Rank formulae from the perspective of orthogonal projectors

For matrices F and G having the same number of rows and the orthogonal projectors P?=?FF ^? and Q?=?GG ^?, with F ^? and G ^? denoting the Moore–Penrose inverses of F and G, respectively, several formulae for ranks of various functions of F, G, P and Q are established. Besides a collection of original characterizations, many of which involve the ranks of F*G and (F?:?G) (which coincide with the ranks of PQ and P?+?Q, respectively), some properties known in the literature are reestablished in a generalized form. The variety of relationships considered shows that the approach utilized in the article, based on the partitioned representations of the projectors, provides a powerful tool of wide applicability.     

Some estimation theory on the sphere

Summary We consider distributions with densities of the formf(μ′x) andf(‖x _v ‖) where μ andx are unit vectors inR ^q and ‖x _v ‖ is the norm of the part ofx in somes dimensional subspaceV ofR ^q . For several loss functions, optimal Bayesian and Pitman estimators of μ andV are given. When uniform priors are used, these estimators are identical. Then the infinitesimal robustness characteristics of several special cases of these estimators are calculated.     

Manifolds modelled over free modules over the double numbers

Manifolds over the algebra of double numbers, which include the case of manifolds equipped with a pair of equidimensional supplementary foliations, are studied. To this end, B-holomorphic functions and B-analytic functions on B ⁿ, where B denotes the algebra of double numbers, are defined and studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

On a strong property of the weak subalgebra lattice

In the present paper, we apply results from [Pió1] to prove that for an arbitrary total and locally finite unary algebra A of finite unary type K, its weak subalgebra lattice uniquely determines its strong subalgebra lattice (recall that in the case of total algebras the strong subalgebra lattice is the well-known lattice of all (total) subalgebras). More precisely, we prove that for every unary partial algebra B of the same unary type K, if weak subalgebra lattices of A and B are isomorphic (with A as above), then the strong subalgebra lattices of A and B are isomorphic, and moreover B is also total and locally finite. At the end of this paper we also show the necessity of all the three conditions for A. Received September 5, 1997; accepted in final form October 7, 1998.     

Approximate multiplication of tensor matrices based on the individual filtering of factors

Algorithms are proposed for the approximate calculation of the matrix product $ \tilde C $ \tilde C ≈ C = A · B, where the matrices A and B are given by their tensor decompositions in either canonical or Tucker format of rank r. The matrix C is not calculated as a full array; instead, it is first represented by a similar decomposition with a redundant rank and is then reapproximated (compressed) within the prescribed accuracy to reduce the rank. The available reapproximation algorithms as applied to the above problem require that an array containing r ^2d elements be stored, where d is the dimension of the corresponding space. Due to the memory and speed limitations, these algorithms are inapplicable even for the typical values d = 3 and r ∼ 30. In this paper, methods are proposed that approximate the mode factors of C using individually chosen accuracy criteria. As an application, the three-dimensional Coulomb potential is calculated. It is shown that the proposed methods are efficient if r can be as large as several hundreds and the reapproximation (compression) of C has low complexity compared to the preliminary calculation of the factors in the tensor decomposition of C with a redundant rank.     

Varieties of finite supersolvable groups with the M. Hall property

The varieties in the title are shown to be precisely the product varieties G_p*Ab(d) for some prime p and some positive integer d dividing p−1. Here G_p denotes the variety of all finite p-groups and Ab(d) the variety of all finite Abelian groups of exponent dividing d. It turns out that these are exactly those varieties H of supersolvable groups for which all finitely generated free pro-H groups are freely indexed in the sense of Lubotzky and van den Dries. Several alternative characterizations of these varieties are presented. Some applications to formal language theory and finite monoid theory are also given. Among these is the determination of all supersolvable solutions H to the equations PH = J*H and J*H = J H which is, to the present date, the most complete solution to a problem raised by Pin. Another consequence of our results is that for each such variety H the monoid variety PH = J*H = J H has decidable membership. The authors gratefully acknowledge the support of NSERC     

The Dimension of the Hilbert Scheme of Special Threefolds #

The Hilbert scheme of 3-folds in ? ⁿ , n ≥  6 , that are scrolls over ? ² or over a smooth quadric surface Q  ? ? ³ or that are quadric or cubic fibrations over ? ¹ is studied. All known such threefolds of degree 7  ≤ d ≤  11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed.     

On the estimates of the order of finitelya k -determined map-germs

In this paper we deal with the estimates of the order of determinacy of finitely determined map-germs. This problem has been noted by mathematicians for more than ten years. But the results given by some authors for the estimates of the order of determinacy are stated with respect toa, while those given in this paper are stated with respect toa _k , wherek may be any non-negative integer, and we havea ₀=a in particular. This paper was completed during the author's visit to the Institute of Mathematics, Now I'd like to express my heart-felt thanks to the Institute for its financial support and Prof. Li Peixin for his instruction.     

On the U-Invariant of P-Adic Function Fields

ABSTRACT

We show that a quadratic form defined over the rational function field ?(x ₁ , …, _{x n} ) of dimension at least 4.2 ⁿ  + 1 is isotropic over all fields ? _p (x ₁ , …, _{x n} ), except for finitely many primes. Partial results concerning the u-invariant of p-adic function fields are also shown.     

连续自映射拓扑压的两个注记

令T:X→ X是紧度量空间(X,d)上的连续映射.该文给出了T的拓扑压和T在非游荡集上的限制的拓扑压相等的不依赖于变分原理的一个直接证明.同时,还讨论了半共轭的两个系统的拓扑压之间的关系,证明了拓扑压在一致有限对一条件下是半共轭不变量.     

The r-cubical Lattice and a Generalization of the cd-index

In this paper we generalize thecd-index of the cubical lattice to anr-cd-index, which we denote byΨ(r). The coefficients ofΨ(r) enumerate augmented Andrér-signed permutations, a generalization of Purtill's work relating thecd-index of the cubical lattice and signed André permutations. As an application we use ther-cd-index to determine that the extremal configuration which maximizes the Möbius function of arbitrary rank selections, where all ther_i's are greater than one, is the odd alternating ranks, {1, 3, 5, ...}.     

On the Beth properties of some intuitionistic modal logics

 Let L be one of the intuitionistic modal logics considered in [4]. As in the classical modal case (see [7]), we define two different forms of the Beth property for L, which are denoted by B ₁ and B ₂ ; in this paper we study the relation among B ₁ ,B ₂ and the interpolation properties C ₁ and C ₂ , introduced in [4]. It turns out that C ₁ implies B ₁ , but contrary to the boolean case, is not equivalent to B ₁ . It is shown that B ₂ and C ₂ are independent, and moreover it comes out that, in contrast to classical case, there exists an extension of the intuitionistic modal logic of S ₄ -type, that has not the property B ₂ . Finally we give two algebraic properties, that characterize respectively B ₁ and B ₂ . Received: 27 September 1999 / Published online: 31 May 2002     

The linear combination of vectors implies the existence of the cross and dot products

Given two vectors u and v, their cross product u × v is a vector perpendicular to u and v. The motivation for this property, however, is never addressed. Here we show that the existence of the cross and dot products and the perpendicularity property follow from the concept of linear combination, which does not involve products of vectors. For our proof we consider the plane generated by a linear combination of uand v. When looking for the coefficients in the linear combination required to reach a desired point on the plane, the solution involves the existence of a normal vector n = u × v. Our results have a bearing on the history of vector analysis, as a product similar to the cross product but without the perpendicularity requirement existed at the same time. These competing products originate in the work of two major nineteen-century mathematicians, W. Hamilton, and H. Grassmann. These historical aspects are discussed in some detail here. We also address certain aspects of the teaching of u × v to undergraduate students, which is known to carry some difficulties. This includes the algebraic and geometric denitions of u × v, the rule for the direction of u × v, and the pseudovectorial nature of u × v.     

On the vector valued Hausdorff-Young inequality

This paper studies Banach space valued Hausdorff-Young inequalities. The largest part considers ways of changing the underlying group. In particular the possibility to deduce the inequality for open subgroups as well as for quotient groups arising from compact subgroups is secured. A large body of results concerns the classical groupsT ⁿ ,R ⁿ andZ _k . Notions of Fourier type are introduced and they are shown to be equivalent to properties expressed by finite groups alone.     

Conjectures around the baker-gammel-wills conjecture

The Baker-Gammel-Wills Conjecture states that if a functionf is meromorphic in a unit diskD, then there should, at least, exist an infinite subsequenceN ⊆N such that the subsequence of diagonal Padé approximants tof developed at the origin with degrees contained inN converges tof locally uniformly inD/{poles off}. Despite the fact that this conjecture may well be false in the general Padé approximation in several respects. In the present paper, six new conjectures about the convergence of diagonal Padé approximants are formulated that lead in the same direction as the Baker-Gammel-Wills Conjecture. However, they are more specific and they are based on partial results and theoretical considerations that make it rather probable that these new conjectures hold true.     

Geometric Characterizations for Variational Minimization Solutions of the 3-Body Problem

Abstract In this paper, we prove that for any given positive masses the variational minimization solutions of the 3-body problem in R ³ or R ² are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular restricted 3-body problem in R ³ or R ² are also planar equilateral triangle circular solutions. *Partially supported by the NNSF and MCME of China, the Qiu Shi Sci. and Tech. Foundation, and Edu. Comm. of Tianjin City. Associate Member of the ICTP. **Partially supported by the NNSF of China     

On Some Finitary Conditions Arising from the Axiomatisability of Certain Classes of Monoid Acts

This article considers those monoids S satisfying one or both of the finitary properties (R) and (r), focussing for the most part on inverse monoids. These properties arise from questions of axiomatisability of classes of S-acts, and appear to be of interest in their own right. If S is weakly right noetherian (WRN), that is, S has the ascending chain condition on right ideals, then certainly (r) holds. Other than this, we show that (R), (r), and (WRN) are independent. Our most detailed results are for Clifford monoids, in which case we completely characterise those S with trivial structure homomorphisms satisfying (R) or (r).     

The distribution of the characteristic roots ofS 1 S 2 −1 under violations in the complex case and power comparisons of four tests

Summary The joint density function of the latent roots ofS ₁ S ₂ ⁻¹ under violations is obtained whereS ₁ has a complex non-central Wishart distributionW _c (p,n ₁,Σ ₁,Ω) andS ₂, an independent complex central Wishart,W _c (p,n ₂,Σ ₂, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.