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!

Iterated Joins of Compact Groups

(Joint work with Alexandru Chirvasitu.) The Borsuk-Ulam theorem in algebraic topology indicates restrictions for equivariant maps between spheres; in particular, there is no odd map from a sphere to another sphere of lower dimension. This idea may be generalized greatly in both the topological and operator algebraic settings for actions of compact (quantum) groups, leading to the the noncommutative Borsuk-Ulam conjectures of Baum, Dabrowski, and Hajac. I will present our recent progress (both positive and negative) toward resolving these conjectures using properties of iterated compact group joins

Atypical Protein Kinase C Dependent Polarized Cell Division is Required for Myocardial Trabeculation

A hallmark of cardiac development is the formation of myocardial trabeculations exclusively from the luminal surface of the primitive heart tube. Although a number of genetic defects in the endocardium (Grego-Bessa et al., 2007; Liu et al., 2010) and cardiac jelly (Camenisch et al., 2000) disrupt myocardial trabeculation, the role of cell polarity machinery in driving this process remains unclear. Herein, we demonstrate that atypical protein kinase C iota (Prkci) and its interacting partners of Par polarity complex are localized to the luminal side of luminal myocardial cells. Remarkably, a subset of these cells undergoes polarized cell division with the mitotic spindle and the cell division plane oriented perpendicular to the heart’s lumen and this polarization requires a normal composition of the cardiac jelly. Disruption of the polarity complex through cardiac specific deletion of Prkci or its downstream interacting partner NuMA, results in aberrant mitotic spindle alignment, loss of polarized cardiomyocyte division, and loss of normal myocardial trabeculation. Collectively these results layout a new mechanism for cardiac morphogenesis where in response to inductive signals, Prkci and its downstream partners direct polarized cell division of luminal myocardial cells to drive trabeculation in the early developing heart

Brady v. NFL: How the Eighth Circuit Saved the 2011 NFL Season by Supporting Negotiation, Not Litigation

The article discusses the United States District Court for the District of Minnesota case Brady v. National Football League (NFL) which deals with the NFL players\u27 attempt to obtain a preliminary injunction to prevent the NFL from locking them out during the NFL\u27s collective bargaining agreement (CBA) negotiations with the NFL Payers Association (NFLPA). The U.S. Sherman Anti-Trust Act is also addressed