Monday, January 30, 2006

misc

some open problems and research directions in the mathematical study of fluid dynamics, by P. Constantin.

050 00 QA911|b.T7413 1995
100 1 Troshkin, O. V
240 10 Netraditsionnye zadachi matematicheskoi gidrodinamiki.
|lEnglish
245 10 Nontraditional methods in mathematical hydrodynamics /
|cO.V. Troshkin
260 Providence, R.I. :|bAmerican Mathematical Society,|cc1995
300 viii, 197 p. :|bill. ;|c26 cm
440 0 Translations of mathematical monographs ;|vv. 144
504 Includes bibliographical references (p. 191- 197
650 0 Hydrodynamics
http://www.math.jussieu.fr/~leila/grothendieckcircle/mathtexts.php
Grothendieck Circle

MOSCOW MATHEMATICAL JOURNAL Volume 3 (2003), Number 2
This issue of the Moscow Mathematical Journal is dedicated to Vladimir I. Arnold on the occasion of his 65th birthday

Vladimir Igorevich Arnold

O. Bogoyavlenskij. Infinite Families of Exact Periodic Solutions to the Navier—Stokes Equations [PDF] [Abstract]

E. Brieskorn, A. Pratoussevitch, and F. Rothenhäusler. The Combinatorial Geometry of Singularities
and Arnold's Series E, Z, Q [PDF] [Abstract]

J. Bruce. On Families of Symmetric Matrices [PDF] [Abstract]

J. Damon. On the Legacy of Free Divisors II: Free* Divisors and Complete Intersections [PDF] [Abstract]

Yu. Drozd, G.-M. Greuel, and I. Kashuba. On Cohen—Macaulay Modules on Surface Singularities [PDF] [Abstract]

I. Dynnikov and S. Novikov. Geometry of the Triangle Equation on Two-Manifolds [PDF] [Abstract]

W. Ebeling and S. Gusein-Zade. Indices of 1-forms on an Isolated Complete Intersection Singularity [PDF] [Abstract]

G. Felder, L. Stevens, and A. Varchenko. Modular Transformations of the Elliptic Hypergeometric Functions, Macdonald Polynomials, and the Shift Operator [PDF] [Abstract]

A. Givental. An−1 Singularities and nKdV Hierarchies [PDF] [Abstract]

V. Goryunov and V. Zakalyukin. Simple Symmetric Matrix Singularities and the Subgroups of Weyl Groups Aμ, Dμ, Eμ [PDF] [Abstract]

Yu. Ilyashenko and V. Moldavskis. Morse—Smale Circle Diffeomorphisms and Moduli of Elliptic Curves [PDF] [Abstract]

S. Natanzon. Effectivisation of a String Solution of the 2D Toda Hierarchy and the Riemann Theorem About Complex Domains [PDF] [Abstract]

D. Novikov and S. Yakovenko. Quasialgebraicity of Picard—Vessiot Fields [PDF] [Abstract]

G. Paternain, L. Polterovich, and K. F. Siburg. Boundary Rigidity for Lagrangian Submanifolds, Non-Removable Intersections, and Aubry—Mather Theory [PDF] [Abstract]

I. Scherbak and A. Varchenko. Critical Points of Functions, sl2 Representations, and Fuchsian Differential Equations with only Univalued Solutions [PDF] [Abstract]

B. Shapiro and A. Vainshtein. Counting Real Rational Functions with All Real Critical Values [PDF] [Abstract]

D. Siersma and M. Tibăr. Deformations of Polynomials, Boundary Singularities and Monodromy [PDF] [Abstract]

S. Tabachnikov. On Skew Loops, Skew Branes and Quadratic Hypersurfaces [PDF] [Abstract]

V. Vladimirov and K. Ilin. Virial Functionals in Fluid Dynamics [PDF] [Abstract]

V. Yudovich. Eleven Great Problems of Mathematical Hydrodynamics [PDF] [Abstract]

Monday, January 16, 2006

Beats, kurtosis and visual coding

Network: Computation in Neural Systems

Publisher: Taylor & Francis

Issue: Volume 12, Number 3 / March 01, 2001

Pages: 271 - 287

URL: Linking Options

DOI: 10.1088/0954-898X/12/3/303
Beats, kurtosis and visual coding

M.G.A. Thomson

Abstract:

Techniques adapted from standard higher-order statistical methods are applied to natural-image data in an attempt to discover exactly what makes `wavelet' representations of natural scenes sparse. Specifically, this paper describes a measure known as the phase-only second spectrum, a fourth-order statistic which quantifies harmonic beat interactions in data, and uses it to show that there are statistical consistencies in the phase spectra of natural scenes. The orientation-averaged phase-only second spectra of natural images appear to show power-law behaviour rather like image power spectra, but with a spectral exponent of approximately -1 instead of -2. They also appear to display a similar form of scale-invariance. Further experimental results indicate that the form of these spectra can account for the observed sparseness of bandpass-filtered natural scenes. This implies an intimate relationship between the merits of sparse neural coding and the exploitation of non-Gaussian `beats' structures by the visual system.


The references of this article are secured to subscribers.

Title: Visual coding and the phase structure of natural scenes
Author(s): Thomson MGA
Source: NETWORK-COMPUTATION IN NEURAL SYSTEMS 10 (2): 123-132 MAY 1999
Document Type: Article
Language: English
Abstract: Although it is now well known that natural images display consistent statistical properties which distinguish them from random luminance distributions, this ecological approach to vision has so far concentrated on those second-order image statistics which are quantified by image power spectra, and it appears to be the image phase spectra which carry the majority of the image-intrinsic information. The present work describes how conventional nth-order statistics can be modified so that they are sensitive to image phase structure only. The modified measures are applied to an ensemble of natural images, and the results show that natural images do have consistent higher-order statistical properties which distinguish them from random-phase images with the same power spectra. An interpretation of this finding in terms of higher-order spectra suggests that these consistent properties arise from the ubiquity of edge structures in natural images, and raises the possibility that the properties of ideal relative-phase-sensitive mechanisms could be determined directly from analyses of the higher-order structure of natural scenes.
KeyWords Plus: STATISTICS; IMAGES; FILTERS

Friday, January 13, 2006

Amazon.com: Foundations of Modern Probability: Books: Olav Kallenberg

Real Analysis and Probability by R. M. Dudley
Foundations of Modern Probability by Olav Kallenberg
Markov Processes from K. Ito's Perspective (AM-155) (Annals of Mathematics Studies) by Daniel W. Stroock
An Introduction to Markov Processes (Graduate Texts in Mathematics) by Daniel W. Stroock
Diffusions, Markov Processes and Martingales (Cambridge Mathematical Library) by L. C. G. Rogers
Probability Theory, an Analytic View (Hardcover)by Daniel W. Stroock

Wednesday, January 11, 2006

S. Kim. Wave SIMULATION

http://www.ms.uky.edu/~skim/WaveSimulation/#FullWave

WAVE SIMULATION (Last edited: October 2004)
Contents
Full Waveform Simulation
Frequency-Domain Solvers
Computer Programs: The source codes are available from GRADE or by asking skim@ms.uky.edu.
Lecture Note: S. Kim, Numerical Methods for Differential Equations NumerPDE.ps NumerPDE.pdf
Dr. Kim's Other Research Branches
Image Processing
Computational Seismology
Full Waveform Simulation
Scope & Goal
It is often desired to design efficient implicit algorithms for the time-domain wave propagation.
Goal: Development of O(N), unconditionally stable algorithms for the wave propagation in 2D/3D heterogeneous media.
On-going Projects: LOD and ADI-II, with applications to the acoustic wave propagation and anisotropic micro-heat transfer.
Publications
H. Lim, S. Kim, and J. Douglas, Jr.. Numerical methods for viscous and nonviscous wave equations. (submitted). Postscript
J. Douglas, Jr., S. Kim, and H. Lim (2003). An improved alternating-direction method for a viscous wave equation. Contemporary Mathematics 329, pp. 99-104. Postscript
J. Douglas, Jr. and S. Kim (2001). Improved accuracy for locally one-dimensional methods for parabolic equations. Math. Models and Methods in Appl. Sci. 11, pp. 1563-1579. Postscript [TopBottom]
Frequency-Domain Solvers
Scope & Goal
The frequency-domain (Helmholtz) problem is hard to solve numerically. In addition to having a complex-valued solution, it is neither Hermitian-symmetric nor coercive, in non-attenuative media. As a consequence, most standard iterative algorithms either fail to converge or converge so slowly as to be impractical. Since many applications need to simulate waves of 20-50 wavelengths and since stability for numerical methods requires at least 10-12 grid points per wavelength, the discrete problem on the coarsest possible mesh is still huge; in realistic simulations, it is required to choose 20-30 points per wavelength for an acceptable accuracy. As Zienkiewicz, a founding father of the finite element method, pointed out, the problem remains unsolved and a completely new method is needed.
Goal: Development of accurate and optimally convergent algorithms for the simulation of 20-50 wavelengths in 3D heterogeneous media; in particular, no objectionable phase lag!!
On-going Project: The High-Frequency Asymptotic Decomposition (HFAD) method and perfect absorbing boundary conditions.
Publications
S. Kim, C.-S. Shin, and J.B. Keller. Asymptotic decomposition methods for the numerical solution of the Helmholtz equation. Appl. Math. Letters (in press). Postscript
S. Kim (2003). Compact Schemes for Acoustics in the Frequency Domain. Mathematical and Computer Modelling 37, pp. 1335-1341. Postscript
S. Kim and Soohyun Kim (2002). Multigrid simulation for high-frequency solutions of the Helmholtz problem in heterogeneous media. SIAM J. Sci. Comput. 24, No. 2, pp. 684-701. PDF
S. Kim (1998). Domain decomposition iterative procedures for solving scalar waves in the frequency domain. Numerische Mathematik 79, pp. 231-259.
S. Kim and W.W. Symes (1998). Multigrid domain decomposition methods for the Helmholtz problem. In: Fourth International Conference on Mathematical and Numerical Aspects of Wave Propagation (Editor: J. A. DeSanto). SIAM, Philadelphia, pp. 617-619.
S. Kim and M. Lee (1996). Artificial damping techniques for scalar waves in the frequency domain. Computers Math. Applic. 31, No. 8, pp. 1-12.
S. Kim (1995). Parallel multidomain iterative algorithms for the Helmholtz wave equation. Appl. Numer. Math. 17, pp. 411-429.
S. Kim (1994). A parallelizable iterative procedure for the Helmholtz problem. Appl. Numer. Math. 14, pp. 435-449.
S. Kim (1994). Numerical Treatments for the Helmholtz Problem by Domain Decomposition Techniques. Contemp. Math. 180, pp. 245-250.
Some Numerical ExamplesIn the following, we compare the standard fourth-order Helmholtz solver and the HFAD technique incorporating a second-order discretization, for the wave propagation in an 1D heterogeneous medium. An impulse source is put at the left edge.
Coarse Mesh
Finer Mesh
Stadard Fourth-Order
HFAD
Note: In the above figure, the dotted curve denotes the true solution, while the solid curve is the real part of the numerical solution. One can see a large phase lag for the standard technique with nx=300 (top left), which corresponds to choosing an average of 10.7 points per wavelength. On the other hand, the solution of HFAD is accurate enough when nx=112 (bottom right), which corresponds to choosing an average of 4 points per wavelength.

Imaging from Wave Propagation

http://www.ima.umn.edu/2005-2006/W10.17-21.05/
IMA Annual Program Year Workshop
Imaging from Wave Propagation
October 17-21, 2005
Organizers:

Margaret Cheney Department of Mathematical SciencesRensselaer Polytechnic Institutehttp://www.rpi.edu/~cheney/

Frank Natterer Fachbereich MathematikUniversitaet Muensterhttp://wwwmath.uni-muenster.de/math/u/natterer/

William W. Symes Department of Computational and Applied MathematicsRice Universityhttp://www.trip.caam.rice.edu/txt/bios/symes/william_symes.html
Schedule
Participants
Program Application
Feedback
Dining Guide
Maps
Abstracts and Talk Materials
Photo Gallery
Description:
This workshop seeks to bring together researchers in disparate fields that involve wave propagation, such as seismic imaging, nondestructive testing, radar imaging, medicalultrasound imaging, medical microwave imaging, etc. These fields have developed independently of each other, and researchers in different fields can scarcely communicate with each other, in spite of the fact that they are working on problems whose mathematical foundations is high-frequency analysis for wave propagation. The goal of our first workshop will be to bring researchers coming from different applications together.

Saturday, January 07, 2006

Lightness Demonstrations

Lightness Perception and Lightness Illusions

interactive movies based on a paper by Edward H. Adelson

The following demonstrations are in Macromedia Flash formats, and therefore suitable for classroom teaching:

To download: Mac users, press and hold mouse button. PC users, press right mouse button.
Click on the link to view.


Simultaneous Contrast Illusion
[ Flash - (69 kb) ]

Vasarely Illusion
[ Flash - (79 kb) ]

Craik-Obrien-Cornsweet Effect

[ Flash - 102 kb) ]

Knill and Kersten's Illusion
[ Flash - (86 kb) ]

The Koffka Ring
[ Flash - (74 kb) ]

Impossible Steps
[ Flash
- (116 kb) ]

Corrugated Plaid
[ Flash - (101 kb) ]

Haze Illusion
[ Flash - (142 kb) ]

White's Illusion
[ Flash - (106 kb)]

Criss-Cross Illusion
[ Flash - (75 kb) ]

The Snake Illusion
[ Flash - (116 kb) ]

2006 CNA Summer School

2006 CNA Summer School

Topics will cover:

  • Essential uses of probability in analysis
  • Quantitative unique continuation for elliptic, parabolic and dispersive equations, and applications
  • Adaptive finite element methods for elliptic PDEs
  • Fully nonlinear stochastic partial differential equations and applications
  • Interacting particle systems and their scaling limits

List of lecturers:

  • Krzysztof Burdzy, University of Washington
  • Carlos Kenig, University of Chicago
  • Ricardo Nochetto, University of Maryland
  • Panagiotis Souganidis, University of Texas at Austin
  • Srinivasa Varadhan, New York University

Organizers:

Irene Fonseca, Giovanni Leoni, Robert Pego, Kavita Ramanann

Graduate students and postdocs are encouraged to contribute a presentation for this conference. The deadline for submission of contributed abstracts is March 1, 2006.

Advanced undergraduate students, graduate students, and postdoctoral fellows are encouraged to apply for financial support. The deadline for applications is March 15, 2006.

Center for Nonlinear Analysis
Carnegie Mellon University
Department of Mathematical Sciences
Pittsburgh, PA 15213

Telephone: 412-268-2545
Fax: 412-268-6380
email: cn0s@math.cmu.edu

This conference is sponsored by the National Science Foundation and the Department of Mathematical Sciences at Carnegie Mellon University.