Department Seminars & Colloquia
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I will discuss the existence and properties of small-scale dependent shock waves to nonlinear hyperbolic systems, with an emphasis on the theory of nonclassical entropy solutions involving undercompressive shocks. Regularization-sensitive structures often arise in continuum physics, especially in flows of complex fluids or solids. The so-called kinetic relation was introduced for van der Waals fluids and austenite-martensite boundaries (Abeyaratne, Knowles, Truskinovsky) and nonlinear hyperbolic systems (LeFloch) to characterize the correct dynamics of subsonic phase boundaries and undercompressive shocks, respectively. The role of a single entropy inequality is essential for these problems and is tied to the regularization associated with higher-order underlying models ---which take into account additional physics and provide a description of small-scale effects. In the last fifteen years, analytical and numerical techniques were developed, beginning with the construction of nonclassical Riemann solvers, which were applied to tackle the initial-value problem via the Glimm scheme. Total variation functionals adapted to nonclassical entropy solutions were constructed. On the other hand, the role of traveling waves in selecting the proper shock dynamics was stressed: traveling wave solutions (to the Navier-Stokes-Korteweg system, for instance) determine the relevant kinetic relation ---as well as the relevant family of paths in the context of nonconservative systems. Several physical applications were pursued: (hyperbolic-elliptic) equations of van der Waals fluids, model of thin liquid films, generalized Camassa-Holm equations, etc. Importantly, finite difference schemes with controled dissipation based on the equivalent equation were designed and the corresponding kinetic functions computed numerically. Consequently, `several shock wave theories' are now available to encompass the variety of phenomena observed in complex flows.
We present a mathematical model of left heart governed by the partial differential equations. This heart is coupled with a lumped model of the whole circulatory system governed by the ordinary differential equations. The immersed boundary method is used to investigate the intracardiac blood flow and the cardiac valve motions of the normal circulation in humans. We investigate the intraventricular velocity field and the velocity curves over the mitral ring and across outflow tract. The pressure and flow are also measured in the left and right heart and the systemic and pulmonary arteries. The simulation results are comparable to the existing measurements.
This talk studies two examples of singular perturbations for particle systems. The first example is based on classical Tichinov theory for ODEs and applied to flocking. The second example uses a new non-classical averaging method and is applied to a KdV-Burgers type equation.
Mathematical logic deals with symbols and strings, which are specialties of computers. So it is natural to use computers in practicing logic in various occasions. For instance, the verification of formal proofs, which are just the strings that follow some specific rules, is quite often tedious and time consuming--it is a job best suited for computers.
In this talk, a formal proof system called "Fitch" is introduced, and demonstrated as implemented at http://www.proofmood.com.
Recently, a new sampling theory called compressive sampling theory was
proposed in signal processing community. According to compressive
sampling, very accurate reconstruction is possible even from very
limited data measurements which breaking Nyquist sampling limit if the
unknown signal is sparse. Furthermore, even if the signal itself is
not sparse, as long as it can be represented sparsely by appropriate
sparsifying transform, compressive sampling can be still very
effectively applied. In this talk, we introduce the basic theory for
compressive sensing, and demonstrate how this can be applied to
various bio-imaging area.
Motivated by a practical application in designing safe control scheme
for automated guided vehicles or robots in industrial settings, graph
braid groups were first proposed and studied by R. Ghrist and A.
Abrams in 1999. We will quickly summarize the history of the theory
including the recent breakthrough on a conjecture by the pioneers.
The relationship between the geometric properties of a Riemannian manifold and those of its unit tangent sphere bundle has been studied for decades by many authors and is still an active research area. If the Ricci operator has 2 eigenvalues of multiplicities (m-1,1) in a contact metric manifold, then we get an eta-Einstein structure.
We study the geometry of a unit tangent sphere bundle of a Riemannian manifold.
In this talk, we shall discuss how the geometric properties of unit tangent sphere bundle influence those of the base manifold.
We study the geometry of a unit tangent sphere bundle of a Riemannian manifold.
In this talk, we shall discuss how the geometric properties of unit tangent sphere bundle influence those of the base manifold.