Hurwitz series as formal functions

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

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.

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.

In the framework of instantaneous approximations to the Bethe-Salpeter formalism for the description of bound states within quantum field theories, depending on the Lorentz structure of the Bethe-Salpeter interaction kernel the solutions of the full Salpeter with some confining interaction may exhibit certain instabilities, which are possibly related to the Klein paradox and signal the decay of states assumed to be bound by the confining interactions. They are observed in numerical (variational) studies of the Salpeter equation. The presumably simplest scenario allowing for the fully analytic investigation of this problem is set by the reduced Salpeter with interaction. In this case, Salpeter's integral simplifies to either an algebraic relation or a second-order homogeneous linear ordinary differential equation, immediately accessible to standard techniques. There one may hope to be able to decide unambiguously whether this setting poses a well-defined eigenvalue problem the solutions of which correspond to stable bound states associated to real energy eigenvalues bounded from below. By analytical spectral analysis the bound-state solutions of this harmonic-oscillator reduced Salpeter equation can be shown to be free of such instabilities.

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.

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.

In 1908 Birkhoff defined a class of series generalizing the Fourier and Sturm-Liouville series which arise from ordinary linear homogeneous differential equations of the second order with linear homogeneous boundary conditions. He starts from a differential system of order n with boundary conditions restricted to be regular and proves that an arbitrary function with a sufficient number of derivatives can be represented in terms of the solutions of this differential system.t Subsequently, Tamarkin extended the discussion to apply to functions integrable in the sense of Lebesgue by the device of comparing the formal Fourier and Birkhoff series for the same function.t Recently the author has published further researches on the problem. ? It is now proposed to complete in some respects our knowledge of regular differential systems of the second order. In our paper cited above, we left certain necessary conditions unstudied and now desire to fill this gap in the theory. 11 In addition, the attempt to discuss the formal expansions for functions integrable in accordance with definitions of integration more extended than that of Lebesgue has led us to results quite different from those hitherto established. We shall indicate the facts for functions integrable in the sense of Denjoy. The situation may be described roughly by saying that, whereas all the regular expansions associated with an arbitrary function integrable in the sense of Lebesgue have virtually the same behavior, the series formed for an arbitrary function integrable in the sense of Denjoy or non-absolutely integrable in the sense of Riemann fall into a non-denumerable infinity of distinct types exhibiting individual behavior. The result is due essentially to the fact that the Riemann-Lebesgue theorem does not hold true for nonabsolutely integrable functions. On the other hand, if the series for an arbi-

One the Solutions of Ordinary Linear Homogeneous Differential Equations of the Third Order

In the case of n functions of a single variable, the vanishing of the wronskian is the most familiar criterion for their linear dependence. The wronskian also plays an important part in connection with the theory of a single ordinary homogeneous linear differential equation of the nth order, even in questions which do not concern the linear dependence of solutions of the equation. It is the purpose of the present paper to generalize the fundamental facts connected with these topics, to the case of functions of several variables. The characteristic property of an ordinary homogeneous linear differential equation is that any solution is expressible linearly, with constant coefficients, in terms of a fundamental set of solutions. The natural generalization to several independent variables is afforded by the completely integrable system of homogeneous linear partial differential equations, in one dependent variable, any solution of which is likewise linearly dependent on a finite number of solutions of the system. It is such systems which are discussed in the latter part of this paper; the subject of the linear dependence of functions of several variables is developed in the first part, chiefly with a view to its subsequent application to completely integrable systems. In our discussion of linear dependence, we consider throughout n functions Yl, Y2, Y n,, of p independent variables u1, u2, * up . Functions and variables may be either real or complex; if any of the variables be complex, we suppose the functions to be analytic in those variables. All of the main theorems, however, are stated and proved under the supposition that the independent variables are real; the modifications usually simplifications and omissions-are easily supplied for cases in which one or more of the independent variables is complex. The n functions, then, of the real variables

We consider a SEIR model for the spread (transmission) of an infectious disease. The model has played an important role due to world pandemic disease spread cases. Our contributions in this paper are three folds. Our first contribution is to provide successive approximation and variational iteration methods to obtain analytical approximate solutions to the SEIR model. Our second contribution is to prove that for solving the SEIR model, the variational iteration and successive approximation methods are identical when we have some particular values of Lagrange multipliers in the variational iteration formulation. Third, we propose a new multistage-analytical method for solving the SEIR model. Computational experiments show that the successive approximation and variational iteration methods are accurate for small size of time domain. In contrast, our proposed multistage-analytical method is successful to solve the SEIR model very accurately for large size of time domain. Furthermore, the order of accuracy of the multistage-analytical method can be made higher simply by taking more number of successive iterations in the multistage evolution.

Picard's method of successive approximations is utilized to address the linear differential equation that governs the behaviour of the damped forced harmonic oscillator under the influence of an arbitrary forcing function. With the time-dependent substitution $\chi(t) = e^{-\gamma t}x(t)$, the first derivative term vanishes and the differential equation reduces to a second-order ODE without damping for $x(t)$. This modified equation takes the form of an inhomogeneous Volterra integral equation of the second type, incorporating two initial conditions. Initiating the iterative process of successive approximations with a meticulously selected zero-order approximation, we demonstrate the emergence of convergent sequences of partial sums, culminating in the derivation of a closed-form solution. The straightforward approach employed in this paper to analyze the damped and forced harmonic oscillator is accessible to undergraduate students in physics, engineering, and mathematics, without requiring sophisticated mathematical tools. Moreover, it facilitates a direct physical analysis of the problem and serves as an insightful example of the application of Taylor series expansion. As specific examples, presented merely as illustrative applications that have not yet been addressed in the literature (at least not in this context for the second example), we consider an oscillator driven by a Gaussian-like force and by a normalized exponential force. These examples can be seen as regularized generalizations of an impulsive, delta-like force.

An analysis is presented for the free vibration of a spinning truncated conical shell. For this purpose, the governing equations and the boundary conditions of the shell are derived by applying Hamilton's principle to the strain and kinetic energies of the shell. The variables in the equations can be written as a summation of the quasi-static components which are independent on time and the dynamic ones. The nonlinear equations governing the finite deformations caused by the spinning of the shell are solved by use of Picard's method of successive approximation, and the linear equations on the vibration about the deformed state are solved by using the transfer matrix method. The method is applied to a spinning truncated conical shell clamped at the small edge and the quasi-static displacement and stress distributions and the frequency parameters are calculated numerically, and the effects of the spinning of the shell on the free vibration are studied.