1,618 research outputs found
Limit operators, collective compactness, and the spectral theory of infinite matrices
In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on
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Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method
We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form where is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations
in the space of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator under consideration, with an emphasis on the function space setting . Firstly, under which conditions is a Fredholm operator, and, secondly, when is the finite section method applicable to
On the Spectra and Pseudospectra of a Class of Non-Self-Adjoint Random Matrices and Operators
In this paper we develop and apply methods for the spectral analysis of
non-self-adjoint tridiagonal infinite and finite random matrices, and for the
spectral analysis of analogous deterministic matrices which are pseudo-ergodic
in the sense of E.B.Davies (Commun. Math. Phys. 216 (2001), 687-704). As a
major application to illustrate our methods we focus on the "hopping sign
model" introduced by J.Feinberg and A.Zee (Phys. Rev. E 59 (1999), 6433-6443),
in which the main objects of study are random tridiagonal matrices which have
zeros on the main diagonal and random 's as the other entries. We
explore the relationship between spectral sets in the finite and infinite
matrix cases, and between the semi-infinite and bi-infinite matrix cases, for
example showing that the numerical range and -norm \eps-pseudospectra
(\eps>0, ) of the random finite matrices converge almost
surely to their infinite matrix counterparts, and that the finite matrix
spectra are contained in the infinite matrix spectrum . We also propose
a sequence of inclusion sets for which we show is convergent to
, with the th element of the sequence computable by calculating
smallest singular values of (large numbers of) matrices. We propose
similar convergent approximations for the 2-norm \eps-pseudospectra of the
infinite random matrices, these approximations sandwiching the infinite matrix
pseudospectra from above and below
Linear Optical Quantum Computing in a Single Spatial Mode
We present a scheme for linear optical quantum computing using time-bin
encoded qubits in a single spatial mode. We show methods for single-qubit
operations and heralded controlled phase (CPhase) gates, providing a sufficient
set of operations for universal quantum computing with the
Knill-Laflamme-Milburn scheme. Our scheme is suited to available photonic
devices and ideally allows arbitrary numbers of qubits to be encoded in the
same spatial mode, demonstrating the potential for time-frequency modes to
dramatically increase the quantum information capacity of fixed spatial
resources. As a test of our scheme, we demonstrate the first entirely single
spatial mode implementation of a two-qubit quantum gate and show its operation
with an average fidelity of 0.84+-0.07.Comment: 5 pages, 4 figures. Updated to be consistent with the published
versio
Homophily-based social group formation in a spin-glass self-assembly framework
Homophily, the tendency of humans to attract each other when sharing similar
features, traits, or opinions has been identified as one of the main driving
forces behind the formation of structured societies. Here we ask to what extent
homophily can explain the formation of social groups, particularly their size
distribution. We propose a spin-glass-inspired framework of self-assembly,
where opinions are represented as multidimensional spins that dynamically
self-assemble into groups; individuals within a group tend to share similar
opinions (intra-group homophily), and opinions between individuals belonging to
different groups tend to be different (inter-group heterophily). We compute the
associated non-trivial phase diagram by solving a self-consistency equation for
'magnetization' (combined average opinion). Below a critical temperature, there
exist two stable phases: one ordered with non-zero magnetization and large
clusters, the other disordered with zero magnetization and no clusters. The
system exhibits a first-order transition to the disordered phase. We
analytically derive the group-size distribution that successfully matches
empirical group-size distributions from online communities.Comment: 6 pages, 5 pages of SI, to appear in Phys. Rev. Let
Voluntary Wheel Running in Old C57BL/6 Mice Reduces Age-Related Inflammation in the Colon but Not in the Brain
Inflammation is considered a possible cause of cognitive decline during aging. This study investigates the influence of physical activity and social isolation in old mice on their cognitive functions and inflammation. The Barnes maze task was performed to assess spatial learning and memory in 3, 9, 15, 24, and 28 months old male C57BL/6 mice as well as following voluntary wheel running (VWR) and social isolation (SI) in 20 months old mice. Inflammatory gene expression was analyzed in hippocampal and colonic samples by qPCR. Cognitive decline occurs in mice between 15 and 24 months of age. VWR improved cognitive functions while SI had negative effects. Expression of inflammatory markers changed during aging in the hippocampus ( Il1a / Il6 / S100b / Iba1 / Adgre1 / Cd68 / Itgam ) and colon ( Tnf / Il6 / Il1ra / P2rx7 ). VWR attenuates inflammaging specifically in the colon ( Ifng / Il10 / Ccl2 / S100b / Iba1 ), while SI regulates intestinal Il1b and Gfap . Inflammatory markers in the hippocampus were not altered following VWR and SI. The main finding of our study is that both the hippocampus and colon exhibit an increase in inflammatory markers during aging, and that voluntary wheel running in old age exclusively attenuates intestinal inflammation. Based on the existence of the gut-brain axis, our results extend therapeutic approaches preserving cognitive functions in the elderly to the colon
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