Mechanical theorem proving in differential geometry

An automated reasoning method, based on Wu’s method and calculus of differential forms, is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry. The method has been used to simplify one of Chem’s theorems: “The non-trivial families of isometric surfaces having the same principal curvatures are W-surfaces.” Some other theorems are also tested by this method. The proofs are generally simpler than those in differential geometry textbooks. Project supported partially by the National Natural Science Foundation of China.     

A Note on the Browder＇s and Weyl＇s Theorem

Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.     

Interpolation theory of anisotropic finite elements and applications

Interpolation theory is the foundation of finite element methods.In this paper,after reviewing some existed interpolation theorems of anisotropic finite element methods,we present a new way to analyse the interpolation error of anisotropic elements based on Newton's formula of polynomial interpolation as well as its applications.     

A Jordan–Brouwer Separation Theorem for Polyhedral Pseudomanifolds

The Jordan Curve Theorem referring to a simple closed curve in the plane has a particularly simple proof in the case that the curve is polygonal, called the “raindrop proof”. We generalize the notion of a simple closed polygon to that of a polyhedral (d−1)-pseudomanifold (d≥2) and prove a Jordan–Brouwer Separation Theorem for such a manifold embedded in ℝ ^d . As a by-product, we get bounds on the polygonal diameter of the interior and exterior of such a manifold which are almost tight. This puts the result within the frame of computational geometry. The research of Y.S. Kupitz was partially supported by the Landau Center at the Mathematics Institute of the Hebrew University of Jerusalem (supported by Minerva Foundation, Germany), and by Deutsche Forschungsgemeinschaft.     

Kantorovich’s majorants principle for Newton’s method

We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s method, the relationship of the majorant function and the non-linear operator under consideration. This approach enables us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new estimate of this rate based on a directional derivative of the derivative of the majorant function. Moreover, the majorant function does not have to be defined beyond its first root for obtaining convergence rate results. The research of O.P. Ferreira was supported in part by FUNAPE/UFG, CNPq Grant 475647/2006-8, CNPq Grant 302618/2005-8, PRONEX–Optimization(FAPERJ/CNPq) and IMPA. The research of B.F. Svaiter was supported in part by CNPq Grant 301200/93-9(RN) and by PRONEX–Optimization(FAPERJ/CNPq).     

A-Browder＇s Theorem and Generalized a-Weyl＇s Theorem

Two variants of the essential approximate point spectrum are discussed. We find for example that if one of them coincides with the left Drazin spectrum then the generalized a-Weyl＇s theorem holds, and conversely for a-isoloid operators. We also study the generalized a-Weyl＇s theorem for Class A operators.     

On the Number of Birch Partitions

Birch and Tverberg partitions are closely related concepts from discrete geometry. We show two properties for the number of Birch partitions: Evenness and a lower bound. This implies the first nontrivial lower bound for the number of Tverberg partitions that holds for arbitrary q, where q is the number of partition blocks. The proofs are based on direct arguments and do not use the equivariant method from topological combinatorics.     

Ordering in mechanical geometry theorem proving

Ordering in mechanical geometry theorem proving is studied from geometric viewpoint and some new ideas are proposed. For Thebault’s theorem which is the most difficult theorem that has ever been proved by Wu’s method, a very simple proof using Wu’s method under a linear order is discovered. Project supported by the National Natural Science Foundation of China.     

Singular values, quasiconformal maps and the Schottky upper bound

Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ϕdimtortion function ϕ _K (r) and the Agard η-distortion function η _K (t) follow. By these results, the explicit quasiconformal Schwan lemma is improved, several properties are obtained for the Schottky upper bound, and a conjecture on the linear distortion function λ (K) is proved to be true.     

On the support of solutions to stochastic differential equations with path-dependent coefficients

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron–Martin space under the flow of mild solutions to a system of path-dependent ordinary differential equations. Our result extends the Stroock–Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on functional Itô calculus.     

Existence of solutions of nonlinear integrodifferential equations of sobolev type with nonlocal condition in Banach spaces

In this paper we prove the existence of mild and strong solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal condition. The results are obtained by using semigroup theory and the Schauder fixed point theorem.     

Variational Approach to the Inviscid Compressible Fluid

It is well-known that not every partial differential equation admits a variational formula. A rigorous proof of the existence of a variational principle is very difficult. In this paper, the semi-inverse method proposed by Ji-Huan He is used to construct a variational principle for a one-dimensional inviscid compressible fluid.      

Analogues of Euler and Poisson summation formulae

Euler-Maclaurin and Poisson analogues of the summations ε _{a <n ≤b} χ(n)f(n), have been obtained in a unified manner, where (χ(n)) is a periodic complex sequence;d(n) is the divisor function andf(x) is a sufficiently smooth function on [a, b]. We also state a generalised Abel’s summation formula, generalised Euler’s summation formula and Euler’s summation formula in several variables.     

Boundedness of solutions for Duffing-type equation

The boundedness of all solutions is shown for Duffing-type equations wherep ₁,p ₂,...,p _2n are of period 1 and of Lipschitzian continuity andp _n+1,...,p _2n are of Zygmundian continuity. This conclusion implies that the boundedness phenomenon for the Duffing-type equations does not require the smoothness in the time-variable, thus answering the question posed by Dieckerhoff and Zehnder.     

On a theorem of Wigner on products of positive matrices

A simple algebraic proof of a theorem due to Wigner on the product of three positive matrices is given. It is shown that the theorem holds for four matrices under an additional condition. The proofs are valid in the more general case of operators in a Hilbert space.     

CALCULUS ON CANTOR TRIADIC SET （I）——DERIVATIVE

For real-valued functions defined on Cantor triadic ,set. a derivative with corresponding formula of Newton-Leihniz‘s type is given In particular, for the self-simltar functions and alter-nately jumping functions defined in this paper, their derivative and exceptional sets are studied ac-curately by using ergodic theory on Е2 and Duffin-Scbaeffer‘s theorem coneerning metric diophan-tine approximation. In addition, Haar basis of L2(Е2) is constructed and Flaar expansion of stan-drd self-similar function is given.     

A Remark on Chen’s Theorem (II)

Let p denote a prime and P2 denote an almost prime with at most two prime factors. The author proves that for sufficiently large x,∑p≤x p＋2=P2 1〉1.13Cx/log^2x,where the constant 1.13 constitutes an improvement of the previous result 1.104 due to J. Wu.     

On a generalization of Pólya’s theorem

We consider generalizations of Pólya’s theorem to the case of entire functions taking algebraic values at natural or integer points.     

Boundary Layer Theory and the Zero-Viscosity Limit of the Navier-Stokes Equation

Abstract A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero. This is particularly important when boundaries are present since vorticity is typically generated at the boundary as a result of boundary layer separation. The boundary layer theory, developed by Prandtl about a hundred years ago, has become a standard tool in addressing these questions. Yet at the mathematical level, there is still a lack of fundamental understanding of these questions and the validity of the boundary layer theory. In this article, we review recent progresses on the analysis of Prandtl's equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes equation. We also discuss some directions where progress is expected in the near future. Also at Courant Institute, New York University     

A Local Limit Theorem for Solutions of BSDEs with Mao’s non-Lipschitz Generator

This paper establishes a local limit theorem for solutions of backward stochastic differential equations with Mao＇s non-Lipschitz generator, which is similar to the limit theorem obtained by [3] under the Lipschitz assumption.