Algebraic Integrability Research Articles

On the existence of periodic orbits and KAM tori in the Sprott A system: a special case of the Nosé–Hoover oscillator

We consider the well-known Sprott A system, which is a special case of the widely studied Nosé–Hoover oscillator. The system depends on a single real parameter a, and for suitable choices of the parameter value, it is shown to present chaotic behavior, even in the absence of an equilibrium point. In this paper, we prove that, for $$a\ne 0,$$ the Sprott A system has neither invariant algebraic surfaces nor polynomial first integrals. For $$a>0$$ small, by using the averaging method we prove the existence of a linearly stable periodic orbit, which bifurcates from a non-isolated zero-Hopf equilibrium point located at the origin. Moreover, we show numerically the existence of nested invariant tori surrounding this periodic orbit. Thus, we observe that these dynamical elements and their perturbation play an important role in the occurrence of chaotic behavior in the Sprott A system.

The Method of Finding Algebraic Integral for First-order Differential Equations

Article is devoted to search of algebraic integrals of the ordinary differential equations in the systems of computer algebra. The main attention is paid to development of practical instructions for work with an original package for Sage called in honor of M. N. Lagutinski. At the beginning of article Beaune’s problem is formulated: for a given differential equation, we need to identify whether it is in the form of rational integral, and if the answer is true, we need to quadrature it. The difficulties of finding the upper bound of the integral order and its value for solving differential equations practically are discussed, bounded Beaune’s problem is formulated. Our work is based on the method of M. N. Lagutinski. The theory and its realization are tested on the problems from Text-Book on Differential Equations by A. F. Filippov. The numerical experiments, which were carried out, show that the method makes it possible to identify the existence of the rational integral without taking much resources and time. However, using the method to calculate integrals is very time-consuming. In conclusion, recommendations on the optimal use of the method of Lagutinski are given. All calculations are executed in the computer algebra system Sage.

The optimal modified variational iteration method for the Lane-Emden equations with Neumann and Robin boundary conditions

An effective analytical technique is proposed for the solution of the Lane-Emden equations. The proposed technique is based on the variational iteration method (VIM) and the convergence control parameter h . In order to avoid solving a sequence of nonlinear algebraic or complicated integrals for the derivation of unknown constant, the boundary conditions are used before designing the recursive scheme for solution. The series solutions are found which converges rapidly to the exact solution. Convergence analysis and error bounds are discussed. Accuracy, applicability of the method is examined by solving three singular problems: i) nonlinear Poisson-Boltzmann equation, ii) distribution of heat sources in the human head, iii) second-kind Lane-Emden equation.

Plane wave scattering by a dielectric loaded slit in a thick impedance screen

The electromagnetic scattering of the plane wave by a slit loaded with dielectric in a thick impedance screen is considered for a rather general case where the surface impedances of the screen planes and vertical sides of the slit have different values. Because of the symmetry of the scattering structure, the image bisection principle is used and the original problem is splitted up into even and odd excitation cases. By introducing the Fourier transform for each case yields a modified Wiener–Hopf equation of the second kind for each excitation. After decoupling these equations by standard transformation, each modified Wiener–Hopf equation is reduced to a pair of coupled Fredholm integral equations of the second kind. The solution of each integral equation involves two sets of infinitely many constants satisfying two infinite systems of linear algebraic equations and Branch-cut integrals which are evaluated approximately. Numerical solutions of these systems are obtained for various values of the parameters such as the surface impedance of the screen, the vertical wall impedance of the slit, the width and the thickness of the slit and the permittivity of the filling material of the slit which permit one to study the effect of these parameters on the diffraction phenomenon. In order to validate the present Wiener–Hopf solution, experimental verification of the dielectric loaded thick metallic slit has been implemented in microwave frequencies with good agreements.

A novel approach to segmentation and measurement of medical image using level set methods

The study proposes a novel approach for segmentation and visualization plus value-added surface area and volume measurements for brain medical image analysis. The proposed method contains edge detection and Bayesian based level set segmentation, surface and volume rendering, and surface area and volume measurements for 3D objects of interest (i.e., brain tumor, brain tissue, or whole brain).Two extensions based on edge detection and Bayesian level set are first used to segment 3D objects. Ray casting and a modified marching cubes algorithm are then adopted to facilitate volume and surface visualization of medical-image dataset. To provide physicians with more useful information for diagnosis, the surface area and volume of an examined 3D object are calculated by the techniques of linear algebra and surface integration. Experiment results are finally reported in terms of 3D object extraction, surface and volume rendering, and surface area and volume measurements for medical image analysis.

Algebraic data integration

AbstractIn this paper, we develop an algebraic approach to data integration by combining techniques from functional programming, category theory, and database theory. In our formalism, database schemas and instances are algebraic (multi-sorted equational) theories of a certain form. Schemas denote categories, and instances denote their initial (term) algebras. The instances on a schemaSform a category,S–Inst, and a morphism of schemasF:S→Tinduces three adjoint data migration functors: ΣF:S–Inst→T–Inst, defined by substitution alongF, which has a right adjoint ΔF:T–Inst→S–Inst, which in turn has a right adjoint ΠF:S–Inst→T–Inst. We present a query language based on for/where/return syntax where each query denotes a sequence of data migration functors; a pushout-based design pattern for performing data integration using our formalism; and describe the implementation of our formalism in a tool we call AQL (Algebraic Query Language).

SOME LINEAR AND NONLINEAR EQUATIONS OF HIGHER ORDER

In this article we consider boundary value problems for some linear and nonlinear differential equations with partial derivatives of the sixth, fifth, fourth and third orders. We write out conditions on equation coefficients for which existence and uniqueness of solutions from Sobolev's space occur. If these conditions on equation coefficients are not valid, then there are given examples when solution is not unique, or is not unstable, or does not belong to Sobolev's space from existence and uniqueness theorem even for analytical coefficients and analytical right side of differential equation. After S.P. Novikov’s fundamental study in 1974 the interest to the nonlinear Korteweg-de Vries equation, Kadomtsev-Petviashvili equation and other nonlinear equations significantly grew. In this study of such equations we used methods of algebraic geometry integration and expansion method. In these studies exact solutions of special nonlinear equations series in partial derivatives play a big role. Solvability of similar equations was also studied in articles of A.I. Kozhanov, N.A. Larkin and other authors. The aim of this article is to find some exact solutions for special series partial differential equations. Solution graphs of such problems for linear equations and for the Korteweg-de Vries, Burgers-Korteweg-deVries, and Kadomtsev-Petviashvili equations are constructed.

The Dirac equation in an external electromagnetic field: symmetry algebra and exact integration

Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are not obtained by separation of variables for several external electromagnetic fields. We have considered an example of crossed electric and magnetic fields of a special type for which the Dirac equation admits a nonlocal symmetry operator.

Algebraic First Integrals of the Polynomial Systems Satisfying the Cauchy–Riemann Conditions

We consider polynomial planar systems of ordinary differential equations that can be written in the form \(\dot{z}={\mathcal {P}}(z), \ z\in {\mathbb {C}}.\) We describe the classes of such systems that have rational first integrals and study some properties of these integrals.

Orbit determination with the two-body integrals: III

We present the results of our investigation on the use of the two-body integrals to compute preliminary orbits by linking too short arcs of observations of celestial bodies. This work introduces a significant improvement with respect to the previous papers on the same subject: Gronchi et al. (2010, 2011). Here we find a univariate polynomial equation of degree 9 in the radial distance $$\rho $$ of the orbit at the mean epoch of one of the two arcs. This is obtained by a combination of the algebraic integrals of the two-body problem. Moreover, the elimination step, which in Gronchi et al. (2010, 2011) was done by resultant theory coupled with the discrete Fourier transform, is here obtained by elementary calculations. We also show some numerical tests to illustrate the performance of the new algorithm.

Infiltration into Two-Layered Soil: The Green–Ampt and Averyanov Models Revisited

Infiltration into a two-layered soil (continuum scale) is studied by the Green–Ampt (GA) model where the rate of infiltration (wetting front propagation) is a non-monotonic and discontinuous function of time. Both a constant ponding depth and a ponding level falling due to evaporation and infiltration are studied theoretically and in column experiments. The triads of saturated conductivity, porosity and front pressure in the two zones, as well as gravity, control vertical infiltration. Another model on the scale of a straight cylindrical capillary uses the Averyanov regime of a film flow, modified to include a front which is a quarter of a torus, rather than a spherical cap in a standard Washburn–Lukas model. The rate of front propagation is controlled by the Newtonian viscosity, tube-film sizes, surface tension, contact angle and gravity. For both scales, Cauchy problems for ODEs with respect to the front position are solved by computer algebra integration. In the field, pedon description and textural analysis of sedimentation-controlled layering are done. Of particular note is the thick cake deposited from a flash flood water pulse, which has very low permeability and can greatly inhibit infiltration. The infiltration rates measured by tension infiltrometers fluctuate, which is attributed to the mosaic and layered special distribution of hydraulic and capillary properties of the accrued sediments. Falling ponding depth laboratory experiments were conducted in four columns with a coarse substratum (sand) subjacent to a layer of thoroughly homogenized repacked dam silt. Imbibition front stopped at the interface between two texturally contrasting strata and the “waiting time” of the capillary barrier breakthrough was measured, as well as further oscillations of the infiltrations rate.

Algebraic integrability for minimum energy curves

Algebraic integrability for minimum energy curves

Automated Theorem Proving in GeoGebra: Current Achievements

GeoGebra is an open-source educational mathematics software tool, with millions of users worldwide. It has a number of features (integration of computer algebra, dynamic geometry, spreadsheet, etc.), primarily focused on facilitating student experiments, and not on formal reasoning. Since including automated deduction tools in GeoGebra could bring a whole new range of teaching and learning scenarios, and since automated theorem proving and discovery in geometry has reached a rather mature stage, we embarked on a project of incorporating and testing a number of different automated provers for geometry in GeoGebra. In this paper, we present the current achievements and status of this project, and discuss various relevant challenges that this project raises in the educational, mathematical and software contexts. We will describe, first, the recent and forthcoming changes demanded by our project, regarding the implementation and the user interface of GeoGebra. Then we present our vision of the educational scenarios that could be supported by automated reasoning features, and how teachers and students could benefit from the present work. In fact, current performance of GeoGebra, extended with automated deduction tools, is already very promising--many complex theorems can be proved in less than 1 second. Thus, we believe that many new and exciting ways of using GeoGebra in the classroom are on their way.

Algebraic structure and integration maps in cocycle models for differential cohomology

of cohomology theories for the graded coefficient vector space V D E R. We always assume E to be rationally even, meaning V 2nC1 D 0 for all n 2 Z. Definition 1.1 A differential extension [4] of E consists of a contravariant functor y E from the category of manifolds to graded abelian groups along with natural linear transformations I;R; a as in the following commutative diagram, which is required to have an exact upper horizontal row.

A Note on Teleparallel Killing Symmetries in Three Dimensional Circularly Symmetric Static Spacetime

In this article we investigate Killing vectors in three dimensional Lorentzian manifold for circularly symmetric static spacetime in the presence of torsion. Teleparallel Killing equations are obtained and then solved by using some algebraic and direct integration techniques. This study reveals that the above spacetime admit two, three, four, five or six teleparallel Killing vectors. We compare our results with the available results in general relativity where the same spacetime admit two, three, four or six Killing vectors. It turns out that the teleparallel Killing vectors are different from Killing vectors in general relativity for the same spacetime.

Relativistic algorithm for time transfer in Mars missions under IAU Resolutions: an analytic approach

With tremendous advances in modern techniques, Einstein's general relativity has become an inevitable part of deep space missions. We investigate the relativistic algorithm for time transfer between the proper time τ of the onboard clock and the Geocentric Coordinate Time, which extends some previous works by including the effects of propagation of electromagnetic signals. In order to evaluate the implicit algebraic equations and integrals in the model, we take an analytic approach to work out their approximate values. This analytic model might be used in an onboard computer because of its limited capability to perform calculations. Taking an orbiter like Yinghuo-1 as an example, we find that the contributions of the Sun, the ground station and the spacecraft dominate the outcomes of the relativistic corrections to the model.

Algebraic Complete Integrability of the Bloch–Iserles System

The goal of this paper is the proof of the algebraic complete integrability of the Bloch-Iserles Hamiltonian system [5]. This result was conjectured in [4], based on its validity in certain special cases.

Algebraic invariant curves and first integrals for Riccati polynomial differential systems

We characterize the algebraic invariant curves for the Riccati polynomial differential systems of the form x′ = 1, y′ = a(x)y2 +b(x)y +c(x), where a(x), b(x) and c(x) are arbitrary polynomials. We also characterize their algebraic first integrals.

The Poincaré problem, algebraic integrability and dicritical divisors

We solve the Poincaré problem for plane foliations with only one dicritical divisor. Moreover, in this case, we give a simple algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case. We also provide an algorithm to compute a rational first integral of prefixed genus g≠1 of any type of plane foliation F. When the number of dicritical divisors dic(F) is larger than 2, this algorithm depends on suitable families of invariant curves. When dic(F)=2, it proves that the degree of the rational first integral can be bounded only in terms of g, the degree of F and the local analytic type of the dicritical singularities of F.

Progress-Curve Analysis Through Integrated Rate Equations and Its Use to Study Cholinesterase Reaction Dynamics

Michaelis and Menten found the direct mathematical analysis of their studied enzyme-catalyzed reaction unrealistic 100 years ago, and hence, they avoided this problem by correct adaptation and analysis of the experiment, i.e., differentiation of the progress-curve data into rates. However, the most elegant and ideal simplification of the evaluation of kinetics parameters from progress curves can be performed when the algebraic integration of the rate equation results in an explicit mathematical equation that describes the dynamics of the model system of the reaction. Recently, it was demonstrated that such an alternative approach can be considered for enzymes that obey the generalized Michaelis–Menten reaction dynamics, although its use is now still limited for cholinesterases, which show kinetics that deviate from saturationlike hyperbolic behavior at high concentrations of charged substrates. However, a mathematical approach is reviewed here that might provide an alternative to the decades-old problemof data analysis of cholinesterase-catalyzed reactions, through the more complex Webb integrated rate equation.