Two results in metric fixed point theory

We establish two fixed point theorems for certain mappings of contractive type. The first result is concerned with the case where such mappings take a nonempty, closed subset of a complete metric space X into X, and the second with an application of the continuation method to the case where they satisfy the Leray–Schauder boundary condition in Banach spaces.     

On Ovoids of Parabolic Quadrics

It is known that every ovoid of the parabolic quadric Q(4, q), q=p ^h , p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p=2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points. We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q>3. We conclude with a 1 mod p result for ovoids of Q(6, q), q=p ^h , p prime.     

Electronic and ESR Spectra of Copper(II) Complexes with N,N-Dialkyl Amino Acids

Five N,N-dialkylamino acids were synthesised [4]: N,N-dimethylisoleucine (DMIL), N,N-dimethyl-alloisoleucine (DMAIL), N,N-dimethylvaline (DMV), N,N-diethyl-alanine (DEA), and N,N-diethylvaline (DEV), as well as their complexes with copper(II): Cu(DMIL)₂, Cu(DMAIL)₂, Cu(DMV)₂, Cu(DMIL)(DMV), Cu(DEA)₂, and Cu(DEV)₂. UV/VIS spectra of the complexes were recorded in CH₂Cl₂ solutions and the ESR spectra in CH₂Cl₂, CD₃OD, and D₂O solutions. By applying the analysis of variance, ESR spectra were shown to be influenced by the solvent and substituent differences, the former effect being related to the degree of covalency of the copper-ligand bonds. The effect of the ligand's spatial configuration (diastereoisomerism) upon the ESR spectrum of the complex was demonstrated and explained.     

Numerical experiments on free-electron lasers without inversion

The inversionless free-electron laser having a drift region consisting of two magnets is analyzed. Performing numerical simulations of electron motion inside wigglers and the drift region, we have shown that this system has a positive mean gain over the entire energy distribution of the electron beam. We study the influence of emittance and the spread of electron energies on the gain.     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

Optimization of top polymer gratings to improve GaN LEDs light transmission

We present a grating model of two-dimensional (2D) rigorous coupled wave analysis (RCWA) to study top diffraction gratings on light-emitting diodes(LEDs). We compare the integrated-transmission of the non-grating,rectangular-grating,and triangular-grating cases for the same grating period of 6μm,and show that the triangular grating has the best performance. For the triangular grating with 6-μmperiod, the LED achieves the highest light transmission at 6-μ gratingbottom width and 2.9-μm grating depth. Compared with the non-grating case, the optimized light transmission improvement is about 74.6%.The simulation agrees with the experimental data of the thin ploymer grating encapsulated flip-chip(FC) GaN-based LEDs for the light extraction improvement.