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

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index

Dynamic Equations on Time Scales: An Introduction with Applications

We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show that the continuous time ODE allows for a better understanding of Nesterov's scheme. As a byproduct, we obtain a family of schemes with similar convergence rates. The ODE interpretation also suggests restarting Nesterov's scheme leading to an algorithm, which can be rigorously proven to converge at a linear rate whenever the objective is strongly convex.

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A differential equation for modeling Nesterov's accelerated gradient method: theory and insights

Jacobi operators: Jacobi operators Foundations of spectral theory for Jacobi operators Qualitative theory of spectra Oscillation theory Random Jacobi operators Trace formulas Jacobi operators with periodic coefficients Reflectionless Jacobi operators Quasi-periodic Jacobi operators and Riemann theta functions Scattering theory Spectral deformations-Commutation methods Completely integrable nonlinear lattices: The Toda system The initial value problem for the Toda system The Kac-van Moerbeke system Notes on literature Compact Riemann surfaces-A review Hergoltz functions Jacobi difference equations with MathematicaR Bibliography Glossary of notations Index.

Jacobi Operators and Completely Integrable Nonlinear Lattices

On Titchmarsh-Weyl M(λ)-functions for linear Hamiltonian systems

Spectral analysis of second order difference equations

Hamiltonian systems of limit point or limit circle type with both endpoints singular

In part I of this work we defined a Titchmarsh-Weyl-coefficient M(λ) for singular 8 hermitian systems of arbitrary deficiency index. This construction proceeded by the method of von Noumann for selfadjoint extensions of symmetric operators. In this part we show how a Titchmarsh-Weyl coefficient M(λ) defined by a limit of Titchmarsh-Weyl coefficients on compact intervals is related to our M(λ). Examples are given in the intermediate deficiency index case which show that put all limits of Titchmarsh - Weyl coefficients on compact intervals give rise to a singular selfadjoint problem.

On the Titchmarsh‐Weyl Coefficients for Singular S‐Hermitian Systems II

For nearly 50 years Lyapunov inequalities have been an important tool in the study of differential equations. In this survey, building on an excellent 1991 historical survey by Cheng, we sketch some new developments in the theory of Lyapunov inequalities and present some recent disconjugacy results relating to second and higher order differential equations as well as Hamiltonian systems.

Don B. Hinton

Papers

On Titchmarsh-Weyl M(λ)-functions for linear Hamiltonian systems

Spectral analysis of second order difference equations

Hamiltonian systems of limit point or limit circle type with both endpoints singular

On the Titchmarsh‐Weyl Coefficients for Singular S‐Hermitian Systems II

Lyapunov Inequalities and their Applications