Speaker: Jakob Kotas, Applied Mathematics at the University of Washington–Seattle
Title: Response-guided dosing
Abstract: Within the broad field of personalized medicine, there has been a recent surge of clinical interest in the idea of response-guided dosing. Roughly speaking, the goal is to devise strategies that administer the right dose to the right patient at the right time. We will present stochastic models that attempt to formalize such optimal dosing problems. Theoretical results about the structure of optimal dosing strategies and associated solution methods rooted in convex optimization, stochastic dynamic programming, robust optimization, and Bayesian learning will be described. Computational results on rheumatoid arthritis will be discussed.
Title: Sums of quadratic functions with two discriminants and Farkas’ identities with quartic characters
Abstract:
In a 1999 paper, Zagier discusses a construction of a function $F_{k,D}(x)$ defined for an even integer $k ge 2$, and a positive discriminant $D$. This construction is intimately related to half-integral weight modular forms. In particular, the average value of this function is a constant multiple of the $D$-th Fourier coefficient of weight $k+1/2$ Eisenstein series constructed by H. Cohen in cite{Cohen}.
In this dissertation, we consider a construction which works both for even and odd positive integers $k$. Our function $F_{k,D,d}(x)$ depends on two discriminants $d$ and $D$ with signs $sign(d)=sign(D)=(-1)^k$, degenerates to Zagier’s function when $d=1$, namely,
$$
F_{k,D,1}(x)=F_{k,D}(x),
$$
and has very similar properties. In particular, we prove that the average value of $F_{k,D,d}(x)$ is again a Fourier coefficient of H. Cohen’s Eisenstein series of weight $k+1/2$, while now the integer $k ge 2$ is allowed to be both even and odd.
In a 2004 paper, Farkas introduces a new arithmetic function and proves an identity involving this function. Guerzhoy and Raji cite{Guerzhoy} generalize this function for primes that are congruent to 3 modulo 4 by introducing a quadratic Dirichlet character and find another identity of the same type. We look at the case when $p equiv 5 Mod 8$ by introducing quartic Dirichlet characters and prove an analogy of their generalization.
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Title: Scattering Theory
Abstract:
Scattering theory is the mathematical formalism of interactions in quantum mechanics. Quantum mechanics being the physical theory that appeared at the turn of the 20th century when it became apparent that not only did light behave as a wave, but matter as well. Further, the waves describing matter were quantized. A classic example of this is the photoelectric effect, where light can only interact with electrons when the light has the same quantized energy as the electron. In this work I will derive mathematically properties of the Schroedinger operator, an unbounded operator on Hilbert space and how its spectra can include both continuous and discrete components. The solutions to the Schroedinger equation are either normalizable or not depending on whether they have eigenvalues in the continuous or discrete spectra respectively. However, the solutions/eigenfunctions for the continuous or scattered spectra are only approximate eigenfunctions, in a sense we will make explicit. Their existence is shown through a rigorous treatment of the Schroedinger operator with no potential – the free state. Next scattering processes for some simple one dimensional cases will be shown, in which the time-independent solutions are related to time-dependent solutions. This will be followed by two dimensional scattering with localized and long range potentials.
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This semester the Logic Seminar will meet on Thursdays, 2:50 – 3:40 pm in Keller 402.
This Thursday we will have a (probably brief) organizational meeting.
Title: Some nonstandard remarks about Egyptian fractions
Abstract: An Egyptian fraction is a finite sum of fractions of the form $1/n$, where $n$ is a natural number. I’ll give simple proofs of some results about such fractions (also about Znám fractions). The proofs only require the compactness theorem from first order logic, though I’ll use the language of nonstandard analysis.
Speaker: Nayantara Bhatnagar (U. Delaware)
Title: Subsequence Statistics in Random Mallows Permutations
Abstract: The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the GUE Tracy-Widom distribution.
We study the length of the LIS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. We prove limit theorems for the LIS for different regimes of the parameter of the distribution. I will also describe some recent results on the longest common subsequence of independent Mallows permutations.
Relevant background for the talk will be introduced as needed.