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

Bolboaca, Sorana Daniela; Jantschi, Lorentz
Fonte: Molecular Diversity Preservation International (MDPI) Publicador: Molecular Diversity Preservation International (MDPI)
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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.

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

Terán Vergara, Fernando de; Martínez Dopico, Froilan C.; Pérez, J.
Fonte: Oxford University Press Publicador: Oxford University Press
Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article; info:eu-repo/semantics/conferenceObject
Relevância na Pesquisa
56.33%
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...

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

Götze, F.; Kösters, H.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.17%
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

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

Bourgade, Paul; Hughes, Chris; Nikeghbali, Ashkan; Yor, Marc
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.22%
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.

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

Webb, Christian
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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

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

Forrester, Peter J.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.17%
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

## Maximum of the characteristic polynomial of random unitary matrices

Arguin, Louis-Pierre; Belius, David; Bourgade, Paul
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.17%
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

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

Randrianarisoa, Tovohery Hajatiana
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.17%
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

## On the characteristic polynomial of a supertropical adjoint matrix

Shitov, Yaroslav
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.22%
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

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

Hatzinikitas, Agapitos
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.17%
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

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

Mewton, C. J.; Ficek, Z.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.16%
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

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

Blaizot, Jean-Paul; Grela, Jacek; Nowak, Maciej A.; Warchoł, Piotr
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.26%
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

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

Damianou, Pantelis A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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

## Counting integral matrices with a given characteristic polynomial

Shah, Nimish A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.16%
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.

## Efficient Computation of the Characteristic Polynomial

Dumas, Jean-Guillaume; Pernet, Clément; Wan, Zhendong
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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.

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

Fürer, Martin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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.

## Factoring the characteristic polynomial of a lattice

Hallam, Joshua; Sagan, Bruce E.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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

## The characteristic polynomial of a multiarrangement

Abe, Takuro; Terao, Hiroaki; Wakefield, Max
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.41%
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

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

Aguiar, Marcelo; Lauve, Aaron
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