A Study of Nonlinear Dynamics in Mathematical Biology

We first discuss some fundamental results such as equilibria, linearization, and stability of nonlinear dynamical systems arising in mathematical modeling. Next we study the dynamics in planar systems such as limit cycles, the Poincaré-Bendixson theorem, and some of its useful consequences. We then study the interaction between two and three different cell populations, and perform stability and bifurcation analysis on the systems. We also analyze the impact of immunotherapy on the tumor cell population numerically

Spectral analysis of semigroups and growth-fragmentation equations

The aim of this paper is twofold: (1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of classical results such as the spectral mapping theorem, some (quantified) Weyl's Theorems and the Krein-Rutman Theorem. Motivated by evolution PDE applications, the results apply to a wide and natural class of generators which split as a dissipative part plus a more regular part, without assuming any symmetric structure on the operators nor Hilbert structure on the space, and give some growth estimates and spectral gap estimates for the associated semigroup. The approach relies on some factorization and summation arguments reminiscent of the Dyson-Phillips series in the spirit of those used in [87,82,48,81]. (2) On the other hand, we present the semigroup spectral analysis for three important classes of ''growth-fragmentation" equations, namely the cell division equation, the self-similar fragmentation equation and the McKendrick-Von Foerster age structured population equation. By showing that these models lie in the class of equations for which our general semigroup analysis theory applies, we prove the exponential rate of convergence of the solutions to the associated remarkable profile for a very large and natural class of fragmentation rates. Our results generalize similar estimates obtained in \cite{MR2114128,MR2536450} for the cell division model with (almost) constant total fragmentation rate and in \cite{MR2832638,MR2821681} for the self-similar fragmentation equation and the cell division equation restricted to smooth and positive fragmentation rate and total fragmentation rate which does not increase more rapidly than quadratically. It also improves the convergence results without rate obtained in \cite{MR2162224,MR2114413} which have been established under similar assumptions to those made in the present work

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Dynamics of the outer planets : 1992 Summer Study Program in Geophysical Fluid Dynamics

The topic this summer was "The Dynamics of the Outer Planets." Andrew Ingersoll gave an excellent review of the current understanding of the strcture of the atmospheres of Jupiter, Neptune, Saturn, and Uranus. He presented the flow structures inferred from the information gathered by the Voyager probes and other observations. The models of the circulations of the interior and of the weather layer - the jets and vortices that we see in the images - were discussed. Jun-Ichi Yano gave further discussions on vortex dynamics in the lab, analytical, and numerical models as applied to the outer planets. Finally, Andy returned with a discussion of thin atmospheres (some so thin that they disappear at night) and new approaches to the dynamics of the interiors. These lectures provided a thorough background in both the data and the theory. As usual, we had talks (or what are sometimes called interactive seminars!) from many visitors during the summer, some directly related to the main topic and others covering other new research in geophysical fluid dynamics. From these, the fellows and staff found new aras for collaborative research and new ideas which they may explore after the summer. Finally, the summer was completed with talks from the fellows on their individual research during the summer. These reports reflect the thought and energy that went into learning new topics and formulating new problems. We look forward to seeing fuller versions of these in journal articles. We gratefully acknowledge the support of the National Science Foundation and the Office of Naval Research. The assistance of Jake Peirson and Barbara Ewing-DeRemer, made the summer, once again, pleasant and easy for all.Funding was provided by the National Science Foundation under Grant No. OCE8901012

Analytical and Numerical Methods for Differential Equations and Applications

The book is a printed version of the Special issue Analytical and Numerical Methods for Differential Equations and Applications, published in Frontiers in Applied Mathematics and Statistic

Entanglement dynamics and chaos in long-range quantum systems

Over the past twenty years, experimental and technological progresses have motivated a renewed attention to the study of non-equilibrium isolated many-body systems, leading to a relatively well-established paradigm in the case of local Hamiltonians. In the present thesis, I have used quantum information theoretical tools to study out-of-equilibrium dynamics, with particular attention on long-range interacting many-body systems. I have explored the dynamics of bipartite and multipartite entanglement in connection to chaos and scrambling in various long-range (clean and disordered) models. The results contained in this thesis contribute to establishing semi-classical tools as powerful techniques for the description of the quantum information spreading in long-range systems. I have further considered a different, yet connected question, concerning the multipartite entanglement structure of chaotic eigenstates and its generic evolution

Theoretical Concepts of Quantum Mechanics

Quantum theory as a scientific revolution profoundly influenced human thought about the universe and governed forces of nature. Perhaps the historical development of quantum mechanics mimics the history of human scientific struggles from their beginning. This book, which brought together an international community of invited authors, represents a rich account of foundation, scientific history of quantum mechanics, relativistic quantum mechanics and field theory, and different methods to solve the Schrodinger equation. We wish for this collected volume to become an important reference for students and researchers

Nonlinear Control of Unmanned Aerial Vehicles : Systems With an Attitude

This thesis deals with the general problem of controlling rigid-body systems through space, with a special focus on unmanned aerial vehicles (UAVs). Several promising UAV control algorithms have been developed over the past decades, enabling truly astounding feats of agility when combined with modern sensing technologies. However, these control algorithms typically come without global stability guarantees when implemented with estimation algorithms. Such control systems work well most of the time, but when introducing the UAVs more widely in society, it becomes paramount to prove that stability is ensured regardless of how the control system is initialized.The main motivation of the research lies in providing such (almost) global stability guarantees for an entire UAV control system. We develop algorithms that are implementable in practice and for which (almost) all initial errors result in perfect tracking of a reference trajectory. In doing so, both the tracking and the estimation errors are shown to be bounded in time along (almost) all solutions of the closed-loop system. In other words, if the initialization is sound and the initial errors are small, they will remain small and decrease in time, and even if the initial errors are large, they will not increase with time.As the field of UAV control is mature, this thesis starts by reviewing some of the most promising approaches to date in Part I. The ambition is to clarify how various controllers are related, provide intuition, and demonstrate how they work in practice. These ideas subsequently form the foundation on which a new result is derived, referred to as a nonlinear filtered output feedback. This represents a diametrically different approach to the control system synthesis. Instead of a disjoint controller/estimator design, the proposed method is comprised of two controller/estimator pairs, which when combined through a special interconnection term yields a system with favorable stability properties.While the first part of the thesis deals with theoretical controller design,Part II concerns application examples, demonstrating how the theory can solve challenging problems in modern society. In particular, we consider the problem of circumnavigation for search and rescue missions and show how UAVs can gather data from radioactive sites to estimate radiation intensity

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics