THE EXISTENCE OF PSEUDOHARMONIC MAPS FOR SMALL HORIZONTAL ENERGY

In this paper,we consider pseudoharmonic heat flow with small initial horizontal energy and give the existence of pseudoharmonic maps from closed pseudo-Hermitian manifolds into closed Riemannian manifolds.

Sequential Propagation of Chaos for Mean-Field BSDE Systems

A new class of backward particle systems with sequential interaction is proposed to approximate the mean-field backward stochastic differential equations.It is proven that the weighted empirical measure of this particle system converges to the law of the McKean-Vlasov system as the number of particles grows.Based on the Wasserstein met-ric,quantitative propagation of chaos results are obtained for both linear and quadratic growth conditions.Finally,numerical experiments are conducted to validate our theoretical results.

Remarks on the Global Existence for Incompressible Navier-Stokes Equations

In this article,the authors use the special structure of helicity for the threedimensional incompressible Navier-Stokes equations to construct a family of finite energy smooth solutions to the Navier-Stokes equations which critical norms can be arbitrarily large.

A Simple Proof of ACC for Minimal Log Discrepancies for Surfaces

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

The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded

Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions,provided that the initial data are compactly supported and sufficiently small in Sobolev norm.In this work,Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions.A natural question then arises whether the time-growth is a true phenomenon,despite the possible conservation of basic energy.In the present paper,we establish that the top-order energy of the solutions in Alinhac theorem remains globally bounded in time.

Using Event-Based Method to Estimate Cybersecurity Equilibrium

Estimating the global state of a networked system is an important problem in many application domains.The classical approach to tackling this problem is the periodic(observation)method,which is inefficient because it often observes states at a very high frequency.This inefficiency has motivated the idea of event-based method,which leverages the evolution dynamics in question and makes observations only when some rules are triggered(i.e.,only when certain conditions hold).This paper initiates the investigation of using the event-based method to estimate the equilibrium in the new application domain of cybersecurity,where equilibrium is an important metric that has no closed-form solutions.More specifically,the paper presents an event-based method for estimating cybersecurity equilibrium in the preventive and reactive cyber defense dynamics,which has been proven globally convergent.The presented study proves that the estimated equilibrium from our trigger rule i)indeed converges to the equilibrium of the dynamics and ii)is Zeno-free,which assures the usefulness of the event-based method.Numerical examples show that the event-based method can reduce 98%of the observation cost incurred by the periodic method.In order to use the event-based method in practice,this paper investigates how to bridge the gap between i)the continuous state in the dynamics model,which is dubbed probability-state because it measures the probability that a node is in the secure or compromised state,and ii)the discrete state that is often encountered in practice,dubbed sample-state because it is sampled from some nodes.This bridge may be of independent value because probability-state models have been widely used to approximate exponentially-many discrete state systems.

Estimation of the Probability of Long-Distance Dispersal: Stratified Diffusion of <i>Spartina alterniflora</i>in the Yangtze River Estuary

The relative contribution of long-distance dispersal and local diffusion in the spread of invasive species has been a subject of much debate. Invasion of the intertidal mudflats by Spartina alterniflora is an ideal example of stratified diffusion, involving both long-distance dispersal of seeds and local diffusion due to clonal growth. In conjunction with experimental data on range radius-versus-time curve, a traveling wave equation-based model is used to investigate the sensitivity of the spread rate of exotic S. alterniflora to parameters of long distance dispersal (c, maximum colonial establishment rate) and local colony diffusion (r, intrinsic growth rate) at two tidal marshes, the Eastern Chongming and the Jiuduansha Islands, at the Yangtze River estuary. Both Eastern Chong ming and Jiuduansha Islands are now national natural reserves in China, which were established in 2005. However, the mudflats and salt marshes in the two reserves are now heavily infested with introduced S. alterniflora, which may threaten the estuarine ecosystems and their biodiversity. S. alterniflora was first found in 1995 on Chongming. For rapid sediment accretion in mudflats in the estuary, S. alterniflora was also intentionally introduced to Jiuduansha in 1997 and Chongming in 2001, which has led to a rapid range expansion in the estuary. Our results show that range expansion of species with stratified diffusion is affected by both long-distance dispersal and local colony diffusion, and that there is a critical c*, below which the spread rate is more influenced by long-distance dispersal than by local diffusion. After applying this model to the invasion of S. alterniflora in the Yangtze River estuary, we derive that c = 1.7 × 10-3, c* = 0.126 and c = 4.8 × 10-3 km-2·yr-1, c* = 0.140 km-2·yr-1 at Chongming and Jiuduansha (Shanghai), respectively. Our results suggest that the range spread of S. alterniflora in the Yangtze River estuary is more influenced by long-distance dispersal than local colony diffusion, and that S. alterniflora generates about 1.7 × 10-3 to 4.8 × 10-3 colonies per square kilometers per year. This study provides important information about dispersal dynamics of S. alterniflora that may be useful for finding optimal control strategies. ·

Gaussian limit for determinantal point processes with J-Hermitian kernels

We show that the central limit theorem for linear statistics over determinantal point processes with J-Hermitian kernels holds under fairly general conditions.In particular,we establish the Gaussian limit for linear statistics over determinantal point processes on the union of two copies of Rdwhen the correlation kernels are J-Hermitian translation-invariant.

Convergence of Gradient Algorithms for Nonconvex C^(1+α) Cost Functions

This paper is concerned with convergence of stochastic gradient algorithms with momentum terms in the nonconvex setting.A class of stochastic momentum methods,including stochastic gradient descent,heavy ball and Nesterov’s accelerated gradient,is analyzed in a general framework under mild assumptions.Based on the convergence result of expected gradients,the authors prove the almost sure convergence by a detailed discussion of the effects of momentum and the number of upcrossings.It is worth noting that there are not additional restrictions imposed on the objective function and stepsize.Another improvement over previous results is that the existing Lipschitz condition of the gradient is relaxed into the condition of H?lder continuity.As a byproduct,the authors apply a localization procedure to extend the results to stochastic stepsizes.

Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal conditions.The generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable z.The existence and uniqueness results are given to these BSDEs.As an application,an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.

Decay of Correlations for Fibonacci Unimodal Interval Maps

We consider a class of generalized Fibonacci unimodal maps for which the central return times {Sn} satisfy that sn= sn-1 ＋ ksh-2 for some k≥ 1. We show that such a unimodal map admits a unique absolutely continuous invariant probability with exactly stretched exponential decay of correlations if its critical order lies in （1, k ＋ 1）.

Global Well-posedness for the 2D Micropolar Rayleigh–Bénard Convection Problem without Velocity Dissipation

In this article,we study the Cauchy problem to the micropolar Rayleigh–Bénard convection problem without velocity dissipation in two dimension.We first prove the local well-posedness of a smooth solution,and then establish a blow up criterion in terms of the gradient of scalar temperature field.At last,we obtain the global well-posedness to the system.

Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach

The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],where the trajectory equation can be obtained and two numerical schemes have been devel-oped based on different dissipative energy laws.It is also proved that the nonlinear scheme,based on f logf as the total energy form of the dissipative law,is uniquely solv-able on an admissible convex set and preserves the corresponding discrete dissipation law.Moreover,under certain smoothness assumption,we have also obtained the sec-ond order convergence in space and the first order convergence in time for the scheme.In this paper,we provide a rigorous proof of the error estimate by a careful higher or-der asymptotic expansion and two step error estimates.The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞norm and a refined estimate is applied to derive the optimal error order.

The Neumann Problem for Parabolic Hessian Quotient Equations

In this paper,we consider the Neumann problem for parabolic Hessian quotient equations.We show that the k-admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of elliptic Hessian quotient equations.Also solutions of the classical Neumann problem converge to a translating solution.

A critical case of Rallis inner product formula

Letπbe a genuine cuspidal representation of the metaplectic group of rank n.We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n+1.We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands L-function ofπtwisted by a character.The bulk of this paper focuses on proving a case of regularised Siegel-Weil formula,on which the Rallis inner product formula is based and whose proof is missing in the literature.

Gradient Convergence of Deep Learning-Based Numerical Methods for BSDEs

The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and backward stochastic differential equations, which is based on time discretization of stochastic differential equations(SDEs for short) and the stochastic approximation method for nonconvex stochastic programming problem. They take the stochastic gradient decent method,quadratic loss function, and sigmoid activation function in the setting of the neural network. Combining classical techniques of randomized stochastic gradients, Euler scheme for SDEs, and convergence of neural networks, they obtain the O(K^(-1/4)) rate of gradient convergence with K being the total number of iterative steps.

Boundary Regularity for Minimal Graphs of Higher Codimensions

In this paper,the authors derive H¨older gradient estimates for graphic functions of minimal graphs of arbitrary codimensions over bounded open sets of Euclidean space under some suitable conditions.

Positivity of Fock Toeplitz Operators via the Berezin Transform

Abstract This paper deals with the relationship between the positivity of the Fock Toeplitz operators and their Berezin transforms. The author considers the special case of the bounded radial function φ（z）=α ＋ be-α｜z｜2 ＋ ce-β｜z｜2, where a, b, c are real numbers and α,β are positive numbers. For this type of φ, one can choose these parameters such that the Berezin transform of is a nonnegative function on the complex plane, but the corresponding Toeplitz operator Tφ is not positive on the Fock space.

The Cartesian analytical solutions for the Ndimensional compressible Navier-Stokes equations with density-dependent viscosity

In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integration theory.Two exact solutions are obtained by solving the reduced systems.

Augmented Lagrangian Methods for Convex Matrix Optimization Problems

In this paper,we provide some gentle introductions to the recent advance in augmented Lagrangian methods for solving large-scale convex matrix optimization problems(cMOP).Specifically,we reviewed two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex matrix optimization problems regularized by nonsmooth spectral functions.Under a mild quadratic growth condition on the dual of cMOP,we further discussed the R-superlinear convergence of the Karush-Kuhn-Tucker(KKT)residuals of the sequence generated by the augmented Lagrangian methods(ALM)for solving convex matrix optimization problems.Implementation details of the ALM for solving core convex matrix optimization problems are also provided.