Peter Shiu

Bio: Peter Shiu is an academic researcher from Loughborough University. The author has contributed to research in topics: Prime (order theory) & Square (algebra). The author has an hindex of 7, co-authored 44 publications receiving 161 citations.

On numbers and games (2nd edn), by J. H. Conway. Pp. 256. £27.00. 2001. ISBN 1 56881 127 6 (A. K. Peters).

Abstract: On numbers and games (2nd edn), by J. H. Conway. Pp. 256. £27.00. 2001. ISBN 1 56881 127 6 (A. K. Peters). First published 25 years ago in the London Mathematical Society Monographs series, ONAG, as the book has since been commonly known, was apparently written in a burst of creative energy released essentially all within one week. The highly original theory being propounded is that a relationship between a new class of transfinite numbers and mathematical games can be set up. Presented in a delightfully whimsical style, the book soon became an exciting phenomenon leading to many research papers and a number of books, including one called Surreal numbers [1] by Donald Knuth.

Practical foundations of mathematics , by Paul Taylor. Pp. 572. £50. 1999. ISBN 0 521 63107 6 (Cambridge University Press).

Abstract: Such nit-picking should not be read as saying that the book is other than an excellent introduction to the techniques of set theory. Besides, I cannot but warm to a book which gives me such a quotable paragraph as: 'Some mathematicians object to the Axiom of Infinity on the grounds that a collection of objects produced by an infinite process ... should not be treated as a completed entity. However, most people with some mathematical training have no difficulty visualising the collection of natural numbers that way.' Note the implied threat in the second sentence; perhaps it can be read as a warning against 'some mathematical training'.

Differential variational inequalities

Abstract: This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with “index” not exceeding two and which have absolutely continuous solutions.

Affine Processes on Positive Semidefinite Matrices

Abstract: This paper provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. These matrix-valued affine processes have arisen from a large and growing range of useful applications in finance, including multi-asset option pricing with stochastic volatility and correlation structures, and fixed-income models with stochastically correlated risk factors and default intensities.

The Role of Convexity in Saddle-Point Dynamics: Lyapunov Function and Robustness

Abstract: This paper studies the projected saddle-point dynamics associated to a convex–concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We examine the role that the local and/or global nature of the convexity–concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics. Under the assumption that the saddle function is twice continuously differentiable, we provide a novel characterization of the omega-limit set of the trajectories of this dynamics in terms of the diagonal blocks of the Hessian. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexity–concavity of the saddle function. When strong convexity–concavity holds globally, we establish three results. First, we identify a Lyapunov function (that decreases strictly along the trajectory) for the projected saddle-point dynamics when the saddle function corresponds to the Lagrangian of a general constrained convex optimization problem. Second, for the particular case when the saddle function is the Lagrangian of an equality-constrained optimization problem, we show input-to-state stability (ISS) of the saddle-point dynamics by providing an ISS Lyapunov function. Third, we use the latter result to design an opportunistic state-triggered implementation of the dynamics. Various examples illustrate our results.