A conjecture of reputable vintage states that c(G)≤b(G) + l for a finite p-group G of class c(G) and breadth b(G). This result has been proved in a medley of special cases and in particular whenever b(G)≤3. We now prove it for b(G) = 4.

The structure of an inverse monoid can be determined by the complete set of Schützenberger graphs of a presentation. Necessary and sufficient conditions for a collection of inverse X-graphs to be the complete set of Schützenberger graphs of some inverse monoid presentation are established and decidability results are obtained. Conditions for a single inverse X-graph to be a Schu¨tzenberger graph for some presentation are also obtained, and both problems are restricted to the case of Clifford monoids and E-unitary inverse monoids. Decidability and undecidability results are obtained for the case of finite graphs. It is also proved that the problem of embedding a finite inverse X-graph in the Cayley graph of a group is undecidable.

In this paper, we study the bounded approximation property for the weighted space $\mathcal{HV}$(U) of holomorphic mappings defined on a balanced open subset U of a Banach space E and its predual $\mathcal{GV}$(U), where $\mathcal{V}$ is a countable family of weights. After obtaining an $\mathcal{S}$-absolute decomposition for the space $\mathcal{GV}$(U), we show that E has the bounded approximation property if and only if $\mathcal{GV}$(U) has. In case $\mathcal{V}$ consists of a single weight v, an analogous characterization for the metric approximation property for a Banach space E has been obtained in terms of the metric approximation property for the space $\mathcal{G}_v$(U).

It is proved that if the normal closure of every element of a group G has rank at most r, then the derived subgroup of G has r-bounded rank.

We prove that all the pure states of the reduced C*-algebra of a free group on an uncountable set of generators are *-automorphism equivalent and extract some consequences of this fact.

Let μ ≠ 0 be an ultradistribution of Beurling type with compact support in the space . We investigate the range of the convolution operator Tμ on the space of non-quasianalytic functions of Beurling type associated with a weight w, in the case the operator is not surjective. It is proved that the range of TM always contains the space of real-analytic functions, and that it contains a smaller space of Beurling type for a weight σ ≥ ω if and only if the convolution operator is surjective on the smaller class.

In its original formulation Lang's theorem referred to a semilinear map on an n-dimensional vector space over the algebraic closure of GF(p): it fixes the vectors of a copy ofV(n, p^h) . In other words, every semilinear map defined over a finite field is equivalent by change of coordinates to a map induced by a field automorphism. We provide an elementary proof of the theorem independent of the theory of algebraic groups and, as a by-product of our investigation, obtain a convenient normal form for semilinear maps. We apply our theorem to classical groups and to projective geometry. In the latter application we uncover three simple yet surprising results.

The main aim of this article is to discuss the relationship between Pontryagin duality and pro-objects. The basic idea arises from K. H. Hofmann's articles [7] and [8] where it is shown that the elementary abelian (Lie) groups are “dense” in the category of locally compact hausdorff abelian groups.

In this paper I construct a reduction formula for the integral

the formula connects any three consecutive members of a set I0, I1, I2, …, Im. We regard m as given, and, in order to avoid wasting time over trivialities, we postulate that the constants a, α, β, γ δ (which are not restricted to be real) have real parts large enough to ensure (i) that the integrals In under consideration and the integrals related to them which will be introduced subsequently are all absolutely convergent, and (ii) that, in all the partial integrations which will be effected, the integrated parts vanish at both limits.

We prove that if X is a real Banach space, with dim X\ge 3, which contains subspace of codimension 1 which is 1-complemented in X and whose group of isometries is almost transitive then X is isometric to a Hilbert space. This partially answers the Banach-Mazur rotation problem and generalizes some recent related results.

The main result in this paper, contained in Theorem 1, is a generalisation of the inequality of the arithmetic-geometric means. A result of a similar character has been proved by Siegel (2). The present result gives an improvement in the inequality in the case when the variables involved are not all distinct, whereas Siegel's result does not. The theorem is used in § 3 to obtain a result in connection with totally real and positive algebraic integers.

Questions about polynomials can be turned into questions about matrices by associating with the polynomial

(over an arbitrary field) its companion matrix

which has p/an as its characteristic polynomial. This technique is often used in stability theory, as indicated in [1]; companion matrices also occur in the theory of the rational canonical form.

The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerning

where T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet series

defines an entire function of s and satisfies the functional equation

The most penetrating statements that have been made on T(n) and LΔ(s)are:

Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

Explicit examples of both hyperelliptic and non-hyperelliptic curves which cannot be defined over their field of moduli are known in the literature. In this paper, we construct a tower of explicit examples of such kind of curves. In that tower there are both hyperelliptic curves and non-hyperelliptic curves.

A sequence of solutions to the Galerkin approximation of a nonstationary magnetohydrodynamic system is proved to converge to a measure-valued solution, in the sense of R. J. DiPerna–A. J. Majda, to the three-dimensional stationary Euler equations.

We prove the Löwner-Heinz inequality, via the Cordes inequality, for elements a,b>0 of a unital hermitian Banach *-algebra A. Letting p be a real number in the interval (0,1], the former asserts that a^p \le b^p if a \le b, a^p < b^p ifa<b , provided that the involution of A is continuous, and the latter that s(a^pb^p) \le s(ab)^p, where s(x)=r(x^*x)^{1/2} andr(x) is the spectral radius of an element x.

Much interest has been shown in determining the range of values of c for which the sequence [n]c contains infinitely many primes. The result is an elementary deduction from the prime number theorem, of course, of 0<c≤l. In 1953, Piatetski–Shapiro [9] showed that

for 1<c<12/11, where xc(X) stands for the number of primes in the set {[nc]n≤x}.

The theory of isoparametric functions and a family of isoparametric hypersurfaces began essentially with E. Cartan in 1930's. He defined a real valued function V defined on a Riemannian space form to be isoparametric if ∥grad υ∥2=TV and ΔV = SV for some real valued functions S, T. Then a family of hypersurfaces Mt, is called isoparametric if Mt,=V-1 (t) where t is a regular value of V.

Let R be a commutative, semi-local ring. Let On,n be the group of linear automorphisms of R2n which preserve the bilinear form . The main result of this paper is the following theorem.

Theorem A. The natural inclusion of Onn into On,n into On+1,n+1induces an isomorphism on the ith homology group if only n is large enough with respect to i.

In this paper we study non-complemented spaces of operators and the embeddability of ℓ∞ in the spaces of operators L(X, Y), K(X, Y) and Kw*(X*, Y). Results of Bator and Lewis [2, 3] (Bull. Pol. Acad. Sci. Math.50(4) (2002), 413–416; Bull. Pol. Acad. Sci. Math.549(1) (2006), 63–73), Emmanuele [8–10] (J. Funct. Anal.99 (1991), 125–130; Math. Proc. Camb. Phil. Soc.111 (1992), 331–335; Atti. Sem. Mat. Fis. Univ. Modena42(1) (1994), 123–133), Feder [11] (Canad. Math. Bull.25 (1982), 78–81) and Kalton [16] (Math. Ann.208 (1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spaces W(X, Y) and K(X, Y) in the space L(X, Y), as well as the complementation of K(X, Y) in W(X, Y). A fundamental result of Drewnowski [7] (Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach space version of the Phillips Lemma.