Analytical Form Research Articles

What is the Exchange Repulsion Energy? Insight by Partitioning into Physically Meaningful Contributions.

It is shown that the exchange repulsion energy, Exr, can be rationalized by partitioning the respective energy expression for two systems with Hartree-Fock orbitals into physically meaningful contributions. A division of Exr into a positive kinetic and a negative potential part is possible, but these contributions correlate only poorly with the actual exchange repulsion energy. A more meaningful partitioning is derived, where all kinetic energy contributions are collected in a term that vanishes for exact Hartree-Fock orbitals due to their stationarity conditions. The remaining terms can be distinguished into an exchange integral contribution as well as contributions to the repulsion energy with two, three and four orbital indices. The forms, relationships and absolute sizes of these terms suggest an intuitive partitioning of the exchange repulsion energy into Molecular Orbital Pair Contributions to the Exchange repulsion energy (MOPCE). Insight into the analytic form and quantitative size of these contributions is provided by considering the state of the H2 molecule, the water dimer, as well as an argon atom interacting with Cl2 and N2.

Anisotropic plasmons in threefold Hopf semimetals

Abstract Threefold Hopf semimetals are a novel type of topological semimetals that possess an internal anisotropy characterized by a dipolar structure of the Berry curvature and an isotropic energy band structure consisting of a Dirac cone and a flat band. In this study, we theoretically investigate the impact of internal anisotropy on plasmons in threefold Hopf semimetals using random-phase approximation. In contrast to the classical intuition that isotropy of the energy band dispersion leads to isotropic plasmons in the classical regime (i.e., in the wavelength limit), we find that plasmons in threefold Hopf semimetals exhibit notable anisotropy even in the long-wavelength limit. We derive an explicit analytical form of the long-wavelength plasmon frequency, and numerically demonstrate the validity of our results in a wide range of situations. Our work reveals that the anisotropy of long-wavelength plasmons can reach 25%, making it experimentally observable.

On Non-Linear Differential Systems with Mixed Boundary Conditions

For the constructive analysis of locally Lipschitzian system of non-linear differential equations with mixed periodic and two-point non-linear boundary conditions, a numerical-analytic approach is developed, which allows one to study the solvability and construct approximations to the solution. The values of the unknown solution at the two extreme points of the given interval are considered as vector parameters whose dimension is the same as the dimension of the given differential equation. The original problem can be reduced to two auxiliary ones, with simple separable boundary conditions. To study these problems, we introduce two different types of parametrized successive approximations in analytic form. To prove the uniform convergence of these series, we use the appropriate technique to see that they form Cauchy sequences in the corresponding Banach spaces. The two parametrized limit functions and the given boundary conditions generate a system of algebraic equations of suitable dimensions, the so-called system of determining equations, which give the numerical values of the introduced unknown parameters. We prove that the system of determining equations define all possible solutions of the given boundary value problems in the domain of definition. We established also the existence of the solution based on the approximate determining system, which can always be produced in practice. The theory was presented in detail in the case of a system of differential equations consisting of two equations and having two different solutions.

Self-similar solutions of the bed deformation

In this paper, a conclusion about the self-similar behavior of the bed surface evolution is made. It is based on the analysis of experimental and numerical studies of the bed surface evolution under the mechanical impact of liquid flow. The bottom wave has a form close to one period of a sinusoidal function with a time-varying wavelength and constant steepness. A method of constructing the automodel dependence of the bed surface on time and spatial coordinate in analytical form is proposed. It was shown that it is enough to select five bottom surfaces with given wavelengths from a series of shapes. Next, the mean values of shear stresses are calculated for them, and the rates of change of wavelengths are found. Then a degree of approximation of the wavelength dependence of its rate of change is determined, and, finally, the exact solution of the corresponding differential equation is obtained. Comparison with experimental data and numerical solutions shows that the solution error does not exceed a few percent and that computational time is reduced by 25–30 times.

Optimal consumption under a drawdown constraint over a finite horizon

This paper studies a finite horizon utility maximization problem on excessive consumption under a drawdown constraint. Our control problem is an extension of the one considered in Angoshtari et al. (2019) to the model with a finite horizon and an extension of the one considered in Jeon and Oh (2022) to the model with zero interest rate. Contrary to Angoshtari et al. (2019), we encounter a parabolic nonlinear HJB variational inequality with a gradient constraint, in which some time-dependent free boundaries complicate the analysis significantly. Meanwhile, our methodology is built on technical PDE arguments, which differs from the martingale approach in Jeon and Oh (2022). Using the dual transform and considering the auxiliary variational inequality with gradient and function constraints, we establish the existence and uniqueness of the classical solution to the HJB variational inequality after the dimension reduction, and the associated free boundaries can be characterized in analytical form. Consequently, the piecewise optimal feedback controls and the time-dependent thresholds for the ratio of wealth and historical consumption peak can be obtained.

APPROXIMATE RELATIONSHIPS FOR DETERMINING THE ENTHALPIES OF WATER AND STEAM IN THE CALCULATIONS OF HEAT EXCHANGER DEVICES OF STEAM TURBINE INSTALLATIONS (PART I – DETERMINATION OF THE ENTHALPY OF WATER)

The work is devoted to the creation of a methodological approach for determining the thermodynamic properties of water and water vapor in the calculation of heat exchange processes for those cases when it is necessary to quickly control the operating modes of steam turbines of large power of thermal and nuclear power plants, but the use of existing tables of properties of working environments in the "manual" mode is not always suppose. In the presented part of the work the determination of the enthalpy of water is considered. Based on the analysis of the operating modes of heat exchangedevices of power units thermal and nuclear power plants of different capacities, the ranges of temperature and pressure changes of the working media during their calculations were determined. They are for water: at a pressure of 1 kPa–30 MPa, at a temperature of 1–300 °С, and for steam, respectively: 1 kPa–6 MPa, 7–450 °С. The analysis of tabular values of the working medium (water) showed that in order to present the enthalpy of water in an analytical form based on approximation equations with the required accuracy, it is advisable to consider their construction in two areas of change: in the area of vacuum (Р < 0.1 MPa) and in the area of excess pressure (0.1–30 MPa), which in turn are divided into 4 ranges. When thepressure changes, this is associated with an additional effect on the temperature enthalpy value, which complicates the choice of approximate dependencies for obtaining enthalpy from the parameters of the working environment while ensuring an acceptable ratio of calculated and tabular values for solving problems. By the authors a system of regression equations was obtained, allowing to calculate the enthalpy of water in a given range of pressures and temperatures with high accuracy. To determine the area in which the measured value is located, the authors proposed the use of Antoine's formula, in which the relationship between the pressure and temperature of the medium at the boundary line of the phase transition is established. To increase its accuracy, approximation equations for temperature corrections at different pressures are proposed, which ensure that the temperature deviates from the tabulated values of thermophysical properties there are no more than ±0.04 °С.

Casting more light in the shadows: dual Somos-5 sequences

Abstract Motivated by the search for an appropriate notion of a cluster superalgebra, incorporating Grassmann variables, Ovsienko and Tabachnikov considered the extension of various recurrence relations with the Laurent phenomenon to the ring of dual numbers. 
Furthermore, by iterating recurrences with specific numerical values, some particular well-known integer sequences, such as the Fibonacci sequence, Markoff numbers, and Somos sequences, were shown to produce associated ``shadow'' sequences when they were extended to the dual numbers. Here we consider the most general version of the Somos-5 recurrence defined over the ring of dual numbers $\mathbb{D}$ with complex coefficients, that is the ring $\mathbb{C}[\varepsilon]$ modulo the relation $\varepsilon^2=0$. We present three different ways to present the general solution of the initial value problem for Somos-5 and its shadow part: in analytic form, using the Weierstrass sigma function with arguments in $\mathbb{D}$; in terms of the solution of a linear difference equation; and using Hankel determinants constructed from $\mathbb{D}$-valued moments, via a connection with a QRT map over the dual numbers.

Efficient relaxation scheme for the SIR and related compartmental models

In this paper, we introduce a novel numerical approach for approximating the Susceptible–Infectious–Recovered (SIR) model in epidemiology. Our method enhances the existing linearization procedure by incorporating a suitable relaxation term to tackle the transcendental equation of nonlinear type. Developed within the continuous framework, our relaxation method is explicit and easy to implement, relying on a sequence of linear differential equations. This approach yields accurate approximations in both discrete and analytical forms. Through rigorous analysis, we prove that, with an appropriate choice of the relaxation parameter, our numerical scheme is non-negativity-preserving; moreover, it is strongly convergent to the true solution. We also extend the applicability of our relaxation method to handle some variations of the traditional SIR model. Finally, we present numerical examples using simulated data to demonstrate the effectiveness of our proposed method.

Построение моделей и статистическая оценка параметров надежности судовых технических средств

A method is proposed to solve the problems that arise when analyzing information obtained from various sources, it is necessary to combine it in a complex of coordinated stochastic models. For most processes of mechanical destruction of machine parts, such as wear, fatigue volumetric or surface destruction, corrosion, etc., the monotony of the implementation of degradation processes leading to failures is characteristic. The scheme of resource allocation formation is given, which corresponds to the main types of mechanical failures occurring in the elements of ship mechanisms. The diagram illustrates a fairly general case when, with an increase in the operating time of an element, its resistance to destruction decreases monotonously, and the damage parameter also increases monotonously. Based on the use of generalized distributions, it is possible to obtain several variants of complexes of mechanical reliability models in which the resource allocation has a fairly simple analytical form. When modeling degradation processes, the DM distribution can be obtained as a special case of distributions and without using the Markov model of random processes. Only the mixing condition of the process implementations is essential. Obtaining various types of operating time distributions before the parameter reaches a fixed level using the obtained dependencies is not associated with any assumptions about the nature of individual implementations of the degradation process, except for their monotony. From the point of view of practical application, the distribution under consideration, obtained on the basis of Weibull’s law, has the advantage of universality, since, unlike the normal law, it can also be used for values of the coefficient of variation significantly exceeding 0.3.

Polarized Semi-Inclusive Deep-Inelastic Scattering at Next-to-Next-to-Leading Order in QCD.

Semi-inclusive hadron production in longitudinally polarized deep-inelastic lepton-nucleon scattering is a powerful tool for resolving the quark flavor decomposition of the proton's spin structure. We present the full next-to-next-to-leading order QCD corrections to the coefficient functions of polarized semi-inclusive deep-inelastic scattering (SIDIS) in analytical form, enabling the use of SIDIS measurements in precision studies of the proton spin structure. The numerical impact of these corrections is illustrated by a comparison with data of polarized single-inclusive hadron spectra from the DESY HERMES and CERN COMPASS experiments.

Time-Like heavy-flavour thresholds for fragmentation functions: the light-quark matching condition at NNLO

Matching conditions are universal ingredients that describe how fragmentation functions change when heavy-flavour thresholds are crossed during the factorisation scale evolution. They are the last missing piece for a consistent description of observables with identified final-state hadrons at next-to-next-to leading order accuracy in quantum chromodynamics. We present an analytical form of the matching condition for light-flavour to hadron fragmentation function at next-to-next-to leading order. The derivation is performed by extending the formalism employed in the extraction of the next-to leading order matching conditions to the subsequent order, making use of e+e-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^+e^-$$\\end{document} annihilation cross sections. We obtain the first non-trivial heavy-quark effect in the light-quark fragmentation functions and provide results in Mellin space.

Engineered Mg-modified biochar-based sorbent for arsenic separation and pre-concentration

The utilization of biochar as a relatively efficient sorbent or stationary phase for the separation and preconcentration of a wide range of analytes represents an innovative approach in current sample pretreatment methods. Appropriate pre- and post-pyrolysis modification of the input precursor and pyrolysis product, respectively, allows targeted design of the physicochemical properties and sorption characteristics of the resulting sorbent. The present work deals with the preparation of pyrolysis materials based on unmodified cattail leaf biomass (BC) and its Mg-modified analogue (MgBC) by a slow pyrolysis process at 500 °C and a residence time of 1 h in a pyrolysis reactor. Physicochemical characterization of BC and MgBC carried out by pH, total C, N, surface size analysis (SSA), 13C NMR, SEM-EDX and XRD confirmed significant morphological and mineralogical differences between the prepared sorbents. By performing sorption experiments using a model anionic analyte (As) and application of Langmuir isotherm, we found that the predicted maximum sorption capacity of MgBC for As is 13.5-fold higher than that of BC. The sorption process of As by both sorbents is best described by the Sips adsorption isotherm (R2 ≥ 0.995) and a pseudo-nth order kinetic model (R2 ≥ 0.997). The optimum pH for As sorption by BC and MgBC sorbents is in the interval 5–6. The presence of competitive phosphate anions (equimolar concentration of 1:1) in the solution significantly reduces the sorption capacity of MgBC for As by 40% for BC by 70%. The presence of Cl- ions showed no significant effect on the sorption capacity of Bc and MgBC for As. Both sorbents were best recovered using 0.1 mol/L NaOH solution when the desorption efficiency for both sorbents was more than 95%. The MgBC sorbent showed 35% retention of As from the real sample in the model SPE column at a flow rate of 0.12 mL/s. Based on the obtained knowledge, it is evident that biochar-based sorbent prepared from Mg-modified precursor represents an effective sorbent for anionic forms of analytes and opens the possibility of its use also in preconcentration and separation techniques.

A robust dipole moment of carbon monoxide (CO) is a permanent puzzle for both spectroscopic and ab initio studies

The asymptotically correct semi-empirical dipole moment function (DMF) for the ground X 1 Σ + state of the CO molecule was redetermined in an analytical form for the entire range of interatomic distances r ∈ [ 0 , + ∞ ] by a global least-squared fitting both experimental intensities and ab initio data. The region r ∈ [ 0.8 , 1.9 ] Å that is governed exclusively by the empirical data was broadened by adding experimental relative intensity ratios measured in the emission from high vibrational levels v. Absolute absorption intensities recently obtained for very high 7-0 overtone and accompanied sub-percent measurements for 3-0 band were also included in the fitting procedure. The present DMF reproduces a vast majority of the experimental intensities up to v ≤ 38 within accuracy of measurements, except ‘abnormally’ sensitive 5-0 band, and systematically overestimates the most previous ab initio and semi-empirical functions at r ≥ 1.5 Å . Possible reasons of the observed discrepancies are throughout discussed. The resulting DMF can be used, along with mass-invariant interatomic potential, to upgrade rovibrational line lists for any isotopologues of the CO molecule up to the dissociation threshold. The corresponding FORTRAN subroutine is provided.

Multiple tsunami waves of the fractional geophysical KdV equation

Exact analytical form and approximate form of different kinds of tsunami wave solutions of the fractional geophysical Korteweg-de-Vries (fgKdV) equation are derived. The fgKdV equation is constructed by replacing the Caputo fractional derivative from the geophysical Korteweg-de-Vries equation. Some important definitions and results of the Caputo fractional derivative are discussed. The Darboux transformation and Residual power series method are used to derive some exact solutions and approximate analytical solutions to the fgKdV equation. Using the Darboux transformation the single solitary tsunami wave, singular periodic tsunami wave, two-soliton tsunami wave, singular solitary tsunami wave, solitary periodic tsunami wave, rational tsunami wave, rational solitary tsunami wave, and rational periodic tsunami wave were obtained. By performing the Residual power series method the approximate analytical solution is obtained. The graphical presentations of several exact solutions are obtained by Darboux transformation and Residual power series method. It is seen that the fractional parameter has a significant effect on the structure of the nonlinear tsunami waves of the fgKdV equation. The results of this study are applicable to understand the dynamics of multiple tsunami wave described by the fgKdV equation in the ocean.

Correlation analysis of multifractal stock price fluctuations based on partition function

Studying the correlation analysis of stock price fluctuations helps to understand market dynamics better and improve the scientific nature of investment decisions and risk management capabilities. Most existing methods use multifractals to explore the correlation between different economic entities. However, the study of multifractals fails to fully consider the weight of each entity’s impact on the market, resulting in an inaccurate description of the overall market dynamics. To address this problem, this paper creatively proposes a weighted multifractal analysis method (WMA). The correlation analysis of government regulation, market supply and demand, and stock price index is performed using the data of A-share listed companies in Shenzhen and Shanghai as samples. First, we consider the amplitude fluctuation information the signal carries and weigh the partition function according to the proportion of variance in the segment for different amplitude changes. Secondly, we derive the theoretical analytical form of the classical multifractal model (SMA) of the scaling indicator under WMA. Finally, through numerical simulation experiments, it is confirmed that WMA is equally effective as SMA. In addition, the re-fractal correlation analysis of real financial time series also confirms that WMA can effectively utilize the amplitude fluctuation information in the series and outperforms the classical SMA method in distinguishing different signals.

Evolving Transparent Credit Risk Models: A Symbolic Regression Approach Using Genetic Programming

Credit scoring is a cornerstone of financial risk management, enabling financial institutions to assess the likelihood of loan default. However, widely recognized contemporary credit risk metrics, like FICO (Fair Isaac Corporation) or Vantage scores, remain proprietary and inaccessible to the public. This study aims to devise an alternative credit scoring metric that mirrors the FICO score, using an extensive dataset from Lending Club. The challenge lies in the limited available insights into both the precise analytical formula and the comprehensive suite of credit-specific attributes integral to the FICO score’s calculation. Our proposed metric leverages basic information provided by potential borrowers, eliminating the need for extensive historical credit data. We aim to articulate this credit risk metric in a closed analytical form with variable complexity. To achieve this, we employ a symbolic regression method anchored in genetic programming (GP). Here, the Occam’s razor principle guides evolutionary bias toward simpler, more interpretable models. To ascertain our method’s efficacy, we juxtapose the approximation capabilities of GP-based symbolic regression with established machine learning regression models, such as Gaussian Support Vector Machines (GSVMs), Multilayer Perceptrons (MLPs), Regression Trees, and Radial Basis Function Networks (RBFNs). Our experiments indicate that GP-based symbolic regression offers accuracy comparable to these benchmark methodologies. Moreover, the resultant analytical model offers invaluable insights into credit risk evaluation mechanisms, enabling stakeholders to make informed credit risk assessments. This study contributes to the growing demand for transparent machine learning models by demonstrating the value of interpretable, data-driven credit scoring models.

Analysis of the third class mechanism using the modeling method in the Mathcad software environment

Purpose. To carry out a kinematic analysis of a flat twelve-link hinged mechanism with three leading links, the basis of which is a structural group of links of the third class of the fourth order with rotational kinematic pairs, which is used in technological equipment. Methodology. The kinematic study of a flat complex mechanism of the third class with three leading links was performed using the mathematical modeling method in the Mathcad software environment. Findings. Using the method of mathematical modeling, the mechanism was presented in the form of free vectors built on its links, for which, taking into account the presence of three leading links, three vector contours were selected, which made it possible to draw up a system of vector equations of their closedness with further solving by a numerical method in the Mathcad software environment. The functions of the rotation angles of the mechanism links, their angular velocities and accelerations were obtained in analytical and graphical form depending on the rotation angle of the first driving crank of the mechanism, the calculation of the angular velocities and accelerations of all the driven links of the mechanism was performed. Originality. A sequence was developed and a kinematic study of a complex planar mechanism of the third class with three leading links was done, the basis of which is a structural group of links of the third class of the fourth order. Schematic modeling of the mechanism with three leading links was performed, a visualization schedule of its kinematic scheme was built, and a valid version of its assembly was obtained from many possible ones, for which the values of kinematic parameters of all its links were determined. Practical value. An analysis of the effect of the kinematic parameters of the three leading links of the third-class mechanism on the movement of the working body of the technological equipment was carried out. From the analysis of the obtained research results for the cycle of the mechanism the reason was found for the incomplete technological stoppage of the link, which should provide the working body of the machine for the required period of time. Recommendations of the elimination were made to identify the deficiency and proposals for the need to improve the drive of this equipment.

Identification of moment equations via data-driven approaches in nonlinear Schrödinger models

IntroductionThe moment quantities associated with the nonlinear Schrödinger equation offer important insights into the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities are amenable to both analytical and numerical treatments.MethodsIn this paper, we present a data-driven approach associated with the “Sparse Identification of Nonlinear Dynamics” (SINDy) to capture the evolution behaviors of such moment quantities numerically.Results and DiscussionOur method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available.

On the issue of the development of adjective degree forms in the Western Lithuanian and Latvian dialects

Abstract In the Lithuanian and Latvian languages, forms of adjective degrees do not have a common etymological basis. The New Curonian (Kuresnieku) language which is a relatively new Baltic geolect has preserved a fairly distinct archaic layer of phonetics and grammar, where one can look not only for the archaisms of the western dialects of the Latvian language but also for traces of the substrate of Old Curonian. Although synthetic forms of the degrees of adjectives with the suffix -āk- are used in the current Latvian language area they are almost unknown in New Curonian. The forms of the comparative degree are made with the particle juo and an adjective of the positive degree. Analytical forms with the pronoun visu and the pronominal form of an adjective express the superlative degree. In the western Latvian and Lithuanian dialects and the Livonian language in Latvia, the relics of the analytical forms of the comparative and superlative degree are recorded in written sources and rarely in modern spoken language. Since a large part of the western dialects of the Lithuanian and Latvian languages take up the former territory of the old tribe, it may be assumed that the analytical forms of adjective degrees are possibly a sign of the substrate of Old Curonian.

On the Implementation of Iterative Methods Without Inverse Updating for Solving Equations in Banach Spaces

The implementation of iterative methods using inverses to solve equations is a computationally expensive or impossible task in general. This is the case, since the analytical form of the inverse is difficult to find in practice. That is why, we replace the inverse by a sum of linear operators which is well defined. The convergence of the resulting hybrid methods is studied based on majorizing sequences under generalized continuity assumptions on the operators involved and in the setting of a Banach space. It is demonstrated by numerical experimentations that the convergence order as well as the number of iterations required to obtain a predetermined error tolerance when comparing the original to the hybrid method is essentially the same.