We consider the scattering of a time-harmonic plane wave incident on a two-scale heterogeneous medium, which consists of scatterers that are much smaller than the wavelength and extended scatterers that are comparable to the wavelength. A generalized Foldy-Lax formulation is proposed to capture multiple scattering among point scatterers and extended scatterers. Our formulation is given as a coupled system, which combines the original Foldy-Lax formulation for the point scatterers and the regular boundary integral equation for the extended obstacle scatterers. An efficient physically motivated Gauss-Seidel iterative method is proposed to solve the coupled system, where only a linear system of algebraic equations for point scatterers or a boundary integral equation for a single extended obstacle scatterer is required to solve at each step of iteration. In contrast to the standard inverse obstacle scattering problem, the proposed inverse scattering problem is not only to determine the shape of the extended obstacle scatterer but also to locate the point scatterers. Based on the generalized Foldy-Lax formulation and the singular value decomposition of the response matrix constructed from the far-field pattern, an imaging function is developed to visualize the location of the point scatterers and the shape of the extended obstacle scatterer.

For each \alpha < \omega_1, let

X_\alpha = \{f : \omega^\alpha \rightarro\powerset_{\omega_1}(\mathbb{R})| f is increasing and continuous}

and \mu_\alpha be a normal fine measure on X_\alpha. We identify X_0 with \powerset_{\omega_1}(R). Martin and Woodin independently showed that these measures exist assuming (ZF + DC_\mathbb{R}) + AD + Every set is Suslin (\mu_0's existence was originally shown by Solovay from AD_\mathbb{R}). We sketch the proof of the derived model construction giving the existence of these measures (+ AD^+) from large cardinals and the Prikry forcing construction which gives back the exact large cardinal strength from AD^+ and the measure. If time allows, we will survey some theorems on the structure theory of the model L(\mathbb{R},\mu_\alpha) assuming the model satisfies \Theta > \omega_2 and \mu_\alpha is a normal fine measure on X_\alpha. Here the main theorem is that our assumption implies L(\mathbb{R},\mu_\alpha) satisfies AD^+

Stochasticity is a hallmark of cellular processes, and different classes of genes show large differences in their cell-to-cell variability (noise). To decipher the sources and consequences of this noise, we systematically measured pairwise correlations between large numbers of genes, including those with high variability. We find that there is substantial pathway variability shared across similarly regulated genes. This induces quantitative correlations in the expression of functionally related genes such as those involved in the Msn2/4 stress response pathway, amino acid biosynthesis, and mitochondrial maintenance. Bioinformatic analyses and genetic perturbations suggest that fluctuations in PKA and Tor signaling contribute to pathway-specific variability. Our results argue that a limited number of well-delineated ‘‘noise regulons’’ operate across a yeast cell and that such coordinated fluctuations enable a stochastic but coherent induction of functionally related genes. We discuss how this principle might be general to stress responses across different organisms and the mechanisms by which stochastic but coherent stress responses strengthen resistance to environmental insults. More broadly, our work shows that pathway noise is a quantitative tool for exploring pathway features and regulatory relationships in un-stimulated systems.

Advisor:  Svetlana Jitomirskaya
Thesis Abstract:
We consider quasi-periodic Jacobi operators with analytic sampling functions. Recent applications in physics of such operators are Graphene or the Quantum Hall effect. Even though much is known for their prototype, the almost Mathieu operator (AMO), not much can be said for models beyond that. The thesis has two main themes: Firstly, to provide a complete understanding of extended Harper's model (EHM), a natural generalization of the AMO proposed by D. J. Thouless, which so far has presented an open problem even from the physics point of view. Secondly, to address aspects of the spectral theory of general quasi-periodic, analytic Jacobi operators: continuity of the Lyapunov exponent, and continuity of spectral properties upon rational frequency approximation. As a result of our investigations, we provide a complete description of the model's ``metal-insulator phase diagram,'' as given by the Lyapunov exponent (LE). The main achievement was to develop a non-duality based approach, able to tackle the self-dual regime in parameter space. The latter is accomplished by developing Avila's ``global theory'' for analytic Jacobi-cocycles. Based on phase-complexification, the spectrum is partitioned into super-critical, sub-critical and critical regime. This provides a refined description of the set of zero LE, going beyond what is known from classical Kotani theory. For EHM, we identify the three regimes, for all values of the coupling and all irrational frequencies. Referring to the second theme of the thesis, we prove continuous dependence of the LE on the cocycle within the analytic category. The main achievement, is that our result allows for cocycles which are not everywhere invertible, a situation that naturally arises for Jacobi operators. Finally, we show how to recover the spectral properties of a quasi-periodic operator from rational approximation of the frequency. Up to sets of zero Lebesgue measure, the absolutely continuous spectrum can be obtained asymptotically from the ``intersection spectrum'' of the periodic operators associated with the continued fraction expansion of the frequency. This proves a conjecture of Y. Last and has not even been known for the AMO. Similarly, from the asymptotics of the ``union spectrum'', one recovers the spectrum.