Scientific computation and functional programming

Using technology in the teaching and learning of mathematics requires, among other things, positive attitudes from students and teachers. In this study, two teachers helped design and then taught two courses of mathematics to four groups of upper secondary school students. Two of the student groups had full access to graphic calculators with symbolic algebra systems and the other two groups had full access to graphing calculators without symbolic algebra software. In this paper we report on the student attitudes and beliefs about using technology in their course within a curriculum that restricts its use in a ’final matriculation examination’. The outcomes suggest that integrated technology teaching impacts in a complex fashion on student attitude. The results of this study suggest that the nature of the assessment system which forms part of the curriculum influences the attitude towards rich usage of the technology.

This paper examines the three central theses of the Copenhagen interpretation of quantum mechanics. (1) Quantum mechanics must make use of concepts of classical physics; (2) descriptions which in classical physis are mutually exclusive are complementary descriptions of the same objects in quantum mechanics; (3) the uncertainty relations are natural laws which determine the limits of the application of classical concepts to micro-physical objects. Usually, these central theses of the Copenhagen interpretation are interpreted in a realistic-ontological manner. This paper criticizes this position, and, by means of a comparative investigation of the function of models in classical physics and quantum mechanics, proposes a model-theoretic interpretation. This interpretation lays special stress on the fictional status of models in quantum mechanics.

Bit-twiddling hacks for gamma matrices

The aim of this communication is to present a Symbolic Linear Algebra package, written in MAPLE V.5 and working also in the release 7, with general but especially educational purpose. Its goal is to run over MAPLE the different Linear Algebra algorithms developped in [OJ]. The implemented algorithms are also valids over finite characteristic and, for educational purpose, may be runned interactively. Three topics are covered by the package: equivalent matrices (echelon matrices, rank, LU decomposition, linear systems,...), similar matrices (Eigenvalues and eigenvectors, Rational or Frobenius form, Irreducible form, Jordan form,...), and congruent matrices (Symmetric and Hermitian matrices, Cholesky and QR decomposition, SVD, Gram-Schmidt over arbitrary euclidean metrics, ortogonal matrices, quadric surfaces...)

The quantum mechanical uncertainty principle for position and momentum plays an important role in many treatments on the (philosophical, physical and other) implications of quantum mechanics. Roughly speaking, the more precisely the momentum (position) of a (quantum mechanical) object is given, the less precisely can one say what its position (momentum) is. This quantum mechanical measurement problem is not just an interpretational difficulty, it raises broader issues as well. The measurement (of a property) of a (quantum mechanical) object determines the existence of the measured. In brief, the quantum mechanical uncertainty principle challenges some fundamental principles of Science and especially the principle of causality. In particular, an independently existing (external) objective reality is denied. As we shall see, that the quantum mechanical uncertainty principle for position and momentum is based on the assumption that 1⩽0, which is a logical contradiction.

Based on the concept of ontological states and their dynamical evolution by permutations, as assumed in the Cellular Automaton Interpretation (CAI) of quantum mechanics, we address the issue of whether quantum-classical hybrids can be described consistently in this framework. We consider chains of ‘classical’ two-state Ising spins and their discrete deterministic dynamics as an ontological model with an unitary evolution operator generated by pair-exchange interactions. A simple error mechanism is identified, which turns them into quantum mechanical objects: chains of qubits. Consequently, an interaction between a quantum mechanical and a ‘classical’ chain can be introduced and its consequences for this quantum-classical hybrid can be studied. We found that such hybrid character of composites, generally, does not persist under interactions and, therefore, cannot be upheld consistently, or even as a fundamental notion à la Kopenhagen interpretation, within CAI.

Chapter Four - Statistical and Perturbation-Based Analysis of Unidirectional Stretch of Rubber-Like Materials

The main aim of this work is to compare various models of rubber elasticity, i.e. neo-Hookean, Mooney-Rivlin, Yeoh, Gent, Arruda-Boyce as well as the Extended Tube model in terms of their application to the probabilistic analysis. Some discussions concerning failure analysis of the rubbers according to these models is provided also. Constitutive relations following these theories are tested for the case of uniaxial tension of the incompressible material, where deformation of a rubber specimen is treated as Gaussian random variable having a priori given expectation and standard deviations varying in some interval with bounds driven by various experimentation techniques. Probabilistic analysis is provided here in two alternative ways—via traditional Monte-Carlo technique as well as using higher order stochastic perturbation method implemented both in the symbolic computer algebra software. An application of non-Gaussian distributions relevant to the considered deformation, like lognormal one for instance has been also considered. This analysis includes computational determination of the first four basic probabilistic characteristics, i.e. expectation, coefficient of variation, skewness and kurtosis, and is provided to verify the resulting probabilistic distribution of the induced stress and its entropy. Some conclusions are drawn for the generalization of this method to other stress softening materials.

Multiplier circuit covers a more extensive area of embedded system application in digital signal processing, cryptography, and multimedia. Nonstandard implementations and custom optimization are being done to reduce the size of multipliers. The circuit became prone to a buggy, and hence the demand for verification increased. Formal verification methods, such as satisfiability (SAT), symbolic computer algebra (SCA), and binary decision diagrams (BDDs) have made massive progress over the last few decades. However, these methods are insufficient to verify the optimized multipliers. SAT-based equivalence checking is computationally expensive. SCA-based backward rewriting is limited to algebraic-friendly multipliers. The complexity of BDDs is exponential with the input size. Although, by allowing an additional variable method, the size of the BDD is limited to 4th degree polynomial of the number of the inputs, this method is not explored to verify optimized multipliers. This article focus on verifying integer multipliers with diverse architectures. We propose an algorithm for the direct construction of BDDs without traversing circuits and generate BDDs up to 1024 bits. We utilize the additional variable method and constructing BDDs using high-to-low variable ordering. We reduce the complexity of BDD size to a 3rd degree polynomial. We generate BDDs and verify the multipliers with various architectures up to 64 bits. We propose a method to verify optimized multipliers by checking equivalence and verifying up to 32-bits optimized multipliers. We do the error tolerance analysis of our approach by inserting bugs in a circuit at various locations.

Many compelling similarities among mechanical phenomena, including quantum-mechanical phenomena and digital-circuit transients provide motivation to initiate a research project to examine whether or not there is any deep-level connection among them. The theory of relativity showed that mass is a form of energy. Why, then, can a digital excitation that carries energy not be considered to be a mass point? Yet such a study would only serve to satisfy pure academic curiosity, if that were its only rationale. There are other incentives to go forward with such a study. Since its introduction, the quantum world has remained a fantasy world, accessible only to those armed with logical reasoning and advanced mathematics. The connection between the digital circuit and quantum mechanics provides a new viewpoint. Since the advent of the microprocessor, digital circuits are everywhere. As a model of the quantum world, they can help make the mystery world more accessible. At this point, a basic question must be asked: is the quantum world really unusual or exotic, or it is just a part of nature that happened to escape our attention until recent years due to a lack of familiar examples? I am inclined to believe that the quantum world is really a part of tangible nature and that it can have many varieties. The function of the human mind has been attributed by several influential authors to quantum-mechanical phenomena (Stapp, 1993; Penrose, 1994). Yet the neuronic systems are more concretely described as a special digital circuit than as a quantum-mechanical object. What then are the functional connections between the two?

Summary form only given. Complementarity is one of the most basic principles of quantum mechanics. A quantum mechanical object, be it a photon or a massive particle, shows position (particle-like) or momentum (wave-like) attributes exclusively depending on the experimental situation. Over the years, Young's two-slit experiment has been emphasized as a good example of complementarity. We report two two-slit experiments in which complementarity is enforced by availability of which-path information at the measurement stage of the quantum, which is different from the traditional understanding. A type-II BBO crystal is pumped by a CW UV laser generating spontaneous parametric down-conversion (SPDC). Entangled signal and idler photons are then sent in different directions. A double slit is placed in the idler beam and the idler detectors are scanned in the transverse direction.

This thesis proposes a novel way of performing type checking, whose results are incremental, depending on the provided local information. This new way of type checking is called co-contextual, where all context information of expressions, methods, classes, etc., is removed. Instead, we introduce corresponding structures using requirements. Standard type systems are translated to the co-contextual ones systematically using dualism as technique. Type systems play an important role to prevent execution errors from occurring during runtime. They are used to check programs statically for potential errors. Programs are type checked against a given set of rules. Depending on these rules programs are well-typed or not. The set of these rules is called typing rules. Each type rule associates types to the constructs of a program given a certain context. There can be different forms of contexts, depending on the features of the typed programming language. Functional languages use a typing context of variables and their types; object-oriented (OO) languages use additional class tables. Class tables are used for example to ensure that method and class declarations are well-typed. Type checking is performed top-down. While traversing the syntax tree of a program, typing contexts are extended with information on the expressions and their types. In case of OO, class tables are extended with clauses from class declarations, including the class members, i.e., fields, methods, or constructors. Contexts are passed through the nodes of the syntax tree in order to coordinate type checking between them. Therefore, while traversing the syntax tree top-down, the type checker creates dependencies between otherwise independent subexpressions. This way, it inhibits incrementalization and parallelization of type checking. That is, a change to a node of the syntax tree would require to redo the type check of the whole syntax tree. In this thesis a novel formulation of type systems is proposed, in order to remove dependencies between subexpressions. We propose a co-contextual formulation of typing rules that depends only on the local program constructs, e.g., expressions, methods, classes. The co-contextual typing rules have as conclusion a type and sets of requirements. That is, contexts and class tables are replaced by the dual concept of context and class table requirements. In addition, operations on contexts and class tables are replaced by new dual operations on requirements. The co-contextual type checker traverses a syntax tree bottom-up and merges context requirements of independently checked subexpressions. We describe a method for systematically constructing a co-contextual formulation of type rules from a regular context-based formulation and we show how co-contextual type rules give rise to incremental type checking. We derive co-contextual type checkers for functional and OO languages. As a representative of functional languages we consider PCF and extensions of it: records, parametric polymorphism, structural subtyping and let-polymorphism. Also, we investigate featherweight java (FJ) as the basis of OO languages and extensions of it: method overloading and generics. We build a1 co-contextual type checker for FJ enabling key features of OO languages: subtype polymorphism, nominal typing and implementation inheritance. The dualism between the co-contextual and contextual type systems preserves the correctness of the contextual calculus. That is, we prove the correctness of the co-contextual calculus via the equivalence between contextual type rules and their co-contextual formulations. We implemented an incremental type checker for PCF along with a performance evaluation showing that co-contextual type checking has performance comparable to standard context-based type checking, and incrementalization can improve performance significantly. Regarding FJ, we implemented a co-contextual type checker with incrementalization and compared its performance against javac on a number of realistic programs. Our performance evaluation shows significant speedups for the co-contextual type checker with incrementalization in comparison to javac.

Today, state-of-the-art computational methods and programs for quantum theory for electron correlation may not be developed entirely manually. They are often constructed with the essential aid of symbolic algebra systems automating lengthy and error-prone mathematical derivation and computer implementation, inevitable in these developments. Recent progress in this new paradigm of chemical theory development—complete automation—is reviewed.

We describe the design and implementation of Yacas, a free computer algebra system currently under development. The system consists of a core interpreter and a library of scripts that implement symbolic algebra functionality. The interpreter provides a high-level weakly typed functional language designed for quick prototyping of computer algebra algorithms, but the language is suitable for all kinds of symbolic manipulation. It supports conditional term rewritingof symbolic expression trees, closures (pure functions) and delayed evaluation, dynamic creation of transformation rules, arbitrary-precision numerical calculations, and flexible user-defined syntax usinginfix notation. The library of scripts currently provides basic numerical and symbolic functionality. The main advantages of Yacas are: free (GPL) software; a flexible and easy-touse programming language with a comfortable and adjustable syntax; crossplatform portability and small resource requirements; and extensibility.

Analytical SN solutions in heterogeneous slabs using symbolic algebra computer programs