Linear orthogonality preservers of Hilbert bundles

Due to the corresponding fact concerning Hilbert spaces, it is natural to ask if the linearity and the orthogonality structure of a Hilbert $C^*$-module determine its $C^*$-algebra-valued inner product. We verify this in the case when the $C^*$-algebra is commutative (or equivalently, we consider a Hilbert bundle over a locally compact Hausdorff space). More precisely, a $\mathbb{C}$-linear map $\theta$ (not assumed to be bounded) between two Hilbert $C^*$-modules is said to be "orthogonality preserving" if \left =0 whenever \left =0. We prove that if $\theta$ is an orthogonality preserving map from a full Hilbert $C_0(\Omega)$-module $E$ into another Hilbert $C_0(\Omega)$-module $F$ that satisfies a weaker notion of $C_0(\Omega)$-linearity (known as "localness"), then $\theta$ is bounded and there exists $\phi\in C_b(\Omega)_+$ such that \left\ =\ \phi\cdot\left, \quad \forall x,y \in E. On the other hand, if $F$ is a full Hilbert $C^*$-module over another commutative $C^*$-algebra $C_0(\Delta)$, we show that a "bi-orthogonality preserving" bijective map $\theta$ with some "local-type property" will be bounded and satisfy \left\ =\ \phi\cdot\left\circ\sigma, \quad \forall x,y \in E where $\phi\in C_b(\Omega)_+$ and $\sigma: \Delta \rightarrow \Omega$ is a homeomorphism

Linear orthogonality preservers of Hilbert C∗C^*-modules over general C∗C^*-algebras

As a partial generalisation of the Uhlhorn theorem to Hilbert $C^*$-modules, we show in this article that the module structure and the orthogonality structure of a Hilbert $C^*$-module determine its Hilbert $C^*$-module structure. In fact, we have a more general result as follows. Let $A$ be a $C^*$-algebra, $E$ and $F$ be Hilbert $A$-modules, and $I_E$ be the ideal of $A$ generated by $\{\langle x,y\rangle_A: x,y\in E\}$. If $\Phi : E\to F$ is an $A$-module map, not assumed to be bounded but satisfying $$\langle \Phi(x),\Phi(y)\rangle_A\ =\ 0\quad\text{whenever}\quad\langle x,y\rangle_A\ =\ 0,$$ then there exists a unique central positive multiplier $u\in M(I_E)$ such that $$\langle \Phi(x), \Phi(y)\rangle_A\ =\ u \langle x, y\rangle_A\qquad (x,y\in E).$$ As a consequence, $\Phi$ is automatically bounded, the induced map $\Phi_0: E\to \overline{\Phi(E)}$ is adjointable, and $\overline{Eu^{1/2}}$ is isomorphic to $\overline{\Phi(E)}$ as Hilbert $A$-modules. If, in addition, $\Phi$ is bijective, then $E$ is isomorphic to $F$.Comment: 15 page

Analysis Of The Suspension Beam In Accelerometer For Stiffness Constant And Resonant Frequency By Using Analytical And Numerical Investigation

Mikro-meterpecut yang digunakan dalam pelbagai penerapan hanya akan tercapai dengan jayanya sekiranya keperluan frekuensi resonans dan kepekaan dapat dipenuhi dan konsisten. A successful and consistent performance of micro-accelerometer which has been applied in various applications can only be achieved when the resonant frequency and the sensitivity requirement are fulfilled

Analysis Of The Suspension Beam In Accelerometer For Stiffness Constant And Resonant Frequency By Using Analytical And Numerical Investigation [TL589.2.A3 W872 2007 f rb].

Mikro-meterpecut yang digunakan dalam pelbagai penerapan hanya akan tercapai dengan jayanya sekiranya keperluan frekuensi resonans dan kepekaan dapat dipenuhi dan konsisten. Berdasarkan syarat-syarat tersebut, analisis struktur pada pekali kekukuhan and frekuensi resonans bagi rasuk ampaian dalam meter pecut dan seterusnya pengoptimuman kepada kepekaan haruslah dilakukan. A successful and consistent performance of micro-accelerometer which has been applied in various applications can only be achieved when the resonant frequency and the sensitivity requirement are fulfilled. In view of this, structural analysis on stiffness constant and resonant frequency for the suspension beam in accelerometer, and subsequently optimization design of accelerometer with respect to sensitivity in term of displacement against acceleration must be performed

Promoting social-emotional learning in Chinese schools : a feasibility study of PATHS implementation in Hong Kong

This paper describes a pilot study of a reduced version of the PATHS Curriculum, a USdeveloped evidence-based SEL program, among schools in Hong Kong SAR (China). Three hundred and sixteen 12th grade students in three elementary schools participated in the study. A limited number of first grade PATHS lessons were adapted and translated into Chinese. Twelve teachers learned and adopted these lessons in their teaching. Students in these classrooms learned about different emotions and practiced self-control. The intervention lasted four months. After the intervention, students showed improvement in emotion understanding, emotion regulation and prosocial behavior. No change was observed in the level of children’s problem behaviors. Over 65% of the teachers reported a high degree of satisfaction and willingness to adopt the intervention. The effects of the intervention varied among schools, with variations in the level of intervention and principal support, but not in the quality of implementation. Discussion is focused on the factors that could shape the adoption and implementation of SEL programs, especially the role of the difference in school systems between Hong Kong and the United States.peer-reviewe

An ab initio Molecular Orbital Study of the Insertions of Difluorocarbene into Substituted Ethenes

Ab initio MO calculations predict the activation energies for the insertions of difluorocarbene into substituted ethenes C2H3X (X = F, Cl, Me) to be 61,68 and 52 kJ mol-1, respectively; these results indicate that the monosubstitution on the ethene has no significant effects on the activation energy of the reaction