In this paper, we study a class of Finsler metrics of cohomogeneity two on $\mathbb{R} \times \mathbb{R}^n$. They are called weakly orthogonally invariant Finsler metrics. These metrics not only contain spherically symmetric Finsler metrics and Marcal-Shen's warped product metrics but also partly contain another "warping" introduced by Chen-Shen-Zhao. We obtain differential equations that characterize weakly orthogonally invariant Finsler metrics with vanishing Douglas curvature, and therefore we provide a unifying frame work for Douglas equations due to Liu-Mo, Mo-Solórzano-Tenenblat and Solórzano. As an application, we obtain a lot of new examples of weakly orthogonally invariant Douglas metrics.

We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric α-stable Lévy processes (1 2 <α ≤ 1), where the drift coefficient is Hölder continuous in space variable, while the noise coefficient is Lipscitz continuous in space variable, and both of them satisfy the Lipschitz condition in distribution variable with respect to Wasserstein distance. If the drift coefficient does not depend on distribution variable, our methodology developed in this paper applies to the case α ∈ (0, 1]. The main tool relies on heat kernel estimates for (distribution independent) stable SDEs and Banach’s fixed point theorem.

In this paper, we consider the Schrödinger type equation -Δu + V(x)u = f(x, u) on the lattice graph $\mathbb{Z}^N$ with indefinite variational functional, where Δ is the discrete Laplacian. Specifically, we assume that V (x) and f(x, u) are periodic in x, f satisfies some growth condition and 0 lies in a finite spectral gap of (-Δ + V). We obtain ground state solutions by using the method of generalized Nehari manifold which has been introduced by Pankov.

In this paper, we study the $p$-Laplacian Choquard equation $$-△_p u+V(x)|u|^{p-2}u=\bigg({\sum_{y\in N^n\atop y\not=x}}\frac{|u(y)|^q}{d(x,\,y)^{n-\alpha}}\bigg)|u|^{q-2}u$$ on a finite lattice graph $N^n$ with $n\in\mathbb{N}_+$, where $p>1,$ $q>1$ and $0\leq\alpha\leq n$ are some constants, $V(x)$ is a positive function on $N^n$. Using the Nehari method, we prove that if 1<p<q<+∞, then the above equation admits a ground state solution. Previously, the $p$-Laplacian Choquard equation on finite lattice graph has not been studied, and our result contains the critical cases $\alpha=0$ and $\alpha=n$, which further improves the study of Choquard equations on lattice graphs.

The main purpose of this article is using the elementary techniques and the properties of the character sums to study the computational problem of one kind products of Gauss sums, and give an interesting triplication formula for them.

This paper develops an Itô-type fractional pathwise integration theory for fractional Brownian motion with Hurst parameters \(H \in (\frac{1}{3}, \frac{1}{2}] \), using the Lyons' rough path framework. This approach is designed to fill gaps in conventional stochastic calculus models that fail to account for temporal persistence prevalent in dynamic systems such as those found in economics, finance, and engineering. The pathwise-defined method not only meets the zero expectation criterion but also addresses the challenges of integrating non-semimartingale processes, which traditional Itô calculus cannot handle. We apply this theory to fractional Black-Scholes models and high-dimensional fractional Ornstein-Uhlenbeck processes, illustrating the advantages of this approach. Additionally, the paper discusses the generalization of Itô integrals to rough differential equations (RDE) driven by fBM, emphasizing the necessity of integrand-specific adaptations in the Itô rough path lift for stochastic modeling.

We study Berry-Esseen bounds and Cramér-type moderate deviations of a jump-type Cox-Ingersoll-Ross (CIR) process driven by a standard Wiener process and a subordinator. In the subcritical case, we obtain the best Berry-Esseen bound of the sample mean and the MLE of the growth rate if the Lévy measure of the subordinator has finite third order moment. Under the Cramér condition, we establish the Cramér-type moderate deviations of the MLE of the growth rate. We first derive a Berry-Esseen bound, a deviation inequality and the Cramér-type moderate deviations for the sample mean of the CIR process by analyzing the asymptotic behaviors of the characteristic function and the moment generating function of the sample mean. Then we analyze a type of additive functional of the jump-type CIR process and use a transformation to study the Berry-Esseen bound and the Cramér-type moderate deviations for the MLE of the growth rate.

We derive a blow-up formula for holomorphic Koszul-Brylinski homologies of compact holomorphic Poisson manifolds. As applications, we investigate the invariance of the E₁-degeneracy of the Dolbeault-Koszul-Brylinski spectral sequence under Poisson blow-ups, and compute the holomorphic Koszul-Brylinski homology for del Pezzo surfaces and two complex nilmanifolds with holomorphic Poisson structures.

The aim of this paper is to investigate the existence of solutions to the prescribing fractional Q-curvature problem on $\mathbb{S}^n$ under some reasonable assumption of the Laplacian sign at the critical point of prescribing curvature function K. Due to the lack of compactness, we choose to return to the basic elements of variational theory and study the deformation along the flow lines. The novelty of the paper is that we obtain the existence without assuming any symmetry and periodicity on K. In addition, to overcome the loss of compactness for high-order operator problem, we need more delicate estimates with the second order cases.

Polya—Carlson theorem asserts that if a power series with integer coefficients and convergence radius 1 can be extended holomorphically out of the unit disc, it must represent a rational function. In this note, we give a generalization of this result to multivariate case and give an application to rationality theorem about D-finite power series.

Cyclotomic multiple zeta values are generalizations of multiple zeta values. In this paper, we establish sum formulas for various kinds of cyclotomic multiple zeta values. As an interesting application, we show that the $\mathbb{Q}$-algebra generated by Riemann zeta values are contained in the $\mathbb{Q}$-algebra generated by unit cyclotomic multiple zeta values of level $N$ for any $N\geq 2$.

For a piecewise monotone function $F$ of height 1, an open question was raised: Does $F$ have an iterative root $f$ of order $n\le N(F)+1$ if the ‘characteristic endpoints condition' is not satisfied? This question was answered partly in the case that $F$ is strictly increasing on its characteristic interval $K(F)$ but $f$ is strictly decreasing on $K(F)$. In this paper we discuss the question for $F$ increasing on $K(F)$ in some remaining cases, giving the necessary and sufficient conditions for the existence of continuous iterative roots $f$ decreasing on $K(F)$ of order $n=N(F)>2$ with $H(f)=n-1$.

In this article, we continue to study K?hler metrics on line bundles over projective spaces to find complete K?hler metrics with positive holomorphic sectional curvatures with two very special properties. These two special kinds of examples were not able to be found in our earlier paper of the first author and Ms. Duan. And therefore, we give a further step toward a famous Yau conjecture with the method in the co-homogeneity one geometry.

In many practical applications, we need to recover block sparse signals. In this paper, we encounter the system model where joint sparse signals exhibit block structure. To reconstruct this category of signals, we propose a new algorithm called block signal subspace matching pursuit (BSSMP) for the block joint sparse recovery problem in compressed sensing, which simultaneously reconstructs the support of block jointly sparse signals from a common sensing matrix. To begin with, we consider the case where block joint sparse matrix $\mathbf{X}$ has full column rank and any $r$ nonzero row-blocks are linearly independent. Based on these assumptions, our theoretical analysis indicates that the BSSMP algorithm could reconstruct the support of $\mathbf{X}$ through at most $K-r+\lceil\frac{r}{L}\rceil$ iterations if sensing matrix $\mathbf{A}$ satisfies the block restricted isometry property of order $L(K-r)+r+1$ with $\delta_{B_{L(K-r)+r+1}}<\max\{\frac{\sqrt{r}}{\sqrt{K+\frac{r}{4}}+\sqrt{\frac{r}{4}}}, \frac{\sqrt{L}}{\sqrt{Kd}+\sqrt{L}}\}$. This condition improves the existing result.

In this paper, we investigate the existence of normalized solutions for a quasilinear elliptic problem as follows \begin{equation*} \left\{\begin{array}{ll} -\Delta_p u+\lambda u^{p-1}=f(u), & x\in \mathbb{R}^N, \\ \displaystyle\int_{\mathbb{R}^N}|u|^p d x=\rho,& u \in W^{1,p}(\mathbb{R}^N), \end{array}\right. \end{equation*} where $-\Delta_p $ is the $p$-Laplace operator, 1<p<N,N≥3,ρ>0 and λ>0. f is a continuous function and satisfies some suitable conditions. Based on a Nehari—Pohozaev manifold, we show the existence of positive normalized solutions by using the minimization method.

In this paper, we introduce a communication-efficient distributed estimation method tailored for massive datasets exhibiting skewness. The data are stored across multiple machines. We construct a surrogate likelihood which only need to transfer subgradient from local machines to approximate higher-order derivatives of the global likelihood. An enhanced EM algorithm is developed for computations. The proposed method not only addresses the non-normality of data by utilizing first-order gradient information in each transmission, ensuring low communication overhead, but also ensures privacy protection. Simulation studies illustrate the superior performance of the proposed methods.

Regression models are often transformed into certain alternative forms in statistical inference theory. In this paper, we assume that a general linear model (GLM) is transformed into two different forms, and our aim is to study some comparison problems under the two transformed general linear models (TGLMs). We first construct a general vector composed of all unknown parameters under the two different TGLMs, derive exact expressions of best linear minimum bias predictors (BLMBPs) by solving a constrained quadratic matrix-valued function optimization problem in the Löwner partial ordering, and describe a variety of mathematical and statistical properties and performances of the BLMBPs. We then approach some algebraic characterization problems concerning relationships between the BLMBPs under two different TGLMs. As applications, two specific cases are presented to illustrate the main contributions in the study.

Using the stratifications of Deligne—Mumford moduli spaces $\overline{\mathcal M}_{g,n}$ indexed by stable graphs, we introduce a partially ordered set of stable graphs by defining a partial ordering on the set of connected stable graphs of genus $g$ with $n$ external edges. By modifying the usual definition of zeta function and Möbius function of a poset, we introduce generalized ($\mathbb Q$-valued) zeta function and generalized ($\mathbb Q$-valued) Möbius function of the poset of stable graphs. We use them to proved a generalized Möbius inversion formula for functions on the poset of stable graphs. Two applications related to duality in earlier work are also presented.

This article deals with the sharp discrete isoperimetric inequalities in analysis and geometry for planar convex polygons. First, the analytic isoperimetric inequalities based on the Schur convex function are established. In the wake of the analytic isoperimetric inequalities, Bonnesen-style isoperimetric inequalities and inverse Bonnesen-style inequalities for the planar convex polygons are obtained.

In this paper, entropy and pressure are investigated for a random dynamical system $\varphi$ over $\mathbb{Z}^k$-actions on a compact metric space. The pressure $P(\varphi, f)$ of $\varphi$ with respect to a random continuous function $f$ and the measure-theoretic entropy $h_\mu(\varphi)$ for a $\varphi$-invariant measure $\mu$ are defined. A variational principle for pressure $P(\varphi, f)$ is established, which states that $P(\varphi, f)$ is the supremum of the sum of $h_\mu(\varphi)$ and the integral of $f$ taken over all invariant measures $\mu$. We also obtain some basic properties for equilibrium states.

We consider the following fractional prescribed curvature problem \begin{align}\label{eq01} (-\Delta)^s u= K(y)u^{2^*_s-1},\quad u>0,y \in \mathbb{R}^N, \end{align} where $s\in(0,\frac{1}{2})$ for $N=3$, $s\in(0,1)$ for $N\geq4$ and $2^*_s=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $K(y)$ has a local maximum point in $r\in(r_0-\delta,r_0+\delta)$. First, for any sufficient large $k$, we construct a $2k$ bubbling solution to (0.1) of some new type, which concentrates on an upper and lower surfaces of an oblate cylinder through the Lyapunov-Schmidt reduction method. Furthermore, a non-degeneracy result of the multi-bubbling solutions is proved by use of various Pohozaev identities, which is new in the study of the fractional problems.

Let $\{X, X_{n}; n \geq 1\}$ be a sequence of i.i.d. non-degenerate real-valued random variables with $\mathbb{E}X^{2} < \infty$. Let $S_{n} = \sum_{i=1}^{n} X_{i}$, $n \geq 1$. Let $g(\cdot): ~[0, \infty) \rightarrow [0, \infty)$ be a nondecreasing regularly varying function with index $\rho \geq 0$ and $\lim_{t \rightarrow \infty} g(t) = \infty$. Let $\mu = \mathbb{E}X$ and $\sigma^{2} = \mathbb{E}(X - \mu)^{2}$. In this paper, on the scale $g(\log n)$, we obtain precise asymptotic estimates for the probabilities of moderate deviations of the form $ \log \mathbb{P}(S_{n} - n \mu > x \sqrt{ng(\log n)} )$, $ \log \mathbb{P}(S_{n} - n \mu < -x \sqrt{ng(\log n)} )$, and $ \log \mathbb{P}(|S_{n} - n \mu | > x \sqrt{ng(\log n)} )$ for all $x > 0$. Unlike those known results in the literature, the moderate deviation results established in this paper depend on both the variance and the asymptotic behavior of the tail distribution of $X$.

Zhao and Xu (2013) constructed a functor from $\mathfrak{o}(n)$-Mod to $\mathfrak{o}(n+2)$-Mod. In this paper, we use the functor successively to obtain full conformal oscillator representation of $\mathfrak{o}(2n+2)$ in $n(n+1)$ variables and determine the corresponding finite-dimensional irreducible module explicitly when the highest weight is dominant integral. We also find an equation of counting the dimension of an irreducible $\mathfrak{o}(2n+2)$-module in terms of certain alternating sum of the dimensions of irreducible $\mathfrak{o}(2n)$-modules, which leads to new combinatorial identities of classical type in the case of the Steinberg modules. One can use the results to study tensor decomposition of finite-dimensional irreducible modules by solving certain first-order linear partial differential equations, and thereby obtain the corresponding physically interested Clebsch—Gordan coefficients and exact solutions of Knizhnik—Zamolodchikov equation in WZW model of conformal field theory.

In this paper, complete convergence and complete moment convergence for maximal weighted sums of $\rho^-$-mixing random variables are investigated, and some sufficient conditions for the convergence are provided. The relationships among the weights of the partial sums, boundary function and weight function are in a sense revealed. Additionally, a Marcinkiewicz-Zygmund type strong law of large {numbers} for maximal weighted sums of $\rho^-$-mixing random variables is established. The results obtained extend the corresponding ones for random variables with independence structure and some dependence structures. As an application, the strong consistency for the tail-value-at-risk (TVaR) estimator in the financial and actuarial fields is established.

We construct an explicit example of a smooth isotopy $\{\xi_t\}_{t \in [0,1]}$ of volume- and orientation-preserving diffeomorphisms on $[0,1]^n$ ($n \geq 3$) that has infinite total kinetic energy. This isotopy has no self-cancellation and is supported on countably many disjoint tubular neighbourhoods of homothetic copies of the isometrically embedded image of $(M,g)$, a "topologically complicated" Riemannian manifold-with-boundary. However, there exists another smooth isotopy that coincides with $\{\xi_t\}$ at $t=0$ and $t=1$ but of finite total kinetic energy.

In this paper, the semirings with invariant basis numbers are investigated. First, we give some properties of a semiring which has an invariant basis number, and then give some necessary and sufficient conditions that the direct sum of two semirings has an invariant basis number. As an application, we prove that division semirings, quasilocal semirings and stably finite semirings have invariant basis numbers, respectively.

In this paper, we introduce the spectral Einstein functional for perturbations of Dirac operators on manifolds with boundary. Furthermore, we provide the proof of the Dabrowski—Sitarz—Zalecki type theorems associated with the spectral Einstein functionals for perturbations of Dirac operators, particularly in the cases of on 4-dimensional manifolds with boundary.

We prove that for any η that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets k^-1((-∞, η]) and k^-1(η), which are the sets of irrational numbers with best constant of Diophantine approximation bounded by η and exactly η respectively, have the same Hausdorff dimension. We also show that, as η varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function.

The aim of this paper is to explore the equivalent characterizations for the boundedness and compactness of C_φ-C_ψ acting from classical (little) Zygmund space $\mathcal{Z}\left(\mathcal{Z}_0\right)$ to (little) Bloch-type space $\mathcal{B}^\alpha\left(\mathcal{B}_0^\alpha\right)$. Especially, we creatively develop a useful lemma, which not only plays a crucial role in the estimations but also offers a sufficient condition for the bounded below property of composition operators.

In this paper, we give the definition of Maslov-type index of the discrete Hamiltonian system, and obtain the relation of Morse index and Maslov-type index of the discrete Hamiltonian system which is a generalization of the case ω = 1 to ω ∈ U degenerate case via direct method which is different from that of the known literatures. Moreover the well-posedness of the splitting numbers S_h,ω^± is proven, then the index iteration theories of Bott and Long are also valid for the discrete case, and those can be also applied to the study of the symplectic algorithm.

The main purpose of this article is to study the lattice structure of quantum logics by using the theory of partially order sets. The goal is achieved by constructing a natural embedding map from the quantum logic posets into the complete lattices. The embedding map needs to be dense (in the order sense) and such complete lattices are so-called extended logics which preserve all essential features of the quantum logic. We obtain that the extended logic is unique up to a lattice isomorphism.

In this paper, we consider $\lambda$-translating solitons in $\mathbb{R}^3$. These surfaces are critical points of the weighted area when the density is a coordinate function. If $\lambda=0$, these surfaces evolve by translations along the mean curvature flow. We give a full classification of $\lambda$-translating solitons that satisfy a linear Weingarten relation between their curvatures. These surfaces are planes, circular cylinders, grim reapers and certain types of cylindrical surfaces. We also prove that planes and circular cylinders are the only $\lambda$-translating soliton with constant squared norm of the second fundamental form.

In this paper, we study asymptotic behavior of small perturbation for path-distribution dependent stochastic differential equations driven simultaneously by a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{2},1)$ and a standard Brownian motion. We establish large and moderate deviation principles by utilising the weak convergence approach.

In this paper, we develop some new variational and analytic techniques to study the multiplicity and concentration of positive solutions for a planar Schrödinger-Poisson system involving competing weight potentials and the nonlinearity $K(x)|u|^{p-2}u$ $(2<p<4)$ in $\mathbb{R}^2$. By Nehari manifold and Ljusternik-Schnirelmann category, we relate the number of positive solutions to the category of the global minima set of a suitable ground energy function. Our results improve and extend the ones in [Du, Weth, Nonlinearity, 30, 3492-3515 (2017)] and [Chen, Tang, J. Differ. Equ., 268, 945-976 (2020)]. In particular, we do not need the assumption $K(x)\equiv1$ and the $C^1$ smoothness of $V(x)$. Furthermore, we do not use the axially symmetric condition of the potential in our second main result. Moreover, we shall show that there is a great difference in our results between $N=2$ and $N\geq3$.

We study the mean orbital pseudo-metric for Polish dynamical systems and its connections with properties of the space of invariant measures. We give equivalent conditions for when the set of invariant measures generated by periodic points is dense in the set of ergodic measures and the space of invariant measures. We also introduce the concept of asymptotic orbital average shadowing property and show that it implies that every non-empty compact connected subset of the space of invariant measures has a generic point.

In this paper we will be concerned with the problem $$ - \Delta u - \frac{1}{2}\Delta(a(x)u^2) u + V(x)u=f(u), x\in \mathbb{R}^2, $$ where $V$ is a potential continuous and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous function with exponential subcritical or exponential critical growth. We use as a main tool the Nehari manifold method in order to show existence of nonnegative solutions and existence of nodal solutions. Our results complement the classical result of Solutions for quasilinear Schr?dinger equations via the Nehari method" due to Jia-Quan Liu, Ya-Qi Wang and Zhi-Qiang Wang in the sense that in this article we are considering nonlinearity of the exponential type.

In this paper, we explore the convergence and convergence rate results for a new methodology termed the half-proximal symmetric splitting method (HPSSM). This method is designed to address linearly constrained two-block non-convex separable optimization problem. It integrates a half-proximal term within its first subproblem to cancel out complicated terms in applications where the subproblem is not easy to solve or lacks a simple closed-form solution. To further enhance adaptability in selecting relaxation factor thresholds during the two Lagrange multiplier update steps, we strategically incorporate a relaxation factor as a disturbance parameter within the iterative process of the second subproblem. Building on several foundational assumptions, we establish the subsequential convergence, global convergence, and iteration complexity of HPSSM. Assuming the presence of the Kurdyka-?ojasiewicz inequality of ?ojasiewicz-type within the augmented Lagrangian function (ALF), we derive the convergence rates for both the ALF sequence and the iterative sequence. To substantiate the effectiveness of HPSSM, sufficient numerical experiments are conducted. Moreover, expanding upon the two-block iterative scheme, we present the theoretical results for the symmetric splitting method when applied to a three-block case.

We provide three types of invariance pressure for uncertain control systems, namely, invariance pressure, strong invariance pressure, and invariance feedback pressure. The first two respectively extend the corresponding pressures for deterministic control systems proposed by Colonius, Cossich, and Santana (2018) and by Nie, Wang, and Huang (2022); and the third generalizes invariance feedback entropy of uncertain control systems presented by Tomar, Rungger, and Zamani (2020), by adding potentials on the control range. Then we prove that (1) an explicit formula for invariance pressure of a controlled invariant set with respect to a potential by the logarithm of the spectral radius of the admissible weighted matrix determined by this potential under some suitable conditions; (2) an explicit formula for pressure of invariant quasi-partitions by maximum mean weight over all irreducible periodic sequences; (3) the invariance feedback pressure of a controlled invariant set is equal to the pressure of an atom partition under some technical assumptions; (4) lower and upper bounds for pressure of invariant quasi-partitions w.r.t. a potential by the logarithm of the spectral radius of the weighted adjacency matrix determined by this potential; (5) a variational principle for strong invariance pressure.

In this paper, we study the vector fields $X$ with a global Poincaré cross-section on a $2 n+1$-dimensional presymplectic manifold ($M, \tilde{\omega}$) under certain conditions. We use ($M, \tilde{\omega}$) to construct a $2 n+2 k$ dimensional symplectic manifold ($\tilde{M}, \Lambda$), on which the vector field $X$ can be extended to a Hamiltonian vector field $\tilde{X}$ with a smooth Hamiltonian $H: \tilde{M} \rightarrow R$. We also consider vector fields $X$ with a first integral $F$ and a Jacobi multiplier $J$ on an $n$-dimensional manifold ($M, \Omega$). On a level set $\Sigma$ of $F$, we get an $n-1$-volume form $\omega_n$ on $\Sigma$ and prove that $X$ is a volume-preserving vector field with respect to $\omega_n$. Specifically, when $X$ is a 3 dimensional devergence-free vector field, the results have been discussed by Lerman in 2019.

For any real number $x,$ $[x]$ denotes the integer part of $x.$ $\mathcal{F}_{1}, \mathcal{F}_{2}$ denote two multiplicative function classes which are small in numerical sense. In this paper, we study the summation $\sum_{n\leq x} f([x/n])$ for $f\in \mathcal{F}_{1}$. As specific cases, we take $d^{(e)}(n), \beta(n), a(n), \mu_{2}(n)$ denoting the number of exponential divisors of $n$, the number of square-full divisors of $n,$ the number of non-isomorphic Abelian groups of order $n,$ and the characteristic function of the square-free integers, respectively. In the case of $\mu_{2}(n),$ we improved the result of Liu, Wu and Yang. The sums shaped like $\Sigma_{n\leq x} f([x/n]+f([x/n]))$ for $f\in \mathcal{F}_{2}$ are also researched.