We show that the total mass, i.e. the sum over all points in the d-dimensional integer lattice of the solution to the parabolic Anderson model with initial function the point mass at the origin goes to zero in the high disorder regime. This talk is basedon joint work with L. Chen, D. Khoshnevisan, and K. Kim.

Mathematical computational biology (MCB) has proven fruitful for all three fields:
mathematics, computation, and biology. One approach to the intersection begins
with symbolic representations of models, so that high-level abstractions and
advantageous problem transformations can be applied computationally before
good numerical methods are called in to do the heavy work of simulation and optimization.
This is the potential advantage of “declarative” modeling. It opens up further connections
between applied mathematics, artificial intelligence (including current trends in hybrid
logical/statistical inference), and foundational mathematics including logic and type theory.
I will illustrate the possibilities with recent work in complex, multiscale computational
biology  including signal transduction in synapses and gene regulation/signaling networks
in  developmental biology, by means of stochastic model reduction, enriched graphical
models expressed using computer algebra, and declarative modeling languages for
multiscale heterogeneous dynamics.

A classical result of Khinchin says that for almost all real numbers α, the geometric mean of the first n digits ai(α) in the continued fraction expansion of α converges to a number K ≈ 2.6854520 . . . (Khinchin’s constant) as n → ∞. On the other hand, for almost all α, the arithmetic mean of the first n continued fraction digits ai(α) approaches infinity as n → ∞. There is a sequence of refinements of the AM-GM inequality, known as Maclaurin’s inequalities, relating the 1/kthpowers of the kth elementary symmetric means of n numbers for 1 ≤ k ≤ n. On the left end (when k = n) we have the geometric mean, and on the right end (k = 1) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves f (n) steps away from either extreme. We also study the limiting behavior of such means for quadratic irrational α.

(Joint work with Francesco Cellarosi, Doug Hensley and Steven J. Miller)