932 research outputs found
Integer symmetric matrices having all their eigenvalues in the interval [-2,2]
We completely describe all integer symmetric matrices that have all their
eigenvalues in the interval [-2,2]. Along the way we classify all signed
graphs, and then all charged signed graphs, having all their eigenvalues in
this same interval. We then classify subsets of the above for which the integer
symmetric matrices, signed graphs and charged signed graphs have all their
eigenvalues in the open interval (-2,2).Comment: 33 pages, 18 figure
Lehmer's conjecture for Hermitian matrices over the Eisenstein and Gaussian integers
We solve Lehmer's problem for a class of polynomials arising from Hermitian
matrices over the Eisenstein and Gaussian integers, that is, we show that all
such polynomials have Mahler measure at least Lehmer's number \tau_0 =
1.17628...
Growth rates of permutation classes: categorization up to the uncountability threshold
In the antecedent paper to this it was established that there is an algebraic
number such that while there are uncountably many growth
rates of permutation classes arbitrarily close to , there are only
countably many less than . Here we provide a complete characterization of
the growth rates less than . In particular, this classification
establishes that is the least accumulation point from above of growth
rates and that all growth rates less than or equal to are achieved by
finitely based classes. A significant part of this classification is achieved
via a reconstruction result for sum indecomposable permutations. We conclude by
refuting a suggestion of Klazar, showing that is an accumulation point
from above of growth rates of finitely based permutation classes.Comment: To appear in Israel J. Mat
The phantom menace in representation theory
Our principal goal in this overview is to explain and motivate the concept of
a phantom in the representation theory of a finite dimensional algebra
. In particular, we exhibit the key role of phantoms towards
understanding how a full subcategory of the category
of all finitely generated left -modules is
embedded into , in terms of maps leaving or entering .
Contents: 1. Introduction and prerequisites; 2. Contravariant finiteness and
first examples; 3. Homological importance of contravariant finiteness and a
model application of the theory; 4. Phantoms. Definitions, existence, and basic
properties; 5. An application: Phantoms over string algebras
Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform
We enumerate the inequivalent self-dual additive codes over GF(4) of
blocklength n, thereby extending the sequence A090899 in The On-Line
Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a
well-known interpretation as quantum codes. They can also be represented by
graphs, where a simple graph operation generates the orbits of equivalent
codes. We highlight the regularity and structure of some graphs that correspond
to codes with high distance. The codes can also be interpreted as quadratic
Boolean functions, where inequivalence takes on a spectral meaning. In this
context we define PAR_IHN, peak-to-average power ratio with respect to the
{I,H,N}^n transform set. We prove that PAR_IHN of a Boolean function is
equivalent to the the size of the maximum independent set over the associated
orbit of graphs. Finally we propose a construction technique to generate
Boolean functions with low PAR_IHN and algebraic degree higher than 2.Comment: Presented at Sequences and Their Applications, SETA'04, Seoul, South
Korea, October 2004. 17 pages, 10 figure
Graph-Based Classification of Self-Dual Additive Codes over Finite Fields
Quantum stabilizer states over GF(m) can be represented as self-dual additive
codes over GF(m^2). These codes can be represented as weighted graphs, and
orbits of graphs under the generalized local complementation operation
correspond to equivalence classes of codes. We have previously used this fact
to classify self-dual additive codes over GF(4). In this paper we classify
self-dual additive codes over GF(9), GF(16), and GF(25). Assuming that the
classical MDS conjecture holds, we are able to classify all self-dual additive
MDS codes over GF(9) by using an extension technique. We prove that the minimum
distance of a self-dual additive code is related to the minimum vertex degree
in the associated graph orbit. Circulant graph codes are introduced, and a
computer search reveals that this set contains many strong codes. We show that
some of these codes have highly regular graph representations.Comment: 20 pages, 13 figure
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12
We consider additive codes over GF(4) that are self-dual with respect to the
Hermitian trace inner product. Such codes have a well-known interpretation as
quantum codes and correspond to isotropic systems. It has also been shown that
these codes can be represented as graphs, and that two codes are equivalent if
and only if the corresponding graphs are equivalent with respect to local
complementation and graph isomorphism. We use these facts to classify all codes
of length up to 12, where previously only all codes of length up to 9 were
known. We also classify all extremal Type II codes of length 14. Finally, we
find that the smallest Type I and Type II codes with trivial automorphism group
have length 9 and 12, respectively.Comment: 18 pages, 4 figure
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