The Torus(Site not responding. Last check: 2007-10-23)
The torus is the orientable surface with Euler characteristic = 0.
There exist an embedding (no self-intersections) of the torus in R^3 and an immersion (self-intersections) which is not regularly homotopic to the embedding.
The next image of the torus is obtained as follows: an ellipse centered at the origin in the xy-plane is tilted 45 degrees.
This can be phrased in the language of algebraic groups, in terms of the existence of a group action of an algebraic torus (equivalently by a multiplicative group of diagonal matrices).
This area is studied under the name of torusembeddings.
In group representation theory, a monomial representation is a particular kind of induced representation.
The orientable (nonorientable) genus of an embedding H~ of a graph H is equal to the minimum g such that a picture of H which respects the clockwise order of H~ can be embedded on an orientable (nonorientable) surface of genus g without edges crossing.
The genus of a orientable embedding (nonorientable embedding) is the smallest g such that the embedding is realizable with g handles (g crosscaps).
The output of embeddings algorithms is a combinatorial embedding (with signs associated with the edges for nonorientable surfaces).
[No title](Site not responding. Last check: 2007-10-23)
Since 3 and 6 are such nice hexagonal numbers, use a hexagonal torus (glue together opposite sides of a regular hexagon).
Here's one embedding, where the vertex sets are {a, b, c, d, e, f} and {x, y, z}.
In a square with the edges identified in the usual way to make a torus, place the vertices like this: y e d f x c a b z and then embed edges as required.
Embedding a Graph Into the Torus in Linear Time - Juvan, Marincek, Mohar (ResearchIndex)(Site not responding. Last check: 2007-10-23)
Abstract: A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus.
The embedding extension problem asks whether it is embedding extension problem possible to extend the embedding of K to an embedding of G in the same...
1 An algorithm for imbedding cubic graphs in the torus (context) - Filotti - 1980
The torus is topologically more simple than the sphere, yet geometrically it is a very complicated manifold indeed.
The "flat" torus, obtained by the action of a finite group on a plane, while topologically identical to the round torus, is not in the same diffeomorphism class.
The round torus metric is most easily constructed via its embedding in a Euclidean space of one higher dimension.
www.rwc.uc.edu /koehler/crg/tori.html (799 words)
Citations: On determining the genus of a graph in O - Filotti, Miller, Reif (ResearchIndex)(Site not responding. Last check: 2007-10-23)
Although the construction of minimum genus embeddings is NP hard (by Minors and embeddings 11 Thomassen [58] Filotti, Miller, and Reif
proved that for every fixed surface S, there is a polynomial time algorithm for embedding graphs in S. For every fixed surface S, Robertson and Seymour s theory gives an O(n 3) algorithm for testing embeddability in S using graph minors [37, 52] Robertson and Seymour recently improved their....
Unfortunately, even for the torus their algorithm has time complexity estimated only by O(n 188) A special polynomial time algorithm for embedding cubic graphs in the torus has been published by Filotti [4] Robertson and Seymour developed an O(n 3) algorithm using graph minors (with....
A quadrangulation of a surface S is an embedded graph G such that every face is of size 4.
A diamond 2-curve is a curve C in S such that C intersects the quadrangulation in a pair of vertices {x,y}.
Conjecture: If G is a graph that has a quadrangular embedding in the torus, then G has a quadrangular embedding in the Klein bottle if and only if the toroidal embedding has an essential diamond 2-curve.
Subject: Re: flat triangulation of torus Date: Mon, 12 Feb 2001 15:42:06 +0530 (IST) From: "Prof.
In December 2000, at Allahabad conference on Low dimensional Topoogy, I gave a talk on `why PL?' In this talk I actually displayed PL models of a torus and a genus 2 orientable closed surface.
My triangulated model of the torus has 9 vertices and I asked the audiance whether any of them know about an embedding of K(7) or some other PL embedding of torus with eight vertices.
www.lehigh.edu /~dmd1/sh211.txt (547 words)
Abstract(Site not responding. Last check: 2007-10-23)
Parallel algorithm for executing fuzzy Kohonen clustering network (FKCN) for image segmentation on distributed computer system with torus topology is presented.
Two approaches for parallelizing computations of global values are investigated — a spanning tree approach and multiplication of computations (a butterfly approach).
For distributed image processing systems having two- and three-measured torus topologies the analysis shows good estimation of the parallel FKCN implementation.
Executing Algorithms with Hypercube Topology on Torus Multicomputers (ResearchIndex)(Site not responding. Last check: 2007-10-23)
However, the scalability of hypercube multicomputers is constrained by the fact that the interconnection cost per node increases with the total number of nodes.
4 Embeddings among Meshes and Tori (context) - Ma, Tao - 1993
2 Embedding Three-Dimensional Meshes in Boolean Cubes by Graph..
sherry.ifi.unizh.ch /28393.html (372 words)
Algorithms(Site not responding. Last check: 2007-10-23)
Theorem (B.M. For every surface S there is a linear time algorithm which, for a given graph G, either finds an embedding of G in S or returns a subgraph of G that is a subdivision of a Kuratowski graph for S
However, already the linear time torus algorithm contains all the main difficulties from [4].
[9] M. Juvan and B. Mohar, A simplified algorithm for embedding a graph into the torus.
Summary for `Fast and Exact Simulation of Large Gaussian Lattice Systems in R^2: Exploring the Limits'(Site not responding. Last check: 2007-10-23)
The circulant embedding technique allows for the fast and exact simulation of stationary and intrinsically stationary Gaussian random fields.
The method uses periodic embeddings and relies on the fast Fourier transform.
However, exact simulations require that the periodic embedding is nonnegative definite, which is frequently not the case for two-dimensional simulations.
is a product of a Hermitian symmetric space and a compact torusembedding satisfying some additional conditions.
In the smooth case these torusembeddings are classified using the description of torusembeddings by systems of cone (``fans") and the theory of Coxeter groups.
KONARSKI, Decompositions of Normal Algebraic Varieties Determined by an Action of a One-dimensional Torus, Bull.
AMCA: Minimal degree sequence of torus knots by Rama Mishra(Site not responding. Last check: 2007-10-23)
In our earlier papers, we found an estimate of degrees of the polynomials defining an embedding of Torus Knots.
In this talk we introduce the concept of minimal degree sequence and obtain the minimal degree sequence for Torus knots.
The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
at.yorku.ca /c/a/l/g/50.htm (223 words)
Executing Algorithms with Hypercube Topology on Torus Multicomputers - Gonz'alez, Valero-Garc'ia, de Cerio ...(Site not responding. Last check: 2007-10-23)
78.6%: Executing Algorithms with Hypercube Topology on Torus..
0.4: Efficient FFT on Torus Multicomputers: A Performance Study - de Cerio, Valero-Garcia..
Gonz'alez, M. Valero-Garc'ia, and L.M. D'iaz de Cerio "Executing Algorithms with Hypercube Topology on Torus Multicomputers" IEEE Trans.
