Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation-invariant operators on multidimensional integer lattices

Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean and scalar covariance matrices. The resulting anisotropy functional is defined for finite power random vectors. Originally, anisotropy was introduced for directionally generic random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated a-anisotropic norm of a matrix is then its maximum root mean square or average energy gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by a≥0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yield mean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields.

A high definition, finite difference time domain method

A high definition, finite difference time domain (HD-FDTD) method is presented in this paper. This new method allows the FDTD method to be efficiently applied over a very large frequency range including low frequencies, which are problematic for conventional FDTD methods. In the method, no alterations to the properties of either the source or the transmission media are required. The method is essentially frequency independent and has been verified against analytical solutions within the frequency range 50 Hz-1 GHz. As an example of the lower frequency range, the method has been applied to the problem of induced eddy currents in the human body resulting from the pulsed magnetic field gradients of an MRI system. The new method only requires approximately 0.3% of the source period to obtain an accurate solution. (C) 2003 Elsevier Science Inc. All rights reserved.

Calculation of electric fields induced by body and head motion in high-field MRI

In modern magnetic resonance imaging (MRI), patients are exposed to strong, nonuniform static magnetic fields outside the central imaging region, in which the movement of the body may be able to induce electric currents in tissues which could be possibly harmful. This paper presents theoretical investigations into the spatial distribution of induced electric fields and currents in the patient when moving into the MRI scanner and also for head motion at various positions in the magnet. The numerical calculations are based on an efficient, quasi-static, finite-difference scheme and an anatomically realistic, full-body, male model. 3D field profiles from an actively shielded 4T magnet system are used and the body model projected through the field profile with a range of velocities. The simulation shows that it possible to induce electric fields/currents near the level of physiological significance under some circumstances and provides insight into the spatial characteristics of the induced fields. The results are extrapolated to very high field strengths and tabulated data shows the expected induced currents and fields with both movement velocity and field strength. (C) 2003 Elsevier Science (USA). All rights reserved.

Influence of magnetically-induced E-fields on cardiac electric activity during MRI: A modeling study

In modern magnetic resonance imaging (MRI), patients are exposed to strong, time-varying gradient magnetic fields that may be able to induce electric fields (E-fields)/currents in tissues approaching the level of physiological significance. In this work we present theoretical investigations into induced E-fields in the thorax, and evaluate their potential influence on cardiac electric activity under the assumption that the sites of maximum E-field correspond to the myocardial stimulation threshold (an abnormal circumstance). Whole-body cylindrical and planar gradient coils were included in the model. The calculations of the induced fields are based on an efficient, quasi-static, finite-difference scheme and an anatomically realistic, whole-body model. The potential for cardiac stimulation was evaluated using an electrical model of the heart. Twelve-lead electrocardiogram (ECG) signals were simulated and inspected for arrhythmias caused by the applied fields for both healthy and diseased hearts. The simulations show that the shape of the thorax and the conductive paths significantly influence induced E-fields. In healthy patients, these fields are not sufficient to elicit serious arrhythmias with the use of contemporary gradient sets. However, raising the strength and number of repeated switching episodes of gradients, as is certainly possible in local chest gradient sets, could expose patients to increased risk. For patients with cardiac disease, the risk factors are elevated. By the use of this model, the sensitivity of cardiac pathologies, such as abnormal conductive pathways, to the induced fields generated by an MRI sequence can be investigated. (C) 2003 Wiley-Liss, Inc.

Simulation of humidity fields in concrete: experimental validation and parameter estimation

The moisture content in concrete structures has an important influence in their behavior and performance. Several vali-dated numerical approaches adopt the governing equation for relative humidity fields proposed in Model Code 1990/2010. Nevertheless there is no integrative study which addresses the choice of parameters for the simulation of the humidity diffusion phenomenon, particularly in concern to the range of parameters forwarded by Model Code 1990/2010. A software based on a Finite Difference Method Algorithm (1D and axisymmetric cases) is used to perform sensitivity analyses on the main parameters in a normal strength concrete. Then, based on the conclusions of the sensi-tivity analyses, experimental results from nine different concrete compositions are analyzed. The software is used to identify the main material parameters that better fit the experimental data. In general, the model was able to satisfactory fit the experimental results and new correlations were proposed, particularly focusing on the boundary transfer coeffi-cient.

An iterative Multiscale Finite-Volume algorithm converging to the exact solution

The multiscale finite volume (MsFV) method has been developed to efficiently solve large heterogeneous problems (elliptic or parabolic); it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. The method essentially relies on the hypothesis that the (fine-scale) problem can be reasonably described by a set of local solutions coupled by a conservative global (coarse-scale) problem. In most cases, the boundary conditions assigned for the local problems are satisfactory and the approximate conservative fluxes provided by the method are accurate. In numerically challenging cases, however, a more accurate localization is required to obtain a good approximation of the fine-scale solution. In this paper we develop a procedure to iteratively improve the boundary conditions of the local problems. The algorithm relies on the data structure of the MsFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. Two variants are considered: in the first, only the MsFV operator is used; in the second, the MsFV operator is combined in a two-step method with an operator derived from the problem solved to construct the conservative flux field. The resulting iterative MsFV algorithms allow arbitrary reduction of the solution error without compromising the construction of a conservative flux field, which is guaranteed at any iteration. Since it converges to the exact solution, the method can be regarded as a linear solver. In this context, the schemes proposed here can be viewed as preconditioned versions of the Generalized Minimal Residual method (GMRES), with a very peculiar characteristic that the residual on the coarse grid is zero at any iteration (thus conservative fluxes can be obtained).

Finite horizon bargaining and the consistent field

This paper explores the relationships between noncooperative bargaining games and the consistent value for non-transferable utility (NTU) cooperative games. A dynamic approach to the consistent value for NTU games is introduced: the consistent vector field. The main contribution of the paper is to show that the consistent field is intimately related to the concept of subgame perfection for finite horizon noncooperative bargaining games, as the horizon goes to infinity and the cost of delay goes to zero. The solutions of the dynamic system associated to the consistent field characterize the subgame perfect equilibrium payoffs of the noncooperative bargaining games. We show that for transferable utility, hyperplane and pure bargaining games, the dynamics of the consistent fields converge globally to the unique consistent value. However, in the general NTU case, the dynamics of the consistent field can be complex. An example is constructed where the consistent field has cyclic solutions; moreover, the finite horizon subgame perfect equilibria do not approach the consistent value.

Interactions among systems of finite size en predictive relativistic mechanics III. Short-range interaction and second-order terms

We compute up to and including all the c-2 terms in the dynamical equations for extended bodies interacting through electromagnetic, gravitational, or short-range fields. We show that these equations can be reduced to those of point particles with intrinsic angular momentum assuming spherical symmetry.

Adsorption kinetics in the presence of external fields

In this article we present a detailed analysis of the kinetics of a class of sequential adsorption models that take into account the effect of externally applied fields (as an electric field, or a shear rate) on the adsorption. The excluded volume interactions related to the finite size of the adsorbing particles are modified by the external fields. As a result, new adsorption mechanisms appear with respect to the ones used to describe the kinetics in a quiescent fluid. In particular, if the adsorbing particles are allowed to roll over preadsorbed ones, adsorption becomes non local even in the simplest geometry. An exact analytic theory cannot be developed, but we introduce a self-consistent theory that turns out to agree with the simulation results over all the range of the parameters.

Adsorption kinetics in the presence of external fields

In this article we present a detailed analysis of the kinetics of a class of sequential adsorption models that take into account the effect of externally applied fields (as an electric field, or a shear rate) on the adsorption. The excluded volume interactions related to the finite size of the adsorbing particles are modified by the external fields. As a result, new adsorption mechanisms appear with respect to the ones used to describe the kinetics in a quiescent fluid. In particular, if the adsorbing particles are allowed to roll over preadsorbed ones, adsorption becomes non local even in the simplest geometry. An exact analytic theory cannot be developed, but we introduce a self-consistent theory that turns out to agree with the simulation results over all the range of the parameters.

Integral cohomology of finite postnikov towers

Depuis le séminaire H. Cartan de 1954-55, il est bien connu que l'on peut trouver des éléments de torsion arbitrairement grande dans l'homologie entière des espaces d'Eilenberg-MacLane K(G,n) où G est un groupe abélien non trivial et n>1. L'objectif majeur de ce travail est d'étendre ce résultat à des H-espaces possédant plus d'un groupe d'homotopie non trivial. Dans le but de contrôler précisément le résultat de H. Cartan, on commence par étudier la dualité entre l'homologie et la cohomologie des espaces d'Eilenberg-MacLane 2-locaux de type fini. On parvient ainsi à raffiner quelques résultats qui découlent des calculs de H. Cartan. Le résultat principal de ce travail peut être formulé comme suit. Soit X un H-espace ne possédant que deux groupes d'homotopie non triviaux, tous deux finis et de 2-torsion. Alors X n'admet pas d'exposant pour son groupe gradué d'homologie entière réduite. On construit une large classe d'espaces pour laquelle ce résultat n'est qu'une conséquence d'une caractéristique topologique, à savoir l'existence d'un rétract faible X K(G,n) pour un certain groupe abélien G et n>1. On généralise également notre résultat principal à des espaces plus compliqués en utilisant la suite spectrale d'Eilenberg-Moore ainsi que des méthodes analytiques faisant apparaître les nombres de Betti et leur comportement asymptotique. Finalement, on conjecture que les espaces qui ne possédent qu'un nombre fini de groupes d'homotopie non triviaux n'admettent pas d'exposant homologique. Ce travail contient par ailleurs la présentation de la « machine d'Eilenberg-MacLane », un programme C++ conçu pour calculer explicitement les groupes d'homologie entière des espaces d'Eilenberg-MacLane. <br/><br/>By the work of H. Cartan, it is well known that one can find elements of arbitrarilly high torsion in the integral (co)homology groups of an Eilenberg-MacLane space K(G,n), where G is a non-trivial abelian group and n>1. The main goal of this work is to extend this result to H-spaces having more than one non-trivial homotopy groups. In order to have an accurate hold on H. Cartan's result, we start by studying the duality between homology and cohomology of 2-local Eilenberg-MacLane spaces of finite type. This leads us to some improvements of H. Cartan's methods in this particular case. Our main result can be stated as follows. Let X be an H-space with two non-vanishing finite 2-torsion homotopy groups. Then X does not admit any exponent for its reduced integral graded (co)homology group. We construct a wide class of examples for which this result is a simple consequence of a topological feature, namely the existence of a weak retract X K(G,n) for some abelian group G and n>1. We also generalize our main result to more complicated stable two stage Postnikov systems, using the Eilenberg-Moore spectral sequence and analytic methods involving Betti numbers and their asymptotic behaviour. Finally, we investigate some guesses on the non-existence of homology exponents for finite Postnikov towers. We conjecture that Postnikov pieces do not admit any (co)homology exponent. This work also includes the presentation of the "Eilenberg-MacLane machine", a C++ program designed to compute explicitely all integral homology groups of Eilenberg-MacLane spaces. <br/><br/>Il est toujours difficile pour un mathématicien de parler de son travail. La difficulté réside dans le fait que les objets qu'il étudie sont abstraits. On rencontre assez rarement un espace vectoriel, une catégorie abélienne ou une transformée de Laplace au coin de la rue ! Cependant, même si les objets mathématiques sont difficiles à cerner pour un non-mathématicien, les méthodes pour les étudier sont essentiellement les mêmes que celles utilisées dans les autres disciplines scientifiques. On décortique les objets complexes en composantes plus simples à étudier. On dresse la liste des propriétés des objets mathématiques, puis on les classe en formant des familles d'objets partageant un caractère commun. On cherche des façons différentes, mais équivalentes, de formuler un problème. Etc. Mon travail concerne le domaine mathématique de la topologie algébrique. Le but ultime de cette discipline est de parvenir à classifier tous les espaces topologiques en faisant usage de l'algèbre. Cette activité est comparable à celle d'un ornithologue (topologue) qui étudierait les oiseaux (les espaces topologiques) par exemple à l'aide de jumelles (l'algèbre). S'il voit un oiseau de petite taille, arboricole, chanteur et bâtisseur de nids, pourvu de pattes à quatre doigts, dont trois en avant et un, muni d'une forte griffe, en arrière, alors il en déduira à coup sûr que c'est un passereau. Il lui restera encore à déterminer si c'est un moineau, un merle ou un rossignol. Considérons ci-dessous quelques exemples d'espaces topologiques: a) un cube creux, b) une sphère et c) un tore creux (c.-à-d. une chambre à air). a) b) c) Si toute personne normalement constituée perçoit ici trois figures différentes, le topologue, lui, n'en voit que deux ! De son point de vue, le cube et la sphère ne sont pas différents puisque ils sont homéomorphes: on peut transformer l'un en l'autre de façon continue (il suffirait de souffler dans le cube pour obtenir la sphère). Par contre, la sphère et le tore ne sont pas homéomorphes: triturez la sphère de toutes les façons (sans la déchirer), jamais vous n'obtiendrez le tore. Il existe un infinité d'espaces topologiques et, contrairement à ce que l'on serait naïvement tenté de croire, déterminer si deux d'entre eux sont homéomorphes est très difficile en général. Pour essayer de résoudre ce problème, les topologues ont eu l'idée de faire intervenir l'algèbre dans leurs raisonnements. Ce fut la naissance de la théorie de l'homotopie. Il s'agit, suivant une recette bien particulière, d'associer à tout espace topologique une infinité de ce que les algébristes appellent des groupes. Les groupes ainsi obtenus sont appelés groupes d'homotopie de l'espace topologique. Les mathématiciens ont commencé par montrer que deux espaces topologiques qui sont homéomorphes (par exemple le cube et la sphère) ont les même groupes d'homotopie. On parle alors d'invariants (les groupes d'homotopie sont bien invariants relativement à des espaces topologiques qui sont homéomorphes). Par conséquent, deux espaces topologiques qui n'ont pas les mêmes groupes d'homotopie ne peuvent en aucun cas être homéomorphes. C'est là un excellent moyen de classer les espaces topologiques (pensez à l'ornithologue qui observe les pattes des oiseaux pour déterminer s'il a affaire à un passereau ou non). Mon travail porte sur les espaces topologiques qui n'ont qu'un nombre fini de groupes d'homotopie non nuls. De tels espaces sont appelés des tours de Postnikov finies. On y étudie leurs groupes de cohomologie entière, une autre famille d'invariants, à l'instar des groupes d'homotopie. On mesure d'une certaine manière la taille d'un groupe de cohomologie à l'aide de la notion d'exposant; ainsi, un groupe de cohomologie possédant un exposant est relativement petit. L'un des résultats principaux de ce travail porte sur une étude de la taille des groupes de cohomologie des tours de Postnikov finies. Il s'agit du théorème suivant: un H-espace topologique 1-connexe 2-local et de type fini qui ne possède qu'un ou deux groupes d'homotopie non nuls n'a pas d'exposant pour son groupe gradué de cohomologie entière réduite. S'il fallait interpréter qualitativement ce résultat, on pourrait dire que plus un espace est petit du point de vue de la cohomologie (c.-à-d. s'il possède un exposant cohomologique), plus il est intéressant du point de vue de l'homotopie (c.-à-d. il aura plus de deux groupes d'homotopie non nuls). Il ressort de mon travail que de tels espaces sont très intéressants dans le sens où ils peuvent avoir une infinité de groupes d'homotopie non nuls. Jean-Pierre Serre, médaillé Fields en 1954, a montré que toutes les sphères de dimension >1 ont une infinité de groupes d'homotopie non nuls. Des espaces avec un exposant cohomologique aux sphères, il n'y a qu'un pas à franchir...

Finite size effects in superconducting nanocomposites

The present Master’s thesis presents theoretical description of the extraodinary behavior of the confined Indium nanoparticles. Superconducting properties of nanoparticles and nanocomposites are extensively reviewed. Special attention has been paid to phase fluctuation, shell and disordered effects. The experimental data has been obtained and provided by Dmitry Shamshur from Ioffe Physical Technical Institute. The investigated material represents a highly ordered system of silicate spheres filled with indium metal, where the In nanoparticles are interconnected between each other. Bulk indium is a superconductor with crititcal superconducting temperature Tc0 = 3:41 K. But indium nanoparticles exhibit different behavior, the critical temperature rise by approximately 20% up to 4.15 K. As well as transition of the indium particles to type-II superconductivity with high critical magnetic fields. Such diversity is explained by finite size effects which originate from nanosize of the samples.

Measure Fields for Function Approximation

The computation of a piecewise smooth function that approximates a finite set of data points may be decomposed into two decoupled tasks: first, the computation of the locally smooth models, and hence, the segmentation of the data into classes that consist on the sets of points best approximated by each model, and second, the computation of the normalized discriminant functions for each induced class. The approximating function may then be computed as the optimal estimator with respect to this measure field. We give an efficient procedure for effecting both computations, and for the determination of the optimal number of components.

A finite element-spherical harmonics model for radiative transfer in inhomogeneous clouds. Part II. some applications

EVENT has been used to examine the effects of 3D cloud structure, distribution, and inhomogeneity on the scattering of visible solar radiation and the resulting 3D radiation field. Large eddy simulation and aircraft measurements are used to create realistic cloud fields which are continuous or broken with smooth or uneven tops. The values, patterns and variance in the resulting downwelling and upwelling radiation from incident visible solar radiation at different angles are then examined and compared to measurements. The results from EVENT confirm that 3D cloud structure is important in determining the visible radiation field, and that these results are strongly influenced by the solar zenith angle. The results match those from other models using visible solar radiation, and are supported by aircraft measurements of visible radiation, providing confidence in the new model.

EQUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS

A classical theorem of H. Hopf asserts that a closed connected smooth manifold admits a nowhere vanishing vector field if and only if its Euler characteristic is zero. R. Brown generalized Hopf`s result to topological manifolds, replacing vector fields with path fields. In this note, we give an equivariant analog of Brown`s theorem for locally smooth G-manifolds where G is a finite group.