Spring, 2008
We present a practical introduction to unsymmetric meshless methods and Schaback's framework for proving error bounds for such methods. We then introduce a high order test discretization for unsymmetric meshless methods, which samples the residual's derivatives in addition to the residual itself. When modified to use this new test discretization, unsymmetric meshless methods can exploit arbitrarily high smoothness in the solution to obtain arbitrarily high convergence orders or convergence in arbitrarily strong Sobolev norms. This is justified using a new sampling inequality within the context of Schaback's framework.
Animal coat pattern formation is an intriguing topic to biologists and mathematicians alike. Using an activator-inhibitor model with Thomas non- linearities, we shall present results from ongoing work pertaining to pattern formation in 2 dimensions.
A Van-Douwen space is a countable, maximally crowded, regular space. Van-Douwen spaces very naturally give rise to n to 1 maps from the Stone-Cech remainder of the naturals onto separable spaces. This talk will be geared towards presenting the Van-Douwen space in a manner that is accessible to graduate students with a semester of topology.
The goal of this talk is to discuss the embeddings of infinite trees in the unit disk of the complex plane. From such embeddings, one can discuss a particular interpolation problem, that is: given a function on the vertices of a tree, can one interpolate to a function on the entire unit disk? We will discuss the answer to this interpolation problem, as well as introduce types of analysis that can be done on such trees.
In many applications, it is not feasible to compute a solution directly. Often of importance is the study of so called Microstructures within the solutions. Unfortunately, it is just as difficult to compute the Microstructures. I plan to give lower bounds on the probability of computing the correct Microstructure homology with a finite number of computations. After finding a lower bound, I will then move to applying this bound for random Fourier series, random polynomials, and the Cahn-Hilliard model.
Toric varieties are an important part of algebraic geometry. This talk will introduce toric varieties and present several important examples in low dimensions. Concepts such as cones, fans, and homogeneous coordinates will be explored as part of the background of this subject.
An antimatroid is an abstract structure with many different interpretations in areas such as convex geometry, graph theory, and combinatorics. We will discuss some of these interpretations as well as some interesting properties of antimatroids in general.
We consider all (commutative unital) minimal ring extensions of an integral domain R. We show that these can be separated, up to R-algebra isomorphism, into 3 nonoverlapping types, the first being domains that are minimal ring extensions of R, and the remaining two being algebraic constructions created from R and maximal ideals of R.
Consider an embedding of a wheel graph in the Euclidean plane as a coin graph. We can show that there is an equation that the radii of the coins must satisfy. These equations correspond to algebraic equations, which in turn correspond to elements of a polynomial ring in n variables. This talk will explore the properties of these polynomials and what the can tell us about the underlying coin graphs.
In their 2002 research monograph, How Is a Graph Like a Manifold? authors Bolker, Guillemin and Holm explore the application of advances in the theory of manifolds to graph theoretic problems. In this talk I will illuminate the value of exploiting the connections between the geometry of manifolds and that of graphs by tracing the transformation of a complex compact manifold into a graph that carries with it information about the manifold that will allow us to count the number of n – k dimensional faces of a simple n-dimensional convex polytope in the graph theoretic realm.
The Hilbert Cube is the space attained from taking a countable product of the closed interval [0,1]. Any regular topological space that satisfies the second axiom of countability can be homeomorphically impeded into a subspace of the Hilbert Cube. The same can be done with a regular topological space that is both metrizable and separable. This talk will cover the basic topological properties of a Hilbert Cube as well as a method of proving each of the previous statements.
This talk serves as both an introduction to the approaches and methodologies of theortetical and computational neuroscience as well as a report on ongoing research.
Shortly after the formulation of the notion of paracompactness and the proof that every metric space is paracompact, Bing, Nagata and Smirnov each independently formulated necessary and sufficient conditions for a topological space to be metrizable. The proof for these three results will be presented in a unified theorem.
Intracranial aneurysms are a serious and surprisingly common medical problem. Deciding when to treat such aneurysms depends largely on whether they might rupture; there are currently several theories of aneurysmal rupture. In this talk, an elastodynamic model for a subclass of intracranial saccular aneurysms is presented and analyzed using both numerical and analytical techniques. While no one has fully characterized the conditions under which these aneurysms are dynamically stable, we discuss certain conditions under which they seem to be.
In the first part of the presentation we will examine the one-dimensional drainage of an incompressible liquid into an initially dry and deformable porous material. Here, we identify numerical solutions that quantify the effects of gravity, capillarity and solid to liquid density ratio on the time required for a finite volume of liquid to drain into a deformable porous material.In the second part of our presentation, we will examine the capillary rise of a non-Newtonian liquid into a rigid porous material. Cases with and without gravity will be discussed. In the presence of gravity the same equilibrium heights are reached for both Newtonian and non-Newtonian cases. The evolution towards the equilibrium solution differs between Newtonian and non-Newtonian cases.
In the third part we will examine capillary rise of a non-Newtonian liquid into a deformable porous material. We obtain analytic results for steady state positions of the wet porous material-dry porous material interface as well as liquid-wet material interface. These are the same as those for the Newtonian case. In the absence of gravity, the time-dependent free-boundary problem is solved numerically and the results compared to the Newtonian case. Here, the liquid rises to an infinite height and the porous material deforms to an infinite depth, following a scaling law that depends on the non-Newtonian index n. The case of non-zero gravity is also briefly discussed.
There are different notions of being disconnected that categorize types of disconnected spaces. The three types this talk will examine are hereditarily disconnected, totally disconnected, and zero dimensional. These notions are nested in this way: zero dimensional implies totally disconnected, and totally disconnected implies hereditarily disconnected. These implications are strictly one way. This talk will display two counterexamples that witness these strict implications. First is an example, due to Erdös, using rational sequences in Hilbert space, l2, which shows a totally disconnected spaces that is not zero dimensional. The second example uses a "leaky tent" of the Cantor set, which is a hereditarily disconnected space that is not totally disconnected. This shows the necessity for the three different notions and how they differ in disconnected spaces.