Rational Equations Research Articles

Finding first order differential invariants through the [formula omitted]-function

Finding first order differential invariants through the [formula omitted]-function

Behavior of some higher order nonlinear rational partial difference equations

Abstract In this paper we give the closed form expressions of some higher order nonlinear rational partial difference equations in the form X n , m = X n − r , m − r Ψ + ∏ i = 1 r X n − i , m − i where n , m ∈ N and the initial values X n, t , X t , m − r are real numbers with t ∈ { 0 , − 1 , − 2 , … , − r + 1 } such that ∏ j = 0 r − 1 X j − r + 1 , i + j − r + 1 ≠ − Ψ and ∏ j = 0 r − 1 X i + j − r + 2 , j − r + 1 ≠ − Ψ , i ∈ N 0 . We will use a new method to prove the results by using what we call ‘piecewise double mathematical induction’ which we introduce here for the first time as a generalization of many types of mathematical induction. As a direct consequences, we investigate and conclude the explicit solutions of some higher order ordinary difference equations.

The weighted Fermat–Torricelli–Menger problem for a given sextuple of edge lengths determining tetrahedra

We introduce the weighted Fermat–Torricelli–Menger problem for a given sextuple of edge lengths in \({\mathbb {R}}^{3}\) which states that: given a sextuple of edge lengths determining tetrahedra and a positive real number (weight) which corresponds to each vertex of every derived tetrahedron find the corresponding weighted Fermat–Torricelli point of these tetrahedra. We obtain a system of three rational equations with respect to three variable distances from the weighted Fermat–Torricelli point to the three vertices of the tetrahedron determined by a given sextuple of edge lengths in the sense of Menger. This system of equations gives a necessary condition to locate the weighted Fermat–Torricelli point at the interior of this class of tetrahedra and allow us to compute the position of the corresponding weighted Fermat–Torricelli point. Furthermore, we give an analytical solution for the weighted Fermat–Torricelli–Menger problem for a given sextuple of equal edge lengths in \({\mathbb {R}}^{3}\) for the case of two pairs of equal weights.

Theory of interaction between vibratory tillage tools and soil

The study deals with mathematical simulation of the interaction between vibratory tillage tools and soil. Most studies on the use of vibrations in tillage processes give conflicting results about their technological and energy efficiency. The issue of mathematical description of the interaction between vibratory tillage tools and soil is quite complicated and primarily determined by a variety of physical and mechanical properties of the object of tillage. This paper studies purely mechanical effects of vibrations as applied to tillage processes. The research goal is to reveal the patterns that characterize the vibration effects on the soil, and to determine the mechanism of the conversion of the resistance of vibratory tillage tools to a viscous type of friction. The analysis of the resistance of vibratory tillage tools is carried out based on the analysis of the rational equation by academician V.P. Goryachkin. The soil is idealized as a viscous and plastic medium. The case of sinusoidal variations of a tillage tool in addition to its translational motion is investigated. The effect of vibrations on all three resistance components of tillage tool (friction, deformation and inertia) is analyzed. The methods of differential and integral calculus are used. It is shown that all the components of the soil resistance change under the effect of vibrations; the effect may be both positive and negative. Resistance reduction is possible in the case when the amplitude of the vibration velocity of tillage tool exceeds its translational velocity. Respectively, the power consumption of tractor traction mechanism may also be reduced. However, the overall power consumption, including the power consumption of the vibration exciter, is not reduced as compared to the operation of conventional tillage tools. These theoretical results are proved by the experimental studies of A.A. Dubrovskiy. The linearization of the traction resistance of the vibratory tillage tools (its dependence on the translational velocity) is explained in this study based on the general laws of mechanics, without using hypotheses on physical change of soil properties.

도로터널의 제연용량 산정을 위한 정체차량대수 산정기법에 관한 연구 - 정체길이를 중심으로

터널 화재시 정체차량에 의한 정체길이가 터널연장을 초과할 경우, 환기저항의 증가에 따른 제연설비 용량의 증가가 발생하게 된다. 그러나 현행 방재지침에는 정체길이에 대한 정의가 없기 때문에 합리적 산출식의 제시가 필요하다. 본 연구에서는 터널 화재시 정체차량 대수에 의한 정체길이의 산정식을 제시하고, 터널연장별 적용성 분석을 수행하였다. 일반적인 터널의 경우, 화재시 정체길이의 과도한 적용을 방지하기 위해서는 터널연장 1,200 m 까지는 정체길이와 터널연장과의 상호비교가 필요한 것으로 분석되었고, 모델터널에 대한 적용성 평가결과 제연용 제트팬의 절감효과가 있는 것으로 분석되었다. 더불어 정체길이의 판별여부를 대형차혼입률과 터널연장의 관계로 설명할 수 있는 정량화 선도를 제시하였다. 결과적으로 제연설비 용량결정시, 정체차량에 의한 정체길이가 터널연장을 초과하는 경우에는 터널연장을 초과하는 차량대수는 차량에 의한 환기저항 산정에서 제외하는 것이 타당한 것으로 분석된다. When the queue length of congestion vehicles in tunnel fire is extended beyond tunnel length, the capacity of smoke control system needs to be increased in line with ventilation resistance. However, the vehicle queue length is not defined, so a rational equation is necessary in current fire prevention guideline. This study is intended to propose an equation to calculate the queue length considering the number of vehicles in queue in tunnel fire and evaluate the applicability by tunnel length as well. When it comes to normal tunnel, it is necessary to compare the vehicle queue length with tunnel length up to the length of 1,200 m in a bid to avoid applying the vehicle queue length excessively in case of fire. As a result of evaluation of applicability to model a tunnel, saving the number of jet fan for smoke control appeared to be effective. Besides, quantitative approach to explain the vehicle queue length through the relationship between the percentage of large vehicles and tunnel length was presented. Consequently, when the queue length of the congestion vehicles exceeds the tunnel length in determining the capacity of smoke control system in case of fire, the number of vehicles beyond the tunnel length needs to be excluded from estimating the ventilation resistance by vehicles.

Traveling Wave Solutions for Nonlinear Differential-Difference Equations of Rational Types

Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous. Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations, the majority of the results deal with polynomial types. Limited research has been reported regarding such equations of rational type. In this paper we present an adaptation of the (G′/G)-expansion method to solve nonlinear rational differential-difference equations. The procedure is demonstrated using two distinct equations. Our approach allows one to construct three types of exact traveling wave solutions (hyperbolic, trigonometric, and rational) by means of the simplified form of the auxiliary equation method with reduced parameters. Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.

Design Equations for Flexural Capacity of Concrete Beams Reinforced with Glass Fiber–Reinforced Polymer Bars

AbstractThe flexural failure mode of concrete beams reinforced with glass fiber-reinforced polymer (GFRP) bars changes from GFRP rupture to concrete crushing as the reinforcement ratio increases. Due to the uncertainties of material strengths, assumptions made in analysis, and variations in locations of reinforcements and dimensions of sections, there is a transition region where both flexural failure modes are possible. An iterative procedure is required when GFRP rupture governs the design. To avoid this iteration, the current American standard adopts a simplified but conservative procedure. In this study, the upper bound of the reinforcement ratio for beams in the transition region is revised. Moreover, a simplified yet rational design equation for calculating the flexural capacity of under-reinforced beams is proposed based on rigorous sectional analyses. Also, alternative design equations based on regression analyses are developed to predict the flexural capacity of beams in the transition region and...

Integration of equivalence theory and ultra-fine-group slowing-down calculation for resonance self-shielding treatment in lattice physics code GALAXY

A new hybrid resonance self-shielding treatment method in reactor physics field is developed by integrating equivalence theory and ultra-fine-group slowing-down calculation from the theoretical point of view. In the conventional equivalence theory, scattering source approximation and taking no account of resonance interference effect cause prediction error of effective cross-section. By reviewing the derivation scheme of neutron flux in the equivalence theory, the essence of the ultra-fine-group treatment is effectively incorporated. A new form of energy-dependent flux is based on multi-term rational equation, but the scattering source can be solved by the way similar to the slowing-down equation. The accurate non-fuel flux is also considered without direct heterogeneous calculation. The new method can also efficiently eliminate the multi-group condensation error by a semi-analytical reaction rate preservation scheme between ultra-fine and multi-group treatments. The present method is implemented in Mitsubishi Heavy Industries, Ltd. lattice physics code GALAXY. From comparisons of neutronics parameters between GALAXY and a continuous energy Monte-Carlo code, applicability of the new method for lattice physics calculations is confirmed. GALAXY achieves high accuracy with short computation time. Therefore, it can be efficiently applied to generation of the nuclear constants used in the nuclear design and safety analysis of commercial light water reactors.

On Generating Integrable Dynamical Systems in 1+1 and 2+1 Dimensions by Using Semisimple Lie Algebras

Abstract We deduce a set of integrable equations under the framework of zero curvature equations and obtain two sets of integrable soliton equations, which can be reduced to some new integrable equations including the generalised nonlinear Schrödinger (NLS) equation. Under the case where the isospectral functions are one-order polynomials in the parameter λ, we generate a set of rational integrable equations, which are reduced to the loop soliton equation. Under the case where the derivative λ t of the spectral parameter λ is a quadratic algebraic curve in λ, we derive a set of variable-coefficient integrable equations. In addition, we discretise a pair of isospectral problems introduced through the Lie algebra given by us for which a set of new semi-discrete nonlinear equations are available; furthermore, the semi-discrete MKdV equation and the Hirota lattice equation are followed to produce, respectively. Finally, we apply the Lie algebra to introduce a set of operator Lax pairs with an operator, and then through the Tu scheme and the binomial-residue representation method proposed by us, we generate a 2+1-dimensional integrable hierarchy of evolution equations, which reduces to a generalised 2+1-dimensional Davey-Stewartson (DS) equation.

Preparation of monodisperse poly(styrene-co-divinylbenzene) microspheres by vibration dispersion method

Monodisperse micron-size poly(styrene-co-divinylbenzene) (PST-DVB) microspheres were successfully prepared by vibration dispersion. A vibrator was used to generate a controlled vibration to jet from a single nozzle of 200 µm to produce uniform droplets. The effects of variations in selected process parameters upon the droplets formation and the diameter of the microspheres were investigated. It was demonstrated that changes in the velocity of oil phase, drive frequency and destabilization amplitude have significant effects on the formation of uniform droplets and the average diameter of the microspheres. Rational polynomial regression equations of the above parameters were established. Based on these equations, we could calculate the operation conditions to produce uniform droplets with the desirable diameter.

Structure Transformation among Deca-, Dodeca- and Tridecavanadates and Their Properties for Thioanisole Oxidation

The transformation of three types of polyoxovanadates, {(n-C4H9)4N}3[H3V10O28], {(n-C4H9)4N}4[V12O32] and {(n-C4H9)4N}3[V13O34] have been investigated according to the rational chemical equations, and the best transformation conditions were reported. By the reaction of [H3V10O28]3− with 0.33 equivalents of {(n-C4H9)4N}OH in acetonitrile at 80 °C, [V12O32]4− was formed with 92% yield. The reaction in nitroethane with 0.69 equivalents of p-toluenesulfonic acid gave [V13O34]3− with 91% yield. The 51V NMR observation of each reaction suggests the complete transformations of [H3V10O28]3− to [V12O32]4− and to [V13O34]3− proceeded without the formation of any byproducts and it provides the reliable synthetic route. Decavanadates were produced by the hydrolysis of [V12O32]4− or [V13O34]3−. While the direct transformation of [V13O34]3− to [V12O32]4− partly proceeded, the reverse one could not be observed. For the thioanisole oxidation, [V13O34]3− showed the highest activity of the three.

Characterization of XR-RV3 GafChromic(®) films in standard laboratory and in clinical conditions and means to evaluate uncertainties and reduce errors.

To investigate the optimal use of XR-RV3 GafChromic(®) films to assess patient skin dose in interventional radiology while addressing the means to reduce uncertainties in dose assessment. XR-Type R GafChromic films have been shown to represent the most efficient and suitable solution to determine patient skin dose in interventional procedures. As film dosimetry can be associated with high uncertainty, this paper presents the EURADOS WG 12 initiative to carry out a comprehensive study of film characteristics with a multisite approach. The considered sources of uncertainties include scanner, film, and fitting-related errors. The work focused on studying film behavior with clinical high-dose-rate pulsed beams (previously unavailable in the literature) together with reference standard laboratory beams. First, the performance analysis of six different scanner models has shown that scan uniformity perpendicular to the lamp motion axis and that long term stability are the main sources of scanner-related uncertainties. These could induce errors of up to 7% on the film readings unless regularly checked and corrected. Typically, scan uniformity correction matrices and reading normalization to the scanner-specific and daily background reading should be done. In addition, the analysis on multiple film batches has shown that XR-RV3 films have generally good uniformity within one batch (<1.5%), require 24 h to stabilize after the irradiation and their response is roughly independent of dose rate (<5%). However, XR-RV3 films showed large variations (up to 15%) with radiation quality both in standard laboratory and in clinical conditions. As such, and prior to conducting patient skin dose measurements, it is mandatory to choose the appropriate calibration beam quality depending on the characteristics of the x-ray systems that will be used clinically. In addition, yellow side film irradiations should be preferentially used since they showed a lower dependence on beam parameters compared to white side film irradiations. Finally, among the six different fit equations tested in this work, typically used third order polynomials and more rational and simplistic equations, of the form dose inversely proportional to pixel value, were both found to provide satisfactory results. Fitting-related uncertainty was clearly identified as a major contributor to the overall film dosimetry uncertainty with up to 40% error on the dose estimate. The overall uncertainty associated with the use of XR-RV3 films to determine skin dose in the interventional environment can realistically be estimated to be around 20% (k = 1). This uncertainty can be reduced to within 5% if carefully monitoring scanner, film, and fitting-related errors or it can easily increase to over 40% if minimal care is not taken. This work demonstrates the importance of appropriate calibration, reading, fitting, and other film-related and scan-related processes, which will help improve the accuracy of skin dose measurements in interventional procedures.

Finding higher order Darboux polynomials for a family of rational first order ordinary differential equations

For the Darbouxian methods we are studying here, in order to solve first order rational ordinary differential equations (1ODEs), the most costly (computationally) step is the finding of the needed Darboux polynomials. This can be so grave that it can render the whole approach unpractical. Hereby we introduce a simple heuristics to speed up this process for a class of 1ODEs. New version program summaryProgram title: LsolverCatalogue identifier: ADZF_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZF_v3_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1831No. of bytes in distributed program, including test data, etc.: 20862Distribution format: tar.gzProgramming language: Maple, release 17.Computer: Any computer running Maple.Operating system: Windows XP, Windows 7.RAM: 20 MbClassification: 4.3.Catalogue identifier of previous version: ADZF_v2_0Journal reference of previous version: Comput. Phys. Comm. 185(2014)1496Does the new version supersede the previous version?: YesNature of problem: We deal with the analytical solving of 1ODEs via a Darboux method.Solution method: The method of solution is based on the standard Prelle–Singer (Darboux) method [1], with extensions of our own [2] which solves 1ODEs with Liouvillian solutions. The first part of the Method depends on the finding of the Darboux polynomials for the ODE. The integrability analysis mentioned above is performed at this stage.Reasons for new version: A great portion of our work has been concentrated on analyzing and solving systems of first and second order differential equations (1ODEs and 2ODEs, respectively) from a numerical point of view [3], using Lie methods [4, 5] and Darboux type approaches [2, 6]. The common factor among the algorithms in the Darbouxian approach has always been that the most costly step is the determination of the associated Darboux polynomials. So, we are hereby trying to improve on this aspect.Summary of revisions: The revision is the introduction of an option to the command Darboux that allows for faster finding of the necessary Darboux polynomials and co-factors, in some cases. In the last version of this package, we have introduced a possibility of using a defined degree for both the Darboux polynomial and co-factor. Previously, one could only change the degree of the Darboux polynomial and the program used the maximum value possible for the degree of the co-factor. It causes the program to be stuck (sometimes) in a very large problem and the package had problems both in time and computer memory management. Here we introduce a similar idea. We allow the program to look for Darboux polynomials with a fixed degree in one variable. We also fix the degree of the polynomial as a whole. This extra liberty (in restricting the problem to be dealt with) allows for the success of the package in solving some 1ODEs that were missed even with the improvement of the last version. In other words, by fixing the degree for the specific variable we divided the search into smaller steps, dividing the problem into smaller ones. This prevents the algorithm entering too large a calculation. In order to properly see the syntax of the usage of the command, please, see the example Maple file. Let us just display the general format of the command Darboux.>Darboux(ode,DPEXT=[[dx,dt],[x,y]])This would produce an output of the form:[[Darboux polynomials],[co-factors]]In the above calling of the command Darboux, the [dx,dt] terms refer to the degrees of x and the total degree of the Darboux polynomial, respectively.It is important to notice that, for some cases, if we do not instruct the Darboux command which degree to use for the variable x (for example) and the total degree of the Darboux polynomial, thus going step-by-step on the algorithm, (via the extra term DPEXT=[[dx,dt],[x,y]]), the command may not be able to solve the 1ODE. Furthermore, the Maple in-built dsolve command cannot solve some of the same class of 1ODEs. So the new improvement, albeit simple to implement, is of great practical value.Restrictions: The finding of the Darboux polynomials and the correspondent co-factors is the most time-demanding task of the method. So, one could say that, for the general case, if the degree for the Darboux polynomials is “4”, for instance, it would make the case hard to deal with. This situation is mitigated (for some 1ODEs) by the improvements hereby introduced.Unusual features: The faster way to find high (>3) degree Darboux polynomials and co-factors for some cases.Running time: This depends strongly on the 1ODE being analyzed. As mentioned above, a degree for the Darboux polynomials “4” or above could (in general) make dealing with the case unpractical. For the degree “3”, the routines may take more than a minute to deal with the task. Generally, the program takes, at most, for the other cases, a few seconds. Now, even for degrees >4 it can sometime only take a couple of seconds.

Improving a family of Darboux methods for rational second order ordinary differential equations

We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most costly step of our methods and algorithms of solution is the determination of Darboux polynomials for the associated differential operators. Here, we are going to apply a procedure to greatly reduce the time expenditure in determining these needed Darboux polynomials for a class of second order differential equations.

Rational and Implicit Equations for Some Polar Curves

SummaryGiven an implicit equation for an algebraic plane curve, we consider when and how it is possible to find rational parametric functions that also draw the curve. Using sequences similar to the Chebyshev polynomials, we detail examples, including Maclaurin sectrices, Ceva sectrices, and polar flowers.

Asymptotic Behavior of a Second-Order Fuzzy Rational Difference Equation

We study the qualitative behavior of the positive solutions of a second-order rational fuzzy difference equation with initial conditions being positive fuzzy numbers, and parameters are positive fuzzy numbers. More precisely, we investigate existence of positive solutions, boundedness and persistence, and stability analysis of a second-order fuzzy rational difference equation. Some numerical examples are given to verify our theoretical results.

Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control

We consider the numerical solution of the generalized Lyapunov and Stein equations in ?n$\mathbb {R}^{n}$, arising respectively from stochastic optimal control in continuous- and discrete-time. Generalizing the Smith method, our algorithms converge quadratically and have an O(n3) computational complexity per iteration and an O(n2) memory requirement. For large-scale problems, when the relevant matrix operators are "sparse", our algorithm for generalized Stein (or Lyapunov) equations may achieve the complexity and memory requirement of O(n) (or similar to that of the solution of the linear systems associated with the sparse matrix operators). These efficient algorithms can be applied to Newton's method for the solution of the rational Riccati equations. This contrasts favourably with the naive Newton algorithms of O(n6) complexity or the slower modified Newton's methods of O(n3) complexity. The convergence and error analysis will be considered and numerical examples provided.

Dynamical behavior of a third-order rational fuzzy difference equation

According to a generalization of division (g-division) of fuzzy numbers, this paper is concerned with the boundedness, persistence and global behavior of a positive fuzzy solution of the third-order rational fuzzy difference equation $$x_{n+1}=A+\frac{x_{n}}{x_{n-1}x_{n-2}},\quad n=0,1,\ldots, $$ where A and initial values $x_{0}$ , $x_{-1}$ , $x_{-2}$ are positive fuzzy numbers. Moreover, some examples are given to demonstrate the effectiveness of the results obtained.

A new Steiner patch based file format for Additive Manufacturing processes

Additive Manufacturing (AM) processes adopt a layering approach for building parts in continuous slices and use the Standard Tessellation Language (STL) file format as an input to generate the slices during part manufacturing. However, the current STL format uses planar triangular facets to approximate the surfaces of the parts. This approximation introduces errors in the part representation which leads to additional errors downstream in the parts produced by AM processes. Recently, another file format called Additive Manufacturing File (AMF) was introduced by ASTM which seeks to use curved triangles based on second degree Hermite curves. However, while generating the slices for manufacturing the part, the curved triangles are recursively sub-divided back to planar triangles which may lead to the same approximation error present in the STL file. This paper introduces a new file format which uses curved Steiner patches instead of planar triangles for not only approximating the part surfaces but also for generating the slices. Steiner patches are bounded Roman surfaces and can be parametrically represented by rational Bezier equations. Since Steiner surfaces are of higher order, this new Steiner file format will have a better accuracy than the traditional STL and AMF formats and will lead to lower Geometric Dimensioning and Tolerancing (GD&T) errors in parts manufactured by AM processes. Since the intersection of a plane and the Steiner patch is a closed form mathematical solution, the slicing of the Steiner format can be accomplished with very little computational complexity. The Steiner representation has been used to approximate the surfaces of two test parts and the chordal errors in the surfaces are calculated. The chordal errors in the Steiner format are compared with the STL and AMF formats of the test surfaces and the results have been presented. Further, an error based adaptive tessellation algorithm is developed for generating the Steiner representation which reduces the number of curved facets while still improving the accuracy of the Steiner format. The test parts are virtually manufactured using the adaptive Steiner, STL and AMF format representations and the GD&T errors of the manufactured parts are calculated and compared. The results demonstrate that the modified Steiner format is able to significantly reduce the chordal and profile errors as compared to the STL and AMF formats.

Excess enthalpies of ternary mixtures of (oxygenated additives + aromatic hydrocarbon) mixtures in fuels and bio-fuels: (Dibutyl-ether + 1-propanol + benzene), or toluene, at T = (298.15 and 313.15) K

New experimental excess molar enthalpy data of the ternary systems (dibutyl ether+1-propanol+benzene, or toluene), and the corresponding binary systems at T=(298.15 and 313.15)K at atmospheric pressure are reported. A quasi-isothermal flow calorimeter has been used to make the measurements. All the binary and ternary systems show endothermic character at both temperatures. The experimental data for the systems have been fitted using the Redlich–Kister rational equation. Considerations with respect the intermolecular interactions amongst ether, alcohol and hydrocarbon compounds are presented.