Some Geometric Properties of the m-Möbius Transformations

Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations fm mapping onto . Even for the simplest entity, the pre-image by fm of a unique point, there is no way of visualization. Pre-images by fm of figures from C are like ghost figures in Cm. This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in Cm are well known and vector calculus in Cm is familiar, yet the pre-image by fm of a vector from C is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in Cm. Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.

CONSOLIDATION HEORY OF UNSATURATED SOIL BASED ON THE THEORY OF MIXTURE(Ⅰ)

Unsaturated soil is a three-phase media and is composed of soil grain,water and gas.In this paper,the consolidation problem of unsaturated soil is investigated based on the theory of mixture.A theoretical formula of effective stress on anisotropic porous media and unsaturated soil is derived.The principle of effective stress and the principle of Curie symmetry are taken as two fundamental constitutive principles of unsaturated soil.A mathematical model of consolidation of unsaturated soil is proposed,which consists of 25 partial differenfial equations with 25 unknowns.With the help of increament linearizing method,the model is reduced to 5 governing equations with 5 unknowns,i.e.,the three displacement components of solid phase,the pore water pressure and the pore gas pressure.7 material parameters are involved in the model and all of them can he measured using soil tests.It is convenient to use the model to engineering practice.The well known Biot's theory is a special case of the model.

An Interacting Gauge Field Theoretic Model for Hodge Theory: Basic Canonical Brackets

We derive the basic canonical brackets amongst the creation and annihilation operators for a two(1 + 1)-dimensional(2D) gauge field theoretic model of an interacting Hodge theory where a U(1) gauge field(Aμ) is coupled with the fermionic Dirac fields(ψ andˉψ). In this derivation, we exploit the spin-statistics theorem, normal ordering and the strength of the underlying six infinitesimal continuous symmetries(and the concept of their generators) that are present in the theory. We do not use the definition of the canonical conjugate momenta(corresponding to the basic fields of the theory) anywhere in our whole discussion. Thus, we conjecture that our present approach provides an alternative to the canonical method of quantization for a class of gauge field theories that are physical examples of Hodge theory where the continuous symmetries(and corresponding generators) provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level.