Prediction of the Frontier Electronic States in a Finite Single-wall Carbon Nanotube

A theoretical model is summarized into the shorter vector principle. It is used to predict the topological structure of wave function and the oscillation rule of energy gap in various types of finite carbon nanotubes (CNTs). The theoretical model indicates that the characteristics of the electronic states only depend on the nanotube size and its symmetry along the shorter vector direction. In this direction, the wave functions of the original 3m (or 3m/2) periodicity are also suitable for armchair, chiral and zigzag finite CNTs with the C2 (Cs), C1 and Cn point groups, respectively. Energy gaps present the oscillation with 3m (or 3m/2) or odd-even n. The first principle calculations for some prototype systems are performed. The results are consistent with the theoretical model.     

The uncertain premium principle based on the distortion function

In this paper, we discuss the premium principle in uncertain environment. First, the net premium principle for uncertain risks is presented within the framework of uncertainty theory. With the help of distortion function, a new uncertain premium principle is derived from the uncertain net premium. Some properties of uncertain distortion premium principle are proved. Furthermore, a sufficient and necessary condition that an uncertain premium principle is an uncertain distortion premium principle has been characterized. Finally, some examples are given to illustrate the calculations of the uncertain distortion premium.     

Muirhead-Rado inequality for compact groups

Muirhead’s majorization inequality was extended by Rado to the case of arbitrary permutation groups. We further generalize this inequality to compact groups and their linear representations over the reals. We characterize saturation of the inequality, and describe the saturation condition in detail for the case of actions on Hermitian operators. Supported in part by the National Science Foundation and the Center for the Mathematics of Information.     

The credit risk+ model with general sector correlations

We consider an enhancement of the credit risk⁺ model to incorporate correlations between sectors. We model the sector default rates as linear combinations of a common set of independent variables that represent macro-economic variables or risk factors. We also derive the formula for exact VaR contributions at the obligor level.     

On the Ehrenfeucht–Mycielski sequence

We introduce the inverted prefix tries (a variation of suffix tries) as a convenient formalism for stating and proving properties of the Ehrenfeucht–Mycielski sequence [A. Ehrenfeucht, J. Mycielski, A pseudorandom sequence—how random is it? American Mathematical Monthly 99 (1992) 373-375]. We also prove an upper bound on the position in the sequence by which all strings of a given length will have appeared; our bound is given by the Ackermann function, which, in light of experimental data, may be a gross over-estimate. Still, it is the best explicitly known upper bound at the moment. Finally, we show how to compute the next bit in the sequence in a constant number of operations.     

Peters v. Roseau: Resolution of a conflict on the classical beach problem

A discrepancy is uncovered between two well documented solutions, to the uniquely solvable classical linearised sloping beach problem, by Peters in 1952 and independently by Roseau also in 1952 (though subsequently adjusted by personal correspondence from Roseau and Lehman to authors Wehausen & Laitone in 1960). The source of the main error (and other more minor errors) is identified, quantified and corrected and a consequence is that a study by Morris in 1976 (on a generalisation where the line source is no longer confined to the origin) will also have suffered from this deficiency. That problem therefore requires a re-visit.     

Ion Exchange Separation of Eu3+ from Different Salt Solutions

Two samples of potassium zinc hexacyanoferrate were prepared using two different ratios of the initial materials. The distribution coefficient of Eu³⁺ on the two samples was determined. The effect of the cation concentration in the aqueous phase on the adsorption of Eu³⁺ was also studied. The mechanism of the exchange process was described on the light of the data obtained, and the equilibrium constant of the exchange reaction was also calculated.     

Software lens compensation applied to athermalization of infrared imaging systems

A different approach, aiming to achieve the constant blur status of point-spread function (PSF) at a certain defocused plane, is described. The correlation between the two PSF is used to control the PSF blur similarity, and simultaneously the Strehl ratio is also used to control the PSF blur minimization. By designing the PSF so that it is significantly insensitive to defocus or related defocus quantity, for example, due to temperature change, all the constantly blurred images can be accurately de-blurred by a simple inverse restoration filter for an adequate range of defocus. We refer to that as “software lens compensation” and apply a design method to solve the athermalization of middle wavelength infrared (MWIR) imaging systems. The resultant PSF is almost invariant in the temperature range from −10 to 50°C at the same focal plane. Consequently, the constant blur spot can be removed by a simple digital signal processing. Thus, clear and sharp de-blurred images at different temperatures are obtained.     

Exact Analytical Solution to Equations of Perihelion Advance in General Relativity

Previously the perihelion advance in binary system was computed approximately. We will present an exact analytical solution to nonlinear differential equation of perihelion advance by method of Jacobian elliptic function and the advanced angle between successive perihelions. Project supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars from Ministry of Education, China (Grant No. [2004]527) and the Natural Science Foundation of Hunan Province, China (Grant No. 06JJ2026)