Essential spectrum of systems of singular differential equations

In this paper we develop a new method to determine the essential spectrum of coupled systems of singular differential equations. Applications to problems from magnetohydrodynamics and astrophysics are given.

Riemannian Submersions with Discrete Spectrum

We prove some estimates on the spectrum of the Laplacian of the total space of a Riemannian submersion in terms of the spectrum of the Laplacian of the base and the geometry of the fibers. When the fibers of the submersions are compact and minimal, we prove that the spectrum of the Laplacian of the total space is discrete if and only if the spectrum of the Laplacian of the base is discrete. When the fibers are not minimal, we prove a discreteness criterion for the total space in terms of the relative growth of the mean curvature of the fibers and the mean curvature of the geodesic spheres in the base. We discuss in particular the case of warped products.

On the generalized Kato spectrum

2010 Mathematics Subject Classification: 47A10.

Butterfly catastrophe for fronts in a three-component reaction–diffusion system

We study the dynamics of front solutions in a three-component reaction–diffusion system via a combination of geometric singular perturbation theory, Evans function analysis, and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numerically observed accelerations and oscillations and pave the way for the analysis of front interactions in a parameter regime where the essential spectrum of a single front approaches the imaginary axis asymptotically.

Spectral Analysis of Bounded Self-adjoint operators - A Linear Algebraic Approach

This thesis Entitled Spectral theory of bounded self-adjoint operators -A linear algebraic approach.The main results of the thesis can be classified as three different approaches to the spectral approximation problems. The truncation method and its perturbed versions are part of the classical linear algebraic approach to the subject. The usage of block Toeplitz-Laurent operators and the matrix valued symbols is considered as a particular example where the linear algebraic techniques are effective in simplifying problems in inverse spectral theory. The abstract approach to the spectral approximation problems via pre-conditioners and Korovkin-type theorems is an attempt to make the computations involved, well conditioned. However, in all these approaches, linear algebra comes as the central object. The objective of this study is to discuss the linear algebraic techniques in the spectral theory of bounded self-adjoint operators on a separable Hilbert space. The usage of truncation method in approximating the bounds of essential spectrum and the discrete spectral values outside these bounds is well known. The spectral gap prediction and related results was proved in the second chapter. The discrete versions of Borg-type theorems, proved in the third chapter, partly overlap with some known results in operator theory. The pure linear algebraic approach is the main novelty of the results proved here.

Sufficiency of Favard's condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on $\ell^\infty(Z,U)$, with $U$ being a complex Banach space, the injectivity of all limit operators of $A$ already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of $A$, which, on the other hand, is often equivalent to the Fredholmness of $A$. As a consequence, for operators $A$ in the Wiener algebra, we can characterize the essential spectrum of $A$ on $\ell^p(Z,U)$, regardless of $p\in[1,\infty]$, as the union of point spectra of its limit operators considered as acting on $\ell^p(Z,U)$.

Spectral bounds and basis results for non-self-adjoint pencils, with an application to Hagen-Poiseuille flow

We obtain eigenvalue enclosures and basisness results for eigen- and associated functions of a non-self-adjoint unbounded linear operator pencil A−λBA−λB in which BB is uniformly positive and the essential spectrum of the pencil is empty. Both Riesz basisness and Bari basisness results are obtained. The results are applied to a system of singular differential equations arising in the study of Hagen–Poiseuille flow with non-axisymmetric disturbances.

SG Pseudo-Differential Operators and Weak Hyperbolicity

2002 Mathematics Subject Classification: 35S05, 47G30, 58J42.

La metrica di Agmon ed il decadimento esponenziale delle soluzioni di equazioni alle derivate parziali

The aim of this master thesis is to study the exponential decay of solutions of elliptic partial equations. This work is based on the results obtained by Agmon. To this purpose, first, we define the Agmon metric, that plays an important role in the study of exponential decay, because it is related to the rate of decay. Under some assumptions on the growth of the function and on the positivity of the quadratic form associated to the operator, a first result of exponential decay is presented. This result is then applied to show the exponential decay of eigenfunctions with eigenvalues whose real part lies below the bottom of the essential spectrum. Finally, three examples are given: the harmonic oscillator, the hydrogen atom and a Schrödinger operator with purely discrete spectrum.

Generation of broad-spectrum disease resistance by overexpression of an essential regulatory gene in systemic acquired resistance

The recently cloned NPR1 gene of Arabidopsis thaliana is a key regulator of acquired resistance responses. Upon induction, NPR1 expression is elevated and the NPR1 protein is activated, in turn inducing expression of a battery of downstream pathogenesis-related genes. In this study, we found that NPR1 confers resistance to the pathogens Pseudomonas syringae and Peronospora parasitica in a dosage-dependent fashion. Overexpression of NPR1 leads to enhanced resistance with no obvious detrimental effect on the plants. Thus, for the first time, a single gene is shown to be a workable target for genetic engineering of nonspecific resistance in plants.

DSM-5 Autism Spectrum Disorder:in search of essential behaviours for diagnosis

Background: Despite initial concerns about the sensitivity of the proposed diagnostic criteria for DSM-5 Autism Spectrum Disorder (ASD; e.g. Gibbs et al., 2012; McPartland et al., 2012), evidence is growing that the DSM-5 criteria provides an inclusive description with both good sensitivity and specificity (e.g. Frazier et al., 2012; Kent, Carrington et al., 2013). The capacity of the criteria to provide high levels of sensitivity and specificity comparable with DSM-IV-TR however relies on careful measurement to ensure that appropriate items from diagnostic instruments map onto the new DSM-5 descriptions.Objectives: To use an existing DSM-5 diagnostic algorithm (Kent, Carrington et .al., 2013) to identify a set of ‘essential’ behaviors sufficient to make a reliable and accurate diagnosis of DSM-5 Autism Spectrum Disorder (ASD) across age and ability level. Methods: Specific behaviors were identified and tested from the recently published DSM-5 algorithm for the Diagnostic Interview for Social and Communication Disorders (DISCO). Analyses were run on existing DISCO datasets, with a total participant sample size of 335. Three studies provided step-by-step development towards identification of a minimum set of items. Study 1 identified the most highly discriminating items (p<.0001). Study 2 used a lower selection threshold than in Study 1 (p<.05) to facilitate better representation of the full DSM-5 ASD profile. Study 3 included additional items previously reported as significantly more frequent in individuals with higher ability. The discriminant validity of all three item sets was tested using Receiver Operating Characteristic curves. Finally, sensitivity across age and ability was investigated in a subset of individuals with ASD (n=190).Results: Study 1 identified an item set (14 items) with good discriminant validity, but which predominantly measured social-communication behaviors (11/14). The Study 2 item set (48 items) better represented the DSM-5 ASD and had good discriminant validity, but the item set lacked sensitivity for individuals with higher ability. The final Study 3 adjusted item set (54 items) improved sensitivity for individuals with higher ability and performance and was comparable to the published DISCO DSM-5 algorithm.Conclusions: This work represents a first attempt to derive a reduced set of behaviors for DSM-5 directly from an existing standardized ASD developmental history interview. Further work involving existing ASD diagnostic tools with community-based and well characterized research samples will be required to replicate these findings and exploit their potential to contribute to a more efficient and focused ASD diagnostic process.