Zur Beschreibung von realen Prozessen werden oft Differentialgleichungen herangezogen. Liegen nun Messdaten von diesen Prozessen vor, so sollen auch die Parameter im mathematischen Modell so gewählt werden, dass diese den Messungen entsprechen. Dieser Vortrag zeigt, wie dies in Mathcad mit der Funktion genfit realisiert werden kann.

We approximate the solution of a quasilinear stochastic partial differential equa- tion driven by fractional Brownian motion B_H(t); H in (0,1), which was calculated via fractional White Noise calculus, see [5].

During the last decade, the use of time scales as a means of unifying and extending results about various types of dynamic equations has proven to be both prolific and fruitful. Many classical results from the theories of differential and difference equations have time scale analogues. In this thesis we derive new oscillation criteria for second order dynamic equations on time scales.

Random graph processes are most often used to investigate theoretical questions about random graphs. A randomised algorithm can be defined specifically for the purpose of finding some structure in a graph, such as a matching, a colouring or a particular kind of sub graph. Properties of the related random graph process then suggest properties, or bounds on properties, of the structure. In this thesis, we use a random graph process to analyse a particular load balancing algorithm from theoretical computer science. By doing so, we demonstrate that random graph processes may also be used to analyse other algorithms and systems of a random nature, from areas such as computer science, telecommunications and other areas of engineering and mathematics. Moreover, this approach can lead to theoretical results on the performance of algorithms that are difficult to obtain by other methods. In the course of our analysis we are also led to some results of the first kind, relating to the structure of the random graph. / The particular algorithm that we analyse is a randomised algorithm for an off-line load balancing problem with two choices. The load balancing algorithm, in an initial stage, mirrors an algorithm which finds the k-core of a graph. This latter algorithm and the related random graph process have been previously analysed by Pittel, Spencer and Wormald, using a differential equation method, to determine the thresholds for the existence of a k-core in a random graph. We modify their approach by using a random pseudograph model due to Bollobas and Frieze, and Chvatal, in place of the uniform random graph. This makes the analysis somewhat simpler, and leads to a shortened derivation of the thresholds and other properties of k-cores.(For complete abstract open document)

在這篇論文當中，我們引用`海岸綠堤--水筆仔'網站上的研究資料並且藉由Matlab程式軟體的幫助建構數學模型，我們討論以下的二階非線性微分方程 (i) u''(t)=f(u(t)), u(t_0)=u_0, u'(t_0)=u_1. (ii) u''(t)=f(u'(t)), u(t_0)=u_0, u'(t_0)=u_1. 我們比較拋物線函數，立方函數，傅立葉和函數，正弦和函數並且從這些函數中選出最好的一個當作我們的模型，我們得到一些主要的結果。 / In this paper, we use the real data from website of `Seacoast Green Bank--Kandelia' and construct mathematical models with the help of Matlab, we discuss the following nonlinear 2nd order differential equation (i) u''(t)=f(u(t)), u(t_0)=u_0, u'(t_0)=u_1. (ii) u''(t)=f(u'(t)), u(t_0)=u_0, u'(t_0)=u_1. We compared with the functions of parabolic, cubic, Fourier summation, sum of sine and choose the best one from them as our model, we have obtained main results.

[pt] Esse trabalho tem como objetivo ser uma introdução ao estudo da existência e unicidade de soluções fracas para equações diferenciais parciais elípticas. Começamos definindo o espaço de Sobolev para, a partir da definição, provarmos algumas propriedades básicas que nos ajudarão no estudo das equações diferenciais parciais elípticas. Finalizamos com o desenvolvimento do Teorema de Lax-Milgram e de Stampacchia que permitirão o uso de técnicas de Análise Funcional para estudarmos alguns exemplos de equações elípticas. / [en] This dissertation aims to be an introduction to the study of the existence and uniqueness of weak solutions for elliptic partial differential equations. We begin by defining the Sobolev spaces and proving some basics properties that will assist in the study of the elliptical equations. Lastly, we develop the Theorems of Lax-Milgram and Stampacchia that allow the use of Functional Analysis for the studying of some examples of elliptic equations.

La these porte sur les corrections mixtes QCD-EW au niveau NNLO a la productionDrell-Yan de bosons Z et W. Le processus Drell-Yan est un processus fondamentalpermettant de tester avec precision le Modele Standard (MS) de physique des partic-ules au sein de collisionneurs hadroniques, car ce dernier presente une section ecaceimportante, une signature experimentale tres propre, ainsi qu'une tres haute sensi-bilite aux proprietes des bosons de jauge. Pour toutes ces raisons, une prediction theorique precise et able, siginant ici que l'on garde sous contr^ole lestermes provenant des corrections perturbatives d'ordre superieur de la section ecaceet des distributions du mecanisme de production de Drell-Yan, est exigee pour menera bien des etudes de physique au niveau de collisionneurs hadroniques.Dans cette thèse , nous étudions les corrections QCD mixtes - EW à Drell - Yan traite à la NNLO . D'un point de vue technique , le calcul d'un tel ensemble de corrections impliquerait le cal-tion de diagrammes de Feynman très compliquées , La plus grande contribution provient des diagrammes dans lesquels la particule de décomposition ( Z ou boson W ) est presque sur - coquille.En utilisant les règles Cutkosky , nous pouvons ré-écrire l'intégration sur l'espace de phase de latermes d'interférence ( une boucle 2 à 2 diagrammes interféré avec le niveau arbre 2 à 2 etarbre 2 ou 3 diagrammes carré ) en termes de combinaison des intégrales de propagationteurs ayant la prescription et propagateurs de causalité droite avec une face .Ces intégrales peuvent être traités de la même manière que les corrections virtuelles . Cette réduction se fait en utilisant l' algorithme Laporta \ " , sur la base del'intégration par parties identités . Le calcul de l' IM est réalisée en utilisant la méthode de la différenceéquations. En conséquence , nous obtenons l' IM exprimée en série de Laurent ,où D est la dimension de l'espace - temps , la multiplication d'un facteur qui prend entenir compte de la limite souple de l'intégrale en D dimensions . / The thesis concerns the NNLO mixed QCD-EW corrections to the Drell-Yan (DY)production of Z andW bosons, via the following reactions: pp(p) Z+X to l + Xand pp to W + X to l + X. This is a fundamental process for an accurate testof the Standard Model (SM) at hadron colliders, since it has a large cross section, aclean experimental signature. In particular, the Drell-Yan production of Ws is important for an accuratedetermination (via transverse mass and pT distributions) of the W mass, mW, aninput parameter of the model. Because of all these reasons, an accurate and reliable theoretical prediction forthe cross section and the distributions of the Drell-Yan production mechanism, thatmeans control on the higher-order perturbative corrections, is demanded for physicsstudies at hadron colliders. In this thesis, we study the mixed QCD-EW corrections to Drell-Yan processes at the NNLO. From a technical point of view, the calculation of such a set of corrections would involve the calcu-lation of very complicated Feynman diagrams, The biggest contribution comes from the diagrams in which the decaying particle(Z or W boson) is nearly on-shell. Using the Cutkosky rules, we can re-write the integration over the phase-space of theinterference terms (one-loop 2 to 2 diagrams interfered with the tree-level 2 to 2 andtree 2 to 3 diagrams squared) in terms of a combination of integrals with propaga-tors having the right causality prescription and propagators with the opposite one.These integrals can be treated in the same way as the virtual corrections. This reduction is done using the \Laporta Algorithm", based onthe Integration-by-Parts Identities. The calculation of the MIs is performed using the method of differentialequations. As a result, we get the MIs expressed as a Laurent series ,where D is the dimension of the space-time, multiplying a factor which takes intoaccount the soft limit of the integral in D dimensions.

abstract: The phycologist, M. R. Droop, studied vitamin B12 limitation in the flagellate Monochrysis lutheri and concluded that its specific growth rate depended on the concentration of the vitamin within the cell; i.e. the cell quota of the vitamin B12. The Droop model provides a mathematical expression to link growth rate to the intracellular concentration of a limiting nutrient. Although the Droop model has been an important modeling tool in ecology, it has only recently been applied to study cancer biology. Cancer cells live in an ecological setting, interacting and competing with normal and other cancerous cells for nutrients and space, and evolving and adapting to their environment. Here, the Droop equation is used to model three cancers. First, prostate cancer is modeled, where androgen is considered the limiting nutrient since most tumors depend on androgen for proliferation and survival. The model's accuracy for predicting the biomarker for patients on intermittent androgen deprivation therapy is tested by comparing the simulation results to clinical data as well as to an existing simpler model. The results suggest that a simpler model may be more beneficial for a predictive use, although further research is needed in this field prior to implementing mathematical models as a predictive method in a clinical setting. Next, two chronic myeloid leukemia models are compared that consider Imatinib treatment, a drug that inhibits the constitutively active tyrosine kinase BCR-ABL. Both models describe the competition of leukemic and normal cells, however the first model also describes intracellular dynamics by considering BCR-ABL as the limiting nutrient. Using clinical data, the differences in estimated parameters between the models and the capacity for each model to predict drug resistance are analyzed. Last, a simple model is presented that considers ovarian tumor growth and tumor induced angiogenesis, subject to on and off anti-angiogenesis treatment. In this environment, the cell quota represents the intracellular concentration of necessary nutrients provided through blood supply. Mathematical analysis of the model is presented and model simulation results are compared to pre-clinical data. This simple model is able to fit both on- and off-treatment data using the same biologically relevant parameters. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2015

abstract: In order to verify the dispersive nature of transverse displacement in a beam, a deep understanding of the governing partial differential equation is developed. Using the finite element method and Newmark’s method, along with Fourier transforms and other methods, the aim is to obtain consistent results across each numerical technique. An analytical solution is also analyzed for the Euler-Bernoulli beam in order to gain confidence in the numerical techniques when used for more advance beam theories that do not have a known analytical solution. Three different beam theories are analyzed in this report: The Euler-Bernoulli beam theory, Rayleigh beam theory and Timoshenko beam theory. A comparison of the results show the difference between each theory and the advantages of using a more advanced beam theory for higher frequency vibrations. / Dissertation/Thesis / Masters Thesis Civil Engineering 2016

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-23T14:42:46Z No. of bitstreams: 1 DissPLS.pdf: 887262 bytes, checksum: e54f2438d019bad9fa31a2f0e8b98d66 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:40:36Z (GMT) No. of bitstreams: 1 DissPLS.pdf: 887262 bytes, checksum: e54f2438d019bad9fa31a2f0e8b98d66 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:40:46Z (GMT) No. of bitstreams: 1 DissPLS.pdf: 887262 bytes, checksum: e54f2438d019bad9fa31a2f0e8b98d66 (MD5) / Made available in DSpace on 2016-09-26T20:40:52Z (GMT). No. of bitstreams: 1 DissPLS.pdf: 887262 bytes, checksum: e54f2438d019bad9fa31a2f0e8b98d66 (MD5) Previous issue date: 2015-04-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / The Rosenau-Hyman, or K(m, n), equations are a generalized version of the Korteweg-de Vries (KdV) equation where the dipersive term is non-linear. Such partial differential equations not always have a specific method by which can be solved, besides the solutions are not always analytical. The Lie symmetry method was applied to look for solutions of these equations. This method consists in finding the most general symmetry group of the equation, wherewith the solution can be found. It was found an expression to the solution and to some particular cases. It was shown that in the case K(2, 2) a new kind of solution, called compacton, with peculiar properties is found. / Equações Rosenau-Hyman, ou K(m, n), são uma versão generalizada da equação Kortewegde Vries (KdV) em que o termo dispersivo é não-linear. Essas equações diferencias nãolineares nem sempre possuem um método específico pelo qual podem ser resolvidas, além de que as soluções nem sempre são analíticas. O método de simetria de Lie foi aplicado para buscar por soluções dessas equações. Esse método consiste em encontrar o grupo de simetria mais geral da equação, por meio do qual a solução pode ser encontrada. Obteve-se uma expressão para a solução e alguns casos particulares. Foi mostrado que para K(2, 2) um novo tipo de solução, chamada compacton, com propriedades peculiares é encontrado.