The Kowalevski top as a reduction of a Hamiltonian system on

We show that the Kowalevski top and Kowalevski gyrostat are obtained as a reduction of a Hamiltonian system on . Therefore the Lax-pair representations for the Kowalevski top and Kowalevski gyrostat are obtained via a direct method by transforming the canonical Lax-pair representation of a system on . Also we show that the nontrivial integral of motion of the Kowalevski top comes from a Casimir function of the Lie-Poisson algebra .

     

Testing polycyclicity of finitely generated rational matrix groups

We describe algorithms for testing polycyclicity and nilpotency for finitely generated subgroups of and thus we show that these properties are decidable. Variations of our algorithm can be used for testing virtual polycyclicity and virtual nilpotency for finitely generated subgroups of .

     

-resultant for rational functions

In this paper we introduce the -resultant of two rational functions and show how it can be used to decide if or if and to find the singularities of the parametric algebraic curve define by . In the course of our work we extend a result about implicitization of polynomial parametric curves to the rational case, which has its own interest.

     

The undecidability of cyclotomic towers

Let be the field obtained by adjoining to all -power roots of unity where is a prime number. We prove that the theory of is undecidable.

     

Numerical radius distance-preserving maps on

Let be a complex Hilbert space, be the algebra of all bounded linear operators on , be the subset of all selfadjoint operators in and or . Denote by the numerical radius of . We characterize surjective maps that satisfy for all without the linearity assumption.

     

Combinatorics of rank jumps in simplicial hypergeometric systems

Let be an integer matrix, and assume that the convex hull of its columns is a simplex of dimension  not containing the origin. It is known that the semigroup ring is Cohen-Macaulay if and only if the rank of the GKZ hypergeometric system equals the normalized volume of for all complex parameters (Saito, 2002). Our refinement here shows that has rank strictly larger than the volume of if and only if lies in the Zariski closure (in  ) of all -graded degrees where the local cohomology is nonzero. We conjecture that the same statement holds even when is not a simplex.

     

A version of Burkholder's theorem for operator-weighted spaces

Let be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space . We prove that if the dyadic martingale transforms are uniformly bounded on for each dyadic grid in , then the Hilbert transform is bounded on as well, thus providing an analogue of Burkholder's theorem for operator-weighted -spaces. We also give a short new proof of Burkholder's theorem itself. Our proof is based on the decomposition of the Hilbert transform into ``dyadic shifts'.

     

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

We consider purely inseparable extensions of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group and a vector space decomposition such that and , where denotes the integral closure. Moreover, is Cohen-Macaulay if and only if is Cohen-Macaulay. Furthermore, is polynomial if and only if is polynomial, and if and only if

where and .

     

Extensions for finite Chevalley groups II

Let be a semisimple simply connected algebraic group defined and split over the field with elements, let be the finite Chevalley group consisting of the -rational points of where , and let be the th Frobenius kernel. The purpose of this paper is to relate extensions between modules in and with extensions between modules in . Among the results obtained are the following: for 2$"> and , the -extensions between two simple -modules are isomorphic to the -extensions between two simple -restricted -modules with suitably ``twisted" highest weights. For , we provide a complete characterization of where and is -restricted. Furthermore, for , necessary and sufficient bounds on the size of the highest weight of a -module are given to insure that the restriction map is an isomorphism. Finally, it is shown that the extensions between two simple -restricted -modules coincide in all three categories provided the highest weights are ``close" together.

     

On a theorem of R. Steinberg on rings of coinvariants

Let be a representation of a finite group over the field . Denote by the algebra of polynomial functions on the vector space . The group acts on and hence also on . The algebra of coinvariants is , where is the ideal generated by all the homogeneous -invariant forms of strictly positive degree. If the field has characteristic zero, then R. Steinberg has shown (this is the formulation of R. Kane) that is a Poincaré duality algebra if and only if is a pseudoreflection group. In this note we explore the situation for fields of nonzero characteristic. We prove an analogue of Steinberg's theorem for the case and give a counterexample in the modular case when .

     

Spectra of operators with Bishop's property

Let be a Banach space and let be the class that consists of all operators such that for every , the range of has a finite-codimension when it is closed. For an integer , we define the class as an extension of . We then study spectral properties of such operators, and we extend some known results of multi-cyclic operators with .

     

Alternating forms and the Brauer group of a geometric field

We compute the theory of for any proabelian group , using a natural isomorphism with the group of continuous alternating forms. We use this to establish a sort of generic behavioral ideal, or role model, for the Brauer group Br of a geometric field of characteristic zero. We show this ideal is attained in several interesting cases.

     

Hermite-Biehler functions with zeros close to the imaginary axis

A Hermite-Biehler function gives rise to a de Branges Hilbert space consisting of entire functions. We are going to show that for Hermite-Biehler functions of sufficiently small growth and a certain distribution of zeros every proper de Branges subspace of coincides for some with the -dimensional linear space of all polynomials of degree at most .

     

Varieties with small discriminant variety

Let be a smooth complex projective variety, let be an ample and spanned line bundle on , defining a morphism and let be its discriminant locus, the variety parameterizing the singular elements of . We present two bounds on the dimension of and its main component relying on the geometry of . Classification results for triplets reaching the bounds as well as significant examples are provided.

     

Brauer groups of genus zero extensions of number fields

We determine the isomorphism class of the Brauer groups of certain nonrational genus zero extensions of number fields. In particular, for all genus zero extensions of the rational numbers that are split by , .

     

A rigidity theorem for holomorphic generators on the Hilbert ball

We present a rigidity property of holomorphic generators on the open unit ball of a Hilbert space . Namely, if is the generator of a one-parameter continuous semigroup on such that for some boundary point , the admissible limit - , then vanishes identically on .

     

Bases for some reciprocity algebras I

For a complex vector space , let be the algebra of polynomial functions on . In this paper, we construct bases for the algebra of all highest weight vectors in , where and for all , and the algebra of highest weight vectors in .

     

Gradient Estimates for the Heat Semigroup on H-Type Groups

By utilizing the Poincaré inequality and representation formulae, it is shown that on the Heisenberg type group, ℍ(2n, m), there exists a constant C > 0 such that

Upper bound for isometric embeddings

The isometric embeddings (, ) over a field are considered, and an upper bound for the minimal is proved. In the commutative case ( ) the bound was obtained by Delbaen, Jarchow and Pełczyński (1998) in a different way.

     

A perturbed elementary operator and range-kernel orthogonality

Let denote the algebra of operators on a Hilbert . If and are commuting normal operators, and and are commuting quasi-nilpotents such that , then define and by , , and . It is proved that and , where is some scalar and is the quasi-nilpotent part of the operator .