An LU-fuzzy calculator for the basic fuzzy calculus

The LU-model for fuzzy numbers has been introduced in [4] and applied to fuzzy calculus in [9]; in this paper we build an LU-fuzzy calculator, in order to explain the use of the LU-fuzzy representation and to show the advantage of the parametrization. The calculator produces the basic fuzzy calculus: the arithmetic operations (scalar multiplication, addition, subtraction, multiplica- tion, division) and the fuzzy extension of many univariate functions (power with integer positive or negative exponent, exponential , logarithm, general power function with numeric or fuzzy exponent, sin, arcsin, cos, arccos, tan, arctan, square root, Gaussian and standard Gaussian functions, hyperbolic sinh, cosh, tanh and inverses, erf error function and complementary erfc error function, cu- mulative standard normal distribution). The use of the calculator is illustrated.Fuzzy Sets, LU-fuzzy Calculator, Fuzzy Calculus

Polynomially Adjusted Saddlepoint Density Approximations

This thesis aims at obtaining improved bona fide density estimates and approximants by means of adjustments applied to the widely used saddlepoint approximation. Said adjustments are determined by solving systems of equations resulting from a moment-matching argument. A hybrid density approximant that relies on the accuracy of the saddlepoint approximation in the distributional tails is introduced as well. A certain representation of noncentral indefinite quadratic forms leads to an initial approximation whose parameters are evaluated by simultaneously solving four equations involving the cumulants of the target distribution. A saddlepoint approximation to the distribution of quadratic forms is also discussed. By way of application, accurate approximations to the distributions of the Durbin-Watson statistic and a certain portmanteau test statistic are determined. It is explained that the moments of the latter can be evaluated either by defining an expected value operator via the symbolic approach or by resorting to a recursive relationship between the joint moments and cumulants of sums of products of quadratic forms. As well, the bivariate case is addressed by applying a polynomial adjustment to the product of the saddlepoint approximations of the marginal densities of the standardized variables. Furthermore, extensions to the context of density estimation are formulated and applied to several univariate and bivariate data sets. In this instance, sample moments and empirical cumulant-generating functions are utilized in lieu of their theoretical analogues. Interestingly, the methodology herein advocated for approximating bivariate distributions not only yields density estimates whose functional forms readily lend themselves to algebraic manipulations, but also gives rise to copula density functions that prove significantly more flexible than the conventional functional type

Inversion for local stress field inhomogeneities

In this thesis, the 1997 Vogtland/NW-Bohemia swarm has been selected for the analysis of inhomogeneities in the stress field because two predominant nearly perpendicular flat zones of seismicity are visible in the hypocentre distribution implying inhomogeneities in the stress field. This is unusual compared to other swarms originating from this area. An existing dataset of waveform data, P- and S-phase picks, and master event locations has been analysed regarding similarity of waveforms, location refinement, and estimation of relative moment tensors. The latter are used together with a regional dataset of 50 single focal mechanisms and 125 focal mechanisms from the 2000 hydraulic fracturing experiment at the KTB for an estimate of the regional homogeneous and the locally inhomogeneous stress field. An automated processing procedure consisting of coherence analysis,precise relocation, relative moment tensor inversion, and stress trajectory determination has been set up. The coherence analysis has been successfully applied using a new method that uses three component seismograms. 457 events are separated into 13 multiplets of similar waveforms of at least size 8. Another result are precise relative arrival time measurements which are fed into the precise relocation program "hypoDD". Two nearly perpendicular structures are found in the hypocentre distribution. 352 moment tensors are estimated using a relative moment tensor inversion. Three different algorithms to distinguish between fault plane and auxiliary plane are successfully applied to them...thesi

Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis

We study two classes of extension problems, and their interconnections: (i) Extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; (ii) In case of Lie groups, representations of the associated Lie algebras $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space (RKHS) $\mathscr{H}_{F}$. Why extensions? In science, experimentalists frequently gather spectral data in cases when the observed data is limited, for example limited by the precision of instruments; or on account of a variety of other limiting external factors. Given this fact of life, it is both an art and a science to still produce solid conclusions from restricted or limited data. In a general sense, our monograph deals with the mathematics of extending some such given partial data-sets obtained from experiments. More specifically, we are concerned with the problems of extending available partial information, obtained, for example, from sampling. In our case, the limited information is a restriction, and the extension in turn is the full positive definite function (in a dual variable); so an extension if available will be an everywhere defined generating function for the exact probability distribution which reflects the data; if it were fully available. Such extensions of local information (in the form of positive definite functions) will in turn furnish us with spectral information. In this form, the problem becomes an operator extension problem, referring to operators in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we have stressed hands-on-examples. Extensions are almost never unique, and so we deal with both the question of existence, and if there are extensions, how they relate back to the initial completion problem.Comment: 235 pages, 42 figures, 7 tables. arXiv admin note: substantial text overlap with arXiv:1401.478

ИНТЕЛЛЕКТУАЛЬНЫЙ числовым программным ДЛЯ MIMD-компьютер

For most scientific and engineering problems simulated on computers the solving of problems of the computational mathematics with approximately given initial data constitutes an intermediate or a final stage. Basic problems of the computational mathematics include the investigating and solving of linear algebraic systems, evaluating of eigenvalues and eigenvectors of matrices, the solving of systems of non-linear equations, numerical integration of initial- value problems for systems of ordinary differential equations.Для більшості наукових та інженерних задач моделювання на ЕОМ рішення задач обчислювальної математики з наближено заданими вихідними даними складає проміжний або остаточний етап. Основні проблеми обчислювальної математики відносяться дослідження і рішення лінійних алгебраїчних систем оцінки власних значень і власних векторів матриць, рішення систем нелінійних рівнянь, чисельного інтегрування початково задач для систем звичайних диференціальних рівнянь.Для большинства научных и инженерных задач моделирования на ЭВМ решение задач вычислительной математики с приближенно заданным исходным данным составляет промежуточный или окончательный этап. Основные проблемы вычислительной математики относятся исследования и решения линейных алгебраических систем оценки собственных значений и собственных векторов матриц, решение систем нелинейных уравнений, численного интегрирования начально задач для систем обыкновенных дифференциальных уравнений

Parametric curve comparison for modelling floating offshore wind turbine substructures

The drive for the cost reduction of floating offshore wind turbine (FOWT) systems to the levels of fixed bottom foundation turbine systems can be achieved with creative design and analysis techniques of the platform with free-form curves to save numerical simulation time and minimize the mass of steel (cost of steel) required for design. This study aims to compare four parametric free-form curves (cubic spline, B-spline, Non-Uniform Rational B-Spline and cubic Hermite spline) within a design and optimization framework using the pattern search gradient free optimization algorithm to explore and select an optimal design from the design space. The best performance free-form curve within the framework is determined using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). The TOPSIS technique shows the B-spline curve as the best performing free-form curve based on the selection criteria, amongst which are design and analysis computational time, estimated mass of platform and local shape control properties. This study shows that free-form curves like B-spline can be used to expedite the design, analysis and optimization of floating platforms and potentially advance the technology beyond the current level of fixed bottom foundations

Computer-Aided Geometry Modeling

Techniques in computer-aided geometry modeling and their application are addressed. Mathematical modeling, solid geometry models, management of geometric data, development of geometry standards, and interactive and graphic procedures are discussed. The applications include aeronautical and aerospace structures design, fluid flow modeling, and gas turbine design