Extensions and block decompositions for finite-dimensional representations of equivariant map algebras

Suppose a finite group acts on a scheme $X$ and a finite-dimensional Lie algebra $\mathfrak{g}$. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from $X$ to $\mathfrak{g}$. The irreducible finite-dimensional representations of these algebras were classified in previous work with P. Senesi, where it was shown that they are all tensor products of evaluation representations and one-dimensional representations. In the current paper, we describe the extensions between irreducible finite-dimensional representations of an equivariant map algebra in the case that $X$ is an affine scheme of finite type and $\mathfrak{g}$ is reductive. This allows us to also describe explicitly the blocks of the category of finite-dimensional representations in terms of spectral characters, whose definition we extend to this general setting. Applying our results to the case of generalized current algebras (the case where the group acting is trivial), we recover known results but with very different proofs. For (twisted) loop algebras, we recover known results on block decompositions (again with very different proofs) and new explicit formulas for extensions. Finally, specializing our results to the case of (twisted) multiloop algebras and generalized Onsager algebras yields previously unknown results on both extensions and block decompositions.Comment: 41 pages; v2: minor corrections, formatting changed to match published versio

Ozone Depletion from Nearby Supernovae

Estimates made in the 1970's indicated that a supernova occurring within tens of parsecs of Earth could have significant effects on the ozone layer. Since that time, improved tools for detailed modeling of atmospheric chemistry have been developed to calculate ozone depletion, and advances have been made in theoretical modeling of supernovae and of the resultant gamma-ray spectra. In addition, one now has better knowledge of the occurrence rate of supernovae in the galaxy, and of the spatial distribution of progenitors to core-collapse supernovae. We report here the results of two-dimensional atmospheric model calculations that take as input the spectral energy distribution of a supernova, adopting various distances from Earth and various latitude impact angles. In separate simulations we calculate the ozone depletion due to both gamma-rays and cosmic rays. We find that for the combined ozone depletion roughly to double the ``biologically active'' UV flux received at the surface of the Earth, the supernova must occur at <8 pc. Based on the latest data, the time-averaged galactic rate of core-collapse supernovae occurring within 8 pc is ~1.5/Gyr. In comparing our calculated ozone depletions with those of previous studies, we find them to be significantly less severe than found by Ruderman (1974), and consistent with Whitten et al. (1976). In summary, given the amplitude of the effect, the rate of nearby supernovae, and the ~Gyr time scale for multicellular organisms on Earth, this particular pathway for mass extinctions may be less important than previously thought.Comment: 24 pages, 4 Postscript figures, to appear in The Astrophysical Journal, 2003 March 10, vol. 58

Target search on a dynamic DNA molecule

We study a protein-DNA target search model with explicit DNA dynamics applicable to in vitro experiments. We show that the DNA dynamics plays a crucial role for the effectiveness of protein "jumps" between sites distant along the DNA contour but close in 3D space. A strongly binding protein that searches by 1D sliding and jumping alone, explores the search space less redundantly when the DNA dynamics is fast on the timescale of protein jumps than in the opposite "frozen DNA" limit. We characterize the crossover between these limits using simulations and scaling theory. We also rationalize the slow exploration in the frozen limit as a subtle interplay between long jumps and long trapping times of the protein in "islands" within random DNA configurations in solution.Comment: manuscript and supplementary material combined into a single documen

Visualizing and quantifying structural diversity around mobile resistance genes.

Understanding the evolution of mobile genes is important for understanding the spread of antimicrobial resistance (AMR). Many clinically important AMR genes have been mobilized by mobile genetic elements (MGEs) on the kilobase scale, such as integrons and transposons, which can integrate into both chromosomes and plasmids and lead to rapid spread of the gene through bacterial populations. Looking at the flanking regions of these mobile genes in diverse genomes can highlight common structures and reveal patterns of MGE spread. However, historically this has been a largely descriptive process, relying on gene annotation and expert knowledge. Here we describe a general method to visualize and quantify the structural diversity around genes using pangraph to find blocks of homologous sequence. We apply this method to a set of 12 clinically important beta-lactamase genes and provide interactive visualizations of their flanking regions at https://liampshaw.github.io/flanking-regions. We show that nucleotide-level variation in the mobile gene itself generally correlates with increased structural diversity in its flanking regions, demonstrating a relationship between rates of mutational evolution and rates of structural evolution, and find a bias for greater structural diversity upstream. Our framework is a starting point to investigate general rules that apply to the horizontal spread of new genes through bacterial populations

Physical and digital phantoms for validating tractography and assessing artifacts

Fiber tractography is widely used to non-invasively map white-matter bundles in vivo using diffusion-weighted magnetic resonance imaging (dMRI). As it is the case for all scientific methods, proper validation is a key prerequisite for the successful application of fiber tractography, be it in the area of basic neuroscience or in a clinical setting. It is well-known that the indirect estimation of the fiber tracts from the local diffusion signal is highly ambiguous and extremely challenging. Furthermore, the validation of fiber tractography methods is hampered by the lack of a real ground truth, which is caused by the extremely complex brain microstructure that is not directly observable non-invasively and that is the basis of the huge network of long-range fiber connections in the brain that are the actual target of fiber tractography methods. As a substitute for in vivo data with a real ground truth that could be used for validation, a widely and successfully employed approach is the use of synthetic phantoms. In this work, we are providing an overview of the state-of-the-art in the area of physical and digital phantoms, answering the following guiding questions: “What are dMRI phantoms and what are they good for?”, “What would the ideal phantom for validation fiber tractography look like?” and “What phantoms, phantom datasets and tools used for their creation are available to the research community?”. We will further discuss the limitations and opportunities that come with the use of dMRI phantoms, and what future direction this field of research might take

Effect of anisotropy and destructuration on behavior of Haarajoki test embankment

This paper investigates the influence of anisotropy and destructuration on the behavior of Haarajoki test embankment, which was built by the Finnish National Road Administration as a noise barrier in 1997 on a soft clay deposit. Half of the embankment is constructed on an area improved with prefabricated vertical drains, while the other half is constructed on the natural deposit without any ground improvement. The construction and consolidation of the embankment is analyzed with the finite-element method using three different constitutive models to represent the soft clay. Two recently proposed constitutive models, namely S-CLAY1 which accounts for initial and plastic strain induced anisotropy, and its extension, called S-CLAY1S which accounts, additionally, for interparticle bonding and degradation of bonds, were used in the analysis. For comparison, the problem is also analyzed with the isotropic modified cam clay model. The results of the numerical analyses are compared with the field measurements. The simulations reveal the influence that anisotropy and destructuration have on the behavior of an embankment on soft clay

On the action potential as a propagating density pulse and the role of anesthetics

The Hodgkin-Huxley model of nerve pulse propagation relies on ion currents through specific resistors called ion channels. We discuss a number of classical thermodynamic findings on nerves that are not contained in this classical theory. Particularly striking is the finding of reversible heat changes, thickness and phase changes of the membrane during the action potential. Data on various nerves rather suggest that a reversible density pulse accompanies the action potential of nerves. Here, we attempted to explain these phenomena by propagating solitons that depend on the presence of cooperative phase transitions in the nerve membrane. These transitions are, however, strongly influenced by the presence of anesthetics. Therefore, the thermodynamic theory of nerve pulses suggests a explanation for the famous Meyer-Overton rule that states that the critical anesthetic dose is linearly related to the solubility of the drug in the membranes.Comment: 13 pages, 8 figure

Simulation-based reachability analysis for nonlinear systems using componentwise contraction properties

A shortcoming of existing reachability approaches for nonlinear systems is the poor scalability with the number of continuous state variables. To mitigate this problem we present a simulation-based approach where we first sample a number of trajectories of the system and next establish bounds on the convergence or divergence between the samples and neighboring trajectories. We compute these bounds using contraction theory and reduce the conservatism by partitioning the state vector into several components and analyzing contraction properties separately in each direction. Among other benefits this allows us to analyze the effect of constant but uncertain parameters by treating them as state variables and partitioning them into a separate direction. We next present a numerical procedure to search for weighted norms that yield a prescribed contraction rate, which can be incorporated in the reachability algorithm to adjust the weights to minimize the growth of the reachable set

Relating a calcium indicator signal to the unperturbed calcium concentration time-course

BACKGROUND: Optical indicators of cytosolic calcium levels have become important experimental tools in systems and cellular neuroscience. Indicators are known to interfere with intracellular calcium levels by acting as additional buffers, and this may strongly alter the time-course of various dynamical variables to be measured. RESULTS: By investigating the underlying reaction kinetics, we show that in some ranges of kinetic parameters one can explicitly link the time dependent indicator signal to the time-course of the calcium influx, and thus, to the unperturbed calcium level had there been no indicator in the cell