Stability of Salpeter Solutions

TABLE OF CONTENTS INTRODUCTION. Historical background. Summary. Notation and terminology. CHAPTER I. ALGEBRAIC MATRIC GROUPS. 1. Reducibility of sets of matrices. 2. Algebraic matric groups. 3. Jordan-Holder-Schreier theorem. 4. Commutator groups. 5. Solvable algebraic matric groups. 6. Anticompact and quasicompact algebraic matric groups. 7. Reducibility to triangular form. 8. Algebraic matric groups with certain types of normal chains. CHAPTER II. SOME P.ESULTS FROM THE THEORY OF ALGEBRAIC DIFFERENTIAL EQUATIONS. 9. Differential rings, fields, and ideals. 10. Differential polynomials. 11. Solutions. 12. Relative isomorphisms. 13. Order. 14. Dependence. 15. Homogeneous linear ordinary differential equations. CHAPTER III. NORMAL DIFFERENTIAL EXTENSION FIELDS. 16. Normal differential extension fields. CHAPTER IV. PICCARD-VESSIOT EXTENSIONS. 17. Picard-Vessiot extensions and their isomorphisms. 18. Normality. 19. Characterization of G. 20. Dimension. 21. Adjunction of new elements. 22. Linear reducibility of L(y). CHAPTER V. LIOUVILLIAN EXTENSIONS. 23. Integrals and exponentials of intgrals. 24. Liouvillian extensions. 25. The principal theorem and some consequences. 26. The proof, first half. 27. The proof, second half. REFERENCES INTRODUCTION

59 - Exact Second Order Equations

In this paper, we present a novel symmetry-enhanced transformℒ(𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥);𝑎𝑎𝑎𝑎) =� 𝑒𝑒𝑒𝑒−𝑎𝑎𝑎𝑎𝑥𝑥𝑥𝑥2+∞−∞𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥)𝑑𝑑𝑑𝑑𝑥𝑥𝑥𝑥=ℬ(𝑎𝑎𝑎𝑎),𝑎𝑎𝑎𝑎>0to evaluate Gaussian integrals commonly used in mathematical and physical domains, particularly in quantum field theory. Additionally, we utilize this original transformation methodology to solve a wide range of second-order linear ordinary differential equations (ODEs) that have variable coefficients, which is a common occurrence in physics. Notable examples encompass Weber,Euler-Cauchy, and Bessel equations, highlighting the broad applicability of our proposed method. Diverging from established transforms like the widely used Laplace transform, our innovative approach introduces a symmetrical model, offering a distinct and founding perspective to the field.

A method for the numerical solution of the n -point boundary value problem for homogeneous linear ordinary differential equations is developed. The method requires two Runge-Kutta integrations over the interval under consideration and the solution of a linear system of equations with n -1 unknowns.

Hurwitz series as formal functions

The eigentensors of an arbitrary second-order tensor and their quality analyses

Complex-valued functions defined on a finite interval [a, b] generalizing power functions of the type (x−x0)n for n ≥ 0 are studied. These functions called Φ-generalized powers, Φ being a given nonzero complex-valued function on the interval, were considered to construct a general solution representation of the Sturm–Liouville equation in terms of the spectral parameter [V. V. Kravchenko and R. M. Porter, Math. Methods Appl. Sci. 33(4), 459–468 (2010)]. The Φ-generalized powers can be considered as natural basis functions for the one-dimensional supersymmetric quantum mechanics systems taking Φ=ψ02, where the function ψ0(x) is the ground state wave function of one of the supersymmetric scalar Hamiltonians. Several properties are obtained such as Φ-symmetric conjugate and antisymmetry of the Φ-generalized powers, a supersymmetric binomial identity for these functions, a supersymmetric Pythagorean elliptic (hyperbolic) identity involving four Φ-trigonometric (Φ-hyperbolic) functions, as well as a supersymmetric Taylor series expressed in terms of the Φ-derivatives. We show that the first n Φ-generalized powers are a fundamental set of solutions associated with nonconstant coefficient homogeneous linear ordinary differential equations of order n + 1. Finally, we present a general solution representation of the stationary Schrödinger equation in terms of geometric series where the Volterra compositions of the first type are considered.

Let be given a homogeneous linear ordinary differential equation with coefficients which are polynomial over a field K. The well known Frobenius’ algorithm [7] computes the fundamental solutions system in the neighborhood of regular singularities. These solutions are from \(\mathop K\limits^ \sim \) [[x]][ln(x)] where \(\mathop K\limits^ \sim \) is an algebraic extension of K. We present a new method based on the Frobenius’ algorithm. This method allows to build solutions with hypergeometric coefficients of Laurent or Puiseux series.

Symmetries and invariants of the systems of two linear second-order ordinary differential equations

The partial Noether operators and first integrals of a general system of two linear second-order ordinary differential equations (ODEs) with variable coefficients are studied by means of a partial Lagrangian. The canonical form for the general system of two second-order ordinary differential equations is invoked and all cases of this system are discussed with respect to partial Noether operators. We also tabulate the results for the special case b(x) = c(x) of the system which was considered elsewhere using a Lagrangian and a partial Lagrangian. The first integrals are obtained explicitly by exploiting a Noether-like theorem with the help of partial Noether operators. This study gives a new way to construct first integrals for systems without a variational principle as not all linear equations have a Lagrangian. Physical applications to conservative and oscillator mechanical systems are given.

Addition theorems for solutions to linear homogeneous constant coefficient ordinary differential equations

In this paper the First Order Linear Fuzzy Ordinary Differential Equations are described. Here coefficients and /or initial condition of said differential equation are taken as the Generalized Triangular Fuzzy Numbers (GTFNs).The solution procedure of this Fuzzy Differential Equation is developed by Lagrange Multiplier Method. An imprecise barometric pressure problem is described.

We explain how the space of linearly recursive sequences over a field can be considered as a Hopf algebra. The algebra structure is that of divided-power sequences, so we concentrate on the perhaps lesser-known coalgebra (diagonalization) structure. Such a sequence satisfies a minimal recursive relation, whose solution space is the subcoalgebra generated by the sequence. We discuss possible bases for the solution space from the point of view of diagonalization. In particular, we give an algorithm for diagonalizing a sequence in terms of the basis of the coalgebra it generates formed by its images under the difference-operator shift. The computation involves inverting the Hankel matrix of the sequence. We stress the classical connection (say over the real or complex numbers) with formal power series and the theory of linear homogeneous ordinary differential equations. It is hoped that this exposition will encourage the use of Hopf algebraic ideas in the study of certain combinatorial areas of mathematics.

We study the linearization of a class of Li ´ enard type nonlinear second-order ordinary differential equations f rom the generalized Sundman transformation viewpoint. The linearizing generalized Sundman transformation for the class of equations is constructed. The transformation is used to map the underlying class of equations into a linear second-order ordinary differential equation which is not in the Laguerre form. The general solution of this class of equations is obtained by integrating the linearized equation and applying the generalized Sundman transformation. Moreover, we apply a Riccati transformation to a general linear third-order variable coefficients ordinary differential equation to extend the underlying class of equations and we also derive the conditions of linearizability of this new class of nonlinear second-order ordinary differential equ ations using the generalized Sundman transformation method and obtain its general solution.

33 - Transformations of Second Order Linear ODEs – 2