Isospectral Mathieu-Hill Operators

In this paper we prove that the spectrum of the Mathieu-Hill Operators with potentials ae^{-i2{\pi}x}+be^{i2{\pi}x} and ce^{-i2{\pi}x}+de^{i2{\pi}x} are the same if and only if ab=cd, where a,b,c and d are complex numbers. This result implies some corollaries about the extension of Harrell-Avron-Simon formula. Moreover, we find explicit formulas for the eigenvalues and eigenfunctions of the t-periodic boundary value problem for the Hill operator with Gasymov's potential

Convergence Radii for Eigenvalues of Tri--diagonal Matrices

Consider a family of infinite tri--diagonal matrices of the form $L+ zB,$ where the matrix $L$ is diagonal with entries $L_{kk}= k^2,$ and the matrix $B$ is off--diagonal, with nonzero entries $B_{k,{k+1}}=B_{{k+1},k}= k^\alpha, 0 \leq \alpha < 2.$ The spectrum of $L+ zB$ is discrete. For small $|z|$ the $n$-th eigenvalue $E_n (z), E_n (0) = n^2,$ is a well--defined analytic function. Let $R_n$ be the convergence radius of its Taylor's series about $z= 0.$ It is proved that $$ R_n \leq C(\alpha) n^{2-\alpha} \quad \text{if} 0 \leq \alpha <11/6.$

Preliminary experiments for the fabrication of thermally actuated bimorph cantilever arrays on non-silicon wafers with vertical interconnects

This paper describes the first steps for the fabrication of low-cost cantilever arrays, developed at RAL, on non-silicon polymer substrates with vertical interconnects, produced at Profactor. The deflection and actuation of these cantilevers is based on the bimorph thermal actuation principle. The fabrication of the cantilever arrays requires many process steps which are presented in this article. The first step is the planarization between the via-holes interconnects with a uniform layer. This was achieved by spin coating of a thick (~58μm) SU-8 layer. In the subsequent step, two thin metal layers of Cr (500Å) and Au (1000Å) were thermally deposited and patterned, using UV lithography with a mask alignment process and wet etching. The following step was the coating of a 1μm structural Au layer, in which the deposited layer had a very poor adhesion. Alternative procedures were explored to overcome this problem in the future. Modifications of the photo masks design and the substrates will be carried out to make the RAL microcantilevers technology more compatible with Profactor substrates.Unión Europea MRTN-CT-2003- 50482

Skew-self-adjoint discrete and continuous Dirac type systems: inverse problems and Borg-Marchenko theorems

New formulas on the inverse problem for the continuous skew-self-adjoint Dirac type system are obtained. For the discrete skew-self-adjoint Dirac type system the solution of a general type inverse spectral problem is also derived in terms of the Weyl functions. The description of the Weyl functions on the interval is given. Borg-Marchenko type uniqueness theorems are derived for both discrete and continuous non-self-adjoint systems too

On the Bohr inequality

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in $\mathbb{D}.$ The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the $n$-dimensional Bohr radius

Physical protection upgrades in Ukraine.

The U.S. DOE is providing nuclear material safeguards assistance in both material control and accountability and in physical protection to several facilities in Ukraine. This paper summarizes the types of physical protection upgrades that have been or are presently being implemented at these facilities. These facilities include the Kiev Institute for Nuclear Research, Kharkov Institute of Physics and Technology, Sevastopol Institute of Nuclear Energy and Industry, and the South Ukraine Nuclear Power Plant. Typical upgrades include: hardening of storage areas; improvements in access control, intrusion detection, and CCTV assessment; central alarm station improvements; and implementation of new voice communication systems. Methods used to implement these upgrades and problems encountered are discussed. Training issues are also discussed

An abstract approach to Bohr's phenomenon

In 1914 Bohr discovered that there exists r is an element of (0, 1) such that if a power series converges in the unit disk and its sum has modulus less than 1, then for \z\ < r the sum of absolute values of its terms is again less than 1. Recently analogous results were obtained for functions of several variables. Our aim here is to present an abstract approach to the problem and show that Bohr's phenomenon occurs under very general conditions

Generalization of a theorem of Bohr for bases in spaces of holomorphic functions of several complex variables

In the first part, we generalize the classical result of Bohr by proving that an analogous phenomenon occurs whenever D is an open domain in C-m (or, more generally, a complex manifold) and (phi (n))(n=0)(infinity) is a basis in the space of holomorphic functions H(D) such that phi (0) = 1 and phi (n)(z(0)) = 0, n greater than or equal to 1, for some z(0) is an element of D. Namely, then there exists a neighborhood U of the point to such that, whenever a holomorphic function on D has modulus less than 1, the sum of the suprema in U of the moduli of the terms of its expansion is less than 1 too. In the second part we consider some natural Hilbert spaces of analytic functions and derive necessary and sufficient conditions for the occurrence of Bohr's phenomenon in this setting. (C) 2001 Academic Pres

Sub-nanosecond delays of light emitted by streamer in atmospheric pressure air: Analysis of N<SUB>2</SUB>(C<SUP>3</SUP>Pi<SUB>u</SUB>) and N<SUB>2</SUB><SUP>+</SUP>(B<SUP>2</SUP>Sigma<SUB>u</SUB><SUP>+</SUP>) emissions and fundamental streamer structure

International audienceTheoretical analysis of ultra-short phenomena occurring during the positive streamer propagation in atmospheric pressure air is presented. Motivated by experimental results obtained with tens-of picoseconds and tens-of-microns precision, it is shown that when the streamer head passes a spatial coordinate, emission maxima from N2 and N2 radiative states follow with different delays. Thesedifferent delays are caused by differences in the dynamics of populating the radiative states, due to different excitation and quenching rates. Associating the position of the streamer head with the maximum value of the self-enhanced electric field, a delay of 160 ps was experimentally found for the peak emission of the first negative system of N2 . A delay dilatation was observed experimentally on early-stage streamers and the general mechanism of this phenomenon is clarified theoretically. In the case of the second positive system of N2, the delay can reach as much as 400 ps. In contrast to the highly nonlinear behavior of streamer events, it is shown theoretically that emission maximum delays linearly depend on the ratio of the streamer radius and its velocity. This is found to be one of the fundamental streamer features and its use in streamer head diagnostics is proposed. Moreover,radially resolved spectra are synthesized for selected subsequent picosecond moments in order to visualize spectrometric fingerprints of radial structures of N2(C3Piu) and N2 (B2Sigma u) populations created by streamer-head electron