Let $(R,\mathfrak{m})$ denote a local Cohen–Macaulay ring and $I$ a non-nilpotent ideal of $R$. The purpose of this article is to investigate Faltings’ finiteness dimension ${{f}_{I}}(R)$ and the equidimensionalness of certain homomorphic images of $R$. As a consequence we deduce that ${{f}_{I}}(R)=\max \{1,\text{ht}I\}$, and if $\text{mAs}{{\text{s}}_{R}}(R/I)$ is contained in $\text{As}{{\text{s}}_{R}}(R)$, then the ring $R/I+{{\bigcup }_{n\ge 1}}(0{{:}_{R}}{{I}^{n}})$ is equidimensional of dimension dim $R-1$. Moreover, we will obtain a lower bound for injective dimension of the local cohomology module $H_{I}^{\text{ht}I}(R)$, in the case where $(R,\mathfrak{m})$ is a complete equidimensional local ring.

We compute the cohomology of the right generalised projective Stiefel manifolds. Following this, we discuss some easy applications of the computations to the ranks of complementary bundles and bounds on the span and immersibility.

In this paper, we give a new definition of radicals of Green’s relations in an ordered semigroup and characterize left regular (right regular), intra regular ordered semigroups by radicals of Green’s relations. We also characterize the ordered semigroups that are unions and complete semilattices of $\text{t}$-simple ordered semigroups.

In this paper, a newnotion of scalar curvature for a Finsler metric $F$ is introduced, and two conformal invariants $Y(M,F)$ and $C(M,F)$ are defined. We prove that there exists a Finsler metric with constant scalar curvature in the conformal class of $F$ if the Cartan torsion of $F$ is sufficiently small and $Y(M,F)C(M,F)<Y({{\mathbb{S}}^{n}})$ where $Y({{\mathbb{S}}^{n}})$ is the Yamabe constant of the standard sphere.

In this paper, double commutativity and the reverse order law for the core inverse are considered. Then new characterizations of the Moore–Penrose inverse of a regular element are given by one-sided invertibilities in a ring. Furthermore, the characterizations and representations of the core and dual core inverses of a regular element are considered.

It follows from earlier work of Silver and Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links. We use this result to provide algorithms for detecting whether a link is the unlink, whether it is split, and whether it is totally split.

It is known that if $E$ is a closed subset of an open Riemann surface $R$ and $f$ is a holomorphic function on a neighbourhood of $E$, then it is “usually” not possible to approximate $f$ uniformly by functions holomorphic on all of $R$. We show, however, that for every open Riemann surface $R$ and every closed subset $E\subset R$, there is closed subset $F\subset E$ that approximates $E$ extremely well, such that every function holomorphic on $F$ can be approximated much better than uniformly by functions holomorphic on $R$.

We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here we consider Schubert problems given by more general isotropic flags, and prove this congruence modulo four for the largest class of Schubert problems that could be expected to exhibit this congruence.

Let $R$ be a commutative ring with non-zero identity. In this paper, we introduce the weakly nilpotent graph of a commutative ring. The weakly nilpotent graph of $R$ denoted by ${{\Gamma }_{w}}(R)$ is a graph with the vertex set ${{R}^{\star }}$ and two vertices $x$ and $y$ are adjacent if and only if $x\,y\in N{{(R)}^{\star }}$, where ${{R}^{\star }}=R\backslash \{0\}$ and $N{{(R)}^{\star }}$ is the set of all non-zero nilpotent elements of $R$. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if ${{\Gamma }_{w}}(R)$ is a forest, then ${{\Gamma }_{w}}(R)$ is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of ${{\Gamma }_{w}}(R)$. Among other results, we show that for an Artinian ring $R$, ${{\Gamma }_{w}}(R)$ is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam $\overline{({{\Gamma }_{w}}(R))}$. Finally, we characterize all commutative rings $R$ for which $\overline{({{\Gamma }_{w}}(R))}$ is a cycle, where $\overline{({{\Gamma }_{w}}(R))}$ is the complement of the weakly nilpotent graph of $R$.

We prove that for $d\in \left\{ 2,3,5,7,13 \right\}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $\chi $, if a prime $p$ is large enough compared to $D$, there is a newform $f\in {{S}_{2}}({{\Gamma }_{0}}(d{{p}^{2}}))$ with sign $(+1)$ with respect to the Atkin–Lehner involution ${{w}_{{{p}^{2}}}}$ such that $L(f\otimes \chi ,1)\ne 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions that generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f\otimes \chi ,\cdot )$ and a Petersson trace formula restricted to Atkin–Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

This paper generalizes the Aleksandrov problem: the Mazur-Ulam theorem on $n-G$-quasi normed spaces. It proves that a one-$n$-distance preserving mapping is an $n$-isometry if and only if it has the zero-$n-G$-quasi preserving property, and two kinds of $n$-isometries on $n-G$-quasi normed space are equivalent; we generalize the Benz theorem to $n$-normed spaces with no restrictions on the dimension of spaces.

Let $\mathbf{S}:={{\{S(t)\}}_{t\ge 0}}$ be a ${{\text{C}}_{0}}$-semigroup of quasinilpotent operators (i.e., $\sigma (S(t))=\{0\}$ for each $t>0$). In dynamical systems theory the above quasinilpotency property is equivalent to a very strong concept of stability for the solutions of autonomous systems. This concept is frequently called superstability and weakens the classical finite time extinction property (roughly speaking, disappearing solutions). We show that under some assumptions, the quasinilpotency, or equivalently, the superstability property of a ${{\text{C}}_{0}}$-semigroup is preserved under the perturbations of its infinitesimal generator.

The notion of coaxers is introduced in a pseudo-complemented distributive lattice. Boolean algebras are characterized in terms of coaxer ideals and congruences. The concept of coaxer lattices is introduced in pseudo-complemented distributive lattices and characterized in terms of coaxer ideals and maximal ideals. Finally, the coaxer lattices are also characterized in topological terms.

In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot{z}=$${{z}^{3}}+{{\epsilon }_{1}}z+{{\epsilon }_{0}}$ for $z\in \mathbb{C}{{\mathbb{P}}^{1}}$, depending on the values of ${{\epsilon }_{1}},{{\epsilon }_{0}}\in \mathbb{C}$. The bifurcation diagram is in ${{\mathbb{R}}^{^{4}}}$, but its conic structure allows describing it for parameter values in ${{\mathbb{S}}^{3}}$. There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points. We also explain the bifurcation of homoclinic connection in terms of the description of Douady and Sentenac of polynomial vector fields.

Let $K$ be an ultraspherical hypergroup associated with a locally compact group $G$ and a spherical projector $\pi$ and let $\text{VN}(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K$. In this note, we show that the set of invariant means on $\text{VN}(K)$ is singleton if and only if $K$ is discrete. Here $K$ need not be second countable. We also study invariant means on the dual of the Fourier algebra ${{A}_{0}}(K)$, the closure of $A(K)$ in the cb-multiplier norm. Finally, we consider generalized translations and generalized invariant means.

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.

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.

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.