Problems in fluid dynamics

A scheme was developed for the parametric differentiation and integration of gas dynamics equations. A numerical integration of the gas dynamics equations is necessarily performed for a specific set of parameter values. The linear variational equations are obtained by differentiating the exact equations with respect to each of the relevant parameters. The resulting matrix of flow quantities is referred to as the Jacobi matrix. The subsequent procedure is then straightforward. The method was tested for two dimensional supersonic flow past an airfoil, with airfoil thickness, camber, and angle of attack varied. This approach has great potential value for rapidly assessing the effect of design changes. The other focus of the work was on problems in fluid stability, bifurcations, and turbulence

Mitigation of Cyclonic Activity

Under realistic estimates of geophysical conditions, two procedures are presented for diminishing the intensity of a hurricane: before it reaches landfall; or quenching it in its incipient stage. We demonstrate that within present-day technology, it is possible to mix the cold deep ocean with the warm surface layer sufficiently, and in a timely manner, in order to decrease the intensity. Two strategies will be presented: (1) In a manner similar to hurricane weakening by landfall, a virtual early landfall is created on the hurricane path, before true landfall; (2) The ocean surface area of an identified tropical depression or storm, with hurricane potential, is tracked and cooled by continued anti-cyclonic mixing until it is no longer a threat. Estimates of the power needed to perform the needed ocean mixing, in a timely manner, shows that this might be accomplished by assembling a sufficient number of high performance submarines. Accomplishment is facilitated by a remarkably high coefficient of performance, O(10^4). Destructive power ~Vm^3 , where Vm is maximal hurricane wind speed, thus even a modest 20% reduction in wind speed produces a ~50% reduction in damage causality. Novel submarine modifications are introduced to achieve the mixing process It is the contention of this paper that a practical framework exists for sensibly reducing the tragedy and devastation caused by hurricanes

The Wigner Transform and Some Exact Properties of Linear Operators

The Wigner transform of an integral kernel on the full line generalizes the Fourier transform of a translation kernel. The eigenvalue spectra of Hermitian kernels are related to the topographic features of their Wigner transforms. Two kernels whose Wigner transforms are equivalent under the unimodular affine group have the same spectrum of eigenvalues and have eigenfunctions related by an explicit linear transformation. Any kernel whose Wigner transform has concentric ellipses as contour lines, yields an eigenvalue problem which may be solved exactly

Kolmogorov Inertial Range for Inhomogeneous Turbulent Flows

The Kolmogorov argument for the existence of an inertial range is reexamined in situations for which neither Fourier modes nor homogeneity and local isotropy are natural assumptions. Scaling arguments are shown which are still valid, and generalizations to the -5/3 law are given for the eigenvalue spectrum of the two-point velocity-correlation matrix. Results from several different numerical simulations are presented. Data derived from simulations of channel and convection flows show that a sensible inertial range appears at very modest Reynolds numbers

Spiking Neurons and the First Passage Problem

We derive a model of a neuron\u27s interspike interval probability density through analysis of the first passage problem. The fit of our expression to retinal ganglion cell laboratory data extracts three physiologically relevant parameters, with which our model yields input-output features that conform to laboratory results. Preliminary analysis suggests that under common circumstances, local circuitry readjusts these parameters with changes in firing rate and so endeavors to faithfully replicate an input signal. Further results suggest that the so-called principle of sloppy workmanship also plays a role in evolution\u27s choice of these parameters

The Approach of a Neuron Population Firing Rate to a New Equilibrium: An Exact Theoretical Result

The response of a noninteracting population of identical neurons to a step change in steady synaptic input can be analytically calculated exactly from the dynamical equation that describes the population\u27s evolution in time. Here, for model integrate-and-fire neurons that undergo a fixed finite upward shift in voltage in response to each synaptic event, we compare the theoretical prediction with the result of a direct simulation of 90,000 model neurons. The degree of agreement supports the applicability of the population dynamics equation. The theoretical prediction is in the form of a series. Convergence is rapid, so that the full result is well approximated by a few terms

Structural Analysis of Biodiversity

Large, recently-available genomic databases cover a wide range of life forms, suggesting opportunity for insights into genetic structure of biodiversity. In this study we refine our recently-described technique using indicator vectors to analyze and visualize nucleotide sequences. The indicator vector approach generates correlation matrices, dubbed Klee diagrams, which represent a novel way of assembling and viewing large genomic datasets. To explore its potential utility, here we apply the improved algorithm to a collection of almost 17000 DNA barcode sequences covering 12 widely-separated animal taxa, demonstrating that indicator vectors for classification gave correct assignment in all 11000 test cases. Indicator vector analysis revealed discontinuities corresponding to species- and higher-level taxonomic divisions, suggesting an efficient approach to classification of organisms from poorly-studied groups. As compared to standard distance metrics, indicator vectors preserve diagnostic character probabilities, enable automated classification of test sequences, and generate high-information density single-page displays. These results support application of indicator vectors for comparative analysis of large nucleotide data sets and raise prospect of gaining insight into broad-scale patterns in the genetic structure of biodiversity

Low rank approximation of multidimensional data

In the last decades, numerical simulation has experienced tremendous improvements driven by massive growth of computing power. Exascale computing has been achieved this year and will allow solving ever more complex problems. But such large systems produce colossal amounts of data which leads to its own difficulties. Moreover, many engineering problems such as multiphysics or optimisation and control, require far more power that any computer architecture could achieve within the current scientific computing paradigm. In this chapter, we propose to shift the paradigm in order to break the curse of dimensionality by introducing decomposition to reduced data. We present an extended review of data reduction techniques and intends to bridge between applied mathematics community and the computational mechanics one. The chapter is organized into two parts. In the first one bivariate separation is studied, including discussions on the equivalence of proper orthogonal decomposition (POD, continuous framework) and singular value decomposition (SVD, discrete matrices). Then, in the second part, a wide review of tensor formats and their approximation is proposed. Such work has already been provided in the literature but either on separate papers or into a pure applied mathematics framework. Here, we offer to the data enthusiast scientist a description of Canonical, Tucker, Hierarchical and Tensor train formats including their approximation algorithms. When it is possible, a careful analysis of the link between continuous and discrete methods will be performed.IV Research and Transfer Plan of the University of SevillaInstitut CarnotJunta de AndalucíaIDEX program of the University of Bordeau

Populations of Tightly Coupled Neurons: the RGC/LGN System

A mathematical model, of general character, for the dynamic description of coupled neural oscillators is presented. The population approach, which is employed, applies equally to individually coupled cells, as well as to populations of such couplings. The formulation includes stochasticity, and preserves the details of precisely firing neurons. Based on the generally accepted view of cortical wiring this formulation is applied to the retinal ganglion cell (RGC)/lateral geniculate nucleus (LGN) relay cell system, of the early mammalian visual system. The smallness of quantal jumps at the retinal level permits a Fokker-Planck ap-proximation for the RGC contribution, however the LGN description requires the use of finite jumps, which in the limit of fast synaptic dynamics appears in the model as jumps in the membrane potential. In-depth analyses of equilibrium spiking behavior for both the deterministic and stochastic case are presented. Use of Green’s function methods form the basis for the asymptotic and exact results that are presented. From these one can determine the spiking ratio (i.e., the number of RGC arrivals per LGN spike), which is the reciprocal 1 of the transfer ratio, under wide circumstance, and criteria for spiking regimes are presented. Under reasonable hypothesis it is shown that the transfer ratio is < 1/2, in the absence of input from other (brain) areas. However, transfer ratios that exceed 1/2 (but < 1) have been recorded in the laboratory. Inclusion of brain stem input has been shown to provide such a signal which elevates the transfer ratio (Ozaki and Kaplan, 2006), and a model which includes this as a contribution is also presented.