Clifford-algebraic random walks on the hypercube

The n-dimensional hypercube is a simple graph on 2ⁿ vertices labeled by binary strings, or words, of length n. Pairs of vertices are adjacent if and only if they differ in exactly one position as binary words; i.e., the Hamming distance between the words is one. A discrete-time random walk is easily defined on the hypercube by “flipping” a randomly selected digit from 0 to 1 or vice-versa at each time step. By associating the words as blades in a Clifford algebra of particular signature, combinatorial properties of the geometric product can be used to represent this random walk as a sequence within the algebra. A closed-form formula is revealed which yields probability distributions on the vertices of the hypercube at any time k ≥ 0 by a formal power series expansion of elements in the algebra. Furthermore, by inducing a walk on a larger Clifford algebra, probabilities of self-avoiding walks and expected first hitting times of specific vertices are recovered. Moreover, because the Clifford algebras used in the current work are canonically isomorphic to fermion algebras, everything appearing here can be rewritten using fermion creation/annihilation operators, making the discussion relevant to quantum mechanics and/or quantum computing.     

Friendship Two-Graphs

A friendship graph is a graph in which every two distinct vertices have exactly one common neighbor. All finite friendship graphs are known, each of them consists of triangles having a common vertex. We extend friendship graphs to two-graphs, a two-graph being an ordered pair G = (G ₀, G ₁) of edge-disjoint graphs G ₀ and G ₁ on the same vertex-set V(G ₀) = V(G ₁). One may think that the edges of G are colored with colors 0 and 1. In a friendship two-graph, every unordered pair of distinct vertices u, v is connected by a unique bicolored 2-path. The pairs of adjacency matrices of friendship two-graphs are solutions to the matrix equation AB + BA = J − I, where A and B are n × n symmetric 0 − 1 matrices, J is an n × n matrix with every entry being 1, and I is the identity n × n matrix. We show that there is no finite friendship two-graph with minimum vertex degree at most two. However, we construct an infinite such graph, and this construction can be extended to an infinite (uncountable) family. Also, we find a finite friendship two-graph, conjecture that it is unique, and prove this conjecture for the two-graphs that have a dominating vertex.     

On quantum mechanics and generalized Clifford algebras

In this note we give elementary examples of the naturalness of generalized Clifford algebras appearance, in some particular quantum mechanical models. First Weyl’s program [1] for quantum kinematics for the case of simplest Galois fieldsZ _n is realized in terms of generalized Clifford algebras. Dynamics might then be introduced, following the ideas of Hanney and Berry [2], as shown in [3]. Second the coherent state picture of the finite dimensional “Z _n — Quantum Mechanics” is presented. In the last part the known coherent states ofq-deformed quantum oscillators (q≡ω) are explicitly shown in the generalized Grassman algebras and the generalized Clifford algebras settings. Presented atThe Polish-Mexican Seminar, Kazimierz Dolny, August 1998 — Poland. 176     

Clifford Algebra-Valued Wavelet Transform on Multivector Fields

This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cl _n,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of \mathbbRⁿ{\mathbb{R}^{n}} . We express the admissibility condition in terms of the Cl _n,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion formula can be established whenever the Clifford admissible wavelet satisfies a particular admissibility condition. As an application we derive a Heisenberg type uncertainty principle for the Clifford algebra Cl _n,0-valued admissible wavelet transform. Finally, we provide some basic examples of these extended wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets.     

On generalized Clifford algebraC 4 (n) andGL q (2;C) quantum group

The non commuting matrix elements of matrices from quantum groupGL _q (2;C) withq≡ω being then-th root of unity are given a representation as operators in Hilbert space with help ofC ₄ ⁽ⁿ⁾ generalized Clifford algebra generators appropriately tensored with unit 2×2 matrix infinitely many times. Specific properties of such a representation are presented. Relevance of generalized Pauli algebra to azimuthal quantization of angular momentum alà Lévy-Leblond [10] and to polar decomposition ofSU _q (2;C) quantum algebra alà Chaichian and Ellinas [6] is also commented. The case ofq∈C, |q|=1 may be treated parallely.     

An algorithm for the determination of longest distances in a graph

This paper presents an algorithm for ranking the vertices of a directed graph. Its space and time requirements are bounded byc ₁ n ² +c ₂, wheren is the number of vertices of the graph andc ₁,c ₂ are positive constants which are independent of the size or other properties of the graph.The algorithm can be easily modified to solve the problem of determining longest distances from a vertex to all other vertices in a positive real valued graph with at mostc ₁ n ² +c ₂ elementary operations; the same result holds for shortest distances in negative real valued graphs.     

Generating planar 4-connected graphs

In this paper, we introduce three operations on planar graphs that we call face splitting, double face splitting, and subdivision of hexagons. We show that the duals of the planar 4-connected graphs can be generated from the graph of the cube by these three operations. That is, given any graphG that is the dual of a planar 4-connected graph, there is a sequence of duals of planar 4-connected graphsG ₀,G ₁, …,G _n such thatG ₀ is the graph of the cube,G _n=G, and each graph is obtained from its predecessor by one of our three operations. Research supported by a Sloan Foundation fellowship and by NSF Grant#GP-27963.     

Existence, uniqueness and graph representation of weighted minimal hypersurfaces

By means of a weight matrix, we introduce the class of weighted minimal hypersurfaces which yield a natural generalisation of minimal surfaces. Generalising a classical result of Radó, we prove existence, uniqueness and graph representation for weighted minimal hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} with prescribed boundary manifold.     

Quantum Fractals on <Emphasis Type="Italic">n</Emphasis>–Spheres. Clifford Algebra Approach

Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (M?bius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by “density matrices” in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two–sphere (octahedron), and on three-dimensional sphere (hypercubetesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we calculate the Radon-Nikodym derivative of the SO(n + 1) invariant measure on Sⁿ under SO(1, n + 1) transformations and discuss the Hamilton’s “icossian calculus” as well as its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. As a by-product of this work we obtain several Clifford algebraic results, such as a characterization of positive elements in a Clifford algebra as generalized Lorentz “spin–boosts”, and their action as M?bius transformation on n-sphere, and a decomposition of any element of Spin⁺(1, n + 1) into a spin–boost and a spin–rotation, including the explicit formula for the pullback of the SO(n + 1) invariant Riemannian metric with respect to the associated M?bius transformation.     

Almost constant sequences and well-distribution in compact groups

We study the properties of sequences (c _n ) in a compact groupG such that (x _n ) being (weakly) well-distributed implies (c _n x _n ) being (weakly) well-distributed and give a complete characterisation in terms of almost constant sequences as well as a generalisation to non-compact locally compact groups.     

Quantum automorphism groups of vertex-transitive graphs of order ≤ 11

We study quantum automorphism groups of vertex-transitive graphs having less than 11 vertices. With one possible exception, these can be obtained from cyclic groups , symmetric groups S _n and quantum symmetric groups , by using various product operations. The exceptional case is that of the Petersen graph, and we present some questions about it.     

On certain locally homogeneous Clifford manifolds

Given a manifoldM, a Clifford structure of orderm onM is a family ofm anticommuting complex structures generating a subalgebra of dimension 2 ^m of End(T(M)). In this paper we investigate the existence of locally invariant Clifford structures of orderm2 on a class of locally homogeneous manifolds. We study the case of solvable extensions ofH-type groups, showing in particular that the solvable Lie groups corresponding to the symmetric spaces of negative curvature carry invariant Clifford structures of orderm2. We also show that for eachm and any finite groupF, there is a compact flat manifold with holonomy groupF and carrying a Clifford structure of orderm.Partially supported by Conicor (Argentina)Partially supported by grants from Conicet, Conicor, SECYTUNg (Argentina), and I.C.T.P. (Trieste)Partially supported by grants from Conicet, Conicor, SECYTUNC (Argentina), T.W.A.S and I.C.T.P. (Trieste)     

Structure of Quantum Doubles of Finite Clifford Monoids

Abstract

In this paper, over a field k, we give the structure theorem of the quantum double of a finite Clifford monoid through bicrossed products and quantum doubles of groups. By this result, it is shown that the quantum double of a finite Clifford monoid is semisimple (resp. von Neumann regular) if and only if the semigroup is a finite group and the characteristic p of k does not divide the order of this group.     

Separating quasiconvex subgroups of right-angled Artin groups

A graph group, or right-angled Artin group, is a group given by a presentation where the only relators are commutators of the generators. A graph group presentation corresponds in a natural way to a simplicial graph, with each generator corresponding to a vertex, and each commutator relator corresponding to an edge. Suppose that G is a graph group whose corresponding graph is a tree and H is a subgroup of G. We show that if H is quasiconvex with respect to either the word metric on G or the CAT(0) metric on the universal cover of the standard complex for G, then H is separable, that is, H is the intersection of finite index subgroups of G. We also discuss some consequences relating to certain 3-manifold groups. Received: 19 July 2000; in final form: 2 March 2001 / Published online: 29 April 2002     

Graphs and their parallel groups

Given an immersion of a manifoldf: M→R ^n+k , dimensionM=n, the parallel groupP(f) off is formed by the diffeomorphisms ofM such that the normalk-planes at points of each orbit are parallel. In [3] we studied the parallel group of a plane closed curve. Here we concentrate on immersionsf: R ⁿ →R ⁿ⁺¹, special attention being paid to graphs of smooth maps fromR toR. Graphs of smooth mapsf: S ⁿ →R ^m are also dealt with and we characterise those maps of which the graph has nontrivial parallel group. To end up we find a sufficient condition for the triviality of the tangent group.     

Distance-Regular (0,α)-Reguli

We introduce distance-regular (0,α)-reguli and show that they give rise to (0,α)-geometries with a distance-regular point graph. This generalises the SPG-reguli of Thas [14] and the strongly regular (α,β)-reguli of Hamilton and Mathon [9], which yield semipartial geometries and strongly regular (α,β)-geometries, respectively. We describe two infinite classes of examples, one of which is a generalisation of the well-known semipartial geometry T_n^*(B) arising from a Baer subspace PG(n, q) in PG(n, q²). Research Fellow supported by the Flemish Institute for the Promotion of Scientific and Technological Research in Industry (IWT), grant no. IWT/SB/13367/Tonesi Research assistant of the Fund for Scientific Research Flanders (FWO-Vlaanderen).