Catálogo de publicaciones - libros

During the past decade, quantum information theory has attracted a lot of interest because of its promise for solving problems that are intractable otherwise. Despite of the recent advancements in understanding the basic principles of quantum information systems, however, there are still a large number of difficulties to be resolved. One of the great challenges concerns for instance the decoherence in quantum systems and how entanglement is lost or transfered between the subsystems, if they are coupled to their enviroment. — To overcome these difficulties, several schemes for studying the decay of quantum states and their interaction with an environment have been developed during recent years, including a large variety of separability and entanglement measures, decoherence-free subspaces as well as (quantum) error correction codes. To support the investigation of entanglement and decoherence phenomena in general N  −qubit quantum systems, we recently developed the Feynman program [1], a computer-algebraic approach within the framework of Maple , which facilitates the symbolic and numerical manipulation of quantum registers and quantum transformations. This program has been designed for studying the dynamics of quantum registers owing to their interaction with external fields and perturbations. In a recent addition to this program [2], moreover, we now implemented also various noise models as well as a number of entanglement measures (and related quantities). In this lecture, I shall display the interactive use of the program by a number of simple but intuitive examples. To make quantum information theory alive , an active (re-) search has been initiated during the past decade to find and explore physical systems that are suitable to produce and control the entanglement in course of their time evolution. In atomic photoionization, for instance, we have shown how the polarization can be transfered from the incoming photons to the emitted photoelectrons, giving rise to a (spin-spin) entanglement between the photoelectron and the remaining (photo-) ion. Detailed computations on the entanglement as function of the energy and polarization of the incoming light have been carried out along various isoelectronic sequences [3]. For the two-photon decay of atomic hydrogen, moreover, we analyzed the geometrical control of the polarization entanglement of the emitted photons.

The algorithmic methods of commutative algebra based on the Gröbner bases technique are briefly sketched out in the context of an application to the constrained finite dimensional polynomial Hamiltonian systems. The effectiveness of the proposed algorithms and their implementation in Mathematica is demonstrated for the light-cone version of the SU (3) Yang-Mills mechanics. The special homogeneous Gröbner basis is constructed that allow us to find and classify the complete set of constraints the model possesses.

Our paper is devoted to the computation of the weight spectra of Conway matrices related to the non-transitive head-or-tail game. We obtained explicit formulas for the spectra containing partial binomial sums. These sums are rather hard to deal with when the methods of classical algebra and number theory are used; but when we used methods of computer algebra, we were able to handle them quite efficiently and could easily produce visualizations. We suggest a recurrence algorithm for efficient calculation of the weight-spectrum matrices, including, as a special case, integer matrices modulo m . The algorithm is implemented with MATHEMATICA and visualizations for some interesting examples are shown.

We derived the quantitative estimates for geometrical parameters of the stability domains of the Lagrange triangle in the restricted three-body problem. We have shown that these domains are plane ellipse-similar figures, extended along a tangent to a circle, on which the Lagrange triangular solutions are located. We have proposed the heuristic algorithm for finding the maximal sizes of the stability domains.

The problem of studying the stability of equilibrium solution of the second order non-autonomous Hamiltonian system, containing a small parameter, is considered. The main steps in solving this problem and application of the computer algebra systems for doing necessary calculations are discussed. As an example, we analyze stability of some equilibrium solutions in the elliptic restricted (2 n  + 1)-body problem. The problem is solved in a strict nonlinear formulation. All calculations are done with the computer algebra system Mathematica.

The paper discusses a technique for investigation of peculiar properties of invariant manifolds of conservative systems. The technique is based on constructing the envelope for the family of first integrals of such systems. Routh–Lyapunov’s method [1] has been applied for obtaining the families of invariant manifolds. With the use of the method of envelope we have analyzed some peculiar properties of families of invariant manifolds in the problems related to rigid body dynamics and vortex theory. For the purpose of solving the computational problems arising in the process of investigations we employed the computer algebra system (CAS) Mathematica . This paper presents a development of our approach [2] to investigation of some qualitative properties of conservative systems.

The expansion base algorithm, which was devised by Abhyankar, Kuo and McCallum is very efficient for analytic factorization of bivariate polynomials. The author had extended it to more than two variables but it was only for polynomials with non-vanishing leading coefficient at the expansion point. In this paper, we improve it to be able to apply to polynomials including the case of vanishing leading coefficient, that is, singular leading coefficient, which comes to a specific problem only for more than two variables.

The algebraic Riccati equation, denoted by ’ARE’ in the paper, is one of the most important equation in the post modern control theory, playing important role for solving H _2 and H _ ∞  optimal control problems. The solution of ARE is given in the form of a matrix, and a typical procedure of computing the solution uses eigenvalues and eigenvectors of matrix H , where H is a matrix determined by a given system. With the aid excellent numerical packages such as “LAPACK” for matrix computations, the procedure is quite efficient for the numerical systems (the systems without unknown parameters). This paper considers a system with an unknown parameter k . In this case, the numerical procedure cannot be applied without fixing parameter k to a constant value. Let us consider some symbolic method to compute the solution of ARE which leaves parameter k symbolic. Letting entries of the solution matrix be unknown variables, ARE can be viewed as a set of m algebraic equations with m variables and parameter k , where m is the number of entries of the unknown matrix. Computing Groebner basis of the algebraic equations with lexicographic ordering, we obtain a polynomial whose roots are the solution of ARE (i.e. the defining polynomial of ARE). Although this method with Groebner basis gives us the defining polynomial of ARE, it is not practical. The method easily collapses when the size of a given system is large because of its heavy numerical complexities. This paper presents a practical algorithm to compute the defining polynomial. The proposed algorithm uses polynomial interpolations, and is easily parallelizable, implying that it is advantageous under multi-CPU environments. Numerical experiments indicate that even in the single CPU environments, the proposed algorithm is much more practical than that with Groebner basis.

We consider discrete dynamical systems and lattice models in statistical mechanics from the point of view of their symmetry groups. We describe a C program for symmetry analysis of discrete systems. Among other features, the program constructs and investigates phase portraits of discrete dynamical systems modulo groups of their symmetries, searches dynamical systems possessing specific properties, e.g., reversibility , computes microcanonical partition functions and searches phase transitions in mesoscopic systems. Some computational results and observations are presented. In particular, we explain formation of moving soliton-like structures similar to “ spaceships ” in cellular automata.

We present an algorithm for finding closed form solutions in elliptic functions of completely integrable systems. First we solve the linear differential equations in spectral parameter of Hermite-Halphen type. The integrability condition of the pair of equations of Hermite-Halphen type gives the large family of completely integrable systems of Lax-Novikov type. This algorithm is implemented on the basis of the computer algebra system MAPLE. Many examples, such as vector nonlinear Schödinger equation, optical cascaded equations and restricted three wave system are considered. New solutions for optical cascaded equations are presented. The algorithm for linear ODE’s with elliptic functions coefficients is generalized to 2×2 matrix equations with elliptic coefficients.