Moufang loops and alternative algebras

Let be the algebra of classical real octonions or the (split) algebra of octonions over the finite field 2$">. Then the quotient loop of the Moufang loop of all invertible elements of the algebra modulo its center is not embedded into a loop of invertible elements of any alternative algebra.

     

Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Let and denote the Gaussian and Poisson measures on , respectively. We show that there exists a unique measure on such that under the Segal-Bargmann transform the space is isomorphic to the space of analytic -functions on with respect to . We also introduce the Segal-Bargmann transform for the Poisson measure and prove the corresponding result. As a consequence, when and have the same variance, and are isomorphic to the same space under the - and -transforms, respectively. However, we show that the multiplication operators by on and on act quite differently on .

     

Degenerating families of rank two bundles

We construct families of rank two bundles on , in characteristic two, where for , is a sum of line bundles, and is non-split. We construct families of rank two bundles on , in characteristic , where for , is a sum of line bundles, and is non-split.

     

Multi-dimensional versions of a theorem of Fine and Wilf and a formula of Sylvester

Let be vectors in which generate . We show that a body with the vectors as edge vectors is an almost minimal set with the property that every function with periods is constant. For the result reduces to the theorem of Fine and Wilf, which is a refinement of the famous Periodicity Lemma.

Suppose is not a non-trivial linear combination of with non-negative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of over , but infinitely many points on each of its hyperfaces are not. For the result reduces to a formula of Sylvester corresponding to Frobenius' Coin-changing Problem in the case of coins of two denominations.

     

Farrell sets for harmonic functions

Let denote a relatively closed subset of the unit ball of . The purpose of this paper is to characterize those sets which have the following property: any harmonic function on which satisfies on (where 0$">) can be locally uniformly approximated on by a sequence of harmonic polynomials which satisfy the same inequality on . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

     

Tight frame oversampling and its equivalence to shift-invariance of affine frame operators

Let generate a tight affine frame with dilation factor , where , and sampling constant (for the zeroth scale level). Then for , oversampling (or oversampling by ) means replacing the sampling constant by . The Second Oversampling Theorem asserts that oversampling of the given tight affine frame generated by preserves a tight affine frame, provided that is relatively prime to (i.e., ). In this paper, we discuss the preservation of tightness in oversampling, where (i.e., and ). We also show that tight affine frame preservation in oversampling is equivalent to the property of shift-invariance with respect to of the affine frame operator defined on the zeroth scale level.

     

Constraints for the normality of monomial subrings and birationality

Let be a field and let be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra and the subring . If the monomials in have the same degree, one of the consequences is a criterion for the -rational map defined by to be birational onto its image.

     

Sur une question de capitulation

Let and be prime numbers such that and . Let , , and let be the 2-Hilbert class field of , the 2-Hilbert class field of and the Galois group of . The 2-part of the class group of is of type , so contains three extensions . Our goal is to study the problem of capitulation of the 2-classes of in , and to determine the structure of .

RSESUM´E. Soient et deux nombres premiers tels que et , , , , le 2-corps de classes de Hilbert de , le 2-corps de classes de Hilbert de et le groupe de Galois de . La 2-partie du groupe de classes de est de type , par suite contient trois extensions . On s'intéresse au problème de capitulation des 2-classes de dans , et à déterminer la structure de .

     

Hyperelliptic jacobians and simple groups

In a previous paper, the author proved that in characteristic zero the jacobian of a hyperelliptic curve has only trivial endomorphisms over an algebraic closure of the ground field if the Galois group of the irreducible polynomial is either the symmetric group or the alternating group . Here 4$"> is the degree of . In another paper by the author this result was extended to the case of certain ``smaller' Galois groups. In particular, the infinite series and were treated. In this paper the case of and is treated.

     

Bernstein-Walsh inequalities and the exponential curve in

It is shown that for the pluripolar set in there is a global Bernstein-Walsh inequality: If is a polynomial of degree on and on , this inequality gives an upper bound for which grows like . The result is used to obtain sharp estimates for .

     

Special values of elliptic functions at points of the divisors of Jacobi forms

The main result of the paper gives an explicit formula for the sum of the values of even order derivatives with respect to of the Weierstrass -function for the lattice (where is in the upper half-plane) extended over the points in the divisor of (where is a meromorphic Jacobi form) in terms of the coefficients of the Laurent expansion of around .

     

Games and general distributive laws in Boolean algebras

The games and are played by two players in -complete and max -complete Boolean algebras, respectively. For cardinals such that or , the -distributive law holds in a Boolean algebra iff Player 1 does not have a winning strategy in . Furthermore, for all cardinals , the -distributive law holds in iff Player 1 does not have a winning strategy in . More generally, for cardinals such that , the -distributive law holds in iff Player 1 does not have a winning strategy in . For regular and , implies the existence of a Suslin algebra in which is undetermined.

     

Invariants of semisimple Lie algebras acting on associative algebras

If is a Lie algebra of derivations of an associative algebra , then the subalgebra of invariants is the set In this paper, we study the relationship between the structure of and the structure of , where is a finite dimensional semisimple Lie algebra over a field of characteristic zero acting finitely on , when is semiprime.

     

A proof of Weinberg's conjecture on lattice-ordered matrix algebras

Let be a subfield of the field of real numbers and let () be the matrix algebra over . It is shown that if is a lattice-ordered algebra over in which the identity matrix 1 is positive, then is isomorphic to the lattice-ordered algebra with the usual lattice order. In particular, Weinberg's conjecture is true.

     

Small covers of the dodecahedron and the -cell

Let be the right-angled hyperbolic dodecahedron or -cell, and let be the group generated by reflections across codimension-one faces of . We prove that if is a torsion free subgroup of minimal index, then the corresponding hyperbolic manifold is determined up to homeomorphism by modulo symmetries of .

     

Reduction numbers and initial ideals

The reduction number of a standard graded algebra is the least integer such that there exists a minimal reduction of the homogeneous maximal ideal of such that . Vasconcelos conjectured that where is the initial ideal of an ideal in a polynomial ring with respect to a term order. The goal of this note is to prove the conjecture.

     

The removal of from some undecidable problems involving elementary functions

We show that in the ring generated by the integers and the functions and defined on it is undecidable whether or not a function has a positive value or has a root. We also prove that the existential theory of the exponential field is undecidable.

     

Moduli of twisted spin curves

In this note we give a new, natural construction of a compactification of the stack of smooth -spin curves, which we call the stack of stable twisted -spin curves. This stack is identified with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible -spaces and -line bundles are given as well. The infinitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves.

We construct representable morphisms from the stacks of stable twisted -spin curves to the stacks of stable -spin curves and show that they are isomorphisms. Many delicate features of -spin curves, including torsion free sheaves with power maps, arise as simple by-products of twisted spin curves. Various constructions, such as the -operator of Seeley and Singer and Witten's cohomology class go through without complications in the setting of twisted spin curves.

     

Approximation of measurable mappings by sequences of continuous functions

Let be a completely regular Hausdorff space, a positive, finite Baire measure on , and a separable metrizable locally convex space. Suppose is a measurable mapping. Then there exists a sequence of functions in which converges to a.e. . If the function is assumed to be weakly continuous and the measure is assumed to be -smooth, then a separability condition is not needed.