references.bib

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@online{Hanquez:2019wk,
	author = {Vincent Hanquez},
	date-added = {2021-03-21 14:38:07 +0900},
	date-modified = {2021-03-21 14:39:01 +0900},
	title = {\texttt{gauge}: small framework for performance measurement and analysis},
	url = {https://hackage.haskell.org/package/gauge},
	urldate = {2021-03-21},
	year = {2019},
	Bdsk-Url-1 = {https://hackage.haskell.org/package/gauge}}

@article{Karczmarczuk:2001ww,
	abstract = {We present a purely functional implementation of the computational differentiation tools---the well known numeric (i.e., not symbolic) techniques which permit one to compute point-wise derivatives of functions defined by computer programs economically and exactly (with machine precision). We show how the use of lazy evaluation permits a transparent and elegant construction of the entire infinite tower of derivatives of higher order for any expressions present in the program. The formalism may be useful in various problems of scientific computing which often demand a hard and ungracious human preprocessing before writing the final code. Some concrete examples are given.},
	author = {Karczmarczuk, Jerzy},
	da = {2001/03/01},
	date-added = {2021-03-11 16:42:15 +0900},
	date-modified = {2021-03-11 16:42:15 +0900},
	doi = {10.1023/A:1011501232197},
	id = {Karczmarczuk2001},
	isbn = {1573-0557},
	journal = {Higher-Order and Symbolic Computation},
	number = {1},
	pages = {35--57},
	title = {Functional Differentiation of Computer Programs},
	ty = {JOUR},
	url = {https://doi.org/10.1023/A:1011501232197},
	volume = {14},
	year = {2001},
	Bdsk-Url-1 = {https://doi.org/10.1023/A:1011501232197}}

@online{haskell.org:2021tt,
	author = {{haskell.org}},
	date-added = {2021-03-11 16:30:47 +0900},
	date-modified = {2021-03-11 16:31:24 +0900},
	title = {{Haskell} Language},
	url = {https://www.haskell.org},
	urldate = {2021-03-11},
	year = {2021},
	Bdsk-Url-1 = {https://www.haskell.org}}

@unpublished{Ishii:2021vw,
	author = {Hiromi Ishii},
	date-added = {2021-03-08 22:01:07 +0900},
	date-modified = {2021-03-08 22:01:44 +0900},
	note = {submitted},
	title = {Automatic Differentiation With Higher Infinitesimals, or Computational Smooth Infinitesimal Analysis in Weil Algebra},
	year = {2021}}

@inproceedings{Ishii:2021ab,
	author = {Hiromi Ishii},
	booksubtitle = {Theory and its Applications},
	booktitle = {Computer Algebra},
	date-modified = {2021-06-27 12:57:55 +0900},
	editor = {Masayo Fujimura},
	eprint = {2103.11615},
	eprinttype = {arXiv},
	issn = {1880-2818},
	pages = {104--112},
	publisher = {Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan},
	series = {{RIMS} {K}\^{o}ky\^{u}roku},
	title = {A Succinct Multivariate Lazy Multivariate Tower AD for Weil Algebra Computation},
	venue = {Research Institute for Mathematical Sciences, Kyoto, Japan},
	volume = {2185},
	year = {2021}}

@inproceedings{Elliott:2018aa,
	abstract = {Automatic differentiation (AD) in reverse mode (RAD) is a central component of deep learning and other uses of large-scale optimization. Commonly used RAD algorithms such as backpropagation, however, are complex and stateful, hindering deep understanding, improvement, and parallel execution. This paper develops a simple, generalized AD algorithm calculated from a simple, natural specification. The general algorithm is then specialized by varying the representation of derivatives. In particular, applying well-known constructions to a naive representation yields two RAD algorithms that are far simpler than previously known. In contrast to commonly used RAD implementations, the algorithms defined here involve no graphs, tapes, variables, partial derivatives, or mutation. They are inherently parallel-friendly, correct by construction, and usable directly from an existing programming language with no need for new data types or programming style, thanks to use of an AD-agnostic compiler plugin.},
	address = {New York, NY, USA},
	articleno = {70},
	author = {Elliott, Conal},
	booktitle = {Proceedings of the ACM on Programming Languages},
	date-added = {2021-01-23 18:12:47 +0900},
	date-modified = {2021-01-23 18:14:21 +0900},
	doi = {10.1145/3236765},
	issue_date = {September 2018},
	journal = {Proc. ACM Program. Lang.},
	keywords = {program calculation, category theory, automatic differentiation},
	month = jul,
	number = {ICFP},
	numpages = {29},
	publisher = {Association for Computing Machinery},
	title = {The Simple Essence of Automatic Differentiation},
	url = {https://doi.org/10.1145/3236765},
	volume = {2},
	year = {2018},
	Bdsk-Url-1 = {https://doi.org/10.1145/3236765}}

@online{Wikipedia:2021aa,
	author = {Wikipedia},
	date-added = {2021-01-12 21:43:28 +0900},
	date-modified = {2021-01-12 21:44:07 +0900},
	title = {Automatic Differentiation},
	url = {https://en.wikipedia.org/w/index.php?title=Automatic_differentiation&oldid=995938170#Automatic_differentiation_using_dual_numbers},
	urldate = {2021-01-12},
	year = {2021},
	Bdsk-Url-1 = {https://en.wikipedia.org/w/index.php?title=Automatic_differentiation&oldid=995938170#Automatic_differentiation_using_dual_numbers}}

@inproceedings{Pearlmutter:2007aa,
	author = {Pearlmutter, Barak and Siskind, Jeffrey},
	date-added = {2021-01-10 00:14:47 +0900},
	date-modified = {2021-01-10 00:15:23 +0900},
	doi = {10.1145/1190216.1190242},
	journal = {Conference Record of the Annual ACM Symposium on Principles of Programming Languages},
	month = {01},
	pages = {155-160},
	title = {Lazy multivariate higher-order forward-mode AD},
	volume = {42},
	year = {2007},
	Bdsk-Url-1 = {https://doi.org/10.1145/1190216.1190242}}

@misc{computational-algebra,
	author = {Hiromi Ishii},
	date-added = {2021-01-09 23:54:49 +0900},
	date-modified = {2021-01-09 23:54:49 +0900},
	keywords = {computaional algebra, software, algebra, computer algebra system},
	title = {computational-algebra},
	url = {https://konn.github.io/computational-algebra},
	year = {2013},
	Bdsk-Url-1 = {http://hackage.haskell.org/package/computational-algebra},
	Bdsk-Url-2 = {https://konn.github.io/computational-algebra}}

@inproceedings{ISHII:2018ek,
	author = {Hiromi Ishii},
	booktitle = {Computer Algebra in Scientific Computing},
	date-added = {2021-01-09 23:51:59 +0900},
	date-modified = {2021-01-09 23:51:59 +0900},
	doi = {10.1007/978-3-319-99639-4_20},
	editor = {Vladimir P. Gerdt and Wolfram Koepf and Werner M. Seiler},
	eprint = {1807.01456},
	eprinttype = {arXiv},
	isbn = {978-3-319-99638-7},
	pages = {288-303},
	publisher = {Springer, Cham},
	series = {Lecture Notes in Computer Science},
	title = {A Purely Functional Computer Algebra System Embedded in {H}askell},
	venue = {Lille, France},
	volume = {11077},
	year = {2018},
	Bdsk-Url-1 = {https://doi.org/10.1007/978-3-319-99639-4_20}}

@article{lawvere1979categorical,
	author = {Lawvere, F William},
	journal = {Topos theoretic methods in geometry},
	pages = {1--28},
	publisher = {Mathematical Institute, Aarhus University Various Publications},
	title = {Categorical dynamics},
	volume = {30},
	year = {1979}}

@book{Moerdijk:1991aa,
	author = {Ieke Moerdijk and Gonzalo E. Reyes},
	date-added = {2020-12-27 00:30:25 +0900},
	date-modified = {2020-12-27 00:35:25 +0900},
	doi = {10.1007/978-1-4757-4143-8},
	isbn = {978-1-4419-3095-8},
	publisher = {Springer-Verlag New York},
	title = {Models for Smooth Infinitesimal Analysis},
	year = {1991},
	Bdsk-Url-1 = {https://doi.org/10.1007/978-1-4757-4143-8}}

@book{CLO:2005,
	abstract = {This book is an introduction to {Gr{\"o}bner} bases and resultants, which are two of the main tools used in computational algebraic geometry and commutative algebra. It also discusses local methods and syzygies, and gives applications to integer programming, polynomial splines and algebraic coding theory.},
	author = {D. Cox and J. Little and D. O'Shea},
	date-added = {2014-01-13 05:22:31 +0000},
	date-modified = {2015-11-11 07:08:57 +0000},
	edition = {2nd},
	keywords = {algebraic geometry, computaional algebra, groebner basis, algorithms, algebra, 代数幾何 , 代数学, 計算機代数, アルゴリズム},
	language = {English},
	length = {558},
	publisher = {Springer},
	refereed = {1},
	title = {Using Algebraic Geometry},
	url = {http://www.cs.amherst.edu/~dac/uag.html},
	year = {2005},
	Bdsk-Url-1 = {http://www.cs.amherst.edu/~dac/uag.html}}

@misc{joyce2016algebraic,
	archiveprefix = {arXiv},
	author = {Dominic Joyce},
	eprint = {1001.0023},
	primaryclass = {math.AG},
	title = {Algebraic Geometry over $C^\infty$-rings},
	year = {2016}}

@inproceedings{Elliott2009-beautiful-differentiation,
	author = {Conal Elliott},
	booktitle = {International Conference on Functional Programming (ICFP)},
	title = {Beautiful differentiation},
	url = {http://conal.net/papers/beautiful-differentiation},
	year = 2009,
	Bdsk-Url-1 = {http://conal.net/papers/beautiful-differentiation}}

@online{Kmett:2010aa,
	author = {Edward A. Kmett},
	date-added = {2020-12-28 17:41:25 +0900},
	date-modified = {2020-12-28 17:42:01 +0900},
	title = {\texttt{ad}: Automatic Differentiation},
	url = {https://hackage.haskell.org/package/ad},
	urlyear = {2020},
	year = {2010},
	Bdsk-Url-1 = {http://hackage.haskell.org/package/algebra}}

@online{Ishii:2020aa,
	author = {Hiromi Ishii},
	date-added = {2020-12-28 17:44:05 +0900},
	date-modified = {2020-12-28 17:45:06 +0900},
	title = {\texttt{smooth}: Computational Smooth Infinitesimal Analysis},
	url = {https://github.com/konn/smooth},
	urldate = {2020-12-28},
	urlyear = {2020},
	year = {2020},
	Bdsk-Url-1 = {https://github.com/konn/smooth}}

@article{Nishimura:2007aa,
	abstract = {Synthetic differential geometry occupies a unique position in topos-theoretic physics. Nevertheless it has appeared somewhat too conceptual to physicists in general, partly because it has appeared to lack computational aspects. Its computational facets are really concerned with computation of the quasi-colimit of a finite diagram of infinitesimal spaces, or equivalently, with computation of the limit of a finite diagram of Weil algebras. Indeed we have been forced to do a highly involved computation of the above kind by hand in our previous papers (Nishimura, H. in Int. J. Theor. Phys. 36:1099--1131, 1997 and Nishimura, H. in Int. J. Theor. Phys. 38:2163--2174, 1999). The principal objective in this paper is to show that Gr{\"o}bner bases techniques provide us with means that relegate such computations to computers.},
	author = {Nishimura, Hirokazu and Osoekawa, Takeshi},
	da = {2007/11/01},
	date-added = {2020-12-28 17:49:22 +0900},
	date-modified = {2020-12-28 17:49:22 +0900},
	doi = {10.1007/s10773-007-9397-z},
	id = {Nishimura2007},
	isbn = {1572-9575},
	journal = {International Journal of Theoretical Physics},
	number = {11},
	pages = {2843--2862},
	title = {General Jacobi Identity Revisited Again},
	ty = {JOUR},
	url = {https://doi.org/10.1007/s10773-007-9397-z},
	volume = {46},
	year = {2007},
	Bdsk-Url-1 = {https://doi.org/10.1007/s10773-007-9397-z}}