On the geometry of the nets in the n-dimensional space of Weyl

Chebyshev and geodesic curvatures of the lines of an arbitrary net belonging to the n-dimensional space of Weyl W_n are introduced. Characteristics of the following special nets in W_n are found: strongly parallel, b-net, c-net and orthogonal (theorems 1.2, 1.3, 1.5, 1.6). Some properties of the Chebyshev nets, orthogonal b-nets and orthogonal c-nets, of the Chebyshev vectors of the second kind, of the orthogonal nets and nets containing Chebyshev subnets are established (theorems 1.1, 1.7, 1.8, 1.4). The fundamental formulae in the case of an orthogonal coordinate net are obtained. The spaces W_n containing one of the following special orthogonal nets — strongly parallel of the first kind, Chebyshev net of the second kind and b-net are defined.     

On the zeros of the Scorer functions

Asymptotic approximations are developed for zeros of the solutions Gi(z) and Hi(z) of the inhomogeneous Airy differential equation . The solutions are also called Scorer functions. Tables are given with numerical values of the zeros.     

Asymptotic expansions of the distributions of the latent roots in MANOVA and the canonical correlations

Asymptotic expansions are given for the density function of the normalized latent roots of S₁S₂^?1 for large n under the assumption of $\text{Ω = O(n)}$, where S₁ and S₂ are independent noncentral and central Wishart matrices having the $\text{W}{}_{\mathrm{p}}\text{(b, Σ; Ω)}$ and W_p(n, Σ) distributions, respectively. The expansions are obtained by using a perturbation method. Asymptotic expansions are also obtained for the density function of the normalized canonical correlations when some of the population canonical correlations are zero.     

Perturbation of the SVD in the presence of small singular values

This paper gives SVD perturbation bounds and expansions that are of use when an m × n, m ? n matrix A has small singular values. The first part of the paper gives subspace bounds that are closely related to those of Wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace. In the second part first and second order approximations are given for perturbed singular values. The subspace bounds are used to show that all approximations retain accuracy when applied to small singular values. The paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation.     

The Influence of Singular Points of the Laplace Image on the Generalized Spectrum of the Original

Integral representations of the Fourier–Jacobi coefficients are constructed in the form of Riemann–Mellin integrals, and on this basis asymptotic formulas for calculating coefficients for large-order numbers are derived.     

Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation

A method for computing the eigenvalues λ _mn (b, c) and the eigenfunctions of the Coulomb spheroidal wave equation is proposed in the case of complex parameters b and c. The solution is represented as a combination of power series expansions that are then matched at a single point. An extensive numerical analysis shows that certain b _s and c _s are second-order branch points for λ _mn (b, c) with different indices n ₁ and n ₂, so that the eigenvalues at these points are double. Padé approximants, quadratic Hermite-Padé approximants, the finite element method, and the generalized Newton method are used to compute the branch points b _s and c _s and the double eigenvalues to high accuracy. A large number of these singular points are calculated.     

On the complement of the Richardson orbit

We consider parabolic subgroups of a general linear group over an algebraically closed field k whose Levi part has exactly t factors. By a classical theorem of Richardson, the nilradical of a parabolic subgroup P has an open dense P-orbit. In the complement to this dense orbit, there are infinitely many orbits as soon as the number t of factors in the Levi part is ≥6. In this paper, we describe the irreducible components of the complement. In particular, we show that there are at most t ? 1 irreducible components. We are also able to determine their codimensions.     

Qualitative investigation of the three-phase solutions of the sine-Laplace equation

Real three-phase solutions of the sine-Laplace equation are constructed. All smooth and singular real doubly periodic solutions are found. The corresponding three-dimensional theta functions are reduced to the elliptic Jacobi functions. Some classes of solutions with symmetries giving possibilities for physical applications are determined. Bibliography: 18 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1996, pp. 199–216.     

On the stability of the J transformation

The stability of a large class of nonlinear sequence transformations is analyzed. Considered are variants of the J transformation [17]. Suitable variants of this transformation belong to the most successful extrapolation algorithms that are known [20]. Similar to recent results of Sidi, it is proved that the _p {J} transformations, the Weniger S transformation, the Levin transformation and a special case of the generalized Richardson extrapolation process of Sidi are S-stable. An efficient algorithm for the calculation of stability indices is presented. A numerical example demonstrates the validity of the approach. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the problem of the critical bound

One new notation, called theA-prompt permitting, is defined in the Turing reducibility, and its basic properties are given, by which we shall show that two r.e. setsA andB are one minimal pairs if and only ifB is notA-prompt permitting. It is proved that there are no maximal minimal pairs. Research partially supported by NSF grand JJ890406 of China. The same results were proved by Harrington and Soare [ta] in 1988, which were reported at the recursion theory meeting at Oberwolfach in Mar. 1989 and at the set theory meeting at Berkeley in Oct. 1990, respectively.     

On the Identification of Sets of Points in the Square Lattice

   Abstract. Identifying codes in the square lattice are considered. The motivation for these codes is the following: if a multiprocessor system is modelled by the square lattice, then we can locate faulty processors in the system with the aid of identifying codes. Constructions, some of which are optimal, are given.     

Uniqueness of the group operation on the lattice of order-automorphisms of the real line

A(R) is the lattice-ordered group (l-group) of all order-automorphisms of the real lineR, with the usual pointwise order and “of course” with composition as the group operation. In fact, what other choices are there for a group operation having the same identity that would give anl-group? Composition in the reverse order would work. But there are no other choices — the group operation can be recognized in the lattice. Several classes of abelianl-groups having a unique group operation have been found by Conrad and Darnel, but this is the first non-abelian example having the minimum of two group operations. “Conversely”, Holland has shown that for the groupA(R) under composition, the only lattice orderings yielding anl-group are the pointwise order and its dual. These results also hold for the rational lineQ.     

On the relationship between the genus and the cardinality of the maximum matchings of a graph

Lower bounds on the cardinality of the maximum matchings of graphs are established in terms of a linear polynomial of p, p⁽¹⁾, p⁽²⁾ and γ whose coefficients are functions of κ, where p is the number of the vertices of a graph, p⁽ⁱ⁾ the number of the vertices of degree i (i = 1,2), γ the genus and κ the connectivity.     

On the explicit solutions of forms of the Sylvester and the Yakubovich matrix equations

In this paper, we consider the explicit solutions of two matrix equations, namely, the Yakubovich matrix equation V−AVF=BW and Sylvester matrix equations AV−EVF=BW,AV+BW=EVF and AV−VF=BW. For this purpose, we make use of Kronecker map and Sylvester sum as well as the concept of coefficients of characteristic polynomial of the matrix A. Some lemmas and theorems are stated and proved where explicit and parametric solutions are obtained. The proposed methods are illustrated by numerical examples. The results obtained show that the methods are very neat and efficient.     

On the classification problem of the quasi-isomorphism classes of free chain algebras

The present paper is devoted to the classification problem of the quasi-isomorphism classes of free differential graded algebras (dgas) over a (P.I.D) R. We introduce the notion of coherent homomorphisms, perfect and quasi-perfect dgas (the Adams-Hilton model of simply connected CW-complex such that H_∗(X,R) is free is a such a dga) and our first main result asserts that two perfect (quasi-perfect) dgas are quasi-isomorphic if and only if their Whitehead exact sequences are coherently isomorphic. Moreover we define the notion of a strong isomorphism between the Whitehead exact sequences and we show that two free R-dgas, of which their Whitehead exact sequences are strongly isomorphic, are quasi-isomorphic.     

Useful properties of the coefficients of the Slevinsky-Safouhi formula for differentiation

Properties of coefficients of the Slevinsky-Safouhi formula are derived. These properties are: matrix and orthogonality properties; the characterization of a zero array; generating functions; asymptotics; and, recurrence relations for computing a sum as efficiently as possible. These properties are useful in a numerical or computational setting.     

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularityinvariants associated to a polynomial f and its germ at 0, respectively.By definition, these zeta functions are rational functions inone variable, and their poles are negative rational numbers.In this paper we study their poles of maximal possible order.When f is non-degenerate with respect to its Newton polyhedron,we prove that its local topological zeta function has at mostone such pole, in which case it is also the largest pole; wegive a similar result concerning the global zeta function. Moreover,for any f we show that poles of maximal possible order are alwaysof the form –1/N with N a positive integer. 1991 MathematicsSubject Classification 14B05, 14E15, 32S50.     

Some characterisations of the ellipsoid

We show that a convex bodyK of dimensiond≧3 is an ellipsoid if it has any of the following properties: (1) the “grazes” of all points close toK are flat, (2) all sections of small diameter are centrally symmetric, (3) parallel (d−1)-sections close to the boundary are width-equivalent, (4)K is strictly convex and all (d−1)-sections close to the boundary are centrally symmetric; the last two results are deduced from their 3-dimensional cases which were proved by Aitchison.     

Wave Invariants of the Spectrum of the G-Invariant Laplacian and the Basic Spectrum of a Riemannian Foliation

Given a compact boundaryless Riemannian manifold upon which a compact Lie group G acts by isometries, recall that the G-invariant Laplacian is the restriction of the ordinary Laplacian on functions to the space of functions which are constant along the orbits of the action. By considering the wave trace of the invariant Laplacian and the connection between G manifolds and Riemannian foliations, invariants of the spectrum of the G-invariant Laplacian can be computed. These invariants include the lengths of certain geodesic arcs which are orthogonal to the principal orbits and contained in the open dense set of principal orbits are associated to the singularities of the wave trace of the G-invariant spectrum. If the action admits finite orbits, then the invariants also include the lengths of certain geodesics arcs connecting the finite orbit to itself. Under additional hypotheses, we obtain partial wave traces. As an application, a partial trace formula for Riemannian foliations with bundle-like metrics is also presented, as well as several special cases where better results are available.     

On the geometry of the space of oriented lines of Euclidean space

We prove that the space of all oriented lines of the n-dimensional Euclidean space admits a pseudo-Riemannian metric which is invariant by the induced transitive action of a connected closed subgroup of the group of Euclidean motions, exactly when n=3 or n=7 (as usual, we consider Riemannian metrics as a particular case of pseudo-Riemannian ones). Up to equivalence, there are two such metrics for each dimension, and they are of split type and complete. Besides, we prove that the given metrics are Kähler or nearly Kähler if n=3 or n=7, respectively.