Explicit Analytical Formulas Research Articles

ANALYTICAL CALCULATION OF A BEAM BASED ON AN ELASTIC WINKLER FOUNDATION WITH RANGE INHOMOGENITY

The aim of the study is the further development of analytical methods for calculating the bending of beams resting on a non-homogeneous continuous Winkler elastic foundation. This paper considers the case when the beam is under the influence of a uniformly distributed constant transverse load, and the inhomogeneity of the elastic foundation is given by a power function with an arbitrary non-negative power exponent . Fundamental functions and a partial solution of the corresponding differential equation of beam bending are found in an explicit closed form. These functions are dimensionless and are represented by absolutely and uniformly convergent power series. In turn, the formulas for the parameters of the stress-strain state of the beam – deflection, angle of rotation, bending moment and transverse force – are expressed through the indicated functions. The unknown constants of integration in these formulas are expressed in terms of the initial parameters, which are after the implementation of the specified boundary conditions. Thus, the calculation of the beam for bending is reduced to the procedure of numerical implementation of explicit analytical formulas for the parameters of the stress-strain state. An example demonstrates the practical application of the obtained solutions. A prismatic concrete beam based on a cubic variable elastic foundation is considered. This case corresponds to the power value . The results of the calculation by the author's method are presented in numerical and graphical formats for the case when the left end of the beam is hinged and the right end is clamped. The numerical values obtained by the author's method are accurate, since the applied calculation method is based on the exact solution of the corresponding differential equation. The availability of such solutions makes it possible to evaluate the accuracy of solutions obtained using various approximate methods by comparison. For the purpose of such a comparison, the paper presents the calculation results obtained by the finite element method (FEM). The absolute error of the FEM method when calculating this design was determined.

Non-linear filtration model with splitting front

We consider filtration of a suspension in a homogeneous porous medium which is described by a macroscopic 1-D model including the mass exchange equation and the kinetic equation for deposit growth. The standard model assumes that suspended particles move at the same speed as the carrier fluid. However, in some experiments, a lag between the front boundary of suspended particles and the front of the carrier fluid was observed. This article proposes a modification of the standard model that provides a description of the separation between the fluid front and the particle front. For this purpose, a non-smooth non-linear function depending on the concentration of the suspension is introduced into the deposit growth equation. In case of a non-smooth suspension function, the concentration of suspended particles at the front decreases to zero in a finite time. At this moment the united front splits into the front of pure injected water and the front of suspended and retained particles. The particle front moves slower than the pure water front. In case of a linear filtration function (Langmuir coefficient), an exact solution is constructed in a closed form. For the filtration problem with a suspension function in the form of a square root, explicit analytical formulae are obtained. In case of a non-smooth filtration function, the filtration time is finite. The curvilinear boundary separates the filtration domain, where concentrations increase with time, from the stabilization domain, where the concentrations of suspended and retained particles have reached their limits. The limit speeds of the stabilization border and of the particle front coincide.

Higher-order phase-space moments for off-diagonal rotating Morse oscillators

The present paper proposes an advancement beyond the recent mathematical formulation [Cherroud O and Yahiaoui S-A 2023 Eur. Phys. J. Plus. 138 534], by extending their approach to the evaluation of off-diagonal matrix elements 〈xˆmpˆl〉n,n′J,J′ for the rotation-vibration Morse oscillators. Calculations are conducted within the framework of phase-space quantum mechanics, considering (r → r eq). This methodology facilitates the derivation of analytical expressions for 〈xˆs〉n,n′J,J′ and 〈pˆs〉n,n′J,J′ matrix elements, where s = 1, 2, …. This approach enables the evaluation of both diagonal ( n=n′,J=J′≠0 ) and off-diagonal ( n≠n′,J=J′≠0 ) matrix elements through a more explicit and compact analytic formula, applicable for cases where J≠J′ as well as for scenarios involving non-rotating effects J=J′=0 .

Pricing derivatives on foreign assets using Markov-modulated cojump-diffusion dynamics

In this study, cross-currency derivatives pricing under stochastic interest rates is analyzed when Markov-modulated cojump-diffusion dynamics with both idiosyncratic and simultaneous jumps drive the underlying foreign equity price and foreign exchange rate processes. More specifically, the noise dynamics of these two assets are captured by the correlated Wiener processes, and the two types of jump events are described as two independent compound Poisson processes with log-normal jump amplitudes. Cojumping addresses the correlation between these two asset price jumps. In an incomplete market setting, the dynamic Esscher transform approach is applied to determine a pricing kernel. According to the resulting dynamics, explicit analytical formulas for evaluating values of European foreign equity call options are provided. Numerical experiments are also described.

On the handling of inconsistent systems of max-min fuzzy relational equations

In this article, we study the inconsistency of systems of max-min fuzzy relational equations. We first compute by an explicit analytical formula the Chebyshev distance (defined by the L-infinity norm) between the second member of an inconsistent system and the set of second members of consistent systems defined with the same matrix: that of the inconsistent system. Then we study the set of Chebyshev approximations of the second member of the inconsistent system, which are the second members of the consistent systems defined with the same matrix as the inconsistent system, whose distance to the second member of the inconsistent system is equal to the computed Chebyshev distance. This set is non-empty and associated to the approximate solutions set of the inconsistent system in the following sense: an approximate solution is a solution vector of a consistent system defined with the same matrix as the inconsistent system, whose second member is a Chebyshev approximation of the second member of the inconsistent system. As main results, we describe both the structure of the approximate solutions set and that of the set of the Chebyshev approximations of the second member of the inconsistent system. As an application of our results, we show that our work can help to find approximate values for the rule parameters of a possibilistic rule-based system.

Tunable optical absorption in undoped graphene sandwiched between multilayer dielectric stacks with mirror symmetry

We theoretically investigate the optical absorption of an undoped graphene monolayer when put in a one-dimensional multilayer stack. Using the transfer matrix method, we perform numerical simulations and derive explicit analytical formulas for the optical absorption of the graphene monolayer at the center of the dielectric stack and find that the optical absorption uniquely depends on repetition number (r) and the unit layers structure. When sandwiched between unit layers structure composed of three dielectric materials (referred to as the ‘ABC’ structure) with even values of r, the graphene monolayer absorbs 2.3% of visible to near-infrared light. This behavior is the same as if graphene were free-standing, not sandwiched between the dielectric stack. In contrast to that situation, in the ABC structure with odd values of r, also when the graphene monolayer is sandwiched between four materials (the ‘ABCD’ structure) with any values of r, we can obtain optical absorption as large as 50% at particular refractive indices (n) of the constituent dielectric materials. The 50% absorption is, in fact, the maximum optical absorption for any undoped monolayer material in the symmetric dielectric stacks. By varying r and n within the ABC or ABCD structures, we can finely adjust the optical absorption of graphene within the range of 0%–50%, facilitating precise control for various optoelectronic applications.

Lower bound of entanglement of formation based on correlation matrix

The entanglement of formation (EOF), which quantifies the required minimally physical resources to prepare a quantum state, plays important roles in many quantum information processes. Due to the convex-roof definition, it is hard to find the explicit analytic formulae of EOF, except for some special quantum systems. The correlation matrix of a quantum state is closely related to entanglement property. In this paper, we use the Lagrange-multiplier approach to obtain an analytic lower bound of EOF for arbitrary dimensional bipartite states, which improves the previous conclusions. Numerical example shows that the bound is also valid for some PPT entangled states.

New analytical results in solid state physics using the Lambert W function

Analytical solutions in physics are always preferred for the sake of mathematical completeness. For this, using the Lambert W function, I derive closed-form analytical expressions for the equilibrium interionic distance in an ionic crystal, the formation energy of a vacancy in a crystal, the zero-temperature energy gap of a clean-limit superconductor and the critical Kosterlitz–Thouless temperature for the phase transition in 2D-XY model. Besides their theoretical interest, some of the present results suggest alternative experimental determinations of the relevant physical quantities. In addition, I similarly derive an explicit analytical formula based on the Lambert W function for a determination of the Boltzmann constant. This formula is proposed here as a theoretical basis for an experimental method to measure this constant. This method is suitable for undergraduate physics students.

On the Lifeline Game of the Inertial Players with Integral and Geometric Constraints

In this paper, we consider a pursuit–evasion game of inertial players, where the pursuer’s control is subject to integral constraint and the evader’s control is subject to geometric constraint. In the pursuit problem, the main tool is the strategy of parallel pursuit. Sufficient conditions are obtained for the solvability of pursuit–evasion problems. Additionally, the main lemma describing the monotonicity of an attainability domain of the evader is proved, and an explicit analytical formula for this domain is given. One of the main results of the paper is the solution of the Isaacs lifeline game for a special case.

A new formulation of the surface charge/surface potential relationship in electrolytes with valence less than three

AbstractSurface complexation models (SCMs) based on Gouy-Chapman theory are often used to describe adsorption of ions onto mineral surfaces. To compensate for the buildup of charge at a solid surface, the composition of the electric diffuse layer next to the surface must balance the surface charge. To calculate the diffuse layer composition, several nonlinear equations and integrals must be solved, usually with an iterative approach. Before convergence, charge balance is typically not fulfilled. One numerical difficulty is that, because of these charge balance errors, the iterative solver may attempt to take the square root of a negative number. Herein, we show that for electrolytes containing only monovalent or divalent ions (i.e., most electrolytes encountered in practice), we can greatly simplify the integrals and eliminate the appearance of complex-valued integrands; it is even possible to derive explicit analytical formulas. Furthermore, using the new method prevents converging to non-physical roots of the Grahame equation, which links surface potential to surface charge. To the best of our knowledge, the presented formulation has not been implemented in geochemical modelling software before, although similar mathematical expressions have been presented in the literature. In Gouy-Chapman theory, ions can only be distinguished by their charges, but this is not consistent with all experimental findings. We present a model that allows for the preferential accumulation of ions in the diffuse layer. The model, which is implemented mathematically by including ion exchange sites with a variable exchange capacity, is flexible and more numerically tractable than the standard models for the diffuse layer.

Machine-learning investigation on the creep-rupture time of heat-resistant steels

Currently, the reliable prediction of creep-rupture times of heat-resistant alloys is a major research area with regard to structural materials. Numerous efforts to predict creep rupture times have been made using phenomenological parameter methods; however, these approaches have only been able to achieve a limited prediction accuracy. Furthermore, the applicability of machine learning to improve the prediction accuracy of creep-rupture time has been investigated. In both studies, nevertheless, approaches have been used where explicit analytical formulas are not available. Because creep rupture is a safety issue, it is essential to use an analytical equation for predicting rupture time. Therefore, in this study, a machine learning method was applied to the analysis of creep-rupture time of heat-resistant steel to obtain an explainable prediction formula. An accurate regression model was obtained using the linear independent descriptor generation (LIDG) regression method of adding independent higher-order terms that were generated from operations between variables. By utilizing LIDG regression, we show that various creep-rupture times could be well-reproduced by different heats using explanatory variables such as creep conditions, chemical compositions, and Vickers hardness. The heat-to-heat variation in creep-rupture time could not be accounted for at all in the evaluation using the conventional parameter method. On the other hand, the LIDG regression could naturally explain the difference in the creep rupture time of various heats. Furthermore, the excellent extrapolation of the LIDG regression model was confirmed, showing its potential for long-time creep-rupture time prediction updating the conventional parametric method.

Comparing two formulas for the Gross–Stark units

Let F be a totally real number field. Dasgupta conjectured an explicit p-adic analytic formula for the Gross–Stark units of F. In a later paper, Dasgupta–Spieß conjectured a cohomological formula for the principal minors and the characteristic polynomial of the Gross regulator matrix associated to a totally odd character of F. Dasgupta–Spieß conjectured that these conjectural formulas coincide for the diagonal entries of Gross regulator matrix. In this paper, we prove this conjecture when F is a cubic field. For a video summary of this paper, please visit https://youtu.be/hlBRUIOke04.

Developments in analytical wall shear stress modelling for water hammer phenomena

New developments in analytical modelling of wall shear stresses during water hammer events are presented and discussed in this paper. These include further development of models based on the quasi-steady and unsteady hypothesis of hydraulic resistances. All the analyzed solutions have been improved so that they have extended ranges of applicability and the simplest possible mathematical form. Exemplary comparative studies were carried out, which showed their advantages and disadvantages. Explicit analytical formulas of wall shear stress were derived for the first time. They will be particularly useful at the stage of design and modernization of the pressurized piping systems (water supply, oil-hydraulics etc.) and enable effective analysis of leakages and other damages (ruptures, fatigue related failures) occurring in industrial pipeline systems. With their help, it will be easier to prevent the increased pulsating pressures occurring during water hammer events (protect against failures), and to verify numerical models used for years in simulations of transient pipe flows.

Differential game with “lifeline” for Pontryagin's control example

The main purpose of this work is to solve one of the main problems of Isaacs, i.e., a game with a “lifeline” for Pontryagin’s control example when both players have the same movement dynamics. To solve this problem, the pursuer is offered a strategy of parallel pursuit (briefly, $\Pi$-strategy), which ensures the fastest convergence of the players and the capture of the evader within a certain closed ball. In addition, for the differential game under consideration, an explicit analytical formula for the players' attainability domain is given and the main lemma is generalized (L.A. Petrosjan's lemma on monotonicity of the players' attainability domain with respect to embedding for a game of simple pursuit). Using this main lemma, we find conditions for the solvability of the game with a “lifeline” for Pontryagin's control example as well. For clarity, at the end of the work, examples are given for some special cases.

Interpolation Formulas for Asymptotically Safe Cosmology

Simple interpolation formulas are proposed for the description of the renormalization group (RG) scale dependences of the gravitational couplings in the framework of the 2-parameters Einstein-Hilbert (EH) theory of gravity and applied to a simple, analytically solvable, spatially homogeneous and isotropic, spatially flat model universe. The analytical solution is found in two schemes incorporating different methods of the determination of the conversion rule k(t) of the RG scale k to the cosmological time t. In the case of the discussed model these schemes turn out to yield identical cosmological evolution. Explicit analytical formulas are found for the conversion rule k(t) as well as for the characteristic time scales tG and tΛ>tG corresponding to the dynamical energy scales kG and kΛ, respectively, arising form the RG analysis of the EH theory. It is shown that there exists a model-dependent time scale td (tG≤td<tΛ) at which the accelerating expansion changes to the decelerating one. It is shown that the evolution runs from a well-identified cosmological fixed point to another one. As a by-product we show that the entropy of the system decreases monotonically in the interval 0<t≤tΛ due to the quantum effects.

Inertia-gravity-wave diffusion by geostrophic turbulence: the impact of flow time dependence

The scattering of three-dimensional inertia-gravity waves by a turbulent geostrophic flow leads to the redistribution of their action through what is approximately a diffusion process in wavevector space. The corresponding diffusivity tensor was obtained by Kafiabad et al. (J. Fluid Mech., vol. 869, 2019, R7) under the assumption of a time-independent geostrophic flow. We relax this assumption to examine how the weak diffusion of wave action across constant-frequency cones that results from the slow time dependence of the geostrophic flow affects the distribution of wave energy. We find that the stationary wave-energy spectrum that arises from a single-frequency wave forcing is localised within a thin boundary layer around the constant-frequency cone, with a thickness controlled by the acceleration spectrum of the geostrophic flow. We obtain an explicit analytic formula for the wave-energy spectrum which shows good agreement with the results of a high-resolution simulation of the Boussinesq equations.

Analytical and Statistical Modelling of a Fast Ion Source Formed by Injection of a Neutral Beam into Magnetically Confined Plasma

Mathematical modelling of heating and current drive as well as yields and distributions of fusion products in a magnetically confined plasma subject to neutral beam injection requires, in turn, modelling of distributions of fast ions, which is a complex task including calculations of the source of suprathermal particles, i.e., the number of fast ions occurring in unit volume during unit time owing to the injection of fast atoms. The knowledge of the magnetohydrodynamic equilibrium, beam injection geometry and spatial distribution of the magnetic field are the necessary prerequisites. Explicit general analytical formulae for the source of fast ions have been obtained by two different methods. In addition, a method of statistical modelling is presented. Calculations of spatial and angular distributions of the fast ion source for a tokamak and verifications of the obtained results have been performed by a number of methods.

DYNAMICS OF A BODY UNDER THE IMPULSE FORCE AND WITH OSCILLATION LIMITER

The paper investigates the dynamics of a body under a piecewise constant periodic force with an arbitrary duty cycle and in the presence of an oscillation limiter. The mathematical model is a strongly nonlinear non-autonomous dynamical system with a truncated phase space along one of the phase coordinates. Using the point mapping method for strongly nonlinear (vibro-impact) dynamical systems, equations for two-dimensional Poincare map are obtained in the form of explicit analytical formulas. The relations obtained for Poincare map made it possible to effectively study the rearrangements of arbitrarily complex periodic motions with both a finite and an infinite number of fixed points, depending on the change in the parameters of the dynamical system. This allowed us to represent for the first time an analytical form of the equations that define in the parameter space the boundaries of the existence and stability domain of periodic motions with an infinite number of fixed points on the Poincare surface (corresponds to the infinitely impact mode of motion). The numerical experiments presented in the paper using a software product developed in the C++ language made it possible to present bifurcation diagrams that clearly demonstrate the ranges of parameter values with chaotic motion modes of the body. A scenario for the emergence of chaos has been established. The transition of body motions with a change in the overload parameter from periodic motion modes to chaos is carried out according to the period doubling. Comparison of numerical analysis with analytical results for different sets of parameters of the system under study (overload parameter) showed their good agreement.

The expected values for the Gutman index, Schultz index, and some Sombor indices of a random cyclooctane chain

AbstractIn this paper, we first introduce the explicit analytical formulas for the expected values of the Gutman and Schultz indices for a random cyclooctane chain . Meanwhile, the explicit formulas of the variances of the Gutman and Schultz indices for a random cyclooctane chain are determined and we prove these two indices are asymptotically subject to normal distribution. Furthermore, we are surprised to find the expected values of the Sombor index , the reduced Sombor index , the modified Sombor index and the reduced and modified Sombor index for a random cyclooctane chain.

Symmetry-resolved Page curves

Given a statistical ensemble of quantum states, the corresponding Page curve quantifies the average entanglement entropy associated with each possible spatial bipartition of the system. In this work, we study a natural extension in the presence of a conservation law and introduce the symmetry-resolved Page curves, characterizing average bipartite symmetry-resolved entanglement entropies. We derive explicit analytic formulae for two important statistical ensembles with a $U(1)$-symmetry: Haar-random pure states and random fermionic Gaussian states. In the former case, the symmetry-resolved Page curves can be obtained in an elementary way from the knowledge of the standard one. This is not true for random fermionic Gaussian states. In this case, we derive an analytic result in the thermodynamic limit based on a combination of techniques from random-matrix and large-deviation theories. We test our predictions against numerical calculations and discuss the sub-leading finite-size corrections.