Generalized Multiple Correlation Coefficient As a Similarity
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Entropy Generation in Gaussian Quantum Transformations: Applying the Replica Method to Continuous-Variable Quantum Information Theory
www.nature.com/npjqi All rights reserved 2056-6387/15 ARTICLE OPEN Entropy generation in Gaussian quantum transformations: applying the replica method to continuous-variable quantum information theory Christos N Gagatsos1, Alexandros I Karanikas2, Georgios Kordas2 and Nicolas J Cerf1 In spite of their simple description in terms of rotations or symplectic transformations in phase space, quadratic Hamiltonians such as those modelling the most common Gaussian operations on bosonic modes remain poorly understood in terms of entropy production. For instance, determining the quantum entropy generated by a Bogoliubov transformation is notably a hard problem, with generally no known analytical solution, while it is vital to the characterisation of quantum communication via bosonic channels. Here we overcome this difficulty by adapting the replica method, a tool borrowed from statistical physics and quantum field theory. We exhibit a first application of this method to continuous-variable quantum information theory, where it enables accessing entropies in an optical parametric amplifier. As an illustration, we determine the entropy generated by amplifying a binary superposition of the vacuum and a Fock state, which yields a surprisingly simple, yet unknown analytical expression. npj Quantum Information (2015) 2, 15008; doi:10.1038/npjqi.2015.8; published online 16 February 2016 INTRODUCTION Gaussian states (e.g., the vacuum state, resulting after amplifica- Gaussian transformations are ubiquitous in quantum physics, tion in a thermal state of well-known -
Projects Applying Linear Algebra to Calculus
PROJECTS APPLYING LINEAR ALGEBRA TO CALCULUS Dr. Jason J. Molitierno Department of Mathematics Sacred Heart University Fairfield, CT 06825-1000 MAJOR PROBLEMS IN TEACHING A LINEAR ALGEBRA CLASS: • Having enough time in the semester to cover the syllabus. • Some problems are long and involved. ◦ Solving Systems of Equations. ◦ Finding Eigenvalues and Eigenvectors • Concepts can be difficult to visualize. ◦ Rotation and Translation of Vectors in multi-dimensioanl space. ◦ Solutions to systems of equations and systems of differential equations. • Some problems may be repetitive. A SOLUTION TO THESE ISSUES: OUT OF CLASS PROJECTS. THE BENEFITS: • By having the students learn some material outside of class, you have more time in class to cover additional material that you may not get to otherwise. • Students tend to get lost when the professor does a long computation on the board. Students are like "scribes." • Using technology, students can more easily visualize concepts. • Doing the more repetitive problems outside of class cuts down on boredom inside of class. • Gives students another method of learning as an alternative to be lectured to. APPLICATIONS OF SYSTEMS OF LINEAR EQUATIONS: (taken from Lay, fifth edition) • Nutrition: • Electrical Currents: • Population Distribution: VISUALIZING LINEAR TRANSFORMATIONS: in each of the five exercises below, do the following: (a) Find the matrix A that performs the desired transformation. (Note: In Exercises 4 and 5 where there are two transformations, multiply the two matrices with the first transformation matrix being on the right.) (b) Let T : <2 ! <2 be the transformation T (x) = Ax where A is the matrix you found in part (a). Find T (2; 3). -
Direct Shear Mapping – a New Weak Lensing Tool
MNRAS 451, 2161–2173 (2015) doi:10.1093/mnras/stv1083 Direct shear mapping – a new weak lensing tool C. O. de Burgh-Day,1,2,3‹ E. N. Taylor,1,3‹ R. L. Webster1,3‹ and A. M. Hopkins2,3 1School of Physics, David Caro Building, The University of Melbourne, Parkville VIC 3010, Australia 2The Australian Astronomical Observatory, PO Box 915, North Ryde NSW 1670, Australia 3ARC Centre of Excellence for All-sky Astrophysics (CAASTRO), Australian National University, Canberra, ACT 2611, Australia Accepted 2015 May 12. Received 2015 February 16; in original form 2013 October 24 ABSTRACT We have developed a new technique called direct shear mapping (DSM) to measure gravita- tional lensing shear directly from observations of a single background source. The technique assumes the velocity map of an unlensed, stably rotating galaxy will be rotationally symmet- ric. Lensing distorts the velocity map making it asymmetric. The degree of lensing can be inferred by determining the transformation required to restore axisymmetry. This technique is in contrast to traditional weak lensing methods, which require averaging an ensemble of background galaxy ellipticity measurements, to obtain a single shear measurement. We have tested the efficacy of our fitting algorithm with a suite of systematic tests on simulated data. We demonstrate that we are in principle able to measure shears as small as 0.01. In practice, we have fitted for the shear in very low redshift (and hence unlensed) velocity maps, and have obtained null result with an error of ±0.01. This high-sensitivity results from analysing spa- tially resolved spectroscopic images (i.e. -
Vol 45 Ams / Maa Anneli Lax New Mathematical Library
45 AMS / MAA ANNELI LAX NEW MATHEMATICAL LIBRARY VOL 45 10.1090/nml/045 When Life is Linear From Computer Graphics to Bracketology c 2015 by The Mathematical Association of America (Incorporated) Library of Congress Control Number: 2014959438 Print edition ISBN: 978-0-88385-649-9 Electronic edition ISBN: 978-0-88385-988-9 Printed in the United States of America Current Printing (last digit): 10987654321 When Life is Linear From Computer Graphics to Bracketology Tim Chartier Davidson College Published and Distributed by The Mathematical Association of America To my parents, Jan and Myron, thank you for your support, commitment, and sacrifice in the many nonlinear stages of my life Committee on Books Frank Farris, Chair Anneli Lax New Mathematical Library Editorial Board Karen Saxe, Editor Helmer Aslaksen Timothy G. Feeman John H. McCleary Katharine Ott Katherine S. Socha James S. Tanton ANNELI LAX NEW MATHEMATICAL LIBRARY 1. Numbers: Rational and Irrational by Ivan Niven 2. What is Calculus About? by W. W. Sawyer 3. An Introduction to Inequalities by E. F.Beckenbach and R. Bellman 4. Geometric Inequalities by N. D. Kazarinoff 5. The Contest Problem Book I Annual High School Mathematics Examinations 1950–1960. Compiled and with solutions by Charles T. Salkind 6. The Lore of Large Numbers by P.J. Davis 7. Uses of Infinity by Leo Zippin 8. Geometric Transformations I by I. M. Yaglom, translated by A. Shields 9. Continued Fractions by Carl D. Olds 10. Replaced by NML-34 11. Hungarian Problem Books I and II, Based on the Eotv¨ os¨ Competitions 12. 1894–1905 and 1906–1928, translated by E. -
Multi-View Metric Learning in Vector-Valued Kernel Spaces
Multi-view Metric Learning in Vector-valued Kernel Spaces Riikka Huusari Hachem Kadri Cécile Capponi Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France Abstract unlabeled examples [26]; the latter tries to efficiently combine multiple kernels defined on each view to exploit We consider the problem of metric learning for information coming from different representations [11]. multi-view data and present a novel method More recently, vector-valued reproducing kernel Hilbert for learning within-view as well as between- spaces (RKHSs) have been introduced to the field of view metrics in vector-valued kernel spaces, multi-view learning for going further than MKL by in- as a way to capture multi-modal structure corporating in the learning model both within-view and of the data. We formulate two convex opti- between-view dependencies [20, 14]. It turns out that mization problems to jointly learn the metric these kernels and their associated vector-valued repro- and the classifier or regressor in kernel feature ducing Hilbert spaces provide a unifying framework for spaces. An iterative three-step multi-view a number of previous multi-view kernel methods, such metric learning algorithm is derived from the as co-regularized multi-view learning and manifold reg- optimization problems. In order to scale the ularization, and naturally allow to encode within-view computation to large training sets, a block- as well as between-view similarities [21]. wise Nyström approximation of the multi-view Kernels of vector-valued RKHSs are positive semidefi- kernel matrix is introduced. We justify our ap- nite matrix-valued functions. They have been applied proach theoretically and experimentally, and with success in various machine learning problems, such show its performance on real-world datasets as multi-task learning [10], functional regression [15] against relevant state-of-the-art methods. -
TEC Forecasting Based on Manifold Trajectories
Article TEC forecasting based on manifold trajectories Enrique Monte Moreno 1,* ID , Alberto García Rigo 2,3 ID , Manuel Hernández-Pajares 2,3 ID and Heng Yang 1 ID 1 Department of Signal Theory and Communications, TALP research center, Technical University of Catalonia, Barcelona, Spain 2 UPC-IonSAT, Technical University of Catalonia, Barcelona, Spain 3 IEEC-CTE-CRAE, Institut d’Estudis Espacials de Catalunya, Barcelona, Spain * Correspondence: [email protected]; Tel.: +34-934016435 Academic Editor: name Version June 8, 2018 submitted to Remote Sens. 1 Abstract: In this paper, we present a method for forecasting the ionospheric Total Electron Content 2 (TEC) distribution from the International GNSS Service’s Global Ionospheric Maps. The forecasting 3 system gives an estimation of the value of the TEC distribution based on linear combination of 4 previous TEC maps (i.e. a set of 2D arrays indexed by time), and the computation of a tangent 5 subspace in a manifold associated to each map. The use of the tangent space to each map is justified 6 because it allows to model the possible distortions from one observation to the next as a trajectory on 7 the tangent manifold of the map. The coefficients of the linear combination of the last observations 8 along with the tangent space are estimated at each time stamp in order to minimize the mean square 9 forecasting error with a regularization term. The the estimation is made at each time stamp to adapt 10 the forecast to short-term variations in solar activity.. 11 Keywords: Total Electron Content; Ionosphere; Forecasting; Tangent Distance; GNSS 12 1. -
MATRICES Part 2 3. Linear Equations
MATRICES part 2 (modified content from Wikipedia articles on matrices http://en.wikipedia.org/wiki/Matrix_(mathematics)) 3. Linear equations Matrices can be used to compactly write and work with systems of linear equations. For example, if A is an m-by-n matrix, x designates a column vector (i.e., n×1-matrix) of n variables x1, x2, ..., xn, and b is an m×1-column vector, then the matrix equation Ax = b is equivalent to the system of linear equations a1,1x1 + a1,2x2 + ... + a1,nxn = b1 ... am,1x1 + am,2x2 + ... + am,nxn = bm . For example, 3x1 + 2x2 – x3 = 1 2x1 – 2x2 + 4x3 = – 2 – x1 + 1/2x2 – x3 = 0 is a system of three equations in the three variables x1, x2, and x3. This can be written in the matrix form Ax = b where 3 2 1 1 1 A = 2 2 4 , x = 2 , b = 2 1 1/2 −1 푥3 0 � − � �푥 � �− � A solution to− a linear system− is an assignment푥 of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by x1 = 1 x2 = – 2 x3 = – 2 since it makes all three equations valid. A linear system may behave in any one of three possible ways: 1. The system has infinitely many solutions. 2. The system has a single unique solution. 3. The system has no solution. 3.1. Solving linear equations There are several algorithms for solving a system of linear equations. Elimination of variables The simplest method for solving a system of linear equations is to repeatedly eliminate variables. -
Karhunen-Loève Analysis for Weak Gravitational Lensing
c Copyright 2012 Jacob T. Vanderplas arXiv:1301.6657v1 [astro-ph.CO] 28 Jan 2013 Karhunen-Lo`eve Analysis for Weak Gravitational Lensing Jacob T. Vanderplas A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2012 Reading Committee: Andrew Connolly, Chair Bhuvnesh Jain Andrew Becker Program Authorized to Offer Degree: Department of Astronomy University of Washington Abstract Karhunen-Lo`eve Analysis for Weak Gravitational Lensing Jacob T. Vanderplas Chair of the Supervisory Committee: Professor Andrew Connolly Department of Astronomy In the past decade, weak gravitational lensing has become an important tool in the study of the universe at the largest scale, giving insights into the distribution of dark matter, the expansion of the universe, and the nature of dark energy. This thesis research explores several applications of Karhunen-Lo`eve (KL) analysis to speed and improve the comparison of weak lensing shear catalogs to theory in order to constrain cosmological parameters in current and future lensing surveys. This work addresses three related aspects of weak lensing analysis: Three-dimensional Tomographic Mapping: (Based on work published in VanderPlas et al., 2011) We explore a new fast approach to three-dimensional mass mapping in weak lensing surveys. The KL approach uses a KL-based filtering of the shear signal to reconstruct mass structures on the line-of-sight, and provides a unified framework to evaluate the efficacy of linear reconstruction techniques. We find that the KL-based filtering leads to near-optimal angular resolution, and computation times which are faster than previous approaches. -
TEC Forecasting Based on Manifold Trajectories
remote sensing Article TEC Forecasting Based on Manifold Trajectories Enrique Monte Moreno 1,* ID , Alberto García Rigo 2,3 ID , Manuel Hernández-Pajares 2,3 ID and Heng Yang 1 ID 1 TALP Research Center, Department of Signal Theory and Communications, Polytechnical University of Catalonia, 08034 Barcelona, Spain; [email protected] 2 UPC-IonSAT, Polytechnical University of Catalonia, 08034 Barcelona, Spain; [email protected] (A.G.R.); [email protected] (M.H.-P.) 3 IEEC-CTE-CRAE, Institut d’Estudis Espacials de Catalunya, 08034 Barcelona, Spain * Correspondence: [email protected]; Tel.: +34-934016435 Received: 18 May 2018; Accepted: 13 June 2018; Published: 21 June 2018 Abstract: In this paper, we present a method for forecasting the ionospheric Total Electron Content (TEC) distribution from the International GNSS Service’s Global Ionospheric Maps. The forecasting system gives an estimation of the value of the TEC distribution based on linear combination of previous TEC maps (i.e., a set of 2D arrays indexed by time), and the computation of a tangent subspace in a manifold associated to each map. The use of the tangent space to each map is justified because it allows modeling the possible distortions from one observation to the next as a trajectory on the tangent manifold of the map. The coefficients of the linear combination of the last observations along with the tangent space are estimated at each time stamp to minimize the mean square forecasting error with a regularization term. The estimation is made at each time stamp to adapt the forecast to short-term variations in solar activity. -
Practical Computer Vision: Theory & Applications
Practical Computer Vision: Theory & Applications Carmen Alonso Montes 23rd-27th November 2015 Contents Course Structure Basic concepts of Computer Vision Pixels Colour vs. Black&White Noise and linear filtering Image enhancement Some practical examples My first programme with images Image denoising Image enhancement Basic commands matlab Carmen Alonso Montes 2 Course Structure This course is an introductory course for students with very few practical knowledge on computer vision Resources available for students through BCAM website Course material: slides, exercises and examples What do you need for this course? The most important thing is your self-motivation! Programming: examples will be done in Matlab, and also, if needed, I will explain how to do it in OpenCV (C++) For those C++ or Phyton programmers, I suggest to use a proper code editor (eclipse, netbeans, …) Some examples of useful tools (not needed) but good to know Versioning: git, SVN Automatic documentation generation: Doxygen Project management: Redmine Carmen Alonso Montes 3 Course Structure: Contents Each class will last 2 hours and a half 50 min Theory (9:30-10:20) 10 min break 50 min Theory (10:30-11:20) 10 min break 30 hour practice (11:30-12:00) Each slide, will contain a blue box with the command in matlab, in case you want to try it during the lessons imread At the end of the slides, you will have a table with the commands used during the lessons, for a quick guide. Carmen Alonso Montes 4 Some good books Digital Image Processing, 3rd Ed. (DIP/3e) by Gonzalez and Woods, (2008) This is the bible in this domain Learning OpenCV, Computer Vision in C++ with the OpenCV Library, by Adrian Kaehler, Gary Bradski. -
Geometric Analysis: Partial Differential Equations and Surfaces
570 Geometric Analysis: Partial Differential Equations and Surfaces UIMP-RSME Lluis Santaló Summer School 2010: Geometric Analysis June 28–July 2, 2010 University of Granada, Spain Joaquín Pérez José A. Gálvez Editors American Mathematical Society Real Sociedad Matemática Española American Mathematical Society Geometric Analysis: Partial Differential Equations and Surfaces UIMP-RSME Lluis Santaló Summer School 2010: Geometric Analysis June 28–July 2, 2010 University of Granada, Spain Joaquín Pérez José A. Gálvez Editors 570 Geometric Analysis: Partial Differential Equations and Surfaces UIMP-RSME Lluis Santaló Summer School 2010: Geometric Analysis June 28–July 2, 2010 University of Granada, Spain Joaquín Pérez José A. Gálvez Editors American Mathematical Society Real Sociedad Matemática Española American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor George Andrews Abel Klein Martin J. Strauss Editorial Committee of the Real Sociedad Matem´atica Espa˜nola Pedro J. Pa´ul, Director Luis Al´ıas Emilio Carrizosa Bernardo Cascales Javier Duoandikoetxea Alberto Elduque Rosa Maria Mir´o Pablo Pedregal Juan Soler 2010 Mathematics Subject Classification. Primary 53A10; Secondary 35B33, 35B40, 35J20, 35J25, 35J60, 35J96, 49Q05, 53C42, 53C45. Library of Congress Cataloging-in-Publication Data UIMP-RSME Santal´o Summer School (2010 : University of Granada) Geometric analysis : partial differential equations and surfaces : UIMP-RSME Santal´o Summer School geometric analysis, June 28–July 2, 2010, University of Granada, Granada, Spain / Joaqu´ın P´erez, Jos´eA.G´alvez, editors p. cm. — (Contemporary Mathematics ; v. 570) Includes bibliographical references. ISBN 978-0-8218-4992-7 (alk. paper) 1. Minimal surfaces–Congresses. 2. Geometry, Differential–Congresses. 3. Differential equations, Partial–Asymptotic theory–Congresses. -
Algorithms for Designing Pop-Up Cards
Algorithms for Designing Pop-Up Cards Zachary Abel∗ Erik D. Demainey Martin L. Demainey Sarah Eisenstaty Anna Lubiwz Andr´eSchulzx Diane L. Souvaine{ Giovanni Vigliettak Andrew Winslow{ Abstract We prove that every simple polygon can be made as a (2D) pop-up card/book that opens to any desired angle between 0 and 360◦. More precisely, given a simple polygon attached to the two walls of the open pop-up, our polynomial-time algorithm subdivides the polygon into a single-degree-of-freedom linkage structure, such that closing the pop-up flattens the linkage without collision. This result solves an open problem of Hara and Sugihara from 2009. We also show how to obtain a more efficient construction for the special case of orthogonal polygons, and how to make 3D orthogonal polyhedra, from pop-ups that open to 90◦, 180◦, 270◦, or 360◦. 1 Introduction Pop-up books have been entertaining children with their surprising and playful mechanics since their mass production in the 1970s. But the history of pop-ups is much older [27], and they were originally used for scientific and historical illustrations. The earliest known example of a \movable book" is Matthew Paris's Chronica Majora (c. 1250), which uses turnable disks (volvelle) to represent a calendar and uses flaps to illustrate maps. A more recent scientific example is George Spratt's Obstetric Tables (1850), which uses flaps to illustrate procedures for delivering babies including by Caesarean section. From the same year, Dean & Sons' Little Red Riding Hood (1850) is the first known movable book where a flat page rises into a 3D scene, though here it was actuated by pulling a string.