This dissertation consists of three parts. In the

We study the Neumann problem for the de Rham complex in a bounded domain of Rn with singularities on the boundary. The singularities may be general enough, varying from Lipschitz domains to domains with cuspidal edges on the boundary. Following Lopatinskii we reduce the Neumann problem to a singular integral equation of the boundary. The Fredholm solvability of this equation is then equivalent to the Fredholm property of the Neumann problem in suitable function spaces. The boundary integral equation is explicitly written and may be treated in diverse methods. This way we obtain, in particular, asymptotic expansions of harmonic forms near singularities of the boundary.

We show that the spectral kernel function of the dbar-Neumann problem on a non-compact strongly pseudoconvex manifold is smooth up to the boundary.

Many problems in the natural world have high contrast properties, like transport in composites, fluid in porous media and so on. These problems have huge numerical difficulties because of the singularities of their solutions. It may be really expensive to solve these problems directly by traditional numerical methods. It is necessary and important to understand these problems more in mathematical aspect first, and then using the mathematical results to simplify the original problems or develop more efficient numerical methods. In this thesis we are going to approximate the Dirichlet to Neumann map for the high contrast two phase composites. The mathematical formulation of our problem is to approximate the energy for an elliptic equation with arbitrary boundary conditions. The boundary conditions may have highly oscillations, which makes our problems very interesting and difficult. We developed a method to divide the domain into two different subdomains, one is close to and the other one is far from the boundary, and we can approximate the energy in these two subdomains separately. In the subdomain far from the boundary, the energy is not influenced that much by the boundary conditions. Methods for approximation of the energy in this subdomain are studied before. In the subdomain near the boundary, the energy depends on the boundary conditions a lot. We used a new method to approximate the energy there such that it works for any kind of boundary conditions. By this way, we can have the approximation for the total energy of high contrast problems with any boundary conditions. In other words, we can have a matrix up to any dimension to approximate the continuous Dirichlet to Neumann map of the high contrast composites. Then we will use this matrix as a preconditioner in domain decomposition methods, such that our numerical methods are very efficient to solve the problems in high contrast composites.

This dissertation consists of three parts. In the

The measure-multiplicity-invariant for masas in II_1 factors was introduced by Dykema, Smith and Sinclair to distinguish masas that have the same Pukanszky invariant. In this dissertation, the measure class (left-right-measure) in the measuremultiplicity- invariant is studied, which equivalent to studying the structure of the standard Hilbert space as an associated bimodule. The focal point of this analysis is: To what extent the associated bimodule remembers properties of the masa. The structure of normaliser of any masa is characterized depending on this measure class, by using Baire category methods (Selection principle of Jankov and von Neumann). Measure theoretic proofs of Chifan's normaliser formula and the equivalence of weak asymptotic homomorphism property (WAHP) and singularity is presented. Stronger notions of singularity is also investigated. Analytical conditions based on Fourier coefficients of certain measures are discussed, that partially characterize strongly mixing masas and masas with nontrivial centralizing sequences. The analysis also provide conditions in terms of operators and L2 vectors that characterize masas whose left-right-measure belongs to the class of product measure. An example of a simple masa in the hyperfinite II1 factor whose left-right-measure is the class of product measure is exhibited. An example of a masa in the hyperfinite II1 factor whose leftright- measure is singular to the product measure is also presented. Unitary conjugacy of masas is studied by providing examples of non unitary conjugate masas. Finally, it is shown that for k greater than/equal to 2 and for each subset S \subseteq N, there exist uncountably many non conjugate singular masas in L(Fk) whose Pukanszky invariant is S u {1}.

For an inclusion N \subseteq M of finite von Neumann algebras, we study the group of normalizers N_M(B) = {u: uBu^* = B} and the von Neumann algebra it generates. In the first part of the dissertation, we focus on the special case in which N \subseteq M is an inclusion of separable II_1 factors. We show that N_M(B) imposes a certain "discrete" structure on the generated von Neumann algebra. By analyzing the bimodule structure of certain subalgebras of N_M(B)'', this leads to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of N_M(B)'' in terms of a unique countable subgroup of N_M(B). We then apply these general techniques to obtain results for inclusions B \subseteq M arising from the crossed product, group von Neumann algebra, and tensor product constructions. Our work also leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N \subseteq M is a regular inclusion of II_1 factors, then N norms M: These new results and techniques develop further the study of normalizers of subfactors of II_1 factors. The second part of the dissertation is devoted to studying normalizers of maximal abelian self-adjoint subalgebras (masas) in nonseparable II_1 factors. We obtain a characterization of masas in separable II_1 subfactors of nonseparable II_1 factors, with a view toward computing cohomology groups. We prove that for a type II_1 factor N with a Cartan masa, the Hochschild cohomology groups H^n(N,N)=0, for all n \geq 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual.

Spiral of silence has been published for 30 years, and been tested in many areas, however these test are not all qualified. This study aims to interpret spiral of silence theory and to inspect if there are any misuse in these test. First, we interpret these theory form the origin of the theory and it¡¦s deducing process, and built an theory model. Then inspect those test based on this model. The result discovered that most of these test stressed on testing people¡¦s willingness to speak out, and misleaded to compare one¡¦s opinion and one¡¦s perception of majority. This comprehension gap may comes from the wrong variable definition in the operational models. For this sake, this study offered a theory model to overcome this gap.

We study the ∂-Neumann operator and the Kobayashi metric. We observe that under certain conditions, a higher-dimensional domain fibered over Ω can inherit noncompactness of the d-bar-Neumann operator from the base domain Ω. Thus we have a domain which has noncompact d-bar-Neumann operator but does not necessarily have the standard conditions which usually are satisfied with noncompact d-bar-Neumann operator. We define the property K which is related to the Kobayashi metric and gives information about holomorphic structure of fat subdomains. We find an equivalence between compactness of the d-bar-Neumann operator and the property K in any convex domain. We also find a local property of the Kobayashi metric [Theorem IV.1], in which the domain is not necessary pseudoconvex. We find a more general condition than finite type for the local regularity of the d-bar-Neumann operator with the vector-field method. By this generalization, it is possible for an analytic disk to be on the part of boundary where we have local regularity of the d-bar-Neumann operator. By Theorem V.2, we show that an isolated infinite-type point in the boundary of the domain is not an obstruction for the local regularity of the d-bar-Neumann operator.

The algebra of matrices M with entries in an abelian von Neumann algebra is a C*-module. C*-modules were originally defined and studied by Kaplansky and we outline the foundations of the theory and particular properties of M. Furthermore, we prove a structure theorem for ultraweakly closed submodules of M, using techniques from the theory of type I finite von Neumann algebras. By analogy with the classical join in topology, the join for operator algebras A and B acting on Hilbert spaces H and K, respectively, was defined by Gilfeather and Smith. Assuming that K is finite dimensional, Gilfeather and Smith calculated the Hochschild cohomology groups of the join. We assume that M is the algebra of matrices with entries in a maximal abelian von Neumann algebra U, A is an operator algebra acting on a Hilbert space K, and B is an ultraweakly closed subalgebra of M containing U. In this new context, we redefine the join, generalize the calculations of Gilfeather and Smith, and calculate the cohomology groups of the join.