The effect of projections on fractal sets and measures in Banach spaces

We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the ‘thickness exponent’ of the set, which was defined by Hunt and Kaloshin (Nonlinearity 12 (1999), 1263–1275). More precisely, let $X$ be a compact subset of a Banach space $B$ with thickness exponent $\tau$ and Hausdorff dimension $d$. Let $M$ be any subspace of the (locally) Lipschitz functions from B to $\mathbb{R}^{m}$ that contains the space of bounded linear functions. We prove that for almost every (in the sense of prevalence) function f in M, the Hausdorff dimension of f(X) is at least min{m,d/(1 + tau)}. We also prove an analogous result for a certain part of the dimension spectra of Borel probability measures supported on X. The factor 1/(1 + tau) can be improved to 1/(1 + tau/2) if B is a Hilbert space. Since dimension cannot increase under a (locally) Lipschitz function, these theorems become dimension preservation results when tau = 0. We conjecture that many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero. We also discuss the sharpness of our results in the case tau > 0

Hold-up power supply for flash memory

A hold-up power supply for flash memory systems is provided. The hold-up power supply provides the flash memory with the power needed to temporarily operate when a power loss exists. This allows the flash memory system to complete any erasures and writes, and thus allows it to shut down gracefully. The hold-up power supply detects when a power loss on a power supply bus is occurring and supplies the power needed for the flash memory system to temporally operate. The hold-up power supply stores power in at least one capacitor. During normal operation, power from a high voltage supply bus is used to charge the storage capacitors. When a power supply loss is detected, the power supply bus is disconnected from the flash memory system. A hold-up controller controls the power flow from the storage capacitors to the flash memory system. The hold-up controller uses feedback to assure that the proper voltage is provided from the storage capacitors to the flash memory system. This power supplied by the storage capacitors allows the flash memory system to complete any erasures and writes, and thus allows the flash memory system to shut down gracefully

Transcriptional delay stabilizes bistable gene networks

Transcriptional delay can significantly impact the dynamics of gene networks. Here we examine how such delay affects bistable systems. We investigate several stochastic models of bistable gene networks and find that increasing delay dramatically increases the mean residence times near stable states. To explain this, we introduce a non-Markovian, analytically tractable reduced model. The model shows that stabilization is the consequence of an increased number of failed transitions between stable states. Each of the bistable systems that we simulate behaves in this manner

Effects of cell cycle noise on excitable gene circuits

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a 3-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.Comment: 15 pages, 8 figure

Circuit for Full Charging of Series Lithium-Ion Cells

An advanced charger has been proposed for a battery that comprises several lithium-ion cells in series. The proposal is directed toward charging the cells in as nearly an optimum manner as possible despite unit-to-unit differences among the nominally identical cells. The particular aspect of the charging problem that motivated the proposal can be summarized as follows: During bulk charging (charging all the cells in series at the same current), the voltages of individual cells increase at different rates. Once one of the cells reaches full charge, bulk charging must be stopped, leaving other cells less than fully charged. To make it possible to bring all cells up to full charge once bulk charging has been completed, the proposed charger would include a number of top-off chargers one for each cell. The top-off chargers would all be powered from the same DC source, but their outputs would be DC-isolated from each other and AC-coupled to their respective cells by means of transformers, as described below. Each top-off charger would include a flyback transformer, an electronic switch, and an output diode. For suppression of undesired electromagnetic emissions, each top-off charger would also include (1) a resistor and capacitor configured to act as a snubber and (2) an inductor and capacitor configured as a filter. The magnetic characteristics of the flyback transformer and the duration of its output pulses determine the energy delivered to the lithium-ion cell. It would be necessary to equip the cell with a precise voltage monitor to determine when the cell reaches full charge. In response to a full-charge reading by this voltage monitor, the electronic switch would be held in the off state. Other cells would continue to be charged similarly by their top-off chargers until their voltage monitors read full charge