On construction ofk-wise independent random variables

A 0–1probability space is a probability space (, 2,P), where the sample space -{0, 1} ⁿ for somen. A probability space isk-wise independent if, whenY _i is defined to be theith coordinate or the randomn-vector, then any subset ofk of theY _i 's is (mutually) independent, and it is said to be a probability spacefor p ₁,p ₂, ...,p _n ifP[Y _i =1]=p _i .We study constructions ofk-wise independent 0–1 probability spaces in which thep _i 's are arbitrary. It was known that for anyp ₁,p ₂, ...,p _n , ak-wise independent probability space of size always exists. We prove that for somep ₁,p ₂, ...,p _n [0,1],m(n,k) is a lower bound on the size of anyk-wise independent 0–1 probability space. For each fixedk, we prove that everyk-wise independent 0–1 probability space when eachp _i =k/n has size (n _k ). For a very large degree of independence —k=[n], for >1/2- and allp _i =1/2, we prove a lower bound on the size of . We also give explicit constructions ofk-wise independent 0–1 probability spaces.This author was supported in part by NSF grant CCR 9107349.This research was supported in part by the Israel Science Foundation administered by the lsrael Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology.     

Boundary regularity ofp-harmonic maps with free and partially constrained boundary conditions

We study the boundary regularity ofp-harmonic maps with free and partially constrained boundary conditions and give estimates on the size of the singular subset of the boundary.     

A continuity property of multivariate Lagrange interpolation

Let be a sequence of interpolation schemes in of degree (i.e. for each one has unique interpolation by a polynomial of total degree and total order . Suppose that the points of tend to as and the Lagrange-Hermite interpolants, , satisfy for all monomials with . Theorem: for all functions of class in a neighborhood of . (Here denotes the Taylor series of at 0 to order .) Specific examples are given to show the optimality of this result.

     

Existence of solutions to BVP for linear differential-functional equations of elliptic type

We consider a homogeneous boundary value problem for a linear partial differential-functional equation of elliptic type. We prove the existence of a unique solution to such problem in a Sobolev space.     

Er-doped hydrogenated amorphous silicon: structural and optical properties

The effect of erbium-doping on the structural and optical properties of hydrogenated amorphous silicon (a-Si:H) is investigated. Optical absorption and Raman spectra indicate that erbium doping introduces defect states, and that above a concentration of 0.27 at.%, induces strong structural disorder. The photoluminescence measurements show that erbium doping introduces non-radiative decay paths for carriers in a-Si:H, leading to decrease in both the Er³⁺ and intrinsic a-Si:H luminescence intensity when the Er concentration is increased to more than 0.04 at.%. The results are compared to that of Er-doped crystalline Si, and the possible excitation mechanisms of Er in a-Si:H are discussed.     

Enhanced frequency upconversion in Er doped fluoroindate glass due to energy transfer from Tm

Energy transfer processes between Er³⁺ and Tm³⁺ were investigated examining the frequency upconversion emissions in a fluoroindate glass pumped at 790 nm. A 60-fold enhancement in the emission at ≈670 nm originating from Er³⁺ was observed when Tm³⁺ at concentration of 2% was introduced in a sample containing 2% of Er³⁺. The results are explained considering the influence of cross-relaxation processes between the active ions.     

Concentration quenching of fluorescence from the (P0 + P1) manifold in heavily-doped Pr: ZBAN glasses

Fluorescence waveforms from the (³P₀ + ³P₁) manifold in Pr³⁺ doped ZBAN glass at wavelengths of 520, 635 and 695 nm were measured for Pr³⁺ concentrations from 4 to 12 mol%. The waveforms were found to be non-exponential with decay rates rapidly increasing with Pr³⁺ concentration and independent of whether the ³P₀ or the ³P₁ level was excited. The multipolar energy transfer model was used to analyse the waveforms and this showed that concentration quenching was due to cross-relaxation by dipole-dipole interaction. The critical concentration, at which the cross-relaxation rate equals the intrinsic decay rate, was found to be of 2.06 × 10²⁶ m⁻³ (1.20 mol%). There was no evidence of excitation diffusion for Pr³⁺ concentrations of up to 12 mol%.     

Nepetanudone, a dimer of the α-pyrone 5,9-dehydronepetalactone

The title compound C₂₀H₂₄O₄ crystallizes in the orthorhombic space group P2₁2₁2, with a = 10.245(3), b = 17.103(3), c = 9.879(3) Å, D _c = 1.260 g cm ^–3, and Z = 4. Nepetanudone is postulated to be formed by a [4+4] cycloaddition of two 5,9-dehydronepetalactone moieties to give an asymmetric dimer in which each pair of atoms is magnetically nonequivalent. The compound was recovered from extracts of Nepeta nuda ssp. albiflora.     

Positron annihilation lifetime study of organic-inorganic hybrid materials prepared by irradiation

Silica based hybrid materials, some containing zirconia, prepared by gamma-irradiation, were studied by positron annihilation lifetime (PAL) spectroscopy, in the temperature range of 30-370 K. One long-lived component was observed in group of samples without zirconia. The behavior of this component as a function of temperature resembles that for polymers. This is explained by the elastomer like structure of the samples and the excellent linking between the inorganic and organic networks. The average radius, R, of the free-volume holes was in the range 0.26-0.42 nm depending on the temperature. In the group of samples containing zirconia two long-lived components were observed. A model with three different free-volume holes was suggested to explain the PAL results. Small closed holes (R ∼ 0.26 nm) and large closed holes introduced by Zr (R ∼ 0.5 nm), the sizes of both holes not changing with the temperature, together with intermediate polymer-like free-volume holes - the same behavior as those present in samples not containing zirconia.