118 research outputs found
The C++0x "Concepts" Effort
C++0x is the working title for the revision of the ISO standard of the C++
programming language that was originally planned for release in 2009 but that
was delayed to 2011. The largest language extension in C++0x was "concepts",
that is, a collection of features for constraining template parameters. In
September of 2008, the C++ standards committee voted the concepts extension
into C++0x, but then in July of 2009, the committee voted the concepts
extension back out of C++0x.
This article is my account of the technical challenges and debates within the
"concepts" effort in the years 2003 to 2009. To provide some background, the
article also describes the design space for constrained parametric
polymorphism, or what is colloquially know as constrained generics. While this
article is meant to be generally accessible, the writing is aimed toward
readers with background in functional programming and programming language
theory. This article grew out of a lecture at the Spring School on Generic and
Indexed Programming at the University of Oxford, March 2010
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Computer algebra techniques in object-oriented mathematical modelling.
This thesis proposes a rigorous object-oriented methodology, supported by computer algebra software, to generate and relate features in a mathematical model. Evidence shows that there is little heuristic or theoretical research into this problem from any of the three principal modelling methodologies: 'case study’, ‘scenario’ and ‘generic’. This thesis comprises two other major strands: applications of computer algebra software and the efficacy of symbolic computation in teaching and learning. Developing the principal algorithms in computer algebra has sometimes been done at the expense of ease of use. Developers have also not concentrated on integrating an algebra engine into other software. A thorough review of quantitative studies in teaching and learning mathematics highlights a serious difficulty in measuring the effect of using computer algebra. This arises because of the disparate nature of learning with and without a computer.
This research tackles relationship formulation by casting the problem domain into object-oriented terms and building an appropriate class hierarchy. This capitalises on the fact that specific problems are instances of generic problems involving prototype physical objects. The computer algebra facilitates calculus operations and algebraic manipulation. In conjunction, I develop an object-oriented design methodology applicable to small-scale mathematical modelling. An object model modifies the generic modelling cycle. This allows relationships between features in the mathematical model to be generated automatically. The software is validated by quantifying the benefits of using the object-oriented techniques, and the results are statistically significant.
The principal problem domain considered is Newtonian particle mechanics. Although the Newtonian axioms form a firm basis for a mathematical description of interactions between physical objects, applying them within a particular modelling context can cause problems. The goal is to produce an equation of motion. Applications to other contexts are also demonstrated.
This research is significant because it formalises feature and equation-generation in a novel way. A new modelling methodology ensures that this crucial stage in the modelling cycle is given priority and automated
Рання історія комп'ютерного навчання математики студентів інженерних спеціальностей у США: 1965-1989
The article discusses ICT development issues in teaching mathematics to engineering students in the United States. The nature of trends in the convergence of information systems in higher technical education and other tendencies in the United States are described in the article. The primary historical stages of computer-assisted mathematics training for engineering students in the United States are defined. The study of historical sources has allowed six stages to be recognized. The use of ICT for teaching mathematics is examined at each stage. It demonstrates the inconsistencies and key elements of using ICT to teach mathematics to engineering students. This article covers the first three stages (1965-1989) of computer-assisted mathematics training for engineering students in the United States.У статті розглядаються питання розвитку ІКТ у навчанні математики студентів технічних спеціальностей у США. Описано характер тенденцій конвергенції інформаційних систем у вищій технічній освіті та інші тенденції в США. Визначено основні історичні етапи розвитку комп'ютерної математичної підготовки студентів інженерних спеціальностей у США. Вивчення історичних джерел дозволило виділити шість етапів. Розглянуто використання ІКТ для навчання математики на кожному етапі. Продемонстровано суперечності та ключові елементи використання ІКТ у навчанні математики студентів технічних спеціальностей. У статті розглядаються перші три етапи (1965-1989 рр.) комп'ютерного навчання математики студентів інженерних спеціальностей у Сполучених Штатах Америки
Scholarly event characteristics in four fields of science : a metrics-based analysis
One of the key channels of scholarly knowledge exchange are scholarly events such as conferences, workshops, symposiums, etc.; such events are especially important and popular in Computer Science, Engineering, and Natural Sciences.However, scholars encounter problems in finding relevant information about upcoming events and statistics on their historic evolution.In order to obtain a better understanding of scholarly event characteristics in four fields of science, we analyzed the metadata of scholarly events of four major fields of science, namely Computer Science, Physics, Engineering, and Mathematics using Scholarly Events Quality Assessment suite, a suite of ten metrics.In particular, we analyzed renowned scholarly events belonging to five sub-fields within Computer Science, namely World Wide Web, Computer Vision, Software Engineering, Data Management, as well as Security and Privacy.This analysis is based on a systematic approach using descriptive statistics as well as exploratory data analysis. The findings are on the one hand interesting to observe the general evolution and success factors of scholarly events; on the other hand, they allow (prospective) event organizers, publishers, and committee members to assess the progress of their event over time and compare it to other events in the same field; and finally, they help researchers to make more informed decisions when selecting suitable venues for presenting their work.Based on these findings, a set of recommendations has been concluded to different stakeholders, involving event organizers, potential authors, proceedings publishers, and sponsors. Our comprehensive dataset of scholarly events of the aforementioned fields is openly available in a semantic format and maintained collaboratively at OpenResearch.org. © 2020, The Author(s)
Algorithmic Contributions to the Theory of Regular Chains
Regular chains, introduced about twenty years ago, have emerged as one of the major
tools for solving polynomial systems symbolically. In this thesis, we focus on different
algorithmic aspects of the theory of regular chains, from theoretical questions to high-
performance implementation issues.
The inclusion test for saturated ideals is a fundamental problem in this theory.
By studying the primitivity of regular chains, we show that a regular chain generates
its saturated ideal if and only if it is primitive. As a result, a family of inclusion tests
can be detected very efficiently.
The algorithm to compute the regular GCDs of two polynomials modulo a regular
chain is one of the key routines in the various triangular decomposition algorithms. By
revisiting relations between subresultants and GCDs, we proposed a novel bottom-up
algorithm for this task, which improves the previous algorithm in a significant manner
and creates opportunities for parallel execution.
This thesis also discusses the accelerations towards fast Fourier transform (FFT)
over finite fields and FFT based subresultant chain constructions in the context of
massively parallel GPU architectures, which speedup our algorithms by several orders
of magnitude
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Towards justifying computer algebra algorithms in Isabelle/HOL
As verification efforts using interactive theorem proving grow, we are in need of certified algorithms in computer algebra to tackle problems over the real numbers. This is important because uncertified procedures can drastically increase the size of the trust base and under- mine the overall confidence established by interactive theorem provers, which usually rely on a small kernel to ensure the soundness of derived results.
This thesis describes an ongoing effort using the Isabelle theorem prover to certify the cylindrical algebraic decomposition (CAD) algorithm, which has been widely implemented to solve non-linear problems in various engineering and mathematical fields. Because of the sophistication of this algorithm, people are in doubt of the correctness of its implementation when deploying it to safety-critical verification projects, and such doubts motivate this thesis.
In particular, this thesis proposes a library of real algebraic numbers, whose distinguishing features include a modular architecture and a sign determination algorithm requiring only rational arithmetic. With this library, an Isabelle tactic based on univariate CAD has been built in a certificate-based way: external, untrusted code delivers solutions in the form of certificates that are checked within Isabelle. To lay the foundation for the multivariate case, I have formalised various analytical results including Cauchy’s residue theorem and the bivariate case of the projection theorem of CAD. During this process, I have also built a tactic to evaluate winding numbers through Cauchy indices and verified procedures to count complex roots in some domains.
The formalisation effort in this thesis can be considered as the first step towards a certified computer algebra system inside a theorem prover, so that various engineering projections and mathematical calculations can be carried out in a high-confidence framework
Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System
Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics
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