Groups definable in local fields and pseudo-finite fields

Using model-theoretic methods we prove: Theorem A If G is a Nash group over the real or p-adic field, then there is a Nash isomorphism between neighbourhoods of the identity of G and of the set of F-rational points of an algebraic group defined over F. Theorem B Let G be a connected affine Nash group over ℝ. Then G is Nash isogeneous with the (real) connected component of the set of real points of an algebraic group defined over ℝ. Theorem C Let G be a group definable in a pseudo-finite field F. Then G is definably virtually isogeneous with the set of F-rational points of an algebraic group defined over F. Both authors supported by NSF grants.     

Resonant and nonresonant patterns in forced oscillators

Uniform oscillations in spatially extended systems resonate with temporal periodic forcing within the Arnold tongues of single forced oscillators. The Arnold tongues are wedge-like domains in the parameter space spanned by the forcing amplitude and frequency, within which the oscillator's frequency is locked to a fraction of the forcing frequency. Spatial patterning can modify these domains. We describe here two pattern formation mechanisms affecting frequency locking at half the forcing frequency. The mechanisms are associated with phase-front instabilities and a Turing-like instability of the rest state. Our studies combine experiments on the ruthenium catalyzed light-sensitive Belousov-Zhabotinsky reaction forced by periodic illumination, and numerical and analytical studies of two model systems, the FitzHugh-Nagumo model and the complex Ginzburg-Landau equation, with additional terms describing periodic forcing.     

Practical considerations in T1 mapping of prostate for dynamic contrast enhancement pharmacokinetic analyses

There are many challenges in developing robust imaging biomarkers that can be reliably applied in a clinical trial setting. In the case of dynamic contrast-enhanced (DCE) MRI, one such challenge is to obtain accurate precontrast T₁ maps for subsequent use in two-compartment pharmacokinetic models commonly used to fit the MR enhancement time courses. In the prostate, a convenient and common approach for this task has been to use the same 3D spoiled gradient-echo sequence used to collect the DCE data, but with variable flip angles (VFAs) to collect data suitable for T₁ mapping prior to contrast injection. However, inhomogeneous radiofrequency conditions within the prostate have been found to adversely affect the accuracy of this technique. Herein we demonstrate the sensitivity of DCE pharmacokinetic parameters to precontrast T₁ values and examine methods to improve the accuracy of T₁ mapping with flip angle-corrected VFA SPGR methods, comparing T₁ maps from such methods with “gold standard” reference T₁ maps generated with saturation recovery experiments performed with fast spin-echo (FSE) sequences.     

Linear ultrafilters

Let X be a k-vector space, and U a maximal proper filter of subspaces of X. Then the ring of endomorphisms of X that are “continuous” with respect to U modulo the ideal of those that are “trivial” with respect to U forms a division ring E(U). (These division rings can also be described as the endomorphism rings of the simple left End(X)-modules.) We study this and the dual construction, based on maximal cofiIters of subspaces of X, in particular, the relation between the constructed division rings and the original field or division ring k. We end by examining a more general construction in which X is a module over a general ring, given with both a filter and a cofilter of submodules.     

The Metabolostasis Network and the Cellular Depository of Aggregation-Prone Metabolites

The vital role of metabolites across all branches of life and their involvement in various disorders have been investigated for decades. Many metabolites are poorly soluble in water or in physiological buffers and tend to form supramolecular aggregates. On the other hand, in the cell, they should be preserved in a pool and be readily available for the execution of biochemical functions. We thus propose that a quality-control network, termed “metabolostasis”, has evolved to regulate the storage and retrieval of aggregation-prone metabolites. Such a system should control metabolite concentration, subcellular localization, supramolecular arrangement, and interaction in dynamic environments, thus enabling normal cellular physiology, healthy development, and preventing disease onset. The paradigm-shifting concept of metabolostasis calls for a reevaluation of the traditional view of metabolite storage and dynamics in physiology and pathology and proposes unprecedented directions for therapeutic targets under conditions where metabolostasis is imbalanced.