We present an extension of the methods of classical Lie group analysis of diierential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau's theory of algebras of generalized functions. We show that under some mild conditions on the diierential equations, symmetries of classical solutions r...

Using a statistical mechanical formalism we calculate the evidence, generalisation error and consistency measure for a linear perceptron trained and tested on a set of examples generated by a non linear teacher. The teacher is said to be unrealisable because the student can never model it without error. Our model allows us to interpolate between the known case of a linear teacher, and an unreal...

Usually the straightforward generalization of explicit Runge{Kutta methods for ordinary diierential equations to half-explicit methods for diierential-algebraic systems of index 2 results in methods of order q 2 ((8]). The construction of higher order methods is simpliied substantially by a slight modiication of the method combined with an improved strategy for the computation of the algebraic ...

We establish some nonlinear integral inequalities for functions defined on a time scale. The results extend some previous Gronwall and Bihari type inequalities on time scales. Some examples of time scales for which our results can be applied are provided. An application to the qualitative analysis of a nonlinear dynamic equation is discussed.

This paper introduces an approach called learning decomposition to analyze what types of practice are most effective for helping students learn a skill. The approach is a generalization of learning curve analysis, and uses non-linear regression to determine how to weight different types of practice opportunities relative to each other. We are able to show that different types of practice differ...

In this work we study the trajectories of test particles in a geometry that is the nonlinear electromagnetic generalization of the ReissnerNordström solution. The studied spacetime is a Einstein-Born-Infeld solution, nonsingular outside a regular event horizon and characterized by three parameters: mass M , charge Q and the Born-Infeld parameter b related to the magnitude of the electric field ...

The method of additional generating conditions is applied for finding new non-Lie ansätze and exact solutions of nonlinear generalizations of the Fisher equation.

We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction g{2}/k+1(ΨΨ){k+1} , as well as a vector-vector self interaction g{2}/k+1(Ψγ{μ}ΨΨγ{μ}Ψ){1/2(k+1)} . We find the exact analytic form for solitary waves for arbitrary k and find that they are a generalization of the exact solutions for the nonlinear Schrödinger equation (NLSE) and reduce to thes...