Joint Brown University Seminar -
Applied Mathematics Pattern and
Mathematics Department Geometry/Topology
Abstract: Surfaces are ubiquitous in mathematics, science, and engineering. With the advent of "circle packing" methods, their conformal properties are now computationally accessible --- often for the first time. This talk will be a largely pictorial tour illustrating the fundamentals of circle packing, the techniques and manipulations available, and emerging applications, from conformal "welding" to conformal "brain flattening." The pleasures of knocking mathematics and applications together in a new sandbox will be a key subtext.
NEW ANNOUNCEMENT
Special Applied Mathematics Pattern Theory Seminar
Abstract: Diffusion tensor magnetic resonance imaging (DT-MRI) probes and quantifies the anisotropic diffusion of water molecules in biological tissues. It is becoming a routine magnetic resonance technique for studying properties of biological tissue, including fiber orientation. I will present a method to match diffusion tensor magnetic resonance images (DT-MRI) through the large deformation metric mapping of vector fields, focusing on the fiber orientations, considered as unit vector fields on the image volume. We define a suitable action of diffeomorphisms on such vector fields, and provide an extension of the Large Deformation Metric Mapping framework to this type of dataset, resulting in optimizing for geodesics on the space of diffeomorphisms connecting two images. Existence of the minimizers under smoothness assumptions on the compared vector fields is proved, and coarse to fine hierarchical strategies are detailed, to reduce both ambiguities and computation load. This is illustrated by numerical experiment on DT-MRI heart images.
Brown University Center for Statistical Sciences Seminar
Department of Interdisciplinary Oncology | |
> From Gene Expression Profiles | |
Abstract: An emerging paradigm of medicine in the 21st century is the prediction of the condition or response of the individual patient from molecular level data, i.e., the translation of laboratory research to the bedside. Key to achieving this goal is the nascent field of Bioinformatics, which is currently a concept having many different interpretations. In this presentation, I will introduce the concept of Translational Bioinformatics, which we have developed at the Moffitt Cancer Center and Research Institute and which we define as the science of measuring the clinical utility of basic science data. I will then introduce a measure of the Translational Utility of Bioinformation. This measure reflects the amount by which the likelihood of a clinical outcome is altered by molecular level data. I will specifically discuss application of this measure to the problem of the prediction of individual response to cancer therapy based on Gene Expression Profiles. A review of the literature provides estimates of Translational Utility of Gene Expression Profiles in several cancer types and illustrates significant limitations and opportunities for future research. Application and development of the methodology for meta-analysis of diagnostic tests will be shown to be critical to this new science.
Applied Mathematics Graduate Student Pizza Seminar
PDE Seminar
Abstract:
New mathematical objects are introduced that characterize
the distribution of stress inside heterogeneous media with
fine scale structure. To fix ideas the stress tensor inside
a multi-phase linearly elastic composite with micro
structure of length scale $\varepsilon$ is denoted by
$\sigma^{\varepsilon}$. One considers the square of the equivalent
stress described mathematically by the positive definite
quadratic form $\Pi(\sigma^{\varepsilon})$. The specific nature
of the quadratic form is determined by the particular
application. For $t > 0$ the Lebesgue measure of the set inside
the composite domain $\Omega$ where $\Pi(\sigma^{\varepsilon}) > t$
is denoted by
$ \lambda^{\varepsilon} (t).$
The analysis focuses on characterizing
$\lim \sup_{\varepsilon \rightarrow 0}\lambda^{\varepsilon}(t).$ New
mathematical quantities, dubbed macrostress modulation functions,
are introduced. The macrostress modulation functions
$f_{p}, 1 \leq p \leq \infty $ capture the excursions of the
local stress fluctuations about the homogenized or macroscopic
stress field. We obtain the weak-$L^{p}$ estimate on
${\Pi(\sigma^{\varepsilon})}_{\varepsilon>0}$ given by
$$\lim \sup_{\varepsilon \rightarrow 0} \lambda^{\varepsilon} (t)\leq t^{-p} \int_{\Omega}\mid f_{p}\mid ^{P} dx.$$ (1)
For $p = \infty$ we introduce the distribution function $\lambda(t, f_{\infty}) $ that gives the measure of the set where $f_{\infty} (x)\geq t.$ For this case the macrostress modulation provides a measure theoretic upper envelope on the oscillatory equivalent stress in the sense that
$$ \lim \sup_{\varepsilon \rightarrow 0} \lambda^{\varepsilon} (t)\leq \lambda (t, f).$$ (2)
In addition one naturally obtains exponential decay for $\lim \sup_{\varepsilon \rightarrow 0} \lambda^{\varepsilon}(t)$ when $f \infty $ is in BMO. These inequalities are used to assess the size of over stressed zones due to reentrant corners or sharp changes in boundary loading. Specific examples illustrating exponential and polynomial decay for $\lim \sup_{\varepsilon \rightarrow 0} \lambda^{\varepsilon} (t) $ are provided and an application to a problem of stress constrained optimal design is given [4].
The theory is based on new homogenization constraints relating the asymptotic distribution of states associated with the sequence $ {\Pi(\sigma^{\varepsilon)}_{\varepsilon > 0}$ to the macrostress modulation functions [1,2,3]. This theory is derived within the general context of G-convergence. The homogenization constraint follows from a new local representation formula for the gradient of G-limits with respect to the constituent elastic properties of the composite [1,2,3,5].
References
1. Lipton R. "Assessment of the local stress state by macroscopic
variables," Philosophical Transactions
R. Soc. Lond. A (2003) 361, 921-946.
2. Lipton R. "Bounds on the distribution of extreme values for
the stress in composite materials."
Journal of the Mechanics and Physics of Solids, 52, 2004, pp.
1053-1069.
3. Lipton R. "Homogenization theory and the assessment of extreme
field values in composites with random microstructure."
To appear in SIAM J. Applied Mathematics, 2004.
4. Lipton R. and Stuebner M. "Optimal design of graded
microstructure through inverse homogenization
for control of point wise stress." Submitted to
International Journal for Numerical Methods in Engineering,
September 2004.
5. Lipton R. "Stress constrained G closure and relaxation of
structural design problems," Quarterly
of Applied Mathematics, 62, 2004, pp. 295-321.
Department of Mathematics Colloquium
<--- 2004 Index