Georgia Institute of Technology College of Sciences

This talk discusses exponential frames and Riesz sequences in L^2 over a set of finite measure. (Roughly speaking, Frames and Riesz sequences are over complete bases and under complete bases, respectively). Intuitively, one would assume that the frequencies of an exponential frame can not be too sparse, while those of an exponential Riesz sequence can not be too dense. This intuition was confirmed in a very general theorem of Landau, which holds for all bounded sets of positive measure. Landau's proof involved a deep study of the eigenvalues of compositions of certain projection operators. Over the years Landaus technique, as well as some relaxed version of it, were used in many different setting to obtain results of a similar nature. Recently , joint with A. Olevskii, we found a surprisingly simple approach to Landau's density theorems, which provides stronger versions of these results. In particular, the theorem for Riesz sequences was extended to unbounded sets (for frames, such an extension is trivial). In this talk we will discuss Landau's results and our approach for studying questions of this type.

A comprehensive mathematical model of β1-adrenergic signaling system for mouse ventricular myocytes is developed. The model myocyte consists of three major compartments (caveolae, extracaveolae, and cytosol) and includes several modules that describe biochemical reactions and electrical activity upon the activation of β1-adrenergic receptors. In the model, β1-adrenergic receptors are stimulated by an agonist isoproterenol, which leads to activation of Gs-protein signaling pathway to a different degree in different compartments. Gs-protein, in turn, activates adenylyl cyclases to produce cyclic AMP and to activate protein kinase A. Catalytic subunit of protein kinase A phosphorylates cardiac ion channels and intracellular proteins that regulate Ca2+ dynamics. Phosphorylation is removed by the protein phosphatases 1 and 2A. The model is extensively verified by the experimental data on β1-adrenergic regulation of cardiac function. It reproduces time behavior of a number of biochemical reactions and voltage-clamp data on ionic currents in mouse ventricular myocytes; β1-adrenergic regulation of the action potential and intracellular Ca2+ transients; and calcium and sodium fluxes during action potentials. The model also elucidates the mechanism of action potential prolongation and increase in intracellular Ca2+ transients upon stimulation of β1-adrenergic receptors.

Given f: \C \times T^1 to itself, an analytic perturbation of a fibered rotation map , we will present two proofs of existence of an analytic conjugation of f to the fibered rotation , on a neighborhood of {0} \times T^1. This talk will be self- contained except for some usual "tricks" from KAM theory and which will be explained better in another talk. In the talk we will discuss carefully the number theoretic conditions on the fibered rotation needed to obtain the theorem.

Flow-cut dualities in network optimization bear a resemblance to topological dualities. Flows are homological in nature, cuts are cohomological in nature, constraints are sheaf-theoretic in nature, and the duality between max flow-values and min cut-values (MFMC) resembles a Poincare Duality. In this talk, we formalize that resemblance by generalizing Abelian sheaf (co)homology for sheaves of semimodules on directed spaces (e.g. directed graphs). Such directed (co)homology theories generalize constrained flows, characterize cuts, and lift MFMC dualities to a directed Poincare Duality. In the process, we can relate the tractability and decomposability of generalized flows to local and global flatness conditions on the sheaf, extending previous work on monoid-valued flows in the literature [Freize].