Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type {\rm II}\sb 1 factors and M_n(\cc)) and symmetric gauge norms on $L^\infty[0,1]$ and \cc^n. As the first application, we obtain that the class of unitarily invariant norms on a type {\rm II}\sb 1 factor coincides with the class of symmetric gauge norms on $L^\infty[0,1]$ and von Neumann's classical result \cite{vN} on unitarily invariant norms on M_n(\cc). As the second application, Ky Fan's dominance theorem \cite{Fan} is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative $L^p$-theory (e.g., non-commutative H$\ddot{\text{o}}$lder's inequality, duality and reflexivity of non-commutative $L^p$-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of \NN(\M), the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor \M. We obtain all extreme points of \NN(M_2(\cc)) and many extreme points of \NN(M_n(\cc)) ($n\geq 3$). For a type {\rm II}\sb 1 factor \M, we prove that if $t$ ($0\leq t\leq 1$) is a rational number then the Ky Fan $t$-th norm is an extreme point of \NN(\M).Comment: 48 pages, final version, to appear in J. Funct. Ana

Investigation of Training Algorithms for Hidden Markov Models Applied to Automatic Speech Recognition

The work presented in this thesis focuses on simulating a speech recognizer which is trained by different people with different speaking styles and investigates how sensitive the training and recognition processes are to the variations in the training data. There are four main parts to this work. The first involves an experiment of weighting methods for training with multiple observation sequences. The second involves the testing of different initial parameters. The third part includes the first experiment involving training with multiple observation sequences. The model\u27s sensitivity to variations in training data was evaluated by comparing the cases of different values of variation. The final part varied the observation vectors with the variation restricted to only one of the eight positions in the sequence. The experiment was repeated for each of eight positions in the observation sequence, and the effect on recognition was evaluated

T-Duality and Homological Mirror Symmetry of Toric Varieties

Let $X_\Sigma$ be a complete toric variety. The coherent-constructible correspondence $\kappa$ of \cite{FLTZ} equates \Perf_T(X_\Sigma) with a subcategory Sh_{cc}(M_\bR;\LS) of constructible sheaves on a vector space M_\bR. The microlocalization equivalence $\mu$ of \cite{NZ,N} relates these sheaves to a subcategory Fuk(T^*M_\bR;\LS) of the Fukaya category of the cotangent T^*M_\bR. When X_\Si is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, DCoh_T(X_\Si)\cong DFuk(T^*M_\bR;\LS), which is an equivalence of triangulated tensor categories. The nonequivariant coherent-constructible correspondence $\bar{\kappa}$ of \cite{T} embeds \Perf(X_\Si) into a subcategory Sh_c(T_\bR^\vee;\bar{\Lambda}_\Si) of constructible sheaves on a compact torus T_\bR^\vee. When X_\Si is nonsingular, the composition of $\bar{\kappa}$ and microlocalization yields a version of homological mirror symmetry, DCoh(X_\Sigma)\hookrightarrow DFuk(T^*T_\bR;\bar{\Lambda}_\Si), which is a full embedding of triangulated tensor categories. When X_\Si is nonsingular and projective, the composition $\tau=\mu\circ \kappa$ is compatible with T-duality, in the following sense. An equivariant ample line bundle \cL has a hermitian metric invariant under the real torus, whose connection defines a family of flat line bundles over the real torus orbits. This data produces a T-dual Lagrangian brane $\mathbb L$ on the universal cover T^*M_\bR of the dual real torus fibration. We prove \mathbb L\cong \tau(\cL) in Fuk(T^*M_\bR;\LS). Thus, equivariant homological mirror symmetry is determined by T-duality.Comment: 34 pages, 2 figures. The previous version of this paper has now been broken into two parts. The other part is available at arXiv:1007.005

THAI STUDENTS’ AWARENESS ATTITUDES OF GLOBAL ENGLISHES PEDAGOGY AND TARGET INTERLOCUTOR

The widespread use of English as a world language has caused changes in the sociolinguistic reality of English speakers, English use, and English-speaking societies. No paradigm has been able to completely capture and account for such changes in the English language given the current changes. In order to reflect the sociolinguistic realities of today, researchers have advocated for a paradigm shift from traditional pedagogy to a new pedagogy that can prepare students to use English in intercultural communication and incorporate multiple varieties of English and cultures. The attitudes of 20 Thai English major students who successfully completed a one-semester Global English language teaching course were investigated using questionnaire and interview data. The analysis included both quantitative and qualitative methods using descriptive statistics and qualitative content analysis, respectively. The results demonstrated that the 15-week GE-informed pedagogy significantly improved student views toward GE awareness. The students further reported that English should be viewed as a language serving varied communicative needs among different multilingual users and not just native users as target interlocutors. These results call for more GE-awareness pedagogical activities for a better understanding of the practical uses of English and in compliance with global linguistic diversifications.  Article visualizations

Coherent control of Snell's law at metasurfaces

It was recently demonstrated that the well-known Snell's law must be corrected for phase gradient metasurfaces to account for their spatially varying phase, leading to normal and anomalous transmission and reflection of light on such metasurfaces. Here we show that the efficiency of normal and anomalous transmission and reflection of light can be controlled by the intensity or phase of a second coherent wave. The phenomenon is illustrated using gradient metasurfaces based on V-shaped and rectangular apertures in a metal film. This coherent control effect can be exploited for wave front shaping and signal routing