Transformations of ray transferences of optical systems to augmented Hamiltonian matrices and the problem of the average system

2007 ◽  
Vol 66 (2) ◽  
Author(s):  
J. R. Cardoso
Keyword(s):  
Optical System ◽  
Optical Systems ◽  
Hamiltonian Matrix ◽  
First Order ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Average System ◽  
Quantitative Analyses ◽  
Logarithm Function

The first-order optical nature of an optical system (including an eye) is completely characterized by a 55 ×  matrix called the ray transference.  It is known  that the image of a ray transference by the matrix logarithm function is an augmented Hamiltonian matrix.  It turns out that there are other ways of transforming transferences into augmented Hamiltonian matrices.  They include Cayley transforms and modified Cayley transforms.  This paper will describe these transforms with a view to finding the most suitable one for quantitative analyses of eyes and other systems in augmented Hamiltonian spaces.  In particular we look at the calculation of average systems.

scholarly journals A dynamical systems view of network centrality

2014 ◽  
Vol 470 (2165) ◽  
pp. 20130835 ◽  
Author(s):  
Peter Grindrod ◽  
Desmond J. Higham
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Dynamic Networks ◽  
Network Evolution ◽  
Mathematical Framework ◽  
The Novel ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Systems View ◽  
Logarithm Function

To gain insights about dynamic networks, the dominant paradigm is to study discrete snapshots , or timeslices , as the interactions evolve. Here, we develop and test a new mathematical framework where network evolution is handled over continuous time, giving an elegant dynamical systems representation for the important concept of node centrality. The resulting system allows us to track the relative influence of each individual. This new setting is natural in many digital applications, offering both conceptual and computational advantages. The novel differential equations approach is convenient for modelling and analysis of network evolution and gives rise to an interesting application of the matrix logarithm function. From a computational perspective, it avoids the awkward up-front compromises between accuracy, efficiency and redundancy required in the prevalent discrete-time setting. Instead, we can rely on state-of-the-art ODE software, where discretization takes place adaptively in response to the prevailing system dynamics. The new centrality system generalizes the widely used Katz measure, and allows us to identify and track, at any resolution, the most influential nodes in terms of broadcasting and receiving information through time-dependent links. In addition to the classical static network notion of attenuation across edges, the new ODE also allows for attenuation over time, as information becomes stale. This allows ‘running measures’ to be computed, so that networks can be monitored in real time over arbitrarily long intervals. With regard to computational efficiency, we explain why it is cheaper to track good receivers of information than good broadcasters. An important consequence is that the overall broadcast activity in the network can also be monitored efficiently. We use two synthetic examples to validate the relevance of the new measures. We then illustrate the ideas on a large-scale voice call network, where key features are discovered that are not evident from snapshots or aggregates.

Atomic Resolution in Crystal Aperture Stem

1990 ◽  
Vol 48 (1) ◽  
pp. 236-237
Author(s):  
J T Fourie
Keyword(s):  
Optical System ◽  
Spherical Aberration ◽  
Three Dimensional ◽  
Transmission Function ◽  
Optical Systems ◽  
Bragg Angle ◽  
Electron Optical System ◽  
Intimate Contact ◽  
Diffraction Effects ◽  
Design Aspect

The attempts at improvement of electron optical systems to date, have largely been directed towards the design aspect of magnetic lenses and towards the establishment of ideal lens combinations. In the present work the emphasis has been placed on the utilization of a unique three-dimensional crystal objective aperture within a standard electron optical system with the aim to reduce the spherical aberration without introducing diffraction effects. A brief summary of this work together with a description of results obtained recently, will be given.The concept of utilizing a crystal as aperture in an electron optical system was introduced by Fourie who employed a {111} crystal foil as a collector aperture, by mounting the sample directly on top of the foil and in intimate contact with the foil. In the present work the sample was mounted on the bottom of the foil so that the crystal would function as an objective or probe forming aperture. The transmission function of such a crystal aperture depends on the thickness, t, and the orientation of the foil. The expression for calculating the transmission function was derived by Hashimoto, Howie and Whelan on the basis of the electron equivalent of the Borrmann anomalous absorption effect in crystals. In Fig. 1 the functions for a g220 diffraction vector and t = 0.53 and 1.0 μm are shown. Here n= Θ‒ΘB, where Θ is the angle between the incident ray and the (hkl) planes, and ΘB is the Bragg angle.

scholarly journals On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

2021 ◽  
Author(s):  
Alice Cortinovis ◽  
Daniel Kressner
Keyword(s):  
Large Scale ◽  
Gaussian Process Regression ◽  
Likelihood Estimation ◽  
Random Vectors ◽  
Symmetric Positive Definite ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Trace Estimation ◽  
Scale Matrix ◽  
Existing Result

AbstractRandomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

FORC analysis of magnetically soft microparticles embedded in a polymeric elastic environment

2022 ◽  
Author(s):  
Dmitry Yu Borin ◽  
Mikhail V Vaganov
Keyword(s):  
Particle Concentration ◽  
Magnetic Hysteresis ◽  
Magnetic Response ◽  
Soft Matrix ◽  
First Order ◽  
Matrix Elasticity ◽  
Intrinsic Coercivity ◽  
The Matrix ◽  
Soft Particles ◽  
Elastomer Composites

Abstract First-order reversal curve (FORC) analysis allows one to investigate composite magnetic materials by decomposing the magnetic response of a whole sample into individual responses of the elementary objects comprising the sample. In this work, we apply this technique to analysing silicone elastomer composites reinforced with ferromagnetic microparticles possessing low intrinsic coercivity. Even though the material of such particles does not demonstrate significant magnetic hysteresis, the soft matrix of the elastomers allows for the translational mobility of the particles and enables their magnetomechanical hysteresis which renders into a wasp-waisted major magnetization loop of the whole sample. It is demonstrated that the FORC diagrams of the composites contain characteristic wing features arising from the collective hysteretic magnetization of the magnetically soft particles. The influence of the matrix elasticity and particle concentration on the shape of the wing feature is investigated, and an approach to interpreting experimental FORC diagrams of the magnetically soft magnetoactive elastomers is proposed. The experimental data are in qualitative agreement with the results of the simulation of the particle magnetization process obtained using a model comprised of two magnetically soft particles embedded in an elastic environment.

A reduction class containing formulas with one monadic predicate and one binary function symbol

1976 ◽  
Vol 41 (1) ◽  
pp. 45-49
Author(s):  
Charles E. Hughes
Keyword(s):  
Normal Form ◽  
Predicate Symbol ◽  
Conjunctive Normal Form ◽  
Function Symbol ◽  
Predicate Calculus ◽  
Binary Function ◽  
First Order ◽  
The Matrix ◽  
Reduction Class ◽  
Order Predicate Calculus

AbstractA new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀x∀yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

Action and agency in complex sentences from present-day Romanian

2020 ◽  
Vol 13 (62) (1) ◽  
pp. 39-56
Author(s):  
Alice BODOC
Keyword(s):  
Natural World ◽  
Qualitative And Quantitative ◽  
Complex Sentences ◽  
The Matrix ◽  
Quantitative Analyses ◽  
Adverbial Clause ◽  
Linguistic Phenomenon ◽  
Matrix Verb

Starting from Alanen’s remark that “actions (exercises of capacities) are found throughout the natural world; and so is agency” (2018, 2), the present paper aims at describing the influence of these two fundamental concepts – action and agency – on the structure of Romanian complex sentences. More precisely, I am interested in providing evidence of a linguistic phenomenon that has received far less attention in the literature, i.e. the semantic restrictions imposed by the matrix verb over the embedded adverbial clause. As concerns the methodology, both qualitative and quantitative analyses will be conducted on an extensive online Romanian corpus (CoRoLa), and will be based on the semantic typologies of the verb included in some of the reference Romanian grammars (GALR 2008, 326; GBLR 2010, 279). One of the most important results of the analysis was the phenomenon of agentivity

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