The main purpose of this paper is using the elementary methods, the number of the solutions of some congruence equations and the properties of the classical Gauss sums to study the calculating problem of the fifth power mean of one kind two-term exponential sums, and give the exact calculating formula for it.

We study some properties of Toeplitz operators with positive operator-valued function symbols on the vector-valued exponential weighted Bergman spaces $A^p_{\varphi}(\mathcal{H})\ (1 ＜ p ＜ \infty)$. Firstly, we discuss when the Bergman projection from $L^p_{\varphi}(\mathcal{H})$ onto $A^p_{\varphi}(\mathcal{H})$ is bounded and get the dual of the vector-valued exponential weighted Bergman spaces. Secondly, we obtain several equivalent descriptions of Carleson condition to characterize the boundedness and compactness of Toeplitz operators on $A^p_{\varphi}(\mathcal{H})$. Finally, we consider the Schatten-$p$ class membership of Toeplitz operators acting on $A^2_{\varphi}(\mathcal{H})$.

Let $\mathbb{F}_{q}$ be the finite field of $q$ elements, and $\mathbb{F}_{q^{n}}$ be its extension of degree $n$. An element $\alpha\in \mathbb{F}_{q^{n}}$ is called a normal element of $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ if $\{\alpha,\alpha^{q},\ldots, \alpha^{q^{n-1}}\}$ constitutes a basis of $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$. Normal elements over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. The minimal polynomial of a normal element is certainly an irreducible polynomial with nonzero trace, while the converse does not hold in general. Using linearized polynomials, we give some necessary and sufficient conditions for this problem, which extend the known results.

In this paper, we study the computational problems of one kind congruent equation modulo $p$, and give some exact computational formulae for them.

In this paper, a nonnegative projection correlation coefficient (NPCC) is proposed to measure the dependence between two random vectors, where the projection direction comes from the standard multivariate normal distribution. The NPCC is nonnegative and is zero if and only if the two random vectors are independent. Also, its estimation is free of tuning parameters and does not require any moment conditions on the random vectors. Based on the NPCC, we further propose a novel feature screening procedure for ultrahigh dimensional data, which is robust, model-free and enjoys both sure screening and rank consistency properties under weak assumptions. Monte Carlo simulation studies indicate that the NPCC-based screening procedure have strong competitive advantages over the existing methods. Lastly, we also use a real data example to illustrate the application of the proposed procedure.

In this paper, we investigate the reciprocal sums of the cubes of odd and even terms in the Fibonacci sequence and we obtain two interesting identities for the Fibonacci numbers.

In this paper, we investigate a class of domains $\Omega^{n+1}_k =\{(z,w)\in \mathbb{C}^n\times \mathbb{C}: |z|^k < |w| < 1\}$ for $k \in \mathbb{Z}^+$ which generalizes the Hartogs triangle. We first obtain the new explicit formulas for the Bergman kernel function on these domains and further give a range of $p$ values for which the $L^p$ boundedness of the Bergman projection holds. This range of $p$ is shown to be sharp.

Based on the complete classification of the torsion subgroup by Mazur, and results of the related diophantine equation, we determine all elliptic curves defined over $\mathbb{Q}$ with a rational point of the order $n\ (n \geq 6, n\neq 11)$ and the conductor $p^{a}q^{b}r^{c}$, where $p, q, r$ are distinct primes, and $a, b, c$ are positive integers. In particular, an upper bound of the minimal discriminant for these elliptic curves are given.

A hole is an induced cycle of length at least 4, a hole of odd length (resp. even length) is called an odd hole (resp. even hole). An HVN is a graph composed by a vertex adjacent to both ends of an edge in $K_4$. Let $H$ be the complement of a cycle on 7 vertices. Chudnovsky et al. in [J. Combin. Theory B, 2010, 100: 313—331] proved that every (odd hole, $K_4$)-free graph is 4-colorable and is 3-colorable if it does not contain $H$ as an induced subgraph. In this paper, we use the idea and proving technique of Chudnovsky et al. to generalize this conclusion to $($odd hole, HVN$)$-free graphs. Let $G$ be an $($odd hole, HVN$)$-free graph. We prove that if $G$ contains $H$ as an induced subgraph, then it either has a special cutset or is in two classes of pre-defined graphs. As its corollary, we show that $\chi(G)\le \omega(G)+1$, and the equality holds if and only if $\omega(G)=3$ and $G$ has $H$ as an induced subgraph.

A $1$-plane graph $D$ of a graph $G$ is a drawing of $G$ in the plane such that each edge is crossed at most once. The crossing number of $G$ is the minimum number of edge crossings in any drawing of $G$ in the plane. Determining the crossing number of a graph is NP-hard, and determining whether a graph is $1$-planar is NP-complete. In this paper, we establish the lower bound on the number of non-crossed edges in $2$-connected locally maximal $1$-plane graphs and locally crossing-optimal maximal $1$-plane graphs, respectively. Consequently, we also determine the upper bound of their crossing numbers in relation to the number of edges.

This paper is concerned with the existence and time-asymptotic nonlinear stability of traveling wave solutions to the Cauchy problem of the one-dimensional compressible planar magnetohydrodynamics system, which governs the motions of a conducting fluid in an electro-magnetic field. Motivated by the relationship between planar magnetohydrodynamics system and Navier—Stokes system, we can prove that the solutions to the compressible planar magnetohydrodynamics system tend time-asymptotically to the traveling wave, provided that the initial disturbance is small and of integral zero.

In this paper, we present a method to study isochronous centers in 3-dimensional polynomial differential systems. Firstly, the isochronous constants of the three dimensional system are defined and recursive formulas to obtain them are given. The conditions for the isochronicity of a center are determined by the computation of isochronous constants for which there is no need to compute center manifolds of the three dimensional systems. Then the isochronous center conditions of two specific systems are discussed as an application of our method. Our method is a generalization of the formal series method proposed by Yirong Liu for determining the order of a fine focus of planar differential systems. This method with the recursive formulas can be easily implemented on a computer using a computer algebra system.

The Lorenz-type maps are piecewise expanding maps with discontinuous points, the discontinuity comes from the singularities of Lorenz equations showing butterfly effect, and the observable statistical properties of such maps are given by the absolutely continuous invariant measures. In this paper, we consider the perturbation $f_t=f+tX\circ f$ of an improved Lorenz-type map $f $, and denote by $\mu_t$ the perturbation of the corresponding absolutely continuous measure $\mu$. We prove that if $X $ takes zero on all image sets of the discontinuous point of $f $, then its sensitivity formula $$\Psi(\lambda)=\sum\limits_{n=0}^\infty \lambda^n \int \mu(dx)X(x)\dfrac{\partial(\varphi(f^nx))}{\partial x},\quad\varphi\in C^1, $$ converges at $ \lambda = 1 $, thus the linear response formula $\frac{d}{dt}|_{t=0}\mu_t(\varphi)=\Psi(1)$ is established.

Research on the Julia sets of meromorphic functions has been one of the hot problems in complex dynamical systems. In the paper, we gave some more accurate estimations of the lower bound of the radial distribution of Julia sets by investigating the growth of solutions of second-order differential equations.

In this paper, we consider the persistence of invariant tori in the following system \begin{equation*} \begin{array}{ll} \left\{\begin{array}{ll} \dot{x}=\omega+y+f(x,y),\\[0.1cm] \dot{y}=g(x,y), \end{array}\right. \end{array} \end{equation*} where $x\in \mathbb{T}^\Lambda$, $y\in\mathbb{R}^\Lambda$, $\Lambda$ is a countable subset of $\mathbb{Z}$, ${\omega}=(\ldots,{\omega}_\lambda,\ldots)_{\lambda\in \Lambda}\in\mathbb{R}^\Lambda$ is the frequency vector, ${\omega}=(\ldots,{\omega}_\lambda,\ldots)_{\lambda\in \Lambda}$ is a bilateral infinite sequence of rationally independent frequency, in other words, any finite segments of ${\omega}=(\ldots,{\omega}_\lambda,\ldots)_{\lambda\in \Lambda}$ are rationally independent, and the perturbations $f, g$ are real analytic functions. We also assume that the above system is reversible with respect to the involution $\mathcal{M }: (x,y) \mapsto (-x,y)$. By the KAM method, we prove the persistence of invariant tori for the above reversible system.

We study a new algorithm to solve a common solution of the split feasibility problem and the fixed point problem involving quasi-nonexpansive mappings in Hilbert spaces. Based on the common solutions of these two classes of problems, we solve the variational inequality problem. Compared with the predecessors, the self-adaptive technique and the inertial iteration method are added, which can speed up the convergence rate of the iterative sequence generated by our algorithms. At the same time, we extend the involving previous nonexpansive mappings to extensive quasi-nonexpansive mappings. In addition, a strong positive bounded operator is added to the algorithm, which extends the original viscous iterative algorithm to a more general viscous iterative algorithm. The effectiveness of the algorithm is verified by numerical examples.

In this paper, we study a generalized comparison question about singular metrics of twisted pluricanonical bundles over complex varieties with canonical singularities. We prove that for an algebraic fibration over the unit disc, the restriction of the twisted relative $m$-Bergman metric on a singular fiber with canonical singularities at worst coincides with the intrinsic twisted $m$-Bergman metric on itself, provided that the singular metric on the twisted line bundle has slope zero at each point belonging to the singular fiber.

The $q$-deformed $W(2,2)$ algebra is a Hom-Lie algebra, denoted by $\ W ^ q$. In this paper, we compute its second cohomology with values in the adjoint module by elementary and direct calculations, and obtain that the second cohomology group $H^2(\ W ^q,\ W ^ q)$ is two-dimensional.

In this paper, we first prove the global existence and exponential decay of small-data analytical solutions to the three-dimensional incompressible Oldroyd-B model in torus. An a priori estimate of viscosity independence will be obtained. Based on such a priori estimate, we then show validity of the inviscid limit of the Oldroyd-B system. The nonlinear quadratic terms have one more order derivative than the linear part and no good structure is found to overcome this derivative loss problem. So we can only build the global-in-time result in the analytical energy functional space rather than the Sobolev space with finite order derivatives.

It has been well established that the predator-induced fear has indirect impact on prey but can have comparable effects on prey population as direct killing. In this paper, a diffusive predator-prey system with nonlocal fear effect is formulated and investigated. We firstly study the existence and boundedness of solutions and then discuss the stability of constant steady states. Steady-state bifurcations are carried out in detail by using the Lyapunov—Schmidt method. Finally, numerical simulations are showed to verify our theoretical results.

Production function is one of the core concepts of neoclassical economics and an important tool for economic analysis. This paper studies quasi-sum production functions from the perspective of geometric invariants. By discussing the constant Gauss curvature equation and the constant mean curvature equation of the corresponding surfaces of quasi-sum production functions, a series of interesting classification results are obtained. The results of this paper not only have certain significance for the study of surface theory in differential geometry, but also provide more alternative types of production models in economic analysis, and promote the development of the theory of production function to a certain extent.

In this paper, we investigate the $\ell$-convex Legendre curves on the plane, which are the natural generalization of strictly convex curves. On the one hand, by using the method of Green and Osher, we obtain the necessary and sufficient condition for the Green-Osher inequality's equality of $\ell$-convex Legendre curves, that is, when $F(x)$ is a strictly convex function in $\mathbb{R}$, the equality holds if and only if the curve $\gamma$ is a circle. On the other hand, we obtain a series of curvature integral inequalities of $\ell$-convex Legendre curves. In particular, when $\gamma$ is a strictly convex curve, the corollaries obtained are the improved forms of Ros inequality, Green-Osher inequality and Gage isoperimetric inequality.

In this paper, we propose a new estimation approach for binary panel data model with error cross-sectional dependence. The estimation approach does not need to estimate the interactive effects in model. The asymptotic property of this proposed estimator is established as long as $N$ is fixed and $T$ goes to infinity. Finally, we present some Monte Carlo studies on the small sample properties of the proposed estimator for binary panel data model with error cross-sectional dependence, showing that our proposed estimator performs well.

We consider the following prescribed curvature problem of fractional operator: \begin{align} (-\Delta)^s u=K(y)u^{2_s^*-1},\quad u> 0,\quad u\in D^s(\mathbb{R}^N), \nonumber \end{align} where $N\geq 3$, $0 ＜ s ＜ 1$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $K(y)$ is a positive function. When $K(y)$ has a sequence of strictly local maximum points moving to infinity, we use the finite dimensional reduction method to prove the existence of any finitely many multi-bubbling solutions to the above problem. These solutions concentrate at $k$ different local maximum points of $K(y)$.

The goal of this paper is to develop new fixed points, best approximation, and Leray—Schauder alternative for single-valued and (quasi) upper semicontinuous (QUSC) set-valued mapping in $p$-vector spaces and locally $p$-convex spaces, where $p \in (0, 1]$. The fixed point theorem established in this paper is a positive answer to Schauder conjecture in $p$-vector spaces and locally $p$-convex spaces; the corresponding best approximation theorem and the principle of Leray—Schauder alternative are also the fundamental tools in nonlinear functional analysis under the framework of $p$-vector spaces and locally $p$-convex spaces. These new results unify and generalize the theoretical results existing in the current mathematical literature, and they are also the continuation and in-depth development of the recent work did by Yuan [$Fixed$ $Point$ $Theory$ $Algorithms$ $Sci$. $Eng$., 2022, 2022: Paper Nos. 20, 26], and related references.

In this paper, we introduce a new Bregman extragradient projection method for solving monotone variational inequalities in real Hilbert spaces. Moreover, we prove a weak convergence theorem for our suggested algorithm under some reasonable assumptions imposed on the parameters. Finally, we give some numerical examples to demonstrate how our algorithm outperforms earlier findings in the literature in terms of convergence performance.

Finding matrix representations of operators is an important part of operator theory. Calculating such a discretization scheme is equally important for the numerical solution of operator equations. Traditionally in both fields, this was done using bases, Hilbert-Schmidt frames have been used here. Firstly, we introduce the concept of generalized cross gram matrix with respect to HS-frame, discuss some basic properties. Then, we give necessary and sufficient conditions for their invertibility and present explicit formulas for the inverse. In particular, the example shows that invertibility of generalized cross Gram matrix is not possible when the associated sequences are HS-frames rather than HS-Riesz bases. Finally, we obtain some stability results. More precisely, it is shown that the invertibility of generalized cross Gram matrices is preserved under small perturbations.

In this paper, a parameterized fast iterative shrinkage-thresholding algorithm with adaptive step size is proposed for nonsmooth optimization problems. The convergence rates of the objective function $O(1/k^{2})$ and the iterative algorithm $o(1/k^{2})$ are studied separately in the real Hilbert space using the degrees of freedom brought by the parameterization strategy, and the strong convergence of the sequence generated by the algorithm is obtained under the condition that the objective function $F$ is uniformly convex. In addition, the connection between the algorithm and inertial dynamical system is established and the related inference of the dynamical system solution trajectory is obtained. Meanwhile, the specific applications and comparisons of the algorithms listed in this paper to the image denoising problem demonstrate their superiority.

For any quasitoric-manifold, $\pi :M^{2n}\to P^{n}$, its cohomology ring is expressed as $H^{\ast}(M^{2n},\mathbb{ Z}) =\mathbb{Z}[F_{1},F_{2},\ldots,F_{m}]/(\mathcal{I}_{P^{n}}+\mathcal{J}_ {P^{n}})$, where $\mathcal{F}(P)=\{F_{1}, F_{2}, \ldots,$ $F_{m}\}$ is the set of all co-one-dimensional surfaces in $P^{n}$. Taking any vertex $\upsilon= F_{i1}\cap F_{i2}\cap\cdots\cap F_{in}$ of $ P^{n}$, we prove that $\langle [F_{i1}F_{i2}\cdots F_{in}],[M^{2n}]\rangle=\pm1$, that is, $[F_{i1}F_{i2}\cdots F_{in}]$ is the generator of $H^{2n}(M^{2n},\mathbb{Z})$. Further we use this conclusion to discuss the rigidity of quasitoric-manifolds, and prove the following conclusions: If $f^{*}:H^{\ast}(M_{1}^{2n},\mathbb{Z})\to H^{\ast}(M_{2}^{2n},\mathbb{Z})$ is a ring isomorphism, then there exists a one-to-one mapping $\tilde{f}:{\rm Fix}(M_{1}^{2n})\to {\rm Fix}(M_{2}^{2n})$, where ${\rm Fix}(M^{2n})$ is the fixed point of $T^{n}$-acting on $M^{2n}$.

Let $\mathcal A$ be a commutative unital ${\rm C}^*$-algebra with the unit element $e$ and $\mathcal M$ be a full Hilbert $\mathcal A$-module. Denote by End$_{\mathcal A}(\mathcal M)$ the algebra of all bounded $\mathcal A$-linear mappings on $\mathcal M$ and by $\mathcal M'$ the set of all bounded $\mathcal A$-linear mappings from $\mathcal M$ into $\mathcal A$. In this paper, we prove that if there exist $x_0$ in $\mathcal M$ and $f_0$ in $\mathcal M'$ such that $f_0(x_0)=e$, then every $\mathcal A$-linear Lie derivation $\delta$ on End$_{\mathcal A}(\mathcal M)$ is standard. That is, $\delta$ can be decomposed into $d+\tau$, where $d$ is a $\mathcal A$-linear derivation, and $\tau$ is a $\mathcal A$-linear mapping of central value such that $\tau(AB)=\tau(BA)$ for any $A,B\in {\rm End}_{\mathcal A}(\mathcal M)$.

The present paper is dedicated to the study of the Cauchy problem for 3D incompressible inhomogeneous asymmetric fluids with only rough density. By exploiting some extra time-weighted energy estimates, and employing the interpolation argument and Lorentz norms for the time variable, we first construct the Lipschitz regularity of the velocity. Based on it, following the duality approach, we finally settle the uniqueness issue of the global weak solution constructed by [Qian,Chen and Zhang,Math.Ann.,2023,386:1555-1593].

In this paper, we give some generalized numerical radius inequalities for Hilbert space bounded linear operators. We also give an improved numerical radius inequality for the sum of two bounded linear operators.

Under investigation in this work is the robust inverse scattering transform of the discrete Hirota equation with nonzero boundary conditions, which is applied to solve simultaneously arbitrary-order poles on the branch points and spectral singularities. Using the inverse scattering transform method, we construct the Darboux transformation but not with the limit progress, which is more convenient than before. Several kinds of rational solutions are derived in detail. These solutions contain $W$-shape solitons, breathers, high-order rogue waves, and various interactions between solitons and breathers. Moreover, we analyze some remarkable characteristics of rational solutions through graphics. Our results are useful to explain the related nonlinear wave phenomena.

In this paper, we use Dudek's [Bull.Aust.Math.Soc., 2019, 99(1): 1-9] method to study the mean of divisor function defined on polynomial rings over finite fields over a quadratic polynomial sequence, and comparing the mean of divisor function defined on integer ring over quadratic arithmetic sequence, we find that the two are the same from the perspective of order.

In this paper, we propose a new quantile feature screening method based on the modified Cholesky decomposition for ultra-high dimensional longitudinal data. Specially, we introduce the optimal quantile estimating equations to cope with potential outliers and heavy-tailed errors. Then, we model the covariance matrix involved in the optimal quantile estimating equations based on the modified Cholesky decomposition, and subsequently propose an iterative feature screening algorithm. Under some regularity conditions, we establish asymptotic properties of the proposed screening method such as consistency of the screening and ranking. Simulation studies and an analysis of the yeast cell-cycle gene expression dataset show that the proposed method not only selects important covariates quickly but also possesses higher screening accuracy.

It is important to study the discounted Hamilton—Jacobi (H-J) equation, because it is a special form of the contact H-J equation. In this article, we provide a definition of the Aubry set in a discounted Hamilton system under certain conditions in the sense of viscosity solution, which is similar to the definition of Aubry set in classical Hamilton systems, and the Aubry set defined by this definition has the properties of minimal action and recurrence in a variational sense.

In this paper, we construct an efficient estimation method for partially linear varying coefficient spatial autoregressive panel model with fixed effects by combining bias correction, variable transformation and quadratic inference functions. Moreover, under some regularity conditions, asymptotic normality of parameter estimators is proved and convergence rate of the estimators of coefficient functions is derived. Lastly, the performance of the proposed method under the finite samples is evaluated by Monte Carlo simulation and real data analysis.

Let $n \geq 2$ be an integer and let $\mathcal{D}_n$ denote the set of vectors $\mathbf{d}=\left(\delta_0, \delta_1, \ldots\right.$, $\left.\delta_{n-1}\right)$, where $\delta_i \in\{*, 0,1\}, i=0,1, \ldots, n-1$. For $\mathbf{d} \in \mathcal{D}_n$ we define the set $$ \mathcal{N}_n(\mathbf{d})=\left\{\sum_{i=0}^{n-1} d_i 2^i: d_i \in\{0,1\} \text { if } \delta_i=*, d_i=\delta_i \text { otherwise }\right\} . $$ Dietmann, Elsholtz and Shparlinski studied the distribution of square-free numbers in $\mathcal{N}_n(\mathbf{d})$. In this paper we will further study the distribution properties of square complement function, square residue function, power function and Smarandache multiplicative function over $\mathcal{N}_n(\mathbf{d})$, and give asymptotic formulas.

In terms of the generators and relations of affine $q$-Schur algebra $\mathcal{S}_{\Delta}(n, r)$, Doty et al. provided a presentation for $n>r$. Deng—Du—Fu gave the presentations for affine $q$-Schur algebra $\mathcal{S}_{\Delta}(r, r)$. The presentation of the affine $q$-Schur algebra $\mathcal{S}_{\Delta}(n, r)$ is more complicated in the case of $n < r$. In this paper, we obtain the monomial basis of affine $q$-Schur algebra $\mathcal{S}_{\Delta}(2, 3)$ and present a new set for generators and relations of $q$-Schur algebra $\mathcal{S}_{\Delta}(2, 3)$ by monomial basis.

Sequences of genus polynomials for what became known as linear (or $H$-linear) families of graphs have been studied for more than 30 years. Most of previous papers concerning them aim to find recursions and expressions for genus (and Euler genus) polynomials of specific families, or try to prove the property of log-concavity. Recently, under some conditions, some researches reveal that the embedding distributions of generalized $H$-linear graph families $\{G_n^\circ \}$ will tend to normal distributions when $n$ tends to infinity (see [19]). Based on this previous work, in this article, we prove that the order of the convergence rate is $\frac{1}{\sqrt{n}}$. We also explain that, for the convergence rate obtained in this paper, it can been considered as optimal. In the end, we use some concrete examples to demonstrate our result.