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## How Good Can the Characteristic Polynomial Be for Correlations?

Fonte: Molecular Diversity Preservation International (MDPI)
Publicador: Molecular Diversity Preservation International (MDPI)

Tipo: Artigo de Revista Científica

Publicado em 30/04/2007
Português

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46.21%

The aim of this study was to investigate the characteristic polynomials resulting from the molecular graphs used as molecular descriptors in the characterization of the properties of chemical compounds. A formal calculus method is proposed in order to identify the value of the characteristic polynomial parameters for which the extremum values of the squared correlation coefficient are obtained in univariate regression models. The developed calculation algorithm was applied to a sample of nonane isomers. The obtained results revealed that the proposed method produced an accurate and unique solution for the best relationship between the characteristic polynomial as molecular descriptor and the property of interest.

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## Backward stability of polynomial root-finding using Fiedler companion matrices

Fonte: Oxford University Press
Publicador: Oxford University Press

Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article; info:eu-repo/semantics/conferenceObject

Publicado em 08/09/2014
Português

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56.33%

#roots of polynomials#eigenvalues#characteristic polynomial#fiedler companion matrices#backward stability#conditioning#Matemáticas

Computing roots of scalar polynomials as the eigenvalues of Frobenius companion matrices using backward stable eigenvalue algorithms is a classical approach. The introduction of new families of companion matrices allows for the use of other matrices in the root-finding problem. In this paper, we analyze the backward stability of polynomial root-finding algorithms via Fiedler companion matrices. In other words, given a polynomial p(z), the question is to determine whether the whole set of computed eigenvalues of the companion matrix, obtained with a backward stable algorithm for the standard eigenvalue problem, are the set of roots of a nearby polynomial or not. We show that, if the coefficients of p(z) are bounded in absolute value by a moderate number, then algorithms for polynomial root-finding using Fiedler matrices are backward stable, and Fiedler matrices are as good as the Frobenius companion matrices. This allows us to use Fiedler companion matrices with favorable structures in the polynomial root-finding problem. However, when some of the coefficients of the polynomial are large, Fiedler companion matrices may produce larger backward errors than Frobenius companion matrices, although in this case neither Frobenius nor Fiedler matrices lead to backward stable computations. To prove this we obtain explicit expressions for the change...

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## On the Second-Order Correlation Function of the Characteristic Polynomial of a Hermitian Wigner Matrix

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/03/2008
Português

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We consider the asymptotics of the second-order correlation function of the
characteristic polynomial of a random matrix. We show that the known result for
a random matrix from the Gaussian Unitary Ensemble essentially continues to
hold for a general Hermitian Wigner matrix. Our proofs rely on an explicit
formula for the exponential generating function of the second-order correlation
function of the characteristic polynomial.; Comment: 20 pages

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## The characteristic polynomial of a random unitary matrix: a probabilistic approach

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/06/2007
Português

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In this paper, we propose a probabilistic approach to the study of the
characteristic polynomial of a random unitary matrix. We recover the Mellin
Fourier transform of such a random polynomial, first obtained by Keating and
Snaith, using a simple recursion formula, and from there we are able to obtain
the joint law of its radial and angular parts in the complex plane. In
particular, we show that the real and imaginary parts of the logarithm of the
characteristic polynomial of a random unitary matrix can be represented in law
as the sum of independent random variables. From such representations, the
celebrated limit theorem obtained by Keating and Snaith is now obtained from
the classical central limit theorems of Probability Theory, as well as some new
estimates for the rate of convergence and law of the iterated logarithm type
results.

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## The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos - The $L^2$-phase

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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46.26%

We study the characteristic polynomial of Haar distributed random unitary
matrices. We show that after a suitable normalization, as one increases the
size of the matrix, powers of the absolute value of the characteristic
polynomial as well as powers of the exponential of its argument converge in law
to a Gaussian multiplicative chaos measure for small enough real powers. This
establishes a connection between random matrix theory and the theory of
Gaussian multiplicative chaos.; Comment: Some changes to the first version: restriction to real powers and
addition of the phase of the characteristic polynomial

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## The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/03/2012
Português

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In classical random matrix theory the Gaussian and chiral Gaussian random
matrix models with a source are realized as shifted mean Gaussian, and chiral
Gaussian, random matrices with real $(\beta = 1)$, complex ($\beta = 2)$ and
real quaternion $(\beta = 4$) elements. We use the Dyson Brownian motion model
to give a meaning for general $\beta > 0$. In the Gaussian case a further
construction valid for $\beta > 0$ is given, as the eigenvalue PDF of a
recursively defined random matrix ensemble. In the case of real or complex
elements, a combinatorial argument is used to compute the averaged
characteristic polynomial. The resulting functional forms are shown to be a
special cases of duality formulas due to Desrosiers. New derivations of the
general case of Desrosiers' dualities are given. A soft edge scaling limit of
the averaged characteristic polynomial is identified, and an explicit
evaluation in terms of so-called incomplete Airy functions is obtained.; Comment: 21 pages

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## Maximum of the characteristic polynomial of random unitary matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/11/2015
Português

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It was recently conjectured by Fyodorov, Hiary and Keating that the maximum
of the characteristic polynomial on the unit circle of a $N\times N$ random
unitary matrix sampled from the Haar measure grows like $CN/(\log N)^{3/4}$ for
some random variable $C$. In this paper, we verify the leading order of this
conjecture, that is, we prove that with high probability the maximum lies in
the range $[N^{1 - \varepsilon},N^{1 + \varepsilon}]$, for arbitrarily small
$\varepsilon$. The method is based on identifying an approximate branching
random walk in the Fourier decomposition of the characteristic polynomial, and
uses techniques developed to describe the extremes of branching random walks
and of other log-correlated random fields. A key technical input is the
asymptotic analysis of Toeplitz determinants with dimension-dependent symbols.
The original argument for these asymptotics followed the general idea that the
statistical mechanics of $1/f$-noise random energy models is governed by a
freezing transition. We also prove the conjectured freezing of the free energy
for random unitary matrices.; Comment: 37 pages

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## The number of matrices over $\mathbb{F}_q$ with irreducible characteristic polynomial

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/02/2014
Português

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Let $\mathbb{F}_q$ be a finite field with $q$ elements. M. Gerstenhaber and
Irving Reiner has given two different methods to show the number of matrices
with a given characteristic polynomial. In this talk, we will give another
proof for the particular case where the characteristic polynomial is
irreducible. The number of such matrices is important to know the efficiency of
an algorithm to factor polynomials using Drinfeld modules.; Comment: 3 pages

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## On the characteristic polynomial of a supertropical adjoint matrix

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/10/2015
Português

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Let $\chi(A)$ denote the characteristic polynomial of a matrix $A$ over a
field; a standard result of linear algebra states that $\chi(A^{-1})$ is the
reciprocal polynomial of $\chi(A)$. More formally, the condition $\chi^n(X)
\chi^k(X^{-1})=\chi^{n-k}(X)$ holds for any invertible $n\times n$ matrix $X$
over a field, where $\chi^i(X)$ denotes the coefficient of $\lambda^{n-i}$ in
the characteristic polynomial $\det(\lambda I-X)$. We confirm a recent
conjecture of Niv by proving the tropical analogue of this result.; Comment: note, 3 pages

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## A Diagrammatic Approach for the Coefficients of the Characteristic Polynomial

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/11/2007
Português

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In this work we provide a novel approach for computing the coefficients of
the characteristic polynomial of a square matrix. We demonstrate that each
coefficient can be efficiently represented by a set of circle graphs. Thus, one
can employ a diagrammatic approach to determine the coefficients of the
characteristic polynomial.; Comment: 4 pages

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## The characteristic polynomial of the next-nearest-neighbour qubit chain for single excitations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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The characteristic polynomial for a chain of dipole-dipole coupled two-level
atoms with nearest-neighbour and next-nearest-neighbour interactions is
developed for the study of eigenvalues and eigenvectors for single-photon
excitations. We find the exact form of the polynomial in terms of the Chebyshev
polynomials of the second kind that is valid for an arbitrary number of atoms
and coupling strengths. We then propose a technique for expressing the roots of
the polynomial as a power series in the coupling constants. The general
properties of the solutions are also explored, to shed some light on the
general properties that the exact, analytic form of the energy eigenvalues
should have. A method for deriving the eigenvectors of the Hamiltonian is also
outlined.; Comment: 19 pages, 8 figures; minor corrections

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## Diffusion in the space of complex Hermitian matrices - microscopic properties of the averaged characteristic polynomial and the averaged inverse characteristic polynomial

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/05/2014
Português

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We show that the averaged characteristic polynomial and the averaged inverse
characteristic polynomial, associated with Hermitian matrices whose elements
perform a random walk in the space of complex numbers, satisfy certain partial
differential, diffusion-like, equations. These equations are valid for matrices
of arbitrary size. Their solutions can be given an integral representation that
allows for a simple study of their asymptotic behaviors for a broad range of
initial conditions.; Comment: 26 pages, 4 figures

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## On the characteristic polynomial of Cartan matrices and Chebyshev polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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46.3%

We explore some interesting features of the characteristic polynomial of the
Cartan matrix of a simple Lie algebra. The characteristic polynomial is closely
related with the Chebyshev polynomials of first and second kind. In addition,
we give explicit formulas for the characteristic polynomial of the Coxeter
adjacency matrix, we compute the associated polynomials and use them to derive
the Coxeter polynomial of the underlying graph. We determine the expression of
the Coxeter and associated polynomials as a product of cyclotomic factors. We
use this data to propose an algorithm for factoring Chebyshev polynomials over
the integers. Finally, we prove an interesting formula which involves products
of sines, the exponents, the Coxeter number and the determinant of the Cartan
matrix.; Comment: 32 pages, 32 references, corrected title

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## Counting integral matrices with a given characteristic polynomial

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/02/2000
Português

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46.16%

#Mathematics - Representation Theory#Mathematics - Number Theory#22E40 (Primary), 11Dxx, 11Hxx, 11P21 (Secondary)

We give a simpler proof of an earlier result giving an asymptotic estimate
for the number of integral matrices, in large balls, with a given monic
integral irreducible polynomial as their common characteristic polynomial. The
proof uses equidistributions of polynomial trajectories on SL(n,R)/SL(n,Z),
which is a generalization of Ratner's theorem on equidistributions of unipotent
trajectories. We also compute the exact constants appearing in the above
mentioned asymptotic estimate.

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## Efficient Computation of the Characteristic Polynomial

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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46.22%

This article deals with the computation of the characteristic polynomial of
dense matrices over small finite fields and over the integers. We first present
two algorithms for the finite fields: one is based on Krylov iterates and
Gaussian elimination. We compare it to an improvement of the second algorithm
of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third
algorithm could improve both complexity and computational time. We use these
results as a basis for the computation of the characteristic polynomial of
integer matrices. We first use early termination and Chinese remaindering for
dense matrices. Then a probabilistic approach, based on integer minimal
polynomial and Hensel factorization, is particularly well suited to sparse
and/or structured matrices.

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## Efficient Computation of the Characteristic Polynomial of a Threshold Graph

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/03/2015
Português

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46.26%

An efficient algorithm is presented to compute the characteristic polynomial
of a threshold graph. Threshold graphs were introduced by Chv\'atal and Hammer,
as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained
from a one vertex graph by repeatedly adding either an isolated vertex or a
dominating vertex, which is a vertex adjacent to all the other vertices.
Threshold graphs are special kinds of cographs, which themselves are special
kinds of graphs of clique-width 2. We obtain a running time of $O(n \log^2 n)$
for computing the characteristic polynomial, while the previously fastest
algorithm ran in quadratic time. Keywords: Efficient Algorithms, Threshold
Graphs, Characteristic Polynomial.

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## Factoring the characteristic polynomial of a lattice

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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46.22%

We introduce a new method for showing that the roots of the characteristic
polynomial of certain finite lattices are all nonnegative integers. This method
is based on the notion of a quotient of a poset which will be developed to
explain this factorization. Our main theorem will give two simple conditions
under which the characteristic polynomial factors with nonnegative integer
roots. We will see that Stanley's Supersolvability Theorem is a corollary of
this result. Additionally, we will prove a theorem which gives three conditions
equivalent to factorization. To our knowledge, all other theorems in this area
only give conditions which imply factorization. This theorem will be used to
connect the generating function for increasing spanning forests of a graph to
its chromatic polynomial. We finish by mentioning some other applications of
quotients of posets as well as some open questions.; Comment: 25 pages, 5 figures. To appear in JCTA

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## The characteristic polynomial of a multiarrangement

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/11/2006
Português

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#Mathematics - Commutative Algebra#Mathematics - Algebraic Geometry#32S22 (Primary) 14N20, 52C35 (Secondary)

Given a multiarrangement of hyperplanes we define a series by sums of the
Hilbert series of the derivation modules of the multiarrangement. This series
turns out to be a polynomial. Using this polynomial we define the
characteristic polynomial of a multiarrangement which generalizes the
characteristic polynomial of an arragnement. The characteristic polynomial of
an arrangement is a combinatorial invariant, but this generalized
characteristic polynomial is not. However, when the multiarrangement is free,
we are able to prove the factorization theorem for the characteristic
polynomial. The main result is a formula that relates `global' data to `local'
data of a multiarrangement given by the coefficients of the respective
characteristic polynomials. This result gives a new necessary condition for a
multiarrangement to be free. Consequently it provides a simple method to show
that a given multiarrangement is not free.; Comment: 12 pages, 2 figures

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## The characteristic polynomial of the Adams operators on graded connected Hopf algebras

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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46.17%

The Adams operators $\Psi_n$ on a Hopf algebra $H$ are the convolution powers
of the identity of $H$. We study the Adams operators when $H$ is graded
connected. They are also called Hopf powers or Sweedler powers. The main result
is a complete description of the characteristic polynomial (both eigenvalues
and their multiplicities) for the action of the operator $\Psi_n$ on each
homogeneous component of $H$. The eigenvalues are powers of $n$. The
multiplicities are independent of $n$, and in fact only depend on the dimension
sequence of $H$. These results apply in particular to the antipode of $H$ (the
case $n=-1$). We obtain closed forms for the generating function of the
sequence of traces of the Adams operators. In the case of the antipode, the
generating function bears a particularly simple relationship to the one for the
dimension sequence. In case H is cofree, we give an alternative description for
the characteristic polynomial and the trace of the antipode in terms of certain
palindromic words. We discuss parallel results that hold for Hopf monoids in
species and $q$-Hopf algebras.; Comment: 36 pages; two appendices

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## The characteristic polynomial and determinant are not ad hoc constructions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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The typical definition of the characteristic polynomial seems totally ad hoc
to me. This note gives a canonical construction of the characteristic
polynomial as the minimal polynomial of a "generic" matrix. This approach works
not just for matrices but also for a very broad class of algebras including the
quaternions, all central simple algebras, and Jordan algebras.
The main idea of this paper dates back to the late 1800s. (In particular, it
is not due to the author.) This note is intended for a broad audience; the only
background required is one year of graduate algebra.; Comment: v2 is heavily revised and somewhat expanded. The product formula for
the determinant on an algebra is proved

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