8,631 research outputs found
Non-triviality of some one-relator products of three groups
In this paper we study a group G which is the quotient of a free product of
three non-trivial groups by the normal closure of a single element. In
particular we show that if the relator has length at most eight, then G is
non-trivial. In the case where the factors are cyclic, we prove the stronger
result that at least one of the factors embeds in G.Comment: 21 pages, 3 figure
Dihedral Gauss hypergeometric functions
Gauss hypergeometric functions with a dihedral monodromy group can be
expressed as elementary functions, since their hypergeometric equations can be
transformed to Fuchsian equations with cyclic monodromy groups by a quadratic
change of the argument variable. The paper presents general elementary
expressions of these dihedral hypergeometric functions, involving finite
bivariate sums expressible as terminating Appell's F2 or F3 series.
Additionally, trigonometric expressions for the dihedral functions are
presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z)
are considered.Comment: 28 pages; trigonometric expressions added; transformations and
invariants moved to arxiv.org/1101.368
Planar functions over fields of characteristic two
Classical planar functions are functions from a finite field to itself and
give rise to finite projective planes. They exist however only for fields of
odd characteristic. We study their natural counterparts in characteristic two,
which we also call planar functions. They again give rise to finite projective
planes, as recently shown by the second author. We give a characterisation of
planar functions in characteristic two in terms of codes over .
We then specialise to planar monomial functions and present
constructions and partial results towards their classification. In particular,
we show that is the only odd exponent for which is planar
(for some nonzero ) over infinitely many fields. The proof techniques
involve methods from algebraic geometry.Comment: 23 pages, minor corrections and simplifications compared to the first
versio
A group-theoretic approach to fast matrix multiplication
We develop a new, group-theoretic approach to bounding the exponent of matrix
multiplication. There are two components to this approach: (1) identifying
groups G that admit a certain type of embedding of matrix multiplication into
the group algebra C[G], and (2) controlling the dimensions of the irreducible
representations of such groups. We present machinery and examples to support
(1), including a proof that certain families of groups of order n^(2 + o(1))
support n-by-n matrix multiplication, a necessary condition for the approach to
yield exponent 2. Although we cannot yet completely achieve both (1) and (2),
we hope that it may be possible, and we suggest potential routes to that result
using the constructions in this paper.Comment: 12 pages, 1 figure, only updates from previous version are page
numbers and copyright informatio
Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups
Classical Hadamard matrices are orthogonal matrices whose elements are ±1. It is well-known that error correcting codes having large minimum distance between codewords can be associated with these Hadamard matrices. Indeed, the success of early Mars deep-space probes was strongly dependent upon this communication technology.
The concept of Hadamard matrices with elements drawn from an Abelian group is a natural generalization of the concept. For the case in which the dimension of the matrix is q and the group consists of the p-th roots of unity, these generalized Hadamard matrices are called “Butson Hadamard Matrices BH(p, q)”, first discovered by A. T. Butson [6].
In this dissertation it is shown that an error correcting code whose codewords consist of real numbers in finite Galois field Gf( p) can be associated in a simple way with each Butson Hadamard matrix BH(p, q), where p \u3e 0 is a prime number. Distance properties of such codes are studied, as well as conditions for the existence of linear codes, for which standard decoding techniques are available.
In the search for cyclic linear generalized Hadamard codes, the concept of an M-invariant infinite sequence whose elements are integers in a finite field is introduced. Such sequences are periodic of least period, T, and have the interesting property, that arbitrary identical rearrangements of the elements in each period yields a periodic sequence with the same least period. A theorem characterizing such M-invariant sequences leads to discovery of a simple and efficient polynomial method for constructing generalized Hadamard matrices whose core is a linear cyclic matrix and whose row vectors constitute a linear cyclic error correcting code.
In addition, the problem is considered of determining parameter sequences {tn} for which the corresponding potential generalized Hadamard matrices BH(p, ptn) do not exist. By analyzing quadratic Diophantine equations, new methods for constructing such parameter sequences are obtained. These results show the rich number theoretic complexity of the existence question for generalized Hadamard matrices
Relative Periodic Solutions of the Complex Ginzburg-Landau Equation
A method of finding relative periodic orbits for differential equations with
continuous symmetries is described and its utility demonstrated by computing
relative periodic solutions for the one-dimensional complex Ginzburg-Landau
equation (CGLE) with periodic boundary conditions. A relative periodic solution
is a solution that is periodic in time, up to a transformation by an element of
the equation's symmetry group. With the method used, relative periodic
solutions are represented by a space-time Fourier series modified to include
the symmetry group element and are sought as solutions to a system of nonlinear
algebraic equations for the Fourier coefficients, group element, and time
period. The 77 relative periodic solutions found for the CGLE exhibit a wide
variety of temporal dynamics, with the sum of their positive Lyapunov exponents
varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8.
Preliminary work indicates that weighted averages over the collection of
relative periodic solutions accurately approximate the value of several
functionals on typical trajectories.Comment: 32 pages, 12 figure
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