43,482 research outputs found
On solving systems of random linear disequations
An important subcase of the hidden subgroup problem is equivalent to the
shift problem over abelian groups. An efficient solution to the latter problem
would serve as a building block of quantum hidden subgroup algorithms over
solvable groups. The main idea of a promising approach to the shift problem is
reduction to solving systems of certain random disequations in finite abelian
groups. The random disequations are actually generalizations of linear
functions distributed nearly uniformly over those not containing a specific
group element in the kernel. In this paper we give an algorithm which finds the
solutions of a system of N random linear disequations in an abelian p-group A
in time polynomial in N, where N=(log|A|)^{O(q)}, and q is the exponent of A.Comment: 13 page
Rank 2 Local Systems, Barsotti-Tate Groups, and Shimura Curves
We develop a descent criterion for -linear abelian categories. Using
recent advances in the Langlands correspondence due to Abe, we build a
correspondence between certain rank 2 local systems and certain Barsotti-Tate
groups on complete curves over a finite field. We conjecture that such
Barsotti-Tate groups "come from" a family of fake elliptic curves. As an
application of these ideas, we provide a criterion for being a Shimura curve
over . Along the way, we formulate a conjecture on the
field-of-coefficients of certain compatible systems.Comment: 30 pages. Part of author's PhD thesis. Comments welcome
Non-binary Unitary Error Bases and Quantum Codes
Error operator bases for systems of any dimension are defined and natural
generalizations of the bit/sign flip error basis for qubits are given. These
bases allow generalizing the construction of quantum codes based on eigenspaces
of Abelian groups. As a consequence, quantum codes can be constructed from
linear codes over \ints_n for any . The generalization of the punctured
code construction leads to many codes which permit transversal (i.e. fault
tolerant) implementations of certain operations compatible with the error
basis.Comment: 10 pages, preliminary repor
New polynomial and multidimensional extensions of classical partition results
In the 1970s Deuber introduced the notion of -sets in
and showed that these sets are partition regular and contain all linear
partition regular configurations in . In this paper we obtain
enhancements and extensions of classical results on -sets in two
directions. First, we show, with the help of ultrafilter techniques, that
Deuber's results extend to polynomial configurations in abelian groups. In
particular, we obtain new partition regular polynomial configurations in
. Second, we give two proofs of a generalization of Deuber's
results to general commutative semigroups. We also obtain a polynomial version
of the central sets theorem of Furstenberg, extend the theory of
-systems of Deuber, Hindman and Lefmann and generalize a classical
theorem of Rado regarding partition regularity of linear systems of equations
over to commutative semigroups.Comment: Some typos, including a terminology confusion involving the words
`clique' and `shape', were fixe
Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties
We study the dynamical behavior of D-dimensional (D >= 1) additive cellular automata where the alphabet is any finite abelian group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [Masanobu Ito et al., 1983; Giovanni Manzini and Luciano Margara, 1999; Giovanni Manzini and Luciano Margara, 1999; Jarkko Kari, 2000; Gianpiero Cattaneo et al., 2000; Gianpiero Cattaneo et al., 2004]. Our main contribution is the proof that topologically transitive additive cellular automata are ergodic. This result represents a solid bridge between the world of measure theory and that of topology theory and greatly extends previous results obtained in [Gianpiero Cattaneo et al., 2000; Giovanni Manzini and Luciano Margara, 1999] for linear CA over Z_m i.e. additive CA in which the alphabet is the cyclic group Z_m and the local rules are linear combinations with coefficients in Z_m. In our scenario, the alphabet is any finite abelian group and the global rule is any additive map. This class of CA strictly contains the class of linear CA over Z_m^n, i.e.with the local rule defined by n x n matrices with elements in Z_m which, in turn, strictly contains the class of linear CA over Z_m. In order to further emphasize that finite abelian groups are more expressive than Z_m we prove that, contrary to what happens in Z_m, there exist additive CA over suitable finite abelian groups which are roots (with arbitrarily large indices) of the shift map.
As a consequence of our results, we have that, for additive CA, ergodic mixing, weak ergodic mixing, ergodicity, topological mixing, weak topological mixing, topological total transitivity and topological transitivity are all equivalent properties. As a corollary, we have that invertible transitive additive CA are isomorphic to Bernoulli shifts. Finally, we provide a first characterization of strong transitivity for additive CA which we suspect it might be true also for the general case
Adding homomorphisms to commutative/monoidal theories or : how algebra can help in equational unification
Two approaches to equational unification can be distinguished. The syntactic approach relies heavily on the syntactic structure of the identities that define the equational theory. The semantic approach exploits the structure of the algebras that satisfy the theory. With this paper we pursue the semantic approach to unification. We consider the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. This class has been introduced by the authors independently of each other as commutative theories (Baader) and monoidal theories (Nutt). The class encompasses important examples like the theories of abelian monoids, idempotent abelian monoids, and abelian groups. We identify a large subclass of commutative/monoidal theories that are of unification type zero by studying equations over the corresponding semiring. As a second result, we show with methods from linear algebra that unitary and finitary commutative/monoidal theories do not change their unification type when they are augmented by a finite monoid of homomorphisms, and how algorithms for the extended theory can be obtained from algorithms for the basic theory. The two results illustrate how using algebraic machinery can lead to general results and elegant proofs in unification theory
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