We determine which isogeny classes of supersingular abelian surfaces over a finite field $k$ of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to $k$-isomorphism, leading to the same zeta function.

We analyze flat ${{S}_{3}}$-covers of schemes, attempting to create structures parallel to those found in the abelian and triple cover theories. We use an initial local analysis as a guide in finding a global description.

In this paper we study lattice properties of operator algebras which are invariant under compact perturbations. It is easy to see that if and are two operator algebras with contained in , then the reverse inclusion holds for their lattices of invariant subspaces. We will show that in certain cases, the assumption thats is contained in , where is the ideal of compact operators, implies that the lattice of is “approximately” contained in the lattice of . In particular, supposed and are reflexive and have commutative subspace lattices containing “enough” finite dimensional elements. We show (Corollary 2.8) that if is unitarily equivalent to a subalgebra of , then there is a unitary operator which carries all “sufficiently large” subspaces in lat into lat .

In a recent paper (5), the author has established the Euclidean spaces of least dimension in which the topological products of finite collections of k-cell-like continua can be embedded. Specifically, it was shown that, for each pair of positive integers k and n, the topological product of any collection of nk-cell-like continua can be embedded in Euclidean space of dimension k(n + 1). This result includes a theorem of Bennett (1) that the topological product of any finite collection of n snakelike continua can be embedded in Euclidean space of dimension n + 1.

If X is a space and , the kth deleted product space Xk* of X is the topological product X ⨯ X ⨯ . . . ⨯ X of copies of X minus the set of all points of the form (x, x, . . . , x), where x ∊ X. In (4), the author shows that the homology groups of Xk*, where X is a tree, produce as much information about trees as counting the orders of vertices.

1. Introduction. Our problem is related to the construction of geographical maps as follows. The reason for using conformai geographical maps is that the scale (viz., the ratio of two corresponding line-elements) does not depend on the direction of these line-elements. In an ideal map the scale, being responsible for the preservation of shape, would also be independent of the points where the line-elements begin.

This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.

Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM $N=4$ theory and are known by previous work to be associated to positroids. We characterize the conditions under which two Wilson loop diagrams give the same positroid, prove that an important subclass of subdiagrams (exact subdiagrams) corresponds to uniform matroids, and enumerate the number of different Wilson loop diagrams that correspond to each positroid cell. We also give a correspondence between those positroids which can arise from Wilson loop diagrams and directions in associahedra.

A finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

Let $\unicode[STIX]{x1D6E4}\leqslant \text{Aut}(T_{d_{1}})\times \text{Aut}(T_{d_{2}})$ be a group acting freely and transitively on the product of two regular trees of degree $d_{1}$ and $d_{2}$. We develop an algorithm that computes the closure of the projection of $\unicode[STIX]{x1D6E4}$ on $\text{Aut}(T_{d_{t}})$ under the hypothesis that $d_{t}\geqslant 6$ is even and that the local action of $\unicode[STIX]{x1D6E4}$ on $T_{d_{t}}$ contains $\text{Alt}(d_{t})$. We show that if $\unicode[STIX]{x1D6E4}$ is torsion-free and $d_{1}=d_{2}=6$, exactly seven closed subgroups of $\text{Aut}(T_{6})$ arise in this way. We also construct two new infinite families of virtually simple lattices in $\text{Aut}(T_{6})\times \text{Aut}(T_{4n})$ and in $\text{Aut}(T_{2n})\times \text{Aut}(T_{2n+1})$, respectively, for all $n\geqslant 2$. In particular, we provide an explicit presentation of a torsion-free infinite simple group on 5 generators and 10 relations, that splits as an amalgamated free product of two copies of $F_{3}$ over $F_{11}$. We include information arising from computer-assisted exhaustive searches of lattices in products of trees of small degrees. In an appendix by Pierre-Emmanuel Caprace, some of our results are used to show that abstract and relative commensurator groups of free groups are almost simple, providing partial answers to questions of Lubotzky and Lubotzky–Mozes–Zimmer.

B. Banaschewski [1] has produced a counterexample to [2, Theorem 6]. As noted in [1, Remark 1], our error occurs in the final paragraph of the purported proof of [2, Theorem 6], for P need not be a subpresheaf of I. Accordingly, it remains an open problem to find an analogue of [2, Proposition 1] in the context of Boolean spaces. We hope that attacks on this problem will be facilitated by the (valid) initial three paragraphs of the argument given for [2, Theorem 6].

The following alterations to [2] are in order. Example 5, being a corollary of Theorem 6, remains doubtful, although the special case noted on pp. 1034-1035 is not affected. In Corollary 7, the assertion that j preserves infectives remains doubtful, although the proof for divisibility of j(M)(G) is valid.

In § I, it is shown that M(G)*, the space of bounded linear functionals on M(G), can be represented as a semigroup of bounded operators on M(G).

Let △ denote the non-zero multiplicative linear functionals on M(G) and let P be the norm closed linear span of △ in M(G)*. In § II, it is shown that P, with the Arens multiplication, is a commutative B*-algebra with identity. Thus P = C(B), where B is a compact, Hausdorff space.

This paper continues the nonstandard duality theory of locally convex, topological vector spaces begun in Section 5 of [3]. In Section 1, we isolate an external property, called the pseudo monad, that appears to be one of the central concepts of the theory (Definition 1.2). In Section 2, we relate the pseudo monad to the Fin operation. For example, it is shown that the pseudo monad of a µ-saturated subset A of *E, the nonstandard model of the vector space E, is the smallest subset of A that generates Fin (A) (Proposition 2.7).

The nonstandard model of a dual system of vector spaces is considered in Section 3. In this section, we use pseudo monads to establish relationships among infinitesimal polars, finite polars (see (3.1) and (3.2)) and the Fin operation (Theorem 3.7).

Let H denote the set of all entire functions of a single complex variable equipped with the topology of convergence uniform on all compact subsets of C, the set of complex numbers. Then an entire function f is mean periodic if the subspace spanned by f and its complex translates is not dense in H. It was shown by Schwartz [13, p. 922] in 1947, to whom this definition is due, that any such function is the limit in H of a certain sequence of exponential polynomials.

Let R be a ring with involution*. In this paper, we study additive subgroups A of R which are invariant under all mappings of the form ϕx : a → xax*. That is, xAx* ⊆ A for all x ∈ R. Obvious examples of such subgroups A are ideals of R, the set of symmetric elements, and the set of skew-symmetric elements. We will prove that when R is *-prime, these examples are essentially the only ones.

The real line will be R, Euclidean n-space will be Rn, the unit ball in Rn will be En, the unit sphere in Rn+1 will be Sn, and real projective n-space will be Pn. The canonical line bundle associated with the double cover Sn → Pn will be ηn. If γ is a vector bundle, E(γ) will be its associated cell bundle, S(γ) its associated sphere bundle, P(γ) its associated projective space bundle (P(γ) = S(γ) / (-1)) and T(γ) = E(γ)/S(γ) its Thorn space.

In a previous paper [1], the authors introduced and studied the notion of special conformally flat spaces and quasi-umbilical hypersurfaces. In that paper, the authors proved that every conformally flat space of codimension one in a Euclidean space is special and, conversely, every special conformally flat space can be isometrically immersed in a Euclidean space as a quasi-umbilical hypersurface.

In the present paper, the authors study the conformally flat spaces of codimension 2 in a Euclidean space. (Manifolds, mappings, functions, etc. are assumed to be sufficiently differentiate and we shall restrict ourselves only to manifolds of dimension n > 3.)

The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.

This viewpoint of studying projective planes was given in my previous paper (12). It is discussed in other papers: Hall (4, 6, 7), Maisano (16), Lombardo-Radice (14, 15), Wagner (19). In particular, we consider how to make identifications in the free plane, or how identifications are forced when one begins with a non-degenerate quadrangle and makes free extensions of this quadrangle of a known plane. We shall continue to develop this topic using the notations and definitions of the previous paper (12). We consider the number of subplanes of certain planes, finding exact values in the cases of the known order-nine planes, and deriving a lower bound in a general case. We prove a theorem concerning the structure of all singly generated planes. We give an example to show that this structure is not definitive. Finally, we prove that a specific Walker-Knuth plane is singly generated.

There are several ways to approach the eigenfunction expansion problem for ordinary differential operators via the spectral theorem for self-ad joint linear operators in Hilbert space. One can examine the resolvent, which requires a detailed study of the Green's function (4, 5, 7), or one can use the spectral theorem for unbounded operators (2, 3, 9). Since the eigenf unction expansion theorem also requires some multiplicity theory, unless one is prepared to use a rather powerful form of the spectral theorem for unbounded operators, as in (2, 9), the proof requires a good deal of work in addition to the spectral theorem.