70 research outputs found
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
where the matrix is diagonal with entries and the matrix
is off--diagonal, with nonzero entries The spectrum of is discrete. For small the
-th eigenvalue is a well--defined analytic
function. Let be the convergence radius of its Taylor's series about 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 (500Ã…) and Au (1000Ã…) 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 , , such that holds whenever in the unit disk
of the complex plane. The exact value of this largest radius,
known as the \emph{Bohr radius}, has been established to be This paper
surveys recent advances and generalizations on the Bohr inequality. It
discusses the Bohr radius for certain power series in as well as
for analytic functions from 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 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 -dimensional Bohr radius
Multirectangular invariants for power Köthe spaces
Using some new linear topological invariants, isomorphisms and quasidiagonal isomorphisms are investigated on the class of first type power Köthe spaces [Proceedings of 7th Winter School in Drogobych, 1976, pp. 101–126; Turkish J. Math. 20 (1996) 237–289; Linear Topol. Spaces Complex Anal. 2 (1995) 35–44]. This is the smallest class of Köthe spaces containing all Cartesian and projective tensor products of power series spaces and closed with respect to taking of basic subspaces (closed linear hulls of subsets of the canonical basis). As an application, it is shown that isomorphic spaces from this class have, up to quasidiagonal isomorphisms, the same basic subspaces of finite (infinite) type
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 such that ; here χ is the characteristic function of the interval [0,∞), the function κ: ℕ → ℕ repeats its values infinitely many times, and . Any of these spaces has the quasi-equivalence property
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