We investigate the perturbation theory of essentially Kato $n$-tuples of commuting operators. Moreover, we give quantitative stability results for the generalized range and generalized kernel. This study has lead to a generalization of some well known results for operators.

Based on McCoy’s theorem on commutative rings, Nielsen called a ring $R$ if the equation $f\left(x\right)g\left(x\right)=0$ implies $f\left(x\right)c=0$ for some nonzero element $c$ in $R$, where $f\left(x\right)$ and $g\left(x\right)$ are nonzero polynomials in $R\left[x\right]$. In this paper, a class of rings is introduced and called ZPZC rings, containing McCoy rings, and then their properties are investigated. Also, associations are found between ZPZC rings and other related rings. Moreover, several extensions of ZPZC rings are studied, including matrix rings, trivial extensions, Hochschild extensions and classical quotient rings.

This article deals with the necessary and sufficient conditions for the oscillation of a class of nonlinear second-order neutral difference equations for four distinct ranges of the neutral coefficient $p\left(n\right)$. Our study allow that the nonlinear function is either strongly sublinear or strongly superlinear. Some examples are given to illustrate our main results.

For any ideal $I$ in a Noetherian local ring or any graded ideal $I$ in a standard graded $K$-algebra over a field $K$, we introduce the socle module $Soc\left(I\right)$, whose graded components give us the socle of the powers of $I$. It is observed that $Soc\left(I\right)$ is a finitely generated module over the fiber cone of $I$. In the case that $S$ is the polynomial ring and all powers of $I\subseteq S$ have linear resolution, we define the module ${Soc}^{\ast }\left(I\right)$, which is a module over the Rees ring of $I$. For the edge ideal of a graph and for classes of polymatroidal ideals, we study the module structure of their socle modules.

We define a duality operation connecting closure operations, interior operations, and test ideals, and describe how the duality acts on common constructions such as trace, torsion, tight and integral closures, and divisible submodules. This generalizes the relationship between tight closure and tight interior demonstrated by Epstein and Schwede (2014) and allows us to extend commonly used results on tight closure test ideals to operations such as those above.

Given a complex Hilbert space $H$, we establish a result on asymptotic energy equipartition for the abstract coupled system

$$\left\{\begin{array}{c}\hfill \\ \hfill \\ u_{tt}+2F\left(S\right)v_t+S^{2}u=0\hfill \\ \hfill \\ \hfill \\ v_{tt}+2F\left(S\right)u_t+S^{2}v=0\hfill \\ \hfill \\ \hfill \\ \hfill \end{array}\right.$$

for $\left(u,v\right):\left[0,\infty \right)\to H\oplus H$ with selfadjoint $S:D\left(S\right)\to H$ and operator-valued damping $F\ge 0$. Both the kinetic and the potential energies of solutions contain interaction terms in the general case and are conveniently weighted to account for the presence of damping. A remarkable feature of the above system is that it is wellposed if and only if $F$ is bounded, a fact that we also prove.

A fast diffusion $p$-Laplace equation with special medium void is considered. Global existence of weak solutions is obtained by using comparison principle and ordinary differential inequalities. Moreover, sufficient conditions for the solutions to vanish in finite time or not are derived with the help of Hardy–Littlewood–Sobolev inequality.

We construct some infinite families of hyperelliptic curves of genus $2$ with exactly two rational points. In the proof, we first show that the Mordell–Weil ranks of these hyperelliptic curves are $0$ and then determine the sets of rational points by using a Lutz–Nagell type theorem for hyperelliptic curves which was proven by Grant.

In this paper, we discussed the stability analysis of a class of Caputo nabla fractional-order delay difference equation. We used Chebyshev norm to obtain Ulam–Hyers–Mittag–Leffler stability. An example is provided to illustrate the theoretical result. Furthermore, Ulam–Hyers–Rassias–Mittag–Leffler stability of the problem is also discussed.

A construction of laminated lattices of full diversity in odd dimensions $d$ with $3\le d\le 15$ is presented. The technique, which uses a combination of number fields and error-correcting codes, consists essentially of two steps: In the first, the Abelian number field $F$ of degree $d$ and prime conductor $p$, where $p$ is a prime congruent to 1 modulo $d$, is considered. In the second, the lattice is obtained as the canonical embedding (Minkowski homomorphism) of a $\mathbb{Z}$-submodule of ${\mathfrak{𝔒}}_F$, the ring of integers of $F$. The submodule is defined by the parity-check matrices of a Reed–Solomon code over $GF\left(p\right)$ and a suitably chosen linear code, typically either binary or over $\mathbb{Z}∕4\mathbb{Z}$, the ring of integers modulo 4.

We study on the second cohomology of the module extension Banach algebra $A{\oplus }_{1}X$, where $A$ is a Banach algebra and $X$ is a Banach $A$-bimodule. We characterize the defined 2-cocycles and 2-coboundaries on $A{\oplus }_{1}X$ and as an interesting result we show that when $H^2\left(A,A\right)=0$, then $H^2\left(A{\oplus }_{1}A,A{\oplus }_{1}A\right)=QM\left(A\right)$, where $QM\left(A\right)$ is the set of quasi-multipliers on $A$. Also, we give some examples related to vanishing and nonvanishing of the second cohomology of various module extensions of Banach algebras.

We determine all nontrivial integer solutions to the equation ${\left(x+r\right)}^2+{\left(x+2r\right)}^2+\cdots +{\left(x+dr\right)}^2=y^n$ for $2\le d\le 10$ and $1\le r\le 10^4$ with $gcd\left(x,y\right)=1$. We make use of a factorization argument and the primitive divisors theorem due to Bilu, Hanrot and Voutier.

In this paper, a patchy predator-prey model with the effect of prey refuge and migration is proposed. The effect of prey refuge and migration on the dynamical behaviors of the proposed model is investigated by applying an analytical approach. The results show the stabilizing and destabilizing effects of prey refuge and migration are observed under some certain conditions. For the effect of prey refuge, the destabilizing impact is found in heterogeneous predation model with Holling II type functional response. For the effect of migration, the dynamical consequences is bistable while the level of prey refuge is less than a threshold value. The destabilizing effect is firstly observed theoretically when the attack rates of predators is proportional to the migration ratio.

We give characterization results of the fractional Hankel transform as well its inverse on some Gelfand–Shilov spaces of type $W$. Furthermore, we derive the boundedness properties of wavelet transforms involving the fractional Hankel transform on certain suitably constructed spaces of type $W$.

We prove the existence of a positive (Solutions Obtained as Limits of Approximations) to the following PDE involving fractional power of Laplacian

$${\left(-\mathrm{\Delta }\right)}^{s}u=\frac{1}{u^{\gamma }}+\lambda u^{2_s^{\ast }-1}+\mu \text{in }\mathrm{\Omega },u>0\text{in }\mathrm{\Omega },u=0\text{in }{ℝ}^N\setminus \mathrm{\Omega }.$$

Here, $\mathrm{\Omega }$ is a bounded domain of ${ℝ}^N$, $s\in \left(0,1\right)$, $2s<N$, $\lambda ,\gamma \in \left(0,1\right)$, $2_s^{\ast }=2N∕N-2s$ is the fractional critical Sobolev exponent and $\mu$ is a nonnegative bounded Radon measure in $\mathrm{\Omega }$.

Let $\mathcal{𝒜}$ be any ring. A nonadditive map $\phi :\mathcal{𝒜}\to \mathcal{𝒜}$ is said to be commuting if $\left[\phi \left(a\right),b\right]=\left[a,\phi \left(b\right)\right]$ holds for all $a,b\in \mathcal{𝒜}$, and is of a standard form if $\phi \left(a\right)=za+f\left(a\right)$ for all $a\in \mathcal{𝒜}$, where $z$ is in the center of $\mathcal{𝒜}$ and $f$ is a map from $\mathcal{𝒜}$ into its center. In this paper, we give a complete characterization of $\phi$ on unital rings $\mathcal{𝒜}$ with a nontrivial idempotent. It is also shown that, under some suitable assumptions on $\mathcal{𝒜}$, $\phi$ has a standard form. As applications, the structures of such maps on some important operator algebras are obtained.

We generalize the result of Rodin (2017) to the case of weighted arithmetic and geometric means.

Weighted Smirnov classes with power weight in bounded and unbounded domains are defined in this work. Nonhomogeneous Riemann problem with a measurable coefficient whose argument is a piecewise continuous function is considered in these classes. A Muckenhoupt type condition is imposed on the power type weight function and the orthogonality condition is found for the solvability of nonhomogeneous problem in weighted Smirnov classes, and the formula for the index of the problem is derived. Some special cases with power type weight function are also considered, and conditions on degeneration order are found.

We prove that there is a one-to-one correspondence between the following three sets: idempotent functions on a set of size $n$, complete exceptional sequences of linear radical square zero Nakayama algebras of rank $n$ and rooted labeled forests with $n$ nodes and height of at most one. Therefore, the number of exceptional sequences is given by the sum ${\sum }_{j=1}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)j^{n-j}$.

Agol proved that hyperbolic cubulated groups are virtually special. The aim of these notes is to make the proof accessible to a wider audience; we retain the underlying ideas and constructions of Agol, but substantially change or add to many parts of the argument to give a more transparent and detailed account.

A numerical semigroup is said to be perfect if it does not contain any isolated gaps. In this paper we show that if $S$ is any numerical semigroup, then for each integer $n\ge 2$ there exists a perfect numerical semigroup $T$ such that $S=\frac{T}{n}$ (that is, $S=\left\{x\in \mathbb{N}:nx\in T\right\}$)

Let $R$ be a commutative Noetherian ring with nonzero identity, $\mathfrak{𝔞}$ an ideal of $R$, and $n$ a nonnegative integer. For an arbitrary $R$-module $X$ which is not necessarily finite, we prove the following results:

(i) $f_{\mathfrak{𝔞}}^n\left(X\right)=inf${$i\in {\mathbb{N}}_0:H_{\mathfrak{𝔞}}^i\left(X\right)$ is not an ${FD}_{<n}$ $R$-module} if ${Ext}_R^i\left(R∕\mathfrak{𝔞},X\right)$ is an ${FD}_{<n}$ $R$-module for all $i$.

(ii) $f_{\mathfrak{𝔞}}^1\left(X\right)=inf${$i\in {\mathbb{N}}_0:H_{\mathfrak{𝔞}}^i\left(X\right)$ is not a minimax $R$-module} if ${Ext}_R^i\left(R∕\mathfrak{𝔞},X\right)$ is finite for all $i$.

(iii) $f_{\mathfrak{𝔞}}^2\left(X\right)=inf${$i\in {\mathbb{N}}_0:H_{\mathfrak{𝔞}}^i\left(X\right)$ is not a weakly Laskerian $R$-module} if $R$ is semilocal and ${Ext}_R^i\left(R∕\mathfrak{𝔞},X\right)$ is finite for all $i$.

(iv) $H_{\mathfrak{𝔞}}^i\left(X\right)$ is $\mathfrak{𝔞}$-cofinite for all $i<f_{\mathfrak{𝔞}}^2\left(X\right)$ and ${Ass}_R\left(H_{\mathfrak{𝔞}}^{f_{\mathfrak{𝔞}}^2\left(X\right)}\left(X\right)\right)$ is finite if ${Ext}_R^i\left(R∕\mathfrak{𝔞},X\right)$ is finite for all $i\le f_{\mathfrak{𝔞}}^2\left(X\right)$.

Here, $f_{\mathfrak{𝔞}}^n\left(X\right)=inf${$f_{\mathfrak{𝔞}{R}_{\mathfrak{𝔭}}}\left(X_{\mathfrak{𝔭}}\right):\mathfrak{𝔭}\in Spec\left(R\right)$ and ${dim}_R\left(R∕\mathfrak{𝔭}\right)\ge n$} is the $n$-th finiteness dimension of $X$ with respect to $\mathfrak{𝔞}$ and $f_{\mathfrak{𝔞}}\left(X\right)=inf${$i\in {\mathbb{N}}_0:H_{\mathfrak{𝔞}}^i\left(X\right)$ is not a finite $R$-module} is the finiteness dimension of $X$ with respect to $\mathfrak{𝔞}$.

We study several variants of Euler sums by using the methods of contour integration and the residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a variant of multiple zeta values of level 2, and we give some identities on relations between these variants of Euler sums and the variant of multiple zeta values.

The sets of functional equations

$$\left\{\begin{array}{ccc}\hfill f^{m_1}+g^{n_1}& =R_1,\hfill & \hfill \\ \hfill f^{m_2}+g^{n_2}& =R_2,\hfill \end{array}\right.$$

can be regarded as the Fermat-type system, where $m_1,m_2,n_1,n_2$ are positive integers, $R_1,R_2$ are nonzero polynomials. The purpose of this paper is to study the existence of solutions for several Fermat-type systems of the difference-differential equations

$$\left\{\begin{array}{ccc}\hfill {\left[f_1^{\prime }\left(z\right)\right]}^2+f_2{\left(z+c\right)}^2& =R_1\left(z\right),\hfill & \hfill \\ \hfill {\left[f_2^{\prime }\left(z\right)\right]}^2+f_1{\left(z+c\right)}^2& =R_2\left(z\right),\hfill \end{array}\right.\left\{\begin{array}{ccc}\hfill {\left[f_1^{\prime }\left(z\right)\right]}^2+{\left[f_2\left(z+c\right)-f_1\left(z\right)\right]}^2& =R_1\left(z\right),\hfill & \hfill \\ \hfill {\left[f_2^{\prime }\left(z\right)\right]}^2+{\left[f_1\left(z+c\right)-f_2\left(z\right)\right]}^2& =R_2\left(z\right),\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{ccc}\hfill {\left[f_1\left(z\right)\right]}^2+{\left[f_2\left(z+c\right)-f_1\left(z\right)\right]}^2& =R_1\left(z\right),\hfill & \hfill \\ \hfill {\left[f_2\left(z\right)\right]}^2+{\left[f_1\left(z+c\right)-f_2\left(z\right)\right]}^2& =R_2\left(z\right).\hfill \end{array}\right.$$

Our results about the existence and the forms of solutions for these Fermat types systems are some improvements of the previous theorems given by Gao, Liu, Yang, Cao, Qi. Meanwhile, we give some examples to utilize the existence and forms of solutions for such systems in each case.

The main aim of this paper is to establish a generalized Schwarz–Pick inequality for quasiconformal mappings satisfying Poisson’s equation from unit disk onto a simply connected and bounded convex domain $\mathrm{\Omega }$. In addition, some applications of this inequality were given.