The problem of the motion of a solid in an ideal fluid. Integration of the Clebsch's case

This paper deals with a geometric approach to the integration of Clebsch's case of equations describing the motion of a solid body in an ideal fluid. This problem is defined by a nonlinear system of 6 differential equations admitting 4 polynomial first integrals. We show that the intersection of surface levels of these integrals can be completed to an abelian surface, i.e., a 2-dimensional algebraic torus. Also, we prove that the problem can be linearized, i.e., can be written in terms of abelian integrals, on a Prym variety of a genus 3 curve obtained naturally. Received August 1998     

Krein space representation and Lorentz groups of analytic Hilbert modules

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H~2(D~2). A closed subspace M in H~2(D~2) is called a submodule if ziM ? M(i = 1, 2). An associated integral operator(defect operator) CM captures much information about M. Using a Kre??n space indefinite metric on the range of CM, this paper gives a representation of M. Then it studies the group(called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup(called little Lorentz group) which turns out to be a finer invariant for M.     

On Cellular Covers with Free Kernels

In this paper we show that every cotorsion-free and reduced abelian group of any finite rank (in particular, every free abelian group of finite rank) appears as the kernel of a cellular cover of some cotorsion-free abelian group of rank 2. This situation is the best possible in the sense that cotorsion-free abelian groups of rank 1 do not admit cellular covers with free kernel except for the trivial ones. This work is motivated by an example due to Buckner?CDugas, and recent results obtained by Fuchs?CG?bel, and G?bel?CRodríguez?CStrüngmann.     

Maximal Abelian von Neumann Algebras and Toeplitz Operators with Separately Radial Symbols

This paper mainly concerns abelian von Neumann algebras generated by Toeplitz operators on weighted Bergman spaces. Recently a family of abelian w^*-closed Toeplitz algebras has been obtained (see [5,6,7,8]). We show that this algebra is maximal abelian and is singly generated by a Toeplitz operator with a “common” symbol. A characterization for Toeplitz operators with radial symbols is obtained and generalized to the high dimensional case. We give several examples for abelian von Neumann algebras in the case of high dimensional weighted Bergman spaces, which are different from the one dimensional case.     

Critical exponent and critical blow-up for quasilinear parabolic equations

B ₂-groups are special (torsion-free) abelian Butler groups. The interest in this class of groups comes from representation theory. A particular functor, also called Butler functor, connects algebraic properties of the category of free abelian groups with (a few) distinguished subgroups with these Butler groups. This helps to understand Butler groups and caused lots of activities on Butler groups. Butler groups were originally defined for finite rank, however a homological connection discovered by Bican and Salce opened the investigation of Butler groups of infinite rank. Despite the fact that classifications of Butler groups are possible under restriction even for infinite rank (see a forthcoming paper by Files and Göbel [Mathematische Zeitschrift]), general structure theorems are impossible. This is supported by the following very special case of the Main Theorem of this paper, showing that any ring with a free additive group is an endomorphism ring of a Butler group. The result implies the existence of large indecomposable or of large superdecomposable Butler groups as well as the existence of counter-examples for Kaplansky’s test problems.     

An answer to Hirasaka and Muzychuk: Every p-Schur ring over Cp3 is Schurian

In Hirasaka and Muzychuk [An elementary abelian group of rank 4 is a CI-group, J. Combin. Theory Ser. A 94 (2) (2001) 339–362] the authors, in their analysis on Schur rings, pointed out that it is not known whether there exists a non-Schurian p-Schur ring over an elementary abelian p-group of rank 3. In this paper we prove that every p-Schur ring over an elementary abelian p-group of rank 3 is in fact Schurian.     

A fast realization of preconditioned Conjugate Gradients for Wiener-Hopf integral equations

We present an efficient implementation of the Conjugate Gradients algorithm for Wiener-Hopf integral equations based on finite rank approximations of the integral operator and the corresponding preconditioner. The resulting algorithm is of linear complexity. Numerical experiments with this implementation of the preconditioned Conjugate Gradients algorithm show significant speed-up in the ill-conditioned case. This algorithm acts on ill-conditioned equations as a regularization algorithm.     

Splitting Mixed Groups of Finite Torsion-Free Rank

Abstract

First, we give a necessary and sufficient condition for torsion-free finite rank subgroups of arbitrary abelian groups to be purifiable. An abelian group G is said to be a strongly ADE decomposable group if there exists a purifiable T(G)-high subgroup of G. We use a previous result to characterize ADE decomposable groups of finite torsion-free rank. Finally, in an extreme case of strongly ADE decomposable groups, we give a necessary and sufficient condition for abelian groups of finite torsion-free rank to be splitting.     

Mapping properties of the heat operator on edge manifolds

We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extending the classical results from smooth manifolds and conical spaces. The estimates, together with strong continuity of the heat operator, yield short‐time existence of solutions to certain semilinear parabolic equations. Our discussion reviews and generalizes earlier work by Jeffres and Loya.     

On Purity and Quasiequality of Abelian Groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module _E(G) G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.     

Class fields,Dirichlet characters, and extended genus fields of global function fields

We study the extended genus field of an abelian extension of a rational function field. We follow the definition of Anglès and Jaulent, which uses the class field theory. First, we show that the natural definition of extended genus field of a cyclotomic function field obtained by means of Dirichlet characters is the same as the one given by Anglès and Jaulent. Next, we study the extended genus field of a finite abelian extension of a rational function field along the lines of the study of genus fields of abelian extensions of rational function fields. In the absolute abelian case, we compare this approach with the one given by Anglès and Jaulent.     

Common eigenfunctions of commuting differential operators of rank 2

Commuting differential operators of rank 2 are considered. With each pair of commuting operators a complex curve called the spectral curve is associated. The genus of this curve is called the genus of the commuting pair. The dimension of the space of common eigenfunctions is called the rank of the commuting operators. The case of rank 1 was studied by I. M. Krichever; there exist explicit expressions for the coefficients of commuting operators in terms of Riemann theta-functions. The case of rank 2 and genus 1 was considered and studied by S. P. Novikov and I.M. Krichever. All commuting operators of rank 3 and genus 1 were found by O. I. Mokhov. A. E. Mironov invented an effective method for constructing operators of rank 2 and genus greater than 1; by using this method, many diverse examples were constructed. Of special interest are commuting operators with polynomial coefficients, which are closely related to the Jacobian problem and many other problems. Common eigenfunctions of commuting operators with polynomial coefficients and smooth spectral curve are found explicitly in the present paper. This has not been done so far.     

A note on mixed A-reflexive groups

This paper presents a Warfield-type characterization of A-reflexive groups in the case that A is a mixed abelian group of torsion-free rank 1.     

The rank of group of cyclotomic units in abelian fields

A formula about the rank of group of cyclotomic units in abelian fields is established. From that formula, a series of equivalent conditions for independence of the system of cyclotomic units in abelian fields is stated and proved. For the particular case of cyclotomic fields, further properties of the rank are researched.     

The Hardy inequality and the heat equation with magnetic field in any dimension

In the Euclidean space of any dimension d, we consider the heat semigroup generated by the magnetic Schrödinger operator from which an inverse-square potential is subtracted to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behavior of the heat semigroup is determined by the eigenvalue problem for a magnetic Schrödinger operator on the (d ? 1)-dimensional sphere whose vector potential reflects the behavior of the magnetic field at the space infinity. From the spectral problem on the sphere, we deduce that in d = 2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions. To prove the results, we establish new magnetic Hardy-type inequalities for the Schrödinger operator and develop the method of self-similar variables and weighted Sobolev spaces for the associated heat equation.     

Positive curvature property for some hypoelliptic heat kernels

In this note, we look at some hypoelliptic operators arising from nilpotent rank 2 Lie algebras. In particular, we concentrate on the diffusion generated by three Brownian motions and their three Lévy areas, which is the simplest extension of the Laplacian on the Heisenberg group H. In order to study contraction properties of the heat kernel, we show that, as in the case of the Heisenberg group, the restriction of the sub-Laplace operator acting on radial functions (which are defined in some precise way in the core of the paper) satisfies a non-negative Ricci curvature condition (more precisely a CD(0,∞) inequality), whereas the operator itself does not satisfy any CD(r,∞) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel.     

On Affine Surface That can be Completed by A Smooth Curve

In C⁶, we consider a non linear system of differential equations with four invariants: two quadrics, a cubic and a quartic. Using Enriques-Kodaira classification of algebraic surfaces, we show that the affine surface obtained by setting these invariants equal to constants is the affine part of an abelian surface. This affine surface is completed by gluing to it a one genus 9 curve consisting of two isomorphic genus 3 curves intersecting transversely in 4 points.     

Non-cancellation for certain classes of groups

For a certain class of groups, which are semidirect products arising from an action of a finite rank free abelian group on another group, we study cancellation of the infinite cyclic group in isomorphic direct products. As an application we obtain a sufficient condition for triviality of the genus of certain nilpotent groups.     

On constructive nilpotent groups

We prove the following: (1) a torsion-free class 2 nilpotent group is constructivizable if and only if it is isomorphic to the extension of some constructive abelian group included in the center of the group by some constructive torsion-free abelian group and some recursive system of factors; (2) a constructivizable torsion-free class 2 nilpotent group whose commutant has finite rank is orderably constructivizable.     

A new upper bound for the cross number of finite abelian groups

In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite abelian groups. Given a finite abelian group, this upper bound appears to depend only on the rank and the number of distinct prime divisors of the exponent. The main theorem of this paper allows us, among other consequences, to prove that a classical conjecture concerning the cross and little cross numbers of finite abelian groups holds asymptotically in at least two different directions.