Railroard curves and earthwork

UANL

Algebraic Geometric Codes and their relation to Cryptography using Elliptic Curves

Communication is the process of transmitting data across channel. Whenever data is transmitted across a channel, errors are likely to occur. Coding theory is a stream of science that deals with finding efficient ways to encode and decode data, so that any likely errors can be detected and corrected. There are many methods to achieve coding and decoding. One among them is Algebraic Geometric Codes that can be constructed from curves. Cryptography is the science ol‘ security of transmitting messages from a sender to a receiver. The objective is to encrypt message in such a way that an eavesdropper would not be able to read it. A eryptosystem is a set of algorithms for encrypting and decrypting for the purpose of the process of encryption and decryption. Public key eryptosystem such as RSA and DSS are traditionally being prel‘en‘ec| for the purpose of secure communication through the channel. llowever Elliptic Curve eryptosystem have become a viable altemative since they provide greater security and also because of their usage of key of smaller length compared to other existing crypto systems. Elliptic curve cryptography is based on group of points on an elliptic curve over a finite field. This thesis deals with Algebraic Geometric codes and their relation to Cryptography using elliptic curves. Here Goppa codes are used and the curves used are elliptic curve over a finite field. We are relating Algebraic Geometric code to Cryptography by developing a cryptographic algorithm, which includes the process of encryption and decryption of messages. We are making use of fundamental properties of Elliptic curve cryptography for generating the algorithm and is used here to relate both.

An attempt on the calculation of relativistic potential energy curves: noble gas difluorides

Total energy SCF calculations were performed for noble gas difluorides in a relativistic procedure and compared with analogous non-relativistic calculations. The discrete variational method with numerical basis functions was used. Rather smooth potential energy curves could be obtained. The theoretical Kr - F and Xe - F bond distances were calculated to be 3.5 a.u. and 3.6 a.u. which should be compared with the experimental values of 3.54 a.u. and 3.7 a.u. Although the dissociation energies are off by a factor of about five it was found that ArF_2 may be a stable molecule. Theoretical ionization energies for the outer levels reproduce the experimental values for KrF_2 and XeF_2 to within 2 eV.

Relativistic Dirac-Fock-Slater program to calculate potential-energy curves for diatomic molecules

A LCAO-MO (linear combination of atomic orbitals - molecular orbitals) relativistic Dirac-Fock-Slater program is presented, which allows one to calculate accurate total energies for diatomic molecules. Numerical atomic Dirac-Fock-Slater wave functions are used as basis functions. All integrations as well as the solution of the Poisson equation are done fully numerical, with a relative accuracy of 10{^-5} - 10{^-6}. The details of the method as well as first results are presented here.

New calculations of P(b) curves for 1s_\sigma excitation in low-Z (Z\le10) ion-atom scattering

Ab initio fully relativistic SCF molecular calculations of energy eigenvalues as well as coupling-matrix elements are used to calculate the 1s_\sigma excitation differential cross section for Ne-Ne and Ne-O in ion-atom collisions. A relativistic perturbation treatment which allows a direct comparison with analogous non-relativistic calculations is also performed.

"Ein Engel ist wie…" – Aussehen, Bedeutung und Wesen eines Engels aus der Sicht von Kindern einer vierten Klasse.

Die Engel erfahren insbesondere in der Gegenwart ein wachsendes Interesse und spielen im Leben vieler Menschen mehr und mehr eine bedeutsame Rolle. Der Engelglaube scheint wiederzukehren. Nicht selten begegnet man Menschen, die an ihren persönlichen Schutzengel glauben und eine intensive Beziehung zu diesem aufbauen. Eine Forsa-Umfrage belegt diese Tendenz und stellt fest, dass mit 66% mehr Menschen an Engel glauben als an Gott (64%). Nicht nur aus diesem Grund stellt Thomas Ruster fest: „Und schon hat sich eine neue Religion herausgebildet, weltweit und in den Herzen unzähliger Menschen fest verankert: die Religion der Engel.“ Auch im Alltag begegnen uns Engel in den unterschiedlichsten Formen und Farben. Es gibt kaum einen Bereich der Alltagskultur, den die Engel noch nicht bevölkert haben. Sie sind allgegenwärtig. Betrachtet man nun diese aktuellen Entwicklungen, ist es eigentlich nicht verwunderlich, wie ein Interesse meinerseits für dieses Thema entstehen konnte. Dabei wies mich diese ununterbrochene Begegnung mit den Engeln vor allem auf eine interessante Gegenläufigkeit zwischen der Häufigkeit dieser Wesen im Alltag und der Seltenheit der tatsächlichen gegenwärtigen theologischen Beschäftigung mit ihnen hin. Wo begegnet man den Engeln also heute noch in christlicher Hinsicht? Außerdem entwickelte sich für mich als angehende Religionslehrerin auch eine andere Fragestellung: Bietet die Aktualität der Engel im alltäglichen Leben der Menschen nicht auch gerade einen Anknüpfungspunkt, um den verloren gegangenen Gottesglauben und die nur noch vereinzelt gemachten religiösen Erfahrungen wiederzubeleben? Die nachfolgende Arbeit ist auf erster Ebene in zwei Bereiche gegliedert. In Teil A soll dabei eine theoretische Grundlegung des anschließenden Forschungsprojektes (Teil B) vorgenommen werden. Diesbezüglich werden zunächst die wichtigsten Aspekte der Engellehre aus biblischer, religionsgeschichtlicher, kunsthistorischer und gegenwartsbezogener Sicht vorgestellt. Im Teil B wird dann die empirische Untersuchung in einer vierten Klasse an einer hessischen Grundschule vorgestellt. Die Arbeit endet mit einer Reflexion der erworbenen Daten und abschließenden Überlegungen.

Statistical treatment of grain-size curves and empirical distributions: densities as compositions?

The preceding two editions of CoDaWork included talks on the possible consideration of densities as infinite compositions: Egozcue and D´ıaz-Barrero (2003) extended the Euclidean structure of the simplex to a Hilbert space structure of the set of densities within a bounded interval, and van den Boogaart (2005) generalized this to the set of densities bounded by an arbitrary reference density. From the many variations of the Hilbert structures available, we work with three cases. For bounded variables, a basis derived from Legendre polynomials is used. For variables with a lower bound, we standardize them with respect to an exponential distribution and express their densities as coordinates in a basis derived from Laguerre polynomials. Finally, for unbounded variables, a normal distribution is used as reference, and coordinates are obtained with respect to a Hermite-polynomials-based basis. To get the coordinates, several approaches can be considered. A numerical accuracy problem occurs if one estimates the coordinates directly by using discretized scalar products. Thus we propose to use a weighted linear regression approach, where all k- order polynomials are used as predictand variables and weights are proportional to the reference density. Finally, for the case of 2-order Hermite polinomials (normal reference) and 1-order Laguerre polinomials (exponential), one can also derive the coordinates from their relationships to the classical mean and variance. Apart of these theoretical issues, this contribution focuses on the application of this theory to two main problems in sedimentary geology: the comparison of several grain size distributions, and the comparison among different rocks of the empirical distribution of a property measured on a batch of individual grains from the same rock or sediment, like their composition

Vector functions and curves

Exercises and solutions about vector functions and curves.

Tendencias y problemas de la Enseñanza de nivel medio en la Europa de hoy : conferencia pronunciada en la Reunión del Profesorado de Granada por el profesor Otto Engel (abril de 1968).

Se transcribe la conferencia titulada Tendencias y problemas de la Enseñanza de nivel medio en la Europa de hoy, pronunciada en la Reunión del Profesorado de Granada por el Profesor Otto Ángel, en abril de 1968. Se habla en primer lugar del espíritu de unión europeo y de las figuras que lo han impulsado, como Adenauer, De Gasperi y Schumann. Posteriormente se adentra en la cuestión de la educación secundaria en Europa. Se destaca como del 20 al 28 de abril de 1967 se celebró en Málaga, una Semana de estudios, organizada por el Gobierno español bajo los auspicios del Consejo de Europa cuyo tema fue La actitud humanista en la enseñanza secundaria para la Europa del mañana. Se precisó como el humanismo no se reduce a las llamadas humanidades de los tiempos pasados, es decir al estudio de las lenguas y literaturas griega y latina, sino que todas las asignaturas tienen un núcleo humanístico. Pero el núcleo central lo integran los siguientes temas: las organizaciones europeas de cooperación pedagógica, las cuestiones de orientación escolar concernientes a la enseñanza secundaria y los métodos de enseñanza contrapuestos. Para terminar se señala que otro tipo de actividad pedagógica moderna lo constituye la llamada forma social-integradora, en la que el profesor da una visión general del trabajo proyectado. Este proyecto es sometido a discusión y a decisión por grupos de alumnos estimulados por el profesor.

Planning rigid body motions using elastic curves

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles.

Finding the point on Bezier Curves with the normal vector passing an external point

Two algorithms for finding the point on non-rational/rational Bezier curves of which the normal vector passes through a given external point are presented. The algorithms are based on Bezier curves generation algorithms of de Casteljau's algorithm for non-rational Bezier curve or Farin's recursion for rational Bezier curve, respectively. Orthogonal projections from the external point are used to guide the directional search used in the proposed iterative algorithms. Using Lyapunov's method, it is shown that each algorithm is able to converge to a local minimum for each case of non-rational/rational Bezier curves. It is also shown that on convergence the distance between the point on curves to the external point reaches a local minimum for both approaches. Illustrative examples are included to demonstrate the effectiveness of the proposed approaches.