Solitary Solution of Discrete mKdV Equation by Homotopy Analysis Method

In this paper, we apply homotopy analysis method to solve discrete mKdV equation and successfully obtain the bell-shaped solitary solution to mKdV equation. Comparison between our solution and the exact solution shows that homotopy analysis method is effective and validity in solving hybrid nonlinear problems, including solitary solution of difference-differential equation.     

A model for the homotopy theory of homotopy theory

We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory', or more precisely that the category of such models has a well-behaved internal hom-object.

     

关于同伦正则态射

该文在点标拓扑空间的范畴中引进了同伦正则态射的概念,研究了它存在的条件,性质以及它与同伦单（满）态,同伦正则单（满）态和同伦等价之间的密切关系。     

建立在同伦基础之上的一种非线性分析方法(2)——单自由度守恒系统周期的通用公式

本文以非线性的单自由度守恒系统为例,描述了一种新的非线性分析方法。此方法建立在光滑映射的基础上,不依赖于小参数,从而特别适合于求解不具有小参数的强非线性问题。本文所给近似公式的精确度对参数值的变化不敏感,对任意大小的参数都是一致有效的。     

A manifold that does not contain a compact core

A core of a (noncompact) manifold is a submanifold with the property that the inclusion of the submanifold into the manifold is a homotopy equivalence. It is shown by example that a manifold may fail to contain a compact core even though the manifold has the homotopy type of a finite complex.     

A homotopy equivalence that is not homotopic to a topological embedding

An open subset W of Sⁿ, n 6 or N = 4, and a homotopy equivalence ƒ: S² × S^{n − 4} → W are constructed having the property that ƒ is not homotopic to any topological embedding.     

On the cost of computing roots of polynomials

Recently Smale has obtained probabilistic estimates of the cost of computing a zero of a polynomial using a global version of Newton's method. Roughly speaking, his result says that, with the exception of a set of polynomials where the method fails or is very slow, the cost grows as a polynomial in the degree. He also asked whether similar results hold for PL homotopy methods. This paper gives such a result for a special algorithm of the PL homotopy type devised by Kuhn. Its main result asserts that the cost of computing some zero of a polynomial of degreen to an accuracy of ε (measured by the number of evaluations of the polynomial) grows no faster than O(n ³ log₂(n/ε)). This is a worst case analysis and holds for all polynomials without exception. This work was supported, in part, by National Science Foundation Grant MCS79-10027 and, in part, by a fellowship of the Guggenheim Foundation.     

HOMOTOPY FORMULA FOR A LOCAL q-CONCAVE WEDGE IN STEIN MANIFOLDS AND IT'S APPLICATIONS

1IntroductionTheauthorhasobtainedaKoppelman-Leray-Norguetformulaforaiccafq-coneavewedgeinndimensionalSteinmanifoldsX[11,inthispaperbasedonthisformulaweobtainahomotopyformulaforalocalq-concavewedge,byusingthisformulaweobtainthesolutionformulafor0-equationonlocalq-concavewedges,anddiscussanextensionproblemonlocalq-concavewedges.Inviewofsavespareweusethenotationsdefinitionsandresultsdirectlywithoutspreadout.airlateruseweonlyintroducetwonewdefinitions[2]:Definition1.1Acolloction(U,pl,''tPN)wi…     

On homotopy varieties

Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties.     

Diagrams and torsors

Suppose that A is a presheaf of categories enriched in simplicial sets on a small Grothendieck site. It is shown that the homotopy theory of enriched A-diagrams of equivalences taking values in simplicial sets can be identified with the homotopy theory of simplicial presheaves fibred over the diagonalized nerve dBA of A. The set [*,dBA] of morphisms in the simplicial presheaf homotopy category is identified with the set of path components of a category of A-torsors, for a suitable definition of A-torsor. These statements are special cases of localized results which hold when the corresponding localized model structures are proper. Examples of the latter include the motivic homotopy category, and so these results lead to a theory of motivic A-torsors which is classifiable up to equivalence by a family of morphisms in the motivic homotopy category. This research was supported by NSERC. Received: March 2006