Fixed point theorems in <Emphasis Type="Italic">b</Emphasis>-metric spaces with applications to differential equations

In this paper, we present some fixed point theorems for a class of contractive mappings in b-metric spaces. We verify the T-stability of Picard’s iteration and the P property for such mappings. We also give an example to support our assertions. In addition, by using our results, we obtain the existence and uniqueness of solution to some ordinary differential equations with initial value conditions. Further, we provide the precise mathematical expressions of solutions to such equations.     

A method of proving non-unitarity of representations of p-adic groups I

In this paper we exhibit a new method of proving non-unitarity of representations, based on semi simplicity of unitarizable representations. Non-unitarity is proved for a half of all irreducible representations of classical p-adic groups whose infinitesimal character is the same as the infinitesimal character of a generalized Steinberg representation (as defined in Tadić, Am J Math 120:159–210, 1998). Only the Steinberg representation and its Aubert dual are expected to be unitary here. In this way we partially generalize a result of Casselman to the case of classical groups. Our argument is completely different from Casselman’s argument (which is hard to extend to this case). It requires a very limited knowledge of the inducing cuspidal representation.     

Some notes on the threshold graphs

In this paper we consider threshold graphs (also called nested split graphs) and investigate some invariants of these graphs which can be of interest in bounding the largest eigenvalue of some graph spectra.     

Cauchy’s residue theorem for a class of real valued functions

Let [a, b] be an interval in ℝ Rand let F be a real valued function defined at the endpoints of [a, b] and with a certain number of discontinuities within [a, b]. Assuming F to be differentiable on a set [a, b] | E to the derivative f, where E is a subset of [a, b] at whose points F can take values ±∞ or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that KH-vt ∝ _a ^b f = F(b) − F(a), where KH-vt denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate the theory.     

Repeated newsvendor game with transshipments under dual allocations

We study a repeated newsvendor game with transshipments. In every period n retailers face a stochastic demand for an identical product and independently place their inventory orders before demand realization. After observing the actual demand, each retailer decides how much of her leftover inventory or unsatisfied demand she wants to share with the other retailers. Residual inventories are then transshipped in order to meet residual demands, and dual allocations are used to distribute residual profit. Unsold inventories are salvaged at the end of the period. While in a single-shot game retailers in an equilibrium withhold their residuals, we show that it is a subgame-perfect Nash equilibrium for the retailers to share all of the residuals when the discount factor is large enough and the game is repeated infinitely many times. We also study asymptotic behavior of the retailers’ order quantities and discount factors when n is large. Finally, we provide conditions under which a system-optimal solution can be achieved in a game with n retailers, and develop a contract for achieving a system-optimal outcome when these conditions are not satisfied.     

Kinetic analysis of the reaction between (V2O5)n=1,2+ and ethylene

A systematic experimental and theoretical investigation of the influence of reactant energy on the reactivity of (V(2)O(5))(n)=1,2(+) clusters with ethylene (Justes, D. R.; Mitri?, R.; Moore, N. A.; Bonaci?-Koutecky, V.; Castleman, A. W., Jr. J. Am. Chem. Soc., 2003, 125, 6289) provided evidence of the rate controlling steps in the reaction. Herein, we present further experimental and theoretical evidence for our recently proposed radical cation mechanism for oxygen atom transfer from (V(2)O(5))(n)=1,2(+) clusters to ethylene. In particular the results of ab initio molecular dynamics simulations are found to further support the radical cation mechanism. Experimental reaction cross sections at the zero pressure limit and rate coefficients show that the energy dependence of the reaction cross section is in accord with the Langevin formula. Evidence is presented that ion-molecule association is the rate determining step, whereas subsequent hydrogen transfer and formation of acetaldehyde proceed without significant barriers. We propose a kinetic model for the reaction cross section that fully accounts for the experimental findings. The model offers the prospect of elucidating the details of the general reaction mechanisms through a study of the energy dependence of the reaction cross sections.     

Hydrogen molecule in the small dodecahedral cage of a clathrate hydrate: quantum five-dimensional calculations of the coupled translation-rotation eigenstates

We report quantum five-dimensional (5D) calculations of the energy levels and wave functions of the hydrogen molecule, para-H2 and ortho-H2, confined inside the small dodecahedral (H2O)20 cage of the sII clathrate hydrate. All three translational and the two rotational degrees of freedom of H2 are included explicitly, as fully coupled, while the cage is treated as rigid. The 5D potential energy surface (PES) of the H2-cage system is pairwise additive, based on the high-quality ab initio 5D (rigid monomer) PES for the H2-H2O complex. The bound state calculations involve no dynamical approximations and provide an accurate picture of the quantum 5D translation-rotation dynamics of H2 inside the cage. The energy levels are assigned with translational (Cartesian) and rotational quantum numbers, based on calculated root-mean-square displacements and probability density plots. The translational modes exhibit negative anharmonicity. It is found that j is a good rotational quantum number, while the threefold degeneracy of the j = 1 level is lifted completely. There is considerable translation-rotation coupling, particularly for excited translational states.     

Defects in ZnO nanorods prepared by a hydrothermal method

ZnO nanorod arrays were fabricated using a hydrothermal method. The nanorods were studied by scanning electron microscopy, photoluminescence (PL), time-resolved PL, X-ray photoelectron spectroscopy, and positron annihilation spectroscopy before and after annealing in different environments and at different temperatures. Annealing atmosphere and temperature had significant effects on the PL spectrum, while in all cases the positron diffusion length and PL decay times were increased. We found that, while the defect emission can be significantly reduced by annealing at 200 degrees C, the rods still have large defect concentrations as confirmed by their low positron diffusion length and short PL decay time constants.