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.

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.

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

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.

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.

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.

Let $q\in(0,\infty]$ and $\varphi$ be a Musielak-Orlicz function with uniformly lower type $p_{\varphi}^-\in(0,\infty)$ and uniformly upper type $p_{\varphi}^+\in(0,\infty)$. In this article, the authors establish various real-variable characterizations of the Musielak-Orlicz-Lorentz Hardy space $H^{\varphi,q}(\mathbb{R}^n)$, respectively, in terms of various maximal functions, finite atoms, and various Littlewood-Paley functions. As applications, the authors obtain the dual space of $H^{\varphi,q}(\mathbb{R}^n)$ and the summability of Fourier transforms from $H^{\varphi,q}(\mathbb{R}^n)$ to the Musielak-Orlicz-Lorentz space $L^{\varphi,q}(\mathbb{R}^n)$ when $q\in(0,\infty)$ or from the Musielak-Orlicz Hardy space $H^{{\varphi}}({\mathbb{R}^n})$ to $L^{\varphi,\infty}(\mathbb{R}^n)$ in the critical case. These results are new when $q\in(0,\infty)$ and also essentially improve the existing corresponding results (if any) in the case $q=\infty$ via removing the original assumption that $\varphi$ is concave. To overcome the essential obstacles caused by both that $\varphi$ may not be concave and that the boundedness of the powered Hardy-Littlewood maximal operator on associated spaces of Musielak-Orlicz spaces is still unknown, the authors make full use of the obtained atomic characterization of $H^{\varphi,q}(\mathbb{R}^n)$, the corresponding results related to weighted Lebesgue spaces, and the subtle relation between Musielak-Orlicz spaces and weighted Lebesgue spaces.

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 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.

We establish the Strassen's law of the iterated logarithm (LIL for short) for independent and identically distributed random variables with $\hat{\mathbb{E}}\left[X_1\right]=\hat{\mathcal{E}}\left[X_1\right]=0$ and $C_{\mathrm{V}}\left[X_1^2\right]<\infty$ under a sub-linear expectation space with a countably sub-additive capacity V. We also show the LIL for upper capacity with $\sigma$=$\bar{\sigma}$ under some certain conditions.

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 this paper, we establish a weighted maximal $L_2$ estimate of operator-valued Bochner-Riesz means. The proof is based on noncommutative square function estimates and a sharp weighted noncommutative Hardy-Littlewood maximal inequality.

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 paper, we derive the optimal Cauchy-Schwarz inequalities on a class of Hilbert and Krein modules over a Clifford algebra, which heavily depend on the Clifford algebraic structure. The obtained inequalities further lead to very general uncertainty inequalities on these modules. Some new phenomena arise, due to the non-commutative nature, the Clifford-valued inner products and the Krein geometry. Taking into account applications, special attention is given to the Dirac operator and the Howe dual pair ${\rm Pin}(m)\times \mathfrak{osp}(1|2)$. Moreover, it is surprisingly to find that the recent highly non-trivial uncertainty relation for triple observables is indeed a direct consequence of our Cauchy-Schwarz inequality. This new observation leads to refined uncertainty relations in terms of the Wigner-Yanase-Dyson skew information for mixed states and other generalizations. These show that the obtained uncertainty inequalities on Clifford modules can be considered as new uncertainty relations for multiple observables.

In this paper, we study the multiplicity and concentration of positive solutions for a Schrödinger-Poisson system involving sign-changing potential and the nonlinearity $K(x)|u|^{p-2}u$ $(2 < p < 4)$ in $\mathbb{R}^3$. Such a problem cannot be studied by variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold since its (PS) sequence may not be bounded. By some new analytic techniques and the Ljusternik-Schnirelmann category theory, we relate the concentration and the number of positive solutions to the category of the global minima set of a suitable ground energy function. Furthermore, we investigate the asymptotic behavior of the solutions. In particular, we do not use Pohozaev equality in this work.

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.

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 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.

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

The classical Mumford stability condition of vector bundles on a complex elliptic curve $X$, can be viewed as a Bridgeland stability condition on $D^b({\rm Coh}\,X)$, the bounded derived category of coherent sheaves on $X$. This point of view gives us infinitely many $t$-structures and hearts on $D^b({\rm Coh}\, X)$. In this paper, we answer the question which of these hearts are Noetherian or Artinian.

Let Ω be a domain of $(\mathbb{R}^n)$ with n ≥ 2 and p(·) be a local Lipschitz funcion in Ω with 1 < p(x) < ∞ in Ω. We build up an interior quantitative second order Sobolev regularity for the normalized p(·)-Laplace equation -Δ_p(·)^Nu = 0 in Ω as well as the corresponding inhomogeneous equation -Δ_p(·)^Nu=f in Ω with f ∈ C⁰(Ω). In particular, given any viscosity solution u to -Δ_p(·)^Nu= 0 in Ω, we prove the following:
(i) in dimension $n=2$, for any subdomain $U \Subset \Omega$ and any $\beta \geq 0$, one has $|D u|^\beta D u \in L_{\text {loc }}^{2+\delta}(U)$ with a quantitative upper bound, and moreover, the map $\left(x_1, x_2\right) \rightarrow|D u|^\beta\left(u_{x_1},-u_{x_2}\right)$ is quasiregular in $U$ in the sense that
$\left|D\left[|D u|^\beta D u\right]\right|^2 \leq-C \operatorname{det} D\left[|D u|^\beta D u\right] \quad$ a.e. in $U$.
(ii) in dimension $n \geq 3$, for any subdomain $U \Subset \Omega$ with $\inf _U p(x)>1$ and $\sup _U p(x)<3+\frac{2}{n-2}$, one has $D^2 u \in L_{\text {loc }}^{2+\delta}(U)$ with a quantitative upper bound, and also with a pointwise upper bound
$\left|D^2 u\right|^2 \leq-C$ $\sum\limits_{1 \le i < j \le n} {} $ $\left[u_{x_i x_j} u_{x_j x_i}-u_{x_i x_i} u_{x_j x_j}\right]$ a.e. in $U$.
Here constants $\delta>0$ and $C \geq 1$ are independent of $u$. These extend the related results obtaind by Adamowicz-Hästö [Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, 10, 1940-1965 (2010)] when $n=2$ and $\beta=0$.

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$\{\Delta(G)+2,12\}$.

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.

Ridge regression is an effective tool to handle multicollinearity in regressions. It is also an essential type of shrinkage and regularization methods and is widely used in big data and distributed data applications. The divide and conquer trick, which combines the estimator in each subset with equal weight, is commonly applied in distributed data. To overcome multicollinearity and improve estimation accuracy in the presence of distributed data, we propose a Mallows-type model averaging method for ridge regressions, which combines estimators from all subsets. Our method is proved to be asymptotically optimal allowing the number of subsets and the dimension of variables to be divergent. The consistency of the resultant weight estimators tending to the theoretically optimal weights is also derived. Furthermore, the asymptotic normality of the model averaging estimator is demonstrated. Our simulation study and real data analysis show that the proposed model averaging method often performs better than commonly used model selection and model averaging methods in distributed data cases.

The eccentricity matrix of a graph is obtained from the distance matrix by keeping the entries that are largest in their row or column, and replacing the remaining entries by zero. This matrix can be interpreted as an opposite to the adjacency matrix, which is on the contrary obtained from the distance matrix by keeping only the entries equal to 1. In the paper, we determine graphs having the second largest eigenvalue of eccentricity matrix less than $1$.

In this paper, we characterize weighted composition operators that preserve frames on the weighted Hardy spaces in the unit disk. In particular, we obtain the symbol properties of the bounded invertible weighted composition operators. Moreover, we establish the equivalence between bounded invertible operators and frame-preserving operators. Furthermore, we show that weighted composition operator preserves frames if and only if it preserves the Riesz bases property. Additionally, we investigate the weighted composition operators that preserve tight or normalized tight frames on the Dirichlet space.

Tree-based models have been widely applied in both academic and industrial settings due to the natural interpretability, good predictive accuracy, and high scalability. In this paper, we focus on improving the single-tree method and propose the segmented linear regression trees (SLRT) model that replaces the traditional constant leaf model with linear ones. From the parametric view, SLRT can be employed as a recursive change point detect procedure for segmented linear regression (SLR) models, which is much more efficient and flexible than the traditional grid search method. Along this way, we propose to use the conditional Kendall's $\tau$ correlation coefficient to select the underlying change points. From the non-parametric view, we propose an efficient greedy splitting method that selects the splits by analyzing the association between residuals and each candidate split variable. Further, with the SLRT as a single-tree predictor, we propose a linear random forest approach that aggregates the SLRTs by a weighted average. Both simulation and empirical studies showed significant improvements than the CART trees and even the random forest.

Measuring and testing tail dependence is important in finance, insurance, and risk management. This paper proposes two tail dependence matrices based on classic rank correlation coefficients, which possess the desired population properties and interpretability. Their nonparametric estimators with strong consistency and asymptotic distributions are derived using the limit theory of $U$-processes. The simulation and application studies show that, compared to the tail dependence matrix based on Spearman's $\rho$ with large deviation, the Kendall-based tail dependence measure has stable variances under different tail conditions; thus, it is an effective approach to testing and quantifying tail dependence between random variables.

Let $f$ be any arithmetic function and define $S_f(x):=\sum_{n\leq x}f([x/n])$. If the function $f$ is small, namely, $f(n)\ll n^\varepsilon,$ then the error term $E_f(x)$ in the asymptotic formula of $S_f(x)$ has the form $O(x^{1/2+\varepsilon}).$ In this paper, we shall study the mean square of $E_f(x)$ and establish some new results of $E_f(x)$ for some special functions.

This paper will deal with a nonlocal geometric flow in centro-equiaffine geometry, which keeps the enclosed area of the evolving curve and converges smoothly to an ellipse. This model can be viewed as the affine version of Gage's area-preserving flow in Euclidean geometry.

We give investigations on the approximation order of translation networks produced by the convolution translation operators defined on a Jacobi cone and the surface cone. We deal with the convolution translation from the view of Fourier analysis, express the translation operator with orthogonal basis and provide a sufficient condition to ensure the density for the translation networks. Based on these facts, we construct with the near best approximation operator and the Gauss integral formula two kinds of translation network operators and show their approximation orders in the best polynomial approximation.

Quantile regression is widely used in variable relationship research for statistical learning. Traditional quantile regression model is based on vector-valued covariates and can be efficiently estimated via traditional estimation methods. However, many modern applications involve tensor data with the intrinsic tensor structure. Traditional quantile regression can not deal with tensor regression issues well. To this end, we consider a tensor quantile regression with tensor-valued covariates and develop a novel variational Bayesian estimation approach to make estimation and prediction based on the asymmetric Laplace model and the CANDECOMP/PARAFAC decomposition of tensor coefficients. To incorporate the sparsity of tensor coefficients, we consider the multiway shrinkage priors for marginal factor vectors of tensor coefficients. The key idea of the proposed method is to efficiently combine the prior structural information of tensor and utilize the matricization of tensor decomposition to simplify the complexity of tensor coefficient estimation. The coordinate ascent algorithm is employed to optimize variational lower bound. Simulation studies and a real example show the numerical performances of the proposed method.

A graph G is edge-$k$-choosable if, for any assignment of lists $L(e)$ of at least $k$ colors to all edges $e\in E(G)$, there exists a proper edge coloring such that the color of $e$ belongs to $L(e)$ for all $e\in E(G)$. One of Vizing's classic conjectures asserts that every graph is edge-$(\Delta+1)$-choosable. It is known since 1999 that this conjecture is true for general graphs with $\Delta\leq4$. More recently, in 2015, Bonamy confirmed the conjecture for planar graph with $\Delta\geq8$, but the conjecture is still open for planar graphs with $5\leq\Delta\leq7$. We confirm the conjecture for planar graphs with $\Delta\ge 6$ in which every 7-cycle (if any) induces a $C_7$ (so, without chords), thereby extending a result due to Dong, Liu and Li.

Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in $\pi_1(M)$, where the integers $n, p\geq2$, $\Gamma$ is a finite abelian group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we prove that there are infinitely many non-contractible closed geodesics of class $[h]$ on the compact space form with $C^r$-generic Finsler metrics, where $4 \leq r \leq \infty$. The conclusion also holds for $C^r$-generic Riemannian metrics for $2 \leq r \leq \infty$. The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.