Let $I$ be an ideal of a commutative Noetherian ring $R$. We determine the set of all attached primes of the top local cohomology module of any finitely generated $R$-module $M$ with respect to the ideal $I$ of $R$ in terms of certain elements of $\text{Supp }M$. Then as a consequence of this result we show that for any pair of finitely generated $R$-modules $M$ and $N$ with $\text{Supp }N\subseteq \text{Supp }M$, if $\text{cd }(I,M)=\text{cd }(I,N)=c\ge 0$, then ${\text{Att }}_R{H}_I^c(N)\subseteq {\text{Att }}_R{H}_I^c(M)$ and $\text{Rad}({\text{Ann }}_R{H}_I^c(M))\subseteq \text{Rad}({\text{Ann }}_R{H}_I^c(N))$. Furthermore, in the case that $R$ is a local ring, we prove some similar results concerning the associated primes of Matlis dual functors of top local cohomology modules.

In order to study the inessential operators Tarafdar introduced the notion of improjective operators. Aiena and González characterize the class of inessential operators in more algebraic terms and they also obtain characterizations of the inessential operators among the improjective operators in terms of the complementability of some subspaces. In the present paper, we extend these characterizations to the general case of linear relations. Firstly, we characterize the inessential linear relations by the nullity and the deficiencies. Secondly, we apply such algebraic results to characterize the inessential linear relations among the improjective linear relations in terms of the complementability of some null and range spaces.

Let $a_0,\dots ,a_d$ be pairwise coprime positive integers, $m_0,\dots ,m_d$ nonnegative integers, and $z_0,\dots ,z_d$ complex numbers. We study expressions of the form

$$\frac{1}{a_0^{m_0+1}}\sum _k\prod _{j=1}^d{\text{cot}}^{(m_j)}\pi \left(a_j\frac{k+z_0}{a_0}-z_j\right),$$

where the summation is taken over all $k(\text{mod }{a}_0)$ for which the summand is well defined. These sums generalize and unify various arithmetic sums introduced and studied by Dedekind, Apostol, Carlitz, Zagier, Berndt, Meyer, Sczech, Dieter and Beck. Special cases of these sums appear in various areas such as analytic and algebraic number theory, topology, algebraic and combinatorial geometry, and algorithmic complexity. In this paper, without any additional assumption on the parameters $a_0,\dots ,a_d$ and $z_0,\dots ,z_d$, we present a simple proof for the reciprocity formula for these generalized Dedekind cotangents sums. We recover the previous known results and improve them. As applications, we give explicit formulae for sums of secant and cosecant values in terms of Apostol–Bernoulli numbers.

For a given family of planar differential systems it is a very difficult problem to determine its limit cycles. We give a family of planar polynomial differential systems of degree 7 with two limit cycles explicitly given, one of them is an algebraic limit cycles and the other one is a nonalgebraic limit cycles. Concrete examples exhibiting the applicability of our results are introduced.

The title refers to the square diagram of bijections between various classes of finite subgroups of classical groups. We survey how Coxeter’s work implies a bijection between complex reflection groups of rank two and real reflection groups in $O(3)$. We also consider this magic square of reflections and rotations in the framework of Clifford algebras: we give an interpretation using (s)pin groups and explore these groups in small dimensions.

We extend the results of Akahori et al. (2019) on ground states. We discuss the existence of ground states to the double-type nonlinear $p$-Laplace problem involving the Sobolev critical exponent in $R^N$:

$$-{\mathrm{\Delta }}_{p}u+|u{|}^{p-2}u=|u{|}^{p^{\ast }-2}u+\lambda |u{|}^{q-2}u,u\in W^{1,p}(R^N),$$

where $N\ge 2$, , , , $p^{\ast }≔Np∕(N-p)$ is the Sobolev critical index, ${\mathrm{\Delta }}_{p}u=\text{div}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)$ is the $p$-Laplace operator. We show that: (i) if , then there exists so that ground states exist for all ; (ii) if , then there exist ground states for all .

We provide a novel discretization method with the help of the new operational matrices and fractal-fractional derivative operator for solving time fractal-fractional partial differential equations. To achieve our target, we consider the Bessel functions of the first kind to get the approximate solution with high precision. For the proposed problem, the basis functions together with their corresponding operational matrices are reduced to a system of algebraic equations. Besides, the error analysis of the method is thoroughly discussed. At last, to confirm the applicability and efficiency of the methodology, we implement several numerical tests. Furthermore, we discuss numerically HIV infection of the model of $\text{CD}{4}^{+}\text{T}$ cells.

We study the topological structure of $N$-soft sets given by Riaz et al. (J. Intell. Fuzzy Syst. 36:6 (2019), 6521–6536). Firstly, we redefine $N$-soft closed sets using the complement operation of $N$-soft sets proposed in (Math. Methods Appl. Sci. 44:8 (2021), 7343–7358) and investigate their basic properties. Then, we introduce the concept of an $N$-soft continuous mapping and also obtain the initial $N$-soft topology determined by a family of $N$-soft mappings. Furthermore, we establish a new concept of $N$-soft topological subspace and analyze some related properties of this concept. Finally, we present some examples to better understand the defined concepts.

The Riccati equation method is used to establish a new comparison theorem for systems of two linear first-order ordinary differential equation. This result is based on the concept of “null-classes”, and is a generalization of Sturm’s comparison theorem.

We investigate the existence of solutions for a class of initial value problems for impulsive Caputo–Hadamard fractional differential inclusions of order $r\in (1,2]$. We apply the concept of lower and upper solutions combined with the Bohnenblust–Karlin fixed point theorem.

Let $f$ denote a normalized primitive holomorphic cusp form of even integral weight $l$ for congruence group ${\mathrm{\Gamma }}_0(N)$ and ${\lambda }_f(n)$ the $n$-th Hecke eigenvalue of $f$. We consider the higher-power moments of $S_f(x,N)≔{\sum }_{n\le x}{\lambda }_f(n)$ and derive asymptotic formulas for the third through seventh power moments, improving the previous results.

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\mathrm{\Gamma }=\text{SL}(2,\mathbb{Z})$. Let ${\lambda }_{f\times f}(n)$ and ${\lambda }_{f\times f\times f}(n)$ be the normalized coefficients of the Dirichlet expansion of the Rankin–Selberg $L$-function and triple product $L$-function attached to $f$, respectively. We establish the asymptotic formulae and the upper bounds for the three-dimensional divisor problems related to these normalized coefficients, respectively.

Let $K$ be an algebraically closed field of characteristic $0$, and let $H_{{\mathbb{𝔸}}_k^n}^{\mu }$ denote the Hilbert scheme of $\mu$ points of the affine space ${\mathbb{𝔸}}_K^n$. An elementary component $E$ of $H_{{\mathbb{𝔸}}_k^n}^{\mu }$ is an irreducible component such that every $K$-point $[I]$ $\in$ $E$ represents a length-$\mu$ closed subscheme $\text{Spec}(k[x_1,\dots ,x_n]∕I)$ $\subseteq$ ${\mathbb{𝔸}}_K^n$ that is supported at one point. In a previous article we found some new examples of elementary components; in this article, we simplify the methods and extend the range of the previous paper to find several more examples. In addition, we present a “plausibility test” that suggests the existence of a vast number of similar examples.

We introduce and study some Hilbert-type operators acting from the function spaces into the sequence spaces. We give some sufficient and necessary conditions for the boundedness and compactness of these Hilbert-type operators. Also, for some special cases, we obtain the sharp estimates for the norms of certain Hilbert-type operators.

The concept of gyrogroups is a generalization of groups that does not explicitly have associativity. Recently, Wattanapan et al. considered the Hartman–Mycielski construction in strongly topological gyrogroups. In this paper, we extend their results in topological gyrogroups. We mainly, among other results, prove that every Hausdorff topological gyrogroup $G$ can be embedded as a closed subgyrogroup of a Hausdorff path-connected and locally path-connected topological gyrogroup $G^{•}$.

The main aim of this paper is to study fixed-point properties for semitopological semigroups ($\mathrm{\Sigma }$) of nonexpansive mappings on a subset of a Banach space which is nonempty, closed and convex or, more generally, a locally convex space, and to establish the fixed-point property of the space of all left uniformly continuous functions by using its amenability for $\mathrm{\Sigma }$ and to establish the fixed-point property for the class of weakly almost periodic functions by using its amenability for $\mathrm{\Sigma }$.

Austin and Virk (2016) showed that every coarse space is induced by a family of metrics defined on the underlying set of that coarse space. Motivated by this, we study the relation between prescribed properties of a coarse space and the corresponding system of metrics on the underlying set. In order to do this, we introduce the notion of coarse directed limits of metric spaces, which is, in general, different from metric space directed limits. We will show that every coarse space having asymptotic dimension dominated by an integer $n$ (respectively, having finite weak coarse decomposition complexity (FWCDC), being locally finite or being uniformly locally finite with property A) is a coarse directed limit of metric spaces which have asymptotic dimensions dominated by $n$ (respectively, have FWCDC, are locally finite or are uniformly locally finite with property A). Moreover, the converse of this statement is true. More generally, we will also introduce and study coarse directed limits of general coarse spaces.

We show that certain relative flags cannot have full exceptional collections. We also prove that some of these flags are categorical representable in dimension zero if and only if they admit a full exceptional collection. As a consequence, these flags are categorical representable in dimension zero if and only if they have $k$-rational points if and only if they are $k$-rational. Moreover, we calculate the categorical representability dimension for the flags under consideration.

We study the existence of solutions for 2D fractional Volterra integral equations, using Petryshyn’s fixed-point theorem and measures of noncompactness in the Banach space $C([0,a]\times [0,b],ℝ)$. Our results generalize results gained from earlier studies under some weaker conditions. We present some examples applying our results.

We study the notion of a quadratic Lie hom-der in a Lie Banach algebra associated with the functional equation

$$f\left(\frac{x+y}{2}+z\right)+f\left(\frac{x+y}{2}-z\right)+f\left(\frac{x-y}{2}+z\right)+f\left(\frac{x-y}{2}-z\right)=f(x)+f(y)+4f(z),$$

which was first introduced by Park, Hong and Kim (2006). We also present a relation between the above functional equation and the quadratic functional equation on certain groups. Finally, we prove some stability results of the quadratic Lie hom-ders in Lie Banach algebras by using Hyers’ direct method.