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.

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.

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.

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.

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

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 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.

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).

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.

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 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.

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 $n\ge 0$.

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.

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.

An outcome dependent sampling (ODS) design is a biased-sampling scheme, which can save the cost and improve the efficiency in studies on large-scale data. We study how to fit the generalized linear models to high-dimensional data collected via ODS design. Inspirited the idea of gradient descent algorithm, we develop two improved adaptive moment estimation algorithms for the computation of the estimator in generalized linear regression with high-dimensional ODS data, and establish the theoretical properties. The proposed algorithms obviate the computation of some high-dimensional matrices and their inverses. We conduct simulation studies and analyze a real data example to illustrate the performance of the proposed algorithms.

Let $\mathbb{N}$ stand for the set of positive integers. Let $\mathbb F_q$ denote the finite field of odd characteristic and ${\mathbb F}^*_q$ its multiplicative group. In this paper, by using the Smith normal form of exponent matrices, we present an explicit formula for the number of rational points on the triangular algebraic variety over $\mathbb F_q$ defined by $\sum_{j=0}^{t_k-1}\sum_{i=1}^{r_{k,j+1}-r_{kj}} a^{(k)}_{r_{kj}+i}x_1^{e_{r_{kj}+i,1}^{(k)}}\cdots x_{n_{k,j+1}}^{e_{r_{kj}+i,n_{k,j+1}}^{(k)}}=b_k, 1\le k\le m$, where $b_k\in \mathbb F_q$, $t_k\in \mathbb N$, $0=r_{k,0}<r_{k,1}<\cdots<r_{k,t_k}$, $a^{(k)}_i\in \mathbb F_q^*$, and $e_{ij}^{(k)}\in \mathbb N$ for $1\le i\le r_{k,t_k}$ and $1\le j\le t_k$, $0<n_{11}<\cdots <n_{1,t_1}<n_{21}<\cdots<n_{2,t_2}<\cdots<n_{m1}<\cdots<n_{m,t_m}$. This generalizes the results obtained previously by J. Wolfmann, Q. Sun, and others. Our result also gives a partial answer to an open problem raised by S.N. Hu, S.F. Hong and W. Zhao in 2015.

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.

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, 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.

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 matched-pair design, relative risk is often used to analyze whether a certain factor has an effect on the occurrence of a certain disease. It is of great importance in epidemiologic studies. In this paper five methods used to construct asymptotic confidence interval of relative risk under multinomial sampling, Delta method, log transformation method, calibrated log transformation method, an improved method based on Fieller's theorem and saddle-point approximation method respectively. We use Monte Carlo simulation to evaluate the five interval estimation methods based on the coverage of interval to relative risk and the average interval length. It is concluded that in the case of a small probability with a small sample size, the saddle point approximation method is the best. Finally, two empirical cases are used to show the different characteristics of five interval estimation methods.

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 this paper, combining non-commutative residues and Lichnerowicz formula, we give the local representation and trace structure in the normal coordinate system of a class of Dirac operators with torsion, and obtain the Einstein-Hilbert action of Dirac operators with torsion.

This paper considers a class of mean-field backward stochastic differential equations (mean-field BSDEs) whose coefficients depend on $(Y,Z)$ and the law of $Y$. Under non-Lipschitz conditions, we prove the existence and uniqueness of strong solutions for such equations. The technique employed is the existence of weak solutions and the pathwise uniqueness of weak solutions. By introducing a new class of backward martingale problems related to this type of mean-field BSDEs and by extending the second-order differential operator to handle the mean-field case appropriately, using the Euler-Maruyama approximation technique, we obtain the existence of weak solutions for these mean-field BSDEs. The proof of pathwise uniqueness of weak solutions is mainly based on the extended Gronwall lemma.

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.

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.

Let $M$ be a $F$-Willmore hypersurface in $S^{n+1}$ with the same mean curvature or the squared length of the second fundamental form of Willmore torus $W_{m,n-m}$ (or Clifford torus $C_{m,n-m}$). In this article the authors proved that if ${\rm Spec}^p(M)={\rm Spec}^p(W_{m,n-m})$ (or ${\rm Spec}^p(M)={\rm Spec}^p(C_{m,n-m})$) for $p=0,1,2$, then $M$ is $W_{m,n-m}$ (or $C_{m,m}$). The $F$-Willmore hypersurface is a critical point of $F$-Willmore functional, where $F$-Willmore functional is a generalization of the well-known classic Willmore functional.

In this paper, we will prove the existence of the solution to Orlicz-Minkowski problem for the discrete measure $\mu$. By the solution of the discrete Minkowski problem and the method of convex body approximation, we obtain the existence of the solution of the Orlicz-Minkowski problem for the general measure $\mu$ under the condition of removing even.

In this paper, we discuss the iterative roots of the flat-top anti-bimodal (briefly: decrease-flat-increase-flat-decrease type) continuous self-maps on the unit interval, and classify the flat-top anti-bimodal continuous self-maps, and obtain the necessary and sufficient conditions for every class of the flat-top anti-bimodal continuous self-maps to have iterative roots of order $n$.

In this paper, we prove that every pair of sufficiently large even integers satisfying some necessary conditions can be represented as a pair of equations involving two squares of primes, four cubes of primes and $k$ powers of $2$ with $k=27$, which largely improves the recent result $k=150$.

Let ${\bf D}$ be a local dendrite with unique branch point and $f:{\bf D}\rightarrow {\bf D}$ be continuous. Denote by $R(f)$ and $\Omega (f)$ the set of recurrent points and the set of non-wandering points of $f$, respectively. Let $\Omega_0 (f)={\bf D}$ and $\Omega_k (f)=\Omega (f|_{\Omega_{k-1} (f)})$ for any positive integer $k$. The minimal $k$ such that $\Omega_{k} (f)=\Omega_{k+1} (f)$ is called the depth of $f$, where $k$ is a positive integer or $\infty$. In this note, we show that $\Omega_2(f)=\overline{R(f)}$ and the depth of $f$ is at most 2.

Consider a class of biharmonic equation with nonsymmetric perturbation functions as follows: \begin{align*} \Delta^2u-\Delta u+u=K(x)|u|^{p-2}u+K(x)|u|^{q-2}u, \ \ x\in\mathbb{R}^N, \end{align*} where $N\geq 5$ and $2＜p＜q＜4^*=\frac{2N}{N-4}$. Firstly, we prove the existence of ground state solution to above equation by establishing a generalized Lieb's compactness theorem. Subsequently, we show the the existence of ground state solution and sign-changing solution of the above equation by means of the sign-changing Nehari manifold, minimax method and Miranda's theorem.

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.

We introduce the notions of invariance entropy points and uniform invariance entropy points for control systems and give some basic properties for these entropy points. For a controlled invariant set with some conditions, it is shown that there exists a countable closed subset of this set such that the invariance entropy of this subset is equal to the invariance entropy of the set.

For $β>1,$ let $T_β$ be the $\beta$-transformation defined on $[0,1].$ We investigate the metric properties of the two-dimensional exact asymptotic approximation sets and exact uniform approximation sets in beta-dynamical systems. As a corollary, for any $0 \leq \hat{v} \leq \infty$, we obtain the Hausdorff dimension of the uniform Diophantine set $$\bigg\{(x,y)\in[0,1]^2:\forall N\gg1, \exists 1\leq n \leq N \text{such that}\! \begin{array}{c} T_{β}^nx <β^{-N \hat{v}} \\ T_{β}^ny< β^{-N \hat{v}} \end{array} \!\! \bigg\} . $$We also determine the Hausdorff dimension of exact multiplicative approximation set $$\{(x,y)\in [0,1]^2: v_{L, β}(x,y)=v \},$$where $v_{L, β}(x,y)$ denotes the supremum of the real numbers $v$ for which the equation $T_β^nx \cdot T_β^ny< \frac{1}{β^{nv}}$ has infinitely many solutions in positive integers $n$.