119,004 research outputs found
On equations over sets of integers
Systems of equations with sets of integers as unknowns are considered. It is
shown that the class of sets representable by unique solutions of equations
using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in
T} and with ultimately periodic constants is exactly the class of
hyper-arithmetical sets. Equations using addition only can represent every
hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can
also be represented by equations over sets of natural numbers equipped with
union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in
T, \: m \geqslant n}. Testing whether a given system has a solution is
-complete for each model. These results, in particular, settle the
expressive power of the most general types of language equations, as well as
equations over subsets of free groups.Comment: 12 apges, 0 figure
Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields
In 1900, the German mathematician David Hilbert proposed a list of 23 unsolved mathematical problems. In his Tenth Problem, he asked to find an algorithm to decide whether or not a given diophantine equation has a solution (in integers). Hilbert's Tenth Problem has a negative solution, in the sense that such an algorithm does not exist. This was proven in 1970 by Y. Matiyasevich, building on earlier work by M. Davis, H. Putnam and J. Robinson. Actually, this result was the consequence of something much stronger: the equivalence of recursively enumerable and diophantine sets (we will refer to this result as "DPRM"). The first new result in the thesis is about Hilbert's Tenth Problem for function fields of curves over valued fields in characteristic zero. Under some conditions on the curve and the valuation, we have undecidability for diophantine equations over the function field of the curve. One interesting new case are function fields of curves over formal Laurent series. The proof relies on the method with two elliptic curves as developed by K. H. Kim and F. Roush and generalised by K. Eisenträger. Additionally, the proof uses the theory quadratic forms and valuations. And especially for non-rational function fields there is some algebraic geometry coming in. The second type of results establishes the equivalence of recursively enumerable and diophantine sets in certain polynomial rings. The most important is the one-variable polynomial ring over a finite field. This is the first generalisation of DPRM in positive characteristic. My proof uses the structure of finite fields and in particular the properties of cyclotomic polynomials. In the last chapter, this result for polynomials over finite fields is generalised to polynomials over recursive algebraic extensions of a finite field. For these rings we don't have a good definition of "recursively enumerable" set, therefore we consider sets which are recursively enumerable for every recursive presentation. We show that these are exactly the diophantine sets. In addition to infinite extensions of finite fields, we also show the analogous result for polynomials over a ring of integers in a recursive totally real algebraic extension of the rationals. This generalises results by J. Denef and K. Zahidi
Shortened Array Codes of Large Girth
One approach to designing structured low-density parity-check (LDPC) codes
with large girth is to shorten codes with small girth in such a manner that the
deleted columns of the parity-check matrix contain all the variables involved
in short cycles. This approach is especially effective if the parity-check
matrix of a code is a matrix composed of blocks of circulant permutation
matrices, as is the case for the class of codes known as array codes. We show
how to shorten array codes by deleting certain columns of their parity-check
matrices so as to increase their girth. The shortening approach is based on the
observation that for array codes, and in fact for a slightly more general class
of LDPC codes, the cycles in the corresponding Tanner graph are governed by
certain homogeneous linear equations with integer coefficients. Consequently,
we can selectively eliminate cycles from an array code by only retaining those
columns from the parity-check matrix of the original code that are indexed by
integer sequences that do not contain solutions to the equations governing
those cycles. We provide Ramsey-theoretic estimates for the maximum number of
columns that can be retained from the original parity-check matrix with the
property that the sequence of their indices avoid solutions to various types of
cycle-governing equations. This translates to estimates of the rate penalty
incurred in shortening a code to eliminate cycles. Simulation results show that
for the codes considered, shortening them to increase the girth can lead to
significant gains in signal-to-noise ratio in the case of communication over an
additive white Gaussian noise channel.Comment: 16 pages; 8 figures; to appear in IEEE Transactions on Information
Theory, Aug 200
On linear equations arising in Combinatorics (Part I)
The main point of this paper is to present a class of equations over integers
that one can check if they have a solution by checking a set of inequalities.
The prototype of such equations is the equations appearing in the well-known
Gale-Ryser theorem
On the Complexity of Hilbert Refutations for Partition
Given a set of integers W, the Partition problem determines whether W can be
divided into two disjoint subsets with equal sums. We model the Partition
problem as a system of polynomial equations, and then investigate the
complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a
given set of integers is not partitionable. We provide an explicit construction
of a minimum-degree certificate, and then demonstrate that the Partition
problem is equivalent to the determinant of a carefully constructed matrix
called the partition matrix. In particular, we show that the determinant of the
partition matrix is a polynomial that factors into an iteration over all
possible partitions of W.Comment: Final versio
Limit and extended limit sets of matrices in Jordan normal form
In this note we describe the limit and the extended limit sets of every
vector for a single matrix in Jordan normal form.Comment: 10 pages, we corrected some typos and we added a questio
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