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 $0<p<1$.

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.

This paper focuses on a class of stochastic functional differential equations (SFDEs) with Hölder continuous coefficients. Under the extended Veretennikov-Khasminskii conditions, this paper proves that such equations admit a unique invariant probability measure and the transition probability of the associated segment process converges exponentially to the equilibrium under a Wasserstein distance.

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.

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.

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, 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.

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 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 $G=(V, E)$ 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 $u: V\to \mathbb{R}$.

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.

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 $\mathbb{F}$ 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 $\mathscr{W}_{L}[\Gamma,s]$.

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.

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 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.

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.

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 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 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.

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.

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.

We focus on the $d$-boundedness of the Bergman kernel function on a class of Forelli-Rudin structure. We use the series method and the Hua Luogeng method to derive the expression of the Bergman kernel function. When $\frac{1}{p_1}, \, \frac{1}{p_2}\in\mathbb{Z}^+$, we provide the finite form of the Bergman kernel function for the domain. Using the two holomorphic invariants of the domain, we obtain that the Bergman kernel function is $d$-bounded with respect to the Bergman metric. As a corollary, we have that the $L^2$ cohomology vanishing theorem is valid on the Forelli-Rudin structure.

We study the Hardy-Stein identities of complex-valued harmonic Hardy and weighted Bergman spaces respectively and observe these two spaces are affine invariant. As an application, an analogous Hardy-Stein identity of composition operators of complex-valued harmonic mappings in the unit disk is established. Meanwhile, we build a relationship between a harmonic mapping and its analytic part in view of Hardy and weighted Bergman spaces. Our results of this paper are extensions of the corresponding known ones in the setting of real-valued harmonic Hardy and weighted Bergman spaces.

Space-time panel data widely exist in various research fields. It is of great theoretical significance and application value to conduct statistical modeling and empirical analysis on space-time panel data. This paper proposes a fixed effects partially linear varying coefficient panel model with separable space-time filters and studies its estimation, testing and application. Based on a local linear dummy variable method, we first propose profile quasi-maximum likelihood estimators of the parameters and varying coefficient functions for the model, as well as an F test statistic for testing varying coefficient effect. Subsequently, the consistency and asymptotic normality of the estimators and asymptotic distribution of the test statistic are demonstrated. Thirdly, Monte Carlo simulations are used to illustrate the finite sample performance of the estimators and test statistic. Finally, the estimation technology and testing method of this model are applied to analyze the driving forces of carbon emission intensity of heavy industry in China.

Plateaued functions are important cryptographic functions due to their various desirable cryptographic characteristics. Having provided a method to produce a set of $s$-gplateaued functions with Walsh-Hadamard transforms having pairwise disjoint support, $p$-ary gbent functions are constructed using gplateaued functions as building blocks. Recursively, the resulting functions are used as building blocks to construct gplateaued functions having pairwise disjoint supports. By a result on cyclotomic fields, a simple proof of $f^{**}(x)=f(-x)$ for plateaued functions and gplateaued functions is presented. Some properties of correlation spectrum of gbent function are presented also in this paper.

Two embeddings $\Pi$ and $\Pi'$ of graph $G$ are called isomorphic embeddings if there exists an automorphism $\psi$ of graph $G$ that bijectively maps the face set $\mathcal{F}(\Pi)$ to the face set $\mathcal{F}(\Pi')$. In this paper, we apply an edge augmentation method to show that the complete bipartite graph $K_{M, N}$ has at least $ \frac{2^{mn}}{8MN}(n!)^{m}(m!)^{n} $ non-isomorphic orientable genus embeddings, and at least $ \frac{2^{mn}}{8MN} \left ( 2^{mn}-1\right ) (n!)^{m}(m!)^{n} $ non-isomorphic non-orientable genus embeddings, where $m=\lfloor(M-2) / 2\rfloor$ and $n=\lfloor(N-2) / 2\rfloor$. The lower bounds obtained in this paper are symmetric with respect to the parameters $M$ and $N$, and improve upon existing results by at least a factor of $(m!)^n(n!)^m$.

This paper presents a Tikhonov regularized primal-dual dynamical system with Hessian-driven damping for solving a separable convex optimization problem in Hilbert spaces. Under some mild conditions, the fast convergence rates of the primal-dual gap, the objective function residual and the feasibility violation along the trajectory generated by the system are first obtained. Subsequently, some convergence rates in the literature are obtained under different assumption on parameters. Furthermore, it is proven that the trajectory generated by the dynamical system converges strongly to the minimal norm solution of the optimization problem. Finally, by virtue of some numerical experiments, it is shown that the Hessian-driven damping term can enable the trajectory converges more smoothly and Tikhonov regularization term can enable the strong convergence of the trajectory.

In the present paper, we generalized the Drinfeld doubles of finite dimensional Lie bialgebras to the cases of infinite dimensional Lie bialgebras, then the Drinfeld doubles of Witt type and Virasoro type Lie bialgebras were given.

Let $0<\alpha<1$, $1<p<\frac{1}{\alpha}$, $\frac{1}{q}=\frac{1}{p}- \alpha$. We first obtain that the function $\omega :\mathbb{Z} \rightarrow (0,\infty)$ belongs to weight class of $\mathcal{A} (1,q)(\mathbb{Z})$ if and only if discrete fractional maximal operator $M_{\alpha}$ or discrete Riesz potential $I_\alpha$ is bounded from $l_{\omega}^{1}(\mathbb{Z})$ to $l_{\omega^q}^{q, {\rm weak}}(\mathbb{Z})$. Then for $q=\infty$, we further obtain that the function $\omega$ belongs to weight class of $\mathcal{A} (p,\infty)(\mathbb{Z})$ if and only if discrete Riesz potential $I_\alpha$ has an upper estimate for a resembling discrete BMO norm. Moreover, we give another simple proof of $I_{\alpha}:l_{\omega ^p}^{p}(\mathbb{Z}) \rightarrow l_{\omega ^q}^{q}(\mathbb{Z})$ for $\omega \in \mathcal{A}(p,q)(\mathbb{Z})$. As applications, more weighted norm inequalities for $M_{\alpha}$ and $I_\alpha$ are established when $\omega \in \mathcal{A}(1,q)(\mathbb{Z})$ or $\omega \in \mathcal{A}(p,\infty)(\mathbb{Z})$, and some of them are new even in continuous setting.

In this paper, we study the following biharmonic Schrödinger equation with potential and mixed nonlinearities \begin{equation*} \left\{\!\!\begin{array}{ll} \Delta^2 u +V(x,y)u+\lambda u =\mu|u|^{p-2}u+|u|^{q-2}u,& (x, y) \in \Omega_r \times \mathbb{T}^n, \\ \displaystyle\int_{\Omega_r\times\mathbb{T}^n}u^2dxdy=\Theta, \end{array} \right. \end{equation*} where $\Omega_r \subset \mathbb{R}^d$ is an open bounded convex domain, $r>0$ is large and $\mu\in\mathbb{R}$. The exponents satisfy $2<p<2+\frac{8}{d+n}<q<4^*=\frac{2(d+n)}{d+n-4}$, so that the nonlinearity is a combination of a mass subcritical and a mass supercritical term. Under some assumptions on $V(x,y)$ and $\mu$, we obtain the several existence results on waveguide manifold. Moreover, we also consider the orbital stability of the solution.

In this paper, we consider a multi-type branching process in independent and identically distributed random environments, and investigate the asymptotic behavior of the estimator for the inner product of the expectation of the conditional mean matrix of the offspring distribution. We establish the law of large numbers, the central limit theorem, and the Berry-Esseen bound for the estimator. To this end, we construct a zero-mean martingale to connect the estimator with a corresponding random walk, and we analyze the moment properties and convergence rate of this martingale sequence.

In this paper, we propose a new forward-reflected-backward algorithm with two inertial extrapolation steps to solve variational inequalities involving quasi-monotone operators in Hilbert spaces. Weak and strong convergence results for the proposed algorithm are established under suitable assumptions on the parameters. The results obtained in this paper extend and improve upon many recent findings in the literature.