Reports and programs from Summer 2004 investigations (preliminary reports: many are still being polished)

• 3x+1 PROBLEM (Sucheta Soundarajan, Bruce Adcock)
  • We show a method to determine arithmetic progressions based on their m-path with respect to the structure theorem for the (3x+1)-Map. This leads us to examine the required properties for m-paths of numbers that would contradict the (3x + 1)-Conjecture. This examination includes describing necessary conditions for the m-paths, and how many such m-paths there could potentially be given certain criterion. Finally, in the appendix we make use of these results with an algorithm to find numbers less than a bound that might contradict the (3x+1)-Conjecture.
  • final report: pdf, zipped files

• RANDOM MATRIX THEORY (John Ramey, John Sinsheimer, Jason Teich)

  • Consider the ensemble of real symmetric palindromic Toeplitz matrices, each independent entry chosen from a fixed probability
    distribution p of mean 0, variance 1, and finite higher moments. The limiting spectral measure (the density of normalized
    eigenvalues) converges, independent of $p$, to what appears to be the Gaussian; the first 10 moments of this distribution are the
    same as the first 10 moments of the Gaussian.
  • preliminary report: pdf, mathematica files

• BIPARTITE EXPANDER GRAPHS (Craig Lennon)

  • Let G be a k-regular bipartite multigraph with 2n vertices, G the disjoint union of I and O, with |I| = |O| = n. Denote by dA = { v : v is a vertex of  O and  (a,v) is in the edge set of  G for some a in A}. Then G is said to be an (n,k,c) expander graph if for any subset A of I, with |A| ≤  n/2, then |dA| ≥ c|A|. We consider the probability that a randomly chosen k-regular bipartite multigraph is an expander. Unless otherwise stated, all graphs below will be multigraphs. We show first that for k < 1/(2-c) there exists an N_k,c such that for n ≥ N_k,c no (n,k,c) expanders exist. We then show that for k ≥ max(4/(2-c),5) we can construct a family C of k-regular bipartite multigraphs such that lim_{n ®  ∞} (number of (n,k,c) expanders in C ) / |C| =1.
  • preliminary report: pdf

• BIPARTITE EXPANDER GRAPHS (Philip Carpenter, Jonathan Kropko)
  • We examined random graphs on n vertices with m edges. We are specifically interested in graphs with large girth and chromatic number, which will mean a small independent set, whose size we call p. We used the probabilistic method: by comparing the number of the graphs with the desired property to the total number of graphs, we can get an idea of whether or not it is likely for such a graph to occur when the number of vertices becomes very large. In Sarnak's book Some Applications of Modular Forms, he used the number of edges, m, of the order n^1+e and the size of the independent set, p, of the order n^1-h. He shows that these graphs, which have an independent set of size p, and containing at most n edges in the independent set, are not likely to be randomly generated in the limit as n becomes large, provided that e  > 2h. We have found that these bounds will work, but are generally wasteful, and the same property can be found using smaller m and p.
  • final report: pdf, zipped files

• C PROGRAMS FOR STUDYING GRAPHS and GRAPH CONFIGURATIONS (Chris Hammond, Mike Tychonievich)
  • C Algorithms for investigating eigenvalues of graphs: zipped files
  • Possible configurations for moments from graphs: configurations.pdf
  • Report on possible random graph ensemble similar to d-regular but faster to generate: report.pdf

• ZEROS OF ELLIPTIC CURVE L-FUNCTIONS (Adam O'Brien)
  • The effect of zeros at the central point on nearby zeros is investigated.
  • preliminary report: pdf, mathematica notebook with histograms

• LOWER ORDER CORRECTIONS TO ZEROS OF ELLIPTIC CURVES (Filip Paun, Colin Deimer)
  • preliminary report: pdf; calculation and references: pdf; programs: program.c

• CIRCLE METHOD (Raciel Valle)
  • It has been almost a century since Hardy and Littlewood developed the Circle Method and applied it to several types of problems.  While it doesn't always produce a proof, it does give some very accurate estimates for certain problems.  What we tried to do here was  to test the accuracy  of the results for large numbers,  and to see if these results indeed improve as we look at larger and larger numbers.  Specifically, we looked at the application of the Circle Method to calculate the number of generalized twin prime pairs (p, p+k both prime) on large intervals (of size 10²⁰ and 10⁴⁰).
  • preliminary report: word file.doc (with some math code) and word_file.doc (without math code) or mathematica notebook.nb; table of some results; zipped mathematica files / programs.

• RANDOM GRAPHS (Mike Tychonievich, Chris Hammond)
  • preliminary report: pdf