The concept of injectivity in classes of algebras can be traced back to Baer's initial results for Abelian groups and modules in [1]. The first results in non-module types of algebras appeared when Halmos [14] described the injective Boolean algebras using Sikorski's lemma on extensions of Boolean homomorphisms [19]. In recent years, there have been several results (see references) describing the injective algebras in other particular equational classes of algebras.

In [10], Eckmann and Schopf introduced the fundamental notion of essential extension and gave the basic relations that this concept had with injectivity in the equational class of all modules over a given ring. They developed the notion of an injective hull (or envelope) which provided every module with a minimal injective extension or equivalently, a maximal essential extension. In [6] and [9], it was noted that these relationships hold in any equational class with enough injectives.

In this paper we study genus 2 curves whose Jacobians admit a polarized (4, 4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion, and we derive the relation their absolute invariants satisfy.

As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered.

Our main tool is a Galois theoretic characterization of genus 2 curves admitting multiple Richelot isogenies.

It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from.

Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of Möbius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.

We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.

There are many concepts which arise naturally in a discussion of injectivity in an equational class; for example, weak injective algebras, absolute subretracts, essential extensions, the congruence extension property, and the amalgamation property (see [3; 9; 17; 18]). It has already been demonstrated in several papers, notably [9; 17; 26; 27; 28], that the study of these concepts is greatly enriched by the assumption that the algebras under consideration have distributive congruence lattices. In this work attention is focused on weak injective algebras (Section 2) and the congruence extension property (Section 3).

In 1944 Whitney proved that any differentiable n-manifold (n ≧ 2) can be (differentiably) immersed in R2n–1[15] and embedded in R2n [14]. Whitney's results are best possible when n = 2r. (One uses a simple argument involving the dual Stiefel-Whitney classes of real projective space Pn. See [9, pp. 14, 15].) However, there is a widely known conjecture that any R-manifold (n ≧ 2) immerses in R2n–α(n) and embeds in R2n–α(n)+1. Here, α(n) denotes the number of ones in the binary expansion of n. We prove (Theorem 5.1) that every compact manifold is cobordant to a manifold that immerses in (2n – α(n))-space and embeds in (2n – α(n) + 1)-space. (See § 1 for the definition of cobordant manifolds.) It is well known that if the conjecture were true it would be the best possible result.

Suppose (P, △) is an undirected graph without loops or multiple edges. We will denote by △ (x) the vertices adjacent to x and . Let (G, P) be a transitive permutation representation of a group G in a, set P, and Δ be a non-trivial self-paired (i.e. symmetric) orbit for the action of G on P X P. We identify △ with the set of all two subsets ﹛x, y﹜ with (x, y) in △. Then we have a graph (P, Δ) with G ≦ Aut (P, △), transitive on both P and △.

We completely determine the residual spectrum of the split exceptional group of type G2, thus completing the work of Langlands and Moeglin-Waldspurger on the part of the residual spectrum attached to the trivial character of the maximal torus. We also give the Arthur parameters for the residual spectrum coming from Borel subgroups. The interpretation in terms of Arthur parameters explains the “bizarre” multiplicity condition in Moeglin-Waldspurger's work. It is related to the fact that the component group of the Arthur parameter is non-abelian.

We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$. In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain ${{\ell }_{1}}$ but none of them is isomorphic to ${{\ell }_{1}}$. We also prove that for any countable set $C$ of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of $C$. In certain cases this ensures that $X$ admits, for each $\alpha \,<\,{{\omega }_{1}}$, a spreading model ${{\left( \tilde{x}_{i}^{\left( \alpha \right)} \right)}_{i}}$ such that if $\alpha \,<\,\beta $ then ${{\left( \tilde{x}_{i}^{\left( \alpha \right)} \right)}_{i}}$ is dominated by (and not equivalent to) ${{\left( \tilde{x}_{i}^{\left( \beta \right)} \right)}_{i}}$. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.

In the theory of Jordan algebras one encounters several definitions of the trace, and it is sometimes unclear whether the different notions are equivalent or not. If we restrict attention to the so–called JB–algebras studied in [2] and their weakly closed analogues JBW–algebras [8], we shall in the present note show that the different concepts are all equivalent for JBW–algebras, and that the conditions not involving projections are equivalent for JB–algebras. Among the seven equivalent conditions we shall consider, the second (ii) was used by Alfsen and Shultz [1] to show that if the JBW–algebra, is the self–adjoint part of a von Neumann algebra, then the condition characterizes traces on the enveloping von Neumann algebra. Condition (iii) appears in Robertson's paper [7] together with the implication (ii) ⇒ (iii). The inequality (iv) is a Jordan analogue of Gardner's inequality | ϕ(x) ≦ ϕ(|x|)|, [3], characterizing traces on Cast;–algebras.

In (3, p. 85) we denned a Baer semigroup to be a multiplicative semigroup with 0 having the property that the left (right) annihilator of every element is a principal left (right) ideal generated by an idempotent. We showed (3, Lemma 1(vi) and Theorem 5, p. 86) that with set inclusion as the partial order, the set of left annihilators and the set of right annihilators of elements of a Baer semigroup form dual isomorphic lattices with 0 and 1.

Using some known results on Gauss sums in a finite field, it is shown that the sum (1.3) defined below can either be evaluated explicitly or expressed in terms of a Kloosterman sum. The same result applies to the more general sum S(α, λ, Q) defined in (5.1). The latter sum also satisfies the reciprocity formula (5.5). Some related sums are discussed in §§6, 7.

We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K0 of a unit-regular ring or even as K0 of a locally matricial algebra. If G is countable, this is well known to be true. Here we provide positive answers in case (a) the cardinality of G is ℵ1, or (b) G is an arbitrary infinite tensor product of the groups considered in (a), or (c) G is the group of all continuous real-valued functions on an arbitrary compact Hausdorff space. In cases (a) and (b), we show that G in fact appears as K0 of a locally matricial algebra. Result (a) is the basis for an example due to de la Harpe and Skandalis of the failure of a determinantal property in a non-separable AF C*-algebra [18, Section 3].

A Steiner triple system (briefly STS) is a pair (S, t) where S is a set and t is a collection of 3-subsets of S (called triples) such that every 2-subset of S is contained in exactly one triple of t. The number |S| is called the order of the STS (S, t). It is well-known that there is an STS of order v if and only if v = 1 or 3 (mod 6). Therefore in saying that a certain property concerning STS is true for all v it is understood that v = 1 or 3 (mod 6).

An involution * of a ring A is a one-one additive mapping of A onto itself such that (xy)* = y*x* and x** = x for all x, y ∊ A. If A is an algebra over a field Φ, one makes the additional requirement that (λx)* = λx* for all λ ∊ Φ, x ∊ A. S will generally denote the set of symmetric elements s* = s, K the set of skew elements , and Z the centre of A.

Let Tt, t > 0, be a strongly continuous semigroup of positive linear contractions on the L1-space of a σ-finite measure space . We denote the integral ∫0tTsƒ ds, ƒ ∈ L1, by S0tƒ, which is denned as the limit of Riemann sums, in the norm topology of L1. It is easy to see that, given ƒ ∈ L1+, there exists a function F on the product space X× (0, ∞), measurable with respect to the usual product σ-field, such that for every t ≧ 0, ∫0tF(·, s) ds gives a representation of S0tƒ. We write S0tƒ(x) for ∫0tF(x, S) ds, with a fixed choice of F.

Let F be the group freely generated by the countably infinite set X = {x1, x2, . . . ,xi, . . . }. Let w(x1, x2, . . . , xn) be a reduced word representing an element of F and let G be an arbitrary group. Then V(w, G) will denote the set

whose elements will be called values of w in G. The subgroup of G generated by V(w, G) will be called the verbal subgroup of G with respect to w and be denoted by w(G).

The idea of one surface covering another is a useful one in the study of regular maps. Coxeter and Moser discuss a particular instance [3, p. 115] and the maps produced by Sherk [6] and Garbe [4] are formed by such coverings, though neither paper mentions that fact. In [5], Sherk explicitly constructs coverings of regular maps on the sphere.

1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

Two formulas for the multiplicity of the fiber cone $F\left( I \right)\,=\,\oplus _{n=0}^{\infty }{{I}^{n}}/\text{m}{{I}^{n}}$ of an $\text{m}$-primary ideal of a $d$-dimensional Cohen–Macaulay local ring $\left( R,m \right)$ are derived in terms of the mixed multiplicity ${{e}_{d-1}}\left( \text{m }|I \right)$, the multiplicity $e\left( I \right)$, and superficial elements. As a consequence, the Cohen–Macaulay property of $F\left( I \right)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of $I$ and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of $\text{m}$-primary ideals with a $d$-generated minimal reduction $J$ satisfying $\ell \left( {{I}^{2}}/JI \right)\,=\,1$ or $l\left( Im \right)Jm) = 1$.