Moments Of Products Research Articles

Fluctuation Moments for Regular Functions of Wigner Matrices

We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of Male et al. (Random Matrices Theory Appl. 11(2):2250015, 2022), showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem given in the recent companion paper (Reker in Preprint, arXiv:2204.03419, 2023). and thus allow identifying the fluctuation around the thermal value in certain thermalization problems.

Convergence in Total Variation for nonlinear functionals of random hyperspherical harmonics

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-dimensional sphere (d≥2). We study the convergence in Total Variation distance for their nonlinear statistics in the high energy limit, i.e., for diverging sequences of Laplace eigenvalues. Our approach takes advantage of a recent result by Bally, Caramellino and Poly (2020): combining the Central Limit Theorem in Wasserstein distance obtained by Marinucci and Rossi (2015) for Hermite-rank 2 functionals with new results on the asymptotic behavior of their Malliavin-Sobolev norms, we are able to establish second order Gaussian fluctuations in this stronger probability metric as soon as the functional is regular enough. Our argument requires some novel estimates on moments of products of Gegenbauer polynomials that may be of independent interest, which we prove via the link between graph theory and diagram formulas.

Moments of the Hurwitz zeta function on the critical line

AbstractWe study the moments $M_k(T;\,\alpha) = \int_T^{2T} |\zeta(s,\alpha)|^{2k}\,dt$ of the Hurwitz zeta function $\zeta(s,\alpha)$ on the critical line, $s = 1/2 + it$ with a rational shift $\alpha \in \mathbb{Q}$ . We conjecture, in analogy with the Riemann zeta function, that $M_k(T;\,\alpha) \sim c_k(\alpha) T (\!\log T)^{k^2}$ . Using heuristics from analytic number theory and random matrix theory, we conjecturally compute $c_k(\alpha)$ . In the process, we investigate moments of products of Dirichlet L-functions on the critical line. We prove some of our conjectures for the cases $k = 1,2$ .

A Computational Study of Solvent and Electric Field Effects on Propylene Oxide Ring‐Opening Reaction

AbstractEthylene glycol (EG) is an important chemical and is widely used in the chemical industry. Hydration of ethylene oxide is a common way to produce EG. We performed density function calculations of propylene oxide (PO) ring‐opening reaction under basic, neutral and acidic conditions. We examined the influence of the solvent and electric field (EF) on the geometry, kinetics and thermodynamics of the ring‐opening. The solvent effect increases/decreases the ring‐opening barriers in the basic/neutral systems and EF exhibits significant influence on barriers and reaction heats in basic condition. All the energetic variation due to solvent and EF can be rationalized with the dipole moments of reactants, transition states, and products.

Isomeric fission yield ratios for odd-mass Cd and In isotopes using the phase-imaging ion-cyclotron-resonance technique

Isomeric yield ratios for the odd-$A$ isotopes of $^{119-127}$Cd and $^{119-127}$In from 25-MeV proton-induced fission on natural uranium have been measured at the JYFLTRAP double Penning trap, by employing the Phase-Imaging Ion-Cyclotron-Resonance technique. With the significantly improved mass resolution of this novel method isomeric states separated by 140 keV from the ground state, and with half-lives of the order of 500 ms, could be resolved. This opens the door for obtaining new information on low-lying isomers, of importance for nuclear structure, fission and astrophysics. In the present work the experimental isomeric yield ratios are used for the estimation of the root-mean-square angular momentum ($J_\mathrm{rms}$) of the primary fragments. The results show a dependency on the number of unpaired protons and neutrons, where the odd-$Z$ In isotopes carry larger angular momenta. The deduced values of $J_\mathrm{rms}$ display a linear relationship when compared with the electric quadrupole moments of the fission products.

On some hybrid power moments of products of generalized quadratic Gauss sums and Kloosterman sums*

In this paper, we investigate hybrid power moments of generalized quadratic Gauss sums weighted with powers of Kloosterman sums and with powers of values of Dirichlet L-functions at 1. We obtain several exact formulas for prime and prime power modulus and some asymptotic formulas.

Testing block‐diagonal covariance structure for high‐dimensional data

A test statistic is developed for making inference about a block‐diagonal structure of the covariance matrix when the dimensionality p exceeds n, where n = N − 1 and N denotes the sample size. The suggested procedure extends the complete independence results. Because the classical hypothesis testing methods based on the likelihood ratio degenerate when p > n, the main idea is to turn instead to a distance function between the null and alternative hypotheses. The test statistic is then constructed using a consistent estimator of this function, where consistency is considered in an asymptotic framework that allows p to grow together with n. The suggested statistic is also shown to have an asymptotic normality under the null hypothesis. Some auxiliary results on the moments of products of multivariate normal random vectors and higher‐order moments of the Wishart matrices, which are important for our evaluation of the test statistic, are derived. We perform empirical power analysis for a number of alternative covariance structures.

Magneto-optical properties of Cu1−xZnxFe2O4 nanoparticles

Zn doped CuFe2O4 nanoparticles (CZF NPs) were synthesized by a citric acid assisted sol–gel auto combustion process. The obtained samples were characterized by powder X-ray diffraction (XRD), Fourier transform infrared spectroscopy (FT-IR), high resolution scanning electron microscopy (HR-SEM), energy dispersive X-ray analysis, UV–Vis diffuse reflectance spectra (DRS), and vibrating sample magnetometer (VSM). XRD results confirmed the formation of cubic spinel-type structure with an average crystallite size in the range of 27–33nm. The variation lattice constants obey the Vegard’s Law. The CZF NPs were evaluated for their magneto-optical characteristics. The CZF NPs exhibit superparamagnetic behavior at room temperature (RT). The average Ms value for CZF NPs is about 57emu/g. The observed magnetic moments of products are in the range of 1.76–2.49μB. The small Mr/Ms ratios (maximum 0.22) specify uniaxial anisotropy in the Cu1−xZnxFe2O4 NPs. The high Ms and magnetic moment values are attributed to cation distribution change. The direct optical band gap (Eg) value of CZF NPs was found as 1.42eV.

Moments of products of automorphic L-functions

Assuming the generalized Riemann hypothesis, we prove upper bounds for moments of arbitrary products of automorphic L-functions and for Dedekind zeta-functions of Galois number fields on the critical line. As an application, we use these bounds to estimate the variance of the coefficients of these zeta- and L-functions in short intervals. We also prove upper bounds for moments of products of central values of automorphic L-functions twisted by quadratic Dirichlet characters and averaged over fundamental discriminants.

RANDOM WALKS, ELLIPTIC INTEGRALS AND RELATED CONSTANTS

Research Doctorate - Doctor of Philosophy (PhD)%%%%In the first quarter of this dissertation, we investigate the problem of how far a walker travels after n unit steps, each taken along a uniformly random direction; the short-step behaviour of this random walk was unknown. Utilising functional equations, we fully analyse the three- and four-step walks, finding the moments and densities of the distance from the origin. Our methods involve a blend of combinatorics, probability, and complex analysis. The derivatives of random walk moments turn out to be Mahler measures. We fruitfully study them using elementary techniques (different to those used by other researchers), namely generating functions of log-sine integrals and trigonometry. On the other hand, some random walk moments can be written as moments of products of complete elliptic integrals. These are studied, culminating in a complete solution for the moments of the product of two elliptic integrals. We also give some results when more elliptic integrals are involved. These endeavours occupy the second quarter of this dissertation. A spectacular application of elliptic integrals is their ability to produce rational series which converge to 1/π, as observed by Ramanujan. Using modular forms and hypergeometric transforms, we produce new classes of 1/π series which involve Legendre polynomials and Apery-like sequences. We give a diverse range of series for related constants, including some based on Legendre's relation. The third quarter of this dissertation is devoted to this topic. In the last quarter we apply experimental methods to better understand a number of areas encountered in our prior investigations. We simplify proofs for some multiple zeta value identities, give new ones and outline how they may be found. We give a method to quickly generate contiguous relations for hypergeometric series. Lastly, we look at orthogonal polynomials, in particular a new application of Gaussian quadrature to multi-dimensional lattice sums.

Helium Nanodroplet Isolation and Infrared Spectroscopy of the Isolated Ion-Pair 1-Ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

The ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide was vaporized at 420 K, and the ion-pair constituents were entrained in a beam of liquid He nanodroplets and cooled to 0.4 K. The vapor pressure was optimized such that each He droplet picked up a single ion-pair from the gas phase. Infrared spectroscopy in the CH stretch region reveals bands that are assigned to intact ion-pairs on the basis of comparisons to ab initio harmonic frequency computations of 23 low energy isomers. The He droplet spectrum is consistent with a weighted sum of the computed harmonic spectra, in which the weights are determined from ab initio computations of the relative free energies at 420 K. Anharmonic resonance polyads in the CH stretch region are treated explicitly, which improves the agreement between the experiment and computed spectra for ion-pairs. For isomers having a strong cation···anion hydrogen bonding interaction, the imidazolium C(2)-H stretch fundamental is shifted to lower energy and into resonance with the overtones and combination bands of the imidazolium ring stretching modes, resulting in a spectral complexity in the CH stretch region that is fully resolved in the He droplet spectrum. The assignment of the infrared spectrum to ion-pairs is confirmed through polarization spectroscopy measurements that reveal the permanent electric dipole moment of the He-solvated species to be 11 ± 2 D. The computed permanent electric dipole moments for the low energy isomers of the [emim(+)][Tf2N(-)] ion-pairs fall in the range 9-13D, whereas the computed dipole moments of decomposition products of the ionic liquid are less than 4.3 D.

EULER–HADAMARD PRODUCTS AND POWER MOMENTS OF SYMMETRIC SQUARE L-FUNCTIONS

In this paper we prove asymptotic formulas for general moments of partial Euler products and the first and the second moments of partial Hadamard products related to central values of the family of L-functions associated to the symmetric square lifts of holomorphic modular forms for SL2(ℤ). Then using a hybrid Euler–Hadamard product formula for the central value, we relate these results with conjectures for general power moments of L-functions in this family and with Random Matrix Theory interpretations. This continues the work done previously by Gonek–Hughes–Keating and Bui–Keating for other families of L-functions.

Representations of the Absolute Value Function and Applications in Gaussian Estimates

We study the expectation of unsigned Gaussian quadratic forms and negative absolute moments of Gaussian products. The main tool we use is the integral representation of the absolute value function.

Moments of products of elliptic integrals

We consider the moments of products of complete elliptic integrals of the first and second kinds. In particular, we derive new results using elementary means, aided by computer experimentation and a theorem of W. Zudilin. Diverse related evaluations, and two conjectures, are also given.

Estimating generalized Lyapunov exponents for products of random matrices

We discuss several techniques for the evaluation of the generalized Lyapunov exponents which characterize the growth of products of random matrices in the large-deviation regime. A Monte Carlo algorithm that performs importance sampling using a simple random resampling step is proposed as a general-purpose numerical method which is both efficient and easy to implement. Alternative techniques complementing this method are presented. These include the computation of the generalized Lyapunov exponents by solving numerically an eigenvalue problem, and some asymptotic results corresponding to high-order moments of the matrix products. Taken together, the techniques discussed in this paper provide a suite of methods which should prove useful for the evaluation of the generalized Lyapunov exponents in a broad range of applications. Their usefulness is demonstrated on particular products of random matrices arising in the study of scalar mixing by complex fluid flows.

On the mean values of L -functions in orthogonal and symplectic families

Hybrid Euler–Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions, in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz–Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with orthogonal and symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L-functions and for the L-functions associated with primitive Hecke eigenforms of weight 2 in terms of partial Euler and Hadamard products. We then prove asymptotic formulae for some moments of these partial products and make general conjectures based on results for the moments of characteristic polynomials of random matrices.

On the moments of Kloosterman sums and fibre products of Kloosterman curves

Let q = p r with p = 3 and r ⩾ 2 . We give a recursion formula for the moments of a Kloosterman sum over the finite field F q , which utilizes known weight formulae for the ternary Melas code M of length q − 1 . The method is illustrated by giving explicit formulae for the moments up to the tenth moment. As an application for the formulae, and for their analogues obtained earlier in case p = 2 , we get the exact number of rational points on fibre products of certain Kloosterman curves. As a corollary we obtain identities between Ramanujan's tau-function, Kronecker class numbers, and Dickson polynomials.

Nano- and Pico-Scale Transport Phenomena in Fluids

A new theoretical approach to describe pre-hydrodynamic stages of evolution in nonequilibrium fluids is presented. The local density, velocity, and temperature fields are expressed as integrals over Green's functions that depend on initial-state ensemble averages of dynamical quantities. For systems in which the initial states are nonuniform in only one spatial direction, explicit expressions for the Green's functions are derived in terms of initial-state ensemble averages of moments of particle displacements and products of particle velocities and particle displacements.

Exteroception and exproprioception by dynamic touch are different functions of the inertia tensor.

When an object is held and wielded, a time-invariant quantity of the wielding dynamics is the inertia tensor Iij. The 3 x 3 quantity Iij is composed of moments of inertia (on the diagonal) and products of inertia (off the diagonal). Examination of Iij as a function of different locations at which a cylindrical object is grasped revealed that the products related systematically to grip position (a direction), and both the products and moments taken together related systematically to the extent of the rod to one side of the hand (a magnitude in a direction). In two experiments, observers wielded an occluded rod that was held at an intermediate point along its length and reproduced both the felt grip position and partial rod length. In both experiments, perceived grip position was a function of the rod's products of inertia and perceived partial rod length was a function of the moments and products. Discussion focuses on the specificity of exteroception and exproprioception to Iij.

Positron annihilation in some organic scintillators: magnetic quenching and three gamma spectroscopy results

We present positron annihilation data in some organic polycrystalline scintillators, obtained through lifetime and three gamma spectroscopies, as well as magnetic quenching measurements. Strong magnetic quenching effects at low fields were discovered, which cannot be explained in terms of simply “swollen” Positronium (qPs): a possible explanation in terms of interactions between the qPs and the magnetic moments of the spur products is proposed.