Level Sets of the Takagi Function: Local Level Sets

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a "generic" full Lebesgue measure set of ordinates y, the level sets are finite sets. Here it is shown for a "generic" full Lebesgue measure set of abscissas x, the level set L(\tau(x)) is uncountable. An interesting singular monotone function is constructed, associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly 3/2.Comment: 32 pages, 2 figures, 1 table. Latest version has updated equation numbering. The final publication will soon be available at springerlink.co

How large are the level sets of the Takagi function?

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figure

Modelling pellet flow in single extrusion with DEM

Plasticating single-screw extrusion involves the continuous conversion of loose solid pellets into a pressurized homogeneous melt that is pumped through a shaping tool. Traditional analyses of the solids conveying stage assume the movement of an elastic solid plug at a fixed speed. However, not only the corresponding predictions fail considerably, but it is also well known that, at least in the initial screw turns, the flow of loose individual pellets takes place. This study follows previous efforts to predict the characteristics of such a flow using the discrete element method. The model considers the development of normal and tangential forces resulting from the inelastic collisions between the pellets and between them and the neighbouring metallic surfaces. The algorithm proposed here is shown to be capable of capturing detailed features of the granular flow. The predictions of velocities in the cross- and down-channel directions and of the coordination number are in good agreement with equivalent reported results. The effect of pellet size on the flow features is also discussed

Universal architecture of bacterial chemoreceptor arrays

Chemoreceptors are key components of the high-performance signal transduction system that controls bacterial chemotaxis. Chemoreceptors are typically localized in a cluster at the cell pole, where interactions among the receptors in the cluster are thought to contribute to the high sensitivity, wide dynamic range, and precise adaptation of the signaling system. Previous structural and genomic studies have produced conflicting models, however, for the arrangement of the chemoreceptors in the clusters. Using whole-cell electron cryo-tomography, here we show that chemoreceptors of different classes and in many different species representing several major bacterial phyla are all arranged into a highly conserved, 12-nm hexagonal array consistent with the proposed “trimer of dimers” organization. The various observed lengths of the receptors confirm current models for the methylation, flexible bundle, signaling, and linker sub-domains in vivo. Our results suggest that the basic mechanism and function of receptor clustering is universal among bacterial species and was thus conserved during evolution

Melting of polymer blends in single-screw extrusion : an experimental study

Melting is a major step in plasticating single screw extrusion, but most of the existing phenomenological know how was gathered by performing Maddock-type experiments with homopolymers. Given the current widespread industrial use of polymer blends, it is worth determining whether the same mechanisms and mathematical models apply, or whether different sequences develop. This work reports the results of Maddock-type experiments using a PA6/PP blend, both in its immiscible and compatibilized varieties. A melting mechanism combining the features of the classical Tadmor mechanism and of the dispersed melting mechanism, also previously reported in the literature, was observed.The authors are grateful to Portuguese Fundacao para a Ciencia e Tecnologia for supporting this work under grant SFRH/BD/19997/2004 and to DSM, the Netherlands, for supplying PA6

Overview of mathematical approaches used to model bacterial chemotaxis I: the single cell

Mathematical modeling of bacterial chemotaxis systems has been influential and insightful in helping to understand experimental observations. We provide here a comprehensive overview of the range of mathematical approaches used for modeling, within a single bacterium, chemotactic processes caused by changes to external gradients in its environment. Specific areas of the bacterial system which have been studied and modeled are discussed in detail, including the modeling of adaptation in response to attractant gradients, the intracellular phosphorylation cascade, membrane receptor clustering, and spatial modeling of intracellular protein signal transduction. The importance of producing robust models that address adaptation, gain, and sensitivity are also discussed. This review highlights that while mathematical modeling has aided in understanding bacterial chemotaxis on the individual cell scale and guiding experimental design, no single model succeeds in robustly describing all of the basic elements of the cell. We conclude by discussing the importance of this and the future of modeling in this area

Screen for Localized Proteins in Caulobacter crescentus

Precise localization of individual proteins is required for processes such as motility, chemotaxis, cell-cycle progression, and cell division in bacteria, but the number of proteins that are localized in bacterial species is not known. A screen based on transposon mutagenesis and fluorescence activated cell sorting was devised to identify large numbers of localized proteins, and employed in Caulobacter crescentus. From a sample of the clones isolated in the screen, eleven proteins with no previously characterized localization in C. crescentus were identified, including six hypothetical proteins. The localized hypothetical proteins included one protein that was localized in a helix-like structure, and two proteins for which the localization changed as a function of the cell cycle, suggesting that complex three-dimensional patterns and cell cycle-dependent localization are likely to be common in bacteria. Other mutants produced localized fusion proteins even though the transposon has inserted near the 5′ end of a gene, demonstrating that short peptides can contain sufficient information to localize bacterial proteins. The screen described here could be used in most bacterial species

The 10th Biennial Hatter Cardiovascular Institute workshop: cellular protection—evaluating new directions in the setting of myocardial infarction, ischaemic stroke, and cardio-oncology

Due to its poor capacity for regeneration, the heart is particularly sensitive to the loss of contractile cardiomyocytes. The onslaught of damage caused by ischaemia and reperfusion, occurring during an acute myocardial infarction and the subsequent reperfusion therapy, can wipe out upwards of a billion cardiomyocytes. A similar program of cell death can cause the irreversible loss of neurons in ischaemic stroke. Similar pathways of lethal cell injury can contribute to other pathologies such as left ventricular dysfunction and heart failure caused by cancer therapy. Consequently, strategies designed to protect the heart from lethal cell injury have the potential to be applicable across all three pathologies. The investigators meeting at the 10th Hatter Cardiovascular Institute workshop examined the parallels between ST-segment elevation myocardial infarction (STEMI), ischaemic stroke, and other pathologies that cause the loss of cardiomyocytes including cancer therapeutic cardiotoxicity. They examined the prospects for protection by remote ischaemic conditioning (RIC) in each scenario, and evaluated impasses and novel opportunities for cellular protection, with the future landscape for RIC in the clinical setting to be determined by the outcome of the large ERIC-PPCI/CONDI2 study. It was agreed that the way forward must include measures to improve experimental methodologies, such that they better reflect the clinical scenario and to judiciously select combinations of therapies targeting specific pathways of cellular death and injury