An Approximate Version of the Jordan von Neumann Theorem for Finite Dimensional Real Normed Spaces

It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite dimensional normed vector spaces over R, we formulate an approximate version of this theorem: if a space approximately satisfies the parallelogram law, then it has a near isometry with Euclidean space. In other words, a small von Neumann Jordan constant E + 1 for X yields a small Banach-Mazur distance with R^n, d(X, R^n) < 1 + B_n E + O(E^2). Finally, we examine how this estimate worsens as the dimension, n, of X increases, with the conclusion that B_n grows quadratically with n.Comment: Version 2 adds contact information for the author and actually states the correct Jordan-von Neumann theorem (oops!

Free Actions on C*-algebra Suspensions and Joins by Finite Cyclic Groups

We present a proof for certain cases of the noncommutative Borsuk-Ulam conjectures proposed by Baum, D\k{a}browski, and Hajac. When a unital $C^*$-algebra $A$ admits a free action of $\mathbb{Z}/k\mathbb{Z}$, $k \geq 2$, there is no equivariant map from $A$ to the $C^*$-algebraic join of $A$ and the compact "quantum" group $C(\mathbb{Z}/k\mathbb{Z})$. This also resolves D\k{a}browski's conjecture on unreduced suspensions of $C^*$-algebras. Finally, we formulate a different type of noncommutative join than the previous authors, which leads to additional open problems for finite cyclic group actions.Comment: 13 pages. To appear in IUMJ. Version 4 restructures the results to address earlier work of Voloviko

Characterization of polyaniline-detonation nanodiamond nanocomposite fibers by atomic force microscopy based technique

Polyaniline (PANI) fibers were synthesized in presence of detonantion nanodiamond (DND) particles by precipitation polymerization technique. Morphological, electrical and mechanical characterizations of the obtained PANI/DND nanocomposited have been performed by different either standard or advanced atomic force microscopy (AFM) based techniques. Morphological characterization by tapping mode AFM supplied information about the structure of fibers and ribbons forming the PANI/DND network. An AFM based technique that takes advantage of an experimental configuration specifically devised for the purpose was used to assess the electrical properties of the fibers, in particular to verify their conductivity. Finally, mechanical characterization was carried out synergically using two different and recently proposed AFM based techniques, one based on AFM tapping mode and the other requiring AFM contact mode, which probed the nanocomposited nature of PANI/DND fiber sample down to different depths. © 2013 Elsevier Ltd. All rights reserved