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.

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 $\vec{b} =(b_{1},b_{2},\dots,b_{m})$ 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.

We prove the asymptotic properties of the solutions to the 3D Navier–Stokes system with singular external force, by making use of Fourier localization method, the Littlewood–Paley theory and some subtle estimates in Fourier–Herz space. The main idea of the proof is motivated by that of Cannone et al. [J. Differential Equations, 314, 316–339 (2022)]. We deal either with the nonstationary problem or with the stationary problem where solution may be singular due to singular external force. In this paper, the Fourier–Herz space includes the function space of pseudomeasure type used in Cannone et al. [J. Differential Equations, 314, 316–339 (2022)]

In this paper, we focus on $p$-sober spaces and prove that (1) the $T_{0}$ space $X$ is $p$-sober if and only if the Smyth power space of $X$ is $p$-sober; (2) the space $X$ has a $p$-sober dcpo model if and only if $X$ is $T_{1}$ and $p$-sober; (3) every non-$p$-sober $T_{0}$ space does not have a $p$-sobrification; (4) the $T_{0}$ space $X$ is sober if and only if $X$ is $p$-sober and $PD$.

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 give a complete real-variable theory of local variable Hardy spaces. First, we present various real-variable characterization in terms of several local maximal functions. Next, the new atomic and the finite atomic decomposition for the local variable Hardy spaces are established. As an application, we also introduce the local variable Campanato space which is showed to be the dual space of the local variable Hardy spaces. Analogous to the homogeneous case, some equivalent definitions of the dual of local variable Hardy spaces are also considered. Finally, we show the boundedness of inhomogeneous Calderón–Zygmund singular integrals and local fractional integrals on local variable Hardy spaces and their duals.

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.

We show that the following properties of the C^*-algebras in a class P are inherited by simple unital C^*-algebras in the class of asymptotically tracially in P: (1) n-comparison, (2) α-comparison (1 ≤ α < ∞).

Multiple change-points estimation for functional time series is studied in this paper. The change-point problem is first transformed into a high-dimensional sparse estimation problem via basis functions. Group least absolute shrinkage and selection operator (LASSO) is then applied to estimate the number and the locations of possible change points. However, the group LASSO (GLASSO) always overestimate the true points. To circumvent this problem, a further Information Criterion (IC) is applied to eliminate the redundant estimated points. It is shown that the proposed two-step procedure estimates the number and the locations of the change-points consistently. Simulations and two temperature data examples are also provided to illustrate the finite sample performance of the proposed method.

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

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.

In this paper, we develop a flexible semiparametric model averaging marginal regression procedure to forecast the joint conditional quantile function of the response variable for ultrahighdimensional data. First, we approximate the joint conditional quantile function by a weighted average of one-dimensional marginal conditional quantile functions that have varying coefficient structures. Then, a local linear regression technique is employed to derive the consistent estimates of marginal conditional quantile functions. Second, based on estimated marginal conditional quantile functions, we estimate and select the significant model weights involved in the approximation by a nonconvex penalized quantile regression. Under some relaxed conditions, we establish the asymptotic properties for the nonparametric kernel estimators and oracle estimators of the model averaging weights. We further derive the oracle property for the proposed nonconvex penalized model averaging procedure. Finally, simulation studies and a real data analysis are conducted to illustrate the merits of our proposed model averaging method.

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 hypergraph $\mathcal{H}$ is an {\em $(n,m)$-hypergraph} if it contains $n$ vertices and $m$ hyperedges, where $n\geq 1$ and $m\geq 0$ are two integers. Let $k$ be a positive integer and let $L$ be a set of nonnegative integers. A hypergraph $\mathcal{H}$ is {\em $k$-uniform} if all its hyperedges have the same size $k$, and $\mathcal{H}$ is {\em $L$-intersecting} if the number of common vertices of every two hyperedges belongs to $L$. In this paper, we propose and investigate the problem of estimating the maximum $k$ among all $k$-uniform $L$-intersecting $(n,m)$-hypergraphs for fixed $n,m$ and $L$. We will provide some tight upper and lower bounds on $k$ in terms of $n,m$ and $L$.

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 $\lambda=(\lambda_1, \ldots,\lambda_n)$ be $\beta$-Jacobi ensembles with parameters $p_1, p_2, n$ and $\beta$ while $\beta$ varying with $n.$ Set $\gamma=\lim_{n\to\infty}\frac{n}{p_1}$ and $\sigma=\lim_{n\to\infty}\frac{p_1}{p_2}.$ In this paper, supposing $\lim_{n\to\infty}\frac{\log n}{\beta n}=0,$ we prove that the empirical measures of different scaled $\lambda$ converge weakly to a Wachter distribution, a Marchenko—Pastur law and a semicircle law corresponding to $\sigma\gamma>0, \sigma=0$ or $\gamma=0,$ respectively. We also offer a full large deviation principle with speed $\beta n^2$ and a good rate function to precise the speed of these convergences. As an application, the strong law of large numbers for the extremal eigenvalues of $\beta$-Jacobi ensembles is obtained.

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.

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.

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.

Let $f_{1}, \ldots, f_{k}$ and $g_{1}, \ldots, g_{k}$ be non-CM newforms of square-free levels. Denote by $\lambda_{\operatorname{sym}^{j}f_{i}}(n)$ the coefficients of the Dirichlet expansion of $L(\operatorname{sym}^{j} f_{i},s)$ and $\nu_{1}, \ldots, \nu_{k}$ the distinct positive integers such that $ \lambda_{\operatorname{sym}^{j}f_{i}}(\nu_{i}) \neq 0$. In this paper, we obtain that there exist infinitely many positive integers $m$ such that $ 0<|\lambda_{\operatorname{sym}^{j}f_{1}}(m+\nu_{1})|<|\lambda_{\operatorname{sym}^{j}f_{2}}(m+\nu_{2})|<\cdots<|\lambda_{\operatorname{sym}^{j}f_{k}}(m+\nu_{k})|. $ For coefficients of the Dirichlet expansion of $L(\operatorname{sym}^{j_{1}} f \times\operatorname{sym}^{j_{2}} g, s)$, we have a similar result.

Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The forcing number $f(G,M)$ of a perfect matching $M$ of $G$ is the smallest cardinality of a subset of $M$ that is contained in no other perfect matching of $G$. Among all perfect matchings $M$ of $G$, the minimum and maximum values of $f(G,M)$ are called the minimum and maximum forcing numbers of $G$, denoted by $f(G)$ and $F(G)$, respectively. Then $f(G)\leq F(G)\leq n-1$. Che and Chen (2011) proposed an open problem: how to characterize the graphs $G$ with $f(G)=n-1$. Later they showed that for a bipartite graph $G$, $f(G)=n-1$ if and only if $G$ is complete bipartite graph $K_{n,n}$. In this paper, we completely solve the problem of Che and Chen, and show that $f(G)=n-1$ if and only if $G$ is a complete multipartite graph or a graph obtained from complete bipartite graph $K_{n,n}$ by adding arbitrary edges in one partite set. For all graphs $G$ with $F(G)=n-1$, we prove that the forcing spectrum of each such graph $G$ forms an integer interval by matching 2-switches and the minimum forcing numbers of all such graphs $G$ form an integer interval from $\lfloor\frac{n}{2}\rfloor$ to $n-1$.

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.

In this paper, we focus on the partially linear varying-coefficient quantile regression with missing observations under ultra-high dimension, where the missing observations include either responses or covariates or the responses and part of the covariates are missing at random, and the ultra-high dimension implies that the dimension of parameter is much larger than sample size. Based on the B-spline method for the varying coefficient functions, we study the consistency of the oracle estimator which is obtained only using active covariates whose coefficients are nonzero. At the same time, we discuss the asymptotic normality of the oracle estimator for the linear parameter. Note that the active covariates are unknown in practice, non-convex penalized estimator is investigated for simultaneous variable selection and estimation, whose oracle property is also established. Finite sample behavior of the proposed methods is investigated via simulations and real data analysis.

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 consider the 3-D Cauchy problem for the diffusion approximation model in radiation hydrodynamics. The existence and uniqueness of global solutions is proved in perturbation framework, for more general gases including ideal polytropic gas. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density, velocity, temperature, and radiation field.

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.

We consider a McKean Vlasov backward stochastic differential equation (MVBSDE) of the form $$ Y_t=-F(t,Y_t,Z_t,[Y_t])\,dt+Z_t \,dB_t, \quad Y_T=\xi,$$ where $[Y_t]$ stands for the law of $Y_t$. We show that if $F$ is locally monotone in $y$, locally Lipschitz with respect to $z$ and law's variable, and the monotonicity and Lipschitz constants $\kappa_N, L_N$ are such that $L^2_N+\kappa_N^+=\mathcal{O}(\log(N))$, then the {\rm MVBSDE} has a unique stable solution.

In this paper we consider the Monge-Ampère type equations on compact almost Hermitian manifolds. We derive C^∞ a priori estimates under the existence of an admissible C-subsolution. Finally, we obtain an existence result if there exists an admissible supersolution.

Let $\mathcal{F}=\{H_1, \ldots, H_k\}$ ($k\ge 1$) be a family of graphs. The Turán number of the family $\mathcal{F}$ is the maximum number of edges in an $n$-vertex $\{H_1, \ldots, H_k\}$-free graph, denoted by ex$(n, \mathcal{F})$ or ex$(n, \{H_1,H_2,\ldots,H_k\})$. The blow-up of a graph $H$ is the graph obtained from $H$ by replacing each edge in $H$ by a clique of the same size where the new vertices of the cliques are all different. In this paper we determine the Turán number of the family consisting of a blow-up of a cycle and a blow-up of a star in terms of the Turán number of the family consisting of a cycle, a star and linear forests with $k$ edges.

We prove the termination of flips for pseudo-effective NQC log canonical generalized pairs of dimension 4. As main ingredients, we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs, and show that the termination of flips for pseudo-effective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions.

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.

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 $f$ be a fixed Maass form for ${\rm SL}_3(\mathbb{Z})$ with Fourier coefficients $A_f(m, n)$. Let $g$ be a Maass cusp form for ${\rm SL}_2(\mathbb{Z})$ with Laplace eigenvalue $\frac14+k^2$ and Fourier coefficient $\lambda_g(n)$, or a holomorphic cusp form of even weight $k$. Denote by $S_X(f\times g, \alpha, \beta)$ a smoothly weighted sum of $A_f(1, n)\lambda_g(n)e(\alpha n^\beta)$ for X< n < 2X, where $\alpha\neq0$ and $\beta>0$ are fixed real numbers. The subject matter of the present paper is to prove non-trivial bounds for a sum of $S_X(f\times g, \alpha, \beta)$ over $g$ as $k$ tends to $\infty$ with $X$. These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec, Luo, and Sarnak.

In order to study the boundedness of some operators in general function spaces which include Lorentz spaces and Orlicz spaces as special examples, Lorentz introduced a new space called rearrangement invariant Banach function spaces, denoted by RIBFS. It is shown in this paper that variation operators of singular integrals and their commutators are bounded on RIBFS whenever the kernels satisfy the L^r-Hörmander conditions. Moreover, we obtain some quantitative weighted bounds in the quasi-Banach spaces and modular inequalities for above variation operators and their commutators.

In this paper we focus on the boundary regularity for a class of $k$-Hessian equations which can be degenerate and (or) singular on the boundary of the domain. Motivated by the case of Monge–Ampère equations, we first construct sub-solutions, then apply the characteristic of the global Hölder continuity for convex functions, and finally use the maximum principle to obtain the boundary Hölder continuity for the solutions of the $k$-Hessian equations. However, finding such sub-solutions is very difficult due to the complexity of the $k$-Hessian operator. In particular, we employ the symmetric mean to overcome the difficulties.

This paper is devoted to the conditions of the existence of CC-center for the generalized Abel equations. Using some new original methods, we obtain extended results of the main theorems in the paper by Llibre and Valls (2020) and the one by Zhou (2020), respectively. The proofs in this paper are much simpler than the previous ones.

We provide an analytical construction of the gluing map for stable affine vortices over the upper half plane with the Lagrangian boundary condition. This result is a necessary ingredient in studies of the relation between gauged sigma model and nonlinear sigma model, such as the closed or open quantum Kirwan map.