420 research outputs found
Rational Convolution Roots of Isobaric Polynomials
In this paper, we exhibit two matrix representations of the rational roots of
generalized Fibonacci polynomials (GFPs) under convolution product, in terms of
determinants and permanents, respectively. The underlying root formulas for
GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier
paper by MacHenry and Tudose, make use of two types of operators. These
operators are derived from the generating functions for Stirling numbers of the
first kind and second kind. Hence we call them Stirling operators. To construct
matrix representations of the roots of GFPs, we use the Stirling operators of
the first kind. We give explicit examples to show how the Stirling operators of
the second kind appear in the low degree cases for the WIP-roots. As a
consequence of the matrix construction, we have matrix representations of
multiplicative arithmetic functions under the Dirichlet product into its
divisible closure.Comment: 13 page
Unified Fock space representation of fractional quantum Hall states
Many bosonic (fermionic) fractional quantum Hall states, such as Laughlin,
Moore-Read and Read-Rezayi wavefunctions, belong to a special class of
orthogonal polynomials: the Jack polynomials (times a Vandermonde determinant).
This fundamental observation allows to point out two different recurrence
relations for the coefficients of the permanent (Slater) decomposition of the
bosonic (fermionic) states. Here we provide an explicit Fock space
representation for these wavefunctions by introducing a two-body squeezing
operator which represents them as a Jastrow operator applied to reference
states, which are in general simple periodic one dimensional patterns.
Remarkably, this operator representation is the same for bosons and fermions,
and the different nature of the two recurrence relations is an outcome of
particle statistics.Comment: 10 pages, 3 figure
Approximating the Permanent of a Random Matrix with Vanishing Mean
We show an algorithm for computing the permanent of a random matrix with
vanishing mean in quasi-polynomial time. Among special cases are the Gaussian,
and biased-Bernoulli random matrices with mean 1/lnln(n)^{1/8}. In addition, we
can compute the permanent of a random matrix with mean 1/poly(ln(n)) in time
2^{O(n^{\eps})} for any small constant \eps>0. Our algorithm counters the
intuition that the permanent is hard because of the "sign problem" - namely the
interference between entries of a matrix with different signs. A major open
question then remains whether one can provide an efficient algorithm for random
matrices of mean 1/poly(n), whose conjectured #P-hardness is one of the
baseline assumptions of the BosonSampling paradigm
A permanent formula for the Jones polynomial
The permanent of a square matrix is defined in a way similar to the
determinant, but without using signs. The exact computation of the permanent is
hard, but there are Monte-Carlo algorithms that can estimate general
permanents. Given a planar diagram of a link L with crossings, we define a
7n by 7n matrix whose permanent equals to the Jones polynomial of L. This
result accompanied with recent work of Freedman, Kitaev, Larson and Wang
provides a Monte-Carlo algorithm to any decision problem belonging to the class
BQP, i.e. such that it can be computed with bounded error in polynomial time
using quantum resources.Comment: To appear in Advances in Applied Mathematic
On the expressive power of planar perfect matching and permanents of bounded treewidth matrices
Valiant introduced some 25 years ago an algebraic model of computation along
with the complexity classes VP and VNP, which can be viewed as analogues of the
classical classes P and NP. They are defined using non-uniform sequences of
arithmetic circuits and provides a framework to study the complexity for
sequences of polynomials. Prominent examples of difficult (that is,
VNP-complete) problems in this model includes the permanent and hamiltonian
polynomials. While the permanent and hamiltonian polynomials in general are
difficult to evaluate, there have been research on which special cases of these
polynomials admits efficient evaluation. For instance, Barvinok has shown that
if the underlying matrix has bounded rank, both the permanent and the
hamiltonian polynomials can be evaluated in polynomial time, and thus are in
VP. Courcelle, Makowsky and Rotics have shown that for matrices of bounded
treewidth several difficult problems (including evaluating the permanent and
hamiltonian polynomials) can be solved efficiently. An earlier result of this
flavour is Kasteleyn's theorem which states that the sum of weights of perfect
matchings of a planar graph can be computed in polynomial time, and thus is in
VP also. For general graphs this problem is VNP-complete. In this paper we
investigate the expressive power of the above results. We show that the
permanent and hamiltonian polynomials for matrices of bounded treewidth both
are equivalent to arithmetic formulas. Also, arithmetic weakly skew circuits
are shown to be equivalent to the sum of weights of perfect matchings of planar
graphs.Comment: 14 page
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