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.

In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on $\mathbb R^{\infty}$ under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given.

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.

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.

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 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.

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.

$\mathfrak{L}_{\nu}$ 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.

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).

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 $\vec{b} =(b_{1},b_{2},\dots,b_{m})$ be a collection of locally integrable functions and $T_{_{\Sigma \vec{b}}}$ the commutator of multilinear singular integral operator $T$. Denote by $\mathbb{L}(\delta)$ and $\mathbb{L}(\delta(\cdot)) $ the Lipschitz spaces and the variable Lipschitz spaces, respectively. The main purpose of this paper is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of multilinear commutator $T_{_{\Sigma \vec{b}}}$ in the context of the variable exponent Lebesgue spaces, that is, the authors give the necessary and sufficient conditions for $b_{j}$ $(j=1,2,\dots,m)$ to be $\mathbb{L}(\delta)$ or $\mathbb{L}(\delta(\cdot)) $ via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. The authors do so by applying the Fourier series technique and some pointwise estimate for the commutators. The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator.

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.

Let $F: \mathbb R^{n}\rightarrow [0,+\infty) $ be a convex function of class $C^{2}(\mathbb R^{n}→\{0\})$ which is even and positively homogeneous of degree 1, and its polar $F^{0}$ represents a Finsler metric on $\mathbb R^{n}$. The anisotropic Sobolev norm in $W^{1,n} (\mathbb{R}^{n} )$ is defined by \begin{equation*} \|u\|_{F}=\bigg(\int_{\mathbb R^{n}}(F^{n}(\nabla u)+|u|^{n})dx\bigg)^{1/n}. \end{equation*} In this paper, the following sharp anisotropic Moser–Trudinger inequality involving $L^{n}$ norm \[ \underset{u\in W^{1,n}( \mathbb{R}^{n}), \Vert u \Vert _{F}\leq 1}{\sup}\int_{ \mathbb{R} ^{n}}\Phi ( \lambda_{n} \vert u \vert ^{\frac{n}{n-1}} ( 1+\alpha \Vert u \Vert _{n}^{n} ) ^{\frac{1}{n-1}} ) dx<+\infty \] in the entire space $\mathbb{R}^n$ for any $0<\alpha<1$ is established, where $\Phi ( t ) ={\rm e}^{t}-\sum_{j=0}^{n-2} \frac{t^{j}}{j!}$, $\lambda_{n}=n^{\frac{n}{n-1}}\kappa_{n}^{\frac{1}{n-1}}$ and $\kappa_{n}$ is the volume of the unit Wulff ball in $\mathbb{R}^n$. It is also shown that the above supremum is infinity for all $\alpha\geq1$. Moreover, we prove the supremum is attained, that is, there exists a maximizer for the above supremum when $\alpha>0$ is sufficiently small.

Using Hartogs' fundamental theorem for analytic functions in several complex variables, we establish a multiple $q$-exponential differential operational identity for the analytic functions in several variables, which can be regarded as a multiple $q$-translation formula. This multiple $q$-translation formula is a fundamental result and play a pivotal role in $q$-mathematics. Using this $q$-translation formula, we can easily recover many classical conclusions in $q$-mathematics and derive some new $q$-formulas. Our work reveals some profound connections between the theory of complex functions in several variables and $q$-mathematics.

We explain how to construct a quantum deformation of a spectral curve associated to a tau-function of the KP hierarchy. This construction is applied to Witten–Kontsevich tau-function to give a natural explanation of some earlier work. We also apply it to higher Weil–Petersson volumes and Witten’s r-spin intersection numbers.

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.

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.

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

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 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.

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 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.

A multiplicative function $f$ is said to be resembling the Möbius function if $f$ is supported on the square-free integers, and $f(p)=\pm 1$ for each prime $p$. We prove $O$- and Ω-results for the summatory function $\sum_{n\leq x} f(n)$ for a class of these $f$, and the point is that these $O$-results demonstrate cancellations better than the square-root saving. It is proved in particular that the summatory function is $O(x^{1/3+ε})$ under the Riemann Hypothesis. On the other hand it is proved to be $\Omega(x^{1/4})$ unconditionally. It is interesting to compare these with the corresponding results for the Möbius function.

Dubrovin establishes a certain relationship between the GUE partition function and the partition function of Gromov–Witten invariants of the complex projective line. In this paper, we give a direct proof of Dubrovin’s result. We also present in a diagram the recent progress on topological gravity and matrix gravity.

We introduce a method to find differential equations for functions which define tables, such that associated billiard systems admit a local first integral. We illustrate this method in three situations: the case of (locally) integrable wire billiards, for finding surfaces in ${\mathbb R}^3$ with a first integral of degree one in velocities, and for finding a piece-wise smooth surface in ${\mathbb R}^3$ homeomorphic to a torus, being a table of a billiard admitting two additional first integrals.

This review is devoted to one of the most interesting and actively developing fields in condensed matter physics—theory of topological insulators. Apart from its importance for theoretical physics, this theory enjoys numerous connections with modern mathematics, in particular, with topology and homotopy theory, Clifford algebras, K-theory and non-commutative geometry. From the physical point of view topological invariance is equivalent to adiabatic stability. Topological insulators are characterized by the broad energy gap, stable under small deformations, which motivates application of topological methods. A key role in the study of topological objects in the solid state physics is played by their symmetry groups. There are three main types of symmetries—time reversion symmetry, preservation of the number of particles (charge symmetry) and PH-symmetry (particle-hole symmetry). Based on the study of symmetry groups and representation theory of Clifford algebras Kitaev proposed a classification of topological objects in solid state physics. In this review we pay special attention to the topological insulators invariant under time reversion.

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 construct the quantum curve for the Baker-Akhiezer function of the orbifold Gromov-Witten theory of the weighted projective line $\mathbb P[r]$. Furthermore, we deduce the explicit bilinear Fermionic formula for the (stationary) Gromov-Witten potential via the lifting operator contructed from the Baker-Akhiezer function.

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, 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.

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.

We propose a conjecture relevant to Galkin’s lower bound conjecture, and verify it for the blow-ups of a four-dimensional quadric at a point or along a projective plane. We also show that Conjecture $\mathcal{O}$ holds in these two cases.

We introduce the notion of ungraded matrix factorization as a mirror of non-orientable Lagrangian submanifolds. An ungraded matrix factorization of a polynomial $W$, with coefficients in a field of characteristic 2, is a square matrix $Q$ of polynomial entries satisfying $Q^2 = W \cdot \mathrm{Id}$. We then show that non-orientable Lagrangians correspond to ungraded matrix factorizations under the localized mirror functor and illustrate this construction in a few instances. Our main example is the Lagrangian submanifold $\mathbb{R}P^2 \subset \mathbb{C}P^2$ and its mirror ungraded matrix factorization, which we construct and study. In particular, we prove a version of Homological Mirror Symmetry in this setting.

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 $M^{n}$.

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.