Commutators of symmetries in characteristic 2

Let V be a nonsingular vector space over a field K of characteristic 2 with |K|>3. Suppose K is perfect and π is an element in the special orthogonal group SO(V)=Ω(V) with dimB(π)=2d. The length of π with respect to the symmetry commutators is d if B(π) is not totally isotropic; otherwise it is d+1.     

Products of quasireflections and transvections over local rings

Let R be a commutative local ring and M a free R-module of rank n. The module M is endowed with a metric structure given by a sesquilinear form or by a quadratic form. We show, every isometry of M is a product of quasireflections or transvections. We determine the minimal number of factors needed in any factorization of if the path of is a subspace. For all other isometries we obtain only an upper bound.Research supported in part by NSERC Canada grant A7251To Helmut Karzel on his 60th birthday     

Equality of orthogonal transvection group and elementary orthogonal transvection group

H. Bass defined orthogonal transvection group of an orthogonal module and elementary orthogonal transvection group of an orthogonal module with a hyperbolic direct summand. We also have the notion of relative orthogonal transvection group and relative elementary orthogonal transvection group with respect to an ideal of the ring. According to the definition of Bass relative elementary orthogonal transvection group is a subgroup of the relative orthogonal transvection group of an orthogonal module with hyperbolic direct summand. Here we show that these two groups are the same in the case when the orthogonal module splits locally.     

On subgroups of orthogonal groups of isotropic quadratic forms

Let a field K be an algebraic extension of a subfield k of characteristic not 2, n an integer, a non-degenerate isotropic form in n variables over K with coefficients in k. We study subgroups of the orthogonal group O_n(K,Q) that contain the derived subgroup Ω_n(k,Q) of the group O_n(k,Q).     

The subregular variety to the variety of special lattices

We recall the basic geometric properties of the full lattice variety, the projective variety parametrizing special lattices over Witt vectors which was introduced in Haboush (2005) [6]. It is an analog in unequal characteristic, of a certain Schubert variety in the affine Grassmannian for , and it is normal and a locally complete intersection (Haboush and Sano, submitted for publication [7], Sano (2004) [15]). In this paper, I prove that the complement of its smooth locus, the subregular variety in it, is also normal and a locally complete intersection. The result is analogous to the geometry of the subregular subvariety of the nilpotent cone.     

On the class group and S-class group of formal power series rings

Let ${\text{Cl}}_t(A)$ denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is injective and if A is a regular UFD, then ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$; $[I]?[{(I.A?X?)}_t]$ is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an $s\in S$ and $a\in I$ such that $sI?aA?I$. The S-class group of A, S-${\text{Cl}}_t(A)$ is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-${\text{Cl}}_t(A)?S$-${\text{Cl}}_t(A?X?)$.     

The combinatorially regular polyhedra of index 2

Summary We investigate polyhedral realizations of regular maps with self-intersections in E³, whose symmetry group is a subgroup of index 2 in their automorphism group. We show that there are exactly 5 such polyhedra. The polyhedral sets have been more or less known for about 100 years; but the fact that they are realizations of regular maps is new in at least one case, a self-dual icosahedron of genus 11. Our polyhedra are closely related to the 5 regular compounds, which can be interpreted as discontinuous polyhedral realizations of regular maps.The author was born on March 5, 1937; so exactly half a century after Otto Haupt.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Rounding error analysis of the classical Gram-Schmidt orthogonalization process

This paper provides two results on the numerical behavior of the classical Gram-Schmidt algorithm. The first result states that, provided the normal equations associated with the initial vectors are numerically nonsingular, the loss of orthogonality of the vectors computed by the classical Gram-Schmidt algorithm depends quadratically on the condition number of the initial vectors. The second result states that, provided the initial set of vectors has numerical full rank, two iterations of the classical Gram-Schmidt algorithm are enough for ensuring the orthogonality of the computed vectors to be close to the unit roundoff level.The work of the second author was supported in part by the US Department of Energy, Office of Basic Energy Science under LAB03-17 initiative, DOE contract No. DE-FG02-03ER25584, and in part by the TeraScale Optimal PDE Simulations (TOPS) SciDAC, DoE Contract No. DE-FC02-01ER25480The work of the third author was supported by the project 1ET400300415 within the National Program of Research ‘‘Information Society’’ and by the GA AS CR under grant No. IAA1030405.     

Sur le tétraèdre régulier dans un espace projectif complexe

Résumé On montre l'existence, dans un espace projectif complexe, d'un tétraèdre régulier ayant un groupe de symétrie isomorphe à celui du tétraèdre régulier euclidien. On précise la classe d'homologie de dimension 2, qui contient le complexe simplicial qui lui est associé.

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We show the existence, in a complex projective space, of a regular tetrahedron which has its symmetry group isomorphic to that of the euclidean regular tetrahedron. We give precisely the two-dimensional homology class, containing the simplicial complexe associated to it.

     

A constructive proof of the Cartan-Dieudonné-Scherk Theorem in the real or complex case

The Cartan-Dieudonné-Scherk Theorem states that for fields of characteristic other than 2, every orthogonality can be written as the product of a certain minimal number of reflections across hyperplanes. The earliest proofs are not constructive, and constructive proofs either do not achieve minimal results or have been restricted to special cases. This paper presents a constructive proof in the real or complex field of the decomposition of a generalized orthogonal matrix into the product of the minimal number of generalized Householder matrices.     

Mod 2 indecomposable orthogonal invariants

Over an algebraically closed base field k of characteristic 2, the ring R^G of invariants is studied, G being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring R of the m-fold direct sum kⁿ⊕?⊕kⁿ of the standard vector representation. It is proved for O(n) (n?2) and for SO(n) (n?3) that there exist m-linear invariants with m arbitrarily large that are indecomposable (i.e., not expressible as polynomials in invariants of lower degree). In fact, they are explicitly constructed for all possible values of m. Indecomposability of corresponding invariants over immediately follows. The constructions rely on analysing the Pfaffian of the skew-symmetric matrix whose entries above the diagonal are the scalar products of the vector variables.     

Solution of augmented linear systems using orthogonal factorizations

We consider solvingx+Ay=b andA ^T x=c for givenb, c andm ×n A of rankn. This is called the augmented system formulation (ASF) of two standard optimization problems, which include as special cases the minimum 2-norm of a linear underdetermined system (b=0) and the linear least squares problem (c=0), as well as more general problems. We examine the numerical stability of methods (for the ASF) based on the QR factorization ofA, whether by Householder transformations, Givens rotations, or the modified Gram-Schmidt (MGS) algorithm, and consider methods which useQ andR, or onlyR. We discuss the meaning of stability of algorithms for the ASF in terms of stability of algorithms for the underlying optimization problems.We prove the backward stability of several methods for the ASF which useQ andR, inclusing a new one based on MGS, and also show under what circumstances they may be regarded as strongly stable. We show why previous methods usingQ from MGS were not backward stable, but illustrate that some of these methods may be acceptable-error stable. We point out that the numerical accuracy of methods that do not useQ does not depend to any significant extent on which of of the above three QR factorizations is used. We then show that the standard methods which do not useQ are not backward stable or even acceptable-error stable for the general ASF problem, and discuss how iterative refinement can be used to counteract these deficiencies.Dedicated to Carl-Eric Fröberg on the occasion of his 75th birthdayThis research was partially supported by NSERC of Canada Grant No. A9236.     

The eigenvalue field is a splitting field

Let K be an algebraically closed field of arbitrary characteristic and F < K a subfield. If is an irreducible semigroup of matrices such that the spectra of all the elements of are contained in F, then is conjugate to a subsemigroup of M _n (F). Research supported in part by the Ministry of Higher Education, Science, and Technology of Slovenia. Received: 6 April 2006     

Google PageRanking problem: The model and the analysis

The spectral and Jordan structures of the Web hyperlink matrix G(c)=cG+(1−c)ev^T have been analyzed when G is the basic (stochastic) Google matrix, c is a real parameter such that 0<c<1, v is a nonnegative probability vector, and e is the all-ones vector. Typical studies have relied heavily on special properties of nonnegative, positive, and stochastic matrices. There is a unique nonnegative vector y(c) such that y(c)^TG(c)=y(c)^T and y(c)^Te=1. This PageRank vector y(c) can be computed effectively by the power method.We consider a square complex matrix A and nonzero complex vectors x and v such that Ax=λx and v^∗x=1. We use standard matrix analytic tools to determine the eigenvalues, the Jordan blocks, and a distinguished left λ-eigenvector of A(c)=cA+(1−c)λxv^∗ as a function of a complex variable c. If λ is a semisimple eigenvalue of A, there is a uniquely determined projection N such that lim_c→1y(c)=Nv for all v; this limit may fail to exist for some v if λ is not semisimple. As a special case of our results, we obtain a complex analog of PageRank for the Web hyperlink matrix G(c) with a complex parameter c. We study regularity, limits, expansions, and conditioning of y(c) and we propose algorithms (e.g., complex extrapolation, power method on a modified matrix etc.) that may provide an efficient way to compute PageRank also with c close or equal to 1. An interpretation of the limit vector Nv and a related critical discussion on the model, on its adherence to reality, and possible ways for its improvement, represent the contribution of the paper on modeling issues.     

Matrix algebras with Multiplicative Decomposition Property

We consider the Multiplicative Decomposition Property (the multiplicative analogue of the Riesz Decomposition Property) for entry-wise ordered matrix algebras over the reals. We characterize for which matrix pairs such a decomposition exists and use this result to present necessary and sufficient conditions for a matrix algebra to enjoy the decomposition property by all its positive matrix pairs. Thus we describe all matrix algebras with the decomposition property.