On the norm of polynomials of two adjoint projections

Letf(X, Y) be a polynomial of two non-commuting variables and letP be an arbitrary nontrivial projection operator in Hilbert space. The class of all polynomialsf(X, Y) for which f(P, P ^*) depends only onf and P are described. In the case when such a dependence exists the explicit formula is obtained. Some applications to singular integral operators are given.     

Constructing convex solutions via Perron’s method

Abstract In this paper we construct convex solutions for certain elliptic boundary value problems via Perron’s method. The solutions constructed are weak solutions in the viscosity sense, and our construction follows work of Ishii (Duke Math. J., 55 (2) 369–384, 1987). The same general approach appears in work of Andrews and Feldman (J. Differential Equations, 182 (2) 298–343, 2002) in which they show existence for a weak nonlocal parabolic flow of convex curves. The time independent special case of their work leads to a one dimensional elliptic result which we extend to two dimensions. Similar results are required to extend their theory of nonlocal geometric flows to surfaces. The two dimensional case is essentially different from the one dimensional case and involves a regularity result (cf. Theorem 3.1), which has independent interest. Roughly speaking, given an arbitrary convex function (which is not smooth) supported at one point by a smooth function of prescribed Hessian (which is not convex), one must construct a third function that is both convex and smooth and appropriately approximates both of the given functions. Keywords: Viscosity solutions, Elliptic partial differential equations, Perron procedure, Convexity, Regularity, Fully nonlinear, Monge-Ampere Mathematics Subject Classification (2000:) 35J60, 53A05, 52A15, 26B05     

Radiation chemistry of polymers

Results of basic and applied research in radiation chemistry of polymers in Russia and Belarus for the last 5–10 years are presented.     

Diffusion-filtration model of methane escape from a coal seam

Methane desorption from a coal seam is theoretically investigated using a model including both the diffusion of methane in coal lumps and its filtration through net-shaped pores and cracks. The methane density distribution along the seam at an arbitrary time instant is found. Explicit dependences of the amount of the methane escaped from the seam on the lump size, open and closed porosity, viscosity and solubility of methane, and pressure and temperature in the seam are determined. An effective diffusion coefficient in lumps containing methane-filled closed pores is found. In the case of hindered diffusion, the methane can be subdivided into the “fast” and “slow” fractions.     

Hypercyclic tuples of operators and somewhere dense orbits

In this paper we prove that there are hypercyclic (n+1)-tuples of diagonal matrices on Cn and that there are no hypercyclic n-tuples of diagonalizable matrices on Cn. We use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions. We then show that on real Hilbert spaces there are tuples with somewhere dense orbits that are not dense, but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit, on a real or complex space, must be dense.     

A Functional Integral Representation for Many Boson Systems II: Correlation Functions

We derive functional integral representations for the partition function and correlation functions of many Boson systems for which the configuration space consists of finitely many points. Submitted: August 20, 2007. Accepted: July 1, 2008.     

Rapid tarnishing of silver nanoparticles in ambient laboratory air

Silver has useful surface-plasmon-resonance properties for many potential applications. However, chemical activity in silver nanoparticles exposed to laboratory air can make interpretation of optical scattering and extinction spectra problematic. We have measured the shift of the plasmon polariton wavelength of arrays of silver nanoparticles with increasing exposure to ambient laboratory air. The resonance peak wavelength shifts 65 nm in 36 h (1.8 nm/h). We show by scanning Auger spectroscopy that the shift is due to contamination from sulfur, most likely chemisorbed on the surface. The rate of corrosion product growth on the nanoparticles is estimated to be 3 nm per day, 7.5 times higher than that of bulk Ag under the same conditions.     

Effects of temperature and oxygen pressure on binary oxide growth using aperture-controlled combinatorial pulsed-laser deposition

In pulsed-laser deposition (PLD), there are many processing parameters that influence film properties such as substrate-target distance, background reactive gas pressure, laser energy, substrate temperature and composition in multi-component systems. By introducing a 12.7-mm diameter circular aperture in front of a 76.2-mm silicon wafer and rotating the substrate while changing conditions during the PLD process, these parameters may be studied in a combinatorial fashion, discretely as a function of processing conditions. We demonstrate the use of the aperture technique to systematically study the effects of oxygen partial pressure on the film stoichiometry and growth rate of VO_x, using Rutherford backscattering spectrometry (RBS). In another example, we discuss the effect of growth temperature on TiO₂ films characterized by X-ray diffraction and Fourier transform far-infrared (Terahertz) absorption spectroscopy. We demonstrate that we have considerable combinatorial control of other processing variables besides composition in our combi-PLD system. These may be used to systematically study film growth and properties.     

Nonlinear Carleman operators on Banach lattices

An operator, not necessarily linear, will be called a Carleman operator if the image of the positive elements in the unit ball are bounded in the universal completion of the range space. For certain Banach lattices, a class of (not necessarily linear) Carleman operators is characterized in terms of an integral representation and in a more general setting as operators satisfying a pointwise finiteness condition. These operators though not linear are orthogonally additive and monotone.