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.

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.

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.

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.

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.

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.

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.

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 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.

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 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.

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, 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.

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.

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.

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 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.

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.

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.

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 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.

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).

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 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 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 $$ \frac{1}{p}+\frac{1}{q}>\frac{2}{3+2 \alpha+\mu}. $$

In this paper, we consider the 3-D steady compressible potential flow of Chaplygin gas. In spherical coordinates, the potential equation is of mixed type in the unit sphere. For the problem of supersonic flow over a delta wing, we establish a comparison principle for elliptic solutions of the equation with a class of mixed boundary conditions. We employ this comparison principle to derive an $L^\infty$ estimate, which is the key to show that the equation is elliptic in the parabolic-elliptic region.

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 $Q$.

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.

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.

In this paper we study the adaptive decomposition for matrix-valued functions in the Hardy space of the unit polydisc by two ways. One uses product-TM systems, and the other uses the Gram-Schmidt orthogonalization of product-Szegö dictionaries. In each step of the decomposition the parameters and the orthogonal projections are adaptively chosen to best match the given matrix-valued functions, and the decomposition we get is of Fourier type. The convergence and the convergence rate are proved under some conditions.

In this paper, we use Lucas sequences and the exponential sums to study the discrepancies of sequence from Koblitz curves and obtain its sharp bound, which can be applied to analyze the uniform distribution of the sequence from Koblitz curves.

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.

A Finsler space (metric) is said to be a Finsler geodesic orbit space (metric) if every geodesic is an orbit of a one-parameter group of isometrics. In this paper, we find many new examples of left invariant Finsler geodesic orbit metrics on compact semi-simple Lie groups.

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.