Moments Of Polynomials Research Articles

Designing shape-preserving descriptors for classifying signals with application to vibrations of large mechanical structures

The main contribution of this paper is introduction of descriptors based on the moments of the Bernstein–Durrmeyer polynomials. As it is demonstrated, their distinctive feature is their ability to preserve knowledge about the shapes of the signals such as monotonicity and convexity, without imposing any a priori constraints. These descriptors act in time-domain as shape-preserving knowledge extractors. For their empirical version, the asymptotic unbiasedness and MSE consistency are also proved without additional pre-filtering.Another significant contribution is the method of designing an autoencoder for selecting the number of descriptors suitable for a whole class of underlying signals. For this purpose, a generalized version of Akaike’s information criterion (AIC) is derived, and its approximate version is proposed and tested. It serves as the main nonlinear part of the autoencoder. Its linear part computes the descriptors, while the decoder part computes the AIC for all particular signals from a considered family.To demonstrate the usefulness of the proposed methodology, we consider the problem of classifying signals based on knowledge about their shapes gathered by the descriptors. First, a general scheme of selecting an appropriate classifier from a predefined list is proposed. Then, the proposed approach is applied to classifying vibrations of a large excavator in an open pit mine.

Characteristic polynomials of orthogonal and symplectic random matrices, Jacobi ensembles & L-functions

Starting from Montgomery’s conjecture, there has been a substantial interest on the connections of random matrix theory and the theory of L-functions. In particular, moments of characteristic polynomials of random matrices have been considered in various works to estimate the asymptotics of moments of L-function families. In this paper, we first consider joint moments of the characteristic polynomial of a symplectic random matrix and its second derivative. We obtain the asymptotics, along with a representation of the leading order coefficient in terms of the solution of a Painlevé equation. This gives us the conjectural asymptotics of the corresponding joint moments over families of Dirichlet L-functions. In doing so, we compute the asymptotics of a certain additive Jacobi statistic, which could be of independent interest in random matrix theory. Finally, we consider a slightly different type of joint moment that is the analogue of an average considered over [Formula: see text] in various works before. We obtain the asymptotics and the leading order coefficient explicitly.

Correction: Relating moments of self-adjoint polynomials in two orthogonal projections

Correction: Relating moments of self-adjoint polynomials in two orthogonal projections

Moments of zeta and correlations of divisor‐sums: Stratification and Vandermonde integrals

AbstractWe refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta‐function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.

Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble

Abstract We find closed formulas for arbitrarily high mixed moments of characteristic polynomials of the Alternative Circular Unitary Ensemble, as well as closed formulas for the averages of ratios of characteristic polynomials in this ensemble. A comparison is made to analogous results for the Circular Unitary Ensemble. Both moments and ratios are studied via symmetric function theory and a general formula of Borodin-Olshanski-Strahov.

The Mathieu conjecture for [formula omitted] reduced to an abelian conjecture

We reduce the Mathieu conjecture for SU(2) to a conjecture about moments of Laurent polynomials in two variables with single variable polynomial coefficients.

An involution on restricted Laguerre histories and its applications

Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations for the moments of Laguerre polynomials via Flajolet's combinatorial theory of continued fractions. In this paper, we first introduce a reflection-like involution on restricted Laguerre histories. Then, we demonstrate its power by composing this involution with three bijections due to Françon-Viennot, Foata-Zeilberger, and Yan-Zhou-Lin, respectively. A host of equidistribution results involving various (multiset-valued) permutation statistics follow from these applications. As byproducts, seven apparently new Mahonian statistics present themselves; new interpretations of known Mahonian statistics are discovered as well. Finally, in our effort to show the interconnections between these Mahonian statistics, we are naturally led to a new link between the variant Yan-Zhou-Lin bijection and the Kreweras complement.

On the moments of moments of random matrices and Ehrhart polynomials

There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size N goes to infinity. These quantities depend on two parameters k and q and when both of them are positive integers it has been shown that these moments are in fact polynomials in the matrix size N. In this paper we classify the integer roots of these polynomials and moreover prove that the polynomials themselves satisfy a certain symmetry property. This confirms some predictions from the thesis of Bailey [7]. The proof uses the Ehrhart-Macdonald reciprocity for rational convex polytopes and certain bijections between lattice points in some polytopes.

Moments of moments of the characteristic polynomials of random orthogonal and symplectic matrices

By using asymptotics of Toeplitz+Hankel determinants, we establish formulae for the asymptotics of the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices, as the matrix size tends to infinity. Our results are analogous to those that Fahs obtained for random unitary matrices in (Fahs B. 2021 Communications in Mathematical Physics 383 , 685–730. (doi: 10.1007/s00220-021-03943-0 )). A key feature of the formulae we derive is that the phase transitions in the moments of moments are seen to depend on the symmetry group in question in a significant way.

Negative moments of orthogonal polynomials

Abstract If a sequence indexed by nonnegative integers satisfies a linear recurrence without constant terms, one can extend the indices of the sequence to negative integers using the recurrence. Recently, Cigler and Krattenthaler showed that the negative version of the number of bounded Dyck paths is the number of bounded alternating sequences. In this paper, we provide two methods to compute the negative versions of sequences related to moments of orthogonal polynomials. We give a combinatorial model for the negative version of the number of bounded Motzkin paths. We also prove two conjectures of Cigler and Krattenthaler on reciprocity between determinants.

On the moments of the moments of the characteristic polynomials of Haar distributed symplectic and orthogonal matrices

We establish formulae for the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices in terms of certain lattice point count problems. This allows us to establish asymptotic formulae when the matrix-size tends to infinity in terms of the volumes of certain regions involving continuous Gelfand–Tsetlin patterns with constraints. The results we find differ from those in the unitary case considered previously.

Relating moments of self-adjoint polynomials in two orthogonal projections

Given two orthogonal projections \(\{P,Q\}\) in a non commutative tracial probability space, we prove relations between the moments of \(P+Q\), of the commutator \(i(PQ-QP)\) and of \(P+QPQ\), and those of the angle operator PQP. Our proofs are purely algebraic and enumerative and does not assume P, Q satisfying Voiculescu’s freeness property or being in general position. As far as the sum and the commutator are concerned, the obtained relations follow from binomial-type formulas satisfied by the orthogonal symmetries associated to P and Q together with the trace property. In this respect, they extend those corresponding to the cases where one of the two projections is rotated by a free Haar unitary operator or more generally by a free unitary Brownian motion. As to the operator \(P+QPQ\), we derive autonomous recurrence relations for the coefficients (double sequence) of the expansion of its moments as linear combinations of those of PQP and determine explicitly few of them. These relations are obtained after a careful analysis of the structure of words in the alphabet \(\{P, QPQ\}\). We close the paper by exploring the connection of our previous results to the so-called Kato’s dual pair. Doing so leads to new identities satisfied by their moments and shows also that the interference operator \(J:=PQ+QP-2QPQ\) and the commutator of P and Q have the same moment sequence.

Random matrix theory and moments of moments of L-functions

In this paper, we give an analytic proof of the asymptotic behavior of the moments of moments of the characteristic polynomials of random symplectic and orthogonal matrices. We therefore obtain alternate, integral expressions for the leading order coefficients previously found by Assiotis, Bailey and Keating. We also discuss the conjectures of Bailey and Keating for the corresponding moments of moments of [Formula: see text]-functions with symplectic and orthogonal symmetry. Specifically, we show that these conjectures follow from the shifted moments conjecture of Conrey, Farmer, Keating, Rubinstein and Snaith.

On the moments of characteristic polynomials

AbstractWe calculate the moments of the characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, at a position t in the bulk of the spectrum, as a series expansion in powers of t. We focus in particular on the Gaussian Unitary Ensemble. We employ a novel approach to calculate the coefficients in this series expansion of the moments, appropriately scaled. These coefficients are polynomials in N. They therefore grow as $N\to\infty$ , meaning that in this limit the radius of convergence of the series expansion tends to zero. This is related to oscillations as t varies that are increasingly rapid as N grows. We show that the $N\to\infty$ asymptotics of the moments can be derived from this expansion when $t=0$ . When $t\ne 0$ we observe a surprising cancellation when the expansion coefficients for N and $N+1$ are formally averaged: this procedure removes all of the N-dependent terms leading to values that coincide with those expected on the basis of previously established asymptotic formulae for the moments. We obtain as well formulae for the expectation values of products of the secular coefficients.

Characteristic polynomials of random truncations: Moments, duality and asymptotics

We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups [Formula: see text], [Formula: see text] and [Formula: see text]. For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal and symplectic cases. Asymptotic expansions in strong and weak non-unitarity regimes are obtained. Using the connection to matrix hypergeometric functions, we establish limit theorems for the log-modulus of the characteristic polynomial evaluated on the unit circle.

On the Critical–Subcritical Moments of Moments of Random Characteristic Polynomials: A GMC Perspective

We study the ‘critical moments’ of subcritical Gaussian multiplicative chaos (GMCs) in dimensions d le 2. In particular, we establish a fully explicit formula for the leading order asymptotics, which is closely related to large deviation results for GMCs and demonstrates a similar universality feature. We conjecture that our result correctly describes the behaviour of analogous moments of moments of random matrices, or more generally structures which are asymptotically Gaussian and log-correlated in the entire mesoscopic scale. This is verified for an integer case in the setting of circular unitary ensemble, extending and strengthening the results of Claeys et al. and Fahs to higher-order moments.

Moments of orthogonal polynomials and exponential generating functions

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ( $$q=1$$ ) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials.

Moments of polynomials with random multiplicative coefficients

For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials $$ P_N(\theta) = \frac1{\sqrt{N}} \sum_{n\leq N} X(n) e(n\theta), $$ and show that the $2k$-th moments on the unit circle $$ \int_0^1 \big| P_N(\theta) \big|^{2k}\, d\theta $$ tend to Gaussian moments in the sense of mean-square convergence, uniformly for $k \ll (\log N / \log \log N)^{1/3}$, but that in contrast to the case of i.i.d. coefficients, this behavior does not persist for $k$ much larger. We use these estimates to (i) give a proof of an almost sure Salem-Zygmund type central limit theorem for $P_N(\theta)$, previously obtained in unpublished work of Harper by different methods, and (ii) show that asymptotically almost surely $$ (\log N)^{1/6 - \varepsilon} \ll \max_\theta |P_N(\theta)| \ll \exp((\log N)^{1/2+\varepsilon}), $$ for all $\varepsilon > 0$.

Recurrence relations for the moments of discrete semiclassical orthogonal polynomials

Recurrence relations for the moments of discrete semiclassical orthogonal polynomials

On the Joint Moments of the Characteristic Polynomials of Random Unitary Matrices

Abstract We establish the asymptotics of the joint moments of the characteristic polynomial of a random unitary matrix and its derivative for general real values of the exponents, proving a conjecture made by Hughes [ 40] in 2001. Moreover, we give a probabilistic representation for the leading order coefficient in the asymptotic in terms of a real-valued random variable that plays an important role in the ergodic decomposition of the Hua–Pickrell measures. This enables us to establish connections between the characteristic function of this random variable and the $\sigma $-Painlevé III’ equation.