Use of calculus of variations to determine the shape of hovering rotors of minimum power and its application to micro air vehicles

In this paper, calculus of variations and combined blade element and momentum theory (BEMT) are used to demonstrate that, in hover, when neither root nor tip losses are considered; the rotor, which minimizes the total power (MPR), generates an induced velocity that varies linearly along the blade span. The angle of attack of every blade element is constant and equal to its optimum value. The traditional ideal twist (ITR) and optimum (OR) rotors are revisited in the context of this variational framework. Two more optimum rotors are obtained considering root and tip losses, the ORL, and the MPRL. A comparison between these five rotors is presented and discussed. The MPR and MPRL present a remarkable saving of power for low values of both thrust coefficient and maximum aerodynamic efficiency. The result obtained can be exploited to improve the aerodynamic behaviour of rotary wing micro air vehicles (MAV). A comparison with experimental results obtained from the literature is presented.

Calculus of the uncertainty in acoustic field measurements: comparative study between the uncertainty propagation method and the distribution propagation method

The new Spanish Regulation in Building Acoustic establishes values and limits for the different acoustic magnitudes whose fulfillment can be verify by means field measurements. In this sense, an essential aspect of a field measurement is to give the measured magnitude and the uncertainty associated to such a magnitude. In the calculus of the uncertainty it is very usual to follow the uncertainty propagation method as described in the Guide to the expression of Uncertainty in Measurements (GUM). Other option is the numerical calculus based on the distribution propagation method by means of Monte Carlo simulation. In fact, at this stage, it is possible to find several publications developing this last method by using different software programs. In the present work, we used Excel for the Monte Carlo simulation for the calculus of the uncertainty associated to the different magnitudes derived from the field measurements following ISO 140-4, 140-5 and 140-7. We compare the results with the ones obtained by the uncertainty propagation method. Although both methods give similar values, some small differences have been observed. Some arguments to explain such differences are the asymmetry of the probability distributions associated to the entry magnitudes,the overestimation of the uncertainty following the GUM

A toolbox with DERIVE: calculus on several variables, Applications of Computer Algebra

A toolbox is a set of procedures taking advantage of the computing power and graphical capacities of a CAS. With these procedures the students can solve math problems, apply mathematics to engineering or simply reinforce the learning of certain mathematical concepts. From the point of view of their construction, we can consider two types of toolboxes: (i) the closed box, built by the teacher, in which the utility files are provided to the students together with the respective tutorials and several worksheets with proposed exercises and problems,

Examples of PDEs all whose points are characteristic

Let π : FM ! M be the bundle of linear frames of a manifold M. A basis Lijk , j < k, of diffeomorphism invariant Lagrangians on J1 (FM) was determined in [J. Muñoz Masqué, M. E. Rosado, Invariant variational problems on linear frame bundles, J. Phys. A35 (2002) 2013-2036]. The notion of a characteristic hypersurface for an arbitrary first-order PDE system on an ar- bitrary bred manifold π : P → M, is introduced and for the systems dened by the Euler-Lagrange equations of Lijk every hypersurface is shown to be characteristic. The Euler-Lagrange equations of the natural basis of Lagrangian densities Lijk on the bundle of linear frames of a manifold M which are invariant under diffeomorphisms, are shown to be an underdetermined PDEs systems such that every hypersurface of M is characteristic for such equations. This explains why these systems cannot be written in the Cauchy-Kowaleska form, although they are known to be formally integrable by using the tools of geometric theory of partial differential equations, see [J. Muñoz Masqué, M. E. Rosado, Integrability of the eld equations of invariant variational problems on linear frame bundles, J. Geom. Phys. 49 (2004), 119-155]

Relational and Allegorical Semantics for Constraint Logic Programming

El cálculo de relaciones binarias fue creado por De Morgan en 1860 para ser posteriormente desarrollado en gran medida por Peirce y Schröder. Tarski, Givant, Freyd y Scedrov demostraron que las álgebras relacionales son capaces de formalizar la lógica de primer orden, la lógica de orden superior así como la teoría de conjuntos. A partir de los resultados matemáticos de Tarski y Freyd, esta tesis desarrolla semánticas denotacionales y operacionales para la programación lógica con restricciones usando el álgebra relacional como base. La idea principal es la utilización del concepto de semántica ejecutable, semánticas cuya característica principal es el que la ejecución es posible utilizando el razonamiento estándar del universo semántico, este caso, razonamiento ecuacional. En el caso de este trabajo, se muestra que las álgebras relacionales distributivas con un operador de punto fijo capturan toda la teoría y metateoría estándar de la programación lógica con restricciones incluyendo los árboles utilizados en la búsqueda de demostraciones. La mayor parte de técnicas de optimización de programas, evaluación parcial e interpretación abstracta pueden ser llevadas a cabo utilizando las semánticas aquí presentadas. La demostración de la corrección de la implementación resulta extremadamente sencilla. En la primera parte de la tesis, un programa lógico con restricciones es traducido a un conjunto de términos relacionales. La interpretación estándar en la teoría de conjuntos de dichas relaciones coincide con la semántica estándar para CLP. Las consultas contra el programa traducido son llevadas a cabo mediante la reescritura de relaciones. Para concluir la primera parte, se demuestra la corrección y equivalencia operacional de esta nueva semántica, así como se define un algoritmo de unificación mediante la reescritura de relaciones. La segunda parte de la tesis desarrolla una semántica para la programación lógica con restricciones usando la teoría de alegorías—versión categórica del álgebra de relaciones—de Freyd. Para ello, se definen dos nuevos conceptos de Categoría Regular de Lawvere y _-Alegoría, en las cuales es posible interpretar un programa lógico. La ventaja fundamental que el enfoque categórico aporta es la definición de una máquina categórica que mejora e sistema de reescritura presentado en la primera parte. Gracias al uso de relaciones tabulares, la máquina modela la ejecución eficiente sin salir de un marco estrictamente formal. Utilizando la reescritura de diagramas, se define un algoritmo para el cálculo de pullbacks en Categorías Regulares de Lawvere. Los dominios de las tabulaciones aportan información sobre la utilización de memoria y variable libres, mientras que el estado compartido queda capturado por los diagramas. La especificación de la máquina induce la derivación formal de un juego de instrucciones eficiente. El marco categórico aporta otras importantes ventajas, como la posibilidad de incorporar tipos de datos algebraicos, funciones y otras extensiones a Prolog, a la vez que se conserva el carácter 100% declarativo de nuestra semántica. ABSTRACT The calculus of binary relations was introduced by De Morgan in 1860, to be greatly developed by Peirce and Schröder, as well as many others in the twentieth century. Using different formulations of relational structures, Tarski, Givant, Freyd, and Scedrov have shown how relation algebras can provide a variable-free way of formalizing first order logic, higher order logic and set theory, among other formal systems. Building on those mathematical results, we develop denotational and operational semantics for Constraint Logic Programming using relation algebra. The idea of executable semantics plays a fundamental role in this work, both as a philosophical and technical foundation. We call a semantics executable when program execution can be carried out using the regular theory and tools that define the semantic universe. Throughout this work, the use of pure algebraic reasoning is the basis of denotational and operational results, eliminating all the classical non-equational meta-theory associated to traditional semantics for Logic Programming. All algebraic reasoning, including execution, is performed in an algebraic way, to the point we could state that the denotational semantics of a CLP program is directly executable. Techniques like optimization, partial evaluation and abstract interpretation find a natural place in our algebraic models. Other properties, like correctness of the implementation or program transformation are easy to check, as they are carried out using instances of the general equational theory. In the first part of the work, we translate Constraint Logic Programs to binary relations in a modified version of the distributive relation algebras used by Tarski. Execution is carried out by a rewriting system. We prove adequacy and operational equivalence of the semantics. In the second part of the work, the relation algebraic approach is improved by using allegory theory, a categorical version of the algebra of relations developed by Freyd and Scedrov. The use of allegories lifts the semantics to typed relations, which capture the number of logical variables used by a predicate or program state in a declarative way. A logic program is interpreted in a _-allegory, which is in turn generated from a new notion of Regular Lawvere Category. As in the untyped case, program translation coincides with program interpretation. Thus, we develop a categorical machine directly from the semantics. The machine is based on relation composition, with a pullback calculation algorithm at its core. The algorithm is defined with the help of a notion of diagram rewriting. In this operational interpretation, types represent information about memory allocation and the execution mechanism is more efficient, thanks to the faithful representation of shared state by categorical projections. We finish the work by illustrating how the categorical semantics allows the incorporation into Prolog of constructs typical of Functional Programming, like abstract data types, and strict and lazy functions.

Could it be possible to replace DERIVE with MAXIMA?

In recent years, a considerable number of teachers in Spain have been using DERIVE to teach math subjects in High Schools and Universities. This software has been used by the authors of this work as a support tool in Mathematics courses for Engineering. Since Texas Instruments does not support DERIVE, we were faced with finding an alternative software product, and considering the possibility of using a public-domain software such as MAXIMA. Here we make a comparative study of DERIVE and MAXIMA as support tools for a Calculus course for first year Engineering students.

Efficient management of backtracking in and-parallelism

A backtracking algorithm for AND-Parallelism and its implementation at the Abstract Machine level are presented: first, a class of AND-Parallelism models based on goal independence is defined, and a generalized version of Restricted AND-Parallelism (RAP) introduced as characteristic of this class. A simple and efficient backtracking algorithm for R A P is then discussed. An implementation scheme is presented for this algorithm which offers minimum overhead, while retaining the performance and storage economy of sequent ial implementations and taking advantage of goal independence to avoid unnecessary backtracking ("restricted intelligent backtracking"). Finally, the implementation of backtracking in sequential and AND-Parallcl systems is explained through a number of examples.

Evaluación del comportamiento mecánico, estructural y frente a la corrosión, de una nueva armadura de acero inoxidable dúplex bajo en níquel

La durabilidad de las estructuras de hormigón armado no es ilimitada, en especial en determinados ambientes. El ingreso de agentes agresivos en el hormigón, fundamentalmente dióxido de carbono e iones cloruros, rebasando el espesor del recubrimiento y alcanzando las armaduras, reducen el alto pH del hormigón hasta alcanzar un umbral crítico, por debajo del cual, el acero queda despasivado. Posteriormente, si existe el suficiente aporte de humedad y oxígeno, el acero se corroe, lo que supone drásticas reducciones de la vida de servicio de estas estructuras y su inevitable reparación. La utilización de armaduras de acero inoxidable es una alternativa que está recibiendo cada vez más consideración. Su resistencia a la corrosión en los ambientes más agresivos, incluso con ataque de cloruros, lo convierte en el material idóneo para prolongar de forma muy considerable la vida útil de la estructura. En este trabajo se ha evaluado el comportamiento mecánico y estructural, y de resistencia a la corrosión, de un nuevo acero inoxidable dúplex de bajo contenido en níquel, el EN 1.4482 (AISI 2001), y se ha comparado con el inoxidable austenítico más utilizado, el EN 1.4301 (AISI 304), con el dúplex EN 1.4362 (AISI 2304) y con el tradicional acero al carbono B-500-SD. El estudio mecánico y estructural se ha realizado en tres niveles diferentes: a nivel de barra, estudiando las propiedades mecánicas y de ductilidad de los cuatro aceros citados; a nivel de sección, estudiando su comportamiento a flexión con diferentes cuantías de armado por medio de los diagramas momento-curvatura; y a nivel de pieza, ensayando una serie de vigas armadas con diferentes aceros y cuantías, y comprobando su comportamiento a desplazamiento y resistencia por medio de los diagramas carga-desplazamiento. El estudio de resistencia a la corrosión se ha realizado embebiendo barras corrugadas, de los tres aceros inoxidables mencionados, en probetas de mortero contaminadas con diferentes cantidades de cloruros, y realizando mediciones electroquímicas durante un periodo de al menos un año. Se han preparado probetas de mortero para dos comparativas diferentes. La primera, manteniendo las probetas en un desecador con el 95 % de humedad relativa durante todo el periodo de mediciones. La segunda, sumergiendo parcialmente las probetas en una solución tampón para carbonatar el mortero. Los resultados de los ensayos mecánicos han demostrado dos aspectos diferentes. Uno, que las armaduras de acero inoxidable tienen un comportamiento muy similar a las de acero al carbono en lo referente a las resistencias alcanzadas, en el límite elástico y en rotura, pero distinto en cuanto al módulo de deformación longitudinal, cuyo valor es claramente inferior al del acero al carbono, por lo que su utilización en las estructuras de hormigón necesita tener en cuenta ese dato en los análisis lineales de cálculo. El segundo aspecto es que las armaduras de acero inoxidable laminadas en caliente presentan una ductilidad muy superior a las de acero al carbono, por lo que ofrecen una mayor seguridad frente a su rotura o al colapso de la estructura, lo que se debe tener en cuenta en el análisis de cálculo plástico. En cambio, las armaduras de acero inoxidable laminadas en frío sólo cumplen con los límites mínimos de ductilidad establecidos en la instrucción EHE-08 para los aceros soldables, y no para los aceros con características especiales de ductilidad. El estudio a nivel de sección refleja la paradoja de obtener secciones menos dúctiles con las armaduras de acero inoxidable laminadas en caliente que con las armaduras de acero al carbono. Para subsanarlo, se definen los conceptos de curvatura última de rotura y ductilidad de la sección en rotura, que tienen en cuenta las altas deformaciones alcanzadas por las armaduras de acero inoxidable. Los resultados a nivel de pieza permiten identificar el comportamiento estructural del hormigón armado con barras corrugadas de acero inoxidable y compararlo con el de las estructuras de hormigón armado convencionales, verificando los resultados experimentales con los teóricos obtenidos con la formulación recogida en la instrucción EHE- 08. Los ensayos de resistencia a la corrosión por cloruros demuestran, durante el primer año y medio de vida de las probetas, un comportamiento muy similar entre el nuevo acero inoxidable dúplex bajo en níquel y el austenítico y el dúplex utilizados para la comparación, incluso para las probetas carbonatadas. Por último, se añade una comparativa económica, realizada sobre dos edificaciones tipo, para cuantificar el sobrecoste que supone la utilización de armaduras de acero inoxidable respecto a las de acero al carbono. El alto coste inicial de las armaduras de acero inoxidable se ve compensado en el coste final de la estructura de muy diferentes formas, principalmente dependiendo del grado de acero elegido y de si se emplean en el total de la estructura o solamente en los elementos más expuestos. The durability of the concrete structures is limited, especially in certain environments. The attack of aggressive agents in the concrete, mainly carbon dioxide and chloride ions, penetrating the thickness of concrete cover and reaching the reinforcements, reduce the high pH of concrete to the point of reaching a critical threshold, under which, the steel despasivates. Therefore, if there is enough humidity and oxygen, the steel corroes, causing drastic reductions in the service life of these structures and its inevitable repair. Despite the high initial cost compared to carbon steel, the usage of stainless steel reinforcements is an alternative with a major consideration nowadays. Its resistance to corrosion in the most aggressive atmospheres, including chlorides attack, makes the stainless steel a suitable material to extend considerably its lifetime. In this study, it’s been evaluated the mechanical and structural behaviour, and the corrosion resistance, of a new low-nickel duplex stainless steel EN 1.4482 (AISI 2001), and it has been compared with the most widely used austenitic type EN 1.4301 (AISI 304), with duplex steel EN 1.4362 (AISI 2304) and with the traditional carbon steel B-500-SD. The mechanical and structural study has been carried out in three different levels: bar level, studying mechanical properties and ductility of the four steels; section level, studying its behaviour when blending with different amounts of reinforcement through the moment-curvature diagrams; and structural element level, testing a series of reinforced beams with different steels and amounts, and checking its sag and resistance through the load-deflection diagrams. The corrosion resistance study was performed by embedding ribbed bars, using the three stainless steel listed, on mortar specimens contaminated with different amounts of chlorides, and taking electrochemical measurements over a period of at least one year. Mortar specimens have been prepared for two different comparisons. The first, keeping the specimens at 95% of relative humidity during the measurement period. The second, immersing the specimens partially in a carbonate buffer solution. The results of those tests have proved two different aspects. Firstly, that stainless steel reinforcements show a very similar behaviour to carbon steel, according to the reached levels of mechanical resistance, yield stress and steel strength, but a different behaviour in Young’s modulus, which value is clearly lower than the carbon steel. Therefore, when using in concrete structures it is need to consider on that point the existing calculus of linear analysis. The second aspect is that stainless steel reinforcement manufactured by hot-rolling process show a very higher ductility than carbon steel, offering a better security on cracks or structure collapse, which it has to be taken into account on plastic calculus analysis. However, the stainless steel reinfor9 cement cold-rolled bars only meet the minimum thresholds of ductility established by EHE-08 for welded steel, and not for steels with special ductility. The results at the section level reflect the paradox of getting less ductile sections with hot rolled stainless steel reinforcement than with carbon steel reinforcements. To overcome that, the concepts of last break curvature and break ductility section have been defined, which take into account the high deformation value achieved by stainless steel reinforcements. The results at the structural element level allow to identify the structural behaviour of reinforced concrete with stainless steel reinforcements and compared with that of conventional steel reinforcement, contrasting the experimental with the theoretical results obtained from the formulation contained in the instruction EHE-08. Tests on resistance of chloride corrosion show during the first year and a half of specimens life, a similar behaviour between the new low nickel duplex stainless steel and austenitic and duplex used for comparison, even for carbonated specimens. Finally, it has been included an economic comparison on two differents building types, to quantify the additional cost involved on the use of stainless steel reinforcement compared to that of carbon steel. The high initial cost of stainless steel reinforcements is offset in the final cost of the structure in many different ways, mainly depending on the chosen steel grade and whether the reinforcement is used in the total structure or only in risky structural elements.