We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential eq...

Preface The nonlinear σ-model is a field theory whose action is a energy functional of maps from a space-time to a Riemannian manifold. This theory is used in particle physics and condensed-matter physics as low-energy effective theories. The solutions of its equation of motion are called harmonic maps, which are important object in geometry. If the dimension of the space-time is 1 + n (this me...

We propose a generalization of the standard geometric formulation of quantum mechanics, based on the classical Nambu dynamics of free Euler tops. This extended quantum mechanics has in lieu of the standard exponential time evolution, a nonlinear temporal evolution given by Jacobi elliptic functions. In the limit where latter’s moduli parameters are set to zero, the usual geometric formulation o...

We present an analysis of how the generalization performance expected test set error relates to the expected training set error for nonlinear learn ing systems such as multilayer perceptrons and radial basis functions The principal result is the following relationship computed to second order between the expected test set and training set errors hEtest i hEtrain i eff peff n Here n is the size ...

In the preface of the book, Cristianini and Shawe-Taylor state that their intention is to present an organic, integrated introduction to support vector machines (SVMs). The authors believe that SVMs are a topic now sufficiently mature that it should be viewed as its own subfield of machine learning. SVMs, first introduced by Vladimir Vapnik, are a type of linear learning machines much like the ...

The Newton method is one of the most powerful tools used to solve systems of nonlinear equations. Its set-valued generalization, considered in this work, allows one to solve also nonlinear equations with geometric constraints and systems of inequalities in a unified manner. The emphasis is given to systems of linear inequalities. The study of the well-posedness of the algorithm and of its conve...

We consider approximation schemes for monotone systems of fully nonlinear second order partial differential equations. We first prove a general convergence result for monotone, consistent and regular schemes. This result is a generalization to the well known framework of Barles-Souganidis, in the case of scalar nonlinear equation. Our second main result provides the convergence rate of approxim...

Support Vector Machines (SVM) are a system for efficiently training linear learning machines in the kernel induced feature spaces, while respecting the insights provided by the generalization theory and exploiting the optimization theory. In this work, Support Vector Machines are employed for the nonlinear regression. The nonlinear regression ability of the Support Vector Machines has been demo...

Stability of the solutions to a nonlinear impulsive Caputo fractional differential equation is studied using Lyapunov like functions. The derivative of piecewise continuous Lyapunov functions among the nonlinear impulsive Caputo differential equation of fractional order is defined. This definition is a natural generalization of the Caputo fractional Dini derivative of a function. Several suffic...

We consider a system of coupled Klein-Gordon equations, which models one-dimensional nonlinear wave processes in two-component media. We find both linear and nonlinear solutions involving the exchange of energy between the different components of the system. The solutions are a continuum generalization of the classical example of energy exchange in Mandelshtam’s system of coupled pendulums. PAC...