Abstract:
We analyze the nonlinear optics of quasi one-dimensional quantum graphs and manipulate their topology and geometry to generate for the first time nonlinearities in a simple system approaching the fundamental limits of the first and second hyperpolarizabilities. Changes in geometry result in smooth variations of the nonlinearities. Topological changes between geometrically-similar systems cause profound changes in the nonlinear susceptibilities that include a discontinuity due to abrupt changes in the boundary conditions. This work may inform the design of new molecules or nano- scale structures for nonlinear optics and hints at the same universal behavior for quantum graph models in nonlinear optics that is observed in other systems.

Abstract:
This work focuses on understanding the nonlinear-optical response of a 1-D quantum wire embedded in 2-D space when quantum-size effects in the transverse direction are minimized using an extremely weighted delta function potential. Our aim is to establish the fundamental basis for understanding the effect of geometry on the nonlinear-optical response of quantum loops that are formed into a network of quantum wires. Using the concept of leaky quantum wires, it is shown that in the limit of full confinement, the sum rules are obeyed when the transverse infinite-energy continuum states are included. While the continuum states associated with the transverse wavefunction do not contribute to the nonlinear optical response, they are essential to preserving the validity of the sum rules. This work is a building block for future studies of nonlinear-optical enhancement of quantum graphs (which include loops and bent wires) based on their geometry. These properties are important in quantum mechanical modeling of any response function of quantum-confined systems, including the nonlinear-optical response of any system in which there is confinement in at leat one dimension, such as nanowires, which provide confinement in two dimensions.

Abstract:
Active control devices, such as active mass dampers, are mainly employed for the reduction of wind-induced vibrations in high-rise buildings, with the final aim of satisfying vibration serviceability limit state requirements and of meeting appropriate comfort criteria. When such active devices, normally operating under wind loads associated with short return periods, are subjected to seismic events, they can experience large amplitude vibrations and exceed stroke limits. This may lead to a reduced performance of the control system that can even worsen the performance of the whole structure. In this paper, a nonlinear control strategy based on a modified direct velocity feedback algorithm is proposed for handling stroke limits of an active mass driver (AMD) system. In particular, a suitable nonlinear braking term proportional to the relative AMD velocity is included in the control law in order to slowdown the device in the proximity of the stroke limits. Experimental and numerical free vibration tests are carried out on a scaled-down five-story frame structure equipped with an AMD to demonstrate the effectiveness of the proposed control strategy. 1. Introduction Active control systems are in principle very effective for the mitigation of the structural response, especially for high-rise buildings and flexible structures that may experience significant wind-induced vibrations [1, 2]. However, their use in practical applications is still limited by the physical bounds of the devices. In the case of strong earthquakes, the limits of the actuators may be exceeded, forcing the system to operate in a nonlinear mode for which it was not designed, thus worsening the performance of the controlled structure. The physical bounds of the actuators include both the control force limits and the stroke limits. The problem of force saturation has been deeply studied in the literature. Some approaches deal with preventing saturation of the control signal by designing the control system to always operate below its limits in the framework of linear control [3]. Another category of control methods accounts for system limitations directly in the control algorithm. Chase et al. [4] modified the H∞ control method through the addition of nonlinear state-dependent terms in order to model the actuators saturation and the uncertainties in the parameters of the system. Indrawan et al. [5] developed the bound-force control method which excludes the control-effort penalty from the performance index defined in the case of LQR control, defines it at the end of each time interval, and

Abstract:
In many biological systems, the interactions that describe the coupling between different units in a genetic network are nonlinear and stochastic. We study the interplay between stochasticity and nonlinearity using the responses of Chinese hamster ovary (CHO) mammalian cells to different temperature shocks. The experimental data show that the mean value response of a cell population can be described by a mathematical expression (empirical law) which is valid for a large range of heat shock conditions. A nonlinear stochastic theoretical model was developed that explains the empirical law for the mean response. Moreover, the theoretical model predicts a specific biological probability distribution of responses for a cell population. The prediction was experimentally confirmed by measurements at the single-cell level. The computational approach can be used to study other nonlinear stochastic biological phenomena.

Abstract:
Quantum graphs have recently emerged as models of nonlinear optical, quantum confined systems with exquisite topological sensitivity and the potential for predicting structures with an intrinsic, off-resonance response approaching the fundamental limit. Loop topologies have modest responses, while bent wires have larger responses, even when the bent wire and loop geometries are identical. Topological enhancement of the nonlinear response of quantum graphs is even greater for star graphs, for which the first hyperpolarizability can exceed half the fundamental limit. In this paper, we investigate the nonlinear optical properties of quantum graphs with the star vertex topology, introduce motifs and develop new methods for computing the spectra of composite graphs. We show that this class of graphs consistently produces intrinsic optical nonlinearities near the limits predicted by potential optimization. All graphs of this type have universal behavior for the scaling of their spectra and transition moments as the nonlinearities approach the fundamental limit.

Abstract:
We exploit theoretically a magneto-controlled nonlinear optical material which contains ferromagnetic nanoparticles with a non-magnetic metallic nonlinear shell in a host fluid. Such an optical material can have anisotropic linear and nonlinear optical properties and a giant enhancement of nonlinearity, as well as an attractive figure of merit.

Abstract:
Quantum Graphity is an approach to quantum gravity based on a background independent formulation of condensed matter systems on graphs. We summarize recent results obtained on the notion of emergent geometry from the point of view of a particle hopping on the graph. We discuss the role of connectivity in emergent Lorentzian perturbations in a curved background and the Bose--Hubbard (BH) model defined on graphs with particular symmetries.

Abstract:
These lecture notes are a personal introduction to signed graphs, concentrating on the aspects that have been most persistently interesting to me. They are just a few corners of signed graph theory; I am leaving out a great deal. The emphasis is on the way signed graphs arise naturally from geometry, especially from the geometry of the classical root systems. Most of the properties I discuss generalize those of unsigned graphs, but the constructions and proofs are often more complicated. My aim is a coherent presentation of the subject, with a few illustrative proofs and adequate references. Hence the arrangement of the notes is topical with only occasional remarks about the historical course of development. Though this is mainly an expository survey, some of the results have not hitherto been published.

Abstract:
It has long been appreciated that transport properties can control reaction kinetics. This effect can be characterized by the time it takes a diffusing molecule to reach a target -- the first-passage time (FPT). Although essential to quantify the kinetics of reactions on all time scales, determining the FPT distribution was deemed so far intractable. Here, we calculate analytically this FPT distribution and show that transport processes as various as regular diffusion, anomalous diffusion, diffusion in disordered media and in fractals fall into the same universality classes. Beyond this theoretical aspect, this result changes the views on standard reaction kinetics. More precisely, we argue that geometry can become a key parameter so far ignored in this context, and introduce the concept of "geometry-controlled kinetics". These findings could help understand the crucial role of spatial organization of genes in transcription kinetics, and more generally the impact of geometry on diffusion-limited reactions.

Abstract:
For a formation $\mathfrak{F}$ of finite groups, tight connections are established between the pro-$\mathfrak{F}$-topology of a finitely generated free group $F$ and the geometry of the Cayley graph $\Gamma(\hat{F_{\mathfrak{F}}})$ of the pro-$\mathfrak{F}$-completion $\hat{F_{\mathfrak {F}}}$ of $F$. For example, the Ribes--Zalesskii-Theorem is proved for the pro-$\mathfrak{F}$-topology of $F$ in case $\Gamma(\hat{F_{\mathfrak F}})$ is a tree-like graph. All these results are established by purely geometric proofs, without the use of inverse monoids which were indispensable in earlier papers, thereby giving more direct and more transparent proofs. Due to the richer structure provided by formations (compared to varieties), new examples of (relatively free) profinite groups with tree-like Cayley graphs are constructed. Thus, new topologies on $F$ are found for which the Ribes-Zalesskii-Theorem holds.