In this paper, we study the $p$-Laplacian Choquard equation $$-△_p u+V(x)|u|^{p-2}u=\bigg({\sum_{y\in N^n\atop y\not=x}}\frac{|u(y)|^q}{d(x,\,y)^{n-\alpha}}\bigg)|u|^{q-2}u$$ on a finite lattice graph $N^n$ with $n\in\mathbb{N}_+$, where $p>1,$ $q>1$ and $0\leq\alpha\leq n$ are some constants, $V(x)$ is a positive function on $N^n$. Using the Nehari method, we prove that if 1<p<q<+∞, then the above equation admits a ground state solution. Previously, the $p$-Laplacian Choquard equation on finite lattice graph has not been studied, and our result contains the critical cases $\alpha=0$ and $\alpha=n$, which further improves the study of Choquard equations on lattice graphs.

This paper develops an Itô-type fractional pathwise integration theory for fractional Brownian motion with Hurst parameters \(H \in (\frac{1}{3}, \frac{1}{2}] \), using the Lyons' rough path framework. This approach is designed to fill gaps in conventional stochastic calculus models that fail to account for temporal persistence prevalent in dynamic systems such as those found in economics, finance, and engineering. The pathwise-defined method not only meets the zero expectation criterion but also addresses the challenges of integrating non-semimartingale processes, which traditional Itô calculus cannot handle. We apply this theory to fractional Black-Scholes models and high-dimensional fractional Ornstein-Uhlenbeck processes, illustrating the advantages of this approach. Additionally, the paper discusses the generalization of Itô integrals to rough differential equations (RDE) driven by fBM, emphasizing the necessity of integrand-specific adaptations in the Itô rough path lift for stochastic modeling.

Using the stratifications of Deligne—Mumford moduli spaces $\overline{\mathcal M}_{g,n}$ indexed by stable graphs, we introduce a partially ordered set of stable graphs by defining a partial ordering on the set of connected stable graphs of genus $g$ with $n$ external edges. By modifying the usual definition of zeta function and Möbius function of a poset, we introduce generalized ($\mathbb Q$-valued) zeta function and generalized ($\mathbb Q$-valued) Möbius function of the poset of stable graphs. We use them to proved a generalized Möbius inversion formula for functions on the poset of stable graphs. Two applications related to duality in earlier work are also presented.

Polya—Carlson theorem asserts that if a power series with integer coefficients and convergence radius 1 can be extended holomorphically out of the unit disc, it must represent a rational function. In this note, we give a generalization of this result to multivariate case and give an application to rationality theorem about D-finite power series.

In this paper, we investigate the existence of normalized solutions for a quasilinear elliptic problem as follows \begin{equation*} \left\{\begin{array}{ll} -\Delta_p u+\lambda u^{p-1}=f(u), & x\in \mathbb{R}^N, \\ \displaystyle\int_{\mathbb{R}^N}|u|^p d x=\rho,& u \in W^{1,p}(\mathbb{R}^N), \end{array}\right. \end{equation*} where $-\Delta_p $ is the $p$-Laplace operator, 1<p<N,N≥3,ρ>0 and λ>0. f is a continuous function and satisfies some suitable conditions. Based on a Nehari—Pohozaev manifold, we show the existence of positive normalized solutions by using the minimization method.

In this article, we continue to study K?hler metrics on line bundles over projective spaces to find complete K?hler metrics with positive holomorphic sectional curvatures with two very special properties. These two special kinds of examples were not able to be found in our earlier paper of the first author and Ms. Duan. And therefore, we give a further step toward a famous Yau conjecture with the method in the co-homogeneity one geometry.

In this paper, entropy and pressure are investigated for a random dynamical system $\varphi$ over $\mathbb{Z}^k$-actions on a compact metric space. The pressure $P(\varphi, f)$ of $\varphi$ with respect to a random continuous function $f$ and the measure-theoretic entropy $h_\mu(\varphi)$ for a $\varphi$-invariant measure $\mu$ are defined. A variational principle for pressure $P(\varphi, f)$ is established, which states that $P(\varphi, f)$ is the supremum of the sum of $h_\mu(\varphi)$ and the integral of $f$ taken over all invariant measures $\mu$. We also obtain some basic properties for equilibrium states.

Let $\{X, X_{n}; n \geq 1\}$ be a sequence of i.i.d. non-degenerate real-valued random variables with $\mathbb{E}X^{2} < \infty$. Let $S_{n} = \sum_{i=1}^{n} X_{i}$, $n \geq 1$. Let $g(\cdot): ~[0, \infty) \rightarrow [0, \infty)$ be a nondecreasing regularly varying function with index $\rho \geq 0$ and $\lim_{t \rightarrow \infty} g(t) = \infty$. Let $\mu = \mathbb{E}X$ and $\sigma^{2} = \mathbb{E}(X - \mu)^{2}$. In this paper, on the scale $g(\log n)$, we obtain precise asymptotic estimates for the probabilities of moderate deviations of the form $ \log \mathbb{P}(S_{n} - n \mu > x \sqrt{ng(\log n)} )$, $ \log \mathbb{P}(S_{n} - n \mu < -x \sqrt{ng(\log n)} )$, and $ \log \mathbb{P}(|S_{n} - n \mu | > x \sqrt{ng(\log n)} )$ for all $x > 0$. Unlike those known results in the literature, the moderate deviation results established in this paper depend on both the variance and the asymptotic behavior of the tail distribution of $X$.

In this paper, we introduce a communication-efficient distributed estimation method tailored for massive datasets exhibiting skewness. The data are stored across multiple machines. We construct a surrogate likelihood which only need to transfer subgradient from local machines to approximate higher-order derivatives of the global likelihood. An enhanced EM algorithm is developed for computations. The proposed method not only addresses the non-normality of data by utilizing first-order gradient information in each transmission, ensuring low communication overhead, but also ensures privacy protection. Simulation studies illustrate the superior performance of the proposed methods.

In this paper, we study asymptotic behavior of small perturbation for path-distribution dependent stochastic differential equations driven simultaneously by a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{2},1)$ and a standard Brownian motion. We establish large and moderate deviation principles by utilising the weak convergence approach.

Zhao and Xu (2013) constructed a functor from $\mathfrak{o}(n)$-Mod to $\mathfrak{o}(n+2)$-Mod. In this paper, we use the functor successively to obtain full conformal oscillator representation of $\mathfrak{o}(2n+2)$ in $n(n+1)$ variables and determine the corresponding finite-dimensional irreducible module explicitly when the highest weight is dominant integral. We also find an equation of counting the dimension of an irreducible $\mathfrak{o}(2n+2)$-module in terms of certain alternating sum of the dimensions of irreducible $\mathfrak{o}(2n)$-modules, which leads to new combinatorial identities of classical type in the case of the Steinberg modules. One can use the results to study tensor decomposition of finite-dimensional irreducible modules by solving certain first-order linear partial differential equations, and thereby obtain the corresponding physically interested Clebsch—Gordan coefficients and exact solutions of Knizhnik—Zamolodchikov equation in WZW model of conformal field theory.

We study the mean orbital pseudo-metric for Polish dynamical systems and its connections with properties of the space of invariant measures. We give equivalent conditions for when the set of invariant measures generated by periodic points is dense in the set of ergodic measures and the space of invariant measures. We also introduce the concept of asymptotic orbital average shadowing property and show that it implies that every non-empty compact connected subset of the space of invariant measures has a generic point.

In this paper, we introduce the spectral Einstein functional for perturbations of Dirac operators on manifolds with boundary. Furthermore, we provide the proof of the Dabrowski—Sitarz—Zalecki type theorems associated with the spectral Einstein functionals for perturbations of Dirac operators, particularly in the cases of on 4-dimensional manifolds with boundary.

In this paper, we develop some new variational and analytic techniques to study the multiplicity and concentration of positive solutions for a planar Schrödinger-Poisson system involving competing weight potentials and the nonlinearity $K(x)|u|^{p-2}u$ $(2<p<4)$ in $\mathbb{R}^2$. By Nehari manifold and Ljusternik-Schnirelmann category, we relate the number of positive solutions to the category of the global minima set of a suitable ground energy function. Our results improve and extend the ones in [Du, Weth, Nonlinearity, 30, 3492-3515 (2017)] and [Chen, Tang, J. Differ. Equ., 268, 945-976 (2020)]. In particular, we do not need the assumption $K(x)\equiv1$ and the $C^1$ smoothness of $V(x)$. Furthermore, we do not use the axially symmetric condition of the potential in our second main result. Moreover, we shall show that there is a great difference in our results between $N=2$ and $N\geq3$.

In this paper we will be concerned with the problem $$ - \Delta u - \frac{1}{2}\Delta(a(x)u^2) u + V(x)u=f(u), x\in \mathbb{R}^2, $$ where $V$ is a potential continuous and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous function with exponential subcritical or exponential critical growth. We use as a main tool the Nehari manifold method in order to show existence of nonnegative solutions and existence of nodal solutions. Our results complement the classical result of Solutions for quasilinear Schr?dinger equations via the Nehari method" due to Jia-Quan Liu, Ya-Qi Wang and Zhi-Qiang Wang in the sense that in this article we are considering nonlinearity of the exponential type.

We construct an explicit example of a smooth isotopy $\{\xi_t\}_{t \in [0,1]}$ of volume- and orientation-preserving diffeomorphisms on $[0,1]^n$ ($n \geq 3$) that has infinite total kinetic energy. This isotopy has no self-cancellation and is supported on countably many disjoint tubular neighbourhoods of homothetic copies of the isometrically embedded image of $(M,g)$, a "topologically complicated" Riemannian manifold-with-boundary. However, there exists another smooth isotopy that coincides with $\{\xi_t\}$ at $t=0$ and $t=1$ but of finite total kinetic energy.

Let $V\cup_{S}W$ be a Heegaard splitting of $M$ with distance $n\geq 2$ and $F$ a boundary component of $\partial_{-}V$. A simple closed curve $J$ in $F$ is called distance degenerating if the distance of $M_{J}=V_{J}\cup_{S}W$ is less than $n$, where $M_{J}$ is obtained by attaching a 2-handle to $M$ along $J$. In this paper, by considering the distance between $J$ and the image of the essential disks of $W$ under the projection map, we obtain a sufficient condition for the diameter of the set of distance degenerating curves in $F$ to be bounded in $C(F)$. Moreover, for $F=\partial M$, an upper bound of the diameter of the set of the boundary reducible curves in $F$ is given under some circumstance.

We study inhomogeneous projective oscillator representations of Lie superalgebras of $Q$-type on supersymmetric polynomial algebras. These representations are infinite-dimensional. We prove that they are completely reducible. Moreover, these modules are explicitly decomposed as direct sums of two irreducible submodules.

In this paper, we explore the convergence and convergence rate results for a new methodology termed the half-proximal symmetric splitting method (HPSSM). This method is designed to address linearly constrained two-block non-convex separable optimization problem. It integrates a half-proximal term within its first subproblem to cancel out complicated terms in applications where the subproblem is not easy to solve or lacks a simple closed-form solution. To further enhance adaptability in selecting relaxation factor thresholds during the two Lagrange multiplier update steps, we strategically incorporate a relaxation factor as a disturbance parameter within the iterative process of the second subproblem. Building on several foundational assumptions, we establish the subsequential convergence, global convergence, and iteration complexity of HPSSM. Assuming the presence of the Kurdyka-?ojasiewicz inequality of ?ojasiewicz-type within the augmented Lagrangian function (ALF), we derive the convergence rates for both the ALF sequence and the iterative sequence. To substantiate the effectiveness of HPSSM, sufficient numerical experiments are conducted. Moreover, expanding upon the two-block iterative scheme, we present the theoretical results for the symmetric splitting method when applied to a three-block case.

In this paper, we study the vector fields $X$ with a global Poincaré cross-section on a $2 n+1$-dimensional presymplectic manifold ($M, \tilde{\omega}$) under certain conditions. We use ($M, \tilde{\omega}$) to construct a $2 n+2 k$ dimensional symplectic manifold ($\tilde{M}, \Lambda$), on which the vector field $X$ can be extended to a Hamiltonian vector field $\tilde{X}$ with a smooth Hamiltonian $H: \tilde{M} \rightarrow R$. We also consider vector fields $X$ with a first integral $F$ and a Jacobi multiplier $J$ on an $n$-dimensional manifold ($M, \Omega$). On a level set $\Sigma$ of $F$, we get an $n-1$-volume form $\omega_n$ on $\Sigma$ and prove that $X$ is a volume-preserving vector field with respect to $\omega_n$. Specifically, when $X$ is a 3 dimensional devergence-free vector field, the results have been discussed by Lerman in 2019.

The $H$-invariant was introduced to compute Perelman’s entropy for Kähler-Ricci flow in a paper of Tian-Zhang-Zhang-Zhu more than ten years ago. It turns out that the $H$-invariant is equal to an earlier invariant by Tian-Zhu in their study on Kähler-Ricci solitons. In this largely expository paper, we will discuss definition of the $H$-invariant, its relation to Tian-Zhu's generalization of the Futaki invariants as well as some of its applications. We will also include some new observations and generalizations of results in existing literature. Several examples will be also provided.

In this paper, we give a slight improvement of El Soufi—Ilias—Ros's upper bound of the first Laplace eigenvalue on a torus in a fixed conformal class. We also optimize Montiel—Ros's argument to obtain a better upper bound of the conformal area for certain rectangular tori.

We provide three types of invariance pressure for uncertain control systems, namely, invariance pressure, strong invariance pressure, and invariance feedback pressure. The first two respectively extend the corresponding pressures for deterministic control systems proposed by Colonius, Cossich, and Santana (2018) and by Nie, Wang, and Huang (2022); and the third generalizes invariance feedback entropy of uncertain control systems presented by Tomar, Rungger, and Zamani (2020), by adding potentials on the control range. Then we prove that (1) an explicit formula for invariance pressure of a controlled invariant set with respect to a potential by the logarithm of the spectral radius of the admissible weighted matrix determined by this potential under some suitable conditions; (2) an explicit formula for pressure of invariant quasi-partitions by maximum mean weight over all irreducible periodic sequences; (3) the invariance feedback pressure of a controlled invariant set is equal to the pressure of an atom partition under some technical assumptions; (4) lower and upper bounds for pressure of invariant quasi-partitions w.r.t. a potential by the logarithm of the spectral radius of the weighted adjacency matrix determined by this potential; (5) a variational principle for strong invariance pressure.

First we investigate relative $n$-regionally proximal tuples. Let $\pi: (X,G)→ (Y,G)$ be a Bronstein extension between minimal systems. It turns out that if $(x_1,\dots, x_n)$ is a minimal point and $(x_{i},x_{i+1})$ is relative regionally proximal for $1\leq i\leq n-1$, then $(x_1,\dots, x_n)$ is relative $n$-regionally proximal. We consider the relative versions of sensitivity, including relative $n$-sensitivity and relative block $F_t$-$n$-sensitivity, where $F_t$ is the family of thick sets. We show that $\pi$ is relatively $n$-sensitive if and only if the relative $n$-regionally proximal relation contains a point whose coordinates are distinct, and the structure of $\pi$ which is relatively $n$-sensitive but not relatively $n+1$-sensitive is determined. We also characterize relatively block $F_t$-$n$-sensitive via relative regionally proximal tuples.

Lower and upper bounds for eigenvalues help estimate the location interval of eigenvalues, which is of practical meanings especially for those problems of which the eigenvalues cannot be exactly obtained. In this paper, we study the lower and upper bounds for linear elasticity eigenvalues by displacement-pressure mixed finite element schemes. By applying expansion identities for the error of eigenvalues, lower and upper numerically computable bounds for the eigenvalues are derived based on certain mathematical hypotheses. For the schemes studied here, roughly speaking, the accuracy loss of the local approximation of the discrete displacement may lead to lower bound and that of pressure to upper bound. By utilizing the min-max principle and perturbation theory for the solution operator, theoretical lower and upper bounds can be controlled by setting proper Lamé parameters.

Left-invariant Riemannian metrics on Lie groups $G_{\mathbb{R}{H}^{n-1}}\times\mathbb{R}$ and $G_{\mathbb{R}{H}^{2}}\times\mathbb{R}^{n-2}$ ($n\geq 3$) have been classified by Hiroshi, Takahara and Tamaru. It is easily seen that $G_{\mathbb{R}{H}^{n-1}}\times\mathbb{R}$ and $G_{\mathbb{R}{H}^{2}}\times\mathbb{R}^{n-2}$ ($n\geq 3$) have the same automorphism group, which is denoted by $L_n$. In this paper, we first classify $n$-dimensional simply connected Lie groups with automorphism group $L_n$, then we classify the left-invariant Riemannian metrics on such Lie groups. As an application, we get the $m$-quasi Einstein metrics.

In the present paper, the authors systematically study the mapping properties of multilinear maximal operators on the Triebel–Lizorkin spaces and Besov spaces. In the global setting, the authors provide a criterion on the boundedness and continuity of a class of multilinear operators on the Triebel–Lizorkin spaces and Besov spaces, which can be used to obtain the boundedness and continuity of the multilinear operators associated to balls, cubes and dyadic cubes, multilinear sharp maximal operator as well as multilinear operators of convolution type on the Triebel–Lizorkin spaces and Besov spaces. The corresponding results for the multilinear maximal operators associated to balls are also proved in the local setting.

Combing the weak KAM method for contact Hamiltonian systems and the theory of viscosity solutions for Hamilton—Jacobi equations, we study the Lyapunov stability and instability of viscosity solutions for evolutionary contact Hamilton—Jacobi equation in the first part. In the second part, we study the existence and multiplicity of time-periodic solutions.

DeBiasio and Krueger showed the following result: For all $0\leq\delta\leq1$ and $\epsilon>0$, there exists $n_0$ such that if $G$ is a balanced bipartite graph on $2n\geq2n_0$ vertices with $\delta(G)=\delta n$, then in every $2$-coloring of G, there exists a monochromatic cycle of order at least $(f(\delta)-\epsilon)n$, where \[f(\delta)=\begin{cases} \delta, & 0\leq\delta\leq\dfrac{2}{3}, \\[3mm] 4\delta-2, & \dfrac{2}{3}<\delta\leq\dfrac{3}{4}, \\[3mm] 1, & \dfrac{3}{4}<\delta\leq1. \end{cases}\] Zhang and Peng (2023) extended the above result to off-diagonal cases when $\delta>\frac{3}{4}$. In this paper, we relax the condition $\delta>\frac{3}{4}$ to $\delta>\frac{2}{3}$. We show the following result: For every $\eta>0$, there exists a positive integer $N_0$ such that for every integer $N>N_0$ the following holds. Let $\frac{2}{3}<\delta\leq\frac{3}{4}$, and let $\alpha_1\geq\frac{\delta \alpha_2}{3\delta-2}>0$ such that $\alpha_1+\alpha_2=1$. Let $G[X, Y]$ be a balanced bipartite graph on $2N$ vertices with $\delta(G)=(\delta+3\eta)N$. Then for each red-blue-edge-coloring of $G$, either there exist red even cycles of each length in $\{4, 6, 8, \dots, 2(2\delta-1)(2-3\eta^2)\alpha_1N\}$, or there exist blue even cycles of each length in $\{4, 6, 8, \dots, 2(2\delta-1)(2-3\eta^2)\alpha_2N\}$. There are constructions of colorings showing that the length of a longest monochromatic cycle is asymptotically tight and the condition $\alpha_1\geq\frac{\delta \alpha_2}{3\delta-2}$ cannot be removed.

In this paper, we prove that four-dimensional hypersurface $M^4_r$ with proper mean curvature vector field (i.e., $\Delta\vec{H}$ is proportional to $\vec{H}$) in pseudo-Riemannian space form $N^5_s(c)$ has constant mean curvature, and give the value or range of this constant. As an application, we obtain that biharmonic hypersurfaces in $N^5_s(c)$ are minimal in some specific case.

Matrix-valued data have found extensive applications in various fields, such as modern biomedical imaging, chemometrics, and economics. In this paper, we address the problem of generalized trace regression involving matrix-valued covariates. To estimate the unknown parameters, we propose a penalty that combines the MCP nuclear norm and two-dimensional spline lasso. This penalty accounts for the potential low-rank and row/column order structures in the matrix-valued covariates. We establish the general theory and explicit statistical convergence rate of the resulting estimator. Through simulations, we demonstrate the advantages of our proposed method compared to other competing methods. Finally, we apply our approach to analyze the brain-image datasets related to Alzheimer's disease, identifying several efficient regions that illustrate the mechanism of Alzheimer.

We study supercritical age-structured branching models starting from a single particle with a random lifetime, where the reproduction law depends on the remaining lifetime of the parent. The lifespan of an individual is decided at its birth and its remaining lifetime decreases at the unit speed. A necessary and sufficient condition is provided for the convergence of the Malthusian normalized random measures. The Malthusian type limit theory in a functional form can be strengthened to hold with probability one under some “L log L” conditions. We further prove a central limit theory with a random normalization factor.

Let $\{X_n\}_{n\geq0}$ be a $p$-type ($p\geq2$) supercritical branching process with immigration and mean matrix $M$. Suppose that $M$ is positively regular and $\rho$ is the maximal eigenvalue of $M$ with the corresponding left and right eigenvectors $\boldsymbol{v}$ and $\boldsymbol{u}$. Let $\rho>1$ and $Y_n=\rho^{-n}[\boldsymbol{u}\cdot X_n -\frac{\rho^{n+1}-1}{\rho-1}( \boldsymbol{u}\cdot \boldsymbol{\lambda})]$, where the vector $\boldsymbol{\lambda}$ denotes the mean immigration rate. In this paper, we will show that $Y_n$ is a martingale and converges to an $r.v.$ $Y$ as $n\rightarrow\infty$. We study the rates of convergence to $0$ as $n\rightarrow\infty$ of $$ P_i\bigg(\bigg|\frac{\boldsymbol{l}\cdot X_{n+1}}{\textbf{1}\cdot X_n}-\frac{\boldsymbol{l}\cdot(X_nM)}{\textbf{1}\cdot X_n}\bigg|>\varepsilon\bigg), P_i\bigg(\bigg|\frac{\boldsymbol{l}\cdot X_n}{\textbf{1}\cdot X_n}-\frac{\boldsymbol{l}\cdot\boldsymbol{v}}{\textbf{1}\cdot \boldsymbol{v} }\bigg|>\varepsilon\bigg), P(|Y_n-Y|>\varepsilon) $$ for any $\varepsilon>0,\, i=1,\dots,p$, $\textbf{1}=(1,\dots,1)$ and $\boldsymbol{l}\in\mathbb{R}^p,$ the $p$-dimensional Euclidean space. It is shown that under certain moment conditions, the first two decay geometrically, while conditionally on the event $Y\geq\alpha\ (\alpha>0)$ supergeometrically. The decay rate of the last probability is always supergeometric under a finite moment generating function assumption.

In this paper, we study the existence and multiplicity of homoclinic solutions for a class of second-order Hamiltonian system: $u''(t)-L(t)u(t) + \nabla V(t,u) = 0$, where $L(t)$ and $V(t,u)$ are not periodic in $t$. First, we introduce the definition of index and establish the corresponding index theory. Then, by using the index theory and critical point theory, we prove our main results under the asymptotic quadratic conditions of the potential function.

Given a graph $H$ and an integer $p\geq 2$, the edge blow-up graph $H^{p+1}$ of $H$ is the graph obtained by replacing each edge in $H$ with a clique of order $p+1$, where the new vertices of the cliques are all distinct. The generalized Turán number ex$(n, K_m, F)$ denote the maximum number of copies of $K_m$ in an $n$-vertex $F$-free graph. Let $C_t$ and $P_t$ denote the cycle and path with $t$ vertices, respectively. In this paper, we obtain the generalized Turán numbers $\operatorname{ex}(n, K_m, P_t^{p+1})$, $\operatorname{ex}(n, K_m, C_t^{p+1})$ and characterize the unique graph for $P_t^{p+1}$ and $C_t^{p+1}$ respectively, when $t\geq3$, $p\geq m\geq 3$ and $n$ is sufficiently large.

In this paper, we establish the sharp local uniform well-posedness of the higher-order nonlinear Schrödinger equations (HNLS) with cubic nonlinear terms i $\partial_t u+\left(-\Delta_g\right)^m u=-|u|^2 u$ on $\mathbb{T}^2$ and $\mathbb{S}^2$. Employing bilinear estimates and lattice point estimates, we prove that the well-posedness thresholds are $s_c\left(\mathbb{T}^2, m\right)=0$ and $s_c\left(\mathbb{S}^2, m\right)=\frac{1}{4}$ for any order $m \in \mathbb{N}$. In contrast, for Euclidean spaces, it has been shown in [Miao, C., Zhang, B.: Discrete Contin. Dyn. Syst., 17, 181-200 (2006)] that $s_c\left(\mathbb{R}^d, m\right)=\frac{d}{2}-m$ if $m<\frac{d}{2}$. These reveal that the geometry of manifolds plays a crucial role in the dynamics of the cubic HNLS.

In this article, we study the limit behavior of solutions to an energy-critical complex Ginzburg-Landau equation. Via energy method, we establish a rigorous theory of the zero-dispersion limit from energy-critical complex Ginzburg-Landau equation to energy-critical nonlinear heat equation in dimensions three and four for both the defocusing and focusing cases. Furthermore, we derive the inviscid limit of energy-critical complex Ginzburg-Landau equation from energy-critical nonlinear Schrödinger equation in dimension four for the focusing case.

Given a finite tensor category $\mathcal{C}$, an exact indecomposable $\mathcal{C}$-module category $\mathcal{M}$, and a tensor subcategory $\mathcal{D} \subseteq \mathcal{C}_{\mathcal{M}}^*$, we describe a way to produce exact commutative algebras in the center $Z(\mathcal{C})$, measuring this inclusion. The construction of such algebras is done in an analogous way as presented by Shimizu [20], but using instead the relative (co)end, a categorical tool developed in [1] in the realm of representations of tensor categories. We provide some explicit computations.

In this article, we study the hyperbolic dynamics of geodesic flows on Riemannian (not necessarily compact) manifolds with no conjugate points. By hyperbolic dynamics, we focus on the Anosov Closing Lemma, the local product structure, and the transitivity of the geodesic flows on the set of rank $1$ non-wandering set $\Omega_1$ under the conditions of bounded asymptote and uniform visibility. As an application, we further discuss on some generic properties of the set of invariant probability measures.

By giving a series of estimates of multiplication operators, investigating Cauchy transforms and using $L^2$-bounded Calderón-Zygmund operators, we provide the corona theorem with countably many functions for multiplier algebras of weighted Dirichlet spaces $\mathcal{D}_K$. Our result thereby gives an extension of a corona theorem with infinitely many functions from the Dirichlet space $\mathcal{D}$ to a weighted Dirichlet space $\mathcal{D}_K$ for a general weight function $K$.