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Planar Domain (planar + domain)

Distribution by Scientific Domains
Mathematics and Statistics60%

Selected Abstracts

Detecting corpus callosum abnormalities in autism subtype using planar conformal mapping

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 2 2010
Ye Duan
Abstract A number of studies have documented that autism has a neurobiological basis, but the anatomical extent of these neurobiological abnormalities is largely unknown. In this paper, we apply advanced computational techniques to extract 3D models of the corpus callosum (CC) and subsequently analyze local shape variations in a homogeneous group of autistic children. Besides the traditional volumetric analysis, we explore additional phenotypic traits based on the oriented bounding rectangle of the CC. In shape analysis, a new conformal parameterization is applied in our shape analysis work, which maps the surface onto a planar domain. Surface matching among different individual meshes is achieved by aligning the planar domains of individual meshes. Shape differences of the CC between autistic patients and the controls are computed using Hotelling T2 two-sample metric followed by a permutation test. The raw and corrected p -values are shown in the results. Additional visualization of the group difference is provided via mean difference map. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Variational principles for symmetric bilinear forms

MATHEMATISCHE NACHRICHTEN, Issue 6 2008
Jeffrey Danciger
Abstract Every compact symmetric bilinear form B on a complex Hilbert space produces, via an antilinear representing operator, a real spectrum consisting of a sequence decreasing to zero. We show that the most natural analog of Courant's minimax principle for B detects only the evenly indexed eigenvalues in this spectrum. We explain this phenomenon, analyze the extremal objects, and apply this general framework to the Friedrichs operator of a planar domain and to Toeplitz operators and their compressions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Convexity in x of the level sets of the first Dirichlet eigenfunction

MATHEMATISCHE NACHRICHTEN, Issue 13-14 2007
Chie-Ping ChuArticle first published online: 7 SEP 200
Abstract On a planar domain which is convex in x, the level sets of the first Dirichlet eigenfunction for Laplacian are also convex in x. This gives an affirmative answer to a conjecture proposed by B. Kawohl. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Detecting corpus callosum abnormalities in autism subtype using planar conformal mapping

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 2 2010
Ye Duan
Abstract A number of studies have documented that autism has a neurobiological basis, but the anatomical extent of these neurobiological abnormalities is largely unknown. In this paper, we apply advanced computational techniques to extract 3D models of the corpus callosum (CC) and subsequently analyze local shape variations in a homogeneous group of autistic children. Besides the traditional volumetric analysis, we explore additional phenotypic traits based on the oriented bounding rectangle of the CC. In shape analysis, a new conformal parameterization is applied in our shape analysis work, which maps the surface onto a planar domain. Surface matching among different individual meshes is achieved by aligning the planar domains of individual meshes. Shape differences of the CC between autistic patients and the controls are computed using Hotelling T2 two-sample metric followed by a permutation test. The raw and corrected p -values are shown in the results. Additional visualization of the group difference is provided via mean difference map. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Shape optimization for low Neumann and Steklov eigenvalues

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2010
Alexandre Girouard
Abstract We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature: they are proved using methods of complex analysis. In particular, we present modernized proofs of the classical inequalities due to Szegö and Weinstock for the first nonzero Neumann and Steklov eigenvalues. We also extend the inequality for the second nonzero Neumann eigenvalue, obtained recently by Nadirashvili and the authors, to nonhomogeneous membranes with log-subharmonic densities. In the homogeneous case, we show that this inequality is strict, which implies that the maximum of the second nonzero Neumann eigenvalue is not attained in the class of simply connected membranes of a given mass. The same is true for the second nonzero Steklov eigenvalue, as follows from our results on the Hersch,Payne,Schiffer inequalities. Copyright © 2009 John Wiley & Sons, Ltd. [source]