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405 results on '"Watt, Stephen M."'

1. A First Look at Chebyshev-Sobolev Series for Digital Ink

Author

Kalhan, Deepak Singh and Watt, Stephen M.

Subjects
Computer Science - Computer Vision and Pattern Recognition
Abstract

Considering digital ink as plane curves provides a valuable framework for various applications, including signature verification, note-taking, and mathematical handwriting recognition. These plane curves can be obtained as parameterized pairs of approximating truncated series (x(s), y(s)) determined by sampled points. Earlier work has found that representing these truncated series (polynomials) in a Legendre or Legendre-Sobolev basis has a number of desirable properties. These include compact data representation, meaningful clustering of like symbols in the vector space of polynomial coefficients, linear separability of classes in this space, and highly efficient calculation of variation between curves. In this work, we take a first step at examining the use of Chebyshev-Sobolev series for symbol recognition. The early indication is that this representation may be superior to Legendre-Sobolev representation for some purposes., Comment: Accepted at MathUI 2024

Published
2024

2. An Abstraction-Preserving Block Matrix Implementation in Maple

Author

Jeffrey, David J. and Watt, Stephen M.

Subjects
Computer Science - Symbolic Computation
Abstract

A Maple implementation of partitioned matrices is described. A recursive block data structure is used, with all operations preserving the block abstraction. These include constructor functions, ring operations such as addition and product, and inversion. The package is demonstrated by calculating the PLU factorization of a block matrix.

Published
2024

3. Computing Clipped Products

Author

Norman, Arthur C. and Watt, Stephen M.

Subjects
Computer Science - Symbolic Computation, Mathematics - Numerical Analysis
Abstract

Sometimes only some digits of a numerical product or some terms of a polynomial or series product are required. Frequently these constitute the most significant or least significant part of the value, for example when computing initial values or refinement steps in iterative approximation schemes. Other situations require the middle portion. In this paper we provide algorithms for the general problem of computing a given span of coefficients within a product, that is the terms within a range of degrees for univariate polynomials or range digits of an integer. This generalizes the "middle product" concept of Hanrot, Quercia and Zimmerman. We are primarily interested in problems of modest size where constant speed up factors can improve overall system performance, and therefore focus the discussion on classical and Karatsuba multiplication and how methods may be combined., Comment: To appear in Computer Algebra in Scientific Computing (CASC 2024)

Published
2024

4. Algorithms for Recursive Block Matrices

Author

Watt, Stephen M.

Subjects
Computer Science - Symbolic Computation
Abstract

We study certain linear algebra algorithms for recursive block matrices. This representation has useful practical and theoretical properties. We summarize some previous results for block matrix inversion and present some results on triangular decomposition of block matrices. The case of inverting matrices over a ring that is neither formally real nor formally complex was inspired by Gonzalez-Vega et al., Comment: To appear in "LALO 60: Matrices and Polynomials in Computer Algebra"

Published
2024

5. A Symbolic Computing Perspective on Software Systems

Author

Norman, Arthur C. and Watt, Stephen M.

Subjects
Computer Science - Symbolic Computation, Computer Science - Mathematical Software, Computer Science - Software Engineering, I.1.3
Abstract

Symbolic mathematical computing systems have served as a canary in the coal mine of software systems for more than sixty years. They have introduced or have been early adopters of programming language ideas such ideas as dynamic memory management, arbitrary precision arithmetic and dependent types. These systems have the feature of being highly complex while at the same time operating in a domain where results are well-defined and clearly verifiable. These software systems span multiple layers of abstraction with concerns ranging from instruction scheduling and cache pressure up to algorithmic complexity of constructions in algebraic geometry. All of the major symbolic mathematical computing systems include low-level code for arithmetic, memory management and other primitives, a compiler or interpreter for a bespoke programming language, a library of high level mathematical algorithms, and some form of user interface. Each of these parts invokes multiple deep issues. We present some lessons learned from this environment and free flowing opinions on topics including: * Portability of software across architectures and decades; * Infrastructure to embrace and infrastructure to avoid; * Choosing base abstractions upon which to build; * How to get the most out of a small code base; * How developments in compilers both to optimise and to validate code have always been and remain of critical importance, with plenty of remaining challenges; * The way in which individuals including in particular Alan Mycroft who has been able to span from hand-crafting Z80 machine code up to the most abstruse high level code analysis techniques are needed, and * Why it is important to teach full-stack thinking to the next generation.

Published
2024

6. Using General Large Language Models to Classify Mathematical Documents

Author

Ion, Patrick D. F. and Watt, Stephen M.

Subjects
Computer Science - Information Retrieval, Computer Science - Computation and Language, Computer Science - Digital Libraries
Abstract

In this article we report on an initial exploration to assess the viability of using the general large language models (LLMs), recently made public, to classify mathematical documents. Automated classification would be useful from the applied perspective of improving the navigation of the literature and the more open-ended goal of identifying relations among mathematical results. The Mathematical Subject Classification MSC 2020, from MathSciNet and zbMATH, is widely used and there is a significant corpus of ground truth material in the open literature. We have evaluated the classification of preprint articles from arXiv.org according to MSC 2020. The experiment used only the title and abstract alone -- not the entire paper. Since this was early in the use of chatbots and the development of their APIs, we report here on what was carried out by hand. Of course, the automation of the process will have to follow if it is to be generally useful. We found that in about 60% of our sample the LLM produced a primary classification matching that already reported on arXiv. In about half of those instances, there were additional primary classifications that were not detected. In about 40% of our sample, the LLM suggested a different classification than what was provided. A detailed examination of these cases, however, showed that the LLM-suggested classifications were in most cases better than those provided.

Published
2024

7. GPU Implementations for Midsize Integer Addition and Multiplication

Author

Oancea, Cosmin E. and Watt, Stephen M.

Subjects
Computer Science - Distributed, Parallel, and Cluster Computing, Computer Science - Mathematical Software, Computer Science - Programming Languages
Abstract

This paper explores practical aspects of using a high-level functional language for GPU-based arithmetic on ``midsize'' integers. By this we mean integers of up to about a quarter million bits, which is sufficient for most practical purposes. The goal is to understand whether it is possible to support efficient nested-parallel programs with a small, flexible code base. We report on GPU implementations for addition and multiplication of integers that fit in one CUDA block, thus leveraging temporal reuse from scratchpad memories. Our key contribution resides in the simplicity of the proposed solutions: We recognize that addition is a straightforward application of scan, which is known to allow efficient GPU implementation. For quadratic multiplication we employ a simple work-partitioning strategy that offers good temporal locality. For FFT multiplication, we efficiently map the computation in the domain of integral fields by finding ``good'' primes that enable almost-full utilization of machine words. In comparison, related work uses complex tiling strategies -- which feel too big a hammer for the job -- or uses the computational domain of reals, which may degrade the magnitude of the base in which the computation is carried. We evaluate the performance in comparison to the state-of-the-art CGBN library, authored by NvidiaLab, and report that our CUDA prototype outperforms CGBN for integer sizes higher than 32K bits, while offering comparable performance for smaller sizes. Moreover, we are, to our knowledge, the first to report that FFT multiplication outperforms the classical one on the larger sizes that still fit in a CUDA block. Finally, we examine Futhark's strengths and weaknesses for efficiently supporting such computations and find out that a compiler pass aimed at efficient sequentialization of excess parallelism would significantly improve performance.

Published
2024

8. Efficient Quotients of Non-Commutative Polynomials

Author

Watt, Stephen M.

Subjects
Computer Science - Symbolic Computation
Abstract

It is shown how to compute quotients efficiently in non-commutative univariate polynomial rings. This extends earlier work where efficient generic quotients were studied with a primary focus on commutative domains. Fast algorithms are given for left and right quotients of polynomials where the variable commutes with coefficients. These algorithms are based on the concept of the ``whole shifted inverse'', which is a specialized quotient where the dividend is a power of the polynomial variable. It is also shown that when the variable does not commute with coefficients, that is for skew polynomials, left and right whole shifted inverses are defined and may be used to compute right and left quotients. In this case their computation is not asymptotically fast, but once obtained, they may be used to compute multiple quotients, each with one multiplication. Examples are shown of polynomials with matrix coefficients, differential operators and difference operators. In addition, a proof-of-concept generic Maple implementations is given.

Published
2023

9. Efficient Generic Quotients Using Exact Arithmetic

Author

Watt, Stephen M.

Subjects
Computer Science - Symbolic Computation
Abstract

The usual formulation of efficient division uses Newton iteration to compute an inverse in a related domain where multiplicative inverses exist. On one hand, Newton iteration allows quotients to be calculated using an efficient multiplication method. On the other hand, working in another domain is not always desirable and can lead to a library structure where arithmetic domains are interdependent. This paper uses the concept of a whole shifted inverse and modified Newton iteration to compute quotients efficiently without leaving the original domain. The iteration is generic to domains having a suitable shift operation, such as integers or polynomials with coefficients that do not necessarily commute., Comment: Reformat of v3

Published
2023

11. Extracting Theory Graphs from Aldor Libraries

Author

Rabe, Florian, Watt, Stephen M., Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Dubois, Catherine, editor, and Kerber, Manfred, editor

Published
2023
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13. Working with Families of Inverse Functions

Author

Jeffrey, David J., Watt, Stephen M., Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Buzzard, Kevin, editor, and Kutsia, Temur, editor

Published
2022
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15. Determining Points on Handwritten Mathematical Symbols

Author

Hu, Rui and Watt, Stephen M.

Subjects
Computer Science - Computer Vision and Pattern Recognition, Computer Science - Computers and Society
Abstract

In a variety of applications, such as handwritten mathematics and diagram labelling, it is common to have symbols of many different sizes in use and for the writing not to follow simple baselines. In order to understand the scale and relative positioning of individual characters, it is necessary to identify the location of certain expected features. These are typically identified by particular points in the symbols, for example, the baseline of a lower case "p" would be identified by the lowest part of the bowl, ignoring the descender. We investigate how to find these special points automatically so they may be used in a number of problems, such as improving two-dimensional mathematical recognition and in handwriting neatening, while preserving the original style., Comment: 16 pages; 19 figures; Conferences on Intelligent Computer Mathematics (CICM2013), July 8-12, 2013, University of Bath, Bath, UK

Published
2013

16. Symbolic Domain Decomposition

Author

Carette, Jacques, Sexton, Alan P., Sorge, Volker, and Watt, Stephen M.

Subjects
Computer Science - Symbolic Computation, Mathematics - Numerical Analysis, 03E99, 05A18, 33F10, I.1.1
Abstract

Decomposing the domain of a function into parts has many uses in mathematics. A domain may naturally be a union of pieces, a function may be defined by cases, or different boundary conditions may hold on different regions. For any particular problem the domain can be given explicitly, but when dealing with a family of problems given in terms of symbolic parameters, matters become more difficult. This article shows how hybrid sets, that is multisets allowing negative multiplicity, may be used to express symbolic domain decompositions in an efficient, elegant and uniform way, simplifying both computation and reasoning. We apply this theory to the arithmetic of piecewise functions and symbolic matrices and show how certain operations may be reduced from exponential to linear complexity., Comment: Calculemus 2010 (17th Symposium on the Integration of Symbolic Computation and Mechanised Reasoning), part of Conferences on Intelligent Computer Mathematics 2010. 17 pages.

Published
2010

19. The Global Digital Mathematics Library and the International Mathematical Knowledge Trust

Author

Ion, Patrick D. F., Watt, Stephen M., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Geuvers, Herman, editor, England, Matthew, editor, Hasan, Osman, editor, Rabe, Florian, editor, and Teschke, Olaf, editor

Published
2017
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20. The Inverse of the Complex Gamma Function

Author

Jeffrey, David J., Watt, Stephen M., Jeffrey, David J., and Watt, Stephen M.

Abstract

We consider the functional inverse of the Gamma function in the complex plane, where it is multi-valued, and define a set of suitable branches by proposing a natural extension from the real case.

Published
2023

21. Hybrid Intervals and Symbolic Block Matrices

Author

Ghesquiere, Mike, Watt, Stephen M., Ghesquiere, Mike, and Watt, Stephen M.

Abstract

Structured matrices with symbolic sizes appear frequently in the literature, especially in the description of algorithms for linear algebra. Recent work has treated these symbolic structured matrices themselves as computational objects, showing how to add matrices with blocks of different symbolic sizes in a general way while avoiding a combinatorial explosion of cases. The present article introduces the concept of hybrid intervals, in which points may have negative multiplicity. Various operations on hybrid intervals have compact and elegant formulations that do not require cases to handle different orders of the end points. This makes them useful to represent symbolic block matrix structures and to express arithmetic on symbolic block matrices compactly. We use these ideas to formulate symbolic block matrix addition and multiplication in a compact and uniform way.

Published
2023

22. Confidence Measures in Recognizing Handwritten Mathematical Symbols

Author

Golubitsky, Oleg, Watt, Stephen M., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Goebel, Randy, editor, Siekmann, Jörg, editor, Wahlster, Wolfgang, editor, Carette, Jacques, editor, Dixon, Lucas, editor, Coen, Claudio Sacerdoti, editor, and Watt, Stephen M., editor

Published
2009
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23. Algorithms for the Functional Decomposition of Laurent Polynomials

Author

Watt, Stephen M., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Goebel, Randy, editor, Siekmann, Jörg, editor, Wahlster, Wolfgang, editor, Carette, Jacques, editor, Dixon, Lucas, editor, Coen, Claudio Sacerdoti, editor, and Watt, Stephen M., editor

Published
2009
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24. Reasoning with Generic Cases in the Arithmetic of Abstract Matrices

Author

Sexton, Alan P., Sorge, Volker, Watt, Stephen M., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Goebel, Randy, editor, Siekmann, Jörg, editor, Wahlster, Wolfgang, editor, Carette, Jacques, editor, Dixon, Lucas, editor, Coen, Claudio Sacerdoti, editor, and Watt, Stephen M., editor

Published
2009
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25. Lightweight Abstraction for Mathematical Computation in Java

Author

Bourdykine, Pavel, Watt, Stephen M., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Gerdt, Vladimir P., editor, Koepf, Wolfram, editor, Mayr, Ernst W., editor, and Vorozhtsov, Evgenii V., editor

Published
2012
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26. Writing on Clouds

Author

Mazalov, Vadim, Watt, Stephen M., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Goebel, Randy, editor, Siekmann, Jörg, editor, Wahlster, Wolfgang, editor, Jeuring, Johan, editor, Campbell, John A., editor, Carette, Jacques, editor, Dos Reis, Gabriel, editor, Sojka, Petr, editor, Wenzel, Makarius, editor, and Sorge, Volker, editor

Published
2012
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27. A Streaming Digital Ink Framework for Multi-party Collaboration

Author

Hu, Rui, Mazalov, Vadim, Watt, Stephen M., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Goebel, Randy, editor, Siekmann, Jörg, editor, Wahlster, Wolfgang, editor, Jeuring, Johan, editor, Campbell, John A., editor, Carette, Jacques, editor, Dos Reis, Gabriel, editor, Sojka, Petr, editor, Wenzel, Makarius, editor, and Sorge, Volker, editor

Published
2012
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29. Type Specialization in Aldor

Author

Dragan, Laurentiu, Watt, Stephen M., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Gerdt, Vladimir P., editor, Koepf, Wolfram, editor, Mayr, Ernst W., editor, and Vorozhtsov, Evgenii V., editor

Published
2010
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31. A Localized Tracing Scheme Applied to Garbage Collection

Author

Chicha, Yannis, Watt, Stephen M., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, and Kobayashi, Naoki, editor

Published
2006
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32. Determining Empirical Characteristics of Mathematical Expression Use

Author

So, Clare M., Watt, Stephen M., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Carbonell, Jaime G., editor, Siekmann, Jörg, editor, and Kohlhase, Michael, editor

Published
2006
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33. An Architecture for Distributed Mathematical Web Services

Author

Smirnova, Elena S., So, Clare M., Watt, Stephen M., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Dough, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Asperti, Andrea, editor, Bancerek, Grzegorz, editor, and Trybulec, Andrzej, editor

Published
2004
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34. Topics of Computer Algebra

Author

Loos, Rüdiger, Giusti, M., Pardo, L. M., Niklasch, Gerhard, Dentzer, Ralf, Landau, Susan, Roy, Marie-Françoise, Pohst, Michael E., Grabmeier, Johannes, Weispfenning, Volker, von zur Gathen, Joachim, Kaltofen, Erich, Gutierrez, Jaime, Kozen, Dexter, Kemper, Gregor, Saunders, B. David, Giesbrecht, Mark, Storjohann, Arne, Villard, Gilles, Buchmann, Johannes, Müller, Volker, Pfahler, Thomas, Zayer, Jörg, Klüners, Jürgen, Heinrich Matzat, B., Zimmer, Horst G., Greuel, Gert-Martin, Becker, Eberhard, Apel, Joachim, Cohen, Arjeh M., Dräxler, Peter, Laßner, Wolfgang, Madlener, Klaus E., Reinert, Birgit, Sims, Charles C., Hiss, Gerhard, Kerber, Adalbert, Laue, Reinhard, Paule, Peter, Strehl, Voker, Bronstein, Manuel, Seiler, W. M., Ulmer, F., Schwarz, F., Hereman, W., Gerdt, V. P., Ollivier, F., Gatermann, K., Hantzschmann, Karl, Rump, Siegfried M., Corless, Robert M., Watt, Stephen M., Bürgisser, Peter, Clausen, Michael, Shokrollahi, M. Amin, Beth, Thomas, Egner, Sebastian, Geiselmann, Willi, Grassl, Markus, Lazic, Dejan, Müller-Quade, Jörn, Schaefer-Lorinser, Frank, Bündgen, Reinhard, Calmet, Jacques, Fortenbacher, Albrecht, Schupp, Sibylle, Gautier, Thierry, Hong, Hoon, Roch, Jean-Louis, Schreiner, Wolfgang, Dewar, Michael, Klappenecker, Andreas, Nückel, Armin, Grabmeier, Johannes, editor, Kaltofen, Erich, editor, and Weispfenning, Volker, editor

Published
2003
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35. Global access to Maple information

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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36. Advanced Examples

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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37. Measuring and improving performance

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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38. Advanced Graphics

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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39. The Maple Programming Language

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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40. Interactive Use of Maple

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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41. Less Simple Maple

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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42. Miscellaneous Packages

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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43. Packages for Geometry

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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44. Packages for Applied Mathematics

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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45. Packages

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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46. Packages for Discrete Mathematics

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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47. The Maple Library

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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48. Main Routines

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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49. Operators

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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50. Internal Representation and Manipulation

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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