Exact Moments Research Articles

Nanorings Resembling Beehives for Ultrasensitive Fluorescence Detection.

In this study, we showcase the fabrication of two nanorings resembling beehives using intricately designed donor-acceptor (D-A) fluorophores. The D-A fluorophores, featuring three twisted fluorene groups on each side of the acceptor group, adopt a bent conformation that promotes the creation of a nanoring morphology upon aggregation. With porosity for maximum binding sites, high emission efficiency, and well-organized arrangements, the nanoring-based hives offer exceptional sensitivity and selectivity in the detection of organic sulfides. Particularly, nanorings formed from benzselenodiazole-containing molecules exhibit heightened sensitivity, achieving a limit of detection (LOD) of 0.2 ppb for dimethyl sulfide and 17 ppb for dimethyl disulfide. Due to its unparalleled sensitivity and selectivity, which was not achievable with previous optical sensors, this technology enables the continuous monitoring of meat spoilage in its early stages on an hourly basis. This provides crucial insights into the exact moments when freshness begins to deteriorate and how long the meat can be stored for.

Distribution of angular momenta ML and MS in non-relativistic configurations: statistical analysis using cumulants and Gram-Charlier series

Abstract The distributions P(ML,MS) of the total magnetic quantum numbers ML and MS for N electrons of angular momentum l, as well as the enumeration of LS spectroscopic terms and spectral lines, are crucial for the calculation of atomic structure and spectra, in particular for the modeling of emission or absorption properties of hot plasmas. However, no explicit formula for P(ML,MS) is known yet. In the present work, we show that the generating function for the cumulants, which characterize the distribution, obeys a recurrence relation, similar to the Newton-Girard identities relating elementary symmetric polynomials to power sums. This enables us to provide an explicit formula for the generating function. We also analyze the possibility of representing the P(ML,MS) distribution by a bi-variate Gram-Charlier series, which coefficients are obtained from the knowledge of the exact moments of P(ML,MS). It is shown that a simple approximation is obtained by truncating this series to the first few terms, though it is not convergent.

Nascent RNA kinetics with complex promoter architecture: Analytic results and parameter inference.

Transcription is a stochastic process that involves several downstream operations which make it difficult to model and infer transcription kinetics from mature RNA numbers in individual cell. However, recent advances in single-cell technologies have enabled a more precise measurement of the fluctuations of nascent RNA that closely reflect transcription kinetics. In this paper we introduce a general stochastic model to mimic nascent RNA kinetics with complex promoter architecture. We derive the exact distribution and moments of nascent RNA using queuing theory techniques, which provide valuable insights into the effect of the molecular memory created by the multistep activation and deactivation on the stochastic kinetics of nascent RNA. Moreover, based on the analytical results, we develop a statistical method to infer the promoter memory from stationary nascent RNA distributions. Data analysis of synthetic data and a realistic example, the HIV-1 gene, verifies the validity of this inference method.

Testing the hypothesis of a nested block covariance matrix structure with applications to medicine and natural sciences

This paper addresses the challenge of testing the hypothesis of what the authors call a nested block circular‐compound symmetric (NBCCS) covariance structure for the population covariance matrix. This is a covariance structure which has an outer block compound symmetric structure, where the diagonal blocks are themselves block circular matrices, while the off‐diagonal blocks are formed by all equal matrices. The NBCCS null hypothesis is decomposed into sub‐hypotheses, allowing this way for a simpler way to obtain a likelihood ratio test and its associated statistic. The exact moments of this statistic are derived, and its distribution is carefully examined. Given the complicated nature of this distribution, highly precise near‐exact distributions were developed. Numerical studies are conducted to assess the proximity between these near‐exact distributions and the exact distribution, highlighting the performance of these approximations, even in the case of very small sample sizes. Furthermore, three datasets, on bone mineral content, metabolic rates of glucose, and soil moisture are used to exemplify the practical application of the methodology derived in this study.

Impact of torque on active Brownian particle: exact moments in two and three dimensions

Abstract In this work, we investigate the effects of torque, accounting for translational diffusion, on active Brownian particles (ABPs) in two (2d) and three (3d) dimensions. The torque is equivalent to chirality in 2d, well known as chiral active Brownian particles (cABPs), whereas in 3d, this is introduced as an external torque on the ABPs. Despite the inherent complexity in solving the Fokker–Planck equation, we demonstrate a Laplace transform method to precisely calculate the temporal evolution of various dynamic moments. Our analysis yields explicit expressions for multiple moments, such as the second and fourth moments of displacement, revealing the impact of persistence and chirality/torque. These moments exhibit oscillatory behavior, and excess kurtosis indicates deviations from the Gaussian distribution during intermediate time intervals.

Exact moments for trapped active particles: inertial impact on steady-state properties and re-entrance

In this study, we investigate the behavior of inertial active Brownian particles in a d-dimensional harmonic trap in the presence of translational diffusion. While the solution of the Fokker–Planck equation is generally challenging, it can be utilized to compute the exact time evolution of all time-dependent dynamical moments using a Laplace transform approach. We present the explicit form for several moments of position and velocity in d-dimensions. An interplay of time scales assures that the effective diffusivity and steady-state kinetic temperature depend on both inertia and trap strength, unlike passive systems. The distance from equilibrium, measured by the violation of equilibrium fluctuation-dissipation and the amount of entropy production, decreases with increasing inertia and trap strength. We present detailed ‘phase diagrams’ using kurtosis of velocity and position, showing possibilities of re-entrance to equilibrium.

Exponential expansions for approximation of probability distributions

AbstractThis work analyses expansions of exponential form for approximating probability density functions, through the utilization of diverse orthogonal polynomial bases. Notably, exponential expansions ensure the maintenance of positive probabilities regardless of the degree of skewness and kurtosis inherent in the true density function. In particular, we introduce novel findings concerning the convergence of this series towards the true density function, employing mathematical tools of functional statistics. In particular, we show that the exponential expansion is a Fourier series of the true probability with respect to a given orthonormal basis of the so called Bayesian Hilbert space. Furthermore, we present a numerical technique for estimating the coefficients of the expansion, based on the first n exact moments of the corresponding true distribution. Finally, we provide numerical examples that effectively demonstrate the efficiency and straightforward implementability of our proposed approach.

Two moments preserving sectional approach for an enzymatic coagulation equation

The coagulation process has found extensive applications in monitoring the evolution of aerosol and granule preparation in pharmaceutical sciences, blood clotting in biology, and cheese manufacturing due to the enzymes in the dairy industry. Among these, modeling the cheese manufacturing process is more challenging due to three indistinguishable sub-mechanisms: (a) enzymatic proteolysis, (b) coagulation, and (c) gelation, which occurs during the enzymatic coagulation of milk. The current study focuses on developing a sectional approach based on the cell average technique for monitoring the evolution of enzyme-induced coagulation of paracasein micelles over time. The proposed technique preserves two integral properties, such as total number and total volume in the system. The mathematical formulation of the proposed technique is very simple, easy to code, and has a robust implementation on any uniform and non-uniform grids. Due to the unavailability of the analytical solutions of the number density functions, the validation of the new proposed approach is done by extracting the new series solutions through the modification of the Homotopy perturbation method [Kaur et al., J. Phys. A 52(38), 385201 (2019)] and exact integral moments for several kernels. It has been shown that the new approach not only estimates the first two integral moments accurately but also computes the second-order moment with high precision without any specific measures. Moreover, domains of varying size grids are taken into account to analyze the convergence behavior of the average-size paracasein micelles formed in the system based on the zeroth and first moments.

Exact Distribution of the Quantal Content in Synaptic Transmission.

During electrochemical signal transmission through synapses, triggered by an action potential (AP), a stochastic number of synaptic vesicles (SVs), called the "quantal content," release neurotransmitters in the synaptic cleft. It is widely accepted that the quantal content probability distribution is a binomial based on the number of ready-release SVs in the presynaptic terminal. But the latter number itself fluctuates due to its stochastic replenishment, hence the actual distribution of quantal content is unknown. We show that exact distribution of quantal content can be derived for general stochastic AP inputs in the steady state. For fixed interval AP train, we prove that the distribution is a binomial, and corroborate our predictions by comparison with electrophysiological recordings from MNTB-LSO synapses of juvenile mice. For a Poisson train, we show that the distribution is nonbinomial. Moreover, we find exact moments of the quantal content in the Poisson and other general cases, which may be used to obtain the model parameters from experiments.

Exact higher-order moments for linear non-homogeneous stochastic differential equation

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

Exact moments and re-entrant transitions in the inertial dynamics of active Brownian particles

In this study, we investigate the behavior of free inertial active Brownian particles in the presence of thermal noise. While finding a closed-form solution for the joint distribution of positions, orientations, and velocities using the Fokker–Planck equation is generally challenging, we utilize a Laplace transform method to obtain the exact temporal evolution of all dynamical moments in arbitrary dimensions. Our expressions in d dimensions reveal that inertia significantly impacts steady-state kinetic temperature and swim pressure while leaving the late-time diffusivity unchanged. Notably, as a function of activity and inertia, the steady-state velocity distribution exhibits a remarkable re-entrant crossover from ‘passive’ Gaussian to ‘active’ non-Gaussian behaviors. We construct a corresponding ‘phase diagram’ using the exact expression of the d-dimensional kurtosis. Our analytic expressions describe steady states and offer insights into time-dependent crossovers observed in moments of velocity and displacement. Our calculations can be extended to predict up to second-order moments for run-and-tumble particles and the active Ornstein–Uhlenbeck process (AOUP). Additionally, the kurtosis shows differences from AOUP.

Exact first moments of the RV coefficient by invariant orthogonal integration

The RV coefficient measures the similarity between two multivariate configurations, and its significance testing has attracted various proposals in the last decades. We present a new approach, the invariant orthogonal integration, permitting to obtain the exact first four moments of the RV coefficient under the null hypothesis.Our proposal can be applied to any multivariate setting endowed with Euclidean distances between the observations. It also covers the weighted setting of observations of unequal importance, where the exchangeability assumption, justifying the usual permutation tests, breaks down.The proposed RV moments express as simple functions of the kernel eigenvalues occurring in the weighted multidimensional scaling of the two configurations (spectral effective dimensionality, spectral skewness and spectral excess kurtosis). The expressions for the third and fourth moments seem original, and explain the marked asymmetry and kurtosis of the RV coefficient. They permit to test the significance of the RV coefficient by Cornish–Fisher cumulant expansion, beyond the normal approximation, as illustrated on a small dataset.The first three moments can be obtained by elementary means, but computing the fourth moment requires a more sophisticated apparatus, the Weingarten calculus for orthogonal groups.

A high order Cartesian grid, finite volume method for elliptic interface problems

We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems with high-contrast and spatially varying coefficients, large discontinuities in the source term, and complex interface geometries. We include a generalized truncation error analysis based on cell-centered Taylor series expansions, which then define stencils in terms of local discrete solution data and geometric information. In the process, we develop a simple method based on Green's theorem for computing exact geometric moments directly from an implicit function definition of the embedded interface. This approach produces stencils with a simple bilinear representation, where spatially-varying coefficients and jump conditions can be easily included and finite volume conservation can be enforced.

Identifiability for mixtures of centered Gaussians and sums of powers of quadratics

AbstractWe consider the inverse problem for the polynomial map that sends an ‐tuple of quadratic forms in variables to the sum of their th powers. This map captures the moment problem for mixtures of centered ‐variate Gaussians. In the first nontrivial case , we show that for any , this map is generically one‐to‐one (up to permutations of and third roots of unity) in two ranges: for and for , thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most . The first result is obtained by the explicit geometry of the tangential contact locus of the variety of sums of cubes of quadratic forms, as described by Chiantini and Ottaviani [SIAM J. Matrix Anal. Appl. 33 (2012), no. 3, 1018–1037], while the second result is accomplished using the link between secant nondefectivity with identifiability, proved by Casarotti and Mella [J. Eur. Math. Soc. (JEMS) (2022)]. The latter approach also generalizes to sums of th powers of ‐forms for and .

Moment analysis for predicting effective transport properties in hierarchical retentive porous media

We report on a new homogenization approach to solve, with drastically improved speed and accuracy, the general advection-diffusion equation in hierarchical porous media with localized diffusion and adsorption/desorption processes, thus opening the way to a much deeper understanding of the band broadening process in chromatographic systems. The proposed robust and efficient moment-based approach allows us to compute the exact local and integral concentration moments and hence provides exact solutions for the effective velocity and dispersion coefficients of migrating solute particles. Innovative to the proposed method is also that it not only produces the exact effective transport parameters of the long-time asymptotic solution, but also their entire transient. The analysis of the transient behaviour can be used, for example, to properly identify the time and length scales needed to achieve the macro-transport conditions. If the hierarchical porous media can be represented as the periodic repetition of a unit lattice cell, the method only requires the solution of the time-dependent advection-diffusion equations for the zeroth order and first-order exact local moments, exclusively on the unit cell. This implies an enormous reduction of the computational efforts and a significant improvement of the accuracy of the results when compared to the direct numerical simulation (DNS) approaches which require flow domains that are long enough to achieve steady-state conditions, and hence often cover tens to hundreds of unit cells. The reliability of the proposed method is verified by comparing its predictions with DNS results, in one, two and three dimensions, in both transient and asymptotic conditions. The influence of top and bottom no-slip walls on the separation performance of chromatographic columns with micromachined porous and nonporous pillars is discussed in detail.

Electromagnetic scattering by arbitrary-shaped magnetic particles and multipole decomposition: Analytical and numerical approaches

We develop a theoretical approach that makes it possible to analyze the role of multipole contributions in the scattering spectra of arbitrarily shaped magnetic particles with the relative permittivity and permeability given in a general tensor-valued form. The method for calculating the exact multipole moments with the inclusion of induced electric polarization and magnetization is suggested and explicitly applied for dipole and quadrupole terms. The final expressions for the dipole and quadrupole moments are obtained in such a form that they can be easily implemented in numerical solvers. Successful verification of the analytical expressions for multipole moments and the multipole decomposition of the scattered field and scattering cross section is provided by comparing the results obtained for differently shaped magnetic particles using the analytical Mie theory, numerical discrete dipole approximation, and comsol multiphysics software. As a particular example, the manifestation of the Faraday effect for fields scattered by a ferrite sphere is discussed in the framework of the multipole decomposition method with the derived magnetization terms. Accounting for the magnetic properties of particles in the multipole analysis of the electromagnetic scattering significantly expands the method's capabilities for studying and modeling metamaterial properties in the spectral ranges where natural materials can have a relative permeability different from unity.

Exact moment analysis of transient/asymptotic dispersion properties in periodic media with adsorbing/desorbing walls

The paper develops a robust and computationally efficient homogenization approach, grounded on exact local and integral moments, to investigate the temporal evolution of effective dispersion properties of solute particles in periodic media possessing absorbing/desorbing walls. Adsorption onto and desorption from active walls allow linear and reversible mass transfer between the solid surface and the fluid phase. The transient analysis reveals some important features of the dispersion process that cannot be captured by asymptotic approaches aimed at determining exclusively the long-range/large-distance dispersion properties. Two case studies are considered: the dispersion of an analyte in a sinusoidal channel with adsorbing/desorbing walls and the retentive pillar array column for liquid chromatography. For both systems, the transient analysis shows how the tortuous fluid motion induced by the sinusoidal walls or by the presence of pillars induces wide and persistent temporal oscillations of the effective velocity and dispersion coefficient even for a steady (non-pulsating) Stokes flow. The adsorption/desorption process strongly amplifies the phenomenon of the overshoot for the effective dispersion coefficient that, on short/intermediate time scales, reaches values significantly larger than the asymptotic one. Moreover, the method proposed allows a detailed analysis of the temporal evolution of the skewness of the marginal distribution of the analyte along the main stream direction. It clearly shows that the time scale for achieving the macro-transport regime, which implies a Gaussian (symmetric) marginal pdf, is largely underestimated if one bases the analysis on the attainment of constant asymptotic values for the effective velocity and for the dispersion coefficient.

The Role of Electrostatic Interactions in IFIT5-RNA Complexes Predicted by the UBDB+EPMM Method.

Electrostatic energy has a significant contribution to intermolecular interaction energy, especially in biological systems. Unfortunately, precise quantum mechanics calculations are not feasible for large biological systems; hence, simpler calculation methods are required. We propose a method called UBDB+EPMM (University at Buffalo Pseudoatom DataBank + Exact Potential Multipole Moments), which shortens computational time without losing accuracy. Here, we characterize electrostatic interactions in selected complexes of IFIT proteins with RNA. IFIT proteins are effectors of the innate immune system, and by binding foreign RNA, they prevent the synthesis of viral proteins in human host cells; hence, they block the propagation of viruses. We show that by using the UBDB+EPMM method it is possible to describe protein-RNA interactions not only qualitatively but also quantitatively. Looking at the charge penetration contribution to electrostatic interactions, we find all amino acid residues with strong local interactions. Moreover, we confirm that electrostatic interaction of IFIT5 with pppRNA does not depend on the sequence of the RNA.

The influence of material on the power performance of Savonius turbines in wind and water applications

Savonius turbines are practical for rivers with low speeds due to their simple design and good self-starting capabilities. Despite these advantages, real implementation is still limited due to their low power efficiency. However, the experimental studies are scarce as they mainly focus on power performance, neglecting the impact of materials exposed to the wind or water medium. Therefore, the current study aims to address this gap by investigating the influence of materials on the power performance of Savonius turbines in wind and water, particularly the effect of their mass moment of inertia (I). Aluminum (ALU) and Polylactic Acid (PLA) were selected as the materials for the turbines as they are the typical materials frequently used to fabricate the turbines. They were tested experimentally in a wind tunnel and a water flume with Reynolds numbers ranging from 52, 000 to 83, 600. The findings discovered that although the turbine materials have the exact mass moments of inertia, they perform differently in water and wind. The turbines exhibit higher power coefficient with increasing mass moment of inertia in the wind at low Reynolds numbers (Re < 70,000), which is most likely due to reduced bearing friction that is more pronounced in the low Reynolds number region. The power coefficient was more likely to be affected in the wind than in water due to the effect of water buoyancy force, which reduces the influence of the increased mass moment of inertia. The results also revealed that the ALU and PLA with the same overall mass moment of inertia had maximum power coefficients of 11% and 18% in wind and water, respectively. There is no significant difference in terms of self-starting speed due to the variation in the moment of inertia. The Reynolds number and the turbine starting angle appeared to have a greater effect on self-starting capabilities than the moment of inertia. The findings of this study help to establish the types of materials that are optimal for enhancing the turbine performance in wind or water applications and could potentially save a lot of costs in generating power.

Exact second moments of strain for composites with isotropic phases

In micromechanical approaches to failure prediction of inhomogeneous microstructures or homogenization of inelastic materials, local field fluctuations beyond the mean field are relevant, but not easily accessible except for resorting to full-field simulations. However, any analytical estimate of the effective strain energy density combined with the Hill–Mandel condition implies some information about the probability distributions of local fields such as stresses and strains. In this paper, analytical equations for the second moments of local fields are derived for particulate composites with isotropic phases. Explicit expressions are provided using the Mori–Tanaka approach, the Hashin–Shtrikman bounds and the singular approximation. The results are compared to full-field simulations of a random sphere assemblage. As the Hashin–Shtrikman covariance is isotropic, it is not well-suited for predicting the behavior under deviatoric loadings. The singular approximation and Mori–Tanaka methods yield sensible approximations for the second moment of matrix strain.