On the error estimate for calculating some singular integrals

We consider some classes of multi-dimensional singular integral operators with Calderon-Zygmund kernels, construct their discrete analogues and give the error estimate for approximation of continual singular integral operator by the discrete ones. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the accelerating property of an algorithm for function minimization without calculating derivatives

It is shown that the alogrithm of Ref. E1, when converging on a uniformly convex function and when technical condition (13) of Ref. E1 is satisfied, has ann-iterationQ-superlinear rate of convergence and a behaviour which is a precursor of every-iterationQ-superlinearity. This result overrides and corrects main result Theorem 3.1 of Ref. E1.     

On the accelerating property of an algorithm for function minimization without calculating derivatives

This paper studies the speed of convergence of a general algorithm for function minimization without calculating derivatives. This algorithm contains Powell's 1964 algorithm as well as Zangwill's second modification of this procedure. The main results are Theorems 3.1 and 4.1 which show that, if the algorithm behaves well, then asymptotically almost conjugate directions are built; therefore, the algorithm has an every-iteration superlinear speed of convergence. The paper hinges on ideas of McCormick and Ritter and Powell.The authors wish to thank the Namur Department of Mathematics, especially its optimization group, for many discussions and encouragements. The authors also thank the reviewer for many helpful suggestions.     

A method of calculating massive Feynman integrals

Scientific-Research Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 89, No. 1, pp. 56–72, October, 1991.     

An algorithm for calculating continued fractions

A new algorithm is described in the paper for calculating continued fractions. The conditions are given under which this algorithm is faster than the hitherto fastest algorithm for handling this problem. Also the interrelation between the suggested algorithm and that for calculating a tridiagonal system of linear equations is investigated.     

Weighted spherical semidesigns and cubature formulae for calculating integrals on a sphere

In this paper we study weighted spherical semidesigns, i.e., systems of points on a sphere of a specific type. We propose a new proof of the necessary and sufficient condition for a system of points on a sphere to be a weighted spherical semidesign. This criterion gives new approaches to the construction of cubature formulae for calculating integrals over a sphere with the degree of accuracy of 5 and 9.     

On an interpolatory product rule for evaluating Cauchy principal value integrals

Convergence results for interpolatory product rules for evaluating Cauchy principal value integrals of the form f _?1 ¹ v(x)f(x)/x ? λ dx wherev is an admissible weight function have been extended to integrals of the form f _?1 ¹ k(x)f(x)/x ? λ dx wherek is an arbitrary integrable function subject to certain conditions. Further, whereas the above convergence results were shown when the interpolation points were the Gauss points with respect to some admissible weight functionw, they are now shown to hold when the interpolation points are Radau or Lobatto points with respect tow.     

An algorithm for calculating logarithms through displacement

A procedure is proposed for the factorial expansion of special real numbers, each of which in turn allows a similar expansion. Problems of executing the program for calculating a natural logarithm are discussed. Approaches to generalizing the method are considered.     

An algorithm for calculating generalized continued fractions

In this paper the generalization of a continued fraction in the sense of the Jacobi-Perron algorithm (called an n-fraction) is considered.Apart from the known algorithms to calculate an n-fraction a new one is derived and the algorithms are compared with respect to the number of operations required and the time to execute these operations.     

On weighted inequalities for singular integrals

In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in then the one-sided condition, , is a sufficient condition for the singular integral to be bounded in , , or from into weak- if . This one-sided condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in . The two-sided version of this result is also obtained: Muckenhoupts condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in or in .

     

On a stopping process for oscillatory integrals

In this paper we introduce a stopping process to analyze oscillatory integral operators with degenerate phases. The resulting bounds give smoothing properties for averages along variable families of curves inR ⁿ with two-sided torsion. Supported in part by the National Science Foundation under Grants DMS-92-04196 and DMS-91-04455.A01.     

On integrals and cointegrals for quasi-Hopf algebras

Using the theory of Frobenius algebras, we study how the antipode of a quasi-Hopf algebra H acts on the space of left and right integrals and cointegrals. We obtain formulas that allow us to find out the explicit form of the integrals and cointegrals for the Drinfeld double $D(H)$, in terms of the integrals and cointegrals of H. This leads to an answer to a conjecture made by Hausser and Nill at the end of the nineties.     

On set-valued stochastic integrals in an M-type 2 Banach space

In a separable Banach space, for set-valued martingale, several equivalent conditions based on the measurable selections are discussed, and then, in an M-type 2 Banach space, at first we define single valued stochastic integral by the differential of a real valued Brownian motion, after that extend it to set-valued case. We prove that the set-valued stochastic integral becomes a set-valued submartingale, which is different from single valued case, and obtain the Castaing representation theorem for the set-valued stochastic integral, which is applicable for set-valued stochastic differential equations.