Surfaces of order three with a peak. II

In [1], we presented a theory of surfaces of order three in real projective three-spaceP ³. In the present paper, we prove that a surfaceF of order three with a peak contains one, two or three lines and there are four types of suchF based upon the configuration of these lines. We describe eachF by determining the existence and the distribution of elliptic, parabolic and hyperbolic points; that is, points ofF not lying on any line contained inF.     

On a Steffensen-Hermite method of order three

In this paper we study a third order Steffensen type method obtained by controlling the interpolation nodes in the Hermite inverse interpolation polynomial of degree 2. We study the convergence of the iterative method and we provide new convergence conditions which lead to bilateral approximations for the solution; it is known that the bilateral approximations have the advantage of offering a posteriori bounds of the errors. The numerical examples confirm the advantage of considering these error bounds.     

On construction of a fourth order rotatable design with three factors

The necessary and sufficient conditions for a set of points to form a rotatable design of order four have been given by Patel and Arap Koske [1], In this paper, a set of 98 points for three factors which satisfies these conditions is given.     

Groups with handles of order different from three

We obtain a test for the unsimplicity of an infinite group.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1030–1042, August, 2004.     

Embeddedsdirk-methods of basic order three

The purpose of this report is to construct 3-stage SDIRK-methods (Singly Diagonally Implicit Runge-Kutta) to be used for the code SIMPLE. The local error control is performed by embedding techniques. Pairs with and without extrapolation are given.Dedicated to Germund Dahlquist, on the occasion of his 60'th birthday.This work was supported by NATO, grant 96.80/D1.     

Classification of Twistor Spaces with a Pencil of Surfaces of Degree 1, Part I

We display a family of compact complex 3-manifolds which yields all twistor spaces containing a pencil of surfaces transversal to the twistor lines. This also yields explicit families of self-dual metrics on connected sums of complex projective planes. This yields a completely alternative approach to a class of such metrics which was recently found by Claude le Brun [11].     

一类三次系统的极限环

主要研究一类三次系统的极限环存在性问题,推广了C.Chicone[2]的结果,给出此类系统极限环存在定理.     

Flat cyclotomic polynomials of order three

We say that a cyclotomic polynomial Φn has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn. For each pair of primes p<q, we give an infinite family of r such that A(pqr)=1. We also prove that A(pqr)=A(pqs) whenever s>q is a prime congruent to .     

On unitary transposable matrices of order three

Mathematical Notes - A matrix A ∈ M n (?) is said to be unitarily transposable if A T = Q*AQ for a certain unitary matrix Q. Every 2 × 2 matrix is unitarily transposable; however,...     

Constructing all magic squares of order three

We find by applying MacMahon's partition analysis that all magic squares of order three, up to rotations and reflections, are of two types, each generated by three basis elements. A combinatorial proof of this fact is given.     

Characteristic polynomial and higher order traces of third order three dimensional tensors

Eigenvalues of tensors play an increasingly important role in many aspects of applied mathematics. The characteristic polynomial provides one of a very few ways that shed lights on intrinsic understanding of the eigenvalues. It is known that the characteristic polynomial of a third order three dimensional tensor has a stunning expression with more than 20000 terms, thus prohibits an effective analysis. In this article, we are trying to make a concise representation of this characteristic polynomial in terms of certain basic determinants. With this, we can successfully write out explicitly the characteristic polynomial of a third order three dimensional tensor in a reasonable length. An immediate benefit is that we can compute out the third and fourth order traces of a third order three dimensional tensor symbolically, which is impossible in the literature.     

Riemann boundary problem with infinite index of logarithmic order on a spiral-form contour. I

We study the Riemann boundary problem with infinite index of logarithmic order on an open rectifiable Jordan spiral-form contour, here the influence of the contour on the solvability of the problem is comparable with the influence of the argument of its coefficient. An explicit solution of the problem is constructured in the class of functions admitting weak-power singularities at the ends of a line of conjugacy.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1509–1517, November, 1990.     

三阶微分方程解的增长性

In this paper,the precise estimation of the order and hyper-order of solutions of a class of three order homogeneous and non-homogeneous linear differential equations are obtained. The results of M. Ozawa (1980), G. Gundersen (1988) and J. K. Langley ( 1986 ) are improved.