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 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 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.

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 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.

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 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 $Q$.

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.

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 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.

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.

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 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.

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.

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.

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.

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.

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

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 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.

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 $X$ 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.

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.

This paper is concerned with the wavefront solutions of general nonlocal diffusion equations with discrete state-dependent delays. Firstly, the existence of wavefront solutions is proved by constructing the upper and lower solutions and invariant set in conjunction with the Schauder's fixed point theorem; then, the strict monotonicity of the wavefront solutions is given; finally, the asymptotics of the wavefront solutions at the minimum wave speed and the existence of the minimum wave speed are proved by using Ikehara's theorem.

Let $G=(V,E)$ be a graph. A $2$-distance $k$-coloring of a graph $G$ is to arrange $k$ colors for all vertices of $G$ such that every pair of vertices with the distance at most $2$ in $G$ is colored differently. A $2$-distance $L$-coloring of a graph $G$ is to give a list assignment $L=\{L(v)\mid v\in V(G)\}$ of $G$ such that $G$ has a $2$-distance coloring $\pi$ with $\pi(v)\in L(v)$ for each $v\in V(G)$. A list $2$-distance $k$-coloring of a graph $G$ is to give any list assignment $L$ of $G$ with $|L(v)|\ge k$ for each $v\in V(G)$ such that $G$ is $2$-distance $L$-colorable. In this paper, we will prove that every planar graph with the maximum degree $\Delta\ge 24$ containing neither $4$-cycles nor $5$-cycles is $2$-distance $(\Delta+3)$-choosable.

Let $X^{H}=\{X^{H}(t),t\in\mathbb{R}_{+}\}$ be a subfractional Brownian motion in $\mathbb{R}$ with index $H\in (0,1)$. We study the oscillation of $X^{H}$ and get the almost sure weak approximation of the occupation measure as an application.

In this paper, based on Popov-type subgradient extragradient method, We construct a new multi-step inertial regularized algorithm for solving the hierarchical variational inequality problem with the generalized Lipschitz mapping over the common solution set of a variational inequality problem and a null point problem in Hilbert spaces. We prove that this algorithm has a strong convergence theorem under certain conditions. Finally, we give some numerical experiments to illustrate the effectiveness and advantages of our new iterative algorithm. The results obtained here extend and improve many recent ones in the literature.

In this paper, we mainly study the representation theory of Hom-Lie algebras. More specifically, using the bosonic oscillators, we construct a two-parameter deformed Virasoro algebra, which is a Hom-Lie algebra. With suitable hypotheses in each case, we construct several kinds of Harish-Chandra modules of the two parameters deformed Virasoro algebra, and classify indecomposable Harish-Chandra module of an intermediate series.

In this paper, we define a family of bounded linear operators on the square integrable Bernoulli functional space $L^{2}(M)$, with the finite power set $\Gamma$ of $\mathbb{N}$ as the index set, which includes QBNs, preserving some properties of QBNs$\{\partial_{k},\partial_{k}^{*};k\geq 0\}$, which is called $\Gamma$-QBNs; the discussion shows that $\Gamma$-QBNs has some new properties, such as the quasi-exchangeability, the quasi-nilpotent property, the absorbed anti-commutation relation, the canonical binomial anti-commutation relation and the multi-indicator absorbed anti-commutation relation, which is a multi-index generalization of the canonical anti-commutation relation of QBNs; in particular, $\Gamma$-QBNs is ``quantum generators" of the full space $L^{2}( M)$. Also, its isochronous mixture product is an ``orthogonal" projection operator on $L^{2}(M)$, and $\Gamma$-QBNs can be represented by the Dirac operator generated by the orthogonal basis of $L^{2}(M)$.

In this paper, we consider the following higher-order Schrödinger equation with critical growth: \[ (-\Delta )^mu+V(y)u=Q(y)u^{m^*-1}, \quad u>0 \ \hbox{in} \ \mathbb{R}^{N}, \ u \in \mathcal{D}^{m,2}\,(\mathbb{R}^{N}), \] where $m^*=\frac{2N}{N-2m},\; N\geq 4m+1$, $m \geq 2$ is an integer, $(y',y'') \in \mathbb{R}^{2} \times \mathbb{R}^{N-2}$ and $V(y) = V(|y'|,y'')$ and $Q(y) = Q(|y'|,y'')$ are bounded non-negative functions in $\mathbb{R}^{+} \times \mathbb{R}^{N-2}$. By using finite dimensional reduction argument and local Pohozaev type identities, we show that if $N \geq 4m+1$, $Q(r,y'')$ has a critical point $(r_0,y_0'')$ satisfying $r_0 >0$, $Q(r_0,y_0'') > 0$, $ D^{\alpha}Q(r_0,y_0'')=0,$ $|\alpha| \leq 2m-1$ and $ {\rm deg} (\nabla (Q(r,y'')), (r_0,y_0'')) \neq 0$, and $\frac{1}{(2m-1)!m^*} \sum_{|\alpha|=2m}D^{\alpha}Q(r_0,y_0'') \int_{\mathbb{R}^N}y^{\alpha}U_{0,1}^{m^*} -m V(r_0,y_0'') \int_{\mathbb{R}^{N}} U_{0,1}^2 < 0$, then the above problem has infinitely many solutions, whose energy can be arbitrarily large. Different from that in [Commun. Contemp. Math., 2022, 24: Paper No. 2050071], in our case, the higher order derivative of $Q(r_0,y_0'')$ plays an important role in the construction of bubble solutions. Besides, it implies that the potential $V(r_0,y_0'') $ can influence the sign of higher order derivative of $Q(r_0,y_0'')$.

The classification of irreducible conformal $\mathcal{S}{{(p)}}$-modules of finite rank was given by H. Chen. In this paper, we use a different way to give a proof of this classification.

The characters of the symmetry group contain rich information on graph embeddings. In his early paper, Stahl [Region distributions of graph embeddings and string numbers, Discrete Mathematics, 1990, 82: 57-78] proposed a conjecture about the upper bound for a class of characters of the symmetry group in the study of the asymptotic estimation of the number of embeddings of some small diameter graphs. In this paper, we disprove the conjecture by giving some counterexamples.

In this paper, we introduce the notions of relative Rota—Baxter operators of weight 1 on Lie—Yamaguti algebras and post-Lie-Yamaguti algebras. A post-Lie-Yamaguti algebras describes an underlying algebraic structure of relative Rota—Baxter operators of weight 1. We clarify the relationship between Lie—Yamaguti algebras and post-Lie-Yamaguti algebras. Besides, we establish the cohomology theory of relative Rota—Baxter operators of weight 1. Consequently, we make use of this cohomology to characterize linear deformations of relative Rota—Baxter operators of weight 1 on Lie—Yamaguti algebras.

In this paper, we study conformal derivations and representations of a class of loop Lie conformal superalgebras $S(a,b)$, where $a, b$ are complex parameters. First, we determine conformal derivations of $S(a,b)$, and show that $S(a,b)$ admits outer conformal derivations if and only if $a=1$. Then, we completely classify the conformal modules of rank $(1+1)$ over $S(a,b)$. Finally, we classify the $\mathbb{Z}$-graded free intermediate series modules over $S(a,b)$ for $a\neq 1$.

With the help of Tauberian theorems for asymptotically almost periodic sequences, the asymptotic behavior of solutions to the difference equations: $$x(n+1)=Tx(n)+y(n),\quad n \in \mathbb{J}\in\{\mathbb{Z},\mathbb{Z}^+\}$$ has been studied in this paper. Here $x(n), y(n) \in X$ and $T$ is a bounded linear operator on a Banach space $X$. We show that if $c_0 \nsubseteq X$, $y$ is an asymptotically almost periodic sequence, and the intersection of the spectrum set $\sigma(T)$ of $T$ with the unit circle $\Gamma$ is finite, then the bounded solution $x$ of the equation is remotely almost periodic sequence (weaker than asymptotically almost periodic sequence). It is worth noting that although the conclusions of the established Tauberian theorem for asymptotically almost periodic sequence and the spectral set determination theorem for difference equations are slightly weaker than asymptotically almost periodic, they completely eliminate the assumption of transitivity in the reference [J. Differential Equations, 1995, 122, 282—301].

Assume that $1\leq p\leq\infty$ and $S_{(\sum \ell_p)_{c_0}}^+=\{x\in(\sum \ell_p)_{c_0}: x\geq 0;\|x\|=1\}$ is the positive unit sphere of $(\sum \ell_p)_{c_0}$. Let $f:S_{(\sum \ell_p)_{c_0}}^+\rightarrow S_{(\sum \ell_p)_{c_0}}^+$ be a norm-additive map (preserving norm of sums), i.e., \begin{equation}\nonumber \|f(x)+f(y)\|=\|x+y\|,\quad \forall x,y\in S_{(\sum \ell_p)_{c_0}}^+. \end{equation} We prove that if $f$ is bijective, then $f$ can be extended to a linear surjective isometry from $(\sum \ell_p)_{c_0}$ onto itself if and only if 1<p≤∞.

Outcome-dependent sampling is a well-known cost-effective sampling design in biomedical and epidemiologic studies. In this article, we propose a two-stage outcome-dependent sampling design in the framework of an accelerated failure time model. We develop the smoothed estimated Gehan estimating equation with the kernel function method to estimate the regression parameters for the data collocated by the proposed design. We establish the consistency and asymptotic normality of the proposed estimator. Simulation studies are conducted to evaluate the finite-sample performance of the proposed estimator and the results show the proposed estimator is more efficient than other competitive estimators. A real data set from the National Wilms' Tumor Study Group is analyzed to illustrate the proposed method.