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.$

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

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

Escherichia coli RuvBL268S: A mutant RuvB protein that exhibits wild-type activities in vitro but confers a UV-sensitive ruv phenotype in vivo

The RuvABC proteins of Escherichia coli process recombination intermediates during genetic recombination and DNA repair. RuvA and RuvB promote branch migration of Holliday junctions, a process that extends heteroduplex DNA. Together with RuvC, they form a RuvABC complex capable of Holliday junction resolution. Branch migration by RuvAB is mediated by RuvB, a hexameric ring protein that acts as an ATP-driven molecular pump. To gain insight into the mechanism of branch migration, random mutations were introduced into the ruvB gene by PCR and a collection of mutant alleles were obtained. Mutation of leucine 268 to serine resulted in a severe UV-sensitive phenotype, characteristic of a ruv defect. Here, we report a biochemical analysis of the mutant protein RuvBL268S. Unexpectedly, the purified protein is fully active in vitro with regard to its ATPase, DNA binding and DNA unwinding activities. It also promotes efficient branch migration in combination with RuvA, and forms functional RuvABC-Holliday junction resolvase complexes. These results indicate that RuvB may perform some additional, and as yet undefined, function that is necessary for cell survival after UV-irradiatio

Combinatorial identities related to Eigen-function decompositions of Hill operators: open questions

We formulate three open questions related to enumerative combinatorics, which arise in the spectral analysis of Hill operators with trigonometric polynomial potentials

On Dragilev type power Köthe spaces

A complete isomorphic classification is obtained for Köthe spaces $X = K(exp[χ(p - κ (i)) - 1/p]a_i)$ such that $X qd_≃ X^2$; here χ is the characteristic function of the interval [0,∞), the function κ: ℕ → ℕ repeats its values infinitely many times, and $a_i → ∞$. Any of these spaces has the quasi-equivalence property