Search results for "Eigenvector"
showing 10 items of 303 documents
Property (R) under perturbations
2012
Property (R) holds for a bounded linear operator $${T \in L(X)}$$ , defined on a complex infinite dimensional Banach space X, if the isolated points of the spectrum of T which are eigenvalues of finite multiplicity are exactly those points λ of the approximate point spectrum for which λI − T is upper semi-Browder. In this paper we consider the permanence of this property under quasi nilpotent, Riesz, or algebraic perturbations commuting with T.
The Fine Spectre of Some Cesàro Generalized Operators Defined onℓp(p> 1)
2004
Abstract The aim of the paper is the study of the fine spectre for a class of Cesaro generalized operators, Rhaly operators, when those operators are defined on the spaces lp, p > 1.
Quantum computing thanks to Bianchi groups
2018
It has been shown that the concept of a magic state (in universal quantum computing: uqc) and that of a minimal informationally complete positive operator valued measure: MIC-POVMs (in quantum measurements) are in good agreement when such a magic state is selected in the set of non-stabilizer eigenstates of permutation gates with the Pauli group acting on it [1]. Further work observed that most found low-dimensional MICs may be built from subgroups of the modular group PS L(2, Z) [2] and that this can be understood from the picture of the trefoil knot and related 3-manifolds [3]. Here one concentrates on Bianchi groups PS L(2, O10) (with O10 the integer ring over the imaginary quadratic fie…
Weak versus strong dominance of shrinkage estimators
2021
We consider the estimation of the mean of a multivariate normal distribution with known variance. Most studies consider the risk of competing estimators, that is the trace of the mean squared error matrix. In contrast we consider the whole mean squared error matrix, in particular its eigenvalues. We prove that there are only two distinct eigenvalues and apply our findings to the James–Stein and the Thompson class of estimators. It turns out that the famous Stein paradox is no longer a paradox when we consider the whole mean squared error matrix rather than only its trace.
Fabric attractors in general triclinic flow systems and their application to high strain shear zones: A dynamical system approach
2007
High strain zones may deform by flow with a triclinic symmetry. This paper describes triclinic flow in a reference frame where Instantaneous Stretching Axes (ISA) are fixed. The operation of triclinic flow is described in two ways: first in terms of flow and the nature of flow eigenvectors and in the second part of the paper in terms of finite strain. In monoclinic flow, at least one of the eigenvectors of the flow coincides with one of the ISA and one or two of the eigenvectors act as attractors of foliation or lineation elements. In triclinic flow some flow eigenvectors are undefined since the two largest eigenvalues (controlling the flow) are imaginary. Imaginary eigenvalues are particul…
Biorthonormal-basis method for the vector description of optical-fiber modes
1998
This paper gives the theoretical basis for the development of real vector modal methods to describe optical-fiber modes. To this end, the vector wave equations, which determine the electromagnetic fields, are written in terms of a pair of linear, nonself-adjoint operators, whose eigenvectors satisfy biorthogonality relations. The key of our method is to obtain a matrix representation of the vector wave equations in a basis that is defined by the modes of an auxiliary system. Our proposed technique can be applied to fibers with any profile, even those with a complex refractive index. An example is discussed to illustrate our approach.
Kernel methods and their derivatives: Concept and perspectives for the earth system sciences.
2020
Kernel methods are powerful machine learning techniques which implement generic non-linear functions to solve complex tasks in a simple way. They Have a solid mathematical background and exhibit excellent performance in practice. However, kernel machines are still considered black-box models as the feature mapping is not directly accessible and difficult to interpret.The aim of this work is to show that it is indeed possible to interpret the functions learned by various kernel methods is intuitive despite their complexity. Specifically, we show that derivatives of these functions have a simple mathematical formulation, are easy to compute, and can be applied to many different problems. We n…
Fast Graph Filters for Decentralized Subspace Projection
2020
A number of inference problems with sensor networks involve projecting a measured signal onto a given subspace. In existing decentralized approaches, sensors communicate with their local neighbors to obtain a sequence of iterates that asymptotically converges to the desired projection. In contrast, the present paper develops methods that produce these projections in a finite and approximately minimal number of iterations. Building upon tools from graph signal processing, the problem is cast as the design of a graph filter which, in turn, is reduced to the design of a suitable graph shift operator. Exploiting the eigenstructure of the projection and shift matrices leads to an objective whose…
Asymptotic and bootstrap tests for subspace dimension
2022
Most linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices, see e.g. Ye and Weiss (2003), Tyler et al. (2009), Bura and Yang (2011), Liski et al. (2014) and Luo and Li (2016). The eigen-decomposition of one scatter matrix with respect to another is then often used to determine the dimension of the signal subspace and to separate signal and noise parts of the data. Three popular dimension reduction methods, namely principal component analysis (PCA), fourth order blind identification (FOBI) and sliced inverse regression (SIR) are considered in detail and the first two moments of subsets of the eigenvalues are used to test…
‘Too interconnected to fail’ financial network of US CDS market: Topological fragility and systemic risk
2012
A small segment of credit default swaps (CDS) on residential mortgage backed securities (RMBS) stand implicated in the 2007 financial crisis. The dominance of a few big players in the chains of insurance and reinsurance for CDS credit risk mitigation for banks' assets has led to the idea of too interconnected to fail (TITF) resulting, as in the case of AIG, of a tax payer bailout. We provide an empirical reconstruction of the US CDS network based on the FDIC Call Reports for off balance sheet bank data for the 4th quarter in 2007 and 2008. The propagation of financial contagion in networks with dense clustering which reflects high concentration or localization of exposures between few parti…