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.

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.

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 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.

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

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.

The Turán number, denoted by ${\rm ex}\,(n,H)$, is the maximum number of edges of a graph on $n$ vertices containing no graph $H$ as a subgraph. Denote by $kC_{\ell}$ the union of $k$ vertex-disjoint copies of $C_{\ell}$. In this paper, we present new results for the Turán numbers of vertex-disjoint cycles. Our first results deal with the Turán number of vertex-disjoint triangles ${\rm ex}\,(n, kC_{3})$. We determine the Turán number ${\rm ex}(n, kC_{3})$ for $n\geq\frac{k^{2}+5k}{2}$ when $k\leq4$, and $n\geq k^{2}+2$ when $k\geq4$. Moreover, we give lower and upper bounds for ${\rm ex}\,(n, kC_{3})$ with $3k\leq n\leq\frac{k^{2}+5k}{2}$ when $k\leq4$, and $3k\leq n\leq k^{2}+2$ when $k\geq4$. Next, we give a lower bound for the Turán number of vertex-disjoint pentagons ${\rm ex}\,(n, kC_{5})$. Finally, we determine the Turán number ${\rm ex}\,(n, kC_{5})$ for $n=5k$, and propose two conjectures for ${\rm ex}\,(n, kC_{5})$ for the other values of $n$.

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.

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.

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.

We are concerned with the boundedness for the equation $x''+f(x,x')+\omega^2x=p(t)$, where $p$ is quasi-periodic function. Since the corresponding system is non-Hamiltonian, we transform the original system into a new reversible one, the Poincar\'{e} mapping of which satisfies the twist theorem for quasi-periodic reversible mappings of low smoothness, or is close to its linear part by normal form theorem. We then obtain results concerning the boundedness of solutions based on the recently work. The above two cases need some smooth and growth assumptions for $f$ and $p$, which are precisely the innovations of this paper.

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.

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 $N\geq 2$.

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.

We study the equation $$-\Delta u=|x|^\alpha u^{p_\alpha^*+\varepsilon}+\lambda_{\varepsilon}|x|^\beta u \quad \text { in } \Omega,$$ under the condition $u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq 5$, which is symmetric respect to $x_1, x_2,\dots,x_N$ and contains the origin, $\alpha>-2$, $-2<\beta< N-4$, $p^*_\alpha = \frac{N+2\alpha+2}{N-2}$, $\varepsilon>0$ is a small parameter and $\lambda_\varepsilon>0$ depends on $\varepsilon$, with $\lambda_\varepsilon\to 0$ as $\varepsilon\to 0$. Our main focus lies in finding positive solutions that take the form of a tower of bubbles of order $\alpha$, exhibiting concentration at the origin as $\varepsilon$ tends to zero. Furthermore, we extend our study to the equation $$-\Delta u=|x|^\alpha u^{p_\alpha^*-\varepsilon}-\lambda_{\varepsilon}|x|^\beta u \quad \text { in } \mathbb{R}^N \backslash B_1,$$ where $B_1$ is the unit ball centered at the origin, under Dirichlet zero boundary condition and an additional vanishing condition at infinity. In this context, we discover positive solutions that take the form of a tower of bubbles of order $\alpha$, progressively flattening as $\varepsilon$ tends to zero.

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.

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.

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 paper, we study the relationship of balanced pairs in a recollement. As an application of balanced pairs, we introduce the notion of the relative tilting objects, and give a characterization of relative tilting objects, which is similar to Bazzoni characterization of $n$-tilting modules. Finally, we investigate the relationship of relative tilting objects in a recollement.

In 2015, a group of mathematicians at the University of Washington, Bothell, discovered the 15th pentagon that can cover a plane, with no gaps and overlaps. However, research on its containment measure theory or geometric probability is limited. In this study, the Laplace extension of Buffon's problem is generalized to the case of the 15th pentagon. In the solving process, the explicit expressions for the generalized support function and containment function of this irregular pentagon are derived. In addition, the chord length distribution function and density function of random distance of this pentagon are obtained in terms of the containment function.

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.

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.

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, 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.

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.

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.

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 $X$.

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.

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.

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, 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.

In this paper we study a heat type equation associated to the curve shortening flow in the plane. We show the solutions become infinitely many times differentiable for a short time. The method of proof is to use the maximum principle following the Bernstein technique.

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.