A commutative ring $R$ is called strongly regular associate if, for any $a,b\in R$, $Ra=Rb$ implies that $a=rb$ and $sa=b$ for some regular elements $r,s\in R$. In this paper, we first give a characterization of strongly regular associate rings. A ring $R$ is said to have regular range 1 if, for any $a,b\in R$, $Ra+Rb=R$ implies that $a+bx$ is a regular for some $x\in R$. We show that the ring of continuous functions $C(X)$ is strongly regular associate if and only if it has regular range 1. Finally, we generalize a theorem of Anderson and Chun, which states that $C([a,b])$ is a strongly regular associate ring.

Let $p$ be an odd prime number, $n\ge 3$ be a rational integer, and let $f(X)=X^{p^n}+aX+a\in \mathbb{Z}[X]$ be an Eisenstein trinomial with respect to $p$. We prove that the Galois group $G$ of $f(X)$ over the field $\mathbb{Q}$ of rational numbers, is either the full symmetric group $S_{p^n}$, or $AGL(1,p^n)\le G\le AGL(n,p)$. We also show that $G\simeq S_{p^n}$, except possibly when $|p^{n{p}^n-1}+\frac{a}{p}{(p^n-1)}^{p^n-1}|$ is a square, and for each prime divisor $\ell$ of $a∕p$, $p$ divides the $\ell$-adic valuation $v_{\ell }(a)$ of the integer $a$.

The Jensen–Mercer inequality, which is well known in the literature, has an important place in mathematics and related disciplines. In this work, we obtain the Hermite–Jensen–Mercer inequality for post-quantum integrals by utilizing Jensen–Mercer inequalities. Then we investigate the connections between our results and those in earlier works. Moreover, we give some examples to illustrate our main results. This is the first paper about Hermite–Jensen–Mercer inequalities for post-quantum integrals.

We consider the real pure quartic number field $K=\mathbb{Q}(\sqrt[4]{p{d}^2})$, where $p$ is a prime number and $d$ is a square-free positive integer such that $d$ is prime to $p$. We compute $r_2(K)$ the $2$-rank of the class group of $K$ and as an application we exhibit all possible forms of $d$ for which the $2$-class group of $K$ is trivial (equivalently: the class number of $K$ is odd), cyclic or isomorphic to $\mathbb{Z}∕2^{n_1}\mathbb{Z}\times \mathbb{Z}∕2^{n_2}\mathbb{Z}$, where $n_i\in {\mathbb{N}}^{\ast }$.

We establish some Newton’s type inequalities in the case of differentiable convex functions through the well-known Riemann–Liouville fractional integrals. Furthermore, we give an example with graph and present the validity of the newly obtained inequalities. Finally, we give some inequalities of Riemann–Liouville fractional Newton’s type for functions of bounded variation.

In our previous work, we introduced a series of new derivation Lie algebras $L_k(V)$ associated to an isolated hypersurface singularity $(V,0)$. These are new analytic invariants of singularities. Here, we investigate $L_2(V)$ for fewnomial isolated singularities and obtain the formula of ${\lambda }_k(V)$ (i.e., the dimension of $L_k(V)$) for trinomial singularities. Furthermore, we prove the sharp upper estimate conjecture for $L_2(V)$. This is a continuation of our previous work (Math. Z. 298:3-4 (2021), 1813–1829). We proposed two new conjectures for ${\tau }_k(V)$ and ${\lambda }_k(V)$ and we prove these conjectures for a large class of singularities.

The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based on this perspective, we introduce the class of ass-decomposable monomial ideals which is a generalization of the class of Stanley–Reisner ideals of vertex decomposable simplicial complexes. The recursive structure of ass-decomposable monomial ideals allows us to find a simple formula for the depth, and in the squarefree case, an upper bound for the regularity of such ideals.

Let $q$ be a prime number and $f(x)=x^{q^s}-a{x}^m-b$ be a monic irreducible polynomial of degree $q^s$ having integer coefficients. Let $K=\mathbb{Q}(𝜃)$ be an algebraic number field with $𝜃$ a root of $f(x)$. We give some explicit conditions involving only $a,b,m,q,s$ for which $K$ is not monogenic. As an application, we show that if $p$ is a prime number of the form $32k+1$, $k\in \mathbb{Z}$ and $𝜃$ is a root of a monic polynomial $f(x)=x^{2^s}-32cp{x}^{2^r}-p\in \mathbb{Z}[x]$ with , then $\mathbb{Q}(𝜃)$ is not monogenic.

We provide a classification of nearly free arrangements of lines in the complex projective plane with nodes and triple points.

About four centuries ago, Johann Faulhaber developed formulas for the power sum $1^n+2^n+\cdots +m^n$ in terms of $m(m+1)∕2$. The resulting polynomials are called the Faulhaber polynomials. We first give a short survey of Faulhaber’s work and discuss the results of Jacobi (1834) and the less known ones of Schröder (1867), which already imply some results published afterwards. We then show, for suitable odd integers $n$, the following properties of the Faulhaber polynomials $F_n$. The recurrences between $F_n$, $F_{n-1}$, and $F_{n-2}$ can be described by a certain differential operator. Furthermore, we derive a recurrence formula for the coefficients of $F_n$ that is the complement of a formula of Gessel and Viennot (1989). As a main result, we show that these coefficients can be expressed and computed in different ways by derivatives of generalized reciprocal Bernoulli polynomials, whose values can also be interpreted as central coefficients. This new approach finally leads to a simplified representation of the Faulhaber polynomials. As an application, we obtain some recurrences of the Bernoulli numbers, which are induced by symmetry properties.

We prove that for every $N\ne 4$ there is only one right triangle that tiles the regular $N$-gon.

We show that

Moreover, the asymptotic behavior of real zeros of $Z(s,a)$ and $P(s,a)$ are studied when $0<a<\frac{1}{4}$. In addition, the complex zeros of these zeta functions are also discussed when $0<a<\frac{1}{2}$ is rational or transcendental.

We investigate the representation theory of the Temperley–Lieb algebra, ${\text{TL}}_n(\delta )$, defined over a field of positive characteristic. The principle question we seek to answer is the multiplicity of simple modules in cell modules for ${\text{TL}}_n$ over arbitrary fields. This provides us with the decomposition numbers for this algebra, as well as the dimensions of all simple modules. We obtain these results from purely diagrammatic principles, without appealing to realisations of ${\text{TL}}_n$ as endomorphism algebras of $U_q({\mathfrak{𝔰}\mathfrak{𝔩}}_2)$ modules. Our results strictly generalise the known characteristic zero theory of the Temperley–Lieb algebras.

We consider a class of quaternion-valued impulsive differential equations (QIDEs). We first use impulsive Cauchy matrices which are essential to explore the stability of QIDEs and whose exponential structure are analyzed in light of eigenvalues of matrices via the distance between impulsive points. Then, a number of sufficient criteria are acquired for the asymptotic stability of linear QIDEs and linear QIDEs with perturbation. In addition, the existence, uniqueness and Ulam–Hyers–Rassias stability of solutions of nonlinear QIDEs are investigated. Most importantly, the results which we obtain are presented in the sense of quaternion-valued and complex-valued, respectively. Finally, examples are given to indicate the utility of our theoretical results.

In recent years there has been significant progress on the problem of representing integers as a sum of three cubes. Most significantly are the relatively new solutions found by Booker and Sutherland involving integers with as many as 21 digits. In this paper, we look closely at the solutions that have been found, associate to them a pair $(\gamma ,s)$, where $\gamma$ is an algebraic number in the associated pure cubic field, and $s$ is a positive integer, and describe an algorithm to compute this pair, which in some cases appears to be competitive with existing methods to solve the original representation problem.

This paper is concerned with the problem of identifying the space-dependent source term and initial value simultaneously for a time-fractional diffusion equation. The inverse problem is ill-posed, and the idea of decoupling it into two operator equations is applied. In order to solve this inverse problem, a fractional Tikhonov regularization method is proposed. Furthermore, the corresponding convergence estimates are presented by using the a priori and a posteriori parameter choice rules. Several numerical examples compared with the classical Tikhonov regularization are constructed for verifying the accuracy and efficiency of the proposed method.

We derive some a priori estimates and then prove the existence and regularity of distributional solutions to degenerate elliptic equations of the form

where the Carathéodory function $\mathcal{𝒜}:\mathrm{\Omega }\times ℝ\times {ℝ}^n\to {ℝ}^n$ satisfies

$$\mathcal{𝒜}(x,s,\xi )\cdot \xi \ge \alpha \frac{|\xi {|}^p}{{(1+|s|)}^{𝜃}},|\mathcal{𝒜}(x,s,\xi )|\le \beta |\xi {|}^{p-1}$$

for $1<p<n$, $0\le 𝜃<p-1$ and $0<\alpha \le \beta <\infty$, with $f$ a Marcinkiewicz function. We obtain a gradient estimate for entropy solutions, and show that the result is optimal by a counterexample. We also consider degenerate elliptic equations with a lower order term,

We show that the presence of the lower order term has regularizing effects on our result.

The generalized Hartogs triangles $H_{\{k_i\},\gamma }^n$ are nonsmooth Reinhardt domains defined by

where $({\tilde{z}}_1,\dots ,{\tilde{z}}_m)=(z_1,\dots ,z_{n-1})$ with ${\tilde{z}}_i\in {ℂ}^{k_i}$, $i=1,\dots ,m$ and $k_1+\cdots +k_m=n-1$. We find some “good” holomorphic automorphisms of $H_{\{k_i\},\gamma }^n$ and use them to obtain necessary conditions for the Toeplitz and Hankel products to be bounded on the Bergman space of $H_{\{k_i\},\gamma }^n$.

We consider difficult problems in solving nonlinear ordinary differential equations with variable coefficients. We use elastic transformation methods (elastic upgrading transformation and elastic reduced transformation) to transform the first and third order equations into the associated Chebyshev equation. Then, according to the general solution of the associated Chebyshev equation, we obtain the general solutions of the first-order and third-order nonlinear ordinary differential equations with variable coefficients, giving curves of general solution. The elastic transformation method provides a new idea and expands the solvable classes for solving nonlinear ordinary differential equation with variable coefficients.

This paper will focus on nonlinear nonautonomous abstract evolution equations with the Riemann–Liouville fractional derivative. The new concepts of classical and mild solutions will be introduced to carry out the study of solvability. Our methods are based on analytic semigroup theory, the probability density function, the fixed point technique and the compactness technique. The main results can be applied to the time variable fractional diffusion equations.