关于RN上的纯量曲率方程

我们证明在RN上纯量方程某种新解的存在性．     

On the Cross Ratio of Solutions of a First-Order Scalar Differential Equation

For a first-order scalar differential equation, a conditionis given which implies that the cross ratio of four solutionsis a monotone function.     

Geometric Transient Solutions of Autonomous Scalar Maps

Smooth autonomous scalar maps with locally asymptotically stable equilibria have families of asymptotically constant solutions which decay geometrically to the equilibria. Locally, all transients converging to the equilibria have this form.     

Higher-Order Method for the Solution of a Nonlinear Scalar Equation

A new iterative method to find the root of a nonlinear scalar function f is proposed. The method is based on a suitable Taylor polynomial model of order n around the current point x _k and involves at each iteration the solution of a linear system of dimension n. It is shown that the coefficient matrix of the linear system is nonsingular if and only if the first derivative of f at x _k is not null. Moreover, it is proved that the method is locally convergent with order of convergence at least n + 1. Finally, an easily implementable scheme is provided and some numerical results are reported.     

Geometric Breathers of the mKdV Equation

In this paper we present some numerical results about the existence of a new family of solutions of the geometric mKdV equation. This equation is characterized by the fact that the curvature evolves according to the focusing mKdV equation. The new type of solutions are similar to the breathers found by M.?Wadati in 1973. We exhibit curves with and without self-intersections, and also some examples of initially simple closed curves that have self-crossings at later times.     

A Geometric Space-Time Multigrid Algorithm for the Heat Equation

We study the time-dependent heat equation on its space-time domain that is discretised by a $k$-spacetree. $k$-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint. The paper presents a full approximation storage geometric multigrid implementation for this setting that combines the smoothing properties of multigrid for the equation's elliptic operator with a multiscale solution propagation in time. While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded, the holistic approach promises to exhibit a better parallel scalability than classical time stepping, adaptive dynamic refinement in space and time fall naturally into place, as well as the treatment of periodic boundary conditions of steady cycle systems, on-time computational steering is eased as the algorithm delivers guesses for the solution's long-term behaviour immediately, and, finally, backward problems arising from the adjoint equation benefit from the the solution being available for any point in space and time.     

中立型标量积分微分方程的周期解

本文考虑中立型标量方程x′(t)=a(t)x(t)+∫     

y-Regularization-Operator Splitting Method for the Numerical Solution of a Scalar Eikonal Equation

In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.     

Degenerate Elliptic Equations and Nonuniqueness of Solutions to the Kolmogorov Equation

Doklady Mathematics - In this paper we propose a new method of constructing examples of nonuniqueness of probability solutions by reducing the stationary Fokker–Planck–Kolmogorov...     

Multiplicity Result for a Scalar Field Equation on Compact Surfaces

Abstract

We consider a scalar field equation on compact surfaces which has variational structure. When the surface is a torus and a physical parameter ρ belongs to (8π,4π²) we show under some extra assumptions that, besides a local minimum, the functional admits at least other two saddle points.     

用于几何非线性分析的增量形式的虚功方程

本文导出了适用于几何非线性有限元分析的增量形式的虚功方程。在这个增量形式的虚功方程中,考虑了积累误差的影响。     

Fundamental Geometric Structures for the Dirac Equation in General Relativity

We present an axiomatic approach to Dirac's equation in General Relativity based on intrinsically covariant geometric structures. Structure groups and the related principal bundle formulation can be recovered by studying the automorphisms of the theory. Various aspects can be most neatly understood within this context, and a number of questions can be most properly addressed (specifically in view of the formulation of QFT on a curved background). In particular, we clarify the fact that the usual spinor structure can be weakened while retaining all essential physical aspects of the theory.     

On the Kolmogorov inequality for fractional derivatives on the half-line

В РАБОтЕ пРИВЕДЕНы НЕ ОБхОДИМыЕ И ДОстАтОЧ НыЕ УслОВИь сУЩЕстВОВАНИь НЕРАВ ЕНстВА НА пОлУпРьМОИ R₊=[0, ∞): $$\left\| {(D^\alpha x)( \cdot )} \right\|_{C(R_ + )} \leqq K\left\| {x( \cdot )} \right\|_{L_2 (R_ + )}^{v_1 } \left\| {(D^n x)( \cdot )} \right\|_{L_2 (R_ + )}^{v_2 } ,$$ гДЕ А-пРОИжВОльНОЕ ВЕ ЩЕстВЕННОЕ ЧИслО,n≧1 — цЕлОЕ Иv _i>0,i=1,2. ДРОБНАь пРОИжВОД НАьD ^α пОНИМАЕтсь В сМыслЕ г. ВЕИль. ВыЧИслЕНА НАИ лУЧшАь (т.Е. НАИМЕНьшАь Иж ВОжМ ОжНых) кОНстАНтАк=к(п, А) В ЁтО М НЕРАВЕНстВЕ И ВыпИс АНА ЁкстРЕМАльНАь ФУНкц Иь, НА кОтОРОИ НЕРАВЕНстВО пРЕВРАЩАЕтсь В РАВЕН стВО.     

A Penalty-Regularization-Operator Splitting Method for the Numerical Solution of a Scalar Eikonal Equation

In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.     

On the Kolmogorov Complexity of Arbitrary Objects

The notion of Kolmogorov program-size complexity (or algorithmic information) is defined here for arbitrary objects. Using a special form of recursive topological spaces, called partition spaces, we define a recursive topology which uses a level of partition for approximation of arbitrary objects instead of the usual metric. It is shown that the formulation for arbitrary objects satisfies most of the previous results obtained usually for natural numbers and for sequences of symbols. Thus we claim the existence of abstract computers formalizes the idea that many real-life objects may, in fact, be calculated (or approximated) effectively. We also show the existence of a universal probability measure for our arbitrary objects.     

m-Dissipativity for Kolmogorov Operator of a Fractional Burgers Equation with Space-time White Noise

This paper is concerned with the essential m-dissipativity of the Kolmogorov operator associated with a fractional stochastic Burgers equation with space-time white noise. Some estimates on the solution and its moments with respect to the invariant measure are given. Moreover we also study the smoothing properties of the transition semigroup and the corresponding fractional Ornstein-Uhlenbeck operator by introducing an auxiliary semigroup and (generalized) Bismut-Elworthy formula. From these results, we prove that the Kolmogorov operator of the problem is m-dissipative and the domain of the infinitesimal generator of the fractional Ornstein-Uhlenbeck operator is a core.