System Of Transcendental Equations Research Articles

On the roots of systems of transcendental equations

On the roots of systems of transcendental equations

Design of Switched Capacitor Multilevel Inverter with Selective Harmonic Elimination

In this paper, Selective Harmonic Elimination (SHE) technique is applied to control multilevel inverters that can be used to eliminate the low order dominant harmonics. This is considered a low frequency technique, in which the switching angles are predetermined based on solving a system of transcendental equations. Newton Raphson is one among the heuristic techniques utilised for selective harmonic elimination. The proposed SHE strategy is simulated along with Sinusoidal Pulse Width Modulation (SPWM) technique and the harmonic distortions are measured and compared for both techniques in MATLAB/Simulink software.

Proposed gear rack cutter for spur gear manufacturing by generating method. Part II: Comparison between the standard and modified profiles

The paper proposes a method of cutting by generating of spur gears using a rack cutter with a modified geometry. The analytical equations of the tooth profile of the modified rack-gear cutter are deduced, both for involute zone and for filleting region, based on the remark that the profile is the envelope of successive positions of the tool. The undercutting radius for the modified profile is obtained from a system of transcendental equations which requires and a numerical method for solving it. Compared to the standard profile, the differences are insignificant. A program was written and the profiles of the teeth are represented for two shifting coefficients, for the standard and the modified tooth profiles.

Ideal Flow Design of Plane-Strain Bending Driven by Springback

The present paper is concerned with the design of the plane-strain bending of wide sheets driven by the sheet’s geometric shape after unloading. The solution is based on the ideal flow theory. It determines the loading path (the dependence between the dimensionless bending moment and the dimensionless tensile force) that produces the desirable shape. The ideal flow theory is valid for the rigid perfectly plastic material model. A method proposed in the literature combines a rigid plastic solution at loading and an elastic solution for springback. The final design solution is practically analytical. A numerical method is only necessary to solve a system of transcendental equations. An illustrative example is provided. This example shows that, in some cases, it is sufficient to use the solution at loading to determine an accurate loading path that produces the desirable sheet’s shape after springback.

Some Systems of Transcendental Equations

Several examples of transcendental systems of equations are considered. Since the number of roots of such systems, as a rule, is infinite, it is necessary to study power sums of the roots of negative degree. Formulas for finding residue integrals, their relation to power sums of a negative degree of roots and their relation to residue integrals (multidimensional analogs of Waring’s formulas) are obtained. Calculations of multidimensional numerical series are given

COMPUTER PROCEDURES OF THE SOFTWARE COMPLEX FOR DETERMINING THE TENSION OF THE THREAD SUPPLY PROCESS ON CIRCULAR KNITTING MACHINES

Determining the intensity of the thread feed process on a circular knitting machine for elastic weaves allows you to initially determine the technological loads that will occur during the interaction of the thread with the guides and working bodies depending on the type of raw material, curvature of guide surfaces, thread feed rate. The tension is changed due to the interaction of the threads with the guides of the circular knitting machine for elastic weaves of large and small curvature, tensioning devices. The development of procedures for the computer determination of thread tension on a circular knitting machine for elastic weaves should be based on the use of object-oriented programming languages. Determining the tension and changes in relative tension in the refueling zones of a circular knitting machine for elastic weaves, taking into account the specifics of the design of the thread feed system, will improve the technology of processing raw materials on a circular knitting machine for elastic weaves. The improvement of technological processes of thread processing on a circular knitting machine for elastic weaves should be based on complex studies of the process of interaction of threads with combined tensioning devices, working guide surfaces of large and small curvature of the feed system. Theoretical dependences for determining the tension will allow to obtain mathematical software used in computer software to calculate the tension and change the relative tension in the refueling zones of circular knitting machine for elastic weaves, reduce the percentage of thread breakage, improve the quality of knitted fabric. You can reduce the tension of the thread feeding process and reduce the number of thread breaks on a circular knitting machine for elastic weaves by using quality raw materials and optimizing the interaction of threads with tensioning devices on a circular knitting machine for elastic weaves, where there is interaction of threads with surfaces large and small curvature. Thread tension relationships are systems of transcendental equations that require numerical methods to solve.

Determination of natural vibration frequencies of reinforced cylindrical shell

Free vibrations of a reinforced cylindrical shell filled with liquid are investigated. The case of an orthotropic shell is considered when the cord filament is placed symmetrically with respect to the meridian of the shell. The motion of a fluid is potential and is described by a wave equation. The fluid moves without separation from the walls of the cylinders. The fluid pressure is taken into account in the equations of motion of the shells, and the velocities of the fluid and the shell are equalized at the boundaries. Representing a solution in a harmonic form reduces to a system of transcendental equations. Comparison of the solutions of the problems without a liquid and with a liquid shows the dependence of the frequency of the system without a liquid at the frequency of the system with the liquid. An inverse method is proposed for solving the equation. The inverse method for solving the problem has made it possible to construct a more accurate frequency spectrum of free oscillations of the system. For some values of the system parameters, the natural frequencies of the cylinder are determined.

Metastable spin-phase diagrams in antiferromagnetic Bose-Einstein condensates

Spinor Bose-Einstein condensates under external magnetic fields exhibit well-characterized spin domains of its ground state due to spin-dependent interactions. At low temperatures, collision-induced spin-mixing instabilities may promote the condensate to dwell into metastable states occurring near the phase boundaries. In this work, we study theoretically the metastable spin-phase diagram of a spin-1 antiferromagnetic Bose-Einstein condensate at zero and finite temperatures. The approach makes use of Hartree-Fock theory and exploits the symmetry of the Hamiltonian and of the order parameters yielding a closed system of transcendental equations for the free energy, fully avoiding the use of selfconsistency. Our results are consistent with recent experiments and allow us to explain qualitatively the different types of observed quench dynamics. In addition, we found that similar phenomena should occur in antiferromagnetic spinor condensates with a sudden change in the temperature. It is shown also that the increase of temperature induces a traceable shift of the Ferromagnetic-Polar transition boundary, behavior previously not noticed by selfconsistent mean-field calculations.

SOFTWARE OF THE ALCHORITHM OF THE DICHOTOMY AT COMPUTER DETERMINATION OF TENSION OF COMPLEX THREADS

Improving the existing technological processes of processing complex threads on knitting machines and looms requires determining the change in relative tension in the areas of refueling. The change of tension occurs due to the interaction of complex threads with guides of large curvature, taking into account and without taking into account the radial coverage of the complex thread by the surface of the guide. The task of determining the tension should be based on the use of dichotomy algorithm software in computer determination of the tension of complex threads. Determining the change in relative tension in the areas of filling of complex threads on knitting machines and looms, taking into account and without taking into account the radial coverage of the complex thread by the guide surface, will improve the technology of production of knitwear and fabrics. Improvement of technological processes of the knitting and textile industry should be based on complex researches of process of interaction of complex threads with directing surfaces of big curvature. Obtaining theoretical dependences will allow to obtain equations for determining the tension of the thread in the area of fabric formation and knitwear, reduce the percentage of thread breaks, improve the quality of finished products. Many studies do not take into account the nonlinear relationship between force and coefficient of friction in obtaining the final equations. All this leads to significant differences between theoretical and experimental dependences, which does not allow the use of the former in specific calculations. Downtime of technological equipment associated with the elimination of cliffs is 75-80% of the total downtime. It is possible to significantly reduce the number of breaks by using quality raw materials and optimizing the conditions of thread processing on knitted and textile machines, where their interaction with the guides and working bodies of great curvature takes place. The tension dependences of complex threads have the form of a system of transcendental equations, for the solution of which it is necessary to use numerical methods. The dichotomy method is the most effective.

Program complex for optimization of surface hardening of steel billets

The aim of the paper is to develop program complex in software MATLAB with integrated numerical 2D nonlinear FLUX model, which is used for solving optimal inductor design and control problems for heating stage of surface induction hardening. Considered program complex is based on alternance method, that allows to write systems of transcendental equations, closed with respect to all unknown design and control parameters of the process. The suggestion for implementation of obtained optimal control algorithm is presented.

A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with inter-related multi-functions and multi-variables, often included in the mathematical modelling of physical systems during problem formulation. Here, we presented detailed steps for using a code-based modelling approach for solving SoTEs that may be encountered in science and engineering problems. A SoTE comprising six functions, including Sine-Gordon wave functions, was used to illustrate the steps. Parametric studies were performed to visualize how a change in the variables affected the superposition of the waves as the independent variable varies from x1 = 1:0.0005:100 to x1 = 1:5:100. The application of the proposed approach in modelling and simulation of photovoltaic and thermophotovoltaic systems were also highlighted. Overall, solutions to SoTEs present new opportunities for including more functions and variables in numerical models of systems, which will ultimately lead to a more robust representation of physical systems.

An Analytic Approximate Solution of the SIR Model

The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure exact (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms; moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the exact equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations; the easy and practical formulation of the final approximate solutions; the mentioned useful feature, never disclosed before. The work’s method and result prove to be robust over a range of values of the well known non-dimensional parameter called basic reproduction ratio, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn’t appear much discussed in analogous works.

Distribution of blood components in the Poiseuille flow

he thermodynamic approach is used to describe the stationary distributions of blood components in a cylindrical Poiseuille flow. Based on it, a method is described for numerical modelling of the distribution of blood components over the blood vessel cross section. Dispersed phases are represented by a combination of cellular components (red blood cells, white blood cells, etc.), blood plasma is considered as a low molecular weight dispersion medium. The distributions are obtained on the basis of the invariance condition of mechanochemical potentials (the sum of the chemical potential and mechanical energy) of the components, which is fulfilled at stationary equilibrium in stationary external fields of different nature. The calculation is reduced to a numerical search for a solution to a system of transcendental equations corresponding to the continuity of the medium. This system has a unique solution that describes the consistent distributions of all the components of a multiphase flow over the cross-section. It is shown that at the Poiseuille flow, the distributions of the individual dispersed phases can have up to three symmetrical extrema, the position of which is determined by the flow conditions (shear rate, vessel diameter, blood composition). As a demonstration of the model’s capabilities, the distribution of blood components (erythrocytes, platelets, lymphocytes) for vessels with a diameter of 0.1 to 5 mm at shear rates from 1 to 100 s–1 is given. It is shown that there is a very complex relationship between flow parameters (shear rate, diameter of a blood vessel) and blood composition. Unlike most hydrodynamic models of two-phase flows, the thermodynamic approach allows one to consider the distributions of an arbitrary set of dispersed and low molecular weight blood components, including foreign ones.

Photoconductivity Relaxation Time in Macroporous Silicon

Macroporous silicon has found applications in sensors, receivers, integrated microchips and solar cells. In this work, we first showed analytically the photoconductivity relaxation time of in macroporous silicon is determined from a system of two transcendental equations. The analytical description of the photoconductivity relaxation model contains the diffusion equation solution both in single crystal substrate and in an effective medium of macroporous layer, as well as boundary conditions recorded for the surfaces of the layer of macroporous silicon and single crystal substrate. We showed that the photoconductivity relaxation time in macroporous silicon rapidly decreases with increasing macropore depth from 0 to 25 μm and reduced thicknesses of the single crystal substrate from 250 to 0 μm. The photoconductivity recombination time in the sample of macroporous silicon is limited by the diffusion of charge carriers from the substrate to the recombination surfaces in the macroporous layer. The system of transcendental equations that we have found will find application in calculating the relaxation time of photoconductivity in macroporous silicon devices such as sensors, receivers, and solar cells.

Evaluation of energy consumption for the object motion optimal control

The energy asymptotic reduction for the optimal control of the “acceleration-deceleration” type (with increasing of the degree of the polynomial that sets the type of optimal control) while realizing the motion purpose is revealed and investigated. For a fixed minimum possible motion time, the object is moved as an absolutely solid or deformable body from the initial state of quiescence to the final state of absolute quiescence. It is shown that the acceptable motion time of an elastic object is found from the moment ratios - the equality to zero of the relative displacement and the relative velocity at the end of the motion. Using the example of a system with one degree of freedom, it is shown that the moment relations are a system of transcendental equations with many mutual roots. The considered class of controls is applicable for the elastic objects optimal motion in modern technology.

STEM-Technology Example of the Computational Problem of a Chain on a Cylinder

An application of the STEM technology to the computational problem of the parameters of a closed chain (with and without load) thrown over a horizontal cylinder is considered. The numerical solution is found and its graphical interpretation is made by compiling a system of transcendental equations, as well as carrying out numerical optimization with constraints. The approximating analytical dependence is determined using the fitting functions. In the process of solving a number of concepts from mathematics, physics, computer science are examined. Some possibilities of using specialized mathematical packages (in particular, Mathcad) and of working on online platforms are shown. Additional problems options for using STEM technology are presented.

Thermo-solutal growth of a dendritic crystal in the form of an elliptical paraboloid with forced convection

The steady-state analytical solutions describing the thermo-solutal growth of a dendritic crystal in the form of an elliptical paraboloid with allowance for a forced convective flow are found. The system of transcendental equations consisting of the undercooling balance and the stability criterion that describes a stable mode of growth of sections of elliptical dendrites is formulated and analyzed. It is shown that the temperature distributions in the melt, the growth and flow Peclét numbers as well as the dendrite tip diameter and tip velocity in the case of elliptical crystals essentially differ from their analogs in the case of symmetric shapes of dendrites in the form of paraboloids of revolution.

Thermal-FIST: A package for heavy-ion collisions and hadronic equation of state

Thermal-FIST11Thermal-FIST — Thermal, Fast and Interactive Statistical Toolkit. is a C++ package designed for convenient general-purpose physics analysis within the family of hadron resonance gas (HRG) models. This mainly includes the statistical analysis of particle production in heavy-ion collisions and the phenomenology of hadronic equation of state. Notable features include fluctuations and correlations of conserved charges, effects of probabilistic decay, chemical non-equilibrium, and inclusion of van der Waals hadronic interactions. Calculations are possible within the grand canonical ensemble, the canonical ensemble, as well as in mixed-canonical ensembles combining the canonical treatment of certain conserved charges with the grand-canonical treatment of other conserved charges. The package contains a fast thermal event generator, which generates particle yields in accordance with the HRG chemistry, and particle momenta based on the Blast Wave model. A distinct feature of this package is the presence of the graphical user interface frontend – QtThermalFIST – which is designed for fast and convenient general-purpose HRG model applications. Program summaryProgram Title:Thermal-FIST, version 1.2Program Files doi:http://dx.doi.org/10.17632/pprr8p4fkp.1Licensing provisions: GPLv3Programming language: C++External routines:Eigen template library for the linear algebra routines [1], MINUIT2 package from CERN ROOT [2], Mersenne Twister random number generator [3], Qt5 framework [4] (for the GUI only), QCustomPlot Qt widget [5] (for the GUI only)Nature of problem: The HRG model and its various modifications constitute a common framework used for modeling of the hadronic equation of state and particle production in heavy-ion collisions. Even the simplest versions of the HRG model require careful considerations of the many details, including the resonance decay feed-down, implementation of charge conservation constraints relevant for heavy-ion collisions, chemical non-equilibrium effects. A notable extra effort is required in order to treat the fluctuations and correlations of various charges , which presently are being extensively studied in the heavy-ion collision experiments and lattice QCD calculations. The inclusion of hadronic interactions, modeled by an excluded-volume (EV) or a van der Waals (vdW) type framework, additionally requires a numerical solution to a system of many transcendental equations.Solution method: The Thermal-FIST package contains a class-based library which calculates relevant HRG observables for a specified setup. The setup includes a particle list, usually to be supplied with an external file, an HRG model specification (statistical ensemble, van der Waals interaction parameters, etc.), a set of thermal parameters, and conservation laws constraints. Whenever necessary, the systems of transcendental equations are solved numerically with the Broyden’s method. The package includes a fitter for extracting thermal parameters from hadron yield data through the χ2 minimization. The HRG model based Monte Carlo event generator is a complementary feature to analytic calculations. A general-purpose thermal analysis is made maximally convenient with QtThermalFIST — a GUI frontend based on the Qt framework where all typical calculations, such as the properties of the equation of state or the thermal fits, can be straightforwardly performed.Additional comments: If the EV/vdW interactions are present, exact analytic calculations are presently only possible within the grand canonical ensemble. Approximate calculations are possible for the strangeness-canonical ensemble on the condition that strange particles form a small subsystem relative to the total system. Effects of probabilistic decays on fluctuation observables are generally included only up to the moments of the 2nd order. The only exception is the ideal HRG model in the grand canonical ensemble, where these effects are included up to the moments of the 4th order. On the other hand, the Monte Carlo event generator is not constrained by the above restrictions.

On computing residue integrals for a class of nonlinear transcendental systems

A method of computing residue integrals with integration over certain cycles for systems of transcendental equations is presented. Such integrals are connected to the power sums of roots for a certain system of equations. The described approach can be used for developing methods for the elimination of unknowns from transcendental systems.

Solving a nonlinear variation of the heat equation: self-similar solutions of the second kind and other results

This paper studies regular self-similar solutions of the following diffusion equation $$\begin{aligned} u_{t}+\gamma \vert u_{t} \vert =\Delta u\quad \text {in}\ \mathbb {R}^{N}\times ]0,\infty [, \end{aligned}$$ where $$-1<\gamma <1$$ . The analysis is focused on radial symmetric solutions $$u(x,t)=t^{-\alpha /2}f(\eta )$$ with $$\alpha >0$$ and $$\eta =\Vert x\Vert /\sqrt{t}$$ . Closed representation is obtained in terms of confluent hypergeometric functions. Employing specific properties of these special functions, oscillatory and symptotic aspects of f are obtained. It is demonstrated that such features are governed by increasing and unbounded sequences of exponents $$\alpha _{0}<\alpha _{1}<\cdots $$ , as in other diffusion equations. These exponents are determined by solving a system of transcendental equations related to specific roots of Kummer and Tricomi functions. As these cannot be determined using dimensional analysis, it is concluded that they are anomalous. For each exponent $$\alpha _{k}$$ , linear approximation when $$\gamma $$ is close to zero is also presented. Finally, relationships with previous results as well as an extension to other fully nonlinear parabolic equations are discussed.