Complex Numbers in n Dimensions

Cohesive sediment, or mud, is encountered in most water bodies throughout the world. Often mud is a valuable resource, synonymous with fertile land, enriching the natural environment and used as an important building material. Yet mud also hinders navigation and consequently, dredging operations have been carried out since ancient times to safeguard navigation. Unfortunately, many mud deposits are now contaminated, endangering the eco-system and increasing the costs of dredging operations. The transport and fate of mud in the environment are still poorly understood and the need for basic research remains. This text contains the proceedings of the INTERCOH-2000 conference on progress in cohesive sediment research. It was the sixth in a series of conferences initially started by Professor Ashish Mehta in 1984 as a "Workshop on Cohesive Sediment Dynamics with Special Reference to the Processes in Estuaries". During these conferences the character of the first workshop has always been maintained, that is, small scale and dedicated to the physical and engineering aspects of cohesive sediments, without parallel sessions, but with ample time for discussions during and after the presentations, and followed by a book of proceedings containing thoroughly reviewed papers. INTERCOH-2000 was integrated with the final workshop of the COSINUS project. This project was carried out as a part of the European MAST-3 programme, and almost all European cohesive sediment workers were involved. INTERCOH-2000 focused on the behaviour and modelling of concentrated benthic suspensions, i.e. high-concentrated near-bed suspensions of cohesive sediment. Special attention was paid to: sediment - turbulence interaction; flocculation and settling velocity; high-concentrated mud suspensions; processes in the bed - consolidation; processes on the bed - erosion; field observations on mud dynamics; instrumentation; and numerical modelling.Two distinct systems of hypercomplex numbers in n dimensions are introduced in this book, for which the multiplication is associative and commutative, and which are rich enough in properties such that exponential and trigonometric forms exist and the concepts of analytic n-complex function, contour integration and residue can be defined.The first type of hypercomplex numbers, called polar hypercomplex numbers, is characterized by the presence in an even number of dimensions greater or equal to 4 of two polar axes, and by the presence in an odd number of dimensions of one polar axis. The other type of hypercomplex numbers exists as a distinct entity only when the number of dimensions n of the space is even, and since the position of a point is specified with the aid of n/2-1 planar angles, these numbers have been called planar hypercomplex numbers.The development of the concept of analytic functions of hypercomplex variables was rendered possible by the existence of an exponential form of the n-complex numbers. Azimuthal angles, which are cyclic variables, appear in these forms at the exponent, and lead to the concept of n-dimensional hypercomplex residue. Expressions are given for the elementary functions of n-complex variable. In particular, the exponential function of an n-complex number is expanded in terms of functions called in this book n-dimensional cosexponential functionsof the polar and respectively planar type, which are generalizations to n dimensions of the sine, cosine and exponential functions.In the case of polar complex numbers, a polynomial can be written as a product of linear or quadratic factors, although it is interesting that several factorizations are in general possible. In the case of planar hypercomplex numbers, a polynomial can always be written as a product of linear factors, although, again, several factorizations are in general possible.The book presents a detailed analysis of the hypercomplex numbers in 2, 3 and 4 dimensions, then presents the properties of hypercomplex numbers in 5 and 6 dimensions, and it continues with a detailed analysis of polar and planar hypercomplex numbers in n dimensions. The essence of this book is the interplay between the algebraic, the geometric and the analytic facets of the relations.