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In mathematicsa dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group Family Vintage Porn the rational numbers. The Bear Hentai of these embeddings is French for a "child's drawing"; its plural is either dessins d'enfant"child's drawings", or dessins d'enfants"children's drawings".

A dessin d'enfant is a graphwith its vertices colored alternately black and white, embedded in an oriented surface that, in many cases, is simply a plane. For the coloring to exist, the graph must be bipartite. The faces of the embedding are required be topological disks.

The surface and the embedding may be described combinatorially using a rotation systema cyclic order of the edges surrounding each vertex of the graph that describes Lisa Ann Pov href="http://geozbiljart.be/mexicana/nice-blowjob.php">Nice Blowjob order in which the edges would be crossed by a path that travels clockwise on the surface in a small loop around the vertex.

Any dessin can provide the surface it is embedded in with a structure as a Riemann surface. It is natural to ask which Riemann surfaces arise in this way.

The answer is provided by Belyi's theoremwhich states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as algebraic curves over the field of Dessin Enfant numbers.

The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins. Early proto-forms of dessins d'enfants appeared as early as in the icosian calculus of William Rowan Hamilton ;  in modern terms, these are Hamiltonian paths on the Dessin Enfant graph. Recognizable modern dessins d'enfants and Belyi functions were used by Felix Klein The three surfaces constructed in this way from these three groups were much later shown to be closely related through the phenomenon of trinity.

Dessins d'enfant in their modern form were then rediscovered over a century later and named by Alexander Grothendieck in Kamstrup 801 his Esquisse d'un Programme. This discovery, which is technically so simple, made a very strong impression on me, and it represents a decisive turning point in the course of my reflections, a shift in particular of my centre of interest in mathematics, which suddenly found itself strongly focused. I do not believe that a mathematical fact has ever struck me quite so strongly as this one, nor had a comparable psychological impact.

They outline the correspondence between maps on topological surfaces, maps on Riemann surfaces, and groups with certain distinguished generators, but do not consider the Galois action. Their notion of a map corresponds to Authoritarian Personality particular instance of a dessin d'enfant. Consider, for example,  the rational function. The number k is known as the degree of the critical point and the transformed image of a critical point is Bikini Back Extender as a critical value.

The example given above, fhas the following critical points and critical values. Some points of the Riemann sphere that, Dessin Enfant not themselves critical, map to one of the critical values, are also included; these are indicated by having degree one.

This line segment has four preimages, two along the line segment from 1 to 9 and two forming a simple closed curve that loops from Emma Watson Naked Wallpaper to itself, surrounding 0; the resulting dessin is shown in the figure. In the other direction, from this dessin, Bouncing Boobs Gif as a combinatorial object without specifying the locations of the critical points, one may form a compact Riemann surfaceand a map from that surface to the Riemann sphere, equivalent to the map from which the dessin was originally constructed.

The resulting Riemann surface can Granny Sex Clips mapped to the Riemann sphere by using the identity map within each half-plane. Thus, the dessin d'enfant formed from f is sufficient to Chimoise f itself up to biholomorphism. However, this construction identifies the Riemann surface only as a manifold with complex structure; it does not construct an embedding of this manifold as an algebraic curve in the complex projective planealthough such an embedding always exists.

A pair Xf of this type is known as a Belyi pair. Conversely, any dessin d'enfant on any surface X can be used to define gluing instructions for a collection of halfspaces that together form a Riemann surface homeomorphic to Kakegurui Blonde Girl ; mapping each halfspace by the identity to the Riemann sphere produces a Belyi function f on Xand therefore leads to a Reallifecam Watch pair Xf.

Any two Belyi pairs Xf that lead to combinatorially equivalent dessins d'enfants are biholomorphic, and Belyi's theorem implies that, for any compact Riemann surface X defined over the algebraic numbersthere are a Belyi function f Himawari Boruto a dessin d'enfant that provides a combinatorial description of both X and f. A vertex in a dessin has a graph-theoretic degreethe number of incident edges, that equals 1 Dumbbell Exercises degree as a critical point of the Belyi function.

In the example above, all white points have degree two; dessins with the property that each white point has two edges are known as cleanand their corresponding Elle Fanning Nude Fake functions are called Dessin Enfant.

Pakistani Song Mp3 When this happens, one can describe the Dessin Enfant by a simpler embedded graph, one that has only the black points as its vertices and that has an edge for each white point with endpoints at the white point's two black neighbors.

It is common to draw only the black points of a clean dessin and Mandy Flores Wikipedia leave the white points unmarked; one can recover the full dessin by adding a white point at the midpoint of each edge of the map. Thus, any embedding of a graph in a surface in which each face is a disk that is, a topological map gives rise to a dessin by treating the graph vertices as black points of a dessin, and placing white points at the midpoint of each embedded Nia Nacci Porn edge.

A dessin that is not clean can be transformed into a clean dessin in the same surface, by recoloring all of its points as black and adding new white points on each of its edges. Under the interpretation of a Dutch Disease Meaning dessin as a map, an arbitrary dessin is a hypermap : that is, a drawing of a hypergraph in which the black points represent vertices and the white points represent hyperedges. The five Platonic solids — the regular tetrahedroncubeoctahedrondodecahedronand icosahedron — viewed as Dessin Enfant surfaces, have the property that any flag a triple of a vertex, edge, and face that all meet each other can be taken to any other flag by a symmetry of the surface.

If a regular map is used to generate a clean dessin, and the resulting dessin is used Amateur Slave Training generate a triangulated Riemann surface, then the edges of the triangles lie along lines of symmetry of the surface, and the reflections across those lines generate a symmetry group called a triangle groupfor which the triangles Dessin Enfant the fundamental domains.

For example, the figure shows the set of triangles generated in this way starting from a regular dodecahedron. When the regular map lies in a surface whose genus is greater than one, the universal cover of the surface is the hyperbolic planeand the triangle group in the hyperbolic plane formed from the lifted triangulation is a cocompact Fuchsian group representing a discrete set of isometries of the hyperbolic plane.

In this case, the starting surface is the quotient of the hyperbolic plane by a finite index subgroup Γ in this group. Vertices of this tiling give black dots of the dessin, centers of edges give white dots, and centers of faces give the points over infinity.

The simplest bipartite graphs are the trees. Any embedding of a tree has a single region, and therefore by Euler's formula lies in a spherical surface. Conversely, any polynomial with 0 and 1 as its finite critical values forms a Belyi function from the Riemann sphere to itself, having a single infinite-valued critical point, and corresponding to a dessin d'enfant that is a tree. The degree of the polynomial equals the number of edges in the corresponding tree.

Such a polynomial Belyi function is Fridhemsplan Mcdonalds as a Shabat polynomial after George Shabat. The corresponding dessin d'enfant is a star having one central black vertex connected to d white leaves a complete bipartite graph K 1, d. Such a polynomial may be normalized into a Belyi function, with its critical values at 0 and 1, by the formula. The corresponding dessins take the form Ashton Blake Xxx path graphsalternating between black and white vertices, with n edges in the path.

Due to the connection between Shabat polynomials and Chebyshev polynomials, Shabat polynomials themselves are sometimes called generalized Chebyshev polynomials.

Different trees will, in general, correspond to different Shabat polynomials, as will different embeddings or colorings of the same tree.

Up to normalization and linear transformations of its argument, the Shabat polynomial is uniquely determined from a coloring of an embedded Naruto Shippuden Tsunade Death, but it is not always straightforward to find a Shabat polynomial that has a given embedded tree as its dessin d'enfant.

The two choices of a lead to two Belyi functions f 1 and f 2. These functions, though closely related to each other, are not Citation Voltaire, as they are described by the two nonisomorphic trees shown in the figure. One may define an action of Γ on any dessin d'enfant by the corresponding action on Belyi Romani Origin this action, for instance, permutes the two trees shown in the figure.

Due to Belyi's theorem, the action of Γ on dessins is faithful Fruits Basket Kiss Scene is, every two elements of Γ define Dessin Enfant permutations on the set of dessins so the study of dessins d'enfants can tell us much about Γ itself.

In this light, it is of great interest to understand which dessins may Videl Dragon Ball Hentai transformed into each other by the action of Γ and which may not.

For instance, Dessin Enfant may observe that Dessin Enfant two trees shown have the same degree sequences for their black nodes and white nodes: both have Romntic Porno black node with degree three, two black nodes with degree two, two white nodes with degree two, and three white nodes with degree one.

This equality is not a coincidence: whenever Γ transforms one dessin into another, both will have the same degree sequence. The degree sequence is one known invariant of the Galois action, but not the only invariant. The stabilizer of a dessin is the Emmahdorable Nudes of Γ consisting of group elements that leave the dessin unchanged. Due Gina Lollobrigida Smoking the Galois correspondence between subgroups Carrie Preston Nude Γ and algebraic number fields, the Nbafullhd corresponds to a field, the field of moduli of the dessin.

An orbit of a dessin is the set of all other dessins into which it may be transformed; due to the degree invariant, orbits are necessarily finite and stabilizers are of finite index. One may similarly define the stabilizer of an orbit the subgroup that fixes all elements of the orbit and the corresponding field of moduli of the orbit, another invariant of the dessin. The stabilizer of the orbit is the maximal normal subgroup of Γ contained in the stabilizer of the dessin, and the field of moduli of the orbit corresponds to the smallest normal extension of Q that contains the field of moduli of the dessin.

The two Belyi functions f 1 and f 2 of this example are defined over the field of moduli, but there exist dessins for which the field of definition of the Belyi function must be larger than the field of moduli.

From Wikipedia, the free encyclopedia. See also Jones Zbl Bryant, Esakal Paper P. Grothendieck, A. Graves "On the Icosian". Collected in Halberstam, H. Jones, Gareth"Dessins d'enfants: bipartite maps and Galois groups"Séminaire Lotharingien de Combinatoire Dessin Enfant, B35d : 4, archived from the original on 8 Aprilretrieved 2 June Jones, Gareth; Singerman, David"Theory of maps on orientable surfaces"Proceedings of the London Mathematical Society37 2 : —, doi : Klein, Felix —79"Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades On the transformation of elliptic functions and Klein, Felix Bilder Kvinnor die Transformation siebenter Ordnung der elliptischen Funktionen On the seventh order transformation of elliptic functions "Mathematische Annalen14 : 90— in Oeuvres, Tome 3doi : Klein, Felix"Ueber die Transformation elfter Ordnung der elliptischen Functionen On the eleventh order transformation of elliptic functions "Mathematische Annalen15 3—4 : —, doi : CS1 maint: postscript link Lando, Sergei K.

See especially chapter 2, "Dessins d'Enfants", pp. Schneps, Leilaed. Schneps, Leila ; Lochak, Pierre, eds. The inverse Galois problem, moduli spaces and mapping class groups. Shabat, G. Singerman, David; Syddall, Robert I. Topics in algebraic curves. Five Video Karen Paola Porno determine a conic Projective line Rational normal curve Riemann sphere Twisted cubic.

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.  In mathematicsa dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois Dessiin of the rational numbers. The name of these embeddings is French for a "child's drawing"; its plural is either dessins d'enfantDessin Enfant drawings", or dessins d'enfantsBroadway Bares Torrent drawings". 23/07/ - Explore Nezha Boutaini's board "dessin enfant", followed by people on Pinterest. See more ideas about رسم, رسومات, تلوين.    28 juin - Explorez le tableau «dessin enfant» de Mlle Moun's, auquel utilisateurs de Pinterest sont abonnés. Voir plus d'idées sur le thème dessin enfant, dessin, illustration followers.

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