Adaptive cubature over a collection of triangles using the d-transformation

We describe an automatic routine to integrate a function over a collection of triangles where on each triangle the function is well behaved, or has singularities of certain types at one or more vertices or edges. The underlying algorithm is globally adaptive and incorporates the d-transformation for extrapolation. Results from performance profile testing indicate that the routine is superior to other published routines when the singularities are located along edges of the triangles.     

Two numerical methods for solving integral equations with a weak singularity

Two approaches are considered to solving two-dimensional Fredholm integral equations of the first kind which arise in spatial problems of optoelectronics. Numerical calculations are given, supporting the efficiency of the proposed methods for complex optoelectronic systems.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 12–20, 1985.     

On the numerical integration of an odd periodic function over a half-period

A method is described for the numerical evaluation of integrals of the form ∫ ₀ ^p f(t)dt, wheref is an odd periodic function with period 2p. The method is based on term-by-term integration of the Fourier sine series forf(t).     

Right triangles with algebraic sides and elliptic curves over number fields

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P _λ of infinite order in the Mordell-Weil group of the elliptic curve Y ² = X ³ − n ² X over ℚ(λ). Research of the rest of authors was supported in part by grant MTM 2006-01859 (Ministerio de Educación y Ciencia, Spain).     

唯一分解整环R上不可约多项式的若干结构

本文在唯一分解整环Ｒ上引入隐含同余和非隐含同余概念．从而获得Ｒ上多项式不可约性的若干结构．一百多年来首次对著名的Ｅｉｓｅｎｓｔｅｉｎ不可约定理作出重要推广     

Geometric convergence of Lagrangian interpolation and numerical integration rules over unbounded contours and intervals

We investigate geometric convergence for rules of numerical integration and the associated Lagrangian interpolation polynomials over unbounded contours and intervals. The results obtained are shown to be substantially best possible.     

Analytic integration of a system of nonlinear ordinary differential equations with an irregular type singularity

Summary We shall discuss how to construct analytic expressions for bounded solutions of nonlinear ordinary differential equations of the form(A) which tend to0 with the order of certain positive powers of x as x approaches the origin through a sector. It will be shown that the opening angle of this sector is maximal. Entrata in Redazione il 1^o luglio 1971.     

Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS²\mathbb{S}^2, we discuss tensor product rules with n ²/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n ³) nodes for numerical integration over spherical caps on \mathbbS²\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that have O(n ^d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n ^d ) nodes and possess a certain regularity property.     

Automorphisms of Chevalley groups of types A l , D l , or E l over local rings with 1/2

In this paper, we prove that every automorphism of an (elementary) Chevalley group of type A _l , D _l , or E _l , l ≥ 2, over a commutative local ring with 1/2 is standard, i.e., is the composition of inner, ring, graph, and central automorphisms.     

Some formulae for the q-Bernoulli and Euler polynomials of higher order

The purpose of this paper is to give a proof of Kummer type congruence for the q-Bernoulli numbers of higher order, which is an answer to a part of the problem in a previous publication (see Indian J. Pure Appl. Math. 32 (2001) 1565-1570).     

Analytic integration of a system of nonlinear ordinary differential equations with an irregular type singularity. - II

Summary We shall study the same differential equations(A) as in our previous paper[5]. The equations(A) admit a formal solution of the form(F) which is expressed by a formal power series of x and φ(x), where φ(x) is a holomorphic general solution of a differential equation of the form(R). Under an additional assumption that the solution φ(x) vanishes identically when the arbitrary constants involved in it are equal to zero, it is already shown in[5] that the formal solution(F _c), which is obtained by formally rearranging(F) in power series of φ(x), is uniformly convergent whenever the values of x and φ(x) satisfy inequalities of the form(0.14). The present paper discusses about how to study an analytic meaning of the formal solution(F) without the additional assumption mentioned above. Dedicated to Professor Hugh.L. Turrittin on the occasion of his retirement. Entrata in Redazione il 27 luglio 1972.     

An efficient numerical scheme for precise time integration of a diffusion-dissolution/precipitation chemical system

A numerical scheme based on an operator splitting method and a dense output event location algorithm is proposed to integrate a diffusion-dissolution/precipitation chemical initial-boundary value problem with jumping nonlinearities. The numerical analysis of the scheme is carried out and it is proved to be of order 2 in time. This global order estimate is illustrated numerically on a test case.

     

Approximation of singularity sets with analytic graphs over the ball in C

Let be a smooth function on the ball in C whose gradient has length less than or equal to 1. We show that if is uniformly near an analytic function on every complex affine one-dimensional slice then it must be near some function analytic on the whole ball. We use this to show the following: a singularity set over the ball which is near the graph of a function with must be near the graph of some analytic function over the ball.

     

Nonlinear integration and Weierstrass integral over a manifold: Connections with theorems on martingales

This paper is a survey on the integrals of the calculus of variations as Weierstrass integrals. While the classic integrals in terms of Lebesgue processes are valid only overAC curves, surfaces, or variety, the corresponding Weierstrass or Burkill integrals are valid in theBV case. This is the immediate advantage of the Weierstrass-Burkill form. Moreover, Cesari defined a very general integration process for set functions, in an abstract setting, which extends the Weierstrass and Burkill integrals. In more detail, we examine here some existence, approximation, representation, and semicontinuity results both in the parametric case and in the nonparametric case.     

An estimate on the error in a numerical integration of a quasilinear equation with a regular degeneracy

For the case of a regular degeneracy of the convexity condition an estimate is obtained for the convergenceO(h In 1/h) of a numerical integration of the quasilinear equation u_t + ((u))_x = 0.Translated from Matematicheskie Zametki, Vol. 18, No. 5, pp. 753–766, November, 1975.The author thanks N. S. Bakhvalov for his attention to this work.     

Second order random fields over l p with homogeneous and isotropic increments

Summary A random field over l _p is a stochastic process X(t), where t is an element of l _p .It is said to have homogeneous and isotropic increments if E(X(t) – X(s)) ² is a function of t-s. The subject of this work is the spectral theory of such processes. The main results are: a representation of the field as a series of filtered, orthogonal processes with a real time parameter; a representation as a white noise integral over l _p ;limit theorems for the average of X over a sphere; and, finally, filtering of the orthogonal components.In particular, we mention: (1) The averages over spheres of increasing dimension converge in quadratic mean for p=2 but not for 0<p<2. (2) The limiting distribution of a fixed coordinate of a point uniformly distributed over the l _p -unit sphere in n-space, n , has the density [2(1/p+1)p ^1/p ]^–1exp(-¦x¦ ^p /P).This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Science Foundation Grants NSF-GP-7378 and NSF-GW-2049.     

Some numerical experience with a globally convergent algorithm for nonlinearly constrained optimization

Global convergence properties are established for a quite general form of algorithms for solving nonlinearly constrained minimization problems. A useful feature of the methods considered is that they can be implemented easily either with or without using quadratic programming techniques. A particular implementation, designed to be both efficient and robust, is described in detail. Numerical results are presented and discussed.This work was carried out by the first author under the support of the Science Research Council (UK), Grant Nos. B/RG/95124 and GR/A/44480, which are gratefully acknowledged.     

Sharp estimates for functions with a pole and logarithmic singularity

We consider functions with a pole and a logarithmic singularity. We obtain sharp estimates for the Schwarzian and the Taylor coefficients of the holomorphic part of such functions. We also describe geometric properties of conformal mappings of the exterior of the unit disc with a cut that connects some boundary point with the point at infinity.