Finite, primitive and euclidean spaces

Euclidean Structure in Finite Dimensional Normed Spaces

Euclidean Quotients of Finite Metric Spaces

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α ≥ 1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embedings int...

Finite analogues of non-Euclidean spaces and Ramanujan graphs

This is a companion paper of “Finite Euclidean graphs and Ramanujan graphs” by the same authors. Finite analogues of the Poincaré upper half plane, i.e., finite upper half plane graphs, were studied by many researchers, including Terras, Evans etc. Finally, it was proved by combining works of A. Weil, P. Deligne, R. Evans, H. Stark, N. Katz, W. Li and many others, that the finite upper half pla...

Explicit Ramsey Graphs and Erdös Distance Problems over Finite Euclidean and Non-Euclidean Spaces

We study the Erdős distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These graphs are shown to be asymptotically Ramanujan graphs. The advantage of using these graphs is twofold. First, we can derive new lower bounds on the Erdős distanc...

The Prime and Primitive Spectra of Multiparameter Quantum Symplectic and Euclidean Spaces

We investigate a class of algebras that provides multiparameter versions of both quantum symplectic space and quantum Euclidean 2n-space. These algebras encompass the graded quantized Weyl algebras, the quantized Heisenberg space, and a class of algebras introduced by Oh. We describe the structure of the prime and primitive ideals of these algebras. Other structural results include normal separ...