On integers of the forms k−2 and k2+1

In this paper we prove that if (r,12)?3, then the set of positive odd integers k such that k^r−2ⁿ has at least two distinct prime factors for all positive integers n contains an infinite arithmetic progression. The same result corresponding to k^r2ⁿ+1 is also true.     

Crack path instabilities in DCDC experiments in the low speed regime

We studied the low speed fracture regime (10⁻⁴-10⁻⁹ m s⁻¹) in different glassy materials (soda-lime glass, glass ceramics) with variable but controlled length scale of heterogeneity. The chosen mechanical system enabled us to work in pure mode I (tensile) and at a fixed load on double cleavage drilled compression specimen. The internal residual stresses of studied samples were carefully relaxed by appropriate thermal treatment. By means of optical and atomic force microscopy techniques fracture surfaces have been examined. We have shown for the first time that the crack front line underwent an out-of-plane oscillating behavior as a result of a reproducible sequence of instabilities. The wavelength of such a phenomenon is in the micrometer range and its amplitude in the nanometer range. These features were observed for different glassy materials providing that a typical length scale characterizing internal heterogeneities was lower than a threshold limit estimated to few nanometers. This effect is the first clear experimental evidence of crack path instabilities in the low speed regime in a uniaxial loading experiment. This phenomenon has been interpreted by referring to the stability criterion for a straight crack propagation as presented by Adda-Bedia et al. [Phys. Rev. Lett. 76 (1996) 1497].     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

The q-numerical radius of weighted shift operators with periodic weights

We deal with the q-numerical radius of weighted unilateral and bilateral shift operators. In particular, the q-numerical radius of weighted shift operators with periodic weights is discussed and computed.     

The effects of O deficiency on the electronic structure of rutile TiO2

Ab initio density functional calculations (plane wave GGA, CASTEP) were performed to determine the effect of O deficiency on the electronic structure of rutile, TiO₂. O deficiency was introduced through either the removal of O or the insertion of interstitial Ti atoms. At physically realistic concentrations of O vacancies in the rutile lattice (i.e. 25% and less) O deficiency results in the population of the bottom of the conduction band, the location of the Ti 3d orbitals in the pure structure, increasingly with increasing vacancy concentration. We propose that this could be confused with the formation and population of gap states especially where O vacancies occur in isolated positions in the lattice. In contrast, Ti interstitials introduce a defect state into the energy gap, without an overall reduction in the size of the energy gap. O vacancies result in a spin polarized solution, whereas Ti interstitials do not.     

Existence of positive solutions for nonlinear three-point problems on time scales

In this paper, by using fixed point theorems in cones, we study the existence of at least one, two and three positive solutions of a nonlinear second-order three-point boundary value problem for dynamic equations on time scales. As an application, we also give some examples to demonstrate our results.     

Ambient connections realising conformal Tractor holonomy

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal Tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connection for which this is the case, and which is torsion free if we start the construction with a C-space, and in addition Ricci-flat if we start with an Einstein manifold. Thus, for a C-space this example leads to an ambient metric in the weaker sense of Čap and Gover, and for an Einstein space to a Ricci-flat ambient metric in the sense of Fefferman and Graham. Current address for first author: Erwin Schr?dinger International Institute for Mathematical Physics (ESI), Boltzmanngasse 9, 1090 Vienna, Austria Current address for second author: Department of Mathematics, University of Hamburg, Bundesstra?e 55, 20146 Hamburg, Germany     

Inherited LU-factorizations of matrices

Various types of LU-factorizations for nonsingular matrices, where L is a lower triangular matrix and U is an upper triangular matrix, are defined and characterized. These types of LU-factorizations are extended to the general m × n case. The more general conditions are considered in the light of the structures of [C.R. Johnson, D.D. Olesky, P. Van den Driessche, Inherited matrix entries: LU factorizations, SIAM J. Matrix Anal. Appl. 10 (1989) 99-104]. Applications to graphs and adjacency matrices are investigated. Conditions for the product of a lower and an upper triangular matrix to be the zero matrix are also obtained.     

Parameterization of all solutions of the three chains completion problem

The family of all solutions of the three chains completion problem with a prescribed tolerance is described explicitly. It is shown that this family can be parameterized by a natural set of contractive upper triangular operators. As an application all solutions to a suboptimal nonstationary Nehari problem are described.     

The synthesis,characterization, and weathering behavior of polycarbonates derived from bis(P-hydroxyphenyl) dimethylsilane (BPSI)

Polycarbonates and copolycarbonates based on BPSi and BPA were easily made using the standard interfacial technique. Molecular weight and NMR data indicate that up to 4 mol % of the BPSi decomposes during polymerization via a base-catalyzed rearrangement of BPSi to arylsiloxanes. Oxygen index studies show an increased flame resistance with increasing BPSi monomer content. DSC measurements show a decrease in glass transition with BPSi monomer concentration while TGA shows an increase in char residue with BPSi monomer content. In addition, TGA indicates no change in either thermal or thermooxidative stability with the relative BPSi/BPA concentrations. Outdoor weathering of BPSi PC and BPA PC films indicates that BPSi PC is ca. 5 times more weathering resistant than BPA PC. This latter result can be attributed to a combination of BPSi PC's lower UV absorptivity, higher water repellency, and lower likelihood for side chain photooxidation. © 1993 John Wiley & Sons, Inc.