There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This tutorial provides an overview of the basic theory of hidden Markov models (HMMs) as originated by L.E. Baum and T. Petrie (1966) and gives practical details on methods of implementation of the theory along with a description of selected applications of the theory to distinct problems in speech recognition. Results from a number of original sources are combined to provide a single source of acquiring the background required to pursue further this area of research. The author first reviews the theory of discrete Markov chains and shows how the concept of hidden states, where the observation is a probabilistic function of the state, can be used effectively. The theory is illustrated with two simple examples, namely coin-tossing, and the classic balls-in-urns system. Three fundamental problems of HMMs are noted and several practical techniques for solving these problems are given. The various types of HMMs that have been studied, including ergodic as well as left-right models, are described. >

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A tutorial on hidden Markov models and selected applications in speech recognition

Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.

Ergodic theory of chaos and strange attractors

Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Holder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.

/pdf/finite-time-stability-of-continuous-autonomous-systems-1e0bmid7w7.pdf

Finite-Time Stability of Continuous Autonomous Systems

Preface * 1 The Evolution of Evolutionary Systems * 2 Dynamical Systems: Basic Theory * 3 Linear Semigroups * 4 Basic Theory of Evolutionary Equations * 5 Nonlinear Partial Differential Equations * 6 Navier Stokes Dynamics * 7 Basic Principles of Dynamics * 8 Inertial Manifolds and the Reduction Principle * Appendices: Basics of Functional Analysis * Bibliography * Notation Index * Subject Index

Dynamics of Evolutionary Equations

Inertial manifolds for nonlinear evolutionary equations

A unique introduction to the theory of linear operators on Hilbert space. The author presents the basic facts of functional analysis in a form suitable for engineers, scientists, and applied mathematicians. Although the Definition-Theorem-Proof format of mathematics is used, careful attention is given to motivation of the material covered and many illustrative examples are presented.

https://www.springer.com/productFlyer_978-0-387-95001-3.pdf?SGWID=0-0-1297-2329529-0

Linear Operator Theory in Engineering and Science

A Spectral Theory for Linear Differential Systems

https://msp.org/pjm/1968/27-2/pjm-v27-n2-p01-s.pdf

George R. Sell

Papers

Dynamics of Evolutionary Equations

Inertial manifolds for nonlinear evolutionary equations

Linear Operator Theory in Engineering and Science

A Spectral Theory for Linear Differential Systems

Growth transformations for functions on manifolds.