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EMBÛCHES TISSUES
Cardboard dodecahedron. I made a cardboard dodecahedron for the needs of a talk. If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetricgroup
REVIEW | EMBÛCHES TISSUES Posts about review written by remyoudompheng. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized MACAULAY | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library WINDOWS | EMBÛCHES TISSUES Posts about Windows written by remyoudompheng. Creating executables for Windows from a Linux box. When writing a program, especially a graphical one, you may feel some compassion towards Windows users and wonder whether it would be possible to give them a chance of using it. FFT | EMBÛCHES TISSUES Posts about FFT written by remyoudompheng. Why can we have a Fast Fourier Transform ? The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of timeseach second.
MATHKABA | EMBÛCHES TISSUES Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. QED | EMBÛCHES TISSUES Saying QED in different languages. I was wondering whether it would be easy to know how to say “QED” in whatever language I could think of Well, it seems (not so unexpectedly after all) that Wikipedia was the right tool to use : so starting from the QED article (not the quantum electrodynamics one !) of the English Wikipedia, the language list allows to switch to the version of the same FEBRUARY | 2010 | EMBÛCHES TISSUES 1 post published by remyoudompheng during February 2010. Changing default applications, for developers. When not using those uber-featured desktop environments (such as KDE or Gnome) out there, it can sometimes be challenging to find a simple, text-based way ofconfiguring things.
HASKELL | EMBÛCHES TISSUES Monads in mathematics 3 : monads from adjunctions. The adjunction property between two functors, and , says that there is a natural bijection between morphisms (in the first category) and (in the second category). Here natural means that these bijection is compatible with composition with morphisms , or and .. Adjunctions are naturally created by the use of monads or operads. APRIL | 2009 | EMBÛCHES TISSUES The tetrahedra have faithful action of the isometry group: there are two sets of five tetrahedron, which are exchanged under signature -1 transformations, and even permutations of the tetrahedra correspond to direct isometries. 6 April 2009 at 12:01 am Leave a comment.EMBÛCHES TISSUES
Cardboard dodecahedron. I made a cardboard dodecahedron for the needs of a talk. If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetricgroup
REVIEW | EMBÛCHES TISSUES Posts about review written by remyoudompheng. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized MACAULAY | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library WINDOWS | EMBÛCHES TISSUES Posts about Windows written by remyoudompheng. Creating executables for Windows from a Linux box. When writing a program, especially a graphical one, you may feel some compassion towards Windows users and wonder whether it would be possible to give them a chance of using it. FFT | EMBÛCHES TISSUES Posts about FFT written by remyoudompheng. Why can we have a Fast Fourier Transform ? The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of timeseach second.
MATHKABA | EMBÛCHES TISSUES Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. QED | EMBÛCHES TISSUES Saying QED in different languages. I was wondering whether it would be easy to know how to say “QED” in whatever language I could think of Well, it seems (not so unexpectedly after all) that Wikipedia was the right tool to use : so starting from the QED article (not the quantum electrodynamics one !) of the English Wikipedia, the language list allows to switch to the version of the same FEBRUARY | 2010 | EMBÛCHES TISSUES 1 post published by remyoudompheng during February 2010. Changing default applications, for developers. When not using those uber-featured desktop environments (such as KDE or Gnome) out there, it can sometimes be challenging to find a simple, text-based way ofconfiguring things.
HASKELL | EMBÛCHES TISSUES Monads in mathematics 3 : monads from adjunctions. The adjunction property between two functors, and , says that there is a natural bijection between morphisms (in the first category) and (in the second category). Here natural means that these bijection is compatible with composition with morphisms , or and .. Adjunctions are naturally created by the use of monads or operads. APRIL | 2009 | EMBÛCHES TISSUES The tetrahedra have faithful action of the isometry group: there are two sets of five tetrahedron, which are exchanged under signature -1 transformations, and even permutations of the tetrahedra correspond to direct isometries. 6 April 2009 at 12:01 am Leave a comment.PROJECTIVE GEOMETRY
Lines in space and the Grassmann variety. This is an English version of my previous post. Schubert calculus is a collection of techniques and formulae used for computations of enumerative, numerical properties of common objects in linear algebra (there are very nice books by William Fulton covering the subject). The example of the set of lines in three-dimensional space is commonly chosen to MACAULAY | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library CARDBOARD ASSOCIAHEDRON After the dodecahedron comes the cardboard associahedron : it represents the combinatorics of moving parentheses to calculate an associative product of five things step by step. The vertices of the associahedron are thus the 14 different binary trees with 5 leaves. Given four factors, there are exactly five ways of multiplying themusing a binary
LOCAL | EMBÛCHES TISSUES Posts about local written by remyoudompheng. Extensions of sheaves and the local-global spectral sequence. GIven a topological space , algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. WINDOWS | EMBÛCHES TISSUES Posts about Windows written by remyoudompheng. Creating executables for Windows from a Linux box. When writing a program, especially a graphical one, you may feel some compassion towards Windows users and wonder whether it would be possible to give them a chance of using it. BOOK | EMBÛCHES TISSUES Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized version of the book (see there).PRIME IDEAL
Posts about prime ideal written by remyoudompheng. Schemes in algebraic geometry 2 : prime spectra and generic points. I just explained how the affine plane could be described by the ring .A point M of the affine plane whose coordinate ring is R is a morphism defined by the assignment , where are the coordinates of M.In the case of points corresponding to morphisms , there is a natural way of PLÜCKER COORDINATES Lines in space and the Grassmann variety. This is an English version of my previous post. Schubert calculus is a collection of techniques and formulae used for computations of enumerative, numerical properties of common objects in linear algebra (there are very nice books by William Fulton covering the subject). The example of the set of lines in three-dimensional space is commonly chosen to QED | EMBÛCHES TISSUES Saying QED in different languages. I was wondering whether it would be easy to know how to say “QED” in whatever language I could think of Well, it seems (not so unexpectedly after all) that Wikipedia was the right tool to use : so starting from the QED article (not the quantum electrodynamics one !) of the English Wikipedia, the language list allows to switch to the version of the sameEMBÛCHES TISSUES
I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name ofthe author.
EMBÛCHES TISSUES
Cardboard dodecahedron. I made a cardboard dodecahedron for the needs of a talk. If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetricgroup
REVIEW | EMBÛCHES TISSUES Posts about review written by remyoudompheng. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized LOGIC | EMBÛCHES TISSUES Laver tables and “unprovable” statements. Laver tables are combinatorial objets whose definition is surprisingly simple. However, they do have somewhat weird properties, which come from their relationship with questions of set theorists (namely, how do elementary embeddings of SOFTWARE | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library CONIC | EMBÛCHES TISSUES Bicircular quartic curves. Working in the Euclidean (projective) plane, a bicircular quartic curve is defined to be a quartic which is singular at the circular points I and J. EXTENSIONS | EMBÛCHES TISSUES Extensions of sheaves and the local-global spectral sequence. GIven a topological space , algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. VARIETY | EMBÛCHES TISSUES Posts about variety written by remyoudompheng. Ten constructions of the cohomology of varieties. When talking about “the” cohomology of mathematical objects, we do not always explicitly mention which cohomology is used, because it is obvious (in cases there is only one possible definition), or because we really don’t care (since as we will see, it is frequent that different definitions GERBES | EMBÛCHES TISSUES Posts about gerbes written by remyoudompheng. Extensions of sheaves and the local-global spectral sequence. GIven a topological space , algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. QUADRICS | EMBÛCHES TISSUES Lines in space and the Grassmann variety. This is an English version of my previous post. Schubert calculus is a collection of techniques and formulae used for computations of enumerative, numerical properties of common objects in linear algebra (there are very nice books by William Fulton covering the subject). The example of the set of lines in three-dimensional space is commonly chosen toEMBÛCHES TISSUES
I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name ofthe author.
EMBÛCHES TISSUES
Cardboard dodecahedron. I made a cardboard dodecahedron for the needs of a talk. If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetricgroup
REVIEW | EMBÛCHES TISSUES Posts about review written by remyoudompheng. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized LOGIC | EMBÛCHES TISSUES Laver tables and “unprovable” statements. Laver tables are combinatorial objets whose definition is surprisingly simple. However, they do have somewhat weird properties, which come from their relationship with questions of set theorists (namely, how do elementary embeddings of SOFTWARE | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library CONIC | EMBÛCHES TISSUES Bicircular quartic curves. Working in the Euclidean (projective) plane, a bicircular quartic curve is defined to be a quartic which is singular at the circular points I and J. EXTENSIONS | EMBÛCHES TISSUES Extensions of sheaves and the local-global spectral sequence. GIven a topological space , algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. VARIETY | EMBÛCHES TISSUES Posts about variety written by remyoudompheng. Ten constructions of the cohomology of varieties. When talking about “the” cohomology of mathematical objects, we do not always explicitly mention which cohomology is used, because it is obvious (in cases there is only one possible definition), or because we really don’t care (since as we will see, it is frequent that different definitions GERBES | EMBÛCHES TISSUES Posts about gerbes written by remyoudompheng. Extensions of sheaves and the local-global spectral sequence. GIven a topological space , algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. QUADRICS | EMBÛCHES TISSUES Lines in space and the Grassmann variety. This is an English version of my previous post. Schubert calculus is a collection of techniques and formulae used for computations of enumerative, numerical properties of common objects in linear algebra (there are very nice books by William Fulton covering the subject). The example of the set of lines in three-dimensional space is commonly chosen toEMBÛCHES TISSUES
I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name ofthe author.
UNDERGRADUATE
Posts about undergraduate written by remyoudompheng. Why can we have a Fast Fourier Transform ? The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions oftimes each second.
PROJECTIVE GEOMETRY
Lines in space and the Grassmann variety. This is an English version of my previous post. Schubert calculus is a collection of techniques and formulae used for computations of enumerative, numerical properties of common objects in linear algebra (there are very nice books by William Fulton covering the subject). The example of the set of lines in three-dimensional space is commonly chosen to MACAULAY | EMBÛCHES TISSUES Posts about Macaulay written by remyoudompheng. Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians. SOFTWARE | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library LOGIC | EMBÛCHES TISSUES Laver tables and “unprovable” statements. Laver tables are combinatorial objets whose definition is surprisingly simple. However, they do have somewhat weird properties, which come from their relationship with questions of set theorists (namely, how do elementary embeddings of MANIFOLD | EMBÛCHES TISSUES Posts about manifold written by remyoudompheng. Ten constructions of the cohomology of varieties. When talking about “the” cohomology of mathematical objects, we do not always explicitly mention which cohomology is used, because it is obvious (in cases there is only one possible definition), or because we really don’t care (since as we will see, it is frequent that different definitionsCOMMUTATIVE ALGEBRA
Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library SYSADMIN | EMBÛCHES TISSUES Posts about sysadmin written by remyoudompheng. Changing default applications, for developers. When not using those uber-featured desktop environments (such as KDE or Gnome) out there, it can sometimes be challenging to find a simple, text-based way ofconfiguring things.
CIRCLE | EMBÛCHES TISSUES Inversion and n-circular curves. Inversion is a transformation of the Euclidean plane closely related to circles. It is seldom taught in high schools (at least in France), but it has special interest when dealing with circles, since it maps circles and lines to circles orlines.
ELLIPSE | EMBÛCHES TISSUESTRANSLATE THIS PAGE Intégrales elliptiques et moyenne arithmético-géométrique. Le calcul de la moyenne arithmético-géométrique est un algorithme simple et très puissant découvert par Gauss permettant de calculer les intégrales elliptiques.L’excellent article de David Cox dans L’enseignement mathématique expose l’historique des découvertes et recherches de Gauss sur le sujet.EMBÛCHES TISSUES
Because mathematics often look like a web of logs. Elliptic curves for high school students. I had to give a talk to high school students about some mathematical notion: I decided to tell them something about elliptic curves, but not the usual speech about cryptography, finite fields and the group law on REVIEW | EMBÛCHES TISSUES Posts about review written by remyoudompheng. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized MACAULAY | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library WINDOWS | EMBÛCHES TISSUES Posts about Windows written by remyoudompheng. Creating executables for Windows from a Linux box. When writing a program, especially a graphical one, you may feel some compassion towards Windows users and wonder whether it would be possible to give them a chance of using it. FFT | EMBÛCHES TISSUES Posts about FFT written by remyoudompheng. Why can we have a Fast Fourier Transform ? The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of timeseach second.
MATHKABA | EMBÛCHES TISSUES Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. QED | EMBÛCHES TISSUES Saying QED in different languages. I was wondering whether it would be easy to know how to say “QED” in whatever language I could think of Well, it seems (not so unexpectedly after all) that Wikipedia was the right tool to use : so starting from the QED article (not the quantum electrodynamics one !) of the English Wikipedia, the language list allows to switch to the version of the same HASKELL | EMBÛCHES TISSUES Monads in mathematics 3 : monads from adjunctions. The adjunction property between two functors, and , says that there is a natural bijection between morphisms (in the first category) and (in the second category). Here natural means that these bijection is compatible with composition with morphisms , or and .. Adjunctions are naturally created by the use of monads or operads. APRIL | 2009 | EMBÛCHES TISSUES 2 posts published by remyoudompheng during April 2009. Cardboard associahedron. After the dodecahedron comes the cardboard associahedron : it represents the combinatorics of moving parentheses to calculate an associative product of five things step by step. FEBRUARY | 2010 | EMBÛCHES TISSUES 1 post published by remyoudompheng during February 2010. Changing default applications, for developers. When not using those uber-featured desktop environments (such as KDE or Gnome) out there, it can sometimes be challenging to find a simple, text-based way ofconfiguring things.
EMBÛCHES TISSUES
Because mathematics often look like a web of logs. Elliptic curves for high school students. I had to give a talk to high school students about some mathematical notion: I decided to tell them something about elliptic curves, but not the usual speech about cryptography, finite fields and the group law on REVIEW | EMBÛCHES TISSUES Posts about review written by remyoudompheng. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized MACAULAY | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library WINDOWS | EMBÛCHES TISSUES Posts about Windows written by remyoudompheng. Creating executables for Windows from a Linux box. When writing a program, especially a graphical one, you may feel some compassion towards Windows users and wonder whether it would be possible to give them a chance of using it. FFT | EMBÛCHES TISSUES Posts about FFT written by remyoudompheng. Why can we have a Fast Fourier Transform ? The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of timeseach second.
MATHKABA | EMBÛCHES TISSUES Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. QED | EMBÛCHES TISSUES Saying QED in different languages. I was wondering whether it would be easy to know how to say “QED” in whatever language I could think of Well, it seems (not so unexpectedly after all) that Wikipedia was the right tool to use : so starting from the QED article (not the quantum electrodynamics one !) of the English Wikipedia, the language list allows to switch to the version of the same HASKELL | EMBÛCHES TISSUES Monads in mathematics 3 : monads from adjunctions. The adjunction property between two functors, and , says that there is a natural bijection between morphisms (in the first category) and (in the second category). Here natural means that these bijection is compatible with composition with morphisms , or and .. Adjunctions are naturally created by the use of monads or operads. APRIL | 2009 | EMBÛCHES TISSUES 2 posts published by remyoudompheng during April 2009. Cardboard associahedron. After the dodecahedron comes the cardboard associahedron : it represents the combinatorics of moving parentheses to calculate an associative product of five things step by step. FEBRUARY | 2010 | EMBÛCHES TISSUES 1 post published by remyoudompheng during February 2010. Changing default applications, for developers. When not using those uber-featured desktop environments (such as KDE or Gnome) out there, it can sometimes be challenging to find a simple, text-based way ofconfiguring things.
MONADS IN MATHEMATICS Monads in mathematics 3 : monads from adjunctions. The adjunction property between two functors, and , says that there is a natural bijection between morphisms (in the first category) and (in the second category). Here natural means that these bijection is compatible with composition with morphisms , or and .. Adjunctions are naturally created by the use of monads or operads. BICIRCULAR QUARTIC CURVES Working in the Euclidean (projective) plane, a bicircular quartic curve is defined to be a quartic which is singular at the circular points I and J. We are usually interested in real curves, so the type of singulaity is the same at I or J. Salmon in his Treatise on Higher Plane Curves and Basset EXTENSIONS OF SHEAVES AND THE LOCAL-GLOBAL SPECTRAL Extensions of sheaves and the local-global spectral sequence. 10 January 2009 at 5:02 pm Leave a comment. GIven a topological space , algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. MONADS IN MATHEMATICS 4 : THE BAR AND COBAR CONSTRUCTIONS Monads in mathematics 4 : the bar and cobar constructions. 4 March 2009 at 10:39 pm Leave a comment. The use of monads and comonads in homological algebra is as old as the theory: Godement’s standard construction refers to the use of monads in cohomology theory, and is said to be the first study of a general method for constructing acyclic resolutions. EXPERIMENTAL ALGEBRAIC GEOMETRY I : THE GRASSMANNIAN I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let's try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians. Macaulay provides a command-line interface using the readline library (like many other command-line programs) : here DECEMBER | 2008 | EMBÛCHES TISSUES 6 posts published by remyoudompheng during December 2008. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a SCHEMES IN ALGEBRAIC GEOMETRY I : THE AFFINE PLANE Schemes in algebraic geometry I : the affine plane. 8 March 2009 at 12:49 pm 2 comments. I think most people blogging around algebraic geometry eventually write about schemes, (as in Rigorous trivialities or algebraic stacks (in the Secret Blogging Seminar), which are traditionnally seen as the main reason (not) to study algebraic geometry today.My turn now. MARCH | 2010 | EMBÛCHES TISSUES 2 posts published by remyoudompheng during March 2010. Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. WHY CAN WE HAVE A FAST FOURIER TRANSFORM ? Why can we have a Fast Fourier Transform ? 12 March 2009 at 3:23 pm 3 comments. The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of times eachsecond.
MATHKABA : MY TOOL FOR MANAGING BIBLIOGRAPHY I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. If I was not really working on the article, I may forgetthe author, or even
EMBÛCHES TISSUES
Cardboard dodecahedron. I made a cardboard dodecahedron for the needs of a talk. If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetricgroup
REVIEW | EMBÛCHES TISSUES Posts about review written by remyoudompheng. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized MACAULAY | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library FFT | EMBÛCHES TISSUES Posts about FFT written by remyoudompheng. Why can we have a Fast Fourier Transform ? The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of timeseach second.
COMBINATORICS
Laver tables and “unprovable” statements. Laver tables are combinatorial objets whose definition is surprisingly simple. However, they do have somewhat weird properties, which come from their relationship with questions of set theorists (namely, how do elementary embeddings of MATHKABA | EMBÛCHES TISSUES Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. QED | EMBÛCHES TISSUES Saying QED in different languages. I was wondering whether it would be easy to know how to say “QED” in whatever language I could think of Well, it seems (not so unexpectedly after all) that Wikipedia was the right tool to use : so starting from the QED article (not the quantum electrodynamics one !) of the English Wikipedia, the language list allows to switch to the version of the same APRIL | 2009 | EMBÛCHES TISSUES The tetrahedra have faithful action of the isometry group: there are two sets of five tetrahedron, which are exchanged under signature -1 transformations, and even permutations of the tetrahedra correspond to direct isometries. 6 April 2009 at 12:01 am Leave a comment. FEBRUARY | 2010 | EMBÛCHES TISSUES 1 post published by remyoudompheng during February 2010. Changing default applications, for developers. When not using those uber-featured desktop environments (such as KDE or Gnome) out there, it can sometimes be challenging to find a simple, text-based way ofconfiguring things.
MAY | 2009 | EMBÛCHES TISSUES 2 posts published by remyoudompheng during May 2009. Operads in Haskell. Mikael Vejdemo Johansson, who was at the Operads conference in Luminy (which I also attended), wrote in only one week a Haskell module computing Gröbner bases for operads. Nice work ! 11 May 2009 at 8:25 pm Leave a commentEMBÛCHES TISSUES
Cardboard dodecahedron. I made a cardboard dodecahedron for the needs of a talk. If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetricgroup
REVIEW | EMBÛCHES TISSUES Posts about review written by remyoudompheng. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized MACAULAY | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library FFT | EMBÛCHES TISSUES Posts about FFT written by remyoudompheng. Why can we have a Fast Fourier Transform ? The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of timeseach second.
COMBINATORICS
Laver tables and “unprovable” statements. Laver tables are combinatorial objets whose definition is surprisingly simple. However, they do have somewhat weird properties, which come from their relationship with questions of set theorists (namely, how do elementary embeddings of MATHKABA | EMBÛCHES TISSUES Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. QED | EMBÛCHES TISSUES Saying QED in different languages. I was wondering whether it would be easy to know how to say “QED” in whatever language I could think of Well, it seems (not so unexpectedly after all) that Wikipedia was the right tool to use : so starting from the QED article (not the quantum electrodynamics one !) of the English Wikipedia, the language list allows to switch to the version of the same APRIL | 2009 | EMBÛCHES TISSUES The tetrahedra have faithful action of the isometry group: there are two sets of five tetrahedron, which are exchanged under signature -1 transformations, and even permutations of the tetrahedra correspond to direct isometries. 6 April 2009 at 12:01 am Leave a comment. FEBRUARY | 2010 | EMBÛCHES TISSUES 1 post published by remyoudompheng during February 2010. Changing default applications, for developers. When not using those uber-featured desktop environments (such as KDE or Gnome) out there, it can sometimes be challenging to find a simple, text-based way ofconfiguring things.
MAY | 2009 | EMBÛCHES TISSUES 2 posts published by remyoudompheng during May 2009. Operads in Haskell. Mikael Vejdemo Johansson, who was at the Operads conference in Luminy (which I also attended), wrote in only one week a Haskell module computing Gröbner bases for operads. Nice work ! 11 May 2009 at 8:25 pm Leave a comment MONADS IN MATHEMATICS Monads in mathematics 3 : monads from adjunctions. The adjunction property between two functors, and , says that there is a natural bijection between morphisms (in the first category) and (in the second category). Here natural means that these bijection is compatible with composition with morphisms , or and .. Adjunctions are naturally created by the use of monads or operads. BICIRCULAR QUARTIC CURVES Working in the Euclidean (projective) plane, a bicircular quartic curve is defined to be a quartic which is singular at the circular points I and J. We are usually interested in real curves, so the type of singulaity is the same at I or J. Salmon in his Treatise on Higher Plane Curves and Basset EXTENSIONS OF SHEAVES AND THE LOCAL-GLOBAL SPECTRAL Extensions of sheaves and the local-global spectral sequence. 10 January 2009 at 5:02 pm Leave a comment. GIven a topological space , algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. MONADS IN MATHEMATICS 4 : THE BAR AND COBAR CONSTRUCTIONS Monads in mathematics 4 : the bar and cobar constructions. 4 March 2009 at 10:39 pm Leave a comment. The use of monads and comonads in homological algebra is as old as the theory: Godement’s standard construction refers to the use of monads in cohomology theory, and is said to be the first study of a general method for constructing acyclic resolutions. EXPERIMENTAL ALGEBRAIC GEOMETRY I : THE GRASSMANNIAN I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let's try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians. Macaulay provides a command-line interface using the readline library (like many other command-line programs) : here DECEMBER | 2008 | EMBÛCHES TISSUES 6 posts published by remyoudompheng during December 2008. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a MARCH | 2010 | EMBÛCHES TISSUES 2 posts published by remyoudompheng during March 2010. Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. SCHEMES IN ALGEBRAIC GEOMETRY I : THE AFFINE PLANE Schemes in algebraic geometry I : the affine plane. 8 March 2009 at 12:49 pm 2 comments. I think most people blogging around algebraic geometry eventually write about schemes, (as in Rigorous trivialities or algebraic stacks (in the Secret Blogging Seminar), which are traditionnally seen as the main reason (not) to study algebraic geometry today.My turn now. WHY CAN WE HAVE A FAST FOURIER TRANSFORM ? Why can we have a Fast Fourier Transform ? 12 March 2009 at 3:23 pm 3 comments. The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of times eachsecond.
MATHKABA : MY TOOL FOR MANAGING BIBLIOGRAPHY I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. If I was not really working on the article, I may forgetthe author, or even
EMBÛCHES TISSUES
Cardboard dodecahedron. I made a cardboard dodecahedron for the needs of a talk. If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetricgroup
REVIEW | EMBÛCHES TISSUES Posts about review written by remyoudompheng. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized MACAULAY | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library FFT | EMBÛCHES TISSUES Posts about FFT written by remyoudompheng. Why can we have a Fast Fourier Transform ? The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of timeseach second.
COMBINATORICS
Laver tables and “unprovable” statements. Laver tables are combinatorial objets whose definition is surprisingly simple. However, they do have somewhat weird properties, which come from their relationship with questions of set theorists (namely, how do elementary embeddings of MATHKABA | EMBÛCHES TISSUES Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. QED | EMBÛCHES TISSUES Saying QED in different languages. I was wondering whether it would be easy to know how to say “QED” in whatever language I could think of Well, it seems (not so unexpectedly after all) that Wikipedia was the right tool to use : so starting from the QED article (not the quantum electrodynamics one !) of the English Wikipedia, the language list allows to switch to the version of the same APRIL | 2009 | EMBÛCHES TISSUES The tetrahedra have faithful action of the isometry group: there are two sets of five tetrahedron, which are exchanged under signature -1 transformations, and even permutations of the tetrahedra correspond to direct isometries. 6 April 2009 at 12:01 am Leave a comment. FEBRUARY | 2010 | EMBÛCHES TISSUES 1 post published by remyoudompheng during February 2010. Changing default applications, for developers. When not using those uber-featured desktop environments (such as KDE or Gnome) out there, it can sometimes be challenging to find a simple, text-based way ofconfiguring things.
MAY | 2009 | EMBÛCHES TISSUES 2 posts published by remyoudompheng during May 2009. Operads in Haskell. Mikael Vejdemo Johansson, who was at the Operads conference in Luminy (which I also attended), wrote in only one week a Haskell module computing Gröbner bases for operads. Nice work ! 11 May 2009 at 8:25 pm Leave a commentEMBÛCHES TISSUES
Cardboard dodecahedron. I made a cardboard dodecahedron for the needs of a talk. If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetricgroup
REVIEW | EMBÛCHES TISSUES Posts about review written by remyoudompheng. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized MACAULAY | EMBÛCHES TISSUES Experimental algebraic geometry I : the grassmannian. I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.. Macaulay provides a command-line interface using the readline library FFT | EMBÛCHES TISSUES Posts about FFT written by remyoudompheng. Why can we have a Fast Fourier Transform ? The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of timeseach second.
COMBINATORICS
Laver tables and “unprovable” statements. Laver tables are combinatorial objets whose definition is surprisingly simple. However, they do have somewhat weird properties, which come from their relationship with questions of set theorists (namely, how do elementary embeddings of MATHKABA | EMBÛCHES TISSUES Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. QED | EMBÛCHES TISSUES Saying QED in different languages. I was wondering whether it would be easy to know how to say “QED” in whatever language I could think of Well, it seems (not so unexpectedly after all) that Wikipedia was the right tool to use : so starting from the QED article (not the quantum electrodynamics one !) of the English Wikipedia, the language list allows to switch to the version of the same APRIL | 2009 | EMBÛCHES TISSUES The tetrahedra have faithful action of the isometry group: there are two sets of five tetrahedron, which are exchanged under signature -1 transformations, and even permutations of the tetrahedra correspond to direct isometries. 6 April 2009 at 12:01 am Leave a comment. FEBRUARY | 2010 | EMBÛCHES TISSUES 1 post published by remyoudompheng during February 2010. Changing default applications, for developers. When not using those uber-featured desktop environments (such as KDE or Gnome) out there, it can sometimes be challenging to find a simple, text-based way ofconfiguring things.
MAY | 2009 | EMBÛCHES TISSUES 2 posts published by remyoudompheng during May 2009. Operads in Haskell. Mikael Vejdemo Johansson, who was at the Operads conference in Luminy (which I also attended), wrote in only one week a Haskell module computing Gröbner bases for operads. Nice work ! 11 May 2009 at 8:25 pm Leave a comment MONADS IN MATHEMATICS Monads in mathematics 3 : monads from adjunctions. The adjunction property between two functors, and , says that there is a natural bijection between morphisms (in the first category) and (in the second category). Here natural means that these bijection is compatible with composition with morphisms , or and .. Adjunctions are naturally created by the use of monads or operads. BICIRCULAR QUARTIC CURVES Working in the Euclidean (projective) plane, a bicircular quartic curve is defined to be a quartic which is singular at the circular points I and J. We are usually interested in real curves, so the type of singulaity is the same at I or J. Salmon in his Treatise on Higher Plane Curves and Basset EXTENSIONS OF SHEAVES AND THE LOCAL-GLOBAL SPECTRAL Extensions of sheaves and the local-global spectral sequence. 10 January 2009 at 5:02 pm Leave a comment. GIven a topological space , algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. MONADS IN MATHEMATICS 4 : THE BAR AND COBAR CONSTRUCTIONS Monads in mathematics 4 : the bar and cobar constructions. 4 March 2009 at 10:39 pm Leave a comment. The use of monads and comonads in homological algebra is as old as the theory: Godement’s standard construction refers to the use of monads in cohomology theory, and is said to be the first study of a general method for constructing acyclic resolutions. EXPERIMENTAL ALGEBRAIC GEOMETRY I : THE GRASSMANNIAN I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let's try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians. Macaulay provides a command-line interface using the readline library (like many other command-line programs) : here DECEMBER | 2008 | EMBÛCHES TISSUES 6 posts published by remyoudompheng during December 2008. Casey’s Treatise on analytical geometry. John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a MARCH | 2010 | EMBÛCHES TISSUES 2 posts published by remyoudompheng during March 2010. Mathkaba : my tool for managing bibliography. I recently decided to systematically conserve a backup copy of articles I am downloading. SCHEMES IN ALGEBRAIC GEOMETRY I : THE AFFINE PLANE Schemes in algebraic geometry I : the affine plane. 8 March 2009 at 12:49 pm 2 comments. I think most people blogging around algebraic geometry eventually write about schemes, (as in Rigorous trivialities or algebraic stacks (in the Secret Blogging Seminar), which are traditionnally seen as the main reason (not) to study algebraic geometry today.My turn now. WHY CAN WE HAVE A FAST FOURIER TRANSFORM ? Why can we have a Fast Fourier Transform ? 12 March 2009 at 3:23 pm 3 comments. The Fourier transform was introduced by Fourier as a tool to solve heat equations, but is now used in its discrete version throughout the computing world more than trillions of times eachsecond.
MATHKABA : MY TOOL FOR MANAGING BIBLIOGRAPHY I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. If I was not really working on the article, I may forgetthe author, or even
EMBÛCHES TISSUES
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------------------------- MATHKABA : MY TOOL FOR MANAGING BIBLIOGRAPHY I recently decided to systematically conserve a backup copy of articles I am downloading. But what if I want to use these backup copy: I usually only remember the title of the article, or the name of the author. If I was not really working on the article, I may forget the author, or even the title, if I only remember that “somewhere it is written that…”. You may know Mendeley : I tried to use it for some time, but it is not open source, has too many features for what I need, and could be better suited to a mathematical use, by supporting, for example, the MSC classification , or by having specific interfaces with MathSciNet orzentralBlatt .
Hence I decided writing my own program to do the job. For the moment, it may probably crash at any time, and seriously lack essential features, but it satisfies my daily purposes, which are: * have a quick overview of the articles I have on my computer * have a way of opening them without having to know where it is * in case it is not stored on my computer, open a suitable URL without having to search through MathSciNet The result is called Mathkaba , and is hosted on GitHub . For the moment, it works by reading metadata which is not stored in a database as usual (I hate databases), but in plain text files along with the PDF files, which should have the same syntax as the ASCII output of ZentralBlatt. MathSciNet can also output entries in the endNote format, which seems equally interesting. Any comments are welcome.Advertisements
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_5 March 2010 at 9:55 am_
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CREATING EXECUTABLES FOR WINDOWS FROM A LINUX BOX When writing a program, especially a graphical one, you may feel some compassion towards Windows users and wonder whether it would be possible to give them a chance of using it. Famous toolkits like GTKand Qt
provide good
support for Windows and an abstraction layer for OS-dependent vital functions, so it becomes easier to write OS-independant code. Next you need a cross-compiler: it’s just a compiler, except that instead of creating executable files you could run, it creates executables in another format (e.g. Win32 .exe files). But since Windows is very different from GNU/Linux, some code or one of its dependencies has to use Windows-specific functions: so you need source headers, and maybe additional libraries.(more…)
_3 March 2010 at 3:34 pm_
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CHANGING DEFAULT APPLICATIONS, FOR DEVELOPERS When not using those uber-featured desktop environments (such as KDE or Gnome) out there, it can sometimes be challenging to find a simple, text-based way of configuring things. For example, I am currently writing a GTK2 program, which uses the method gtk_show_uri to open files using default programs: on the computer I am using at work, this results in opening Acrobat Reader over an XDMCP connection, which is really distasteful. Even using GNOME’s control center, there does not seem to be a way to change that. What should I look at?(more…)
_24 February 2010 at 7:33 pm_
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MERRY CHRISTMAS FROM A DYING BLOG To anyone still reading me, I wish a merry Christmas. May it be free from any gratuitous consumerist madness and proprietary software. I may still have several things to talk about in the near future. _25 December 2009 at 10:58 am_
_ 1 comment
_
OPERADS IN HASKELL
Mikael Vejdemo Johansson, who was at the Operads conferencein Luminy (which I
also attended), wrote in only one week a Haskell module computing Gröbner bases for operads. Nice work ! _11 May 2009 at 8:25 pm_
_ Leave a comment
_
ELLIPTIC CURVES FOR HIGH SCHOOL STUDENTS I had to give a talk to high school students about some mathematical notion: I decided to tell them something about elliptic curves, but not the usual speech about cryptography, finite fields and the group law on a cubic curve… Instead, I talked about the perhaps less known appearances of elliptic functions as solutions of classical ODEs (even if I don’t really know much about these myself). The simplest mechanical system whose motion is governed by an elliptic curve is the pendulum: the reason for this is that the ODE which classically describes the time evolution of the angle of the pendulum is best rewritten in terms of the altitude of the pendulum: the law of energy conservation is then written as where 0 and _2l_ are the extremal values of the altitude , is the highest altitude which can be reached with a given energy (even if , which corresponds to the pendulum make full rotations around its axis), and is the vertical momentum of the pendulum. In this setting, there are classical Hamilton relations , , so the differential form turns out to be the canonical non-vanishing abelian differential on the elliptic curve. This explains why the period of the pendulum is an elliptic integral, which can be calculed by an arithmetic-geometric mean, and why the position of the pendulum at can be deduced from its position at times and by the classicalsecant-tangent law.
The notes for the talk (in French) are available here.
_11 May 2009 at 8:08 pm_
_ Leave a comment
_
CARDBOARD ASSOCIAHEDRON After the dodecahedron comes the cardboard associahedron : it represents the combinatorics of moving parentheses to calculate an associative product of five things step by step. The vertices of the associahedron are thus the 14 different binary trees with 5 leaves. Given four factors, there are exactly five ways of multiplying them using a binary operation: by moving parentheses according to the associativity rule, you go through five different trees in a cyclic way. This is known as MacLane’s pentagon coherence rule, which states that in not too weak notions of monoid, checking coherence for pentagon diagrams ensures that the definition of the product iswell-behaved.
(more…)
_8 April 2009 at 12:11 am_
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