No description available.

yes / Highlights: Non-additive probabilities and Choquet integrals in a classical context. The use of Choquet integrals in a quantum context. Approximation of partially known density matrices with Choquet integrals.

When making an estimate of the total of some quantity, sampling at carefully selected points will frequently be preferable to employing a method which involves randomization. The estimation of the total stand of timber on a given area or the amount of energy being released in a given time and space are examples of problems where specified points for sampling should result in a reduction the error of estimate. Problems such as these naturally lead us to to numerical integration methods. In the case of single integrals, the Newton-Cotes formulae can be applied directly to experimentally determined ordinates at equally-spaced abscissa points and are of great practical importance. Gauss’ formulae yield maximum efficiency with respect to controlling the polynomial error and can be used appropriately when an analytical expression for the curve in question is available but defies exact integration, or if for some other reason the statistical error is of minor importance. Tchebichef’s a formulae give maximum efficiency with respect to controlling the statistical or observational error. The basic elements in the development of numerical integration formulae like Newton-Cotes, Gauss' and Tchebichef's, can be extended to developing formulae for the approximate evaluation of multiple integrals. In the case of double integrals, an eight point and a thirteen point formula for fifth degree accuracy and a twelve point and a twenty-one point formula for seventh degree accuracy have been developed for integrating over a rectangle and similar formulae have been developed for integrating over areas bounded by a parabola and a straight line or two parabolas. Formulae for the numerical evaluation of triple integrals taken over a rectangular parallelepiped are developed, including a twenty-one point formula with fifth degree accuracy. It is shown that comparable formulae can be developed for integrating functions of more than three variables and a 2n /- 1 point formula with third degree accuracy for integrating a function of n variables over a rectangular n-space is obtained. In many problems involving statistical estimation, the dominant source of inaccuracies will be the error of observation. The magnitude of this error can be estimated by subjecting the observations to an orthogonal transformation which will isolate the trends and leave the residual variance free from these effects. This treatment is most easily carried out in terms of orthogonal polynomials and it is shown that this type analysis can be extended to functions of several variables. / Ph. D.

The object of this paper is to present the properties and some of the applications of the Elliptic Integrals.

Title: Nonabsolutely convergent integrals Author: Kristýna Kuncová Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Jan Malý, DrSc., Department of Mathematical Analysis Abstract: Our aim is to introduce an integral on a measure metric space, which will be nonabsolutely convergent but including the Lebesgue integral. We start with spaces of continuous and Lipschitz functions, spaces of Radon measures and their dual and predual spaces. We build up the so-called uniformly controlled integral (UC-integral) of a function with respect to a distribution. Then we investigate the relationship between the UC-integral with respect to a measure and the Lebesgue integral. Then we introduce another kind of integral, called UCN-integral, based on neglecting of small sets with respect to a Hausdorff measure. Hereafter, we focus on the concept of n-dimensional metric currents. We build the UC-integral with respect to a current and then we proceed to a very general version of Gauss-Green Theorem, which includes the Stokes Theorem on manifolds as a special case. Keywords: Nonabsolutely convergent integrals, Multidimensional integrals, Gauss-Green Theorem 1

Another extension of Nonlinear Integral, Upper and Lower Nonlinear Integrals, which is a pair of extreme nonlinear integrals to contain all types of Nonlinear Integrals in the same scheme, is also proposed. It can give a set of upper and lower bounds which include all types of Nonlinear Integrals. We tried to find a solution with the smallest distance between the upper and lower bounds and the smallest error which is a NP hard problem. So we use the multi-objective optimization method to find a set of results for the regression model based on the Upper and Lower Nonlinear Integrals. We can just select one or more optimal solution(s) for a specific problem from the set of results. A weather predictor based on this model has been constructed to predict the next days temperature changing trend and range. / Finally, a NI based data mining framework has been established for identifying the chance of developing liver cancer based on the Hepatitis B Virus DNA sequence data. We have shown that the framework obtains the best diagnosing performance amongst many existing classifiers. / Nonlinear Integral (NI) is a useful integration tool. It has been applied to many areas including classification and regression. The classical method relies on a large number of training data, which lead to large time and space complexity. Moreover, the classical Nonlinear Integral has many limitations. For dealing with different situation, we propose Double Nonlinear Integrals and Nonlinear Integrals with Polynomial Kernel to deal with the problems transversely and longitudinally. / The classical Nonlinear Integrals implement projection along a line with respect to the features. But in many cases the linear projection cannot achieve good performance for classification or regression due to the limitation of the integrand. The linear function used for the integrand is just a special type of polynomial functions with respect to the features. We propose Nonlinear Integral with Polynomial Kernel (NIPK) in which a polynomial function is used as the integrand of Nonlinear Integral. It enables the projection to be along different types of curves on the virtual space, so that the virtual values gotten by the Nonlinear Integrals with Polynomial Kernel can be better regularized and easier to deal with. Experiments show that there is evident improvement of performance for NIPK compared to classical NI. / When the data to be classified have special distribution in the data space, the projection may overlap and the classification accuracy will be lowered. For example, when one group of the data is surrounded by the data of another group, or the number of classes for the data is large. To handle this kind of problems; we propose a new classification model based on the Double Nonlinear Integrals. Double Nonlinear Integral means projecting to a 2-Dimensional space by using the Nonlinear Integral twice in succession and classifying the virtual values in the 2-D space corresponding to the original data. Double Nonlinear Integrals can lessen loss of information due to the intersection of different classes on real axis. Accuracy will also be increased accordingly. / Wang, Jinfeng. / Advisers: Kwong Sak Leung; Kin Hong Lee. / Source: Dissertation Abstracts International, Volume: 72-01, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 139-151). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

The purpose of this thesis is to examine certain questions concerning the Cantor ternary set. The second chapter deals with proving that the Cantor ternary set is equivalent to the middle thirds set of [0,1], closed, compact, and has Lebesgue measure zero. Further a proof that the Cantor ternary set is a locally compact, Hausdorff topological group is given. The third chapter is concerned with establishing the existence of a Haar integral on certain topological groups. In particular if G is a locally compact and Hausdorff topological group, then there is a non-zero translation invariant positive linear form on G. The fourth chapter deals with proving that for any Haar integral I on G there exists a unique Haar measure on G that represents I.

The present work presents a number of contributions to the areas of numerical integration, singular integrals, and boundary element methods. The first contribution is an elemental distortion technique, based on the Duffy transformation, used to improve efficiency for the numerical integration of near hypersingular integrals. Results show that this method can reduce quadrature expense by up to 75 percent over the standard Duffy transformation. The second contribution is an improvement to integration of weakly singular integrals by using regularization to smooth weakly singular integrals. Errors show that the method may reduce errors by several orders of magnitude for the same quadrature order. The final work investigated the use of regularization applied to hypersingular integrals in the context of the boundary element method in three dimensions. This work showed that by using the simple solutions technique, the BEM is reduced to a weakly singular form which directly supports numerical integration. Results support that the method is more efficient than the state-of-the-art.

The purpose of this thesis is to present a study of the Lebesgue definite integral, defined in four different ways.

In leading up to the proof, methods for constructing fields and finitely additive set functions are introduced with an application involving the Tagaki function given as an example. Also, non-absolutely continuous set functions are constructed using Banach limits and maximal filters.