113 research outputs found
Ternary numbers and algebras. Reflexive numbers and Berger graphs
The Calabi-Yau spaces with SU(m) holonomy can be studied by the algebraic way
through the integer lattice where one can construct the Newton reflexive
polyhedra or the Berger graphs. Our conjecture is that the Berger graphs can be
directly related with the -ary algebras. To find such algebras we study the
n-ary generalization of the well-known binary norm division algebras, , , , , which helped to discover the
most important "minimal" binary simple Lie groups, U(1), SU(2) and G(2). As the
most important example, we consider the case , which gives the ternary
generalization of quaternions and octonions, , , respectively. The
ternary generalization of quaternions is directly related to the new ternary
algebra and group which are related to the natural extensions of the binary
algebra and SU(3) group. Using this ternary algebra we found the
solution for the Berger graph: a tetrahedron.Comment: Revised version with minor correction
The hyperbolic geometry of random transpositions
Turn the set of permutations of objects into a graph by connecting
two permutations that differ by one transposition, and let be the
simple random walk on this graph. In a previous paper, Berestycki and Durrett
[In Discrete Random Walks (2005) 17--26] showed that the limiting behavior of
the distance from the identity at time has a phase transition at .
Here we investigate some consequences of this result for the geometry of .
Our first result can be interpreted as a breakdown for the Gromov hyperbolicity
of the graph as seen by the random walk, which occurs at a critical radius
equal to . Let be a triangle formed by the origin and two points
sampled independently from the hitting distribution on the sphere of radius
for a constant . Then when , if the geodesics are suitably
chosen, with high probability is -thin for some , whereas
it is always O(n)-thick when . We also show that the hitting
distribution of the sphere of radius is asymptotically singular with
respect to the uniform distribution. Finally, we prove that the critical
behavior of this Gromov-like hyperbolicity constant persists if the two
endpoints are sampled from the uniform measure on the sphere of radius .
However, in this case, the critical radius is .Comment: Published at http://dx.doi.org/10.1214/009117906000000043 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A permutational triadic approach to jazz harmony and the chord/scale relationship
This study provides an original triadic theory that combines existing jazz theory, in particular the chord/scale relationship, and mathematical permutation group theory to analyze repertoire, act as a pedagogical tool, and provide a system to create new music. Permutations are defined as group actions on sets, and the sets used here are the constituent consonant triads derived from certain scales. Group structures provide a model by which to understand the relationships held between the triadic set elements as defined by the generating functions. The findings are both descriptive and prescriptive, as triadic permutations offer new insights into existing repertoire. Further, the results serve as an organizational tool for the improviser and composer/arranger. In addition to the ability to describe individual triadic musical events as group actions, we also consider relationships held among the musical events by considering subgroups, conjugacy classes, direct products and semidirect products. As an interdisciplinary study, it is hoped that this work helps to increase the discourse between those in the music subdisciplines of mathematical music theory and jazz studies
Random Sorting Networks
A sorting network is a shortest path from 12...n to n...21 in the Cayley
graph of S_n generated by nearest-neighbour swaps. We prove that for a uniform
random sorting network, as n->infinity the space-time process of swaps
converges to the product of semicircle law and Lebesgue measure. We conjecture
that the trajectories of individual particles converge to random sine curves,
while the permutation matrix at half-time converges to the projected surface
measure of the 2-sphere. We prove that, in the limit, the trajectories are
Holder-1/2 continuous, while the support of the permutation matrix lies within
a certain octagon. A key tool is a connection with random Young tableaux.Comment: 38 pages, 12 figure
Embedding Schemes for Interconnection Networks.
Graph embeddings play an important role in interconnection network and VLSI design. Designing efficient embedding strategies for simulating one network by another and determining the number of layers required to build a VLSI chip are just two of the many areas in which graph embeddings are used. In the area of network simulation we develop efficient, small dilation embeddings of a butterfly network into a different size and/or type of butterfly network. The genus of a graph gives an indication of how many layers are required to build a circuit. We have determined the exact genus for the permutation network called the star graph, and have given a lower bound for the genus of the permutation network called the pancake graph. The star graph has been proposed as an alternative to the binary hypercube and, therefore, we compare the genus of the star graph with that of the binary hypercube. Another type of embedding that is helpful in determining the number of layers is a book embedding. We develop upper and lower bounds on the pagenumber of a book embedding of the k-ary hypercube along with an upper bound on the cumulative pagewidth
IST Austria Thesis
This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph.
For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton.
In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars
Group invariant machine learning by fundamental domain projections
We approach the well-studied problem of supervised group invariant and
equivariant machine learning from the point of view of geometric topology. We
propose a novel approach using a pre-processing step, which involves projecting
the input data into a geometric space which parametrises the orbits of the
symmetry group. This new data can then be the input for an arbitrary machine
learning model (neural network, random forest, support-vector machine etc).
We give an algorithm to compute the geometric projection, which is efficient
to implement, and we illustrate our approach on some example machine learning
problems (including the well-studied problem of predicting Hodge numbers of
CICY matrices), in each case finding an improvement in accuracy versus others
in the literature. The geometric topology viewpoint also allows us to give a
unified description of so-called intrinsic approaches to group equivariant
machine learning, which encompasses many other approaches in the literature.Comment: 21 pages, 4 figure
On the Coarse Geometry of Infinite Regular Translation Surfaces
Wir untersuchen eine gewisse Familie unendlicher Translationsflächen, genannt reguläre Translationsflächen. Diese sind unendliche Translationsflächen, die eine gegebene endliche Translationsfläche regulär überlagern. Jede reguläre Translationsfläche kann aus einer endlich erzeugten Gruppe und einem geeigneten euklidischen Polygon konstruiert werden. Wir präsentieren viele Beispiele von regulären Translationsflächen und studieren ihre Grobgeometrie. Insbesondere beschreiben wir, wie der Quasi-isometrie-Typ der Fläche mit dem Quasi-Isometrie-Typ der entsprechenden Gruppe zusammenhängt. Wir beweisen, dass in vielen Fällen die reguläre Translationsfläche quasi-isometrisch zu einer Quotientengruppe von ist. Allerdings gibt es auch Beispiele von Flächen, die nicht quasi-isometrisch zu einer endlich erzeugten Gruppe sind. Wir beweisen ein allgemeines Resultat, nach welchem jede reguläre Translationsfläche zu einem Cayleygraphen von bezüglich eines unendlichen Erzeugendensystems quasi-isometrisch ist
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