Let u(t, x) be the solution to the parabolic Anderson model on ℝ+ × ℝ driven by a space-time white Gaussian noise. In this paper, we prove the central limit theorem and almost sure central limit theorem for product of spatial average of the form ∏_k=1^n S_k as n → ∞ for fixed t > 0, where S_k = ∫_0^k u(t, x)dx .
The trace range of products of three projections in a finite factor (ℳ, τ), with their trace equal to λ is studied in this article. When λ≤1/2 , the corresponding boundary curve of the trace range is shown to be a degenerate elliptic curve. For n ≥ 3, when ℳ is of type In and λ=n-1/n , it is shown that the corresponding boundary curve of the trace range is an algebraic curve in the complex plane.
In 2022, Li, Ye and Yu introduced multivariate sensitivity version of notions of mean m-Sensitivity and m-sensitivity in the mean for m ≥ 2. In this manuscript, we mainly focus on the investigation of multivariate sensitivity in mean forms. First, we prove that for a linear system, the equivalence between mean m-sensitivity and m-sensitivity in the mean holds without any more conditions. Subsequently, we introduce the notion of m-equicontinuity in the mean, and obtain an Auslander–Yorke type dichotomy between m-equicontinuity in the mean and m-sensitivity in the mean for minimal systems. As a consequence, we demonstrate that the equivalence between mean m-sensitivity and m-sensitivity in the mean is valid under the condition of minimality for a general system, thereby affirmatively resolving the conjecture proposed in [Li, J., Ye, X. D., Yu, T.: Equicontinuity and sensitivity in mean forms. J. Dynam. Differential Equations, 34, 133–154 (2022)].
In this work, we prove an L^p_1× L^p_2× L^p_3→ L^p estimate for a trilinear pseudo-differential operators with flag-type symbols taking the form of a(x, ξ1, ξ2, ξ3)b(x, ξ1, ξ2 + ξ3). The study of such operators is motivated by investigations on the corresponding flag Fourier multipliers studied by Muscalu [Rev. Mat. Iberoam., 23, 705–742 (2007); Contemp. Math., 505, 131–151 (2010)], Germain, Masmoudi, Shatah [J. Math. Pures Appl., 97, 505–543 (2012); Ann. of Math., 175, 691–754 (2012)] and Miyachi, Tomita [Math. Z., 282, 577–613 (2016)].
In this paper we study the analytical smoothing effect of Cauchy problem for the incompressible MHD-Boussinesq equations. Precisely, we used the Fourier Gevrey space method to show that the Sobolev H1-solution to the incompressible MHD-Boussinesq equations in periodic domain is analytical for both spatial variable and time variable.
In this paper we introduce a new geometric flow consisting of the Yamabe flow coupled with harmonic heat flow of a function on a closed manifold M, or shortly Yamabe-harmonic flow. It is define by ∂∂ tg = -(R - ϕ)g, ϕ = (n - 1)Δ_g ϕ. To begin with we establish the short-time extsience via conformal transformation for any smooth initial data. Furthermore we derive interior-in-time derivative estimates for the Riemannian curvature and Lapse function and obtain the global existence of Yamabe-harmonic flow. Moreover, compact gradient Yamabe-harmonic solitons are proved to be manifolds with R = ϕ = Constant. Eventually we obtain the classification of singularities by blow-up rate of AC curvature, which is analogous to that of Ricci-harmonic flow and Ricci-connection flow. The relationship between singularity models and solitons shall appear in forthcoming work.
The paper is mainly concerned with the multifractal analysis for smooth dynamical systems in dimension one without uniform hyperbolicity. We characterize the Hausdorff dimension of the saturated set obtained from the empirical measure by the local dimensions of hyperbolic measures for a topologically mixing C2 map modeled by an abstract dynamical system.
For the topological dynamical system with infinite entropy and the specification property, in the context of continuous and affine deformations of empirical measures, the set of divergence points is either empty, or its Bowen and packing metric mean dimensions equal those of the full phase space. We also establish the variational principles of Bowen and packing metric mean dimension for the sup set.
In this paper, we investigate the uniqueness of meromorphic functions sharing four values. We show that if f(z) and g(z) are distinct non-constant meromorphic functions sharing 0, 1, c IM and ∞ CM, then either f(z) and g(z) share 0, 1, c, ∞ CM or 1/13 T(r, f) - 2N̅(r, f) ≤ N_E^(1)(r, 0) + N_E^(1)(r, 1) + N_E^(1)(r, c) + S(r, f) ≤4/3 T(r, f) - 2/3N̅(r, f) holds.
The aim of this article is to introduce a boson-fermionic quantum Weyl algebra W_q(2(m | n)) of type A(m∣n), which is the quantum differential operators (QDO) algebra (also denoted by Diffq(Ωq)) defined on Ωq(m∣n), the quantum Grassmann super-algebra Ωq(m∣n) we defined. Here we develop a constructive approach of Radford–Majid bosonization theory to yield some pointed Hopf algebras via QDO. This addresses a basic problem due to Manin, namely, any QDO approach might help to yield new Hopf algebras (see §2.2. Basic problem, p. 1010 of Manin’s paper in 1992). Remarkably, W_q(2(m | n)) itself is not a Hopf algebra, however, it contains some interesting pointed Hopf algebras as its subquotient objects, like the bosonization A_q(2(m | n)) of the quantum Manin (m∣n)-superspace A ∣ , the bosonization G_q(2(m | n)) of Ωq(m∣n), the multi-rank Taft algebras of 2(m∣n)-type, the homomorphic image of pointed Hopf algebra U_q(gl(m | n)) over Ωq(m∣n) or Ω ! (m∣n) (the quantum dual Grassmann superalgebra), etc.
In this paper, we study Legendrian submanifolds of the Sasakian space form 𝕊^2n+1(c̃) with constant sectional curvature and conformal Maslov form. As main results, we first classify the 2-dimensional Legendrian submanifolds of 𝕊^5(c̃) with constant sectional curvature and conformal Maslov form; then we establish a complete classification of the Legendrian submanifolds of 𝕊^2n+1(c̃) with constant sectional curvature and C-parallel mean curvature vector field.
It is our pleasure to present this Special Issue on Discrete Probability and Mathematical Physics in the Acta Mathematica Sinica,English Series.This volume brings together a collec-tion of original research articles and surveys that reflect recent advances at the vibrant interface between probability theory on discrete structures and problems arising in mathematical physics.
We initiate the study of bulk/boundary quotients of Gaussian multiplicative cascades measures, for which we establish preliminary joint moment bounds. This is a preliminary result towards studying the bulk/boundary quotients of Gaussian multiplicative chaos measures coming from the theory of boundary Liouville conformal field theory.
There are two parts in the Fujita conjecture of 1987. One is the freeness part which since 1995 has been proved with various weaker bounds. The other is the very ampleness part, which remains open even with weaker bounds. Here we will present a proof of the very ampleness part of the Fujita conjecture with a weaker bound by the method of multiplier ideal sheaves of higher order. Moreover, we prove the following more general main result. For an ample line bundle L on a compact complex manifold X of complex dimension n, any prescribed (qj − 1)-jets at prescribed points Pj of X for 1 ≤ j ≤ k can be achieved by a global holomorphic section of mL + KX for m ≥ m0 with m0 explicitly computable from n,q1,…,qk. This article is a write-up of my talk given on April 21, 2024 in the Conference on Several Complex Variables and Complex Geometry at Beijing Normal University. This write-up keeps the conversational style of my talk.
We consider the axisymmetric incompressible Euler equations without swirl in ℝd or in a cylinder domain for d ≥ 3. For 3 ≤ d ≤ 6, we prove the global regularity under the following conditions: u_0∈ L^2(ℝ^d), ω_0 r^d-2∈ L^d d-2,∞(ℝ^d) and min{1,r^3-d}ω_0 r^α∈ L^∞(ℝ^d) for some α ∈ (0, 1). Moreover, if the domain is a cylinder or if ω0 is single-signed, we prove the same global regularity result for all d ≥ 3. Additionally, for 3 ≤ d ≤ 6, we obtain the same growth bounds as in [Lim and Jeong, Arch. Ration. Mech. Anal., 249, Paper No. 32, 31 pp. (2025)], without assuming compact support on the initial data.
The H-invariant was introduced to compute Perelmans entropy for Kähler–Ricci flow in a paper of Tian–Zhang–Zhang–Zhu more than ten years ago. It turns out that the H-invariant is equal to an earlier invariant by Tian–Zhu in their study on Kähler–Ricci solitons. In this largely expository paper, we will discuss definition of the H-invariant, its relation to Tian–Zhu’s generalization of the Futaki invariants as well as some of its applications. We will also include some new observations and generalizations of results in existing literature. Several examples will be also provided.
We study clustering dynamics for the infinite Cucker-Smale(ICS)model and its connection to the spectrum of the graph Laplacian.For the ICS model,we overcome the challenge of estimating the velocities of particles and derive a system of dissipative differential inequalities(SDDI)in terms of infinite norms.As in the finite ensemble,we show that mono-cluster flocking emerges exponentially fast,and additionally establish a sufficient framework for algebraic multi-cluster flocking.Moreover,for the CS model with a finite system size,we offer a complete spectral characterization of multi-cluster flocking.Specifically,the emergence of n-cluster behavior corresponds to the limit of the n-th eigenvalue of the time-varying Laplacian approaching zero.In contrast,the lower bound of the(n+1)-th eigenvalue remains strictly positive.Furthermore,we extend this framework to the ICS model by characterizing weak n-cluster flocking via spectral asymptotics,where the n-th eigenvalue tends to zero,while both the(n+1)-th eigenvalue and the infimum of the essential spectrum remain strictly positive.Our results bridge spectral analysis and clustering dynamics,providing indirect evidence for the fast emergence of mono-cluster flocking and the slow relaxation of multi-cluster patterns in both finite and infinite particle systems.
The investigation of nilpotent structures aims to uncover the underlying nilpotent information in general dynamical systems, as well as the dynamical behaviors of the systems with respect to the nilpotent structures. These structures play crucial roles in ergodic theory and topological dynamics, and their applications across several fields of mathematics, including combinatorial number theory. In this paper we will review the results related to the nilpotent structures, with a particular emphasis on topological nilpotent structures of ℤ-actions. Furthermore, we present applications and state open questions for future research.
This paper is a review of K-theory methods in solid state physics. Our goal is to demonstrate that K-theory is a natural language for the mathematical description of solid bodies. Our main tool is the K-theory of C*-algebras. We follow Kitaev’s idea that the symmetry algebras of solid bodies belong to the class of Clifford algebras which reduces the quantization problem for solid states to the representation theory of Clifford algebras.
Conventional statistical analysis methods encounter challenges when addressing precise medical treatment for high-dimensional survival data. Overcoming these challenges involves accounting for heterogeneity to mitigate estimation bias and avoiding model overfitting for enhanced interpretability. This paper introduces a regularization technique based on the heterogeneous Cox model, enabling simultaneous variable selection, subgroup identification, and parameter estimation. The approach is entirely data-driven and capable of handling exponential increases in covariate dimension with sample size, as well as divergent numbers of significant variables. Under mild assumptions, we establish asymptotic properties for the proposed estimator, including the oracle property of variable selection, consistency in subgroup identification, and asymptotic normality. Leveraging the coordinate descent method and ADMM algorithm, we propose an efficient MCD-ADMM algorithm for optimization. Simulation studies further validate the effectiveness of our approach, complemented by an analysis of ovarian cancer data for illustrative purposes.
