By comprehensive utilizing of the geometry structure of 2D Burgers equation and the stochastic noise, we find the decay properties of the solution to the stochastic 2D Burgers equation with Dirichlet boundary conditions. Consequently, the expected ergodicity for this turbulence model is established.

Let $\{X_{v}:v\in\mathbb{Z}^d\}$ be i.i.d. random variables. Let $S(\pi)=\sum_{v\in\pi}X_v$ be the weight of a self-avoiding lattice path $\pi$. Let \[M_n=\max\{S(\pi):\pi\text{ has length }n\text{ and starts from the origin}\}.\] We are interested in the asymptotics of $M_n$ as $n\to\infty$. This model is closely related to the first passage percolation when the weights $\{X_v:v\in\mathbb{Z}^d\}$ are non-positive and it is closely related to the last passage percolation when the weights $\{X_v,v\in\mathbb{Z}^d\}$ are non-negative. For general weights, this model could be viewed as an interpolation between first passage models and last passage models. Besides, this model is also closely related to a variant of the position of right-most particles of branching random walks. Under the two assumptions that $\exists\alpha>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+\alpha}<+\infty$ and that $E[X_0^{-}]<+\infty$, we prove that there exists a finite real number $M$ such that $M_n/n$ converges to a deterministic constant $M$ in $L^{1}$ as $n$ tends to infinity. And under the stronger assumptions that $\exists\alpha>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+\alpha}<+\infty$ and that $E[(X_0^{-})^4]<+\infty$, we prove that $M_n/n$ converges to the same constant $M$ almost surely as $n$ tends to infinity.

In this article, we consider a modified version of minimal $L^2$ integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points, and obtain a concavity property of the modified version. As applications, we give characterizations for the concavity degenerating to linearity on open Riemann surfaces and on fibrations over open Riemann surfaces.

Let $\pi:(X,T)\rightarrow (Y,S)$ be a factor map between two topological dynamical systems, and $\mathcal{F}$ a Furstenberg family of $\mathbb{Z}$. We introduce the notion of $relative broken$ $\mathcal{F}-sensitivity$. Let $\mathcal{F}_{s}$ (resp. $\mathcal{F}_{\rm pubd},\mathcal{F}_{\rm inf}$) be the families consisting of all syndetic subsets (resp. positive upper Banach density subsets, infinite subsets). We show that for a factor map $\pi:(X,T)\rightarrow (Y,S)$ between transitive systems, $\pi$ is relatively broken $\mathcal{F}$-sensitive for $\mathcal{F}=\mathcal{F}_{s}\ \text{or}\ \mathcal{F}_{\rm pubd}$ if and only if there exists a relative sensitive pair which is an $\mathcal{F}$-recurrent point of $(R_\pi, T^{(2)})$; is relatively broken $\mathcal{F}_{\rm inf}$-sensitive if and only if there exists a relative sensitive pair which is not asymptotic. For a factor map $\pi:(X,T)\rightarrow (Y,S)$ between minimal systems, we get the structure of relative broken $\mathcal{F}$-sensitivity by the factor map to its maximal equicontinuous factor.

This research paper deals with an extension of the non-central Wishart introduced in 1944 by Anderson and Girshick, that is the non-central Riesz distribution when the scale parameter is derived from a discrete vector. It is related to the matrix of normal samples with monotonous missing data. We characterize this distribution by means of its Laplace transform and we give an algorithm for generating it. Then we investigate, based on the method of the moment, the estimation of the parameters of the proposed model. The performance of the proposed estimators is evaluated by a numerical study.

$\mathfrak{L}_{\nu}$ operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of $\mathfrak{L}^{2}_{\nu}$ operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of $\mathfrak{L}^{2}_{\nu}$ operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, minimal submanifolds on the Euclidean space, submanifolds on the unit sphere and projective spaces. In particular, we get a universal inequality with respect to the $\mathcal{L}_{II}$ operator on the translating solitons. Usually, it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore, this work can be viewed as a new contribution to universal estimate.

How to analyze flocking behaviors of a multi-agent system with local interaction functions is a challenging problem in theory. Motsch and Tadmor in 2011 also stressed the significance to assume that the interaction function is rapidly decaying or cut-off at a finite distance (cf. Motsch and Tadmor in J. Stat. Phys. 2011). In this paper, we study the flocking behavior of a Cucker–Smale type model with compactly supported interaction functions. Using properties of a connected stochastic matrix, together with an elaborate analysis on perturbations of a linearized system, we obtain a sufficient condition imposed only on model parameters and initial data to guarantee flocking. Moreover, it is shown that the system achieves flocking at an exponential rate.

Let $G$ be a symplectic or orthogonal group defined over a finite field with odd characteristic and let $D\leq G$ be a Sylow $2$-subgroup. In this paper, we classify the essential $2$-subgroups and determine the essential $2$-rank of the Frobenius category $F_D(G)$. Together with the results of An–Dietrich and Cao–An–Zeng, this completes the work of essential subgroups and essential ranks of classical groups.

Let $\mathcal A$ be a unital C$^*$-algebra and $\mathcal B$ a unital C$^*$-algebra with a faithful trace $\tau$. Let $n$ be a positive integer. We give the definition of weakly approximate diagonalization (with respect to $\tau$) of a unital homomorphism $\phi: \mathcal{A} \to M_n(\mathcal{B})$. We give an equivalent characterization of McDuff II$_1$ factors. We show that, if $\mathcal A$ is a unital nuclear C$^*$-algebra and $\mathcal B$ is a type II$_1$ factor with faithful trace $\tau$, then every unital $*$-homomorphism $\phi: \mathcal A \to M_n(\mathcal B)$ is weakly approximately diagonalizable. If $\mathcal{B}$ is a unital simple infinite dimensional separable nuclear C$^*$-algebra, then any finitely many elements in $M_n(\mathcal B)$ can be simultaneously weakly approximately diagonalized while the elements in the diagonals can be required to be the same.

This is a continuation of our previous work (Ann. Sc. Norm. Super. Pisa Cl. Sci.,20, 1295–1324, 2020). Let $(\Sigma,g)$ be a closed Riemann surface, where the metric $g$ has conical singularities at finite points. Suppose $\mathbf{G}$ is a group whose elements are isometries acting on $(\Sigma,g)$. Trudinger–Moser inequalities involving $\mathbf{G}$ are established via the method of blow-up analysis, and the corresponding extremals are also obtained. This extends previous results of Chen (Proc. Amer. Math. Soc., 1990), Iula–Manicini (Nonlinear Anal., 2017), and the authors (2020).

In this paper, with $(\Sigma,g)$ being a closed Riemann surface, we analyze the possible concentration behavior of a heat flow related to the Trudinger–Moser energy. We obtain a long time existence for the flow. And along some sequence of times $t_k\rightarrow +\infty$, we can deduce the convergence of the flow in $H^2(\Sigma)$. Furthermore, the limit function is a critical point of the Trudinger–Moser functional under certain constraint.

In this paper, the notion of $C$-semigroup of continuous module homomorphisms on a complete random normal (RN) module is introduced and investigated. The existence and uniqueness of solution to the Cauchy problem with respect to exponentially bounded $C$-semigroups of continuous module homomorphisms in a complete RN module are established.

In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of $G=\text{GL}_n(\mathbb{C})$ possessing the minimal Gelfand-Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of $G$ of type $(n-1,1)$. We give the transition matrix between the two bases for the corresponding coherent families.

In this paper we prove that isometries with respect to the Kobayashi metric between certain domains having boundary points at which the boundary is infinitely flat extend continuously to the boundary. The strategy is to reestablish the Gehring-Hayman-type Theorem for these complex domains. Furthermore, the regularity of boundary extension map is given.

We completely characterize the boundedness of area operators from the Bergman spaces A^p_α(??_n) to the Lebesgue spaces L^q(??_n) for all $0＜p,q＜∞$. For the case $n=1$, some partial results were previously obtained by Wu in [Wu, Z.: Volterra operator, area integral and Carleson measures, Sci. China Math., 54, 2487–2500 (2011)]. Especially, in the case $q＜p$ and $q＜s$, we obtain some characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to $n$-complex dimension.

Following Shokurov’s idea, we give a simple proof of the ACC conjecture for minimal log discrepancies for surfaces.

Let $\mathfrak{g}$ be a classical complex simple Lie algebra and $\mathfrak{q}$ be a parabolic subalgebra. Let $M$ be a generalized Verma module induced from a one dimensional representation of $\mathfrak{q}$. Such $M$ is called a scalar generalized Verma module. In this paper, we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand-Kirillov dimension of the corresponding highest weight modules.

In this paper we consider the theta correspondence over a non-Archimedean local field. Using the homological method and the theory of derivatives, we show that under a mild condition the big theta lift is irreducible.

In this paper, we construct a superfermionic representation as well as a vertex representation for twisted general linear affine Lie superalgebras. We also establish a module isomorphism between them, which generalizes the super boson-fermion correspondence of type $B$ given by Kac-van de Leur. Based on this isomorphism, we determine explicitly the irreducible components of these two representations. Particularly, we obtain in this way two kinds of systematic construction of level $1$ irreducible integrable highest weight modules for twisted general linear affine Lie superalgebras.

In this article, we study deformations of conjugate self-dual Galois representations. The study is twofold. First, we prove an R=T type theorem for a conjugate self-dual Galois representation with coefficients in a finite field, satisfying a certain property called rigid. Second, we study the rigidity property for the family of residue Galois representations attached to a symmetric power of an elliptic curve, as well as to a regular algebraic conjugate self-dual cuspidal representation.

In the spirit of Morse homology initiated by Witten and Floer, we construct two $\infty$-categories $\mathcal{A}$ and $\mathcal{B}$. The weak one $\mathcal{A}$ comes out of the Morse-Smale pairs and their higher homotopies, and the strict one $\mathcal{B}$ concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we build up a weak $\infty$-functor $\mathcal{F}: \mathcal{A}\rightarrow \mathcal{B}$. Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory.

In a previous paper, the author and his collaborator studied the problem of lifting Hamiltonian group actions on symplectic varieties and Lagrangian subvarieties to their graded deformation quantizations and apply the general results to coadjoint orbit method for semisimple Lie groups. Only even quantizations were considered there. In this paper, these results are generalized to the case of general quantizations with arbitrary periods. The key step is to introduce an enhanced version of the (truncated) period map defined by Bezrukavnikov and Kaledin for quantizations of any smooth symplectic variety $X$, with values in the space of Picard Lie algebroid over $X$. As an application, we study quantizations of nilpotent orbits of real semisimple groups satisfying certain codimension condition.

We study the topology of closed, simply-connected, $6$-dimensional Riemannian manifolds of positive sectional curvature which admit isometric actions by ${\rm SU}(2)$ or ${\rm SO}(3)$. We show that their Euler characteristic agrees with that of the known examples, i.e., $S^6$, $\mathbb{C P}^3$, the Wallach space ${\rm SU}(3)/T^2$ and the biquotient ${\rm SU}(3)//T^2$. We also classify, up to equivariant diffeomorphism, certain actions without exceptional orbits and show that there are strong restrictions on the exceptional strata.

In this article, we construct free centroid hom-associative algebras and free centroid hom-Lie algebras. We also construct some other relatively free centroid hom-associative algebras by applying the Gröbner–Shirshov basis theory for (unital) centroid hom-associative algebras. Finally, we prove that the "Poincaré-Birkhoff-Witt theorem" holds for certain type of centroid hom-Lie algebras over a field of characteristic 0, namely, every centroid hom-Lie algebra such that the eigenvectors of the map $\beta$ linearly generates the whole algebra can be embedded into its universal enveloping centroid hom-associative algebra, and the linear basis of the universal enveloping algebra does not depend on the multiplication table of the centroid hom-Lie algebra under consideration.

Let $G$ be a special linear group over the real, the complex or the quaternion, or a special unitary group. In this note, we determine all special unipotent representations of $G$ in the sense of Arthur and Barbasch-Vogan, and show in particular that all of them are unitarizable.

We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine $\mathfrak{sl}_n$ and $\mathfrak{gl}_n$. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the embedding on the idempotent version with that on the quantum affine Schur algebra level. A ${\hat{\mathfrak{gl}}}_n$-variant of the embedding is also established.

Using methods from complex analysis in one variable, we define an integral operator that solves $\bar\partial$ with supnorm estimates on product domains in $\mathbb{C}^n$.

In this paper, the entropy of discrete Heisenberg group actions is considered. Let $\alpha$ be a discrete Heisenberg group action on a compact metric space $X$. Two types of entropies, $\widetilde{h}(\alpha)$ and $h(\alpha)$ are introduced, in which $\widetilde{h}(\alpha)$ is defined in Ruelle's way and $h(\alpha)$ is defined via the natural extension of $\alpha$. It is shown that when $X$ is the torus and $\alpha$ is induced by integer matrices then $\widetilde{h}(\alpha)$ is zero and $h(\alpha)$ can be expressed via the eigenvalues of the matrices.

In this paper, the dynamics (including shadowing property, expansiveness, topological stability and entropy) of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view. It is shown that (1) if $f$ is a hyperbolic endomorphism then for each $\varepsilon>0$ there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that the induced set-valued map $F_{f,\mathcal{U}}$ has the $\varepsilon$-shadowing property, and moreover, if $f$ is an expanding endomorphism then there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that the induced set-valued map $F_{f,\mathcal{U}}$ has the Lipschitz shadowing property; (2) when a set-valued map $F$ is generated by finite expanding endomorphisms, it has the shadowing property, and moreover, if the collection of the generators has no coincidence point then $F$ is expansive and hence is topologically stable; (3) if $f$ is an expanding endomorphism then for each $\varepsilon>0$ there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that $h(F_{f,\mathcal{U}}, \varepsilon)=h(f)$; (4) when $F$ is generated by finite expanding endomorphisms with no coincidence point, the entropy formula of $F$ is given. Furthermore, the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.

We introduce notions of continuous orbit equivalence and its one-sided version for countable left Ore semigroup actions on compact spaces by surjective local homeomorphisms, and characterize them in terms of the corresponding transformation groupoids and their operator algebras. In particular, we show that two essentially free semigroup actions on totally disconnected compact spaces are continuously orbit equivalent if and only if there is a canonical abelian subalgebra preserving $C^*$-isomorphism between the associated transformation groupoid $C^*$-algebras. We also give some examples of orbit equivalence, consider the special case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of the associated group actions.

In this paper, we study the projectively Ricci-flat general $(\alpha, \beta)$-metrics within to a spray framework and also bring out the rich variety of behaviour displayed by an important projective invariant. Projective Ricci curvature is one of the essential projective invariant in Finsler geometry which has been introduced by Z. Shen. The projective Ricci curvature is defined as Ricci curvature of a projective spray associated with a given spray $G$ on $M^{n}$ with a volume form $dV$ on $M^{n}$.

We prove a converse theorem for split even special orthogonal groups over finite fields. This is the only case left on converse theorems of classical groups and the difficulty is the existence of the outer automorphism. In this paper, we develop new ideas and overcome this difficulty.

Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory. In the present paper, we investigate the relationship between fractional calculus and fractal functions, based only on fractal dimension considerations. Fractal dimension of the Riemann-Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves. Meanwhile fractal dimension of the Riemann-Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist. After further discussion, fractal dimension of the Riemann-Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann-Liouville fractional differential is at most linearly increasing for the Hölder continuous functions. Investigation about other fractional calculus, such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary. This work is helpful to reveal the mechanism of fractional calculus on continuous functions. At the same time, it provides some theoretical basis for the rationality of the definition of fractional calculus. This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.

In this paper, we propose a class of robust independence tests for two random vectors based on weighted integrals of empirical characteristic functions. By letting weight functions be probability density functions of a class of special distributions, the proposed test statistics have simple closed forms and do not require moment conditions on the random vectors. Moreover, we derive the asymptotic distributions of the test statistics under the null hypothesis. The proposed testing method is computationally feasible and easy to implement. Based on a data-driven bandwidth selection method, Monte Carlo simulation studies indicate that our tests have a relatively good performance compared with the competitors. A real data example is also presented to illustrate the application of our tests.

In this work, we give some criteria of the weakly compact sets and a representation theorem of Riesz’s type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function. Finally, as an immediate consequence of the criteria considered in this paper, the criteria of the weakly compact sets of Orlicz sequence spaces are deduced.

The penalized variable selection methods are often used to select the relevant covariates and estimate the unknown regression coefficients simultaneously, but these existing methods may fail to be consistent for the setting with highly correlated covariates. In this paper, the semi-standard partial covariance (SPAC) method with Lasso penalty is proposed to study the generalized linear model with highly correlated covariates, and the consistencies of the estimation and variable selection are shown in high-dimensional settings under some regularity conditions. Some simulation studies and an analysis of colon tumor dataset are carried out to show that the proposed method performs better in addressing highly correlated problem than the traditional penalized variable selection methods.

Given a simple graph $G$ and a proper total-$k$-coloring $\phi$ from $V(G)\cup E(G)$ to $\{1,2,\ldots,k\}$. Let $f(v)=\phi(v)\prod_{uv\in E(G)}\phi(uv)$. The coloring $\phi$ is neighbor product distinguishing if $f(u)\neq f(v)$ for each edge $uv\in E(G)$. The neighbor product distinguishing total chromatic number of $G$, denoted by $\chi_{\Pi}^{\prime\prime}(G)$, is the smallest integer $k$ such that $G$ admits a $k$-neighbor product distinguishing total coloring. Li et al. conjectured that $\chi_{\Pi}^{\prime\prime}(G)\leq \Delta(G)+3$ for any graph with at least two vertices. Dong et al. showed that conjecture holds for planar graphs with maximum degree at least 10. By using the famous Combinatorial Nullstellensatz, we prove that if $G$ is a planar graph without 5-cycles, then $\chi_{\Pi}^{\prime\prime}(G)\leq$ max$\{\Delta(G)+2,12\}$.

In this paper, we prove that for each positive $k\equiv 1$ mod $m$ there exists a $P$-symmetric $km\tau$-periodic solution $x_k$ for asymptotically linear $m\tau$-periodic Hamiltonian systems, which are nonautonomous and endowed with a $P$-symmetry. If the $P$-symmetric Hamiltonian function is semi-positive, one can prove, under a new iteration inequality of the Maslov-type $P$-index, that $x_{k_1}$ and $x_{k_2}$ are geometrically distinct for $k_1/k_2\geq(2n+1)m+1$; and $x_{k_1},x_{k_2}$ are geometrically distinct for $k_1/k_2\geq m+1$ provided $x_{k_1}$ is non-degenerate.

The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup, which is defined by the backward differential equation. We provide a proof of the assertion of Rhyzhov and Skorokhod (Theory Probab. Appl., 1970) on the uniqueness of the solutions to the equation, which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.

We extend the $\imath$Hall algebra realization of $\imath$quantum groups arising from quantum symmetric pairs, which establishes an injective homomorphism from the universal $\imath$quantum group of Kac-Moody type to the $\imath$Hall algebra associated to an arbitrary $\imath$quiver (not necessarily virtually acyclic). This generalizes Lu-Wang's result.