path: root/documentation/research/math.md

# Prime number

1. [The History of the Primality of One: A Selection of Sources](https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf)

# Trigonometry

1. [Approximating the Sine Function](http://datagenetics.com/blog/july12019/index.html)

# Period

1. https://en.wikipedia.org/wiki/Ring_of_periods
1. [PERIODS](http://www.ihes.fr/~maxim/TEXTS/Periods.pdf)

# Proof theory

1. [Mathematical Components](https://math-comp.github.io/mcb/)

# Statistics

1. https://www.statisticsdonewrong.com/

# Topology

1. [What is Topology?](https://medium.com/cantors-paradise/what-is-topology-963ef4cc6365)
1. [Pointless topology](https://en.wikipedia.org/wiki/Pointless_topology)
1. [ELEMENTARY APPLIED TOPOLOGY](https://www.math.upenn.edu/~ghrist/notes.html)

# Library

1. https://github.com/ZacharyPatten/Towel

# Polynomials

1. https://brainbomb.org/0-1-efficient-metholds-for-evaluting-polynomials/
1. https://en.wikipedia.org/wiki/Horner%27s_method

# Chaos Theory

1. http://ncase.me/attractors/

# Number Theory

1. https://en.wikipedia.org/wiki/Dual_number
1. https://en.wikipedia.org/wiki/Division_algebra
1. [Division algebras](https://www.youtube.com/watch?v=3BZyds_KFWM&list=PLNxhIPHaOTRZMO1VjJcs7_3dgyJ2qU1yZ)
1. https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720
1. http://illustratedtheoryofnumbers.com/
1. [Topology of Numbers](http://pi.math.cornell.edu/~hatcher/TN/TNpage.html)

# Quaternions

1. [Dual Quaternions for Mere Mortals](https://www.jeremyong.com/math/2019/08/05/dual-quaternions-for-mere-mortals/)
1. [Stepping into a New Dimension: Using Quaternions to See the Invisible](https://medium.com/@vieyrasoftware/stepping-into-a-new-dimension-using-quaternions-to-see-the-invisible-478087c9ebbf)
1. [How Quaternions Produce 3D Rotation](https://penguinmaths.blogspot.com/2019/06/how-quaternions-produce-3d-rotation.html)
1. [APPLICATION OF DUAL QUATERNIONS ON SELECTED PROBLEMS](https://otik.uk.zcu.cz/bitstream/11025/28563/1/phd_Application%20of%20dual%20quaternions%20on%20selected%20problems.pdf)
1. https://www.3dgep.com/understanding-quaternions/
1. https://probablydance.com/2017/08/05/intuitive-quaternions/
1. [Quaternion algebras](https://math.dartmouth.edu/~jvoight/quat.html)

# _Compendium of resources_

1. [ALL IN ONE MATHEMATICS CHEAT SHEET](https://ourway.keybase.pub/mathematics_cheat_sheet.pdf)
1. https://github.com/llSourcell/learn_math_fast
1. https://www.algorithm-archive.org/
1. [3b1b featured creators #1](https://www.youtube.com/watch?v=VcgJro0sTiM)
1. [An infinite descent into pure mathematics](https://infinitedescent.xyz/)
1. https://hackernoon.com/be-a-better-programmer-with-these-40-mathematics-courses-d8ca48a2f8a2
1. https://www.intmath.com/
1. https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics

# Bézier curves

1. [Circles and lines vs. polynomial splines](https://wordsandbuttons.online/circles_and_lines_vs_polynomial_splines.html)
1. https://cormullion.github.io/blog/2018/06/21/bezier.html
1. https://pomax.github.io/bezierinfo/
1. https://99designs.com/blog/engineering/math-draw-vector-curves/
1. https://medium.freecodecamp.org/nerding-out-with-bezier-curves-6e3c0bc48e2f
1. https://pomax.github.io/bezierinfo/#curvature
1. https://raphlinus.github.io/curves/2018/12/28/bezier-arclength.html
1. https://astiopin.github.io/2019/01/04/qbez-parabola.html
1. http://dlacko.org/blog/2016/10/19/approximating-bezier-curves-by-biarcs/
1. https://bosker.wordpress.com/2013/11/13/beyond-bezier-curves/

# Number systems

1. https://en.wikipedia.org/wiki/Negative_base

# Discrete mathematics

1. [Notes on Discrete Mathematics](http://www.cs.yale.edu/homes/aspnes/classes/202/notes.pdf)
1. [The system of integer functions, an efficient version of discrete mathematical analysis](https://arxiv.org/abs/1710.00676)
1. [Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra](http://math.sfsu.edu/beck/papers/ccd.pdf)
1. [Discrete Mathematics: An Open Introduction](http://discrete.openmathbooks.org/dmoi2/frontmatter.html)

# Linear Algebra

1. [Don’t invert that matrix](https://www.johndcook.com/blog/2010/01/19/dont-invert-that-matrix/)
1. [“Don’t invert that matrix” – why and how](https://civilstat.com/2015/07/dont-invert-that-matrix-why-and-how/)
1. [Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra](https://kam.mff.cuni.cz/~matousek/stml-53-matousek-1.pdf)
1. [Covectors are Different than Vectors, Usually](http://infohost.nmt.edu/~parendt/Phys509/notes/2diracnot/node4.html)
1. [Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares](https://web.stanford.edu/~boyd/vmls/vmls.pdf)
1. [Linear Algebra Done Wrong.](http://www.math.brown.edu/~treil/papers/LADW/LADW.html)
1. [Linear Algebra Done Right](http://linear.axler.net/)
1. [Down with Determinants!](http://www.axler.net/DwD.html)
1. https://github.com/jacobbogers/blasjs
1. https://graphicallinearalgebra.net/
1. [Wild Linear Algebra](https://www.youtube.com/watch?v=yAb12PWrhV0&list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp &&& https://www.youtube.com/watch?v=nJdWFoliRnc&list=PLIljB45xT85DF5G-Hq0xPsVUnR5_wJRSw)
1. https://github.com/ferreiradaselva/mathc
1. http://www.fast.ai/2017/07/17/num-lin-alg/
1. https://github.com/bluemathsoft/bm-linalg
1. [The Matrix Cookbook](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf)
1. https://dhruvp.netlify.com/2018/12/31/matrices/
1. https://www.nalgebra.org/
1. http://immersivemath.com/ila/index.html
1. [Linear Logic and Linear Algebra](http://www.cs.bham.ac.uk/~drg/bll/steve.pdf)
1. https://www.math3ma.com/blog/matrices-probability-graphs
1. [Essence of linear algebra](https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)
1. https://www.dhruvonmath.com/2018/12/31/matrices/
1. http://blog.ezyang.com/2019/05/a-short-note-about-functional-linear-maps/
1. https://www.math3ma.com/blog/matrices-as-tensor-network-diagrams
1. [Convolution is outer product](https://arxiv.org/abs/1905.01289)
1. [Graphical Calculus for products and convolutions](https://arxiv.org/abs/1903.01366)

# Category Theory

1. [Categorical Query Language](https://www.categoricaldata.net/)
1. [Abstract and Concrete Categories: The Joy of Cats](http://katmat.math.uni-bremen.de/acc/acc.pdf)
1. https://bartoszmilewski.com/
1. [YOW! Lambda Jam 2018 - Mark Hopkins -All of Basic Category Theory](https://www.youtube.com/watch?v=9u1q3lH7VbE)
1. [Categorifying cardinal arithmetic](http://www.math.jhu.edu/~eriehl/arithmetic.pdf)
1. https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/
1. [Elements of ∞-Category Theory](http://www.math.jhu.edu/~eriehl/elements.pdf)
1. [Category Theory: A Gentle Introduction](https://www.logicmatters.net/resources/pdfs/GentleIntro.pdf)
1. [Applied Category Theory (@ MIT 2019)](https://www.youtube.com/playlist?list=PLhgq-BqyZ7i5lOqOqqRiS0U5SwTmPpHQ5)
1. https://jvns.ca/blog/2019/03/26/what-are-monoidal-categories/
1. [Daniela Sfregola - A Pragmatic Introduction to Category Theories - Code Mesh 2017](https://www.youtube.com/watch?v=8XGFFMPHG0o)
1. [Seven Sketches in Compositionality: An Invitation to Applied Category Theory](http://xahlee.info/math/i/category_theory_brendan_fong_david_spivak_2018-03.pdf)

# Direct Logic

1. [Strong Types for Direct Logic](https://drive.google.com/file/d/1relKFB4aOMD2VZHM9qj3wWMXOq-q2AJr/view)

# Geometric Algebra | Clifford Algebra

1. [Geometric Algebra for Computer Graphics](https://slides.com/enkimute/siggraph/#/)
1. https://bivector.net/
1. https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/
1. [Vectors and Beyond: Geometric Algebra and its Philosophical Significance](http://www.tara.tcd.ie/bitstream/handle/2262/61825/Vectors%20and%20Beyond%20as%20Printed.pdf?sequence=1&isAllowed=y)
1. [Let's remove Quaternions from every 3D Engine](http://marctenbosch.com/quaternions/)
1. [Geometric Algebra Playlist](https://www.youtube.com/playlist?list=PLLvlxwbzkr7igd6bL7959WWE7XInCCevt)
1. https://github.com/wolftype/versor
1. [Exploring Physics with Geometric Algebra, Book 1](http://peeterjoot.com/archives/math2015/gabookI.pdf)
1. [Joan Lasenby on Applications of Geometric Algebra in Engineering](https://www.youtube.com/watch?v=ikCIUzX9myY)
1. https://www.amazon.com/Geometric-Algebra-Computer-Graphics-Vince/dp/1846289963
1. https://www.amazon.com/Geometric-Algebra-Computer-Science-Revised/dp/0123749425
1. https://www.amazon.com/Space-Time-Algebra-David-Hestenes/dp/3319184121
1. https://www.amazon.com/Introduction-Clifford-Algebras-Spinors/dp/0198782926
1. https://gist.github.com/pervognsen/b204bf31af0809af078617af1609c8b8
1. [Fundamentals of Grassmann Algebra](http://www.terathon.com/gdc12_lengyel.pdf)
1. https://github.com/chakravala/Grassmann.jl

# Abstract Algebra

1. [A Brief Guide to A Few Algebraic Structures](https://argumatronic.com//posts/2019-06-21-algebra-cheatsheet.html)
1. [DUALITY FOR GROUPS](https://projecteuclid.org/download/pdf_1/euclid.bams/1183515045)
1. https://gowers.wordpress.com/2011/11/20/normal-subgroups-and-quotient-groups/
1. [Abstract Algebra: Theory and Applications](http://abstract.ups.edu/)
1. [Group Theory](http://birdtracks.eu/version9.0/GroupTheory.pdf)
1. [VISUAL GROUP THEORY](http://web.bentley.edu/empl/c/ncarter/vgt/index.html)
1. [THE GEOMETRY OF REFLECTION GROUPS](http://people.mpim-bonn.mpg.de/geordie/mpg.pdf)
1. https://kubuszok.com/2018/algebras-we-love/
1. [Notes on Computational Group Theory](https://www.math.colostate.edu/~hulpke/CGT/cgtnotes.pdf)

# Logic

1. http://openlogicproject.org/

# Calculus

1. [Calculus on Manifolds](http://strangebeautiful.com/other-texts/spivak-calc-manifolds.pdf)
1. [Calculus BLUE](https://www.youtube.com/watch?v=Jes5jwLl1q8&list=PL8erL0pXF3JYm7VaTdKDaWc8Q3FuP8Sa7)
1. [Extending the Algebraic Manipulability of Differentials](https://arxiv.org/abs/1801.09553)
1. https://en.wikipedia.org/wiki/Fractional_calculus
1. https://medium.com/@notaredpanda/fractional-calculus-48192f4e9c9f
1. [Fractional Calculus: Theory and Applications](https://www.mdpi.com/books/pdfview/book/755)
1. [EXTENDING THE ALGEBRAIC MANIPULABILITYOF DIFFERENTIALS](http://online.watsci.org/abstract_pdf/2019v26/v26n3a-pdf/4.pdf)
1. [Finite Calculus: A Tutorial for Solving Nasty Sums](https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20finite%20calculus.pdf)
1. http://jliszka.github.io/2013/10/24/exact-numeric-nth-derivatives.html

# Octonion

1. http://math.ucr.edu/home/baez/octonions/

# Differential Geometry

1. [differential geometry reconstructed: a unified systematic framework](http://www.geometry.org/tex/conc/dgstats.php)
1. [Differential geometry reconstructed: a unified systematic framework](http://www.topology.org/tex/conc/dg.html)

# Information Theory

1. [Functional Bits: Lambda Calculus based Algorithmic Information Theory](https://tromp.github.io/cl/LC.pdf)

# Graph Theory

1. [Graphs and Geometry](http://web.cs.elte.hu/~lovasz/bookxx/geomgraphbook/geombook2019.01.11.pdf)
1. [Graph-Based Algorithms for Boolean Function Manipulation](http://mtv.ece.ucsb.edu/courses/ece156B_14/randy_obdd86.pdf)
1. [better geometry through graph theory](http://ideolalia.com/2018/08/28/artifex.html)

# Division by Zero

1. https://www.hillelwayne.com/post/divide-by-zero/
1. https://www.1dividedby0.com/

# Partial Fractions

1. https://jaydaigle.net/blog/calculus/easier-partial-fractions/

# Non-commutative Algebra

1. [An Invitation to Noncommutative Algebra](https://arxiv.org/abs/1808.03172)

# Lambda Calculus

1. https://medium.com/@maiavictor/the-abstract-calculus-fe8c46bcf39c

# _Software Library_

1. https://github.com/arguiot/TheoremJS
1. https://github.com/arguiot/Descartes

# Set Theory

1. [Set Theory and Algebra in Computer Science: A Gentle Introduction to Mathematical Modeling](https://pdfs.semanticscholar.org/d106/6b6de601c1d7d5af25af3f7091bc7ad3ad51.pdf)

# Uncertainty Theory

1. http://faculty.math.tsinghua.edu.cn/~bliu/ut.pdf
1. [An Introduction to the Mathematics of Uncertainty: including Set Theory, Logic, Probability, Fuzzy Sets, Rough Sets, and Evidence Theory](https://www.creighton.edu/fileadmin/user/CCAS/programs/fuzzy_math/docs/MOU.pdf)

# Computational Geometry

1. [CMPT 813: Computational Geometry (Fall 2011)](http://www.cs.sfu.ca/~binay/813.2011/)
1. https://www.johndcook.com/blog/2018/09/26/polygon-area/
1. http://s2geometry.io/devguide/s2cell_hierarchy.html
1. http://alienryderflex.com/polygon/
1. https://libigl.github.io/

# Finitism

1. ["REAL" ANALYSIS Is A DEGENERATE CASE of DISCRETE ANALYSIS](http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf)

# Linear programming

1. https://en.wikibooks.org/wiki/GLPK
1. https://white.ucc.asn.au/2018/05/28/Optimizing-your-diet-with-JuMP.html

# Combinatorics

1. http://andy.kitchen/combinations.html
1. [Combinatorial Algorithms](http://www2.denizyuret.com/bib/kreher/donald1999combinatorial/combinatorialA.pdf)

# Vector

1. [A Student's Guide to Vectors and Tensors (Student's Guides)](https://www.amazon.com/Students-Guide-Vectors-Tensors-Guides/dp/0521171903)
1. https://www.researchgate.net/publication/327989714_Coordinate_Free_Vector_Algebra_in_R2

# Knot theory

1. [The Knot Book](http://math.harvard.edu/~ctm/home/text/books/adams/knot_book/knot_book.pdf)

# Axiom

1. [INTRODUCTION TO AXIOMATIC REASONING](http://www.math.harvard.edu/~mazur/papers/Axiomatic.pdf)

# Differential equations

1. [ELEMENTARY DIFFERENTIAL EQUATIONS](http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_FREE_DIFFEQ_I.PDF)
1. [Lecture Notes on Difference Equations](http://people.math.aau.dk/~matarne/11-imat/notes2011a.pdf)

# Algebraic geometry

1. http://nautil.us/issue/69/patterns/the-math-that-takes-newton-into-the-quantum-world
1. [Algebraic Geometry](http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2014/main.pdf)
1. https://en.wikipedia.org/wiki/Superformula

# Physics

1. https://www.researchgate.net/publication/309533409_Beyond_Einstein_non-local_physics
1. https://www.theoretical-physics.com/dev/index.html

# Spherical trigonometry

1. https://en.wikipedia.org/wiki/Haversine_formula

# Hyperbolic geometry

1. [Hyperbolic Functions and Non-Hyperbolic Claims](https://elliptigon.com/hyperbolic-functions-explained/)

# **Temp Cache**

1. [Quadratic splines are useful too](https://wordsandbuttons.online/quadric_splines_are_useful_too.html)
1. [Derivations of Applied Mathematics](http://www.derivations.org/)
1. https://www.juliahomotopycontinuation.org/
1. https://marginalrevolution.com/marginalrevolution/2018/05/one-parameter-equation-can-exactly-fit-scatter-plot.html
1. https://plato.stanford.edu/entries/mathematics-inconsistent/
1. http://immersivemath.com/ila/index.html?m=1
1. https://stacks.math.columbia.edu/
1. http://www.gutenberg.org/ebooks/17384
1. http://illustratedtheoryofnumbers.com/
1. [Construction of the Transreal Numbers and Algebraic Transfields](http://www.iaeng.org/IJAM/issues_v46/issue_1/IJAM_46_1_03.pdf)
1. http://simit-lang.org/language
1. http://wykobi.com/tutorial0.html
1. https://pron.github.io/computation-logic-algebra
1. https://github.com/rossant/awesome-math
1. [Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Engineering](https://www.cis.upenn.edu/~jean/math-basics.pdf)
1. http://math.chapman.edu/~jipsen/structures/doku.php/
1. http://stacks.math.columbia.edu/browse
1. https://bitworking.org/news/ga/2d
1. https://www.youtube.com/watch?v=OmJ-4B-mS-Y
1. https://www.youtube.com/watch?v=ZihywtixUYo
1. https://www.youtube.com/user/njwildberger
1. http://www.hedonisticlearning.com/posts/category-theory-syntactically.html
1. https://www.youtube.com/watch?v=EA1FFtTGAfw&list=PLSekr_gm4hWLRNHB5-sbajyUzN5tLOZmx
1. https://www.artofproblemsolving.com/wiki/index.php/Modular_arithmetic/Introduction
1. [Mathematics for Computer Science](https://courses.csail.mit.edu/6.042/spring17/mcs.pdf)
1. http://www.paultaylor.eu/ASD/foufct/
1. [Toward the Unification of Physics and Number Theory](http://www.quantumgravityresearch.org/wp-content/uploads/2017/02/Toward-the-Unification-of-Physics2-1.pdf)
1. https://jeremykun.com/2014/03/13/programming-with-finite-fields/
1. https://courses.cs.washington.edu/courses/cse590b/13au/
1. http://magma.maths.usyd.edu.au/magma/
1. [A VIEW OF MATHEMATICS](http://www.alainconnes.org/docs/maths.pdf)
1. [Introduction to Tensor Analysis and the Calculus of Moving Surfaces](https://www.amazon.com/gp/product/1461478669)
1. https://en.wikipedia.org/wiki/Wheel_theory
1. http://wordsandbuttons.online/interactive_guide_to_homogeneous_coordinates.html
1. https://a-guide-to-the-purescript-numeric-hierarchy.readthedocs.io/en/latest/index.html
1. http://blog.wolfram.com/2016/12/22/the-semantic-representation-of-pure-mathematics/
1. https://www.youtube.com/watch?v=psSyM1zp82k
1. [Viviane Pons - Experimental pure mathematics using Sage - PyCon 2015](https://www.youtube.com/watch?v=3LZiZKgVjaU)
1. http://www.sagemath.org/
1. [The axioms of constructive geometry](https://www.sciencedirect.com/science/article/pii/0168007295000052)
1. https://www.quora.com/What-is-it-like-to-understand-advanced-mathematics-Does-it-feel-analogous-to-having-mastery-of-another-language-like-in-programming-or-linguistics/answers/873950?share=1
1. http://www.tricki.org/tricki/map
1. https://maxow.github.io/posts/computational-geometry-set-operations-on-polytopes.html
1. http://blog.robertelder.org/fast-multiplication-using-fourier-transform/
1. [Graphical Models in a Nutshell](https://ai.stanford.edu/~koller/Papers/Koller+al:SRL07.pdf)