13,320 research outputs found
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
Bounding Stochastic Dependence, Complete Mixability of Matrices, and Multidimensional Bottleneck Assignment Problems
We call a matrix completely mixable if the entries in its columns can be
permuted so that all row sums are equal. If it is not completely mixable, we
want to determine the smallest maximal and largest minimal row sum attainable.
These values provide a discrete approximation of of minimum variance problems
for discrete distributions, a problem motivated by the question how to estimate
the -quantile of an aggregate random variable with unknown dependence
structure given the marginals of the constituent random variables. We relate
this problem to the multidimensional bottleneck assignment problem and show
that there exists a polynomial -approximation algorithm if the matrix has
only columns. In general, deciding complete mixability is
-complete. In particular the swapping algorithm of Puccetti et
al. is not an exact method unless . For a
fixed number of columns it remains -complete, but there exists a
PTAS. The problem can be solved in pseudopolynomial time for a fixed number of
rows, and even in polynomial time if all columns furthermore contain entries
from the same multiset
Specht Polytopes and Specht Matroids
The generators of the classical Specht module satisfy intricate relations. We
introduce the Specht matroid, which keeps track of these relations, and the
Specht polytope, which also keeps track of convexity relations. We establish
basic facts about the Specht polytope, for example, that the symmetric group
acts transitively on its vertices and irreducibly on its ambient real vector
space. A similar construction builds a matroid and polytope for a tensor
product of Specht modules, giving "Kronecker matroids" and "Kronecker
polytopes" instead of the usual Kronecker coefficients. We dub this process of
upgrading numbers to matroids and polytopes "matroidification," giving two more
examples. In the course of describing these objects, we also give an elementary
account of the construction of Specht modules different from the standard one.
Finally, we provide code to compute with Specht matroids and their Chow rings.Comment: 32 pages, 5 figure
From quantum mechanics to classical statistical physics: generalized Rokhsar-Kivelson Hamiltonians and the "Stochastic Matrix Form" decomposition
Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar-Kivelson
(RK) points, first presented in the context of quantum dimer models, are
defined by their representations in preferred bases in which their ground state
wave functions are intimately related to the partition functions of
combinatorial problems of classical statistical physics. We show that all the
known examples of quantum Hamiltonians, when fine-tuned to their RK points,
belong to a larger class of real, symmetric, and irreducible matrices that
admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that
are SMF decomposable are shown to be in one-to-one correspondence with
stochastic classical systems described by a Master equation of the matrix type,
hence their name. It then follows that the equilibrium partition function of
the stochastic classical system partly controls the zero-temperature quantum
phase diagram, while the relaxation rates of the stochastic classical system
coincide with the excitation spectrum of the quantum problem. Given a generic
quantum Hamiltonian construed as an abstract operator defined on some Hilbert
space, we prove that there exists a continuous manifold of bases in which the
representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a
(continuous) manifold of distinct stochastic classical systems related to the
same quantum problem. Finally, we illustrate with three examples of
Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model,
the quantum eight-vertex model, and the quantum three-coloring model on the
honeycomb lattice, how they can be understood within our framework, and how
this allows for immediate generalizations, e.g., by adding non-trivial
interactions to these models.Comment: 32 pages, 4 figures - accepted for publication in Annals of Physics,
Elsevie
Calculation of Relaxation Spectra from Stress Relaxation Measurements
Application of stress on materials increases the energy of the system. After removal of stress,
macromolecules comprising the material shift towards equilibrium to minimize the total
energy of the system. This process occurs through molecular rearrangements or “relaxation”
during which macromolecules attain conformations of a lower energetic state. The time,
however, that is required for these rearrangements can be short or long depending on the
interactions between the macromolecular species that consist the material. When
rearrangements occur faster than the time of observation (experimental timescale) then
molecular motion is observed (flow) and the material is regarded as viscou
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