<p>Over the last few years it became clear that the students struggle with the basic concepts of logarithms and inequalities, let alone logarithmic inequalities due to the lack of exposure of these concepts at high school. In order to fully comprehend logarithmic inequalities, a good understanding of the logarithmic graph is important. Thus, the opportunity was seen to change the method of instruction by introducing the graphical method to solve logarithmic inequalities. It was decided to use an mathematical software program, Omnigraph, in this research.</p>

Orientador: Antônio Carlos Patrocinio / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T12:22:05Z (GMT). No. of bitstreams: 1 Pereira_JaksonDaCruz_M.pdf: 1462332 bytes, checksum: 393b1f36be156bf1cdedb16da1cc3fcd (MD5) Previous issue date: 2014 / Resumo: O presente trabalho se dedica ao estudo das médias aritmética, geométrica e harmônica. Inicialmente, definimos cada uma das médias e trabalhamos suas aplicações através da resolução de problemas. Posteriormente destacamos as desigualdades entre as médias e suas aplicações / Abstract: This dissertation is dedicated to the study of the arithmetic, geometric and harmonic means. Initially, we defined each of the means and applied its uses through problem resolution. Afterwards, we gave emphasis to the inequalities between the means and its uses / Mestrado / Matemática em Rede Nacional / Mestre em Matemática em Rede Nacional

Magister Scientiae - MSc / Over the last few years it became clear that the students struggle with the basic concepts of logarithms and inequalities, let alone logarithmic inequalities due to the lack of exposure of these concepts at high school. In order to fully comprehend logarithmic inequalities, a good understanding of the logarithmic graph is important. Thus, the opportunity was seen to change the method of instruction by introducing the graphical method to solve logarithmic inequalities. It was decided to use an mathematical software program, Omnigraph, in this research. / South Africa

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A general algorithm for tracing the path of optima of inequality constrained optimization problems as a function of a parameter was developed in this research. The algorithm is an active set algorithm using a homotopy method to trace the path. A new feature of the algorithm is a capability of handling the transition points between segments in a routine way. The algorithm locates the transition points, and finds an active set for the next segment by considering all possible sets of active constraints. The nonoptimal sets are eliminated on the basis of the Lagrange multipliers and the derivatives of the optimal solutions with respect to the parameter. The algorithm was implemented for three different problems. The first application, a spring-mass problem, was used to illustrate various kinds of transition events between segments. The second application, a well known ten-bar truss structural optimization problem, was used to validate the algorithm, since the numerical results for this problem have been obtained by other methods. The third application, bi-objective control-structure optimization, had an important engineering application. The numerical results obtained in this application could be used in the design process — they allowed selection of the best designs and provided some insight into behavior of the structure. The sufficient conditions for persistence of the minima were given using the results of the stability theory, and the connection was shown between these results and classical optimization theory. For the standard nonlinear programming problem these conditions are equivalent to the Mangasarian-Fromovitz criterion and the standard second order sufficient optimality condition. A method for computational verification of the conditions for persistence of the minima was proposed. By using Motzkin’s Transposition Theorem, the Mangasarian-Fromovitz criterion can be reduced to the problem of finding a feasible point for a linear optimization problem. In this form it can be easily solved by Phase I of the simplex method. It was shown that the linear programming problem can be transformed to the form of general nonlinear problem in such a way that the regularity of the constraints is preserved. Therefore for linear problems necessary and sufficient condition for solvability of the perturbed system can be verified computationally. The results of bifurcation theory were used to characterize the possible points of discontinuity of the path. The necessary conditions for discontinuity of the path of optima can be checked by the developed algorithm with no extra work. The results of bifurcation theory were also used to describe the possible singularities of the path of optima and the behavior of the path near singular points. / Ph. D.

Assume that {φn} is an orthonormal uniformly bounded (ONB) sequence of complex-valued functions de ned on a measure space (Ω,Σ,µ), and f ∈ L1(Ω,Σ,µ). Let be the Fourier coefficients of f with respect to {φn} . R.E.A.C. Paley proved a theorem connecting the Lp-norm of f with a related norm of the sequence {cn}. Hardy and Littlewood subsequently proved that Paley’s result is best possible within its context. Their results were generalized by Dikarev, Macaev, Askey, Wainger, Sagher, and later by Tikhonov, Li yand, Booton and others.The present work continues the generalization of these results. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2014. / FAU Electronic Theses and Dissertations Collection

by Yan Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 78-80). / Abstract also in Chinese. / Chapter 1 --- Admissible Points of Convex Sets --- p.7 / Chapter 1.1 --- Introduction and Notations --- p.7 / Chapter 1.2 --- The Main Result --- p.7 / Chapter 1.2.1 --- The Proof of Theoreml.2.1 --- p.8 / Chapter 1.3 --- An Application --- p.10 / Chapter 2 --- A Generalization on The Theorems of Admissible Points --- p.12 / Chapter 2.1 --- Introduction and Notations --- p.12 / Chapter 2.2 --- Fundamental Lemmas --- p.14 / Chapter 2.3 --- The Main Result --- p.16 / Chapter 3 --- Introduction to Variational Inequalities --- p.21 / Chapter 3.1 --- Variational Inequalities in Finite Dimensional Space --- p.21 / Chapter 3.2 --- Problems Which Relate to Variational Inequalities --- p.25 / Chapter 3.3 --- Some Variations on Variational Inequality --- p.28 / Chapter 3.4 --- The Vector Variational Inequality Problem and Its Relation with The Vector Optimization Problem --- p.29 / Chapter 3.5 --- Variational Inequalities in Hilbert Space --- p.31 / Chapter 4 --- Vector Variational Inequalities --- p.36 / Chapter 4.1 --- Preliminaries --- p.36 / Chapter 4.2 --- Notations --- p.37 / Chapter 4.3 --- Existence Results of Vector Variational Inequality --- p.38 / Chapter 5 --- The Generalized Quasi-Variational Inequalities --- p.44 / Chapter 5.1 --- Introduction --- p.44 / Chapter 5.2 --- Properties of The Class F0 --- p.46 / Chapter 5.3 --- Main Theorem --- p.53 / Chapter 5.4 --- Remarks --- p.58 / Chapter 6 --- A set-valued open mapping theorem and related re- sults --- p.61 / Chapter 6.1 --- Introduction and Notations --- p.61 / Chapter 6.2 --- An Open Mapping Theorem --- p.62 / Chapter 6.3 --- Main Result --- p.63 / Chapter 6.4 --- An Application on Ordered Normed Spaces --- p.66 / Chapter 6.5 --- An Application on Open Decomposition --- p.70 / Chapter 6.6 --- An Application on Continuous Mappings from Order- infrabarreled Spaces --- p.72 / Bibliography

He Yiran. / Includes index. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (p. 74-80) and index.. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

This thesis will present new results involving Hardy and Hardy-Sobolev-Maz'ya inequalities for fractional integrals. There are two key ingredients to many of these results. The first is the conformal transformation between the upper halfspace and the unit ball. The second is the pseudosymmetric halfspace rearrangement, which is a type of rearrangment on the upper halfspace based on Carlen and Loss' concept of competing symmetries along with certain geometric considerations from the conformal transformation. After reducing to one dimension, we can use the conformal transformation to prove a sharp Hardy inequality for general domains, as well as an improved fractional Hardy inequality over convex domains. Most importantly, the sharp constant is the same as that for the halfspace. Two new Hardy-Sobolev-Maz'ya inequalities will also be established. The first will be a weighted inequality that has a strong relationship with the pseudosymmetric halfspace rearrangement. Then, the psuedosymmetric halfspace rearrangement will play a key part in proving the existence of the standard Hardy-Sobolev-Maz'ya inequality on the halfspace, as well as some results involving the existence of minimizers for that inequality.

M. Tech. Mathematical Technology. / Discusses the main problems in terms of how to derive mathematical models for a free, a forced and a damped spring pendulum and determining numerical solutions using a computer algebra system (CAS), because exact analytical solutions are not obvious. Hence this mini-dissertation mainly deals with how to derive mathematical models for the spring pendulum using the Euler-Lagrange equations both in the Cartesian and polar coordinate systems and finding solutions numerically. Derivation of the equations of motion are done for the free, forced and damped cases of the spring pendulum. The main objectives of this mini-dissertation are: firstly, to derive the equations of motion governing the oscillatory and rotational components of the spring pendulum for the free, the forced and damped cases of the spring pendulum ; secondly, to solve these equations numerically by writing the equations as initial value problems (IVP); and finally, to introduce a novel way of incorporating nonlinear damping into the Euler-Lagrange equations of motion as introduced by Joubert, Shatalov and Manzhirov (2013, [20]) for the spring pendulum and interpreting the numerical solutions using CAS-generated graphics.