Completeness of the Coulomb wave functions in quantum mechanics

We give an explicit, direct, and fairly elementary proof that the radial energy eigenfunctions for the hydrogen atom in quantum mechanics, bound and scattering states included, form a complete set. The proof uses only some properties of the confluent hypergeometric functions and the Cauchy residue theorem from analytic function theory; therefore it would form useful supplementary reading for a graduate course on quantum mechanics.

Algorithms for Colouring Random k-colourable Graphs

The k-colouring problem is to colour a given k-colourable graph with k colours. This problem is known to be NP-hard even for fixed k greater than or equal to 3. The best known polynomial time approximation algorithms require n(delta) (for a positive constant delta depending on k) colours to colour an arbitrary k-colourable n-vertex graph. The situation is entirely different if we look at the average performance of an algorithm rather than its worst-case performance. It is well known that a k-colourable graph drawn from certain classes of distributions can be ii-coloured almost surely in polynomial time. In this paper, we present further results in this direction. We consider k-colourable graphs drawn from the random model in which each allowed edge is chosen independently with probability p(n) after initially partitioning the vertex set into ii colour classes. We present polynomial time algorithms of two different types. The first type of algorithm always runs in polynomial time and succeeds almost surely. Algorithms of this type have been proposed before, but our algorithms have provably exponentially small failure probabilities. The second type of algorithm always succeeds and has polynomial running time on average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work as long as p(n) greater than or equal to n(-1+is an element of) where is an element of is a constant greater than 1/4.

Superconductivity in the 100–120 K region in oxides of the Tl-Ca-Ba-Cu-O system

Oxides with different cation ratios 2122, 2212, 2213 and 2223 in the Ti-Ca-Ba-Cu-O system exhibit onset of superconductivity in the 110–125 K range with zero-resistance in the 95–105 K range. Electron microscopic studies show dislocations, layered morphology and other interesting features. These oxides absorb electromagnetic radiation (9.11 GHz) in the superconducting phase.

The Alfvén Wave Equation for Magnetospheric Plasmas

It is shown that besides the continuous spectrum which damps away as inverse power of time, the coupled Alfvén wave equation, which gives coupling between a shear Alfvén wave and a surface wave, can also admit a well behaved harmonic solution in the closed form for a set of initial conditions. This solution, though valid for finite time intervals, points out that the Alfvén surface waves can have a band of frequency (instead of a monochromatic frequency for a nonsheared magnetic field) within which the local field line resonance frequency can lie, and thus can excite magnetic pulsations with latitude-dependent frequency. By considering magnetic fields not only varying in magnitude but also in direction, it is shown that the time interval for the validity of the harmonic solution depend upon the angle between the magnetic field directions on either side of the magnetopause. For small values of the angle the time interval can become appreciably large.

Thermodynamic measurements of b.c.c. Cr-Mn solid solutions at 1373 K

Thermodynamics of Cr-Mn alloys have been studied by Eremenko et al (l) using a fused salt e.m.f.technique. Their results indicate positive deviations from ideality at 1023 K. Kaufman (2) has independently estimated negative enthaipy and excess entropy for the b.c.c. Cr-Mn alloys, such that at high temperatures, the entropy term predominates over the enthalpy term giving positive deviations from ideality. Recently the thermodynamic properties of the alloys have been measured by 3acob (3) using a Knudsen cell technique in the temperature range of 1200 to 1500 K. The results indicate mild negative deviations from ideality over the entire composition range. Because of the differences in the reported results and Mn being a volatile component in the alloys which leads to surface depletion under a dynamic set up, an isopiestic technique is used to measure the properties of the alloys.

Topological defects in crystals: A density-wave theory

A new approach for describing dislocations and other topological defects in crystals, based on the density wave theory of Ramakrishnan and Yussouff is presented. Quantitative calculations are discussed in brief for the order parameter profiles, the atomic configuration and the free energy of a screw dislocation with Burgers vector b = (a/2, a/2,a/2 ) in a bcc solid. Our results for the free energy of the dislocation in a crystal of sizeR, when expressed as (λb 2/4π) ln (αR/|b|) whereλ is the shear elastic constant, yield, for example, the valueα ⋍ 1·85 for sodium at its freezing temperature (371°K). The density distribution in the presence of the dislocation shows that the dislocation core has a columnar character. To our knowledge, this study represents the first calculation of dislocation structure, including the core, within the framework of an order parameter theory incorporating thermal effects.

Practical Proposals for Specifying k-Nearest Neighbours Weights Matrices

In this article we introduce and evaluate testing procedures for specifying the number k of nearest neighbours in the weights matrix of spatial econometric models. The spatial J-test is used for specification search. Two testing procedures are suggested: an increasing neighbours testing procedure and a decreasing neighbours testing procedure. Simulations show that the increasing neighbours testing procedures can be used in large samples to determine k. The decreasing neighbours testing procedure is found to have low power, and is not recommended for use in practice. An empirical example involving house price data is provided to show how to use the testing procedures with real data.

Tsoi Yuen Resistance and Post-80s Generation: A new wave of domestic social movement

Hong Kong was once a British colony and has been under the sovereignty of People’s Republic of China (PRC) since 1997. However, some of the unjust practices and colonial legacies are infiltrated into the development ideology as well as the social structures. The construction of intercity express railway project announced in 2008 causing the demolishment of Tsoi Yuen Tsuen, a “non-indigenous” agricultural village in Hong Kong, was one of the current examples. Tsoi Yuen village was established under the former colonial sovereignty sixty years ago. Approximately 450 populations were affected that they had to relocate their homeland involuntarily. However, these villagers were very attached to their homelands and were unwilling to move, and meanwhile they found that they were absent in the government’s consultation and decision-making process. Soon they began their resistance and demanded for “No Move! No Demolish!”. Their movement was strongly supported by a group of “Post-80s generation” and turned into the most important social movement of the city in recent years. In fact, demolition of Tsoi Yuen Village for city development is not an isolated case in the city. Meanwhile the situation is getting worse in Mainland China. I chose the case study of Tsoi Yuen Resistance from 2008 to 2011 for revelation of the complicated colonial history and postcolonial era of Hong Kong. I focused on discussing the Tsoi Yuen Resistance and the Post-80s movement, and how they have exposed the tension between top-down urban planning and development and public movements fighting for a more democratic process in choosing their way of living. Through the study of a village movement which as well as the rationale behind the Post-80s’ support, I hoped to illustrate how this movement has awaken a different sense of living for the new generations in the midst of the high-sounding urban development. It is an opportunity to examine Hong Kong’s colonial epoch in a different perspective: through studying the Tsoi Yuen Village, let them (subalterns) speak for themselves. Furthermore, the significance of this resistance, taking place eleven years after the handover to the PRC, is an important fact that I shall not miss in later discussion. Last but not least, during the resistance, advanced technology and social networks such as Facebook, Twitter, iPhone were used by Post 80s generation to spread the latest information in order to attract public’s concern and participation. Therefore, apart from studying Tsoi Yuen Resistance as a local social movement, I also regard it as a part of the global movement in perusing ecological lifestyle and civil society. How Post 80s’ generation manipulates the global idea in a local context will also be examined.

Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

System of kinematical conservation laws (KCL) govern evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication K. R. Arun, P. Prasad, 3-D kinematical conservation laws (KCL): evolution of a surface in R-3-in particular propagation of a nonlinear wavefront, Wave Motion 46 (2009) 293-311] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. The 7 x 7 system of equations of this KCL based 3-D weakly nonlinear ray theory (WNLRT) is quite complex and explicit expressions for its two nonzero eigenvalues could not be obtained before. In this short note, we use two different methods: (i) the equivalence of KCL and ray equations and (ii) the transformation of surface coordinates, to derive the same exact expressions for these eigenvalues. The explicit expressions for nonzero eigenvalues are important also for checking stability of any numerical scheme to solve 3-D WNLRT. (C) 2010 Elsevier Inc. All rights reserved.

Uniformly valid analytical solution to the problem of a decaying shock wave

An explicit representation of an analytical solution to the problem of decay of a plane shock wave of arbitrary strength is proposed. The solution satisfies the basic equations exactly. The approximation lies in the (approximate) satisfaction of two of the Rankine-Hugoniot conditions. The error incurred is shown to be very small even for strong shocks. This solution analyses the interaction of a shock of arbitrary strength with a centred simple wave overtaking it, and describes a complete history of decay with a remarkable accuracy even for strong shocks. For a weak shock, the limiting law of motion obtained from the solution is shown to be in complete agreement with the Friedrichs theory. The propagation law of the non-uniform shock wave is determined, and the equations for shock and particle paths in the (x, t)-plane are obtained. The analytic solution presented here is uniformly valid for the entire flow field behind the decaying shock wave.

Electromagnetic wave propagation at the interface between two polar semiconductors

The interface between two polar semiconductors can support three types of phonon-plasmon-polariton modes propagating in three well-defined frequency windows ??1?[min(?1,?3),?R1], ??2?[max(?2,?4),?R2], and ??3?[min(?2,?4),?R3]. The limiting frequencies ?1,2,3,4 are defined by ?1(?)=0, ?2(?)=0, and ?R1,2,3 by ?1(?)+?2(?)=0, where ?i(?) are dielectric functions of the two media with i=1,2. The dispersion, decay distances, and polarization of the three modes are discussed. The variation of the limiting frequencies with the interface plasma parameter ???p22/?p12 reveals an interesting feature in the dispersion characteristics of these modes. For the interfaces for which the bulk coupled phonon-plasmon frequencies of medium 1 are greater than the LO frequency or are less than the TO frequency of medium 2, there exist two values of ?=?1 and ?2(