Quadratic initial ideals of root systems

Let be one of the root systems , , and and write for the set of positive roots of together with the origin of . Let denote the Laurent polynomial ring over a field and write for the affine semigroup ring which is generated by those monomials with , where if . Let denote the polynomial ring over and write for the toric ideal of . Thus is the kernel of the surjective homomorphism defined by setting for all . In their combinatorial study of hypergeometric functions associated with root systems, Gelfand, Graev and Postnikov discovered a quadratic initial ideal of the toric ideal of . The purpose of the present paper is to show the existence of a reverse lexicographic (squarefree) quadratic initial ideal of the toric ideal of each of , and . It then follows that the convex polytope of the convex hull of each of , and possesses a regular unimodular triangulation arising from a flag complex, and that each of the affine semigroup rings , and is Koszul.     

On the triple jump of the set of atoms of a Boolean algebra

We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let be a computable Boolean algebra with infinitely many atoms and be the Turing degree of the atom relation of . If is a c.e. degree such that , then there is a computable copy of where the atom relation has degree . In particular, for every c.e. degree , any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree .

     

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 .

     

On polynomial products in nilpotent and solvable Lie groups

We are dealing with Lie groups which are diffeomorphic to , for some . After identifying with , the multiplication on can be seen as a map . We show that if is a polynomial map in one of the two (sets of) variables or , then is solvable. Moreover, if one knows that is polynomial in one of the variables, the group is nilpotent if and only if is polynomial in both its variables.

     

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.

     

Codes over and and Hermitian lattices over imaginary quadratic fields

We introduce a family of bi-dimensional theta functions which give uniformly explicit formulae for the theta series of hermitian lattices over imaginary quadratic fields constructed from codes over and , and give an interesting geometric characterization of the theta series that arise in terms of the basic strongly modular lattice . We identify some of the hermitian lattices constructed and observe an interesting pair of nonisomorphic 3/2 dimensional codes over that give rise to isomorphic hermitian lattices when constructed at the lowest level 7 but nonisomorphic lattices at higher levels. The results show that the two alphabets and are complementary and raise the natural question as to whether there are other such complementary alphabets for codes.

     

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.

     

A multiplicity theorem for the Neumann problem

Here is a particular case of the main result of this paper: Let be a bounded domain, with a boundary of class , and let be two continuous functions, , with 0$">, , with n$">. If

and if the set of all global minima of the function has at least connected components, then, for each 0$"> small enough, the Neumann problem

admits at least strong solutions in .

     

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.

     

Homogeneous polynomials on strictly convex domains

We consider a circular, bounded, strictly convex domain with boundary of class . For any compact subset of we construct a sequence of homogeneous polynomials on which are big at each point of . As an application for any circular subset of type we construct a holomorphic function which is square integrable on and such that where denotes unit disc in .

     

An extremal problem of quasiconformal mappings

In this paper, the following problem is studied. Let and be two domains in the complex plane with . Suppose that are two quasiconformal mappings satisfying . Let be the mapping in defined by (). If both and are uniquely extremal, is always uniquely extremal? It is shown in this paper that the answer to this problem is no.

     

A splitting theorem for degrees

We prove that, for any , and with _{T}A\oplus U$"> and r.e., in , there are pairs and such that ; ; and, for any and from and any set , if and , then . We then deduce that for any degrees , , and such that and are recursive in , , and is into , can be split over avoiding . This shows that the Main Theorem of Cooper (Bull. Amer. Math. Soc. 23 (1990), 151-158) is false.

     

Strong comparison principle for solutions of quasilinear equations

Let , , be a bounded smooth connected open set and be a map satisfying the hypotheses (H1)-(H4) below. Let with , in and with be two weak solutions of

Suppose that in . Then we show that u_1$"> in under the following assumptions: either u_1$"> on , or on and in . We also show a measure-theoretic version of the Strong Comparison Principle.

     

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.

     

On the Loewner problem in the class

Loewner's theorem on boundary interpolation of functions is proved under rather general conditions. In particular, the hypothesis of Alpay and Rovnyak (1999) that the function , which is to be extended to an function, is defined and continuously differentiable on a nonempty open subset of the real line, is replaced by the hypothesis that the set on which is defined contains an accumulation point at which satisfies some kind of differentiability condition. The proof of the theorem in this note uses the representation of functions in terms of selfadjoint relations in Pontryagin spaces and the extension theory of symmetric relations in Pontryagin spaces.

     

On the residual finiteness and other properties of (relative) one-relator groups

A relative one-relator presentation has the form where is a set, is a group, and is a word on . We show that if the word on obtained from by deleting all the terms from has what we call the unique max-min property, then the group defined by is residually finite if and only if is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form (Theorem 6).

     

The dual spectral set conjecture

Suppose that where are real numbers such that and The union is not assumed to be disjoint. It is shown that the translates , , tile the real line for some bounded measurable set if and only if the exponentials , , form an orthogonal basis for some bounded measurable set

     

Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations

Let be a bounded starshaped domain. In this note we consider critical points of the functional

where of class satisfies the natural growth

for some and 0$">, is suitably rank-one convex and in addition is strictly quasiconvex at . We establish uniqueness results under the extra assumption that is stationary at with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Sverák (2003).

     

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.