21 research outputs found
Spectra of definite type in waveguide models
We develop an abstract method to identify spectral points of definite type in
the spectrum of the operator . The method is
applicable in particular for non-self-adjoint waveguide type operators with
symmetries. Using the remarkable properties of the spectral points of definite
type, we obtain new results on realness of weakly coupled bound states and of
low lying essential spectrum in the -symmetric
waveguide. Moreover, we show that the pseudospectrum has a normal tame behavior
near the low lying essential spectrum and exclude the accumulation of non-real
eigenvalues to this part of the essential spectrum. The advantage of our
approach is particularly visible when the resolvent of the unperturbed operator
cannot be explicitly expressed and most of the mentioned spectral conclusions
are extremely hard to prove using direct methods.Comment: 15 pages, 4 figures, submitte
Spectral bounds for infinite dimensional polydiagonal symmetric matrix operators on discrete spaces
In this thesis, we prove a variety of discrete Agmon Kolmogorov inequalities and apply them to prove Lieb Thirring inequalities for discrete Schrodinger operators on ℓ[superscript 2](ℤ). We generalise these results in two ways: Firstly, to higher order difference operators, leading to spectral bounds for Tri-, Penta- and Polydiagonal Jacobi-type matrix operators. Secondly, to ℓ[superscript 2]-spaces on higher dimensional domains, specifically on ℓ[superscript 2](ℤ[superscript 2]), ℓ[superscript 2](ℤ[superscript 3]) and finally ℓ[superscript 2](ℤ[superscript d]).
In the Introduction we discuss previous work on Landau Kolmogorov inequalities on a variety of Banach Spaces, Lieb Thirring inequalities in ℓ[superscript 2](ℝ[superscript d]), and the use of Jacobi Matrices in relation to the discrete Schrodinger Operator. We additionally give our main results with some introduction to the notation at hand.
Chapters 2, 3 and 4 follow a similar structure.
We first introduce the relevant difference operators and examine their properties. We then move on to prove the Agmon Kolmogorov and Generalised Sobolev inequalities over ℤ of order 1, 2 and σ respectively. Furthermore, we prove the Lieb Thirring inequality for the respective discrete Schrodinger-type operators, which we subsequently lift to arbitrary moments. Finally we apply this inequality to obtain spectral bounds for tri-, penta- and polydiagonal matrices.
In Chapter 5, we prove a variety of Agmon Kolmogorov inequalities on ℓ[superscript 2](ℤ[superscript 2]) and ℓ[superscript 2](ℤ[superscript 3]). We use these intuitive ideas to obtain 2[superscript d-1] Agmon Kolmogorov inequalities on ℓ[superscript 2](ℤ[superscript d]). We continue from here in the same manner as before and prove the discrete Generalised Sobolev and Lieb Thirring inequalities for a variety of exponent combinations on ℓ[superscript 2](ℤ[superscript d]).Open Acces
Permutation of elements in double semigroups
Double semigroups have two associative operations related by
the interchange relation: . Kock \cite{Kock2007} (2007) discovered a
commutativity property in degree 16 for double semigroups: associativity and
the interchange relation combine to produce permutations of elements. We show
that such properties can be expressed in terms of cycles in directed graphs
with edges labelled by permutations. We use computer algebra to show that 9 is
the lowest degree for which commutativity occurs, and we give self-contained
proofs of the commutativity properties in degree 9.Comment: 24 pages, 11 figures, 4 tables. Final version accepted by Semigroup
Forum on 12 March 201
}T$.
The spectral properties of two products AB and BA of possibly unbounded operators A and B in a Banach space are considered. The results are applied in the comparison of local spectral properties of the operators
Generation of Neuronal Trees by a New Three Letters Encoding
A neuronal tree is a rooted tree with n leaves whose each internal node has at least two children; this class not only is defined based on the structure of dendrites in neurons, but also refers to phylogenetic trees or evolutionary trees. More precisely, neuronal trees are rooted-multistate phylogenetic trees whose size is defined as the number of leaves. In this paper, a new encoding over an alphabet of size 3 (minimal cardinality) is introduced for representing the neuronal trees with a given number of leaves. This encoding is used for generating neuronal trees with n leaves in A-order with constant average time and O(n) time complexity in the worst case. Also, new ranking and unranking algorithms are presented in time complexity of O(n) and O(n log n), respectively