Invariants, Kronecker Products, and Combinatorics of Some Remarkable Diophantine Systems (Extended Version)

This work lies across three areas (in the title) of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link consists of a single formal power series with a multifaced interpretation. The deeper exploration of this link yielded results as well as methods for solving some numerical problems in each of these separate areas.Comment: 33 pages, 5 figure

The Formal Laplace-Borel Transform of Fliess Operators and the Composition Product

The formal Laplace-Borel transform of an analytic integral operator, known as a Fliess operator, is defined and developed. Then, in conjunction with the composition product over formal power series, the formal Laplace-Borel transform is shown to provide an isomorphism between the semigroup of all Fliess operators under operator composition and the semigroup of all locally convergent formal power series under the composition product. Finally, the formal Laplace-Borel transform is applied in a systems theory setting to explicitly derive the relationship between the formal Laplace transform of the input and output functions of a Fliess operator. This gives a compact interpretation of the operational calculus of Fliess for computing the output response of an analytic nonlinear system. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved

Intelligent systems for efficiency and security

As computing becomes ubiquitous and personalized, resources like energy, storage and time are becoming increasingly scarce and, at the same time, computing systems must deliver in multiple dimensions, such as high performance, quality of service, reliability, security and low power. Building such computers is hard, particularly when the operating environment is becoming more dynamic, and systems are becoming heterogeneous and distributed. Unfortunately, computers today manage resources with many ad hoc heuristics that are suboptimal, unsafe, and cannot be composed across the computer’s subsystems. Continuing this approach has severe consequences: underperforming systems, resource waste, information loss, and even life endangerment. This dissertation research develops computing systems which, through intelligent adaptation, deliver efficiency along multiple dimensions. The key idea is to manage computers with principled methods from formal control. It is with these methods that the multiple subsystems of a computer sense their environment and configure themselves to meet system-wide goals. To achieve the goal of intelligent systems, this dissertation makes a series of contributions, each building on the previous. First, it introduces the use of formal MIMO (Multiple Input Multiple Output) control for processors, to simultaneously optimize many goals like performance, power, and temperature. Second, it develops the Yukta control system, which uses coordinated formal controllers in different layers of the stack (hardware and operating system). Third, it uses robust control to develop a fast, globally coordinated and decentralized control framework called Tangram, for heterogeneous computers. Finally, it presents Maya, a defense against power side-channel attacks that uses formal control to reshape the power dissipated by a computer, confusing the attacker. The ideas in the dissertation have been demonstrated successfully with several prototypes, including one built along with AMD (Advanced Micro Devices, Inc.) engineers. These designs significantly outperformed the state of the art. The research in this dissertation brought formal control closer to computer architecture and has been well-received in both domains. It has the first application of full-fledged MIMO control for processors, the first use of robust control in computer systems, and the first application of formal control for side-channel defense. It makes a significant stride towards intelligent systems that are efficient, secure and reliable

Characterizations of recognizable picture series

AbstractThe theory of two-dimensional languages as a generalization of formal string languages was motivated by problems arising from image processing and pattern recognition, and also concerns models of parallel computing. Here we investigate power series on pictures. These are functions that map pictures to elements of a semiring and provide an extension of two-dimensional languages to a quantitative setting. We assign weights to different devices, ranging from picture automata to tiling systems. We will prove that, for commutative semirings, the behaviours of weighted picture automata are precisely alphabetic projections of series defined in terms of rational operations, and also coincide with the families of series characterized by weighted tiling or weighted domino systems

Abelian Conformal Field theories and Determinant Bundles

The present paper is the first in a series of papers, in which we shall construct modular functors and Topological Quantum Field Theories from the conformal field theory developed in [TUY]. The basic idea is that the covariant constant sections of the sheaf of vacua associated to a simple Lie algebra over Teichm\"uller space of an oriented pointed surface gives the vectorspace the modular functor associates to the oriented pointed surface. However the connection on the sheaf of vacua is only projectively flat, so we need to find a suitable line bundle with a connection, such that the tensor product of the two has a flat connection. We shall construct a line bundle with a connection on any family of pointed curves with formal coordinates. By computing the curvature of this line bundle, we conclude that we actually need a fractional power of this line bundle so as to obtain a flat connection after tensoring. In order to functorially extract this fractional power, we need to construct a preferred section of the line bundle. We shall construct the line bundle by the use of the so-called $bc$-ghost systems (Faddeev-Popov ghosts) first introduced in covariant quantization [FP]. We follow the ideas of [KNTY], but decribe it from the point of view of [TUY].Comment: A couple of typos correcte