The object of this paper is to extend a result of Agnew in [1] for single series to double series, and so enable Tauberian constants between matrix transformations of double sequences to be calculated using the technique that many authors have used previously for single sequences (see, for instance, [2, 3, 8]). This method (see below) gives the best possible Tauberian constants in the form of certain upper limits, and we attempt to calculate these for Cesàro summability with various known Tauberian conditions. In Section 2, we introduce our notation and review earlier work. In Section 3, we prove our main result, and we discuss applications to Cesàro summability in Section 4.

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.

Every subalgebra of [Lscr ]([Hscr ]) which is isometrically weak*-homeomorphic with H∞(Ω) is reflexive, when Ω is a finitely connected region in the complex plane.

The structure of a collineation group G preserving a unital in a projective plane of order m2 with m ≡ 1(4) is investigated and several results are obtained about G, when G is a 2-group. These results are then used to investigate the structure of G/O(G) in the general case and, in particular, that of F*(G/O(G)).

Let K be the field of real or complex numbers. Let $(X \cong {\bf K}^{2n}, \omega)$ be a symplectic vector space and take $0 < k < n,N = ({2n \atop 2k})$. Let $L_1,\ldots,L_N \subset X$ be $2k$-dimensional linear subspaces which are in a sufficiently general position. It is shown that if $F : X \longrightarrow X$ is a linear automorphism which preserves the form $\omega^k$ on all subspaces $L_1,\ldots,L_N$, then $F$ is an $\epsilon_k$-symplectomorphism (that is, $F^*\omega = \epsilon_k\omega$, where $\epsilon_k^k = 1$). In particular, if ${\bf K} = \R$ and $k$ is odd then $F$ must be a symplectomorphism. The unitary version of this theorem is proved as well. It is also observed that the set ${\cal A}_{l,2r}$ of all $l$-dimensional linear subspaces on which the form $\omega$ has rank $\le 2r$ is linear in the Grassmannian $G(l,2n)$, that is, there is a linear subspace $L$ such that ${\cal A}_{l,2r} = L \cap G(l, 2n)$. In particular, the set ${\cal A}_{l,2r}$ can be computed effectively. Finally, the notion of symplectic volume is introduced and it is proved that it is another strong invariant.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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 $\Omega^{\varepsilon}(n,q)$.

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.

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.

Let [Bscr ]2 denote the family of all circular discs in the plane. It is proved that the discrepancy for the family {B1 × B2 [ratio ] B1, B2 ∈ [Bscr ]2} in R4 is O(n1/4+ε) for an arbitrarily small constant ε > 0, that is, it is essentially the same as that for the family [Bscr ]2 itself. The result is established for the combinatorial discrepancy, and consequently it holds for the discrepancy with respect to the Lebesgue measure as well. This answers a question of Beck and Chen. More generally, we prove an upper bound for the discrepancy for a family {Πki=1Ai[ratio ] Ai∈Ai, i = 1, 2, …, k}, where each Ai is a family in Rdi, each of whose sets is described by a bounded number of polynomial inequalities of bounded degree. The resulting discrepancy bound is determined by the ‘worst’ of the families Ai , and it depends on the existence of certain decompositions into constant-complexity cells for arrangements of surfaces bounding the sets of Ai. The proof uses Beck's partial coloring method and decomposition techniques developed for the range-searching problem in computational geometry.

For a linear and bounded operator T from a Banach space X into a Banach space Y, let ϱ(T[mid ][Iscr ]n, [Rscr ]n) and ϱ(T[mid ][Iscr ]n, [Gscr ]n) denote the Rademacher and Gaussian cotype 2 norm of T computed with n vectors, respectively. It is shown that the sequence ϱ(T[mid ][Iscr ]n, [Rscr ]n) has submaximal behaviour if and only if ϱ(T[mid ][Iscr ]n, [Gscr ]n) has. This means that

[formula here]

Moreover, the class of these operators coincides with the class of operators preserving copies of ln∞ uniformly. The tool connecting these concepts is the equal norm Rademacher cotype of operators.