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Topological Crystals

July 26, 2016

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Mathematics
Algebraic Topology
Combinatorics
Algebraic Topology
Combinatorics
Sunada's work on crystallography emphasizes the role of the "maximal abelian cover" of a graph XX. This is a covering space of XX for which the group of deck transformations is the first homology group H1(X,Z)H_1(X,\mathbb{Z}). An embedding of...

On some Lie groups containing spin group in Clifford algebra

July 25, 2016

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Physics
Mathematics
Mathematical Physics
Mathematical Physics
Mathematical Physics
Mathematical Physics
In this paper we consider some Lie groups in complexified Clifford algebras. Using relations between operations of conjugation in Clifford algebras and matrix operations we prove isomorphisms between these groups and classical matrix groups...

Relaxed Highest Weight Modules from D\mathcal{D}-Modules on the Kashiwara Flag Scheme

July 21, 2016

Mathematics
Representation Theory
Algebraic Geometry
The relaxed highest weight representations introduced by Feigin et al. are a class of representations of the affine Kac-Moody algebra sl2^\hat{\mathfrak{sl}_2}, which do not have a highest (or lowest) weight. We formulate a generalization of ...

The Complexity of Drawing Graphs on Few Lines and Few Planes

July 21, 2016

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Computer Science
Computational Geometry
Computational Complexity
It is well known that any graph admits a crossing-free straight-line drawing in R3\mathbb{R}^3 and that any planar graph admits the same even in R2\mathbb{R}^2. For a graph GG and d∈{2,3}d \in \{2,3\}, let ρd1(G)\rho^1_d(G) denote the smallest numb...

Large Deviations of a Network of Neurons with Dynamic Sparse Random Connections

July 19, 2016

Mathematics
Probability
In this work we determine a process-level Large Deviation Principle (LDP) for a model of interacting particles indexed by a lattice Zd\mathbb{Z}^d. The connections are random, sparse and unscaled, so that the system converges in the large s...

Multiparameter Fuss--Catalan numbers with application to algebraic equations

July 14, 2016

Mathematics
Combinatorics
We present an exposition on the Fuss--Catalan numbers, which are a generalization of the well known Catalan numbers. The literature on the subject is scattered (especially for the case of multiple independent parameters, as will be explaine...

Circuit partitions and signed interlacement in 4-regular graphs

July 14, 2016

Mathematics
Combinatorics
Let FF be a 4-regular graph. Each circuit partition PP of FF has a corresponding touch-graph Tch(P)Tch(P); the circuits in PP correspond to vertices of Tch(P)Tch(P), and the vertices of FF correspond to edges of Tch(P)Tch(P). We discuss the connec...

The denominators of harmonic numbers (Revised)

July 11, 2016

Mathematics
Number Theory
The denominators dnd_n of the harmonic number 1+12+13+⋯+1n1+\frac12+\frac13+\cdots+\frac1n do not increase monotonically with~nn. It is conjectured that dn=Dn=LCM(1,2,…,n)d_n=D_n={\rm LCM}(1,2,\ldots,n) infinitely often. For an odd prime pp, the set $\{n:pd_n|D_n\...

Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns

July 7, 2016

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Physics
Chaotic Dynamics
We propose a general method for constructing a minimal cover of high-dimensional chaotic attractors by embedded unstable recurrent patterns. By minimal cover we mean a subset of available patterns such that the approximation of chaotic dyna...

Feature Selection Library (MATLAB Toolbox)

July 5, 2016

Computer Science
Computer Vision and Pattern Recognition
The Feature Selection Library (FSLib) signifies a notable progression in machine learning and data mining for MATLAB users, emphasizing the critical role of Feature Selection (FS) in enhancing model efficiency and effectiveness by pinpointi...