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Determining the absorption in anisotropic media
Get PDFAbstract. The problem in Optical Tomography of determining the spatially dependent absorption coefficient in an anisotropic medium with a-priori known strong
scattering is considered. The problem is modelled by the diffusion approximation of the Radiative Transfer Equation and the time-harmonic case is studied. In this
particular situation the diffusion approximation leads to an elliptic second order partial differential equation with complex variable coefficients which allows to treat the
problem equivalently to the inverse conductivity problem in Electrical Impedance Tomography (EIT). Results of uniqueness and stability for the absorption coefficient are
proven by using the approach of the work in SIAM J. Math. Anal. 33 (2001), no. 1, 153�171 for the inverse conductivity problem in EIT
A logical characterization of coherence for imprecise probabilities
Get PDFWhilst supported by compelling arguments, the consensus on representing uncertainty by means of (subjective) probability is not unanimous. A substantial part of the relevant criticisms point to its alleged inadequacy for representing ignorance as opposed to uncertainty. The purpose of this paper is to show how a strong justification for taking belief as probability, namely the Dutch Book argument, can be extended naturally so as to provide a logical characterization of coherence for imprecise probability, a framework which is widely believed to accommodate some fundamental features of reasoning under ignorance. The appropriate logic for our purposes is an algebraizable logic whose equivalent algebraic semantics is a variety of MV-algebras with
an additional internal unary operation representing upper probability (these algebras will be called UMV-algebras)
A Schanuel property for exponentially transcendental powers
Get PDFAbstract. We prove the analogue of Schanuel�s conjecture for raising to the
power of an exponentially transcendental real number. All but countably many
real numbers are exponentially transcendental. We also give a more general
result for several powers in a context which encompasses the complex case
Hermitian Matrix Polynomials with Real Eigenvalues of Definite Type. Part I: Classification
Get PDFThe spectral properties of Hermitian matrix polynomials with real eigenvalues
have been extensively studied, through classes such as the definite or
definitizable pencils, definite, hyperbolic, or quasihyperbolic matrix
polynomials, and overdamped or gyroscopically stabilized quadratics.
We give a unified treatment of these and related classes that uses the
eigenvalue type (or sign characteristic) as a common thread. Equivalent
conditions are given for each class in a consistent format. We show that these
classes form a hierarchy, all of which are contained in the new class of
quasidefinite matrix polynomials. As well as collecting and unifying existing
results, we make several new contributions.
We propose a new characterization of hyperbolicity in terms of the distribution
of the eigenvalue types on the real line. By analyzing their effect on
eigenvalue type, we show that homogeneous rotations allow results for matrix
polynomials with nonsingular or definite leading coefficient to be translated
into results with no such requirement on the leading coefficient, which is
important for treating definite and quasidefinite polynomials. We also give a
sufficient condition for a quasihyperbolic matrix polynomial to be
diagonalizable by structure preserving congruence, and show that this condition
is always satisfied in the quadratic case and for any hyperbolic matrix
polynomial, thereby identifying an important new class of diagonalizable matrix
polynomials
Particle size segregation in granular avalanches: A brief review of recent progress.
No full textHazardous natural flows such as snow avalanches, debris-flows, lahars and pyroclastic flows are part of a much wider class of granular avalanches, that frequently occur in industrial processes and in our kitchens! Granular avalanches are very efficient at sorting particles by size, with the smaller ones percolating down towards the base and squeezing the larger grains up towards the free-surface, to create inversely-graded layers. This paper provides a short introduction and review of recent theoretical advances in describing segregation and remixing with relatively simple hyperbolic and parabolic models. The derivation from two phase mixture theory is briefly summarized and links are drawn to earlier models of Savage & Lun and Dolgunin & Ukolov. The more complex parabolic version of the theory has a diffusive force that competes against segregation and yields S-shaped steady-state concentration profiles through the avalanche depth, that are able to reproduce results obtained from particle dynamics simulations. Time-dependent exact solutions can be constructed by using the Cole-Hopf transformation to linearize the segregation-remixing equation and the nonlinear surface and basal boundary conditions. In the limit of no diffusion, the theory is hyperbolic and the grains tend to separate out into completely segregated inversely graded layers. A series of elementary problems are used to demonstrate how concentration shocks, expansion fans, breaking waves and the large and small particles paths can be computed exactly using the model. The theory is able to capture the key features of the size distribution observed in stratification experiments, and explains how a large particle rich front is connected to an inversely graded avalanche in the interior. The theory is simple enough to couple it to the bulk flow field to investigate segregation-mobility feedback effects that spontaneously generate self-channelizing leveed avalanches, which can significantly enhance the total run-out distance of geophysical mass flow
Point-Line Collinearity Graphs of two Sporadic Minimal Parabolic Geometries
Get PDFThe disc structure of the point-line collinearity graph for the rank two minimal parabolic geometries of the Thompson and Harada-Norton simple groups are investigated. Additionally details of the sub-orbits of these two groups in their conjugation action upon an involution conjugacy class is given
The Canonical Generalized Polar Decomposition
Get PDFThe polar decomposition of a square matrix has been generalized by several authors to scalar products on or given by a bilinear or sesquilinear form. Previous work has focused mainly on the case of square matrices, sometimes with the assumption of a Hermitian scalar product. We introduce the canonical generalized polar decomposition , defined for general matrices , where is a partial -isometry and is -selfadjoint with nonzero eigenvalues lying in the open right half-plane, and the nonsingular matrices and define scalar products on and , respectively. We derive conditions under which a unique decomposition exists and show how to compute the decomposition by matrix iterations. Our treatment derives and exploits key properties of -partial isometries and orthosymmetric pairs of scalar products, and also employs an appropriate generalized Moore--Penrose pseudoinverse. We relate commutativity of the factors in the canonical generalized polar decomposition to an appropriate definition of normality. We also consider a related generalized polar decomposition , defined only for square matrices and in which is an automorphism; we analyze its existence and the uniqueness of the selfadjoint factor when is singular
Geometric structure in the representation theory of p-adic groups II
Get PDFThis expository note will state the ABP (Aubert-Baum-Plymen) conjecture. The conjecture can be stated at four levels:
1. K-theory of C*-algebras
2. Periodic cyclic homology of finite type algebras
3. Geometric equivalence of finite type algebras
4. Representation theory.
The emphasis in this note will be on representation theory
IT in university level mathematics teaching and learning: a mathematician's point of view
Get PDFUniversity mathematicians are often selective in their approaches to the use of IT in teaching. Although mathematicians systematically use specialist software in direct teaching of mathematics, as means of delivery e-learning technologies have so far been less widely used. This article is an attempt to explain the rationale for this selectivity, and proposes a ‘wish list’ of future developments for e-learning technologies
GPU-enabled steady-state solution of large Markov models
Get PDFWe describe a novel parallel steady-state solver that uses NVIDIA's Compute Unified Device Architecture (CUDA) library to perform calculations on a graphics processing unit (GPU). We demonstrate speed-ups of over 8 times compared with a CPU-only solver. We also discuss a parallel implementation which runs on multiple GPUs on separate machines, and explain how we deal with allocating appropriate amounts of work to heterogeneous computing resources