Abstract:
The problem of flow and consolidation of fine-grained bulk solids through a vessel comprised of cylinders, cones and a cylinder-cone combination is modeled in this paper. Unlike coarse-grained bulk solids, the pressure of the gases trapped between the solid particles is substantial when the powder is fine. Therefore, the flow of powder is treated as a two-phase interaction between gases and solids. To simplify the modeling of the problem, spatial averaging is adopted. This formulation, even though 1-D in space, includes geometry effects of the container. The model is derived using mass continuity and force equilibrium resulting in three partial differential equations that describe the density of the solid, the velocity of the solid and the pressure of the gas. Apart from the three partial differential equations, the model also contains an ordinary differential equation that describes the height of the powder in the vessel, or the free boundary, in time. Furthermore, a numerical method based on finite difference approximations in space and a Runge-Kutta method in time is implemented and computational results are presented to discuss the validity of this approach.Abstract:
Graffiti is a computer program whose early conjectures inspired results by many mathematicians including Alon, Bollobas, Chung, Erdos, Lovasz, and Kleitman. In the past few years the program made conjectures in carbon chemistry suggesting that stable fullerenes minimize their independence numbers and that they tend to be good expanders. A set of vertices of a graph is independent, if no two of them are adjacent. The independence number of a graph is the number of vertices of a maximum independent set. I will define the expanding properties of graphs and shortly discuss the origin of this concept (related to the subject of this talk) and its relation to some classical mathematical problems and problems in theoretical computer science. Neither the independence number nor the expanding properties were discussed before in carbon chemistry and initially some chemists were openly critical about Graffiti's stability conjectures. Nevertheless, thanks to extensive computation of Craig Larson, it appears now that there is a strong statistical evidence for the stability-independence hypothesis and that the stability-expanding hypothesis is consistent with the accepted theories of stability of benzenoids.Abstract:
Given vector spaces $V$ and $W$ over a field $F$, a linear transformation from $V$ to $W$ is a function $T: V \to W$ preserving vector addition and scalar multiplication in the sense that $T(\alpha +\beta) = T(\alpha)+ T(\beta)$ and $T(c\alpha)=cT(\alpha)$ for all $\alpha, \beta\in V$ and all scalars $c\in F$. The set of all linear transformations from $V$ to $V$ is a ring under the operations of pointwise addition and composition of functions which has the identity function on $V$ as multiplicative identity. These concepts generalize to the notions of (unital) $R$--module and $R$--module homomorphism where $R$ is a ring with a multiplicative identity element: the only difference is that the scalars need not come from a field, just from a ring $R$ with identity. Analogous to vector spaces, given an $R$--module $G$, the set of all $R$--module homomorphisms $\varepsilon: G \to G$ is a ring, $E_R(G)$, called the endomorphism ring of $G$. The talk will focus on settings in which the term $R$--module endomorphism coincides with with $R$--homogeneous function. A function $f: G\to G$ is said to be $R$--homogeneous if $f(ca)=cf(a)$ for all $a\in G$ and all $c\in R$. The set of all $R$--homogeneous functions from $G$ to $G$ is a near--ring, ${M}_R(G)$, containing the endomorphism ring of $G$. Usually, $E_R(G) \neq {M}_R(G)$. Several open problems will be presented.Abstract:
In this talk I will explain how the motion of a viscous incompressible fluid - say, of water in a container heated through a net of wires - can be mathematically modelled. Then I will describe a strategy for rigorous mathematical proofs.Abstract:
We begin with the basic classification of partial differential equations (PDE), in which many time-dependent problems are of hyperbolic type; their solutions are characterized by wave propagation, finite domain of dependence, and focussing of singularities. The mathematical tools (which will NOT be the focus of this talk) developed to analyse linear and semilinear hyperbolic equations do not help much with nonlinear problems. By contrast, in elliptic PDE (which typically govern time-independent problems), the quasilinear theory is a relatively simple variant of the linear theory. Despite impressive recent advances in the theory of hyperbolic conservation laws in a single space variable, there is little theory for multidimensional conservation laws. One approach to this problem is to study self-similar solutions. It turns out that one ends up with quasilinear elliptic equations --- and can take advantage of the advanced state of elliptic theory. The talk will conclude with some recent results of Suncica Canic, Eun Heui Kim and myself related to weak shock reflection and the von Neumann paradox.Abstract:
The accurate simulation of remote manipulator systems like the articulated arms of the Space Shuttle and of the International Space Station requires taking dry friction into consideration. This can be done in a relatively simple way if one accepts to replace differential equations by differential inequalities. The resulting mathematical problems and their numerical analysis are sufficiently simple to be understood by senior undergraduate and 1st year graduate students with a good background in real and/or functional and/or applicable analysis. One of our motivations at presenting this dry friction problem is that it will give us an opportunity to "visualize" a weak convergence phenomenon, showing thus that this phenomenon can take place in real life related problems.Abstract:
It is now well known that deterministic dynamical systems can behave "chaotically". But what does this actually mean and how can we measure it? In this talk we explore what is meant by terms such as "random" and show how simple deterministic systems can have statistics indistinguishable from coin-tossing. The talk will include a visual component where we will show some of the intricate structure that can be embedded within chaotic systems.Abstract:
The set of continuous complex-valued functions C(X) on a compact Hausdorff space X is an example of a commutative C*-algebra. We will examine how this algebra completey describes the topological structure of X, and vice versa. Major results in this direction include Gelfand's theorem and the Banach-Stone theorem. We will emphasize this connection between topology and algebra for its own beauty and use it to discuss some generalizations of the classical Banach-Stone theorem. This theorem demonstrates that the topological structure of X is encoded in the linear structure of C(X). Ultimately We hope to discuss some recent extensions of this result to the noncommutative setting of general C*-algebras.Abstract:
We will present some new challenging topics in wavelet design for Image Processing stemming form the study of human vision. We will focus on certain characteristics of human vision and show how they can be incorporated in the design of non-separable multi-scale analysis.Abstract:
There is a paradox concerning symmetry: symmetric causes can have asymmetric effects. This paradox is called spontaneous symmetry-breaking. In recent years researchers have shown that it plays a major role in pattern formation in physical systems. Chaos rides on the back of another paradox: deterministic mathematical models can produce random behavior. Finally, the way in which symmetry and chaos co-exist is a third paradox: symmetry suggests order and regularity while chaos suggests disorder and randomness. The combination leads to a striking series of pictures and to a notion of pattern on average.Abstract:
The mathematical models in air quality typically involve highly nonlinear coupled partial differential equations whose solution poses computational challenges. In this talk we review problems arising from important issues related to mathematical modeling of air quality, and computational methods to address them quantitatively. We present examples of scientific advances made possible by a close interaction between atmospheric science and mathematics, and draw conclusions whose validity should transcend the examples.Abstract:
The numerical solution of partial differential equations in an unbounded domain requires the introduction of artificial boundaries to limit the region of computation. One needs boundary conditions at these artificial boundaries in order to guarantee a unique and well posed solution to the differential equation. In turn these boundary conditions are necessary to guarantee stable difference approximations. We would like these artificial boundaries and corresponding boundary conditions to affect the solution in a manner such that they closely approximate the free space solution that exists in the absence of these boundaries. In particular, one would like to minimize the amplitude of waves reflected from these artificial boundaries.
The Perfectly Matched Layer (PML) is a technique of free space simulation for solving unbounded electromagnetic problems. This technique is based on the the use of an absorbing layer especially designed to absorb, without reflection, the electromagnetic waves. Designed by J. P. Berenger in 1994, the PML technique has been demonstrated to be the most effective free space simulation technique that has been developed for linear electromagnetic wave propagation so far.
In this talk, I will describe the PML technique for two dimensional problems, and then discuss the numerical implementation of this model. Examples will be presented that demonstrate the effectiveness of the model.
The talk is related to the materials covered in our courses on Applied Analysis, PDEs and Optimization theory.
Abstract:
Many basic questions in mathematics and science take the form "What is the optimal way to ...", or "What is the state of a system which minimizes some energy (or related) functional?" For example the famous 9 volume "Course on Theoretical Physics" by Landau and Lifshitz made an effort to describe most topics in terms of variational principles governing the behavior of the system under consideration.
It turns out that variational principles not only provide good motivation for deriving physical laws and relationships but they lead to good mathematical formulations of the problems that lead to analytical results and good methods for numerical computation and simulation.
In this talk I will illustrate this by describing the mathematical formulation of two basic scientific problems whose mathematical analysis is far from complete. These analyses are very important as they resolve issues that have arisen in the development of numerical methods and computational codes for simulation of these problems.
The first problem is to characterize the equilibrium state of a chemical reaction involving gaseous reactants. This is an issue in problems of pollution, devising good fuels and in the petrochemical industry. The second problem is what data is needed to solve Maxwell's equations of electromagnetism in a region with various types of boundaries? Attempts at numerical solution of Maxwell's equations have made us realize how many basic mathematical questions about the equations have not been solved.