Suppose that G is a nite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use ${p}^{M(G)}$ and ${p}^{m(G)}$ to denote the maximum and minimum of the orders of the non-normal subgroups of G; respectively. In this paper, we classify groups G such that $M(G)\langle \mathrm{2}m(G)-\mathrm{1}$: As a by-product, we also classify p-groups whose orders of non-normal subgroups are ${p}^{k}$ and ${p}^{k+\mathrm{1}}$:

Almost automorphic is a particular case of the recurrent motion, which has been studied in differential equations for a long time. We introduce square-mean pseudo almost automorphic and some of its properties, and then study the pseudo almost automorphic solution in the distribution sense to stochastic differential equation driven by Lévy process.

Let k be an integer with $k\ge \mathrm{6}$: Suppose that ${{\displaystyle \lambda}}_{\mathrm{1}},{{\displaystyle \lambda}}_{\mathrm{2}},\mathrm{...},{{\displaystyle \lambda}}_{\mathrm{5}}$ be nonzero real numbers not all of the same sign, satisfying that ${{\displaystyle \lambda}}_{\mathrm{1}}/{{\displaystyle \lambda}}_{\mathrm{2}}$ is irrational, and suppose that $\eta $ is a real number. In this paper, for any $\epsilon \rangle \mathrm{0}$; we consider the inequality $\left|{{\displaystyle \lambda}}_{\mathrm{1}}{{\displaystyle p}}_{\mathrm{1}}+{{\displaystyle \lambda}}_{\mathrm{2}}{{\displaystyle p}}_{\mathrm{2}}^{\mathrm{2}}+{{\displaystyle \lambda}}_{\mathrm{3}}{{\displaystyle p}}_{\mathrm{3}}^{\mathrm{3}}+{{\displaystyle \lambda}}_{\mathrm{4}}{{\displaystyle p}}_{\mathrm{4}}^{\mathrm{4}}+{{\displaystyle \lambda}}_{\mathrm{5}}{{\displaystyle p}}_{\mathrm{5}}^{\mathrm{5}}+\eta \right|\langle {({{\displaystyle \mathrm{max}?p}}_{j})}^{-\sigma (k)+\epsilon}$ has innitely many solutions in prime variables $${p}_{\mathrm{1}},{p}_{\mathrm{2}},\mathrm{...},{p}_{\mathrm{5}},$$ where $\sigma (k)$ depends on k: Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can rene some results on Diophantine approximation by unlike powers of primes, and get the related problem.

We consider the Euler-Maruyama discretization of stochastic volatility model $${{\displaystyle dS}}_{t}={{\displaystyle \sigma}}_{t}{{\displaystyle S}}_{t}{{\displaystyle dW}}_{t},{{\displaystyle d\sigma}}_{t}=\omega {{\displaystyle \sigma}}_{t}{{\displaystyle \mathrm{d}Z}}_{t},t\in [\mathrm{0},T]$$ which has been widely used in nancial practice, where ${{\displaystyle W}}_{t},{{\displaystyle Z}}_{t},t\in [\mathrm{0},T]$ are two uncorrelated standard Brownian motions. Using asymptotic analysis techniques, the moderate deviation principles for log S_{n} (or log $\left|{{\displaystyle S}}_{n}\right|$ in case S_{n} is negative) are obtained as $n\to \infty $ under different discretization schemes for the asset price process S_{t} and the volatility process ${{\displaystyle \sigma}}_{t}$: Numerical simulations are presented to compare the convergence speeds in different schemes.

A ring is said to be right (resp., left) regular-duo if every right (resp., left) regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of (one-sided) regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.

We discuss the uniqueness and the perturbation analysis for sparse non-negative tensor equations arriving from data sciences. By two different techniques, we may get better ranges of parameters to guarantee the uniqueness of the solution of the tensor equation. On the other hand, we present some perturbation bounds for the tensor equation. Numerical examples are given to show the effciency of the theoretical results.

For a recollement (A ;ℬ; C ) of abelian categories, we show that torsion pairs in A and C can induce torsion pairs in ℬ; and the converse holds true under certain conditions.

We define the {i}-inverse (i = 1; 2; 5) and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formulas of block tensors. We use the {1}-inverse of tensors to give the solutions of a multilinear system represented by tensors. The representations for the {1}-inverse and group inverse of some block tensors are established.

Employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coeffcients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coeffcients are polynomial growth with respect to the delay variables.

We introduce two new types of tensors called the strictly semi-monotone tensor and the range column Sufficient tensor and explore their structure properties. Based on the obtained results, we make a characterization to the solution of tensor complementarity problem.

The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible ${?}_{+}$-modules over the near-group fusion ring K(${?}_{\mathrm{3}}$, 3) are explicitly classified. It turns out that there are only four inequivalent irreducible ${?}_{+}$-modules of rank 2 and two inequivalent irreducible ${?}_{+}$-modules of rank 4 over K(${?}_{\mathrm{3}}$, 3).

We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type (a, b, 0) class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein's identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inated data sets. With the aid of ratio regression, we employ maximum likeli-hood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.

For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson structure on a polynomial algebra S, we first construct a Poisson algebra R, then prove that the Poisson enveloping algebra of S is isomorphic to the specialization of the quantized universal enveloping algebra of R, and therefore, is a deformation quantization of R.