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, 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, 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 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 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 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, 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.

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 $\beta=0$.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

In the paper we generalize some classic results on limit cycles of Liénard system \[\dot x=\phi(y)-F(x), \quad \dot y=-g(x)\] having a unique equilibrium to that of the system with several equilibria. As applications, we strictly prove the number of limit cycles and obtain the distribution of limit cycles for three classes of Liénard systems, in which we correct a mistake in the literature.

Let $G$ be a finite group. We denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. In this paper, we characterize the structure of finite groups $G$ with lower bounds $\frac{1}{p}$, $\frac{p^2+8}{9p^2}$ and $\frac{p+3}{4p}$ on $\nu(G)$, where $p$ is a prime divisor of $|G|$.

In this paper, we classify the finite non-abelian $p$-groups all of whose non-abelian proper subgroups have centers of the same order.

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.

In this paper, we first give characterizations of weighted Besov spaces with variable exponents via Peetre's maximal functions. Then we obtain decomposition characterizations of these spaces by atom, molecule and wavelet. As an application, we obtain the boundedness of the pseudo-differential operators on these spaces.

Let $G$ be a graph and $d_i$ denote the degree of a vertex $v_i$ in $G$, and let $f(x,y)$ be a real symmetric function. Then one can get an edge-weighted graph in such a way that for each edge $v_iv_j$ of $G$, the weight of $v_iv_j$ is assigned by the value $f(d_i, d_j)$. Hence, we have a weighted adjacency matrix $\mathcal A_f(G)$ of $G$, in which the $ij$-entry is equal to $f(d_i,d_j)$ if $v_iv_j\in E(G)$ and $0$ otherwise. This paper attempts to unify the study of spectral properties for the weighted adjacency matrix $\mathcal A_f(G)$ of graphs with a degree-based edge-weight function $f(x,y)$. Some lower and upper bounds of the largest weighted adjacency eigenvalue $\lambda_1$ are given, and the corresponding extremal graphs are characterized. Bounds of the energy $\mathcal E_f(G)$ for the weighted adjacency matrix $\mathcal A_f(G)$ are also obtained. By virtue of the unified method, this makes many earlier results become special cases of our results.

After reviewing Grunsky operator and Faber operator acting on Dirichlet space, we discuss the boundedness of Faber operator on BMOA, a new subject which turns out to be closely related to the BMO theory of the universal Teichmüller space. In particular, we show that the Faber operator acts as a bounded operator on BMOA if the symbol conformal map stays nearly to the base point in the BMO-Teichmüller space. Meanwhile, we obtain several results on quasiconformal mappings, BMO-Teichmüller space and chord-arc curves as well. As by-products, this provides a complex analysis approach to the boundedness of the Cauchy integral acting on BMO functions on a chord-arc curve near to the unit circle in the BMO-Teichmüller space.

In this paper, we consider the reducibility of three-dimensional skew symmetric systems. We obtain a reducibility result if the base frequency is high-dimensional weak Liouvillean and the parameter is sufficiently small. The proof is based on a modified KAM theory for 3-dimensional skew symmetric systems.

A model space is a subspace of the Hardy space which is invariant under the backward shift, and a truncated Toeplitz operator is the compression of a Toeplitz operator on some model space. In this paper we prove a necessary and sufficient condition for the commutator of two truncated Toeplitz operators on a model space to be compact.

Let C be a triangulated category. We define $m$-term subcategories on C induced by $n$-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on $2$-term subcategories $\mathcal{G}$ and support $\tau$-tilting subcategories on an abelian quotient of $\mathcal{G}$. If an $m$-term subcategory is induced by a co-t-structure, then we have a one-to-one correspondence between cotorsion pairs on it and cotorsion pairs on C under certain conditions.

This paper concerns two-player zero-sum stochastic differential games with nonanticipative strategies against closed-loop controls in the case where the coefficients of mean-field stochastic differential equations and cost functional depend on the joint distribution of the state and the control. In our game, both the (lower and upper) value functions and the (lower and upper) second-order Bellman--Isaacs equations are defined on the Wasserstein space $\mathcal{P}_2(\mathbb{R}^n)$ which is an infinite dimensional space. The dynamic programming principle for the value functions is proved. If the (upper and lower) value functions are smooth enough, we show that they are the classical solutions to the second-order Bellman--Isaacs equations. On the other hand, the classical solutions to the (upper and lower) Bellman--Isaacs equations are unique and coincide with the (upper and lower) value functions. As an illustrative application, the linear quadratic case is considered. Under the Isaacs condition, the explicit expressions of optimal closed-loop controls for both players are given. Finally, we introduce the intrinsic notion of viscosity solution of our second-order Bellman--Isaacs equations, and characterize the (upper and lower) value functions as their viscosity solutions.

With the advent of modern devices, such as smartphones and wearable devices, high-dimensional data are collected on many participants for a period of time or even in perpetuity. For this type of data, dependencies between and within data batches exist because data are collected from the same individual over time. Under the framework of streamed data, individual historical data are not available due to the storage and computation burden. It is urgent to develop computationally efficient methods with statistical guarantees to analyze high-dimensional streamed data and make reliable inferences in practice. In addition, the homogeneity assumption on the model parameters may not be valid in practice over time. To address the above issues, in this paper, we develop a new renewable debiased-lasso inference method for high-dimensional streamed data allowing dependences between and within data batches to exist and model parameters to gradually change. We establish the large sample properties of the proposed estimators, including consistency and asymptotic normality. The numerical results, including simulations and real data analysis, show the superior performance of the proposed method.

To learn the subgroup structure generated by multidimensional interaction, we propose a novel multiview subgroup integration technique based on tensor decomposition. Compared to the traditional subgroup analysis that can only handle single-view heterogeneity, our proposed method achieves a greater level of homogeneity within the subgroups, leading to enhanced interpretability and predictive power. For computational readiness of the proposed method, we build an algorithm that incorporates pairwise shrinkage-encouraging penalties and ADMM techniques. Theoretically, we establish the asymptotic consistency and normality of the proposed estimators. Extensive simulation studies and real data analysis demonstrate that our proposal outperforms other methods in terms of prediction accuracy and grouping consistency. In addition, the analysis based on the proposed method indicates that intergenerational care significantly increases the risk of chronic diseases associated with diet and fatigue in all provinces while only reducing the risk of emotion-related chronic diseases in the eastern coastal and central regions of China.

Combining p-values is a well-known issue in statistical inference. When faced with a study involving $m$ p-values, determining how to effectively combine them to arrive at a comprehensive and reliable conclusion becomes a significant concern in various fields, including genetics, genomics, and economics, among others. The literature offers a range of combination strategies tailored to different research objectives and data characteristics. In this work, we aim to provide users with a systematic exploration of the p-value combination problem. We present theoretical results for combining p-values using a logarithmic transformation, which highlights the benefits of this approach. Additionally, we propose a combination strategy together with its statistical properties utilizing the gold section method, showcasing its performance through extensive computer simulations. To further illustrate its effectiveness, we apply this approach to a real-world scenario.

In this paper, we first prove that the retract of a consonant space (or co-consonant space) is consonant (co-consonant). Simultaneously, we consider the co-consonance of two powerspace constructions and proved that (1) the co-consonance of the Smyth powerspace $P_{S}(X)$ implies the co-consonance of $X$ if $X$ is strongly compact; (2) the co-consonance of $X$ implies the co-consonance of the Smyth powerspace under some conditions; (3) if the lower powerspace $P_{H}(X)$ is co-consonant, then $X$ is co-consonant; (4) for a continuous poset $P$, the lower powerspace $P_{H}(\Sigma P)$ is co-consonant.

Let $M^n$ be an embedded closed submanifold of $\mathbb{R}^{k+1}$ or a smooth bounded domain in $\mathbb{R}^{n}$, where $n\geq 3$. We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time, provided that the initial map $u_{0}$ is in a suitable nontrivial homotopy class with energy small enough.

On Hom-Lie algebras and superalgebras, we introduce the notions of biderivations and linear commuting maps, and compute them for some typical Hom-Lie algebras and superalgebras, including the $q$-deformed $W(2,2)$ algebra, the $q$-deformed Witt algebra and superalgebra.

The paper considers the Cauchy problem with small initial values for semilinear wave equations with weighted nonlinear terms. Similar to Strauss exponent $p_0(n)$ which is the positive root of the quadratic equation $1+\frac{1}{2}( {n + 1} )p - \frac{1}{2}( {n - 1} ){p^2}=0$, we get smaller critical exponents $p_m(n), p_m^{*}(n)$ and have global existence in time when $p>p_{m}(n)$ or $p>p_{m}^{*}(n)$. In addition, for the blow-up case, the introduction of the spacial weight shows the optimality of new upper and lower bound.