Exact Expressions Research Articles

Axisymmetric Cylindrical Green’s Function for the Steady-State Advection-Diffusion Operator with Uniform Velocity

The advection-diffusion equation has a wide range of applications as it governs energy and mass transport in a moving medium. Analytical solutions of the advection-diffusion equation for classes of boundary conditions amounts to finding exact expressions of the Green’s function of the advection-diffusion operator. In this work, the Green’s function of the steady-state advection-diffusion operator is obtained for an axisymmetric cylindrical problem with uniform velocity along the axis of the cylinder. The solution is exact, as axial diffusion (or conduction) is not neglected. The method of solution utilizes a formalism relating the Green’s function of the diffusion operator to that of the steady-state advection-diffusion operator with uniform velocity. The Green’s function obtained is applied to a Dirichlet boundary value problem. The distribution, (either particle concentration, or temperature), is plotted for various values of Péclet numbers. This work demonstrates how to invert the steady-state advection-diffusion operator with uniform velocity in a non-trivial geometry, namely the cylinder.

Approach to stationarity for the KPZ fixed point with boundaries

Current fluctuations for the one-dimensional totally asymmetric exclusion process (TASEP) connected to reservoirs of particles, and their large scale limit to the KPZ fixed point in finite volume, are studied using exact methods. Focusing on the maximal current phase for TASEP, corresponding to infinite boundary slopes for the KPZ height field, we obtain for general initial condition an exact expression for the late time correction to stationarity, involving extreme value statistics of Brownian paths. In the special cases of stationary and narrow wedge initial conditions, a combination of Bethe ansatz and numerical conjectures alternatively provide fully explicit exact expressions.

Reliability and Security Performances Analysis of Rate Splitting Based VLC System With HARQ

This work first attempts to investigate the reliability and physical layer security (PLS) performances of a rate splitting (RS) based visible light communication (VLC) system with one trusted user and one untrusted user. Specifically, the hybrid automatic repeat request (HARQ) protocol is adopted at the trusted user and the movement of users is assumed to be subject to the random way point (RWP) model with four decoding strategies considered at receiver side. The exact closed-form expressions of outage probability (OP) for both users are then obtained, and the analytical expressions of secrecy outage probability (SOP), reliable and secure transmission probability (RSP), and effective secrecy throughput (EST) of trusted user are also mathematically derived. Through simulations, it is found that to guarantee the reliability and PLS performances, the trusted user should first decode the untrusted user’s private stream and the untrusted user should first decode the trusted user’s private stream. Besides, the reliability and PLS performances of RS-VLC system could be enhanced with the appropriate increase in the maximum transmission round of HARQ with the best decoding strategy. Moreover, they could be further enhanced by the proper common stream power allocation coefficient. This work will benefit the research and development of indoor VLC system.

Harmonic chain driven by active Rubin bath: transport properties and steady-state correlations

Characterizing the properties of an extended system driven by active reservoirs is a question of increasing importance. Here, we address this question in two steps. We start by investigating the dynamics of a probe particle connected to an ‘active Rubin bath’: a linear chain of overdamped run-and-tumble particles. We derive exact analytical expressions for the effective noise and dissipation kernels, acting on the probe and show that the active nature of the bath leads to a modified fluctuation–dissipation relation. In the next step, we study the properties of an activity-driven system, modelled by a chain of harmonic oscillators connected to two such active reservoirs at the two ends. We show that the system reaches a non-equilibrium stationary state (NESS), remarkably different from that generated due to a thermal gradient. We characterize this NESS by computing the kinetic temperature profile, spatial and temporal velocity correlations of the oscillators, and the average energy current flowing through the system. It turns out that the activity drive leads to the emergence of two characteristic length-scales, proportional to the activities of the reservoirs. Strong signatures of activity are also manifest in the anomalous short-time decay of the velocity autocorrelations. Finally, we find that the energy current shows a non-monotonic dependence on the activity drive and reversal in direction, corroborating previous findings.

Exchange-biased quantum tunneling of magnetization in a nanomagnetic dimer via a Josephson current

We investigate a Josephson junction where the weak link consists of a nanomagnetic dimer, featuring two large, antiferromagnetically coupled spins under simple easy-axis anisotropy. In this setup, the exchange coupling constant serves as a transverse anisotropy constant, and the quantum tunneling of the magnetization is influenced by the supercurrent, which behaves like an external magnetic field applied along the hard axis. Our results demonstrate that the superconducting current induces non-Kramers freezing and unfreezing of the magnetic moment’s tunneling within a biaxial spin model driven by exchange interaction. We derive an exact expression for the quenching condition caused by the exchange coupling. Additionally, we extend our analysis to tunneling phenomena in two antiferromagnetically or ferromagnetically coupled spins biased by the anisotropic exchange interaction, revealing more varied cases of tunnel splitting.

Classicality of stochastic noise away from quasi-de Sitter inflation

It is well known that a coarse-grained scalar field living on a de Sitter (dS) background exhibits classical stochastic behavior, driven by a noise whose amplitude is set by the Hubble constant H. The coarse-graining is achieved by discarding wave numbers larger than a cutoff σaH and demanding that σ ≪ 1. Similar results hold for quasi-dS space, where the equation of state parameter w is close to -1. Here we present exact expressions for the noise amplitude of a free massless field on an inflationary background with constant w < -1/3. We find that a classical stochastic behavior can emerge for -5/3 < w < -1/3. Furthermore, as we move away from w = -1 and approach w = -1/3, the constraint σ ≪ 1 is relaxed and larger cutoffs (σ ∼ 1) become feasible, too. However, in general the amplitude of the noise depends on σ, except in the quasi-dS regime w ≈ -1.

An analog of the pendulum/rotor system with elementary solutions

Abstract We explore a novel one-dimensional model potential, V(x), which mimics many aspects of the classic pendulum/rotor system, including low- and high-energy limits which are exactly solvable, as well as a divergent period separating the two. Using energy methods, the classical period, T(E), is evaluated in closed form for all values of the energy (0 &lt; E &lt; ∞) in terms of simple logarithmic functions, in contrast to the pendulum/rotor system where the results require elliptic integrals. We find an unexpected duality between the exact expressions for the period on either side of the divergence. Using both energy methods, and a Newton’s law differential equation approach, we present closed form expressions for trajectory solutions, x(t) and p(t) = mv(t) (or the linear momentum), in terms of hyperbolic functions, which explicitly give the same values for T(E).

Kirchhoff index of some networks based on the cluster of graph

Abstract The Kirchhoff index represents the sum of resistance distances between all pairs of nodes in a network. It reflects the integrity and connectivity of the network. In this paper, we propose three classes of network models, all generated based on the cluster of graphs. We derive exact expressions for their Kirchhoff indices through an iterative methodology. Leveraging the derived formulas, we compared the Kirchhoff indices of the network families corresponding to three classes of networks generated under the same base graph. This enhances our evaluation of network characteristics, potentially serving as a critical tool in the practical design of networks.

Explicit bivariate simplicial depth

The simplicial depth (SD) is a celebrated tool defining elements of nonparametric and robust statistics for multivariate data. While many properties of SD are well-established, and its applications are abundant, explicit expressions for SD are known only for a handful of the simplest multivariate probability distributions. This paper deals with SD in the plane. It (i) develops a one-dimensional integral formula for SD of any properly continuous probability distribution, (ii) gives exact explicit expressions for SD of uniform distributions on (both convex and non-convex) polygons in the plane or on the boundaries of such polygons, and (iii) discusses several implications of these findings to probability and statistics: (a) An upper bound on the maximum SD in the plane, (b) an implication for a test of symmetry of a bivariate distribution, and (c) a connection of SD with the classical Sylvester problem from geometric probability.

Scattering of a spinning dielectric sphere to polarized plane waves

The exact expression of the wave vector inside a spinning homogeneous dielectric sphere illuminated by polarized plane waves is derived utilizing the “instantaneous rest-frame” hypothesis and Minkowski’s theory. On this basis, the analytical expressions of the electromagnetic field in the rotation sphere system are attained. The asymmetry of the system is discussed, in which the cause is emphasized. The influence of the polarization states and rotation angular velocity on the scattering are analyzed, including the optical rotation effect and photonic hook (PH). The results of this manuscript have extensive application prospects in optical tweezers, particle manipulation, and antenna design.

On Detour Index of Join of Hamilton-connected Graphs

In this paper we contribute to the literature of computational chemistry by providing exact expressions for the detour index of joins of Hamilton-connected ( H C ) graphs. This improves upon existing results by loosening the requirement of a molecular graph being Hamilton-connected and only requirement certain subgraphs to be Hamilton-connected.

Active fluctuations in the harmonic chain: phonons, entropons and velocity correlations

Non-equilibrium random fluctuations of a non-thermal nature are a defining feature of active matter. In this work, we study the collective excitations of active systems at high density, focusing on a one-dimensional chain of elastically coupled inertial particles, where activity is modeled by an Ornstein–Uhlenbeck process. The excitation spectrum reveals two types of fluctuations: thermally excited phonons, analogous to those in passive crystals, and entropons, which are associated with entropy production due to active forces. These fluctuations exhibit distinct properties: only entropons generate spatial velocity correlations and violate the standard fluctuation-response relation. We derive exact expressions for equal-time velocity and displacement correlations, as well as for the structure factor, identifying the contributions from both phonons and entropons. Finally, we explore the dynamical properties of these excitations through steady-state two-time correlations, such as the intermediate scattering function and mean-square displacement. Both phonon and entropon fluctuations are characterized by a long-wavelength overdamped regime and a short-wavelength underdamped regime. In the large persistence case, entropons decay more slowly than phonons, and activity generally suppresses the oscillations typical of the underdamped regime.

Moments of doubly truncated multivariate normal mean-mixture distributions

Recently, considerable interest has been in deriving explicit expressions for moment of different truncated multivariate distributions. This is due to their usefulness in developing efficient EM-algorithm for the maximum likelihood estimation of the corresponding distributional parameters. One rich family of distributions is the multivariate normal mean-mixture distribution as it includes a number of important multivariate distributions as special cases. In this article, we derive exact expressions for the first two moments of doubly truncated multivariate normal mean-mixture distributions, from which results for various special cases are then deduced.

ADHM wilson line defect indices

The Coulomb and Higgs indices of the 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 U(N) ADHM theories can be decorated by line defect operators as the line defect correlators. We obtain exact closed-form expressions and various non-trivial algebraic relations for the correlators of the Wilson lines in the fundamental and (anti)symmetric representations by means of the Hall-Littlewood expansion, the Fermi-gas method and the residue calculation. From the large N limit of the correlators we obtain the single particle gravity indices which are expected to encode the spectra of fluctuation modes on the gravity duals of line operators in M2-brane SCFTs.

Long-time behavior and quasi-density function for a stochastic epidemic model with relapse and reinfection

A stochastic susceptible–infected–recovered–infected (SIRI) epidemic model with relapse and reinfection is established in this paper. First, we prove that the solution to the epidemic model is unique and globally positive. Next, we determine some sufficient conditions for the extinction of the disease when [Formula: see text] and for the persistence in mean in the case of [Formula: see text]. Furthermore, we prove the existence of at least one ergodic stationary distribution of the stochastic model if [Formula: see text]. Additionally, by solving the corresponding three-dimensional Fokker–Planck equation, it is theoretically shown that the epidemic model has a log-normal probability density function when [Formula: see text], then we obtain the exact expression of density function of the stationary distribution. Finally, we give some numerical simulations to support our theoretical results.

Representation computation for the hypergeometric function of a Hermitian matrix argument

We establish the exact expressions for the hypergeometric function of a Hermitian matrix argument. This result allows for the eigenvalues of the matrix argument to occur with arbitrary multiplicities and can be used for numerical computation. These exact expressions are particularly important since they provide the key ingredient which allows many results which involve this function to be useful from a practical engineering perspective.

Quasi-likelihood estimation of the arrival time of an ultra-wideband quasi-radio signal with unknown amplitude and initial phase

The problem of estimating the time of arrival of a signal is relevant in the fields of radio and hydrolocation, radio astronomy, remote observation and sensing, etc. The useful signal is often described by a quasi-deterministic function of time, which can be regular or discontinuous. In recent years, problems of processing ultra-wideband signals have been of significant interest. Among the many UWB signals, it is customary to distinguish a separate class of ultra-wideband quasi-radio signals, the structure of which is similar to the structure of narrowband radio signals, but the condition of relative narrowband for which may not be met. Algorithms for estimating the arrival time of regular ultra-wideband quasi-radio signals have been studied in the existing literature. In this case, the synthesis and analysis of an algorithm for quasi-plausible estimation of the arrival time of a discontinuous signal of finite duration and arbitrary shape is of practical interest. The purpose of the work is to synthesize and analyze a quasi-likelihood algorithm for estimating the time of arrival of a discontinuous ultra-wideband quasi-radio signal with an unknown amplitude and initial phase against a background of white Gaussian noise. In this work, closed asymptotically exact expressions for the bias and scattering of the estimate are obtained, which make it possible to study the effectiveness of the synthesized algorithm, as well as compare its effectiveness with special cases of signal models - narrowband and wideband radio and video signals. The results of the study make it possible to determine losses in the accuracy of estimation the signal arrival time due to a priori ignorance of the initial phase and shape of the modulating function of the signal. Depending on the required accuracy of estimate the time of arrival of an ultra-wideband quasi-radio signal, it is possible to optimize the parameters of the transmitted signals and determine the limits of applicability of the designed radio system.

On the Stability of Planetary Motions During Stellar Approaches

The problem of the spatial motion of a passively gravitating body during an to the central body of a perturbing body – a test star – is considered. Using the exact expression of the force function, an integral invariant relationship – a quasi-integral – was found. Due to the quasi-integral, the regions of possible motion of the passively gravitating body, the surfaces of minimal energy (a generalization of the zero velocity surfaces), and the singular points of these surfaces were determined. The stability of planetary motion according to Hill during the approach of a test star to the Solar System was investigated. Criteria for the possibility, as well as the impossibility of capturing the passively gravitating body by the test star, were established. According to the Hill stability criteria, critical values of the orbital parameters of the test star were established, at which the planets of the Solar System either become satellites of the test star or leave the bounds of the Solar System.

Three Cases of Complex Eigenvalue/Vector Distributions of Symmetric Order-Three Random Tensors

Abstract Random tensor models have applications in a variety of fields, such as quantum gravity, quantum information theory, mathematics of modern technologies, etc., and studying their statistical properties, e.g. tensor eigenvalue/vector distributions, is interesting and useful. Recently some tensor eigenvalue/vector distributions have been computed by expressing them as partition functions of 0D quantum field theories. In this paper, using this method, we compute three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors, where the three cases can be characterized by the Lie-group invariances, $O(N,\mathbb {R})$, $O(N,\mathbb {C})$, and $U(N,\mathbb {C})$, respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the “signed” distribution, which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-N limit by computing the edge of the last signed distribution, obtaining agreement with an earlier numerical result in the literature.

Mathematical analysis of isothermal study of reverse roll coating using Micropolar fluid

This article demonstrates a mathematical model and theoretical analysis of the Micropolar fluid in the reverse roll coating process. It is important because micropolar fluids account for the microstructure and microrotation of particles within the fluid. These characteristics are significant for accurately describing the behavior of complex fluids such as polymer solutions, biological fluids, and colloidal suspensions. First, we modeled the flow equations using basic laws of fluid dynamics. The flow equations are made modified using low Reynolds number theory. The simplified equations are solved analytically. The exact expression for velocity and pressure gradient are obtained, while pressure is calculated numerically using Simpson Rule. Graphical depictions are carried out to comprehend the impact of the newly emerged physical constraints. The influence of micropolar and microrotation parameters on the velocity, pressure and pressure gradient are elaborated with the help of different graphs.