Algebraic Integrability Research Articles

Killing Motion of Static Cylindrically Symmetric Spacetimes in the f(R) Gravity

In this study we have studied "Killing Motion of Static Cylindrically Symmetric Spacetimes in f(R) Gravity" by using algebraic and direct integration techniques. This study investigates the Killing motions of static cylindrically symmetric spacetimes with in framework of f(R) gravity, a generalization of Einstein’s General Relativity. We explore the existence of Killing vector fields to understand the symmetries and conserved quantities in such spacetimes. By analysing the modified field equations, we determine the constraints imposed by f(R) gravity on the geometry and dynamics of cylindrically symmetric spacetimes. These contribute to understanding the interplay between symmetry properties and gravitational theories beyond General Relativity. The results have implications for astrophysical and cosmological models influenced by alternative gravity theories. We discussed four cases and found that the dimension of Killing vector fields is either three, four or ten.

Algebraic Complete Integrability of the $a_4^{(2)}$ Toda Lattice

The aim of this work is focused on the investigation of the algebraic complete integrability of the Toda lattice associated with the twisted affine Lie algebra $a_4^{(2)}$. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description of these abelian surfaces and the divisor at infinity.

Estudio comparativo de los paradigmas de programación orientada a objetos y programación reactiva en la resolución de Integrales Algebraicas

The article evaluates the performance in terms of execution times in solving algebraic integrals using both the object-oriented programming (OOP) and reactive programming (RP) paradigms. The problem addressed is the lack of scientific evidence that allows us to determine which paradigm offers the best results in terms of execution times in solving these integrals. The approach of the study involves the implementation of two versions in Java, each built following the principles of the aforementioned paradigms. Subsequently, through controlled experimental scenarios and integrated Java methods, the execution time of each application is measured. The results reveal that reactive programming demonstrates greater efficiency in terms of performance. This research focuses on the resolution of linear and polynomial algebraic integrals and points out the need for more extensive research in this field. In conclusion, the study shows that reactive programming significantly outperforms object-oriented programming, demonstrating notably lower execution times.

A Collocation Numerical Method for Highly Oscillatory Algebraic Singular Volterra Integral Equations

The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis.

Study of algebra integration and analysis in teaching mathematics and its applications

Study of algebra integration and analysis in teaching mathematics and its applications

Quadratic Starlike Trees

We introduce the notion of a quadratic graph, which is a graph whose eigenvalues are integral or quadratic algebraic integral, and we determine nine infinite families of quadratic starlike trees, which are just all the quadratic starlike trees including integral starlike trees. Thus, the quadratic starlike trees are completely characterized. The expressions for characteristic polynomials of quadratic starlike trees are also given.

Construction of wave solutions for stochastic Radhakrishnan-Kundu-Lakshmanan equation using modified extended direct algebraic technique

In this article, stochastic solutions of Radhakrishnan-Kundu-Lakshmanan equation (RKLE) with multiplicative noise are investigated. Studying is conducted with the aid of the modified extended direct algebraic integration technique which depended on the extended Riccati equation. To aid in our evaluation, a graphic representation of noise’s impact on the retrieved solutions is presented using MATLAB software packages.

AN ALGEBRAIC INTERPRETATION OF THE SUPER CATALAN NUMBERS

AbstractWe extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields $\mathbb {F}$ of characteristic zero as a normalised $\mathbb {F}$ -linear functional on $\mathbb {F}[\alpha _1, \alpha _2]$ that maps polynomials that evaluate to zero on C to zero and is $\mathrm {SO}(2,\mathbb {F})$ -invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers $$ \begin{align*} S(m,n) = \frac{(2m)!(2n)!}{m!n!(m+n)!} \end{align*} $$ an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $\alpha _1^{2m}\alpha _2^{2n}$ .

Symbolic algebra integration of soil elastoplastic models

This work analyses the time integration of elastoplastic models using implementation platforms with symbolic algebra capabilities. In such platforms, the variables and state functions involved in solving boundary value problems are considered entities continuous in time, thus the time integration of stresses and hardening parameters is always based on implicit schemes. The latter confers to the computation an accuracy that is inspected in this work by analysing the integration of the Clay and Sand Model. Finally, the advantages of using integration schemes in which the linearisation of the yield function is imposed to be zero when calculating the plastic multiplier are illustrated.

Non-Associative Structures and Their Applications in Differential Equations

This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between these theories is the nonlinear spectral theory developed for algebra and homogeneous differential equations. A nonlinear spectral method is used to prove the existence of an algebraic first integral, interpretations of various phase zones, and the separatrices construction for ODEs. In algebra, the same methods exploit subalgebra construction and explain fusion rules. In conclusion, perturbation methods may also be interpreted for near-Jordan algebra construction.

Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of , this provides an explicit connection with the generic superintegrable model on the -dimensional sphere and the related Racah algebra R(n). In particular, we show explicitly how the models on the two-sphere and three-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of and , respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.

Tropical Monte Carlo quadrature for Feynman integrals

We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The method can be improved further by exploiting the geometric structure of the underlying integrand. As an illustration of this, we give a specialized integration algorithm for a class of integrands that exhibit the form of a generalized permutahedron. This class includes integrands for scattering amplitudes and parametric Feynman integrals with tame kinematics. A proof-of-concept implementation is provided with which Feynman integrals up to loop order 17 can be evaluated.

Динамика перманентного движения твердого тела в псевдоевклидовом пространстве

The conditions for the existence and properties of the permanent motion of an absolutely rigid body rotating around a fixed pole are studied in the pseudo-Euclidean space. The body is affected by a force vector-moment, which is constant relative to the coordinate system, invariably associated with the body, as well as a system of gyroscopic forces with a resulting vector-moment, which is linear relative to the angular velocity of the body. The first algebraic integrals of a dynamical system are found, as well as the necessary conditions for the existence of a permanent motion of a body. The equations of the moving hodograph of the angular velocity in this motion are obtained. The structural-kinetic properties of the axes of permanent rotation and the geometry of their distribution in space relative to the body are studied.

Conservative finite difference schemes for dynamical systems

The article presents the implementation of one of the approaches to the integration of dynamical systems, which preserves algebraic integrals in the original fdm for Sage system. This approach, which goes back to the paper by del Buono and Mastroserio, makes it possible, based on any two explicit difference schemes, including any two explicit Runge-Kutta schemes, to construct a new numerical algorithm for integrating a dynamical system that preserves the given integral. This approach has been implemented and tested in the original fdm for Sage system. Details and implementation difficulties are discussed. For testing, two Runge-Kutta schemes were taken having the same order, but different Butcher tables, which does not complicate the method due to paralleling. Two examples are considered - a linear oscillator and a Jacobi oscillator with two quadratic integrals. The second example shows that the preservation of one integral of motion does not lead to the conservation of the other. Moreover, this method allows us to propose a practical application of the well-known ambiguity in the definition of Butcher tables.

The algebraic dynamics of the pentagram map

AbstractThe pentagram map, introduced by Schwartz [The pentagram map. Exp. Math.1(1) (1992), 71–81], is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev’s proof of complex integrability [F. Soloviev. Integrability of the pentagram map. Duke Math. J.162(15) (2013), 2815–2853]. In the course of the proof, we construct the moduli space of twisted n-gons, derive formulas for the pentagram map, and calculate the Lax representation by characteristic-independent methods.

The influence of geometric algebra in surgical practice of sleeve gastrectomy-single center experience.

Laparoscopic surgery could be considered as an art of geometric algebra. However, very little is studied in the context of bariatric surgery. The current study aims to explore the possible influence concept of geometric algebra on the surgical process in the overweight and obese patients in the setting of laparoscopic sleeve gastrectomy (LSG). During the study period, clinical data of subjects who underwent LSG was retrospectively analyzed. Parameters examined include body mass index (BMI), umbilical-xiphoidal interval (U-X) and umbilical-fundus (U-F) interval. In this study, LSG was performed via central view approach (C) and left view approach (L). In both groups, the body surface projection points of viewing hole (V), main and accessory operating holes (O1 and O2) and surface display of fundus (F) were connected to form a geometric figure. The accessibility of the surgical instrument into the fundus, the need for elongated instruments and related intra- and post-operative parameters were noted. The overweight and obese subjects showed a significant increased U-X and U-F interval compared to the non-obese subjects. The length of both U-X and U-F interval were correlated with the BMI. The geometric figure is quite different between L and C approach with significant increase of area of quadrangle. Significant longer O1-F, O2-F and V-F interval was calculated in C approach of patients and thus the elongated instruments were frequently required. The integration of the concept geometric algebra with the proper selection of troca may provide a better surgical experience and smooth surgical process.

Algebraic Integrability of Planar Polynomial Vector Fields by Extension to Hirzebruch Surfaces

We study algebraic integrability of complex planar polynomial vector fields \(X=A (x,y)(\partial /\partial x) + B(x,y) (\partial /\partial y) \) through extensions to Hirzebruch surfaces. Using these extensions, each vector field X determines two infinite families of planar vector fields that depend on a natural parameter which, when X has a rational first integral, satisfy strong properties about the dicriticity of the points at the line \(x=0\) and of the origin. As a consequence, we obtain new necessary conditions for algebraic integrability of planar vector fields and, if X has a rational first integral, we provide a region in \({\mathbb {R}}_{\ge 0}^2\) that contains all the pairs (i, j) corresponding to monomials \(x^i y^j\) involved in the generic invariant curve of X.

Multiple color image encryption based on cascaded quaternion gyrator transforms

In this paper, we propose a novel encryption algorithm using cascaded quaternion gyrator transforms that allows protecting multiple color images with efficiency at once. The originality is reflected in the integration of quaternion algebra with a cascaded encryption structure. First, image RGB channels are constituted into three imaginary parts of a quaternion matrix. A cascaded encryption structure is constructed to cooperate with the vortex phase mask, which is easy to transmit and saves storage. Then, a quaternion gyrator transform is conducted, after which the phase is extracted to serve as the decryption key while the amplitude is used as the real part of the following quaternion matrix. Finally, the amplitude of the last gyrator transform is scrambled to obtain the ciphertext. All phase matrices are further modulated by a secondary image phase mask for additional security. Numerical simulations have been conducted to verify the validity of the proposed algorithm. Experiments on several data test set demonstrate the robustness of the proposed method to various attacks.

Cubic–quartic optical solitons in fiber Bragg gratings with anti-cubic nonlinearity using the modified extended direct algebraic method

Objective:This paper obtains cubic–quartic solitons in fiber Bragg gratings with anti-cubic nonlinearity. Methods:The extended direct algebraic integration algorithm is adopted. Results:Dark, bright and singular solitons are recovered along with their respective parameter constraints for their existence. Conclusion:The integration tool yields solutions in terms of Jacobi’s elliptic functions which led to the emergence of optical solitons from the limiting operation applied to the modulus of ellipticity. The results are thus extremely promising to move ahead with this approach to study additional forms of self-phase modulation coupled with other forms of dispersive effects.

A Green’s basis for the bosonic SMEFT to dimension 8

We present a basis of dimension-eight Green’s functions involving Standard Model (SM) bosonic fields, consisting of 86 new operators. Rather than using algebraic identities and integration by parts, we prove the independence of these interactions in momentum space, including discussion on evanescent bosonic operators. Our results pave the way for renormalising the SM effective field theory (SMEFT), as well as for performing matching of ultraviolet models onto the SMEFT, to higher order. To demonstrate the potential of our construction, we have implemented our basis in matchmakereft and used it to integrate out a heavy singlet scalar and a heavy quadruplet scalar up to one loop. We provide the corresponding dimension-eight Wilson coefficients. Likewise, we show how our results can be easily used to simplify cumbersome redundant Lagrangians arising, for example, from integrating out heavy fields using the path-integral approach to matching.