Problems in the classical calculus of variations

This dissertation concerns convergence analysis for nonparametric problems in the calculus of variations and sufficient conditions for weak local minimizer of a functional for both nonparametric and parametric problems. Newton's method in infinite-dimensional space is proved to be well-defined and converges quadratically to a weak local minimizer of a functional subject to certain boundary conditions. Sufficient conditions for global converges are proposed and a well-defined algorithm based on those conditions is presented and proved to converge. Finite element discretization is employed to achieve an implementable line-search-based quasi-Newton algorithm and a proof of convergence of the discretization of the algorithm is included. This work also proposes sufficient conditions for weak local minimizer without using the language of conjugate points. The form of new conditions is consistent with the ones in finite-dimensional case. It is believed that the new form of sufficient conditions will lead to simpler approaches to verify an extremal as local minimizer for well-known problems in calculus of variations.

A Calculus of Evolving Objects

The demands of developing modern, highly dynamic applications have led to an increasing interest in dynamic programming languages and mechanisms. Not only applications must evolve over time, but the object models themselves may need to be adapted to the requirements of different run-time contexts. Class-based models and prototype-based models, for example, may need to co-exist to meet the demands of dynamically evolving applications. Multi-dimensional dispatch, fine-grained and dynamic software composition, and run-time evolution of behaviour are further examples of diverse mechanisms which may need to co-exist in a dynamically evolving run-time environment How can we model the semantics of these highly dynamic features, yet still offer some reasonable safety guarantees? To this end we present an original calculus in which objects can adapt their behaviour at run-time to changing contexts. Both objects and environments are represented by first-class mappings between variables and values. Message sends are dynamically resolved to method calls. Variables may be dynamically bound, making it possible to model a variety of dynamic mechanisms within the same calculus. Despite the highly dynamic nature of the calculus, safety properties are assured by a type assignment system.

A Calculus of Evolving Objects

The demands of developing modern, highly dynamic applications have led to an increasing interest in dynamic programming languages and mechanisms. Not only must applications evolve over time, but the object models themselves may need to be adapted to the requirements of different run-time contexts. Class-based models and prototype-based models, for example, may need to co-exist to meet the demands of dynamically evolving applications. Multi-dimensional dispatch, fine-grained and dynamic software composition, and run-time evolution of behaviour are further examples of diverse mechanisms which may need to co-exist in a dynamically evolving run-time environment. How can we model the semantics of these highly dynamic features, yet still offer some reasonable safety guarantees? To this end we present an original calculus in which objects can adapt their behaviour at run-time. Both objects and environments are represented by first-class mappings between variables and values. Message sends are dynamically resolved to method calls. Variables may be dynamically bound, making it possible to model a variety of dynamic mechanisms within the same calculus. Despite the highly dynamic nature of the calculus, safety properties are assured by a type assignment system.

Local Stereology of Tensors of Convex Bodies

In this paper, we present local stereological estimators of Minkowski tensors defined on convex bodies in ℝ d . Special cases cover a number of well-known local stereological estimators of volume and surface area in ℝ3, but the general set-up also provides new local stereological estimators of various types of centres of gravity and tensors of rank two. Rank two tensors can be represented as ellipsoids and contain information about shape and orientation. The performance of some of the estimators of centres of gravity and volume tensors of rank two is investigated by simulation.

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

Calculus of variations. Supplementary notes and exercises to 1945-1946 notes, at New York University.

Reproduced from typewritten copy.

Calculus of variations 1945-1946,

Bibliography: leaf 169.

Aspects of the calculus of variations /

Reproduced from type-written copy.

A treatise on the differential and integral calculus : and the calculus of variations /

Mode of access: Internet.

Statistical matrix techniques and their application to the directional correlation of radiations /

"NBS Project 0436."