Оптимальные формулы типа Эйлера-Маклорена для численного интегрирования в пространстве Соболева

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L2(m)(0,1) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number N ≥ m-3 and for any m ≥ 4 using S. L. Sobolev method which is based on the discrete analogue of the differential operator d2m/dx2m. In particular, for m = 4 and m = 5 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m=6 new optimal quadrature formulas are obtained. At the end of this work some numerical results are presented. В настоящей статье рассматривается задача построения оптимальных квадратурных формул в смысле Сарда в пространстве L2(m)(0,1). Здесь квадратурная сумма состоить из значений подынтегральной функции в узлах и значений первой и третьей производных подынтегральной функции на концах интервала интегрирования. Найдены коэффициенты оптимальных квадратурных формул и вычислена норма оптимального функционала погрешности для любого натурального числа N ≥ m-3 и для любого m ≥ 4, используя метод С. Л. Соболева который основывается на дискретный аналог дифференциального оператора d2m/dx2m. В частности, при m = 4 и m = 5 получен оптимальность классической формулы Эйлера-Маклорена. Начиная с m = 6 получены новые оптимальные квадратурные формулы. В конце работы приведаны некоторые численные результаты.

Objects generated by an arbitrary natural number

The set \underline{Set}(n), generated by an arbitrary natural number n, is defined. Some arithmetic functions, defined over its elements are introduced. Some of the arithmetic, set-theoretical and algebraic properties of the new objects are studied.

Permutations that preserve asymptotically null sets and statistical convergence

The main result of this article is a characterization of the permutations ?: N ? N that map a set with zero asymptotic density into a set with zero asymptotic density; a permutation has this property if and only if the lower asymptotic density of Cp tends to 1 as p ? ? where p is an arbitrary natural number and Cp = {l : ?-1(l)? lp}. We then show that a permutation has this property if and only if it maps statistically convergent sequences into statistically convergent sequences.

Vanishing theorems for higher-order Killing and Codazzi

A Killing p-tensor (for an arbitrary natural number p ≥ 2) is a symmetric p-tensor with vanishing symmetrized covariant derivative. On the other hand, Codazzi p-tensor is a symmetric p-tensor with symmetric covariant derivative. Let M be a complete and simply connected Riemannian manifold of nonpositive (resp. non-negative) sectional curvature. In the first case we prove that an arbitrary symmetric traceless Killing p-tensor is parallel on M if its norm is a Lq -function for some q > 0. If in addition the volume of this manifold is infinite, then this tensor is equal to zero. In the second case we prove that an arbitrary traceless Codazzi p-tensor is equal to zero on a noncompact manifold M if its norm is a Lq -function for some q  1 .

Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations

The Kadomtsev-Petviashvili and Boussinesq equations (uxxx -6uux)x -utx ±uyy = 0; (uxxx - 6uux)x +uxx ±utt = 0; are completely integrable, and in particular, they possess the three-soliton solution. This article aims to expose a uniqueness property of the Kadomtsev-Petviashvili (KP) and Boussinesq equations in the integrability theory. It is shown that the Kadomtsev-Petviashvili and Boussinesq equations and their dimensional reductions are the only integrable equations among a class of generalized Kadomtsev-Petviashvili and Boussinesq equations (ux1x1x1 - 6uux1 )x1 + ΣMi;j=1aijuxixj = 0; where the aij’s are arbitrary constants and M is an arbitrary natural number, if the existence of the three-soliton solution is required

Quantum Multiple Construction of Subfactors

We construct the quantum s-tuple subfactors for an AFD II1 subfactor with finite index and depth, for an arbitrary natural number s. This is a generalization of the quantum multiple subfactors by Erlijman and Wenzl, which in turn generalized the quantum double construction of a subfactor for the case that the original subfactor gives rise to a braided tensor category. In this paper we give a multiple construction for a subfactor with a weaker condition than braidedness of the bimodule system.

On Orbits of the Ring Zmn under Action of the Group SL(m, Zn)

We consider the action of the finite matrix group SL(m,Zn ) on the ring Zmn. We determine orbits of this action for n arbitrary natural number. It is a generalization of the task which was studied by A. A. Kirillov for m = 2 and n prime number.

A note on Baker's method

Let …, αn, (n ≧ 2) be (fixed) multiplicatively independent non zero algebraic numbers and set M(H) = min |β1log α1+…+βn log αn| the minimum taken over all algebraic numbers bgr;1,…βn not all equal to zero, of degrees not exceeding a fixed natural number d0, and heights not exceeding an arbitrary natural number H. Then an important result [1] of Baker states that for every fixed ε > 0 and an explicit constant . It may be remarked that Baker deduces his general result from the special case where βn is fixed to be —1. The following straight forward generalization might be of some interest since it shows that the exponent n+1+ε need not be the best, and that the best exponent obtainable by his method has some chance of being 1 + ε (see the corollary to the Theorem).