Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1995. / Includes bibliographical references (p. 118-119).

Thesis (doctoral)--Universität Bonn, 1993. / Includes bibliographical references (p. 51).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1997. / Includes bibliographical references (p. 72-74).

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 72-74).

The Cauchy–Riemann equations admit a bilinear multiplication of solutions, since the product of two holomorphic functions is again holomorphic. This multiplication plays the role of a nonlinear superposition principle for solutions, allowing for construction of new solutions from already known ones, and it leads to the exceptional property of the Cauchy–Riemann equations that all solutions can locally be built from power series of a single solution z = x + iy ∈ C. In this thesis we have found a differential algebraic characterization of linear first order systems of partial differential equations admitting a bilinear ∗-multiplication of solutions, and we have determined large new classes of systems having this property. Among them are the already known quasi-Cauchy–Riemann equations, characterizing integrable Newton equations, and the gradient equations ∇f = M∇g with constant matrices M. A systematic description of linear systems of PDEs with variable coefficients have been given for systems with few independent and few dependent variables. An important property of the ∗-multiplication is that infinite families of solutions can be constructed algebraically as power series of known solutions. For the equation ∇f = M∇g it has been proved that the general solution, found by Jodeit and Olver, can be locally represented as convergent power series of a single simple solution similarly as for solutions of the Cauchy–Riemann equations.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-17Bitstream added on 2014-06-13T18:06:51Z : No. of bitstreams: 1 roque_mrd_me_sjrp.pdf: 11300361 bytes, checksum: 634655b9889665fb4488c7076d5db292 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Nesta dissertação, estuda-se estruturas matemáticas relacionadas à álgebra dos sedenions de Cayley-Dickson. O conceito de funções sedeniônicas do tipo f(z) = zn, z 2 S e n 2 N, é desenvolvido a partir da distância jf(y)¡f(x)j, com o objetivo de obter-se uma generalização. A este tipo de mapeamentos trata-se por funções quaseconformes, ou seja, mapeamentos que não preservam a magnitude dos ângulos. Em particular, através de métodos de resolução, apresenta-se e discute-se polinômios de 2n graus com coeficientes sedeniônicos com o intuito de enfatizar o valor da k-dilatação causada quando trabalha-se com o número sedeniônico em coordenadas esféricas. Por fim, ilustra-se geometricamente os cortes produzidos em hiperesferas B(x; r) quando submetidas às transformações do tipo z2 e z3. / In this work, we propose to study the mathematical construction related with algebra of Cayley-Dickson sedenions. We will present the concept of sedenions functions of f(z) = zn type, z 2 S and n 2 N, developing jf(y) ¡ f(x)j distance, with the objective of creating a generalization. This type of mappings is known as quasiconformal functions, that is, mapping that don't preserve the magnitude of angles. Specially, by means of resolution methods, we will discuss polynomials of 2n degrees with sedenions coefficients focused on highlighting the value of the k-dilation caused when we work with the sedenion number in spherical coordinates. Finally, it is illustrated geometrically the cuts produced in hiperspheres B(x; r) when submitted to the transformations of the type z2 and z3.

Orientador: Manoel Ferreira Borges Neto / Banca: Masoyoshi Tsuchida / Banca: José Arnaldo Frutuoso Roveda / Resumo: Nesta dissertação, estuda-se estruturas matemáticas relacionadas à álgebra dos sedenions de Cayley-Dickson. O conceito de funções sedeniônicas do tipo f(z) = zn, z 2 S e n 2 N, é desenvolvido a partir da distância jf(y)¡f(x)j, com o objetivo de obter-se uma generalização. A este tipo de mapeamentos trata-se por funções quaseconformes, ou seja, mapeamentos que não preservam a magnitude dos ângulos. Em particular, através de métodos de resolução, apresenta-se e discute-se polinômios de 2n graus com coeficientes sedeniônicos com o intuito de enfatizar o valor da k-dilatação causada quando trabalha-se com o número sedeniônico em coordenadas esféricas. Por fim, ilustra-se geometricamente os cortes produzidos em hiperesferas B(x; r) quando submetidas às transformações do tipo z2 e z3. / Abstract: In this work, we propose to study the mathematical construction related with algebra of Cayley-Dickson sedenions. We will present the concept of sedenions functions of f(z) = zn type, z 2 S and n 2 N, developing jf(y) ¡ f(x)j distance, with the objective of creating a generalization. This type of mappings is known as quasiconformal functions, that is, mapping that don't preserve the magnitude of angles. Specially, by means of resolution methods, we will discuss polynomials of 2n degrees with sedenions coefficients focused on highlighting the value of the k-dilation caused when we work with the sedenion number in spherical coordinates. Finally, it is illustrated geometrically the cuts produced in hiperspheres B(x; r) when submitted to the transformations of the type z2 and z3. / Mestre