The fifth meeting of the Carolina Mathematics Seminar took place at the University of South Carolina Sumter, Sumter, SC. This is a list of our speakers, titles and abstracts.
Jason Holt (USC Lancaster)
Title: Non-Random Perturbations of the Anderson Hamiltonian and Cwikel-Lieb-Rozenblum Type Estimates.
Abstract: We will consider the Anderson Hamiltonian H0 = ∆ + V (x;ω) where V is a random potential and \omega belongs to a probability space (ΩF; P). The main object of the present work is the perturbed operator H =
H0 -W where W(x)≥ 0 decays at infinity. It is known that the spectrum of H below 0 is discrete consisting
only of eigenvalues and that the total number N0(W) of eigenvalues below 0 is a random variable for which P{N0(W) < ∞} = 1, or P{N0(W) < ∞ } = 0. We develop general conditions on V and W to guarantee P-a.s one case or the other and present several examples demonstrating the borderline decay in W. In particular, it will be shown that if V has a Bernoulli structure, then the borderline between finitely and infinitely many eigenvalues is obtained with a decay in W as O(c0 ln-2 |x|) where c0 is a determined positive constant.
Joshua Cooper (USC Columbia)
Title: Tree reconstruction and a Waring-type problem on partitions.
Abstract: The ``line graph'' of a graph G is a new graph L(G) whose vertices are the edges of G, with a new edge in L(G) from e to f if e and f were incident in G. Graham's Tree Reconstruction Conjecture says that, if T is a tree (a connected, acyclic graph), then the sequence of sizes of the iterated line graphs of T uniquely determine T. That is, T can be reconstructed from | L (j) (G)|∞j=0 , where L (0) (G) = G and L (j+1) (G) = L(L (j) (G)). Call two trees equivalent if they yield the same sequence; we call the resulting equivalence classes ``Graham classes.'' Clearly, the conjecture is equivalent to the statement that the number of Graham classes of n-vertex trees is equal to the number of isomorphism classes of such trees, which is known to be about 2.955765n. We show that the number of Graham classes is at least superpolynomial in n (namely, exp(c log n 3/2)) by converting the question into the following Waring-type problem on partitions. For a partition λ = {λ_1,...,λ_k} of the integer n and a degree d polynomial f ∈ Z[x], define f(λ) = ∑ k j=1
f(λj). We show that the range of f(λ) over all partitions λ of n grows as Ω(n d-1). The proof employs a well-known family of solutions to the Prouhet-Tarry-Escott problem. Strong evidence suggests the conjecture that the size of the range is actually Θ(nd). Joint work with Bill Kay of USC.
Kwan Lam (Benedict College)
Title: The formation of Turing pattern on networks of complete.
Abstract: In this talk, we will discuss the formation of Turing pattern in networks of homogeneous coupled reactors with a focus on the a
specific reaction diffusion model - Lengyel-Epstein kinetics. Special attention will be paid to the formation of bimodal pattern on the complete graph. We will use it as a building block to construct the linear and circular chains of the complete graphs with bimodal patterns.
Rigoberto Florez (USC Sumter)
Title: Some open questions.
Abstract: In this talk I am going to discuss some open questions in number theory and combinatorics. If the time allows us, I also go to discuss some potential research problems.
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