The symbol map (of Goncharov) takes multiple polylogarithms to a tensor product space where calculations are easier, but where important differential and combinatorial properties of the multiple polylogarithm are retained. Finding linear combinations of multiple polylogarithms in the kernel of the symbol map is an effective way to attempt finding functional equations. We present and utilise methods for finding new linear combinations of multiple polylogarithms (and specifically harmonic polylogarithms) that lie in the kernel of the symbol map. During this process we introduce a new pictorial construction for calculating the symbol, namely the hook-arrow tree, which can be used to easier encode symbol calculations onto a computer. We also show how the hook-arrow tree can simplify symbol calculations where the depth of a multiple polylogarithm is lower than its weight and give explicit expressions for the symbol of depth 2 and 3 multiple polylogarithms of any weight. Using this we give the full symbol for I_{2,2,2}(x,y,z). Through similar methods we also give the full symbol of coloured multiple zeta values. We provide introductory material including the binary tree (of Goncharov) and the polygon dissection (of Gangl, Goncharov and Levin) methods of finding the symbol of a multiple polylogarithm, and give bijections between (adapted forms of) these methods and the hook-arrow tree.

Several linear embeddings of the logistic equation, xn+1=axn(1-xn) are considered, the goal being to establish a connection between linear embedding and the Poincaré Functional Equation. In particular, we consider linear embedding schemes in a classical Hardy space.

The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s,q) is a generalization of ζ(s). We modify Apostolʼs methods to find values of the derivatives of ζ(s,q) with respect to s at s = 0. As a consequence, we obtain relations among certain important constants, the generalized Stieltjes constants. We also give numerical estimates of several values of the derivatives of ζ(s,q).

No description available.

在這篇論文中，我們將利用值分佈的理論來探討下列函數方程解的存在性與其性質： \[\sum_{j=1}^pa_j(z)f_j(z)^{k_j}=1,\] 其中 $a_1(z),\cdots ,a_p(z)$ 為半純函數。對某些特殊方程，除了文獻裡已知的結果外，我們亦提供其它的例子。一般而言，我們探討解存在的必要條件。另外，我們證明了某一類半純函數之零點與極點之分佈的結果。 / In this thesis, we use the theory of value distribution to study the existence of solution of the following functional equation: \[\sum_{j=1}^pa_j(z)f_j(z)^{k_j}=1,\] where $a_1(z),\cdots ,a_p(z)$ are meromorphic functions. For some special case, new and old examples of the solutions are given. For the general case, a necessary condition for the existence of solution is considered. Moreover, we obtain a result on the distribution of zeros and poles of a class of meromorphic functions.

Submitted by (lucia.rodrigues@ufrpe.br) on 2017-03-29T13:17:55Z No. of bitstreams: 1 Emerson de Oliveira Dantas.pdf: 321375 bytes, checksum: e7a1908ccaf8bcb9ee0df5fdb308d1d3 (MD5) / Made available in DSpace on 2017-03-29T13:17:55Z (GMT). No. of bitstreams: 1 Emerson de Oliveira Dantas.pdf: 321375 bytes, checksum: e7a1908ccaf8bcb9ee0df5fdb308d1d3 (MD5) Previous issue date: 2014-08-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is motivated by the Cauchy Functional Equation f (x + y) = f (x) .f (y), characteristic of the exponential function. To arrive at this equation we will begin our study of the definitions and statements of the Exponent Properties Real Power, particularly in the case in which the power exponent is irrational, besides doing a pedagogical proposal on teaching potentiation, Characterization of the Exponential Function and Functional Equation Linear Cauchy. / Este trabalho tem por motivação a Equação Funcional de Cauchy f(x + y) = f(x).f(y), característica da Função Exponencial. Para chegarmos a essa equação iniciaremos o nosso estudo pelas definições e demonstrações das Propriedades da Potência de Expoente Real, destacando o caso em que a Potência tem Expoente Irracional, além de fazermos uma proposta pedagógica sobre o ensino de Potenciação, Caracterização da Função Exponencial e Equação Funcional Linear de Cauchy

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-06-08T11:47:22Z No. of bitstreams: 1 gabrieldelimaesilva.pdf: 258088 bytes, checksum: bfc041ce4de1b43fe1a8880438b1efad (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-06-26T17:48:13Z (GMT) No. of bitstreams: 1 gabrieldelimaesilva.pdf: 258088 bytes, checksum: bfc041ce4de1b43fe1a8880438b1efad (MD5) / Made available in DSpace on 2017-06-26T17:48:13Z (GMT). No. of bitstreams: 1 gabrieldelimaesilva.pdf: 258088 bytes, checksum: bfc041ce4de1b43fe1a8880438b1efad (MD5) Previous issue date: 2012-08-21 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho discutimos o modelo de Heisenberg isotrópico bidimensional do ponto de vista de sistema vinculado. Nós apresentamos este sistema como uma teoria que não tem, a princípio, invariância de calibre. O conceito de invariância de calibre é muito útil em física teórica, uma vez que permite a compreensão profunda de sistemas físicos já que permite uma escolha arbitrária de um referencial a cada instante de tempo. Na verdade, todas as teorias que descrevem as interações fundamentais são teorias de calibre. No método de Dirac todos os vínculos obtidos para o modelo de Heisenberg isotrópico bidimensional são de segunda classe, isso significa que, em princípio, o modelo não apresenta invariância de calibre. Isto será verificado através da aplicação do método simplético. Neste contexto, o potencial simplético (hamiltoniana) será obtido e para uma escolha de fator-ordenação particular, a saber, que as funções dos campos devem ficar à esquerda do operador momento, nós escrevemos a equação de Schrõdinger funcional correspondente. Esta equação não será resolvida explicitamente aqui, no entanto, isso poderia ser feito aplicando o método do cálculo funcional assim seria obtido o espetro de energias do sistema. / In this work we discuss the two-dimensional isotropic Heisenberg model from the constrained systems point of view. We present this system as a theory which has not gauge invariance. The concept of gauge invariance is very useful in theoretical physics, since it allows a deep understanding of physical systems and an arbitrary choice of a reference at each instant of time. In fact, all theories describing the fundamental interactions are gauge theories. In the method of Dirac all constraints obtained for the two-dimensional isotropic Heisenberg model are second class, this means, at first, the model has no gauge invariance. This will be checked by applying the symplectic method. In this context, the symplectic potential (Hamiltonian) will be obtained and a choice of particular factor-ordering, namely that the functions of the fields should be left to the operators, we write the associated functional equation of Schrodinger. This equation will not be solved explicitly here, however, this could be done by applying the method of functional calculation, and so we should obtain the energy spectrum of the system.

No description available.

Disertační práce formuluje asymptotické odhady řešení tzv. sublineárních a superlineárních diferenciálních rovnic se zpožděním. V těchto odhadech vystupuje řešení pomocných funkcionálních rovnic a nerovností. Dále práce pojednává o kvalitativních vlastnostech diferenčních rovnic se zpožděním, které vznikly diskretizací studovaných diferenciálních rovnic. Pozornost je věnována souvislostem asympotického chování řešení rovnic ve spojitém a diskrétním tvaru, a to v obecném i v konkrétních případech. Studována je rovněž stabilita numerické diskretizace vycházející z $\theta$-metody. Práce obsahuje několik příkladů ilustrujících dosažené výsledky.