The paper characterizes the boundedness and compactness of Toeplitz operators by means of the method of dual spaces. When $1 \leq p<\infty$, the paper completely describes the Schatten-p class of Toeplitz operators on the Békollé weighted Bergman space over the upper half-plane. In addition, by presenting the atomic decomposition of this Bergman space, the paper investigates the Schatten-p class of Toeplitz operators when $0<p<1$.

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 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.

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.

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 $k=150$.

This paper proposes a novel class of inertial-type algorithms for obtaining numerical solutions to monotone inclusion problems associated with maximally monotone operators, through difference discretization of a second-order dynamical system with variable damping. Compared with traditional inertia algorithms, the convergence of the iterative algorithm proposed in this paper only requires the inertia coefficient to be in (0,1), which greatly weakens the traditional convergence conditions; in addition, the algorithm also weakens the requirements for damping parameters in the dynamics system. Finally, a numerical example is given to verify the effectiveness of the algorithm.

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 $n\ge 0$.

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.

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.

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.

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, 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.

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 $n$.

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.

Let $H$ be a positive integer and $p$ be an odd prime. For $1\le x,y,z\le H$, we study the distribution of consecutive $r$-free integers of the type $x^2+y^2+z^2+k$, $x^2+y^2+z^2+k+1$ using the properties of certain Salié sums. Additionally, we provide an asymptotic formula for this distribution.

In this paper, we show the boundedness for the Dunkl-Calderón-Zygmund operators and their maximal operators on the Dunkl-Lebesgue spaces with variable exponents $L^{p(\cdot)}(\mathbb R^n, dw)$. The key tools are the Dunkl sharp function, the Cotlar's inequality in the Dunkl setting and the $L^{p(\cdot)}(\mathbb R^n, dw)$-boundedness of Dunkl-Hardy-Littlewood maximal function. The paper is perhaps the first attempt at a study of the Dunkl harmonic analysis in the variable exponents setting.

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 $n$.

For a vertex $x$ of a digraph $D$, $|N_{D}^{+}(x)|$ is the number of vertices at distance 1 from $x$ and $|N_{D}^{++}(x)|$ is the number of vertices at distance 2 from $x$. In 1990, Seymour conjectured that for any oriented graph $D$ there exists a vertex $x$ such that $|N_{D}^{+}(x)| \leq |N_{D}^{++}(x)|$, where $x$ is called Seymour vertex. In 2018, Dara et al. conjectured that in every oriented graph with no sink, there are at least two Seymour vertices. In this paper, we investigate the existence of a Seymour vertex in line digraph and give a sufficient and necessary condition for line digraph to have a Seymour vertex. In particular, the result that line digraph of oriented graph has a Seymour vertex is obtained. Moreover, we give a sufficient and necessary condition for jump digraph (complement of line digraph) of digraph to have a Seymour vertex or at least two Seymour vertices, respectively.

Let $\mathbb{F}$ be a field of characteristic 0, $\Gamma$ an additive subgroup of $\mathbb{F}$, $s\in \mathbb{F}$ satisfying $s\notin \Gamma$ and $2s\in \Gamma$. We define a class of infinite-dimensional Lie algebras which are called generalized extended loop Schrödinger-Virasoro algebras $\mathscr{W}_{L}[\Gamma,s]$. In this paper, derivation algebras of $\mathscr{W}_{L}[\Gamma,s]$ are completely determined. As a by-product, we also obtain derivation algebras of the universal central extension of $\mathscr{W}_{L}[\Gamma,s]$.

This paper examines the global well-posedness and large time behavior to the 3D anisotropic magnetohydrodynamics (MHD) equations with mixed partial dissipation and magnetic diffusion. We first obtain the global existence and uniqueness of solutions to this system with initial data small. When the initial data also belongs to the negative Sobolev spaces, we establish the optimal decay estimates for the aforementioned global solutions. In particular, the enhanced decay estimates of the third component of velocity and the first component of magnetic field are derived.

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 article, I give a definition of topological entropy of random dynamical systems associated to an infinite countable discrete amenable group action. I obtain a variational principle between the topological entropy and measure fiber entropy of a random dynamical system.

The computational problem of one kind hybrid power mean involving the quartic Jacobsthal sum and Kloosterman sum is studied. Using elementary method and the properties of Gauss sums and character sums, an interesting linear recurrence and an explicit formula are derived.

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.

We introduce a new inertial projected reflected gradient algorithm for solving variational inequality problems in Hilbert spaces. Moreover, we prove a weak convergence theorem for our proposed algorithm under some suitable assumptions imposed on the parameters. The results obtained in this paper extend and improve many recent ones in the literature.

In this paper, we study the semilinear Moore-Gibson-Thompson (MGT) equations with nonlinearities $|\partial_t^k\psi|^p$ for $k=0,1$ in a compact Lie group. By using iteration method with slicing procedure for an unbounded multiplier, we prove blow-up of energy solutions for any $p>1$ and $k=0,1$, even in the viscous or inviscid models. As a byproduct, we also derive upper bound estimates for the lifespans. This result indicates different influences of the viscous dissipation in the Euclidean space and compact Lie groups.

This paper gives three kinds of variational formula for the first nontrivial eigenvalues of single birth processes on a finite state space, from which, the explicit upper and lower bounds for the eigenvalues are obtained. Additionally, using the first hitting time, a new formula for the corresponding eigenfunction is presented.

We consider a class of elliptic equations with variable exponents and degenerate coercivity. The key feature is that the coefficient function of the first-order term belongs to an appropriate Lorentz space. Using the Marcinkiewicz estimate within the framework of variable exponents and degenerate coercivity, we obtain a Lorentz estimate for the sequences of solutions. By selecting suitable test functions, we prove the strong convergence of the truncation sequences in the energy space.

In this paper, we study the following $p$-Laplacian problem $$ \left\{\begin{array}{l} -\Delta_p u+V(x)|u|^{p-2}u=|u|^{{p^*}-2}u+a(x)|u|^{q-2}u, \quad x\in\mathbb{R}^N,\\ u\in W^{1,p}(\mathbb{R}^N), \end{array}\right. $$ where $N>p^2$, $1<p<q<p^*$ and $p^*=\frac{Np}{N-p}$ is the Sobolev critical exponent, $\Delta_p:=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator, $V$ is a nonnegative function. Under appropriate assumptions on $V$ and $a$, by using barycenter function, quantitative deformation lemma and Brouwer degree theory, we prove that this problem has at least two distinct positive solutions.

In this paper, we mainly study the structure theory of Hom-Lie algebra. Hom-Lie algebra is an extension of usual Lie algebra, which can be obtained by deforming Lie algebra and its endomorphism. Using the endomorphism of the Lie algebra $W(2,2)$, we construct $W(2,2)$-type Hom-Lie algebra $({\mathcal L},[,],\phi)$ and determine the transposed Hom-Poisson algebra structures on $W(2,2)$-type Hom-Lie algebra $({\mathcal L},[,],\phi)$.

In this paper, we focus on the algorithm for higher order Melnikov functions which can be used to deal with general planar perturbed piecewise Hamiltonian systems separated by a straight line. We propose a new formula for any order Melnikov function which is more symmetric. Furthermore, we apply the formula to a problem on the number of limit cycles for above piecewise linear differential systems.

Let $\mathcal H$ be a complex Hilbert space with dimension $n\ge 3 $, $\mathcal{P} (\mathcal{H})$ the set of projections on $\mathcal{H}$, and $\varphi :\mathcal{P} (\mathcal{H})\to \mathcal{P} (\mathcal{H})$ is a surjective map. If $\varphi$ preserves the joint spectrum of any pair of projections, then $\varphi$ preserves the unitary equivalence of the projections, and $\varphi$ is a lattice isomorphism on $\mathcal{P} (\mathcal{H}) $, we obtain that $\varphi$ can be induced by a semi-linear isomorphism. If $\psi :\mathcal{P} (\mathcal{H})\to \mathcal{P} (\mathcal{H})$ is a surjective map which preserves the joint spectrum of the identity operator $I$ and any two projections, then $\psi$ preserves the orthogonality, thus $\psi$ can be induced by a unitary or anti-unitary.

Let $G=(V, E)$ be a locally finite graph, $\Omega \subset V$ be a connected finite subset. In this paper, we consider the following nonlinear Dirichlet problem \begin{align*} \begin{cases} -\Delta_{p}u(x)=f(x, u), &\mathrm{in} \ \Omega,\\ u(x)=0, & \mathrm{on} \ \partial \Omega, \end{cases} \end{align*} where $\Delta_{p}$ denotes the $p$-Laplace operator. By using the method of Morse theory and local linking, we prove that for any $p>1$, the above equation admits at least two nontrivial solutions, provided that $f(x, s)$ satisfies certain assumptions. Moreover, similar method was used to obtain multiple solutions of the following $p$-bi-harmonic equation \begin{align*} \begin{cases} \Delta_{p}^{2}u(x)=g(x, u), & \mathrm{in} \ \Omega,\\ u(x)=\Delta u(x)=0, & \mathrm{on} \ \partial \Omega, \end{cases} \end{align*} where $\Delta^{2}_{p}u=\Delta(\vert \Delta u \vert^{p-2} \Delta u)$ is the $p$-bi-harmonic operator of $u: V\to \mathbb{R}$.

This paper considers Liouville theorems for semi-convex solutions to a class of mixed Hessian equations. In particular, this paper proves that any semi-convex solution in $\mathbb{R}^n$ to ${\sigma_2(D^2u)}/{\sigma_1(D^2u)}=1$ is quadratic.

In this paper, we consider an expanding flow of smooth, closed, $(\eta,k)$-convex hypersurfaces in Euclidean $\mathbb{R}^{n+1}$ with speed $u^{\alpha}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))$, where $u, \rho$ are the support function and radical function of the hypersurface, respectively, $\alpha,\delta\in\mathbb{R}^1$, $\beta>0$, $k$ is an integer and $1 \leq k \leq n$, $\eta=Hg-h$, the first Newton transformation of the second fundamental form $h$, $\lambda(\eta)$ denote the eigenvalues of $g^{-1}\eta$. For $\alpha+\delta+\beta\leq 1$, we prove that the flow has a unique smooth and $(\eta,k)$-convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for $\alpha+\delta+\beta< 1$, we prove that the flow with the speed $fu^{\alpha}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))$ exists for all time and converges smoothly after normalisation to a soliton which is a solution of $fu^{\alpha-1}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))=\gamma$ provided that $f$ is a smooth positive function on $\mathbb{S}^n$ and $\gamma>0$ is a constant. What's more, we can also use a more general flow to prove the existence of solution to a class of Hessian quotient equations again.

In this paper, a new iterative algorithm is proposed in Hilbert space to solve the equilibrium problem, the fixed point problem of a family of nonexpansive mappings and the split variational inclusion problem. Under appropriate parameter restriction conditions, it is proved that this algorithm converges strongly to the common solution of the above three types of problems, and numerical examples are given to illustrate the effectiveness of this algorithm.

In this paper, we study the properties of the intersection local times of two independent real valued spherical Gaussian random fields. By means of the mean square increment and the strong local nondeterminism of spherical Gaussian random fields, the existence of the intersection local times is proved. Moreover, the joint continuity and Hölder conditions of the intersection local times are obtained by using the occupancy density theory and the moment method. These results extend the cases of Gaussian fields in Euclidean space to spherical Gaussian fields, and further improve the sample paths properties of the more complex spherical Gaussian random fields.

Product systems over left cancellative small categories are introduced and studied in this paper. We also introduce the notion of compactly aligned product systems over finite aligned left cancellative small categories and its Nica covariant Toeplitz representations. Furthermore, the existence of co-universal $C^*$-algebras for injective, gauge-compatible, Nica covariant Toeplitz representations of compactly aligned product systems over finite aligned subcategories of groupoids is proved in this paper.