On the Darboux transformation. I

Automorphisms of the family of all Sturm-Liouville equationsy=qy are investigated. The classical Darboux transformation arises as a particular case of a general results.     

On the stability of the J transformation

The stability of a large class of nonlinear sequence transformations is analyzed. Considered are variants of the J transformation [17]. Suitable variants of this transformation belong to the most successful extrapolation algorithms that are known [20]. Similar to recent results of Sidi, it is proved that the _p {J} transformations, the Weniger S transformation, the Levin transformation and a special case of the generalized Richardson extrapolation process of Sidi are S-stable. An efficient algorithm for the calculation of stability indices is presented. A numerical example demonstrates the validity of the approach. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the generalization of the Cole-Hopf transformation

The Cole-Hopf transformation, which reduces the Burgers equantion to the linear heat equation can be generalized and applied to systems of nonlinear equations displaying several interesting features such as dispersion, dissipation and coupling.

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Zusammenfassung Die Cole-Hopf-Transformation, welche die Burgers-Gleichung in die lineare Wärmeleitungsgleichung überführt, wird verallgemeinert und bei Systemen von nichtlinearen Gleichungen angewendet, die verschiedene Eigenschaften wie Dispersion, Dissipation und Kopplung aufweisen.

     

On a transformation operator

We prove the existence of a transformation operator that takes the solution of the equationy″=λ²ⁿ y to the solution of the equation

On meaningful transformation equations

Aequationes mathematicae - The meaningfulness condition, applied to scientific or geometric laws, requires that the mathematical form of an equation does not change when we change the units of its...     

On the generalization of probabilistic transformation method

The probabilistic transformation method with the finite element analysis is a new technique to solve random differential equation. The advantage of this technique is finding the “exact” expression of the probability density function of the solution when the probability density function of the input is known. However the disadvantage is due to the characteristics (geometrics and materials) of the analyzed structure included in the random differential equation.

In this paper, a developed formula is used to generalize this technique by obtaining the “exact” joint probability density function of the solution in any situations, as well as the proposed technique for the non-linear case.     

On the Hankel transformation of order zero

In this paper we write the Hankel transform of order zero of a function as a composition of Fourier transforms, and a new proof is given for Hankel's theorem on Hankel transformation of order zero.     

On the determination of an orthogonal transformation

Let A and B be n × m-matrices with m >- n. An iterative algorithm for the determination of the orthogonal P minimizing /vbPA - B /vb_F² is suggested and analyzed. Convergence is proved under natural assumptions.     

On the similarity transformation to tridiagonal form

For every real square matrix A there exists a nonsingular real matrix X, for which X^-1AX is a tridiagonal matrix, and under certain conditions X is uniquely determined. Simple proofs are presented for the existence and uniqueness of this transformation.     

On condition numbers of the Shanks transformation

Quasi-linear functions generate sequence transformation methods whose conditioning depends upon the nature of the sequence to be accelerated. These methods are often well conditioned when they are applied to alternating sequences; however, they are relatively ill-conditioned in case of monotonic convergence. The condition numbers of the Shanks transformation e_k(s_n) are given in order to prove that the closely related ε-algorithm to a such transformation is ill-conditioned when performed on the set of totally monotonic sequences. In the same way, we show that this algorithm is well conditioned on the set of totally oscillating sequences.     

On the convergence rate of the unscented transformation

Nonlinear state-space models driven by differential equations have been widely used in science. Their statistical inference generally requires computing the mean and covariance matrix of some nonlinear function of the state variables, which can be done in several ways. For example, such computations may be approximately done by Monte Carlo, which is rather computationally expensive. Linear approximation by the first-order Taylor expansion is a fast alternative. However, the approximation error becomes non-negligible with strongly nonlinear functions. Unscented transformation was proposed to overcome these difficulties, but it lacks theoretical justification. In this paper, we derive some theoretical properties of the unscented transformation and contrast it with the method of linear approximation. Particularly, we derive the convergence rate of the unscented transformation.