We extend the classical notion of an outer action $\alpha $ of a group $G$ on a unital ring $A$ to the case when $\alpha $ is a partial action on ideals, all of which have local units. We show that if $\alpha $ is an outer partial action of an abelian group $G$, then its associated partial skew group ring $A\,{{\star }_{\alpha }}\,G$ is simple if and only if $A$ is $G$-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

In [1], Solovay constructed a model of ZFC in which every set of reals in OD (R) is Lebesgue measurable. Here we construct a model in which every equivalence class of sets of reals modulo null sets that is in OD(R) consists of Lebesgue measurable sets. This result immediately implies Solovay's, since the equivalence class of any set of reals in OD (R) is itself OD (R). As a consequence, one cannot provably explicitly define a nonmeasurable set modulo null sets within ZFC. We do not know whether this holds in the model Solovay uses (where an inaccessible cardinal is collapsed to ω 1). Instead, our model is a generic extension of his model.

In the model we construct, a somewhat stronger statement holds: every set in OD(R) of sets of reals which has < 2C inequivalent elements modulo null sets, consists entirely of Lebesgue measurable sets of reals.

Let E be a completely regular space, let C* denote the Banach algebra of continuous bounded real-valued functions on E, and let A* denote a Banach subalgebra which contains the constant function 1. Let

The purpose of this note is to discuss some properties of the subsets L(A*) of the ring C of continuous real-valued functions on E. They arose out of an unsuccessful attempt to distinguish C* from its Banach subalgebras.

Let K be a number field. For which primes p does there exist an elliptic curve $E / K$ admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.

The “combinatorial line conjecture” states that for all q ≧ 2 and there exists such that if , X is a q-element set, and A is any subset of X n (= cartesian product of n copies of X) with more than elements (that is, A has density greater than ), then A contains a combinatorial line. (For a definition of combinatorial line, together with statements and proofs of many results related to the combinatorial line conjecture, including all those results mentioned below, see [5]. Since we are not directly concerned with combinatorial lines in this paper, we do not reproduce the definition here.)

This conjecture (which is a strengthened version of a conjecture of Moser [7]), if true, would bear the same relation to the Hales-Jewett theorem that Szemerédi's theorem bears to van der Waerden's theorem.

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.

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.

For nonlinear equations of the form

I

there has been considerable interest in determining the asymptotic forms of nonoscillatory solutions. We assume r(t) is continuous and positive on [0, ∞), and f(t, x) is continuous on [0, ∞) × R, and f(t, x) ≥ 0 for x ≠ 0. For n = 2, equation (I) was studied by Kusano and Naito [3], who found necessary and sufficient conditions for the existence of minimal and maximal nonoscillatory solutions. The former are the bounded solutions, while the later are those asymptotic to the function

1.1

Their method consisted of writing (I) in the form of an integral operator and applying the Schauder fixed point theorem. For arbitrary n, but for r(t) = 1, Kreith [2] found necessary and sufficient conditions for the existence of maximal solutions.

Residual torsion-free nilpotence has proved to be an important property for knot groups with applications to bi-orderability and ribbon concordance. Mayland proposed a strategy to show that a two-bridge knot group has a commutator subgroup which is a union of an ascending chain of para-free groups. This paper proves Mayland’s assertion and expands the result to the subgroups of two-bridge link groups that correspond to the kernels of maps to $\mathbb{Z}$. We call these kernels the Alexander subgroups of the links. As a result, we show the bi-orderability of a large family of two-bridge link groups. This proof makes use of a modified version of a graph-theoretic construction of Hirasawa and Murasugi in order to understand the structure of the Alexander subgroup for a two-bridge link group.

Let $E/F$ be a quadratic extension of number fields. In this paper, we show that the genus formula for Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the 2-rank of the Hilbert kernel of $E$ provided that the 2-primary Hilbert kernel of $F$ is trivial. However, since the original genus formula is not explicit enough in a very particular case, we first develop a refinement of this formula in order to employ it in the calculation of the 2-rank of $E$ whenever $F$ is totally real with trivial 2-primary Hilbert kernel. Finally, we apply our results to quadratic, bi-quadratic, and tri-quadratic fields which include a complete 2-rank formula for the family of fields $\mathbb{Q}(\sqrt{2},\sqrt{\delta )}$ where $\delta $ is a squarefree integer.

Given a sequence $\varrho =(r_n)_n\in [0,1)$ tending to $1$, we consider the set ${\mathcal {U}}_A({\mathbb {D}},\varrho )$ of Abel universal functions consisting of holomorphic functions f in the open unit disk $\mathbb {D}$ such that for any compact set K included in the unit circle ${\mathbb {T}}$, different from ${\mathbb {T}}$, the set $\{z\mapsto f(r_n \cdot )\vert _K:n\in \mathbb {N}\}$ is dense in the space ${\mathcal {C}}(K)$ of continuous functions on K. It is known that the set ${\mathcal {U}}_A({\mathbb {D}},\varrho )$ is residual in $H(\mathbb {D})$. We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it is not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at $0$ are dense in ${\mathcal {C}}(K)$ for any compact set $K\subset {\mathbb {T}}$ different from ${\mathbb {T}}$. Moreover, we prove that the class of Abel universal functions is not invariant under the action of the differentiation operator. Finally, an Abel universal function can be viewed as a universal vector of the sequence of dilation operators $T_n:f\mapsto f(r_n \cdot )$ acting on $H(\mathbb {D})$. Thus, we study the dynamical properties of $(T_n)_n$ such as the multiuniversality and the (common) frequent universality. All the proofs are constructive.

In [5] Pall denned a partitioning of a quadratic space over a field of characteristic not 2 to be a collection of disjoint (except for ﹛0﹜ ) maximal totally isotropic subspaces whose union formed the set of isotropic vectors. Clearly isometric quadratic spaces simultaneously do or do not have partitionings. Pall exhibited the existence of partitionings for the spaces associated with

over formally real fields for n = 1, 2, 4, 8 and over Z/(p), p prime, for n = 1, 2. Using the latter, he was able to find a new proof for Jacobi's formula for the number of representations of a positive integer as the sum of four integral squares.

The main purpose of this note is to prove a special case of the following conjecture.

Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).

Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functions

is in Hp(B2) for 0 < p < 3/2.

We consider the Randers space $({{V}^{n}},\,{{F}_{b}})$ obtained by perturbing the Euclidean metric by a translation, ${{F}_{b}}\,=\,\alpha \,+\,\beta $, where $\alpha $ is the Euclidean metric and $\beta $ is a 1-form with norm $b,\,0\,\le \,b\,<\,1$. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces $({{V}^{3}},\,{{F}_{b}})$ of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when $b\,=\,0$. It also reduces to the equation that characterizes the minimal rotational surfaces in $({{V}^{3}},\,{{F}_{b}})$ when $H\,=\,0$, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for $0\,<\,b\,<\,\frac{\sqrt{3}}{3}$. Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical systemand by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.

Let En be the n-dimensional euclidean real space and the group of unimodular affine transformations which operates on it. It is known that the sets of linear h-spaces Lh (0 < h < n) have no invariant measure with respect to u (5). We wish now to consider sets of elements

1.1

composed by q parallel subspaces of dimensions h1, h2, … , hq which transform transitively by .

Soit $G$ le groupe $\text{GL}\,\left( N,\,F \right)$, où $F$ est un corps localement compact et non archimédien. En utilisant la théorie des types simples de Bushnell et Kutzko, ainsi qu'une idée originale d'Henniart, nous construisons des pseudo-coefficients explicites pour les représentations de la série discrète de $G$. Comme application, nous en déduisons des formules inédites pour la valeur du charactère d'Harish- Chandra de certaines telles représentations en certainséléments elliptiques réguliers.

The purpose of this paper is to introduce a generalization of metric space that arises naturally out of the notion of signal function as it occurs, for example, in (5). In §§ 2-5, the basic definitions and motivation are given. In §§ 6 and 7 several elementary topological properties are proved, and in §§ 8 and 9 an important example from special relativity is developed.

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.

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.

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).