Catálogo de publicaciones - libros

Given the equations of the surfaces, our goal is to construct a univariate polynomial one of the zeroes of which coincides with the square of the distance between these surfaces. To achieve this goal we employ the Elimination Theory methods.

The study of linear ordinary differential equations (ODEs) with parametric coefficients is an important topic in robust control theory. A central problem is to determine parameter ranges that guarantee certain stability properties of the solution functions. We present a logical framework for the formulation and solution of problems of this type in great generality. The function domain for both parametric functions and solutions is the differential ring D of complex exponential polynomials. The main result is a quantifier elimination algorithm for the first-order theory T of D in a language suitable for global and local stability questions, and a resulting decision procedure for T . For existential formulas the algorithm yields also parametric sample solution functions. Examples illustrate the expressive power and algorithmic strengh of this approach concerning parametric stability problems. A contrasting negative theorem on undecidability shows the boundaries of extensions of the method.

The solution of Partial Differential Equations (PDEs) is one of the most important problems of mathematics, and has an enormous area of applications. One of the methods for extending the range of analytically solvable PDEs consists in transformations of PDEs and the corresponding transformations of their solutions. Thus, based on the fact that a second-order equation can be solved if one of its factorizations is known, the famous method of Laplace Transformations suggests a certain sequence of transformations of a given equation. Then, if at a certain step in this transformation process an equation becomes factorizable, an analytical solution of this transformed equation — and then of the initial one — can be found. The aim of this talk is a description of some old and new developments and generalizations of analytical approaches to the solution of PDEs and the corresponding algebraic theory of differential operators. Recently we have introduced the notion of obstacle for the factorization of a differential operator, i.e. conditions preventing a given operator from being factorizable. These obstacles give rise to a ring of obstacles and furthermore to a classification of operators w.r.t. to their factorization properties. From obstacles we can also get (Laplace) invariants of operators w.r.t. to certain (gauge) transformations. We have shown how such systems of invariants can be extended to full systems of invariants for certain low order operators. Another related problem is the description of the structure of families of factorizations. For operators of order 3 it has been shown that a family of factorizations depends on at most 3 or 2 parameters, each of these parameters being a function on one variable.

One of the most important features of authentication is the non-repudiation property, implemented by digital signatures. This useful feature of authentication may, however, not be necessary in some cases, such as e-voting, and should indeed be avoided. In this paper, by a combined use of public-key encryption, digital signatures, coding, quadratic residues, and randomness, a new scheme for deniable/repudiable authentication is proposed and analyzed, and a complete example of the scheme is given. The security of the scheme is based on the intractability of the quadratic residuosity problem.

We propose a novel algorithm to select a model that is consistent with the time series of observed data. In the first step, the kinetics for describing a biological phenomenon is expressed by a system of differential equations, assuming that the relationships between the variables are linear. Simultaneously, the time series of the data are numerically fitted as a series of exponentials. In the next step, both the system of differential equations with the kinetic parameters and the series of exponentials fitted to the observed data are transformed into the corresponding system of algebraic equations, by the Laplace transformation. Finally, the two systems of algebraic equations are compared by an algebraic approach. The present method estimates the model’s consistency with the observed data and the determined kinetic parameters. One of the merits of the present method is that it allows a kinetic model with cyclic relationships between variables that cannot be handled by the usual approaches. The plausibility of the present method is illustrated by the actual relationships between specific leaf area, leaf nitrogen and leaf gas exchange with the corresponding simulated data.

In the present paper multiresolutional representation of differential operator in basis of periodized coiflets was constructed. The properties of differential operator coefficients have been investigated. Taking into consideration the behavior of Coifman scaling function, in considered multiresolutional representation of differential operator, coefficients of wavelet-approximation are substituted by evaluations of the function at dyadic points. For sufficiently smooth function the convergence rate of the considered approximations to derivative is stated. For 1-periodic function the formula of numeric differentiation is obtained, and also the error estimate is stated. The application of multiresolution representation of differential operator for numerical solution of ordinary differential equation with periodic boundary conditions is considered.