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

2002
*
Lecture Notes in Computer Science
*

We use filters

doi:10.1007/3-540-45470-5_23
fatcat:6pnliaqog5af3nbcn7wvzi4jfe
*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. ...##
###
The meaning of infinity in calculus and computer algebra systems

2005
*
Journal of symbolic computation
*

We use filters

doi:10.1016/j.jsc.2004.12.002
fatcat:xxvqu7fkynamvftd3gkk2qurqq
*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. ...##
###
"Region calculus" : generalization of Newton calculus

2019
*
International Journal of Mathematical Analysis
*

It is only

doi:10.12988/ijma.2019.9742
fatcat:3b7bv4sdhretfaibie23mmavea
*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. ...##
###
The review: Approaches in teaching calculus

2022
*
figshare.com
*

A few approaches used

doi:10.6084/m9.figshare.19009382.v1
fatcat:jpaym6omm5annaevm3u4n26mna
*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*...##
###
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]

2017
*
arXiv
*
pre-print

We mention

arXiv:quant-ph/0612199v5
fatcat:f2gtq7b63bhcpnakhm5rde4764
*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 ...##
###
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]

2002
*
Networking the Learner
*

Though they have not been designed for this usage,

doi:10.1007/978-0-387-35596-2_43
fatcat:5oouel6hnveshmsnmublik4yrm
*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. ...##
###
Pure Condition of Both 6R Serial and 3-PRS Parallel Robots Using Grassmann-Cayley Algebra

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. ...

##
###
Contents

2005
*
Journal of symbolic computation
*

. . . . . . 503 Beeson, M.

doi:10.1016/s0747-7171(05)00160-4
fatcat:a3hcsoc2orad3fh2g2mvgsxm6q
*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 ...##
###
Grassmann's legacy
[chapter]

2010
*
From Past to Future: Graßmann's Work in Context
*

(I use capitals to emphasize specific

doi:10.1007/978-3-0346-0405-5_22
fatcat:ikv66aq5svgcvjbob5uvz4nzz4
*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 ...##
###
Euler's definition of the derivative

2007
*
Bulletin of the American Mathematical Society
*

Rather, it calls for rewriting

doi:10.1090/s0273-0979-07-01174-3
fatcat:ndz6d5lfp5aafb3nlvaea2e6ja
*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. ...##
###
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

2010
*
Proceedings of the 2010 ACM Symposium on Applied Computing - SAC '10
*

Assuring Quality

doi:10.1145/1774088.1774517
dblp:conf/sac/MengB10
fatcat:tzd5t3pjyrdsfptz2upewuldsy
*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. ...##
###
Semantics of a Typed Algebraic Lambda-Calculus

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. ...

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