The algebraic trace form (as defined by O. Loos) of an element (x, y) of a (complex) Banach Jordan pair V, where x or y is in the socle, is equal to the sum of the products of all spectral values and their multiplicity. The trace form is calculated for two examples, the Banach Jordan pair of bounded linear operators between two Banach spaces, and the Banach Jordan pair of a quadratic form. Using analytic multifunctions, it is also shown that the complement of the socle of a Banach Jordan pair V is either dense or empty. In the last case, V has finite capacity.

Denote by f(n) the number of subgroups of the symmetric group Sym(n) of degree n, and by ftrans(n) the number of its transitive subgroups. It was conjectured by Pyber [9] that almost all subgroups of Sym(n) are not transitive, that is, ftrans(n)/f(n) tends to 0 when n tends to infinity. It is still an open question whether or not this conjecture is true. The difficulty comes from the fact that, from many points of view, transitivity is not a really strong restriction on permutation groups, and there are too many transitive groups [9, Sections 3 and 4]. In this paper we solve the problem in the particular case of permutation groups of prime power degree, proving the following result.

The simplest example of the sort of representation formula that we shall study is the following familiar inequality for a smooth, real-valued function f(x) defined on a ball B in N-dimensional Euclidean space IRN:

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where ∇f denotes the gradient of f, fB is the average [mid ]B[mid ]−1∫ Bf(y)dy, [mid ]B[mid ] is the Lebesgue measure of B, and C is a constant which is independent of f, x and B. This formula can be found, for example, in [4] and [12]; see also the closely related estimates in [20, pp. 228{231]. Indeed, such a formula holds in any bounded convex domain.

In order to present the results of this note, we begin with some definitions. Consider a differential system

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where I⊆ℝ is an open interval, and f(t, x), (t, x)∈I×ℝn, is a continuous vector function with continuous first derivatives δfr/δxs, r, s=1, 2, …, n.

Let Dxf(t, x), (t, x)∈I×ℝn, denote the Jacobi matrix of f(t, x), with respect to the variables x1, …, xn. Let x(t, t0, x0), t∈I(t0, x0) denote the maximal solution of the system (1) through the point (t0, x0)∈I×ℝn.

For two vectors x, y∈ℝn, we use the notations x>y and x[Gt ]y according to the following definitions:

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An n×n matrix A=(ars) is called reducible if n[ges ]2 and there exists a partition

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(p[ges ]1, q[ges ]1, p+q=n) such that

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The matrix A is called irreducible if n=1, or if n[ges ]2 and A is not reducible.

The system (1) is called strongly monotone if for any t0∈I, x1, x2∈ℝn

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holds for all t>t0 as long as both solutions x(t, t0, xi), i=1, 2, are defined. The system is called cooperative if for all (t, x)∈I×ℝn the off-diagonal elements of the n×n matrix Dxf(t, x) are nonnegative.

The structural stability of constrained polynomial differential systems of the form a(x, y)x′+b(x, y)y′=f(x, y), c(x, y)x′+d(x, y)y′=g(x, y), under small perturbations of the coefficients of the polynomial functions a, b, c, d, f and g is studied. These systems differ from ordinary differential equations at ‘impasse points’ defined by ad−bc=0. Extensions to this case of results for smooth constrained differential systems [7] and for ordinary polynomial differential systems [5] are achieved here.

Two-sided bounds for the heat kernel of the Schrödinger operator with a singular potential are obtained with explicit constants.

Let S be an elementary operator on a C*-algebra with minimal length [lscr ](S)[ges ]1. Then S is completely positive if and only if S is n-positive for some natural number n[ges ]([lscr ](S)−1)/2.

In a previous paper, the second author introduced a compact topology τr on the space of closed ideals of a unital Banach algebra A. If A is separable, then τr is either metrizable or else neither Hausdorff nor first countable. Here it is shown that τr is Hausdorff if A is C1[0, 1], but that if A is a uniform algebra, then τr is Hausdorff if and only if A has spectral synthesis. An example is given of a strongly regular, uniform algebra for which every maximal ideal has a bounded approximate identity, but which does not have spectral synthesis.

The purpose of this note is to establish a new version of the local Steiner formula and to give an application to convex bodies of constant width. This variant of the Steiner formula generalizes results of Hann [3] and Hug [6], who use much less elementary techniques than the methods of this paper. In fact, Hann asked for a simpler proof of these results [4, Problem 2, p. 900]. We remark that our formula can be considered as a Euclidean analogue of a spherical result proved in [2, p. 46], and that our method can also be applied in hyperbolic space.

For some remarks on related formulas in certain two-dimensional Minkowski spaces, see Hann [5, p. 363].

For further information about the notions used below, we refer to Schneider's book [9]. Let [Kscr ]n be the set of all convex bodies in Euclidean space IRn, that is, the set of all compact, convex, non-empty subsets of IRn. Let Sn-1 be the unit sphere. For K∈[Kscr ]n, let NorK be the set of all support elements of K, that is, the pairs (x, u)∈IRn×Sn−1 such that x is a boundary point of K and u is an outer unit normal vector of K at the point x. The support measures (or generalized curvature measures) of K, denoted by Θ0(K, ·), …, Θn−1(K, ·), are the unique Borel measures on IRn×Sn−1 that are concentrated on NorK and satisfy

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for all integrable functions f[ratio ]IRn→IR; here λ denotes the Lebesgue measure on IRn. Equation (1), which is a consequence and a slight generalization of Theorem 4.2.1 in Schneider [9], is called the local Steiner formula. Our main result is the following.

A series of complexes Qp indexed by all primes p is constructed with catQp=2 and catQp×Sn=2 for either n[ges ]2 or n=1 and p=2. This disproves Ganea's conjecture on Lusternik–Schnirelmann (LS) category.

Let Ratk(CPn) denote the space of based holomorphic maps of degree k from the Riemannian sphere S2 to the complex projective space CPn. The basepoint condition we assume is that f(∞)=[1, …, 1]. Such holomorphic maps are given by rational functions:

Ratk(CPn) ={(p0(z), …, pn(z))[ratio ]each pi(z) is a monic, degree-k polynomial and such that there are no roots common to all pi(z)}. (1.1)

The study of the topology of Ratk(CPn) originated in [10]. Later, the stable homotopy type of Ratk(CPn) was described in [3] in terms of configuration spaces and Artin's braid groups. Let W(S2n) denote the homotopy theoretic fibre of the Freudenthal suspension E[ratio ]S2n→ ΩS2n+1. Then we have the following sequence of fibrations: Ω2S2n+1→ W(S2n)→S2n→ ΩS2n+1. A theorem in [10] tells us that the inclusion Ratk(CPn)→ Ω2kCPn≃ Ω2S2n+1 is a homotopy equivalence up to dimension k(2n−1). Thus if we form the direct limit Rat∞(CPn)= limk→∞ Ratk(CPn), we have, in particular, that Rat∞(CPn) is homotopy equivalent to Ω2S2n+1.

If we take the results of [3] and [10] into account, we naturally encounter the following problem: how to construct spaces Xk(CPn), which are natural generalizations of Ratk(CPn), so that X∞(CPn) approximates W(S2n). Moreover, we study the stable homotopy type of Xk(CPn).

The purpose of this paper is to give an answer to this problem. The results are stated after the following definition.

The purpose of this note is to provide a new proof for the explicit formulas of the heat kernel on hyperbolic space. By definition, the hyperbolic space IHn is a (unique) simply connected complete n-dimensional Riemannian manifold with a constant negative sectional curvature −1.

The aim of this note is to give a sharp upper bound on the ratio

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where ϕ is a nonconstant eigenfunction for the Laplace–Beltrami operator on a connected compact Riemannian manifold without boundary. This ratio is always positive, since maxϕ>0 and minϕ<0 for every nonconstant eigenfunction. We assume that maxϕ[ges ]−minϕ, in order to simplify the notation. For the case of a two-dimensional manifold with nonnegative Ricci curvature, our theorem implies that the above ratio is less than the ratio of the maximum divided by the absolute value of the minimum of the Bessel function of order zero.

The proof is based on a gradient estimate from a previous paper of the author (see [5]), which in turn was proved using the maximum principle technique. In contrast to the standard applications of gradient estimates, which are based on integration along geodesics, we arrive at a contradiction by integrating the gradient estimate over small spheres centred at a point where the absolute value of the eigenfunction attains its maximum.

The main motivation for our work is that the ratio of the maximum and the minimum of an eigenfunction plays a role in estimates of the corresponding eigenvalues (see [5] and [7]). More precisely, our theorem implies that there are minimizing sequences of compact manifolds such that the first eigenvalues of the manifolds approach the corresponding lower bound for the first eigenvalue obtained in [5, Theorem 2] for every possible ratio of the maximum and the minimum of the corresponding eigenfunction.