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The Meaning of Infinity in Calculus and Computer Algebra Systems [chapter]

Michael Beeson, Freek Wiedijk
2002 Lecture Notes in Computer Science  
We use filters of open sets to provide a semantics justifying formally the use of infinity in informal limit calculations in calculus, and in the same kind of calculations in computer algebra.  ...  The computer algebra method under discussion in this paper is the use of rewrite rules to evaluate limits involving infinity.  ...  Nonstandard analysis has been used in (5) to help in the computation of limits in a computer algebra system. Filters, lifting, refinement and limits Definition: Let X be a topological space.  ... 
doi:10.1007/3-540-45470-5_23 fatcat:6pnliaqog5af3nbcn7wvzi4jfe

The meaning of infinity in calculus and computer algebra systems

Michael Beeson, Freek Wiedijk
2005 Journal of symbolic computation  
We use filters of open sets to provide a semantics justifying formally the use of infinity in informal limit calculations in calculus, and in the same kind of calculations in computer algebra.  ...  The computer algebra method under discussion in this paper is the use of rewrite rules to evaluate limits involving infinity.  ...  But in fact, in Mathematica there is a complete 'calculus of infinity' (and some related symbols): In[1]:= 1/(Infinity + 1) Other computer algebra systems implement similar calculi.  ... 
doi:10.1016/j.jsc.2004.12.002 fatcat:xxvqu7fkynamvftd3gkk2qurqq

"Region calculus" : generalization of Newton calculus

Ranjit Biswas
2019 International Journal of Mathematical Analysis  
It is only the algebraic system region at minimum which has the capability to validate such elementary and higher level computations.  ...  The same is true for Region Calculus too, where the concept of infinity is also a generalized concept in the context of the new algebraic structure region.  ... 
doi:10.12988/ijma.2019.9742 fatcat:3b7bv4sdhretfaibie23mmavea

The review: Approaches in teaching calculus

Kinley .
A few approaches used in teaching calculus that focus specifically on differential and integral calculus are reviewed: algebraic, graphical, contextual or realistic, and computer-assisted approaches.  ...  Difficulty of learning calculus was faced by many students in higher secondary schools because the teaching was teacher centered and usually focused on explaining the notations and symbols used in calculus  ...  The themes of calculus reform movement include: involving students in doing mathematics instead of lecturing at them; stressing conceptual understanding rather than only computation; developing meaningful  ... 
doi:10.6084/m9.figshare.19009382.v1 fatcat:jpaym6omm5annaevm3u4n26mna

Page 130 of Mathematical Reviews Vol. , Issue 84a [page]

1984 Mathematical Reviews  
whose solution x= x(t) has a pole (simple or critical) at infinity ‘=o, and we perform a qualitative investigation in the large of these systems in the case where a and 5, , are real.”  ...  .; Lamnabhi, M. 84a:34011 Algebraic computation of the solution of some nonlinear differential equations. Computer algebra ( Marseille, 1982), pp. 204-211, Lecture Notes in Comput.  ... 

Lineal: A linear-algebraic Lambda-calculus [article]

Pablo Arrighi, Gilles Dowek
2017 arXiv   pre-print
We mention the perspectives of this work in the field of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus.  ...  We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra.  ...  Acknowledgments The authors would like to thank Alejandro Díaz-Caro, Evelyne Contejean, Philippe Jorrand, Jean-Pierre Jouannaud, Claude Marché, Simon Perdrix, Benoît Valiron and Lionel Vaux for some enlightening  ... 
arXiv:quant-ph/0612199v5 fatcat:f2gtq7b63bhcpnakhm5rde4764

Page 3212 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews  
Thus the calculus can be established without involving the unpleasant calculation of infinity.  ...  This view suggests the use of equational specifications of many- sorted algebras as a means for specifying data types.  ... 

User Centred Design in a Learning Environment Based on a Computer Algebra System [chapter]

Dominique Lenne, Jean-Michel Gélis
2002 Networking the Learner  
Though they have not been designed for this usage, computer algebra systems are increasingly used in leaming situations.  ...  We present the main features ofthis environment, focusing on three points: algebraic expressions manipulation, conjectures and proofs.  ...  That is the case with Computer Algebra Systems (CAS), such as Maple, Derive or Mathematica, which are increasingly used in mathematics teaching and learning.  ... 
doi:10.1007/978-0-387-35596-2_43 fatcat:5oouel6hnveshmsnmublik4yrm

Pure Condition of Both 6R Serial and 3-PRS Parallel Robots Using Grassmann-Cayley Algebra

Luc Djimon Clément Akonde, Alain Adomou, Tognon Clotilde Guidi
2018 Modern Mechanical Engineering  
In this work we require in Projective Space both Twist System (TS) and Global Wrench System (GWS) respectively for serial and parallel manipulators which represent the Jacobian Matrix (J) in symbolic approach  ...  This research presents Pure Condition approach, which has used in analyzing simultaneously the singularity configuration and the rigidity of mechanism.  ...  Acknowledgements Conflicts of Interest The authors declare no conflicts of interest regarding the publication of this paper.  ... 
doi:10.4236/mme.2018.84016 fatcat:tapicxj575ezrcsfspqzrzfmum


2005 Journal of symbolic computation  
. . . . . . 503 Beeson, M. and Wiedijk, F., The meaning of infinity in calculus and computer algebra systems .. . . . . . . . . . . . . . . . . 523 Gottliebsen, H., Kelsey, T. and Martin, U., Hidden  ...  , J.R., Computation of the topology of real algebraic space curves . . . . . . . . . . . . . . . . . . . 719 Volume 40 Number 1 Reduction Strategies in Rewriting and Programming special issue Gramlich  ... 
doi:10.1016/s0747-7171(05)00160-4 fatcat:a3hcsoc2orad3fh2g2mvgsxm6q

Grassmann's legacy [chapter]

David Hestenes
2010 From Past to Future: Graßmann's Work in Context  
(I use capitals to emphasize specific meanings for the terms 'geometric algebra and calculus' employed here.)  ...  I was a graduate student in 1959 intensively studying the alternative mathematical systems used in physics, including the Feynmann trace calculus in quantum electrodynamics and tensor calculus in general  ... 
doi:10.1007/978-3-0346-0405-5_22 fatcat:ikv66aq5svgcvjbob5uvz4nzz4

Euler's definition of the derivative

Harold M. Edwards
2007 Bulletin of the American Mathematical Society  
Rather, it calls for rewriting the difference quotient in a way that remains meaningful when the denominator is zero.  ...  Euler's method of defining the derivative of a function is not a failed effort to describe a limit.  ...  operations of calculus have real meaning for them.  ... 
doi:10.1090/s0273-0979-07-01174-3 fatcat:ndz6d5lfp5aafb3nlvaea2e6ja

Page 3739 of Mathematical Reviews Vol. , Issue 96f [page]

1996 Mathematical Reviews  
The basic idea of exception or specification as suggested in this pa- per means that a number of subsorts are introduced representing exceptional situations.  ...  They prove some fundamental results about their approach, such as initiality results and the soundness of the calculus.  ... 

Towards the introduction of QoS information in a component model

Sun Meng, Luis S. Barbosa
2010 Proceedings of the 2010 ACM Symposium on Applied Computing - SAC '10  
Assuring Quality of Service (QoS) properties is critical in the development of component-based distributed systems.  ...  Such constraints are formally captured through the concept of a Q-algebra which, in its turn, can be smoothly integrated in the definition of component combinators.  ...  This definition means that the computation of an action in a component will not simply produce an output and a continuation state, but a B-structure of such pairs.  ... 
doi:10.1145/1774088.1774517 dblp:conf/sac/MengB10 fatcat:tzd5t3pjyrdsfptz2upewuldsy

Semantics of a Typed Algebraic Lambda-Calculus

Benoît Valiron
2010 Electronic Proceedings in Theoretical Computer Science  
In this paper we propose a semantic analysis of a general simply-typed lambda-calculus endowed with a structure of vector space. We sketch the relation with two established vectorial lambda-calculi.  ...  Then we study the problems arising from the addition of a fixed point combinator and how to modify the equational theory to solve them.  ...  Acknowledgments I would like to thank Gilles Dowek for introducing me to algebraic calculi. I would also like to thank Pablo Arrighi and the research group CAPP in Grenoble for helpful discussions.  ... 
doi:10.4204/eptcs.26.14 fatcat:vvlq5y2tozbgvblqnzf3jglaaq
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