Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)

On computable numbers

We present an approach to construct reachable set overapproximations for continuous-time dynamical systems controlled using neural network feedback systems. Feedforward deep neural networks are now widely used as a means for learning control laws through techniques such as reinforcement learning and data-driven predictive control. However, the learning algorithms for these networks do not guarantee correctness properties on the resulting closed-loop systems. Our approach seeks to construct overapproximate reachable sets by integrating a Taylor model-based flowpipe construction scheme for continuous differential equations with an approach that replaces the neural network feedback law for a small subset of inputs by a polynomial mapping. We generate the polynomial mapping using regression from input-output samples. To ensure soundness, we rigorously quantify the gap between the output of the network and that of the polynomial model. We demonstrate the effectiveness of our approach over a suite of benchmark examples ranging from 2 to 17 state variables, comparing our approach with alternative ideas based on range analysis.

https://dl.acm.org/doi/pdf/10.1145/3302504.3311807

Reachability analysis for neural feedback systems using regressive polynomial rule inference

Reset control systems are a special type of impulsive hybrid systems in which the time evolution depends both on continuous dynamics between resets and the discrete dynamics corresponding to the reset instants. In this work, stability of reset control systems is approached by using an equivalent (time-varying) discrete time system, introducing new sufficient stability conditions that explicitly depend on the reset times. The stability conditions overcome previous results such as the -condition and can be also applied to reset systems with unstable base system. In addition, stabilization of reset control systems is approached for systems with stable base system, resulting in that a reset control system can be always stabilized if reset intervals are forced to be large enough, avoiding resets at some crossings if necessary.

Reset Times-Dependent Stability of Reset Control Systems

http://sqig.math.ist.utl.pt/pub/campagnolom/06-BCGH-GPAC.pdf

Polynomial differential equations compute all real computable functions on computable compact intervals

Dynamical systems allow to modelize various phenomena or processes by only describing their local behaviour It is however useful to understand the behaviour in a more global way Checking the reachability of a point for example is a fundamental problem In this document we will show that this problem that is undecidable in the general case is in fact decidable for a natural class of continuous-time dynamical systems: linear systems For this, we will use results from the algebraic numbers theory such as Gelfond-Schneider's theorem

/pdf/reachability-in-linear-dynamical-systems-e0s9viskze.pdf

Reachability in Linear Dynamical Systems

Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. 

In a different and orthogonal work, we proposed a minimalization schema that allows to provide a class of real recursive functions that corresponds to extensions of computable functions over the integers.

Mixing the two approaches we prove that computable functions over the real numbers in the sense of recursive analysis can be characterized as the smallest class of functions that contains some basic functions, and closed by composition, linear integration, minimalization and limit schema.

https://members.loria.fr/EHainry/papers/fi2005.pdf

Recursive Analysis Characterized as a Class of Real Recursive Functions

In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPAC-computability in a natural way, we compute exactly all real computable functions (in the sense of computable analysis). Moreover, since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions can be defined by such models.

/pdf/the-general-purpose-analog-computer-and-computable-analysis-9ftiov5dap.pdf

The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation

We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema.We generalize this result to all higher levels of the Grzegorczyk Hierarchy.This paper improves several previous partial characterizations and has a dual interest: • Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. • Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers and provide new insights for understanding the relations between several analog computational models.

/pdf/elementarily-computable-functions-over-the-real-numbers-and-4xwj3hv078.pdf

Emmanuel Hainry

Papers

Polynomial differential equations compute all real computable functions on computable compact intervals

Reachability in Linear Dynamical Systems

Recursive Analysis Characterized as a Class of Real Recursive Functions

The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation

Elementarily computable functions over the real numbers and R-sub-recursive functions