Fundamental limits to optical response in absorptive systems

At visible and infrared frequencies, metals show tantalizing promise for strong subwavelength resonances, but material loss typically dampens the response. We derive fundamental limits to the optical response of absorptive systems, bounding the largest enhancements possible given intrinsic material losses. Through basic conservation-of-energy principles, we derive geometry-independent limits to per-volume absorption and scattering rates, and to local-density-of-states enhancements that represent the power radiated or expended by a dipole near a material body. We provide examples of structures that approach our absorption and scattering limits at any frequency, by contrast, we find that common "antenna" structures fall far short of our radiative LDOS bounds, suggesting the possibility for significant further improvement. Underlying the limits is a simple metric, $|\chi|^2 / \operatorname{Im} \chi$ for a material with susceptibility $\chi$, that enables broad technological evaluation of lossy materials across optical frequencies.Comment: 21 pages and 6 figures (excluding appendices, references

Random Matrices

Large complex systems tend to develop universal patterns that often represent their essential characteristics. For example, the cumulative effects of independent or weakly dependent random variables often yield the Gaussian universality class via the central limit theorem. For non-commutative random variables, e.g. matrices, the Gaussian behavior is often replaced by another universality class, commonly called random matrix statistics. Nearby eigenvalues are strongly correlated, and, remarkably, their correlation structure is universal, depending only on the symmetry type of the matrix. Even more surprisingly, this feature is not restricted to matrices; in fact Eugene Wigner, the pioneer of the field, discovered in the 1950s that distributions of the gaps between energy levels of complicated quantum systems universally follow the same random matrix statistics. This claim has never been rigorously proved for any realistic physical system but experimental data and extensive numerics leave no doubt as to its correctness. Since then random matrices have proved to be extremely useful phenomenological models in a wide range of applications beyond quantum physics that include number theory, statistics, neuroscience, population dynamics, wireless communication and mathematical finance. The ubiquity of random matrices in natural sciences is still a mystery, but recent years have witnessed a breakthrough in the mathematical description of the statistical structure of their spectrum. Random matrices and closely related areas such as log-gases have become an extremely active research area in probability theory. This workshop brought together outstanding researchers from a variety of mathematical backgrounds whose areas of research are linked to random matrices. While there are strong links between their motivations, the techniques used by these researchers span a large swath of mathematics, ranging from purely algebraic techniques to stochastic analysis, classical probability theory, operator algebra, supersymmetry, orthogonal polynomials, etc

Flow control design inspired by linear stability analysis

In the recent literature, a growing number of research papers have been dedicated to applying the techniques of global stability and sensitivity analysis to the design of flow controls. The controls that are designed in this way are mainly passive or open-loop controls. Among those, we consider here controls that are aimed at linearly stabilizing flow configurations which would be otherwise globally unstable. In particular, a review of the literature on flow controls designed on the basis of stability and sensitivity analysis is presented. The mentioned methods can be rigorously applied to relatively simple flow regimes, typically observed at low values of the Reynolds number. In this respect, the recent literature also demonstrates a large interest in the application of the same methods for the control of coherent large-scale flow structures in turbulent flows, as, for instance, the quasiperiodic shedding of vortices in turbulent wakes. The papers dedicated to this subject are also reviewed here. Finally, all the described methods imply the solution of eigenvalue problems which are at the state-of-the-art for computational complexity. On the one hand, there are attempts to reduce the complexity of the involved computational problems by applying local stability analysis, and some examples are illustrated. On the other hand, recent advances in numerical methods, also concisely reviewed here, allow the manipulation of large eigenvalue problems and greatly simplify the development of numerical tools for stability and sensitivity analysis of complex flow models, often built using existing fluid dynamics codes

On the Variational Principles of Linear and Nonlinear Resolvent Analysis

Despite decades of research, the accurate and efficient modeling of turbulent flows remains a challenge. However, one promising avenue of research has been the resolvent analysis framework pioneered by McKeon and Sharma (2010) which interprets the nonlinearity of the Navier-Stokes equations (NSE) as an intrinsic forcing to the linear dynamics. This thesis contributes to the advancement of both the linear and nonlinear aspects of resolvent analysis (RA) based modeling of wall bounded turbulent flows. On the linear front, we suggest an alternative definition of the resolvent basis based on the calculus of variations. The proposed formulation circumvents the reliance on the inversion of the linear operator and is inherently compatible with any arbitrary choice of norm. This definition, which defines resolvent modes as stationary points of an operator norm, allows for more tractable analytical manipulation and leads to a straightforward approach to approximate the resolvent (response) modes of complex flows as expansions in any arbitrary basis. The proposed method avoids matrix inversions and requires only the spectral decomposition of a matrix of significantly reduced size as compared to the original system, thus having the potential to open up RA to the investigation of larger domains and more complex flow configurations. These analytical and numerical advantages are illustrated through a series of examples in one and two dimensions. The nonlinear aspects of RA are addressed in the context of Taylor vortex flow. Highly truncated and fully nonlinear solutions are computed by treating the nonlinearity not as an inherent part of the governing equations but rather as a triadic constraint which must be satisfied by the model solution. Our results show that as the Reynolds number increases, the flow undergoes a fundamental transition from a classical weakly nonlinear regime, where the forcing cascade is strictly down scale, to a fully nonlinear regime characterized by the emergence of an inverse (up scale) forcing cascade. It is shown analytically that this is a direct consequence of the structure of the quadratic nonlinearity of the NSE formulated in Fourier space. Finally, we suggest an algorithm based on the energy conserving nature of the nonlinearity of the NSE to reconstruct the phase information, and thus higher order statistics, from knowledge of solely the velocity spectrum. We demonstrate the potential of the proposed algorithm through a series of examples and discuss the challenges and potential applications to the study and simulation of turbulent flows.</p

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao

Cosmic Microwave Background and Large Scale Structure: Cross-Correlation as seen from Herschel and Planck satellites

As well as providing us with a snapshot of the Universe at the time of recombination, the cosmic microwave backround (CMB) radiation carries a wealth of information about the later evolution of the Universe through the so-called CMB secondary anisotropies that originates from the interaction between CMB photons and the Large Scale Structure (LSS). This thesis deals with two of these effects: the CMB lensing and the kinematic Sunyaev-Zel'dovich (kSZ). In particular, we present the first cross-correlation analysis between the CMB lensing maps reconstructed by Planck team and the angular position of galaxies from the Herschel H-ATLAS survey, the highest redshift sample exploited for cross-correlation analysis to date. By splitting the galaxy catalog in two redshift bins, we also attempt a tomographic analysis of the signal and reconstruct the galaxy bias evolution over cosmic time. On the other hand, the kSZ effect measures the integrated free electron momentum up to high redshift, thus being sensitive to the cosmic flows and the reionization history. Here we study its capabilities in constraining theories of modified gravity