We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map u : X → R such that for every interval I ⊂ R, the connected components of u−1(I) have diameter ≤ const ·diam(I), then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples [Laa00] bilipschitz embed in L1, though they do not embed i...

Let A be an abelian variety defined over a non-prime finite field Fq that has embedding degree k with respect to a subgroup of prime order r. In this paper we give explicit conditions on q, k, and r that imply that the minimal embedding field of A with respect to r is Fqk . When these conditions hold, the embedding degree k is a good measure of the security level of a pairing-based cryptosystem...

We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces and adjointable maps, preserving adjoint morphisms and all fini...

In 1961 G. Higman proved a remarkable theorem establishing a deep connection between the logical notion of recursiveness and questions about finitely presented groups. The basic aim of the present paper is to provide the reader with a rigorous and detailed proof of Higman’s Theorem. All the necessary preliminary material, including elements of group theory and recursive functions theory, is sys...