epshteyn (at) math.utah.edu)
August 22 (Welcome Back and Group Photo!)
Speaker: Davit Harutyunyan, Department of Mathematics, The University of Utah
Title: Quantitative Wulff and Brunn-Minkowski inequalities for convex sets
Abstract: In this lecture we revisit the anisotropic isoperimetric (Wulff) and the Brunn-Minkowski inequalities for convex sets. The best know constant C(n)=Cn^8.5 depending on the space dimension n in both inequalities is due to Figalli, Maggi and Pratelli, 2010. We improve that constant to Cn^6 for convex sets and even better in some cases. We also conjecture, that the best constant in both inequalities must be of the form Cn^2, i.e., quadratic in n. The tools are the Brenier's mapping from the theory of optimal mass transportation combined with new sharp geometric-arithmetic mean and some algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.
Speaker: Jared Whitehead, Department of Mathematics, BYU
Title: Variations in the heat transport of Rayleigh Benard convection
Abstract: Rayleigh Benard convection is the canonical system where a fluid is heated from below and/or cooled from above, yielding an unstably driven system. When this temperature difference is sufficiently large, buoyancy effects induce convective motion within the fluid. As this driving force increases, the flow becomes turbulent. The fundamental question asked of this system is how the volume averaged heat transport depends on the driving force and material properties of the fluid. After reviewing the sometimes contradictory experimental evidence, we consider how different boundary conditions, heat sources, and variations in the kinematic properties of an incompressible fluid affect the heat transport in the turbulent, convective regime using rigorous upper bounds derived via variational techniques.
Speaker: Alexander Kurganov, Department of Mathematics, Tulane University
Speaker: Noa Kraitzman, Department of Mathematics, The University of Utah
Speaker: Michael Ryvkin, The Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University
Title: ANALYSIS OF NON-PERIODIC STRESS STATE IN PERIODIC MATERIALS. Applications to fracture and optimization.
Abstract: Many man-made materials have a periodic microstructure, periodic materials are widely met also in nature. Study of overall elastic properties of such materials and their optimization is a well-developed topic. The corresponding problems are characterized by a periodic stress state, however, in many cases of interest the stress state in periodic material is non-periodic. The non-periodicity can result from a non-periodic applied loading, from presence of cracks, inclusions and other flaws, and from the finite dimensions of the sample to be considered.
In these cases a direct numerical simulation is computationally expensive due to a large number of degrees of freedom to be involved, but reducing the analysis domain to a single repetitive cell is not straightforward. This goal is achieved by applying the discrete Fourier transform, casted as the representative cell method. As a result, one has to resolve a number of representative(repetitive) cell problems in the transforms space and can obtain the sought elastic field by the inverse transformation. The important feature of these problems is that they are independent and, consequently, can be treated by the use of parallel computing. It is shown how to plug-in the method into an efficient multiscale analysis scheme for arbitrary shaped sample of periodic material.
The suggested approach is employed for the fracture analysis of beam lattices: two-dimensional honeycombs and spatial open cell Kelvin foam, both cracks nucleation and propagation problems are addressed. Solid periodically voided and composite materials with flaws are considered as well, the optimal parameter combinations maximizing the fracture toughness are determined.
Speaker: Orly Alter, Departments of Bioengineering and Human Genetics, The University of Utah
Title: Cancer Diagnostics and Prognostics from Comparative Spectral Decompositions of Patient-Matched Genomic Profiles
Abstract: I will, first, briefly review our matrix and tensor modeling of large-scale molecular biological data, which, as we demonstrated, can be used to correctly predict previously unknown physical, cellular, and evolutionary mechanisms that govern the activity of DNA and RNA. Second, I will describe our recent generalized singular value decomposition (GSVD) and tensor GSVD comparisons of the genomes of tumor and normal cells from the same sets of astrocytoma brain and, separately, ovarian cancer patients, which uncovered patterns of DNA copy-number alterations that are correlated with a patient's survival and response to treatment. Third, I will present our higher-order GSVD, the only mathematical framework that can create a single coherent model from, i.e., simultaneously find similarities and dissimilarities across multiple two-dimensional datasets, by extending the GSVD from two to more than two matrices.
Speaker: Lajos Horvath, Department of Mathematics, The University of Utah
Title: Eigenvalue analysis of large dimensional matrices
Abstract: Testing for stability in linear factor models has become an important topic in both the statistics and econometrics research communities. The available methodologies address testing for changes in the mean/linear trend, or testing for breaks in the covariance structure by checking for the constancy of common factor loadings. In such cases when an external shock induces a change to the stochas tic structure of high dimensional data, it is unclear whether the change would be reflected in the mean, the covariance structure, or both. We develop a test for structural stability of linear factor models that is based on monitoring for changes in the largest eigenvalue of the sample covariance matrix. The asymptotic distribu tion of the proposed test statistic is established under the null hypothesis that the mean and covariance structure of the cross sectional units remain stable during the observation period. We show that the test is consistent assuming common breaks in the mean or factor loadings. These results are investigated by means of a Monte Carlo simulation study, and their usefulness is demonstrated with an application to U.S. treasury yield curve data, in which some interesting features of the 2007-2008 subprime crisis are illuminated.
Joint work with Gregory Rice.
Speaker: Vianey Villamizar, Department of Mathematics, BYU
November 28 (Student Talk)
Speaker: Qing Xia, Department of Mathematics, The University of Utah
epshteyn (at) math.utah.edu).
Past lectures: Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Spring 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001, Fall 2000, Spring 2000, Fall 1999, Spring 1999, Fall 1998, Spring 1998, Winter 1998, Fall 1997, Spring 1997, Winter 1997, Fall 1996, Spring 1996, Winter 1996, Fall 1995.