General Analytical Expression Research Articles

Energetics of particle-size segregation

We introduce a continuum framework for the energetics of particle-size segregation in bidisperse granular flows. Building on continuum segregation equations and a recent segregation flux model, the proposed framework offers general analytical expressions to study the physics of granular flows from a mechanical energy perspective. We demonstrate the framework's applicability by examining the energetics of shear-driven granular flows. Numerical experiments with varying frictional coefficients and particle-size ratios reveal two distinct phases in the energetics, marked by the separate onset of particle segregation and diffusive remixing. Furthermore, our numerical simulations alongside previous experimental results show that the bulk Richardson number $Ri$ , defined as the potential energy to kinetic energy ratio at steady state, follows the scaling relationship $Ri\equiv \hat {E}^{(s)}_{gp} / \hat {E}^{(s)}_{k} \propto Pe^{-1/2}_{sr}$ for $0.1 \leq Ri\leq ~10^{3}$ and $10^{-4} \leq Pe_{sr} \leq ~300$ , the segregation–rheology Péclet number. Finally, we present a Péclet-number-dependent theoretical expression for the degree of mixing (or segregation), validated by the compiled numerical and experimental dataset. Our findings hint that the bulk segregation–mixing state can be predicted and controlled using the segregation Péclet number $Pe$ and $Pe_{sr}$ , both determined from known system parameters, providing an instrumental tool for engineering and geophysical applications.

Genuine N-partite entanglement in Schwarzschild-de Sitter black hole spacetime

Complex quantum information tasks in a gravitational background require multipartite entanglement for effective processing. Therefore, it is necessary to investigate the properties of multipartite entanglement in a relativistic setting. In this paper, we study genuine N-partite entanglement of massless Dirac fields in the Schwarzschild-de Sitter (SdS) spacetime, characterized by the presence of a black hole event horizon (BEH) and a cosmological event horizon (CEH). We obtain the general analytical expression of genuine N-partite entanglement shared by n observers near BEH and m (n+m=N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+m=N$$\\end{document}) observers near CEH. It is shown that genuine N-partite entanglement monotonically decreases with the decrease of the mass of the black hole, suggesting that the Hawking effect of the black hole destroys quantum entanglement. It is interesting to note that genuine N-partite entanglement is a non-monotonic function of the cosmological constant, meaning that the Hawking effect of the expanding universe can enhance quantum entanglement. This result contrasts with multipartite entanglement in single-event horizon spacetime, offering a new perspective on the Hawking effect in multi-event horizon spacetime.

Effect of leakage current on magnetoelectric effect of 0-3 multiferroic composites based on an equivalent circuit model

In 0-3 multiferroic composites, the content of ferromagnetic particles with conductive properties significantly influences their magnetoelectric (ME) performance. The occurrence of leakage current (Lc) at high contents is a crucial factor affecting the ME properties of such composites. Establishing a rational and effective theoretical model to analyze and predict the impact of Lc on their ME performance is of paramount theoretical significance for expanding the applications of these composites. In pursuit of this objective, by introducing conduction currents, magneto-electric-mechanical/electro-magnetic-mechanical coupling factor, and resistive components, an equivalent circuit model is built to analyze the direct and converse ME effects of this composites. Meanwhile, the general analytical expressions of various effective properties are obtained. By introducing factors with combined exponential and linear decay characteristics into the piezoelectric coefficient, our theoretically calculated results quantitatively agree with experimental data. On this basis and considering the effects of displacement current and conduction current, we utilized our equivalent circuit model to calculate the ME coefficients, which showed high consistency with experimental results. Especially at high volume fractions, our computed results accurately reflected the sharp decline characteristics of the ME coefficients caused by Lc. This demonstrates the accurate and reasonable explanation provided by our model for such ME problems. In addition, the optimal set of volume fractions and magnetic fields that maximize the direct ME coefficient is obtained. This study fills a theoretical gap in the equivalent circuit method for 0-3 multiferroic composites and gives an analytical solution for the ME coefficients and provides theoretical guidance and support for the application of this composites.

Analysing Flexural Response in RC Beams: A Closed-Form Solution Designer Perspective from Detailed to Simplified Modelling

This paper presents a detailed analytical approach for the bending analysis of reinforced concrete beams, integrating both structural mechanics principles and Eurocode 2 provisions. The general analytical expressions derived for the curvature were applied for the transverse displacement analysis of a simply supported reinforced concrete beam under four-point loading, focusing on key limit states: the initiation of cracking, the yielding of tensile reinforcement and the compressive failure of concrete. The displacement’s results were validated through experimental testing, showing a high degree of accuracy in the elastic and crack propagation phases. Deviations in the yielding phase were attributed to the conservative material assumptions within the Eurocode 2 framework, though the analytical model remained reliable overall. To streamline the computational process for more complex structures, a simplified model utilising a non-linear rotational spring was further developed. This model effectively captures the influence of cracking with significantly reduced computational effort, making it suitable for serviceability limit state analyses in complex loading scenarios, such as seismic impacts. The results demonstrate that combining detailed analytical methods with this simplified model provides an efficient and practical solution for the analysis of reinforced concrete beams, balancing precision with computational efficiency.

Bosonic and fermionic coherence of N-partite states in the background of a dilaton black hole

We study the N-partite coherences of GHZ and W states for free bosonic and fermionic fields when any n observers hover near the event horizon of a Garfinkle-Horowitz-Strominger (GHS) dilaton black hole. We derive the more general analytical expressions for N-partite coherence, encompassing both physically accessible and inaccessible coherences in the context of the dilaton black hole. It has been found that the coherence of the bosonic field is greater than that of the fermionic field, while the entanglement of the fermionic field is greater than that of the bosonic field in dilaton spacetime. Additionally, the coherence of the W state is greater than that of the GHZ state, whereas the entanglement of the GHZ state is greater than that of the W state in curved spacetime. These results suggest that we should utilize suitable quantum resources and different types of particles for relativistic quantum information tasks.

Gaussian model of fluctuating membrane and its scattering properties.

A mathematical model is developed to jointly analyze elastic and inelastic scattering data of fluctuating membranes within a single theoretical framework. The model builds on a nonhomogeneously clipped time-dependent Gaussian random field. This specific approach provides one with general analytical expressions for the intermediate scattering function for any number of sublayers in the membrane and arbitrary contrasts. The model is illustrated with the analysis of small-angle x-ray and neutron scattering as well as with neutron spin-echo data measured on unilamellar vesicles prepared from phospholipids extracted from porcine brain tissues. The parameters fitted on the entire data set are the lengths of the chain and the head of the molecules that make up the membrane, the amplitude and lateral sizes of the bending deformations, the thickness fluctuation, and a single parameter characterizing the dynamics.

An exact analytical solution for the weakly magnetized flow around an axially symmetric paraboloid, with application to magnetosphere models

Rotationally symmetric bodies with longitudinal cross sections of parabolic shape are frequently used to model astrophysical objects, such as magnetospheres and other blunt objects, immersed in interplanetary or interstellar gas or plasma flows. We discuss a simple formula for the potential flow of an incompressible fluid around an elliptic paraboloid whose axis of symmetry coincides with the direction of incoming flow. Prescribing this flow, we derive an exact analytical solution to the induction equation of ideal magnetohydrodynamics for the case of an initially homogeneous magnetic field of arbitrary orientation being passively advected in this flow. Our solution procedure employs Euler potentials and Cauchy's integral formalism based on the flow's stream function and isochrones. Furthermore, we use a particular renormalization procedure that allows us to generate more general analytical expressions modeling the deformations experienced by arbitrary scalar or vector-valued fields embedded in the flow as they are advected first toward and then past the parabolic obstacle. Finally, both the velocity field and the magnetic field embedded therein are generalized from incompressible to mildly compressible flow, where the associated density distribution is found from Bernoulli's principle.

Simplified Universal Equations for Ionic Conductivity and Transference Number

Nernst-Einstein equation can provide a reasonable estimate of the ionic conductivity of dilute solutions. For concentrated solutions, alternate methods such as Green–Kubo relations and Einstein relations are more suitable to account for ion-ion interactions. Such computations can be expensive for multicomponent systems. Simplified mathematical expressions like the Nernst-Einstein equation do not exist for concentrated multicomponent mixtures. Newman’s treatment of multicomponent concentrated solutions yields a conductivity relation in terms of species concentration and Onsager phenomenological coefficients. However, the estimation of these phenomenological coefficients is not straightforward. Here, mathematical formulations that relate the phenomenological coefficients with the friction coefficients are developed, leading to simplified, ready-to-use expressions of conductivity and transference numbers that can be used for a wide range of ionic mixtures. This approach involves spectral decomposition of the matrix of Onsager phenomenological coefficients. The general analytical expressions for conductivity and transference number are simplified for binary electrolytes, and numerical solutions are provided for ternary and quaternary mixtures with ion dissociation.

Analytical study on the upward laser beam propagation in the turbulent atmosphere

The upward laser beam propagation in the turbulent atmosphere is studied analytically, where the nonlinear self-focusing, the atmospheric turbulence, and the atmospheric extinction effects are all taken into account. The analytical propagation expression of intensity is derived. The initial beam defocusing and the optimal beam power are two methods to improve the beam quality at the target. The analytical expressions of the focal length considering the initial beam defocusing and the corresponding target intensity are derived. Moreover, the analytical expressions of optimal beam power and the corresponding target intensity are also derived, and the target intensity is independent of the atmospheric extinction. In particular, the criterion for which method (the initial beam defocusing or the optimal beam power) to use is derived to optimize the target beam quality. The choice of the two methods depends on the laser beam parameters and the turbulent atmosphere parameters.

Минимизация вероятности ошибки при взвешенном приеме сообщения с двухпозиционной импульсно-кодовой манипуляцией в условиях межсимвольной интерференции

In order to increase the noise immunity of data transmission systems, it is proposed to reduce the level of intersymbol interference by means of orthogonalizing the channel symbols. Orthogonalization is done through determining the orthogonality weight. Presented below is a calculation of the E-criterion for assessing the level of intersymbol interference for the proposed data transmission systems using the pulse-code modulation. For both systems, it was assumed that normalized Butterworth low-pass filters would be used as the shaping filter. It is shown that when signal-to-noise ratios exceed a certain critical value, weighted signal processing provides a lower bit error probability compared to the conventional processing technique. A general analytical expression for the orthogonality weight of channel symbols is given, which ensures a minimum bit error probability.

Accretion of matter by a charged dilaton black hole

Considering accretion onto a charged dilaton black hole, the fundamental equations governing accretion, general analytic expressions for critical points, critical velocity, critical speed of sound, and ultimately the mass accretion rate are obtained. A new constraint on the dilation parameter coming from string theory is found and the case for polytropic gas is delved into a detailed discussion. It is found that the dialtion and the adiabatic index of accreted material have deep effects on the accretion process.

Autofocusing circular symbolic umbilic beams

In this paper, we report a new class of autofocusing beams, circular symbolic umbilic beams (CSUBs), based on the catastrophe theory. The analytical propagation expression and the evolution of the 2+1-D SUBs are firstly introduced and then the autofocusing properties of CSUBs are theoretically and experimentally studied in detail, including the amplitude distribution of the light field, the autofocusing capability and the focusing length. Compared with the previous circular Airy, Pearcey or swallowtail beams, CSUBs can focus twice without extra phase term or optical component. The autofocusing capabilities in two focal planes are remarkable as well as comparable and the two focusing lengths are symmetrically distributed around a fixed position. The influences of related parameters on CSUBs are also analyzed. Furthermore, by applying a chirp phase on CSUB, the beam exhibits a thrice focusing behavior during propagation. Our experimental results match well with theoretical predictions.

Atom-photon dressed states in a waveguide-QED system with multiple giant atoms.

We study the properties of bound states in waveguide-QED systems consisting of multiple giant atoms coupled to a coupled-resonator waveguide. Based on the general analytical expressions for these states and the corresponding energy spectra, we analyze in detail the threshold conditions for the appearance of bound states and the photon-mediated interactions between dressed atoms for different configurations. In addition, when multiple giant atoms are coupled to the waveguide, different types of interacting atomic chain can be obtained by manipulating the coupling configurations. Accordingly, the energy spectra of the bound states form metaband structures in the photonic band gaps. This makes the system a useful platform for quantum simulation and quantum information processing.

Study on the construction of twisted cosine partially coherent beams and their propagation characteristics

We propose a novel Schell model source for generating twisted partially coherent beams with an initial radius of curvature, which is called a twisted flat-topped cosine Gaussian Schell-model (TFCGSM) source. The TFCGSM beam comprises a wavefront phase and a flat-top structure, with the source degree of coherence determined by two cosine functions. Based on the Huygens–Fresnel principle, the general analytical expression of the cross-spectral density function of the TFCGSM beam propagating through the paraxial ABCD optical system is derived, and then its propagation properties are studied. The results show that the conversion of the array of the beam and the non-uniform structure can be realized by adjusting the parameters in the source plane. As the propagation distance of the TFCGSM beam increases, it rotates around the axis and increases the intensity of the array distribution. Surprisingly, the initial radius of curvature can cause the beam to rotate. The unique shape and properties of the TFCGSM beam create new possibilities for optical communication and enhanced optical functions.

General Exact Analytical Expressions for Rotation and Displacement of a Timoshenko Beam Under Variable Loads with Validation

The researchers involved in the field of applied mathematics mostly project the efficient and viable solutions of those problems in which have practical applications in science and engineering. The Timoshenko beam model (TBM) is described by a system of ordinary differential equations where the expressions of rotation and displacement of the beam are ultimately required. Most of the work on analytical solutions of beam problems in literature focuses the elastic and Euler-Bernoulli beams, whereas for the TBM the numerical solutions are usually preferred. The analytical expressions for the TBM exist for only very basic load cases and are also not general for any load function. In this paper, we attempt to suggest a novel protocol based on expressing a load function by a polynomial or its power series development, and then use it to develop general analytical expressions for the rotation and displacement of a fixed TBM which is not load specific. The proposed general equations can provide ease of access and handling for the practitioners working in applied mathematics, structural engineering and mechanical vibrations as the developed equations can be used with only constant inputs to tailor particular expressions of rotation and displacement profiles of a fixed TBM under any variable load. For performance evaluations of the proposed general equations, we have also obtained particular expressions for some important variable loads, like: linearly varying loads ((LVLs): triangular and trapezoidal and quadratically varying loads (QVLs): parabolic/square and circular loads. Finally, the proposed protocol for generalization has been validated for fixed elastic beams under uniformly distributed loads (UDLs), and the results match exactly with those expressions available in literature. The contributions of this study, on one hand provide ready, direct and exact general expressions for the rotation and displacement profiles of a fixed TBM, while on the other hand the solution of such problem is achieved with quite negligible computational overhead, execution time and software implementations.

The majoron coupling to charged leptons

The particle spectrum of all Majorana neutrino mass models with spontaneous violation of global lepton number include a Goldstone boson, the so-called majoron. The presence of this massless pseudoscalar changes the phenomenology dramatically. In this work we derive general analytical expressions for the 1-loop coupling of the majoron to charged leptons. These can be applied to any model featuring a majoron that have a clear hierarchy of energy scales, required for an expansion in powers of the low-energy scale to be valid. We show how to use our general results by applying them to some example models, finding full agreement with previous results in several popular scenarios and deriving novel ones in other setups.

Exploring approximate analytical expression for magnetic skyrmion structure based on symbolic regression method

Magnetic skyrmion is a kind of nontrivial topological magnetic structure, which can exist stably in chiral magnet with Dzyaloshinskii-Moriya (DM) interaction, and its static and dynamic properties are closely related to its structural characteristics. However, there are no general analytical expressions for skyrmion profiles. Therefore, many researchers have provided approximate solutions. In this paper, a new approach to exploring magnetic skyrmion structures is introduced by using a symbolic regression approach. Considering the influence of DM interaction and external magnetic field on magnetic skyrmion structure, two suitable approximate expressions are obtained through symbolic regression algorithms. The applicability of these expressions depends on the dominant interaction. The research results in this work validate the powerful capability of symbolic regression algorithms in exploring the magnetic skyrmion profiles. So, the present study provides a new method for finding the analytical expressions for magnetic structure.

Problems for combinatorial numbers satisfying a class of triangular arrays

Numbers satisfying a class of triangular arrays, defined by a bivariate first-order linear difference equation with linear coefficients, include a wide range of combinatorial numbers: binomial coefficients, Morgan numbers, Stirling numbers of the first and the second types, non-central Stirling numbers, Eulerian numbers, Lah numbers, and their generalizations. In this work, we derive the general analytic expression of the numbers satisfying a class of triangular arrays and propose problems (both teaching and unsolved ones) for undergraduates studying probability theory and analytical combinatorics subjects in the study programs of the fields of mathematics and computer science. Some of the unsolved challenges can also be used as the basis for a thesis.

Genuinely accessible and inaccessible entanglement in Schwarzschild black hole

The genuine entanglement of Dirac fields for an N-partite system is investigated in Schwarzschild spacetime and the analysis is carried out using the single-mode approximation. Due to the Hawking effect, quantum entanglement is divided into two parts physically accessible and inaccessible entanglement. We obtain a general analytic expression of genuine N-partite entanglement that includes all accessible and inaccessible entanglement in a Schwarzschild black hole. Unlike bosonic entanglement, the accessible N-partite entanglement of Dirac fields monotonically decreases to a nonzero value with the Hawking temperature. Interestingly, the inaccessible N-partite entanglement is a monotonic or non-monotonic function of the Hawking temperature, depending on the ratio between accessible and inaccessible modes, in contrast to bipartite or tripartite entanglement that is only a monotonic function of the Hawking temperature. Finally, we obtain two restrictive relationships for the quantum information of the black hole. This conclusion provides a new understanding of Hawking effect of the black hole.

Theoretical study of nonlinear absorption of a strong electromagnetic wave in infinite semi-parabolic Plus Semi-inverse Squared Quantum Wells by using quantum kinetic equation

General analytic expressions for the total absorption coefficient of strong electromagnetic waves caused by confined electrons in Infinite semi-parabolic plus Semi-inverse Squared Quantum Wells (ISPSISQW) are obtained by using the quantum kinetic equation for electrons in the case of electron-optical phonon scattering. A second-order multi-photon process is included in the result. The dependence of the total absorption coefficient on the intensity E0, the photon energy ħΩ of an SEMW, and the temperature T for a specific GaAs/GaAsAl ISPPSISQW is achieved by using a numerical method. The computational results demonstrate that the total absorption coefficient’s dependence on photon energy can be utilized for optically detecting the electric sub-bands in an ISPPSISQW. Besides, we also give theoretical rules on the dependence of the Full Width at Half Maximum on important external parameters such as temperature and perpendicular magnetic field. Furthermore, the obtained results are consistent with prior theoretical and experimental findings.