as well as their interpolation properties. Further, by using a linear transformation, we generate a new class of rational functions and state the necessary conditions for guaranteeing their orthogonality.

This paper focuses on the log-convexity of the sequence $\{D_r(n)\}_{n\ge r}$ of $r$-derangement numbers, where $r\ge 2$ is a positive integer. We mainly prove that $\{D_2(n)\}_{n\ge 2}$ and $\{D_3(n)\}_{n\ge 7}$ are log-convex. In addition, we also show that $\{\sqrt {D_2(n)}\}_{n\ge 2}$ and $\{\sqrt [3]{D_3(n)}\}_{n\ge 7}$ are log-balanced.