The first person to consider the discrepancy of sequences of the type (αnσ)n[ges ]1 (where 0<σ<1) was H. Behnke. The subject was taken up again by one of the authors of this paper, who gave a detailed description of the discrepancy's behaviour if either 0<σ<½ or σ=½ and α2∉ℚ or σ=½ and α−2∈ℕ. In this paper, we study the case of sequences (α√n)n[ges ]1 where α>0 and α2∈ℚ. Both

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are expressed as maxima of finitely many numbers which involve class numbers of imaginary quadratic fields in many cases.

It is proved that the number of countable models of a countable supersimple theory is either 1 or infinite. This result is an extension of Lachlan‘s theorem on a superstable theory.

This paper defines the n-fold central Haagerup tensor product [otimes ]ni=1[Lscr ]h[Rscr ] of a von Neumann algebra [Rscr ], and shows that the map θz[ratio ][otimes ]ni =1[Lscr ]h[Rscr ]→CB([Rscr ]n−1, [Rscr ]) given by

formula here

is an isometry.

Let $A$ be a Banach algebra and let $p$ and $q$ be two positive integers. We show that if

$A$ has a left bounded sequential approximate identity $(e_n)_{n\ge1}$ such that ${\rm lim}\,{\rm inf}_{n\to+\infty}\|e^p_n-e^{p+q}_n\| < ({p \over {p+q}})^{p\over q}{q\over{p+q}}$ then

$A$ has a left-bounded sequential identity $(f_n)_{n\ge1}$ such that $f^2_n = f_n$ for $n\ge1$. A simple example shows that the constant $({p\over {p+q}})^{p\over q}{q\over{p+q}}$ is best possible.

This result is based on some algebraic or integral formulae which associate an idempotent to elements of a Banach algebra satisfying some inequalities involving polynomials or entire functions.

The first examples of triangular AF algebras to be studied were the refinement algebra and the standard algebra. Both are analytic algebras with the property that the cocycle can be taken to be constant on the matrix units of the algebra. The latter property, called local constancy, is quite special and is still ill-understood. In the present paper we examine the class of nest algebras [Tscr ] in AF C*-algebras which share the distinctive features of the refinement algebra:

(1) [Tscr ] is a nest algebra in which the nest generates the diagonal,

(2) [Tscr ] admits a locally constant cocycle.

The paper concerns sufficiently saturated structures M over a countable language with a unary predicate P. It is shown that if $P(M)$ is stably embedded and there are no Vaughtian pairs with respect to P, then an infinite group is interpretable over M (in an infinitary sense of ‘interpretable’). Also, it is shown that if M is $\omega$-categorical, $f\,{:}\,D\,{\longrightarrow}\,P$ is a 0-definable map with finite fibres, and $P(M)$ is stably embedded but D is not, then some infinite group is first-order interpretable over M.

Waldspurger's involution for the genuine irreducible supercuspidal representations of $\widetilde{\mathrm{SL}_{2}(F)}$ is parametrized in terms of types in the case $F$$p$-adic with $p$ odd. In particular, it is shown that the in-volution is given by conjugating by an element of $\widetilde{\mathrm{GL}_{2}(F)}$ and twisting one of the defining parameters of an associated type by a quadratic character, the relevant parameter being a character on the norm one elements of a quadratic extension.

We construct a one-parameter family of sets in ℝ2 generated by a disjoint union of open squares. We study the spectral counting function associated to a variational Dirichlet eigenvalue problem on a set from this family and show that the spectral asymptotics depend not only on the Minkowski dimension of the boundary, but also on whether the values of a specific function of the parameter are rational or irrational. Furthermore, we significantly sharpen the results in the rational case.

McMichael proved that the convolution with the (euclidean) arclength measure supported on the curve t [xrarr ] (t, t2, …, tn), 0 < t < 1, maps Lp(ℝn) boundedly into Lp′(ℝn) if and only if 2n(n+1)/(n2+n+2) [les ] p [les ] 2. In proving this, a uniform estimate on damping oscillatory integrals with polynomial phase was crucial. In this paper, a remarkably simple proof of the same estimate on oscillatory integrals is presented. In addition, it is shown that the convolution operator with the affine arclength measure on any polynomial curve in ℝn maps Lp(ℝn) boundedly into Lp′(ℝn) if p = 2n(n+1)/(n2+n+2).

We prove that the permutation representation of the finite orthogonal group $\Omega^{\varepsilon}(n,q)$, where $\varepsilon=+$ or $-$, on the set of anisotropic lines is multiplicity-free, if q is a power of 2 and $n\ge 6$ is even. This result is established by giving a description of orbitals of this action. The rank of this action is $(q^2+2q)/2$ if $\varepsilon=+$ and $n=6$, and $(q^2+2q+2)/2$ otherwise. Moreover, we compute the subdegrees of the orbitals of $\Omega^{\varepsilon}(n,q)$.

Let G be a locally compact Abelian group and consider the group algebra L1(G) of all complex-valued absolutely integrable functions on G. Some invariants are found that are necessarily shared by groups with isomorphic group algebras, such as dimension, ranks or minimal divisible extensions in the case of torsion-free groups. This sheds some light on the question of finding out to what extent the algebra structure of L1(G) reflects the topological group structure of G. It is shown in particular that some classes of torsion- free locally compact Abelian groups are characterized by their group algebra.

Let A be a commutative ring. A graded A-algebra U = [oplus ]n[ges ]0 is a standard A-algebra if U0 = A and U = A[U1] is generated as an A-algebra by the elements of U1. A graded U-module F = [oplus ]n[ges ]0Fn is a standard U-module if F is generated as a U-module by the elements of F0, that is, Fn = UnF0 for all n [ges ] 0. In particular, Fn = U1Fn−1 for all n [ges ] 1. Given I, J, two ideals of A, we consider the following standard algebras: the Rees algebra of I, [Rscr ](I) = [oplus ]n[ges ]0Intn = A[It] ⊂ A[t], and the multi-Rees algebra of I and J, [Rscr ](I, J) = [oplus ]n[ges ]0([oplus ]p+q=nIpJqupvq) = A[Iu, Jv] ⊂ A[u, v]. Consider the associated graded ring of I, [Gscr ](I) = [Rscr ](I) [otimes ] A/I = [oplus ]n[ges ]0In/In+1, and the multi-associated graded ring of I and J, [Gscr ](I, J) = [Rscr ](I, J) [otimes ] A/(I+J) = [oplus ]n[ges ]0([oplus ]p+q= nIpJq/(I+J)IpJq). We can always consider the tensor product of two standard A-algebras U = [oplus ]p[ges ]0Up and V = [oplus ]q[ges ]0Vq as a standard A-algebra with the natural grading U [otimes ] V = [oplus ]n[ges ]0([oplus ]p+q=nUp [otimes ] Vq). If M is an A-module, we have the standard modules: the Rees module of I with respect to M, [Rscr ](I; M) = [oplus ]n[ges ]0InMtn = M[It] ⊂ M[t] (a standard [Rscr ](I)-module), and the multi-Rees module of I and J with respect to M, [Rscr ](I, J; M) = [oplus ]n[ges ]0([oplus ]p+q=nIpJqMupvq) = M[Iu, Jv] ⊂ M[u, v] (a standard [Rscr ](I, J)-module). Consider the associated graded module of M with respect to I, [Gscr ](I; M) = [Rscr ](I; M) [otimes ] A/I = [oplus ]n[ges ]0InM/In+1M (a standard [Gscr ](I)-module), and the multi-associated graded module of M with respect to I and J, [Gscr ](I, J; M) = [Rscr ](I, J; M) [otimes ] A/(I+J) = [oplus ]n[ges ]0([oplus ]p+q= nIpJqM/ (I+J)IpJqM) (a standard [Gscr ](I, J)-module). If U, V are two standard A-algebras, F is a standard U-module and G is a standard V-module, then F [otimes ] G = [oplus ]n[ges ]0([oplus ]p+q= nFp [otimes ] Gq) is a standard U [otimes ] V-module.

Denote by π[ratio ][Rscr ](I) [otimes ] [Rscr ](J; M) → [Rscr ](I, J; M) and σ[ratio ][Rscr ](I, J; M) → [Rscr ](I+J; M) the natural surjective graded morphisms of standard [Rscr ](I) [otimes ] [Rscr ](J)-modules. Let ϕ[ratio ][Rscr ](I) [otimes ] [Rscr ](J; M) → [Rscr ](I+J; M) be σ∘π. Denote by &πmacr;[ratio ][Gscr ](I) [otimes ] [Gscr ](J; M) → [Gscr ](I, J; M) and &σmacr;[ratio ][Gscr ](I, J; M) → [Gscr ](I+J; M) the tensor product of π and σ by A/(I+J); these are two natural surjective graded morphisms of standard [Gscr ](I) [otimes ] [Gscr ](J)-modules. Let &ϕmacr;[ratio ][Gscr ](I) [otimes ] [Gscr ](J; M) → [Gscr ](I+J; M) be &σmacr;∘&πmacr;. The first purpose of this paper is to prove the following theorem.

It is determined when a block for the reduced enveloping algebra of a classical Lie algebra corresponding to a character in Levi form has finite representation type. These results refine earlier work of Premet.

It is proved that there is, up to isotopy, a unique irreducible Heegaard splitting in an orientable, closed, connected Seifert 3-manifold with an orientable elliptical or euclidean orbifold basis. Using Hamilton's and Lawson's results, the topological uniqueness is obtained of closed orientable minimal surfaces of a given genus g [ges ] 2, embedded in a closed orientable Riemannian 3-manifold with strictly positive Ricci curvature.

The concept of Yang–Baxter bases is useful in interpreting Young's constructions for the symmetric group. This concept is extended first to any Weyl group, and then to any Coxeter group.

Unless otherwise explicitly stated all mappings and tensors in the paper are C∞. A Poisson structure on a (C∞) manifold M is a bracket operation (f, g) [map ] {f, g}, on the set of functions on M, which gives to this set a Lie algebra structure and which verifies the relation

formula here

An equivalent way to get such a structure is to give a 2-vector (that is, an antisymmetric two times contravariant tensor) P satisfying

formula here

where [,] is the Schouten bracket [7]. We then have

formula here

This paper is devoted to the local classification of these structures. The decomposition theorem of A. Weinstein [8, 9] reduces the problem to the case where P is a Poisson structure on Rn which vanishes at zero.

In this paper we will denote by PS(n) the set of germs at zero of Poisson structures on Rn vanishing at the origin.

A class of rational functions and sufficient conditions on it such that the Julia set is a Jordan arc, respectively no Jordan arc, are introduced. Further, it is shown that any rational function which has a Jordan arc as Julia set belongs up to Möbius conjugation to this class.

The most important object in real singularity theory is the C∞ map germ and the most important equivalence relation among them is C∞ right left equivalence. In [7], we presented a new systematic method for the classification of C∞ map germs by characterising C∞ right left equivalence. This paper is a topological version of [7].

Two C∞ map germs f, g:(Rn, 0) → (Rp, 0) are said to be topologically equivalent if there exist homeomorphism map germs s:(Rn, 0) → (Rn 0) and t:(Rp, 0) → (Rp, 0) such that f(x) = t∘g∘s(x). The notion of topological equivalence, although it seems to be unnatural, is also important since we know the existence of C∞ moduli for the classification of C∞ map germs with respect to C∞ right left equivalence. However, we had only one method to obtain topological equivalence for two given C∞ map germs, as stated in the following.

For two given C∞ map germs f, g:(Rn, 0) → (Rp, 0), take an appropriate one-parameter family F:(Rn×[0, 1], {0}×[0, 1]) → (Rp, 0) such that F(x, 0) = f(x) and F(x, 1) = g(x). Then prove that F is in fact topologically trivial.(*)

Two C∞ map germs f, g:(Rn, 0) → (Rp, 0) are said to be [Kscr ]-equivalent if there exist a C∞ diffeomorphism map germ s:(Rn, 0) → (Rn, 0) and a C∞ map germ M:(Rn, 0) → (GL(p, R), M(0)) such that f(x) = M(x)g(s(x)). The notion of [Kscr ]-equivalence was introduced by Mather [4, 5] in order to classify the C∞ stable map germs, and we know that generally in a [Kscr ]-orbit there are uncountably many C∞ right left orbits.

Hence it is significant to give an alternative systematic method for the topological classification even in a single [Kscr ]-orbit, which is the purpose of this paper. One of our results (Theorem 1.2) yields the following well-known theorem [2] as a trivial corollary.

A family [Fscr ] of functions from ℝn to ℝ is k-point separating if, for every k-subset S of ℝn, there is a function f∈[Fscr ] such that f is one-to-one on S. The paper shows that, if the functions are required to be linear (or smooth), then a minimum k-point separating family [Fscr ] has cardinality n(k−1). In the linear case, this result is extended to a larger class of fields including all infinite fields as well as some finite fields (depending on k and n). Also, some partial results are obtained for continuous functions on ℝn, including the case when k is infinite. The proof of the main result is based on graph theoretic results that have some interest in their own right. Say that a graph is an n-tree if it is a union of n edge-disjoint spanning trees. It is shown that every graph with k[ges ]2 vertices and n(k−1) edges has a non-trivial subgraph which is an n-tree. A determinantal criterion is also established for a graph with k vertices and n(k−1) edges to be an n-tree.