Let E/F be a Galois extension of number fields with Galois group G. The purpose of this paper is to place limitations on the structure of a Sylow 2-subgroup of G in the case when the extension E/F is a minimal counterexample to Artin’s Conjecture on the holomorphy of L-series. More specifically, assume for some s0 ∈ C − {1} and some irreducible character χ of G that the Artin L-series L(s, χ,E/...

Let $G=\operatorname {SO}^\circ (n,1) \times \operatorname (n,1)$ and $X=\mathbb {H}^{n}\times \mathbb {H}^{n}$ for $n\ge 2$. For a pair $(\pi _1, \pi _2)$ of non-elementary convex cocompact representations finitely generated group $\Sigma$ into $\operatorname (n,1)$, let $\Gamma =(\pi _1\times _2)(\Sigma )$. Denoting the bottom $L^2$-spectrum negative Laplacian on {\setminus } X$ by $\lambda _...

Let ‖ · ‖ be the euclidean norm on R and γn the (standard) Gaussian measure on R with density (2π)e 2/2. Let θ (≃ 1.3489795) be defined by γ1([−θ/2, θ/2]) = 1/2 and let L be a lattice in R n generated by vectors of norm ≤ θ. Then, for any closed convex set V in R with γn(V ) ≥ 1 2 and for any a ∈ R, (a + L) ∩ V 6= φ. The above statement can be viewed as a “nonsymmetric” version of Minkowski The...

In this paper we provide a rigorous toolkit for extending convex risk measures from L∞ to L, for p ≥ 1. Our main result is a one-to-one correspondence between law-invariant convex risk measures on L∞ and L. This proves that the canonical model space for the predominant class of law-invariant convex risk measures is L. Some significant counterexamples illustrate the many pitfalls with convex ris...

Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let {Ti}i=1 be N nonexpansive self-mappings of K with F = ∩i=1F (Ti) ̸= ∅ (here F (Ti) denotes the set of fixed points of Ti). Suppose that one of the mappings in {Ti}i=1 is semi-compact. Let {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1) and {βn} ⊂ [τ, 1] for some τ ∈ (0, 1]. For arbitrary x0 ∈ K, let the sequence {x...

A set A of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. The class of sets B which contain the same information as A under Turing computability (</=T) is the (Turing) degree of A, and a degree is c.e. if it contains a c.e. set. The extension of embedding problem for the c.e. degrees R = (R, ...

Let G be a finitely generated residually finite group and let an(G) denote the number of index n subgroups of G. If an(G) ≤ nα for some α and for all n, then G is said to have polynomial subgroup growth (PSG, for short). The degree of G is then defined by deg(G) = limsup log an(G) log n . Very little seems to be known about the relation between deg(G) and the algebraic structure of G. We derive...

Let X be an infinite-dimensional real reflexive Banach space with dual space X∗ and G⊂ X open and bounded. Assume that X and X∗ are locally uniformly convex. Let T : X ⊃ D(T) → 2X be maximal monotone and C : X ⊃ D(C) → X∗ quasibounded and of type (S̃+). Assume that L ⊂ D(C), where L is a dense subspace of X , and 0 ∈ T(0). A new topological degree theory is introduced for the sum T +C. Browder’s...

In a complete simply connected Riemannian manifold X of pinched negative curvature, we give a sharp criterion for a subset C to be the ǫ-neighbourhood of some convex subset of X, in terms of the extrinsic curvatures of the boundary of C. 1

The aim of this paper is to present some existence results for nearest-point and farthestpoint problems, in connection with some geometric properties of Banach spaces. The idea goes back to Efimov and Stečkin who, in a series of papers (see [28, 29, 30, 31]), realized for the first time that some geometric properties of Banach spaces, such as strict convexity, uniform convexity, reflexivity, an...