For regular semigroups, the appropriate analogue of the concept of a variety seems to be that of an e(xistence)-variety, developed by Hall [6,7,8]. A class V of regular semigroups is an e-variety if it is closed under taking direct products, regular subsemigroups and homomorphic images. For orthodox semigroups, this concept has been introduced under the term “bivariety” by Kaďourek and Szendrei [12]. Hall showed that the collection of all e-varieties of regular semigroups forms a complete lattice under inclusion. Further, he proved a Birkhoff-type theorem: each e-variety is determined by a set of identities. For e-varieties of orthodox semigroups a similar result has been proved by Kaďourek and Szendrei. At variance with the case of varieties, prima facie the free objects in general do not exist for e-varieties. For instance, there is no free regular or free orthodox semigroup. This seems to be true for most of the naturally appearing e-varieties (except for cases of e-varieties which coincide with varieties of unary semigroups such as the classes of all inverse and completely regular semigroups, respectively). This is true if the underlying concept of free objects is denned as usual. Kaďourek and Szendrei adopted the definition of a free object according to e-varieties of orthodox semigroups by taking into account generalized inverses in an appropriate way. They called such semigroups bifree objects. These semigroups satisfy the properties one intuitively expects from the “most general members” of a given class of semigroups. In particular, each semigroup in the given class is a homomorphic image of a bifree object, provided the bifree objects exist on sets of any cardinality. Concerning existence, Kaďourek and Szendrei were able to prove that in any class of orthodox semigroups which is closed under taking direct products and regular subsemigroups, all bifree objects exist and are unique up to isomorphism. Further, similar to the case of varieties, there is an order inverting bijection between the fully invariant congruences on the bifree orthodox semigroup on an infinite set and the e-varieties of orthodox semigroups. Recently, Y. T. Yeh [22] has shown that suitable analogues to free objects exist in an e-variety V of regular semigroups if and only if all members of V are either E-solid or locally inverse.

The Cauchy problem of ut, = ∆uα + uβ, where 0 < α < l and α>1, is studied. It is proved that if 1< β<α + 2/n then every nontrivial non-negative solution is not global in time. But if β>α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t–1/(β–1) when t →∞. Thus the famous Fujita result on ut = ∆u + up is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W1,s (W2.s) with S > 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W1.s (W2.s) as t → 0.

In this paper we establish bounds constraining the effective conductivity tensor of composites made of an arbitrary number n of possibly anisotropic phases in prescribed volume fractions. The bounds are valid in any spatial dimension d≧2. The bounds have a very simple and concise form and include those previously obtained by Hashin and Shtrikman, Murat and Tartar, Lurie and Cherkaev, Kohn and Milton, Avellaneda, Cherkaev, Lurie and Milton and Dell'Antonio and Nesi.

A basic problem in population dynamics is that of finding criteria for the long-term coexistence of interacting species. An important aspect of the problem is determining how coexistence is affected by spatial dispersal and environmental heterogeneity. The object of this paper is to study the problem of coexistence for two interacting species dispersing through a spatially heterogeneous region. We model the population dynamics of the species with a system of two reaction–diffusion equations which we interpret as a semi-dynamical system. We say that the system is permanent if any state with all components positive initially must ultimately enter and remain within a fixed set of positive states that are strictly bounded away from zero in each component. Our analysis produces conditions that can be interpreted in a natural way in terms of environmental conditions and parameters, by combining the dynamic idea of permanence with the static idea of studying geometric problems via eigenvalue estimation.

An upper bound for the “sausage catastrophe” of dense sphere packings in 4-space is given.

A basic problem in the theory of finite packing is to determine, for a given positive integer k, the minimal volume of all convex bodies into which k translates of the unit ball Bd of the Euclidean d-dimensional space Ed can be packed ([5]). For d = 2 this problem was solved by Groemer ([6]).

Let f(x) be a monic polynomial of degree n with complex coefficients, which factors as f(x) = g(x)h(x), where g and h are monic. Let be the maximum of on the unit circle. We prove that , where β = M(P0) = 1 38135 …, where P0 is the polynomial P0(x, y) = 1 + x + y and δ = M(P1) = 1 79162…, where P1(x, y) = 1 + x + y - xy, and M denotes Mahler's measure. Both inequalities are asymptotically sharp as n → ∞.

Let D be an atomic integral domain (i.e., a domain in which each nonzero nonunit of D can be written as a product of irreducible elements) and k any positive integer. D is known as a half factorial domain (HFD) if for any irreducible elements α1, …, αn, β1, …, βm of D the equality α1… αn = β1… βm implies that n = m. In [5] the present authors define D to be a k-half factorial domain (k-HFD) if the equality above along with the fact that n or m ≤ k implies that n = m. In this paper we consider the k-HFD property in Dedekind domains with small class group and prove the following Theorem: if D is a Dedekind domain with class group of order less than 16 then D is k-HFD for some integer k > 1, if, and only if, D is HFD.

Given ξ in [0, 1] let ξ = [0, a1, a2…] denote a simple continued fraction expansion of ξ Given the expansion ξ = [0, a1, a2, …] let

Thus with the conventions p1 = q0 = 1 and q−1 = p0 = 0 we have

Let K be an algebraic number field, [ K: ] = KΣ. Most of what we shall discuss is trivial when K = , so that we assume that K ≥ 2 from now onwards. To describe our results, we consider the classical device [2] of Minkowski, whereby K is embedded (diagonally-) into the direct product MK of its completions at its (inequivalent) infinite places. Thus MK is -algebra isomorphic to , and is to be regarded as a topological -algebra, dimRMK = K, in which K is everywhere dense, while the ring Zx of integers of K embeds as a discrete -submodule of rank K. Following the ideas implicit in Hecke's fundamental papers [6] we may measure the “spatial distribution” of points of MK (modulo units of κ) by means of a canonical projection onto a certain torus . The principal application of our main results (Theorems I–III described below) is to the study of the spatial distribution of the which have a fixed norm n = NK/Q(α). In §2 we shall show that, with suitable interpretations, for “typical” n (for which NK/Q(α) = n is soluble), these α have “almost uniform” spatial distribution under the canonical projection onto TK. Analogous questions have been considered by several authors (see, e.g., [5, 9, 14]), but in all cases, they have considered weighted averages over such n of a type which make it impossible to make useful statements for “typical” n.

Let K be a convex body in Euclidean space Rd, d≥2, with volume V(K) = 1, and n ≥ d +1 be a natural number. We select n independent random points y1, y2, …, yn from K (we assume they all have the uniform distribution in K). Their convex hull co {y1, y2, …, yn} is a random polytope in K with at most n vertices. Consider the expected value of the volume of this polytope

It is easy to see that if U: Rd → Rd is a volume preserving affine transformation, then for every convex body K with V(K) = 1, m(K, n) = m(U(K), n).

§1. Introduction and main results. A map f: A → R (A ⊂ R) is called piecewise contractive if there is a finite partition A = A1∪ … ∪ An such that the restriction f| Ai is a contraction for every i = 1, …, n. According to a theorem proved by von Neumann in [3], every interval can be mapped, using a piecewise contractive map, onto a longer interval. This easily implies that whenever A, B are bounded subsets of R with nonempty interior, then A can be mapped, using a piecewise contractive map, onto B (see [6], Theorem 7.12, p. 105). Our aim is to determine the range of the Lebesgue measure of B, supposing that the number of pieces in the partition of A is given. The Lebesgue outer measure will be denoted by λ. If I is an interval then we write |I| = λ(I).

Let A and B be two compact, convex sets in ℝn, each symmetric with respect to the origin 0. L is any (n - l)-dimensional subspace. In 1956 H. Busemann and C. M. Petty (see [6]) raised the question: Does vol (A ⌒ L) < vol (B ⌒ L) for every L imply vol (A) < vol(B)? The answer in case n = 2 is affirmative in a trivial way. Also in 1953 H. Busemann (see [4]) proved that if A is any ellipsoid the answer is affirmative. In fact, as he observed in [5], the answer is still affirmative if A is an ellipsoid with 0 as center of symmetry and B is any compact set containing 0.

The second theorem of Minkowski establishes a relation between the successive minima and the volume of a 0-symmetric convex body. Here we show corresponding inequalities for arbitrary convex bodies, where the successive minima are replaced by certain successive diameters and successive widths.

We further give some applications of these results to successive radii, intrinsic volumes and the lattice point enumerator of a convex body.

We show that under certain circumstances quasi self-similar fractals of equal Hausdorff dimensions that are homeomorphic to Cantor sets are equivalent under Hölder bijections of exponents arbitrarily close to 1. By setting up algebraic invariants for strictly self-similar sets, we show that such sets are not, in general, equivalent under Lipschitz bijections.

Let k ≤ 2 be an integer, and set

Ek (X) = |{n ≤ X, n ≠ mk, n not a sum of a prime and a k-th power}|.

We prove that there exists δ = δ(k) > 0 such that Ek (X)Ek(X)≪kX1−δ.

The Shirshov-Cohn theorem asserts that in a Jordan algebra (with 1), any subalgebra generated by two elements (and 1) is special. Let J be a Jordan algebra with 1, a, b elements of J and let a1, a2, …, an be invertible elements of J such that

Where

are Jordan polynomials. In [2, p. 425] Jacobson conjectured that for any choice of the Pi the subalgebra of J generated by 1, a, b, a1…, an is special.

In [7] the notion of minimal pairs of convex compact subsets of a Hausdorff topological vector space was introduced and it was conjectured, that minimal pairs in an equivalence class of the Hörmander-Rådström lattice are unique up to translation. We prove this statement for the two-dimensional case. To achieve this we prove a necessary and sufficient condition in terms of mixed volumes that a translate of a convex body in ℝn is contained in another convex body. This generalizes a theorem of Weil (cf. [10]).