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

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.

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.

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 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.

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 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.

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.

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.

Let $X$ and $Y$ be real finite-dimensional Banach spaces with the same dimension and $f:X\rightarrow Y$ be a mapping. In this note, we show that if $X$ is smooth, then $f$ satisfies $\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\},\; x, y\in X$, if and only if $f$ is phase equivalent to a linear surjective isometry.

In this paper, we prove a Talagrand's ${\bf T_2}$ transportation cost-information inequality for the law of the space-time fractional stochastic heat equation with fractional noise on the continuous path space with respect to the weighted $L^2$-norm.

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.

A topological space is called dense-separable if each dense subset of it is separable. Therefore, each dense-separable space is separable. This paper is devoted to establishing some basic properties of dense-separable topological groups. We prove that each separable space with a countable tightness is dense-separable, and give a dense-separable topological group which is not hereditarily separable. We also prove that, for a Hausdorff locally compact group, it is locally dense-separable iff it is metrizable. Moreover, we study dense-subgroup-separable topological groups. We prove that, for each locally compact abelian group, it is dense-subgroup-separable iff it is dense-separable iff it is metrizable. Finally, we discuss some applications in $d$-independent topological groups and related structures. We prove that each regular dense-subgroup-separable abelian semitopological group with $r_{0}(G)\geq\mathfrak{c}$ is $d$-independent. We also prove that, for each regular dense-subgroup-separable bounded paratopological abelian group $G$ with $|G|>1$, it is $d$-independent iff it is a nontrivial $M$-group iff each nontrivial primary component $G_{p}$ of $G$ is $d$-independent. Applying this result, we prove that a separable metrizable almost torsion-free paratopological abelian group $G$ with $|G|=\mathfrak{c}$ is $d$-independent. Further, we prove that each dense-subgroup-separable MAP abelian group with a nontrivial connected component is also $d$-independent.

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.

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.

With the development of big data technology, the dimensionality of spatial data is becoming higher and higher, and the endogeneity and heterogeneity of data often exist simultaneously. In this paper, we propose a quantile regression model of high-dimensional spatial dependent data with endogenous spatial weight matrix so as to analyze high-dimensional spatial dependent data robustly. We then develop a three-step penalized quantile estimation procedure through combining the instrumental variable method, variable selection method with robust statistic method, and establish the consistency and the asymptotic normality of the corresponding estimators. In addition, the oracle theoretical properties of variable selection are derived under some mild conditions. At last, we investigate the effectiveness and robustness of the proposed model and method through simulations and an application to housing prices in 284 prefecture-level cities across the country.

We extend Davie's trick (it Int. Math. Res. Not., 2007, 2007(1): 1-26) from stochastic differential equations with bounded measurable drifts to the ones in which the drifts are square integrable in time variable and Hölder continuous in space variable, and obtain the gradient estimates as well as the uniformly local quasi-Lipschitz estimates for strong solutions. As applications, we prove the unique strong solvability for stochastic transport equations driven by Wiener noise with square integrable drift as well as the uniformly local quasi-Lipschitz estimates for stochastic strong solutions, which partially solves the open problem posed by Fedrizzi and Flandoli (J. Funct. Anal., 2013, 264(6): 1329-1354).

In this paper，our purpose is to study the following Hénon type Choquard system $$ \left\{\begin{array}{l} -\Delta u=\int_{\mathbb{R}^3} \frac{|x|^\alpha|y|^\alpha v^p(y)}{|x-y|^{3-\mu}} d y \cdot v^{p-1} \text { in } \mathbb{R}^3, \\ -\Delta v=\int_{\mathbb{R}^3} \frac{|x|^\alpha|y|^\alpha u^q(y)}{|x-y|^{3-\mu}} d y \cdot u^{q-1} \text { in } \mathbb{R}^3, \end{array}\right. $$ where $0<\mu<3, \alpha>0$. We will show that there are no positive classical solutions in three dimension-space $\mathbb{R}^3$ for $p, q>2$ and $$ \frac{1}{p}+\frac{1}{q}>\frac{2}{3+2 \alpha+\mu}. $$

We establish the equivalence between invertible and preserved frames of weighted composition operators on $H_{\gamma}$. Moreover, we prove that $W_{\psi, \varphi}$ is invertible is equivalent its adjoint is invertible if $W_{\psi, \varphi}$ is bounded on $A_{\alpha}^{2}$. Additionally, we find the connection between dynamical sampling structures of weighted composition operators and frame preserving.

This article studies the trajectory statistical solution and its properties of the 3D tropical climate model. Firstly, the authors establish that the 3D tropical climate model with damping terms possesses a trajectory attractor, and use this trajectory attractor and generalized Banach limit to construct the trajectory statistical solution. Then they prove that the trajectory statistical solution has degenerate regularity provided that the associated generalized Grashof number is small enough. Finally they verify that the trajectory statistical solution converges to that of the 3D tropical climate model without damping term when the damping coefficients tends to zero.

In this paper, we consider a one-dimensional Nosé—Hoover system: $\dot{q}=p^{2m+1},$ $\dot{p}=-q^{2n+1}-\frac{\xi}{Q} p,$ $\dot{\xi}=p^{2m+2}-\beta^{-1},$ where $p, q, \xi\in \mathbb{R}$ are one-dimensional variables, $m,n\geq 0$ are integers and $Q, \beta$ are parameters. For $Q$ large enough, by using the averaging method we prove the existence of a linearly stable periodic solution. In addition, based on Moser's twist theorem we give a proof for the existence of invariant tori surrounding the periodic orbit for large $Q$.

Using the chord power integral and its inequalities of a convex body, we establish inequalities about moments for $\mu$-random chord length, $\nu$-random chord length, and $\lambda$-random chord length in $\mathbb{R}^n$. Based on the relationship between the chord power integral and containment function of a convex body, we obtain a new expression for moments of three kinds of random chord length mentioned above. By utilizing the properties of the distribution function and probability density function of $\mu$-random chord length, we get the calculation formulas for the distribution function and probability density function of $\nu$-random chord length, and the distribution function and probability density function of $\lambda$-random chord length, respectively. Further, we establish the relationships among three kinds of distribution functions. On this basis, taking a rhombus, regular pentagon, and regular hexagon as examples in $\mathbb{R}^2$, we give the expressions of their 1-order moment for three kinds of random chord length and the distribution function of $\nu$-random chord length.

This paper deals with a non-local curvature flow which preserves convexity and the modified elastic energy $\int^{L}_{0}\kappa^{2}ds-\epsilon L (\epsilon\ge0)$ of the evolving curve. We show that the flow exists globally, the length of the evolving curve is non-increasing, and the evolving curve converges to a finite circle in $C^{\infty}$ topology as time goes to infinity. As an application of this flow, we prove two new geometric inequalities.

This paper studied the limit properties of exceedances point processes for weakly dependent stationary random fields subject to random missing. By using the obtained results, this paper got the limit properties of extreme order statistics for the random fields and the limit properties of exceedances point processes for Gaussian order random fields and $\chi$ random fields.

The least square Lasso estimator for high-dimensional sparse linear model may arise several limitations in practical applications due to the dependence of the tuning parameter on the variance of model error. The square root Lasso estimator is proposed to make the tuning parameter free of the variance of the model error, which however exhibits some weakness from the perspective of robustness. Furthermore, the least absolute deviation Lasso estimator achieves some robustness, but it requires that the density of model error is bounded away from zero at some specific point. We propose a novel pairwise square root Lasso estimator for high-dimensional sparse linear model which only assumes that distribution of the model error is symmetric. The proposed estimator enjoys the advantage of tuning-free parameter and enables to address much heavier tailed model errors than the least absolute deviation Lasso estimator. We establish the error bound and consistency of variable selection for pairwise square root Lasso approach. Simulation studies demonstrate some favorable and compelling performances of the proposed method in some typical scenarios. A real example is analyzed to show the practical effectiveness of the proposed method.

In this paper we study strictly convex curves in the plane, that is curves with positive curvature. By applying the Wirtinger inequality we prove new integral inequalities of curvature. Furthermore, by applying the higher-order Wirtinger inequality, we prove a new inverse isoperimetric inequality.

Let $(\Sigma, g)$ be a compact Riemannian surface without boundary, and $\psi, h$ be two smooth functions on $\Sigma$ with $\int_{\Sigma} \psi d v_g \neq 0$ and $0 \leq h \not \equiv 0$. %$$ %\lambda_1^{\psi}(\Sigma)=\inf _{\int_{\Sigma} \psi u d v_g=0, %\int_{\Sigma} u^2 d v_g=1} \int_{\Sigma} |\nabla_g u |^2 d v_g . %$$ In this paper, we study the existence of generalized Kazdan-Warner equation $$ \left\{\begin{array}{l} \Delta_g u-\alpha u=8\pi\bigg(\displaystyle\frac{ h \mathrm{e}^u}{\int_{\Sigma} h \mathrm{e}^u d v_g}-\displaystyle\frac{\psi}{\int_{\Sigma} \psi d v_g}\bigg), \\ \displaystyle\int_{\Sigma} \psi u d v_g=0 \end{array}\right. $$ on $(\Sigma, g)$, where $\alpha < \lambda_1^{\psi}(\Sigma)$. In a previous work [Sci. China Math., 2018, 61(6): 1109-1128], Yang and Zhu obtained a sufficient condition under which the Kazdan-Warner equation has a solution when $h>0$ and $\psi = 1 $. We generalize this result to non-negative prescribed function $h$ and general function $\psi$. Our main contribution is the proof of that the blow-up points are not in the set of the zero of $h$.

In this paper, we introduce the concepts of a short arc and quasi-isometric mapping in quasi-hyperbolic metric spaces, and obtain some geometric characterizations of Gromov hyperbolicity for quasi-hyperbolic metric spaces in terms of the properties of short arc and quasi-isometry mapping.

For bounded linear operators $A$ and $B$ defined on the same space, if $AB=BA^{*}$, then $A$ and $B$ are said to be skew commutative. In this paper, we give some necessary and sufficient conditions for skew commutativity of two Toeplitz operators on the Hardy space of unit disk, and we also give some necessary and sufficient characterizations for two Hankel operators under some given conditions being skew commutative.

In this paper we study the following boundary value problem for fractional $p$-Laplace equations \begin{equation*}\left\{\begin{array}{ll} (-\Delta)_{p}^{s}u =f(x)u^{-\gamma }-g(x,u) , \ \ & x\in \Omega,\\ u >0, ~&x\in \Omega,\\ u =0,~&x\in \mathbb{R} ^{N}\setminus \Omega, \end{array}\right. \end{equation*} where $\Omega $ is a bounded smooth domain of $\mathbb{R} ^{N}$. Different from the general singular problem based on the variational method, this paper considers the strong singular case, that is $\gamma >1$. By defining two new manifolds, using Ekeland's variational principle, we obtain the existence of the solution of above problem. Due to the special structure of the equation, we also get the uniqueness of the solution.

This paper aims to establish the small time large deviation principle for the reflected stochastic heat equation driven by multiplicative noise. The main difficulty is dealing with space-time white noise and the singularity generated by reflection terms. In this paper, we adopt a new sufficient condition for weak convergence method similar to that proposed by A. Matoussi et al.

Over a field of characteristic $p>2,$ the low-dimensional cohomology groups of the special linear Lie superalgebra A(1,0) with coefficients in Hamiltonian Lie superalgebra $H(m,n)$ are computed by means of a direct sum decomposition of submodules and the weight space decomposition of $H(m,n)$ viewed as A(1,0)-module.

In this paper, we consider random upper bounds of star discrepancy for Hilbert space filling curve-based sampling and its applications. This problem stems from multivariate integration approximation. The main idea is the stratified random sampling method, and the strict condition for sampling number of classical jittered sampling is removed, the convergence order of the upper bound of probabilistic star discrepancy is $O(N^{-\frac{1}{2}-\frac{1}{2d}}\cdot \ln^{\frac{1}{2}}{N})$. Secondly, by obtaining the upper bound of probability, we derive the expected upper bound, which improves the existing results numerically. In the end, we apply the results to the uniform integral approximation of the function in the weighted function space and the generalized Koksma$-$Hlawka inequality.

In recent years, many people have paid attention to the enumeration problem of permutation polynomials over finite fields. In this paper, we construct a new enumeration formula for permutation polynomials over finite fields and provide a criterion for the existence of permutation polynomials. Our results solve a problem proposed by Qiang Wang.

Let $c_{k,j}(n)$ be the number of $(k,j)$-colored partitions of $n$. In 2021, Keith proved the following results: For $j=2,5,8,9$, we have $c_{9,j}(3n+2)\equiv 0\pmod {27}$ for all integers $n\ge 0$. For $j\in\{3,6\}$, we have $c_{9,j}(9n+2)\equiv 0\pmod {27}$ for all integers $n\ge 0$. Let $a,b$ be coprime positive integers. Recently, the authors gave the necessary and sufficient conditions for $c_{9,j}(an+b)\equiv 0\pmod {27}$ for all integers $n\ge 0$. In particular, for $j=1,4,7$, there does not exist coprime positive integers $a,b$ such that $c_{9,j}(an+b)\equiv 0\pmod {27}$ for all integers $n\ge 0$. In this paper, we study the congruences of $c_{4,j}(n)$. For $1\le j\le 3$, we determine all coprime positive integers $a,b$ such that $c_{4,j}(an+b)\equiv 0\pmod {8}$ for all integers $n\ge 0$.

In this article, we obtain the generalized Fekete and Szegö inequality for a subclass of quasi-convex mappings (including quasi-convex mappings of type $\mathbb{A}$ and type $\mathbb{B}$) defined on the unit ball of complex Banach spaces and the unit polydisc in $\mathbb{C}^n$. We also establish the successive homogeneous expansions difference bounds for the above mappings defined on the corresponding domains as applications of the main results. These obtained results not only reduce to the classical result in one complex variable but also generalize some known results in several complex variables.

In this paper, the concepts of $\delta$-BiHom-Jordan Lie supertriple systems and the definitions of generalized derivations, quasiderivations and central derivations are introduced, and some basic properties of generalized derivation algebra, quasiderivation algebra and central derivation algebra of $\delta$-BiHom-Jordan Lie supertriple systems are obtained. Particularly, it is proved that the quasiderivations of\ $\delta$-BiHom-Jordan Lie supertriple system can be embedded as a derivation in another $\delta$-BiHom Jordan Lie supertriple system, and when the central derivations of former are zero, the direct sum decomposition of later derivation can be obtained.

In 2007, Fan A. H. et al. introduced the Sylvester continued fraction expansions of real numbers and investigated the metric properties of the digits occurring in these expansions. In this paper, we will consider the analogous expansions over the field of formal Laurent series and discuss the related metric properties of the polynomial digits in these new continued fraction expansions.