268 research outputs found
Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem
In this paper, we introduce properly-invariant diagonality measures of
Hermitian positive-definite matrices. These diagonality measures are defined as
distances or divergences between a given positive-definite matrix and its
diagonal part. We then give closed-form expressions of these diagonality
measures and discuss their invariance properties. The diagonality measure based
on the log-determinant -divergence is general enough as it includes a
diagonality criterion used by the signal processing community as a special
case. These diagonality measures are then used to formulate minimization
problems for finding the approximate joint diagonalizer of a given set of
Hermitian positive-definite matrices. Numerical computations based on a
modified Newton method are presented and commented
Symplectic determinant laws and invariant theory
We introduce the notion of by analogy
with Chenevier's definition of determinant laws. Symplectic determinant laws
are a way to define pseudorepresentations for symplectic representations of
algebras with involution over arbitrary -algebras. We
prove that this notion satisfies the properties expected from a good theory of
pseudorepresentations, and we compare it to Lafforgue's
-pseudocharacters. In the process, we compute generators of the
invariant algebras and over an arbitrary commutative
ring when ,
generalizing results of Zubkov
A Double-Strand Elastic Rod Theory
Abstract.: Motivated by applications in the modeling of deformations of the DNA double helix, we construct a continuum mechanics model of two elastically interacting elastic strands. The two strands are described in terms of averaged, or macroscopic, variables plus an additional small, internal or microscopic, perturbation. We call this composite structure a birod. The balance laws for the macroscopic configuration variables of the birod can be cast in the form of a classic Cosserat rod model with coupling to the internal balance laws through the constitutive relations. The internal balance laws for the microstructure variables also take a mathematical form analogous to that for a Cosserat rod, but with coupling to the macroscopic system through terms corresponding to distributed force and couple load
TERNARY QUARTIC APPROACH FOR POSITIVE 4TH ORDER DIFFUSION TENSORS REVISITED
International audienceIn Diffusion Magnetic Resonance Imaging (D-MRI), the 2nd order diffusion tensor has given rise to a widely used tool â Diffusion Tensor Imaging (DTI). However, it is known that DTI is limited to a single prominent diffusion direction and is inaccurate in regions of complex fiber structures such as crossings. Various other approaches have been introduced to recover such complex tissue micro-geometries, one of which is Higher Order Cartesian Tensors. Estimating a positive diffusion function has also been emphasised mathematically, since diffusion is a physical quantity. Recently there have been efforts to estimate 4th order diffusion tensors from DiffusionWeighted Images (DWIs), which are capable of describing crossing configurations with the added property of a positive diffusion function. We take up one such, the Ternary Quartic approach, and reformulate the estimation equation to facilitate the estimation of the non-negative 4th order diffusion tensor. With our modified approach we test on synthetic, phantom and real data and confirm previous results
A fast algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems
In this paper, we develop a new algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems. Numerical experiments are given in order to illustrate the validity and efficiency of our algorithm.The authors would like to thank the supports of the Portuguese Funds through FCTâFundação para a CiĂȘncia e a Tecnologia, within the Project UID/MAT/00013/2013
An algebraic condition and an algorithm for the internal contact between two ellipsoids
4th International Conference on Discrete Element Methods, Brisbane, AUSTRALIA, AUG, 2007International audiencePurpose - The purpose of this paper is to present a new method for the detection and resolution of the contact point between two ellipsoids. Numerical simulations of ellipsoidal particles in a rotary cylinder are also presented. Desigri/methodology/approach - An algebraic condition is developed for the internal contact between two ellipsoids and an efficient contact detection algorithm for overlapping ellipsoids is implemented. Findings - This method was found to have the advantages of effectiveness and speed in the detection and resolution of the contact point. Originality/value - The dynamics of granular materials are of great importance in many industries dealing with powders and grains, such as pharmaceutical, chemical, and food industries. The main difficulty of such simulations is the excessive CPU time required for a large number of particles. In the discrete element method, contact detection between grains is the most expensive step in solving a nonlinear system for determination of the contact point, the normal vector and the overlap distance between ellipsoids. The numerical behavior and the optimization of the new algorithm presented in this paper are important also
Numerical convergence and stability of mixed formulation with X-FEM cut-off
National audienceSee http://hal.archives-ouvertes.fr/docs/00/59/26/89/ANNEX/r_KJ9QU562.pd
Characterizing Distances of Networks on the Tensor Manifold
At the core of understanding dynamical systems is the ability to maintain and
control the systems behavior that includes notions of robustness,
heterogeneity, or regime-shift detection. Recently, to explore such functional
properties, a convenient representation has been to model such dynamical
systems as a weighted graph consisting of a finite, but very large number of
interacting agents. This said, there exists very limited relevant statistical
theory that is able cope with real-life data, i.e., how does perform analysis
and/or statistics over a family of networks as opposed to a specific network or
network-to-network variation. Here, we are interested in the analysis of
network families whereby each network represents a point on an underlying
statistical manifold. To do so, we explore the Riemannian structure of the
tensor manifold developed by Pennec previously applied to Diffusion Tensor
Imaging (DTI) towards the problem of network analysis. In particular, while
this note focuses on Pennec definition of geodesics amongst a family of
networks, we show how it lays the foundation for future work for developing
measures of network robustness for regime-shift detection. We conclude with
experiments highlighting the proposed distance on synthetic networks and an
application towards biological (stem-cell) systems.Comment: This paper is accepted at 8th International Conference on Complex
Networks 201
Conformational analysis of nucleic acids revisited: Curves+
We describe Curves+, a new nucleic acid conformational analysis program which is applicable to a wide range of nucleic acid structures, including those with up to four strands and with either canonical or modified bases and backbones. The program is algorithmically simpler and computationally much faster than the earlier Curves approach, although it still provides both helical and backbone parameters, including a curvilinear axis and parameters relating the position of the bases to this axis. It additionally provides a full analysis of groove widths and depths. Curves+ can also be used to analyse molecular dynamics trajectories. With the help of the accompanying program Canal, it is possible to produce a variety of graphical output including parameter variations along a given structure and time series or histograms of parameter variations during dynamic
Numerical convergence and stability of mixed formulation with X-fem cut-off
International audienceIn this paper we are concerned with the mathematical and numerical analysis of convergence and stability of the mixed formulation for incompressible elasticity in cracked domains. The objective is to extend the X-FEM cut-off analysis done in the case of compressible elasticity to the incompressible one. A mathematical proof of the inf-sup condition of the discrete mixed formulation with X-FEM is established for some enriched fields. We also give a mathematical result of quasi-optimal error estimate. Finally, we validate these results with numerical tests
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