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Automatic real-time fault detection for industrial assets using metasensors
Get PDFLarge-scale industrial plants require physical sensors to continuously measure quantities such as temperatures or pressures. A large number of sensors is required to accurately describe the operating state of the plant, which unfortunately makes it very difficult for them to be effectively monitored by human operators. In this work we present a method to construct so-called metasensors, virtual sensors that compress the information from several sensors in an optimal manner. These metasensors are used as inputs to a novel anomaly detection system that automatically alerts operators to abnormal operation behaviour
Principal pivot transforms of quasidefinite matrices and semidefinite Lagrangian subspaces
Get PDFLagrangian subspaces are linear subspaces that appear naturally in control theory applications, and especially in the context of algebraic Riccati equations. We introduce a class of \emph{semidefinite} Lagrangian subspaces and show that these subspaces can be represented by a subset and a Hermitian matrix with the property that the submatrix is negative semidefinite and the submatrix is positive semidefinite. A matrix with these definiteness properties is called -semidefinite and it is a generalization of a quasidefinite matrix. Under mild hypotheses which hold true in most applications, the Lagrangian subspace associated to the stabilizing solution of an algebraic Riccati equation is semidefinite, and in addition we show that there is a bijection between Hamiltonian and symplectic pencils and semidefinite Lagrangian subspaces; hence this structure is ubiquitous in control theory.
The (symmetric) principal pivot transform (PPT) is a map used by Mehrmann and Poloni [\textit{SIAM J.\ Matrix Anal.\ Appl.}, 33(2012), pp.\ 780--805] to convert between two different pairs and representing the same Lagrangian subspace. For a semidefinite Lagrangian subspace, we prove that the symmetric PPT of an -semidefinite matrix is a -semidefinite matrix , and we derive an implementation of the transformation that both makes use of the definiteness properties of and guarantees the definiteness of the submatrices of in finite arithmetic.
We use the resulting formulas to obtain a semidefiniteness-preserving version of an optimization algorithm introduced by Mehrmann and Poloni to compute a pair (\mathcal{I}_{\opt},X_{\opt}) with M = \max_{i,j} \abs{(X_{\opt})_{ij}} as small as possible. Using semidefiniteness allows one to obtain a stronger inequality on with respect to the general case
Locally finite groups containing a -element with Chernikov centralizer
Get PDFSuppose that a locally finite group has a -element
with Chernikov centralizer. It is proved that if the involution in has nilpotent centralizer, then has a soluble subgroup of finite index
Chebyshev rootfinding via computing eigenvalues of colleague matrices: when is it stable?
Get PDFComputing the roots of a scalar polynomial, or the eigenvalues of a matrix polynomial, expressed in the Chebyshev basis is a fundamental problem that arises in many applications.
In this work, we analyze the backward stability of the polynomial rootfinding problem solved with colleague matrices.
In other words, given a scalar polynomial or a matrix polynomial expressed in the Chebyshev basis, the question is to determine whether the whole set of computed eigenvalues of the colleague matrix, obtained with a backward stable algorithm, like the QR algorithm, are the set of roots of a nearby polynomial or not.
In order to do so, we derive a first order backward error analysis of the polynomial rootfinding algorithm using colleague matrices adapting the geometric arguments in [A. Edelman and H. Murakami, \emph{Polynomial roots for companion matrix eigenvalues}, Math. Comp. 210, 763--776, 1995] to the Chebyshev basis.
We show that, if the absolute value of the coefficients of (respectively, the norm of the coefficients of ) are bounded by a moderate number, computing the roots of (respectively, the eigenvalues of ) via the eigenvalues of its colleague matrix using a backward stable eigenvalue algorithm is backward stable.
This backward error analysis also expands on the very recent work [Y. Nakatsukasa and V. Noferini, \emph{On the stability of computing polynomial roots via confederate linearizations}, To appear in Math. Comp.] that already showed that this algorithm is not backward normwise stable if the coefficients of the polynomial do not have moderate norms
The noncommutative geometry of inner forms of p-adic special linear groups
Get PDFLet be any reductive -adic group. We conjecture that every Bernstein component in the space of irreducible smooth -representations can be described as a
"twisted extended quotient" of the associated Bernstein torus by the associated finite group. We also pose some conjectures about L-packets and about the structure
of the Schwartz algebra of in these noncommutative geometric terms. Ultimately, our conjectures aim to reduce the classification of irreducible representations to
that of supercuspidal representations, and similarly for the local Langlands correspondence. These conjectures generalize earlier versions, which are only expected to hold for quasi-split groups.
We prove our conjectures for inner forms of general linear and special linear groups over local non-archimedean fields. This relies on our earlier study of Hecke algebras for types in these groups. We also make the relation
with the local Langlands correspondence explicit
Conjectures about p-adic groups and their noncommutative geometry
Get PDFLet G be any reductive p-adic group. We discuss several conjectures, some of them new, that involve the representation theory and the geometry of G.
At the heart of these conjectures are statements about the geometric structure of Bernstein components for G, both at the level of the space of irreducible representations and at the level of the associated Hecke algebras. We relate this to two well-known conjectures: the local Langlands correspondence and the Baum-Connes conjecture for G. In particular, we present a strategy to reduce the local Langlands correspondence for irreducible G-representations to the local Langlands correspondence for supercuspidal representations of Levi subgroups
SophiaBeads Datasets Project Documentation and Tutorials
Get PDFThe SophiaBeads Datasets [4] are real microCT datasets, acquired specifically for implementing, testing and comparing iterative reconstruction algorithms. The main motivations for the SophiaBeads Datasets Project are providing real datasets for researchers, and introducing a framework for designing experiments and choosing appropriate reconstruction methods via fair comparisons. Our aim with this report is to provide the reader with enough information to work with the project codes [3] so the reader can reconstruct the datasets. Additionally, we include a quantification tutorial so the readers are able to reproduce our results presented in [5,6]
An algorithm to compute the polar decomposition of a 3x3 matrix
Get PDFWe propose an algorithm for computing the polar decomposition of a
3 x 3 real matrix that is based on the connection between orthogonal
matrices and quaternions.
An important application is to 3D transformations in
the level 3 Cascading Style Sheets specification used in web browsers.
Our algorithm is numerically reliable and requires fewer arithmetic
operations than the alternative of computing the polar decomposition
via the singular value decomposition
Functoriality and K-theory for GL_n(R)
Get PDFWe investigate base change and automorphic induction C/R at the level of K-theory for the general linear group GL_n(R). In the course of this study, we compute in detail the C*-algebra K-theory of this disconnected group. We investigate the interaction of base change with the Baum-Connes correspondence for GL_n(R) and GL_n(C). This article is the archimedean companion of our previous article in the Journal of
Noncommutative Geometry
Embedding and Time Series Analysis
Get PDFThe 1970's and 80's saw a tremendous wave of interest---across the sciences and beyond---in the subject of nonlinear dynamics. Under the heading of `chaos theory' the subject even gripped the public imagination, leading to popular books and television programmes and even a mention in the film \emph{Jurassic Park}. One of the central ideas driving this interest was the realization that the complex, unpredictable behaviour known as chaos might be widespread in physical systems, and possibly even in biological, economic and social systems as well. This kind of behaviour, with its characteristic \emph{sensitive dependence on initial conditions}, had been shown to occur in a range of simple mathematical systems, many of which were conceived as models of physical or biological phenomena. As well as triggering much work on the mathematical theory of nonlinear dynamical systems, this also raised the intriguing question of whether chaotic behaviour could actually be observed in the broad range of experimental situations that the simple models hinted at. But while physicists and engineers were very familiar with experiments designed to investigate the various periodicities within a system, how should they treat the experimental data (or devise the experiments themselves) so as to reveal the characteristic features of chaos, such as the aforementioned sensitive dependence on initial conditions, or the strange attractors, with their fractal structures, that live in the state spaces of some chaotic systems