First mean return time in decoherent quantum walks
|
|
- Tyler Arnold
- 8 years ago
- Views:
Transcription
1 First mean return time in decoherent quantum walks Péter Sinkovicz, János K. Asbóth, Tamás Kiss Wigner Research Centre for Physics Hungarian Academy of Sciences, April 0.
2 Problem statement Example: N=,,,... A ij = A ji transition amplitude First mean return time in decoherent quantum walks /
3 Problem Our numerical results Conclusion Arrival by absorption: 0i := ψ0 i First mean return time in decoherent quantum walks /
4 Problem Our numerical results Conclusion Arrival by absorption: 0i := ψ0 i First mean return time in decoherent quantum walks U 0i /
5 Problem Our numerical results Conclusion Arrival by absorption: 0i := ψ0 i U 0i or p := h0 U 0i First mean return time in decoherent quantum walks ψ i := [I 0ih0 ] U 0i q := hψ ψ i /
6 Conditional wave function: evolve these parts, which haven t come back where ψ t+ := [I 0 0 ] U ψ t p t := 0 U ψ t q t+ := ψ t+ ψ t+ p t probability: Prob(X t = 0 X n 0 if n < t) q t probability: Prob(X n 0 if n t) First mean return time in decoherent quantum walks /
7 Mean return time T := tp t q t := Trρ(t) t= t=0 t=0 where q t+ + p t+ = q t and q 0 = F. A. Grünbaum,A. H. Werner, R. F. Werner, (0) First mean return time in decoherent quantum walks /
8 Mean return time T := tp t q t := Trρ(t) t= t=0 t=0 where q t+ + p t+ = q t and q 0 = pt N= example p t : t pt t F. A. Grünbaum,A. H. Werner, R. F. Werner, (0) First mean return time in decoherent quantum walks /
9 Mean return time T := tp t q t := Trρ(t) t= t=0 t=0 where q t+ + p t+ = q t and q 0 = pt N= example p t : t pt t T is an integer number equal with the graph size independent of A ij transition amplitude T = N F. A. Grünbaum,A. H. Werner, R. F. Werner, (0) First mean return time in decoherent quantum walks /
10 Classical case : symmetric, regular graph T Kemeny, G.; Snell, L. (90) First mean return time in decoherent quantum walks /
11 Our interest? decoherence x First mean return time in decoherent quantum walks 7 /
12 Numerical study in order to find out?? T = N I) homogeneous decoherence II) Kraus representation III) Master equation First mean return time in decoherent quantum walks 8 /
13 Homogeneous decoherence For see a transition between Grünbaum et al. s and the Classical random walk we stick in a simplest decoherence. So one step of the process is the following: coherent time step decoherence C[ρ] = UρU measurement D[ρ] xy = dρ xy + ( d)ρ xx δ xy M[ρ] = [I 0 0 ] ρ [I 0 0 ] First mean return time in decoherent quantum walks 9 /
14 T numerical result U + homogeneous decoherence T = N N for d = unitary time evolution, and d = 0 is the classical case d 0 where d ρ t+ = M[T [ρ t ]] = M[D[C[ρ t ]]] d and D[ρ] = ρ dρ dρ... dρ ρ First mean return time in decoherent quantum walks 0 /
15 Numerical study in order to find out?? T = N I) homogeneous decoherence II) Kraus representation III) Master equation First mean return time in decoherent quantum walks /
16 Kraus representation In order to map all quantum channel we can use Kraus representation for the decoherence channel d ρ := D[ρ] = K µ ρk µ µ= ρ m,µ = D m,n µ,ν ρ n,ν trace preserving (probability preserving): d K µk µ = I D stochastic µ= may we will get some redundancy, because the measure and the unitary time evolution can be defined by Kraus operators First mean return time in decoherent quantum walks /
17 numerical guess unital map T [ N I] = N I T = N T [ ] = D[C[ ]] measure less process is unital (leave the totally mixed state invariant) only if d K µ K µ = I D T stochastic µ= trace preserving and untial map quantum walk ρ T [ρ] classical walk λ t Wλ t First mean return time in decoherent quantum walks /
18 Numerical study in order to find out?? T = N I) homogeneous decoherence II) Kraus representation III) Master equation First mean return time in decoherent quantum walks /
19 Master equation For check we formulated our guess in another language: T [ ] can be substitute with the time evolution which has generated by the t ρ = i [H, ρ] + L(ρ) L(ρ) = i,j κ i,j [ i j ρ j i ( i i ρ + ρ j j ) ] Master equation. First mean return time in decoherent quantum walks /
20 Master equation For check we formulated our guess in another language: T [ ] can be substitute with the time evolution which has generated by the t ρ = i [H, ρ] + L(ρ) L(ρ) = i,j κ i,j [ i j ρ j i ( i i ρ + ρ j j ) ] Master equation. numerical guess unital map L(I) = 0 κ i,j = κ j,i T = N First mean return time in decoherent quantum walks /
21 Conclusion I) numerically studied systems Kraus representation unitary quantum walk homogeneous decoherence inhomogeneous, but symmetric decoherence uni-stochastic map Master equation L(I) = 0 (unital map) II) analytically we need proof for numerical guess T [I] = I unital T = N First mean return time in decoherent quantum walks /
LECTURE 4. Last time: Lecture outline
LECTURE 4 Last time: Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Lecture outline Stochastic processes Markov chains Entropy rate Random
More informationChapter 9. Systems of Linear Equations
Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables
More informationQuantum Computing Lecture 7. Quantum Factoring. Anuj Dawar
Quantum Computing Lecture 7 Quantum Factoring Anuj Dawar Quantum Factoring A polynomial time quantum algorithm for factoring numbers was published by Peter Shor in 1994. polynomial time here means that
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
More informationThree Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009
Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,
More informationQuantum Algorithms in NMR Experiments. 25 th May 2012 Ling LIN & Michael Loretz
Quantum Algorithms in NMR Experiments 25 th May 2012 Ling LIN & Michael Loretz Contents 1. Introduction 2. Shor s algorithm 3. NMR quantum computer Nuclear spin qubits in a molecule NMR principles 4. Implementing
More informationThe Limits of Adiabatic Quantum Computation
The Limits of Adiabatic Quantum Computation Alper Sarikaya June 11, 2009 Presentation of work given on: Thesis and Presentation approved by: Date: Contents Abstract ii 1 Introduction to Quantum Computation
More informationRate of convergence towards Hartree dynamics
Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson
More informationHow to Gamble If You Must
How to Gamble If You Must Kyle Siegrist Department of Mathematical Sciences University of Alabama in Huntsville Abstract In red and black, a player bets, at even stakes, on a sequence of independent games
More informationTopic: Special Products and Factors Subtopic: Rules on finding factors of polynomials
Quarter I: Special Products and Factors and Quadratic Equations Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials Time Frame: 20 days Time Frame: 3 days Content Standard:
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation
More informationSystem Identification for Acoustic Comms.:
System Identification for Acoustic Comms.: New Insights and Approaches for Tracking Sparse and Rapidly Fluctuating Channels Weichang Li and James Preisig Woods Hole Oceanographic Institution The demodulation
More informationDO WE REALLY UNDERSTAND QUANTUM MECHANICS?
DO WE REALLY UNDERSTAND QUANTUM MECHANICS? COMPRENONS-NOUS VRAIMENT LA MECANIQUE QUANTIQUE? VARIOUS INTERPRETATIONS OF QUANTUM MECHANICS IHES, 29 janvier 2015 Franck Laloë, LKB, ENS Paris 1 INTRODUCTION
More informationSimplification of Radical Expressions
8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of
More informationTime dependence in quantum mechanics Notes on Quantum Mechanics
Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:37-05:00 Copyright 2003 Dan
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationQuantum Computing and Grover s Algorithm
Quantum Computing and Grover s Algorithm Matthew Hayward January 14, 2015 1 Contents 1 Motivation for Study of Quantum Computing 3 1.1 A Killer App for Quantum Computing.............. 3 2 The Quantum Computer
More informationSPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BI-DIAGONAL REPRESENTATIONS FOR PHASE TYPE DISTRIBUTIONS AND MATRIX-EXPONENTIAL DISTRIBUTIONS
Stochastic Models, 22:289 317, 2006 Copyright Taylor & Francis Group, LLC ISSN: 1532-6349 print/1532-4214 online DOI: 10.1080/15326340600649045 SPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BI-DIAGONAL
More informationLecture 4: Thermodynamics of Diffusion: Spinodals
Materials Science & Metallurgy Master of Philosophy, Materials Modelling, Course MP6, Kinetics and Microstructure Modelling, H. K. D. H. Bhadeshia Lecture 4: Thermodynamics of Diffusion: Spinodals Fick
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationQuantum Mechanics and Representation Theory
Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30
More informationSimplifying Algebraic Fractions
5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationPolynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range
THEORY OF COMPUTING, Volume 1 (2005), pp. 37 46 http://theoryofcomputing.org Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range Andris Ambainis
More information~ EQUIVALENT FORMS ~
~ EQUIVALENT FORMS ~ Critical to understanding mathematics is the concept of equivalent forms. Equivalent forms are used throughout this course. Throughout mathematics one encounters equivalent forms of
More informationMaster equation for retrodiction of quantum communication signals
Master equation for odiction of quantum communication signals Stephen M. Barnett 1, David T. Pegg 2, John Jeffers 1 and Ottavia Jedrkiewicz 3 1 Department of Physics and Applied Physics, University of
More informationChemical group theory for quantum simulation
Title JDWhitfield@gmail.com 1/19 Chemical group theory for quantum simulation James Daniel Whitfield U. Ghent September 28, 2015 Title JDWhitfield@gmail.com 2/19 1. Computational chemistry 2. Symmetry
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationHomework set 4 - Solutions
Homework set 4 - Solutions Math 495 Renato Feres Problems R for continuous time Markov chains The sequence of random variables of a Markov chain may represent the states of a random system recorded at
More informationThe Australian Journal of Mathematical Analysis and Applications
The Australian Journal of Mathematical Analysis and Applications Volume 7, Issue, Article 11, pp. 1-14, 011 SOME HOMOGENEOUS CYCLIC INEQUALITIES OF THREE VARIABLES OF DEGREE THREE AND FOUR TETSUYA ANDO
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More information1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let
Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as
More informationWhat Has Quantum Mechanics to Do With Factoring? Things I wish they had told me about Peter Shor s algorithm
What Has Quantum Mechanics to Do With Factoring? Things I wish they had told me about Peter Shor s algorithm 1 Question: What has quantum mechanics to do with factoring? Answer: Nothing! 2 Question: What
More informationModeling and Performance Evaluation of Computer Systems Security Operation 1
Modeling and Performance Evaluation of Computer Systems Security Operation 1 D. Guster 2 St.Cloud State University 3 N.K. Krivulin 4 St.Petersburg State University 5 Abstract A model of computer system
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationELECTRON SPIN RESONANCE Last Revised: July 2007
QUESTION TO BE INVESTIGATED ELECTRON SPIN RESONANCE Last Revised: July 2007 How can we measure the Landé g factor for the free electron in DPPH as predicted by quantum mechanics? INTRODUCTION Electron
More informationThe Ramsey Discounting Formula for a Hidden-State Stochastic Growth Process / 8
The Ramsey Discounting Formula for a Hidden-State Stochastic Growth Process Martin L. Weitzman May 2012 Bergen Conference Long-Term Social Discount Rates What is Approach of This Paper? Increasing fuzziness
More informationResearch Article The General Traveling Wave Solutions of the Fisher Equation with Degree Three
Advances in Mathematical Physics Volume 203, Article ID 65798, 5 pages http://dx.doi.org/0.55/203/65798 Research Article The General Traveling Wave Solutions of the Fisher Equation with Degree Three Wenjun
More informationpath tracing computer graphics path tracing 2009 fabio pellacini 1
path tracing computer graphics path tracing 2009 fabio pellacini 1 path tracing Monte Carlo algorithm for solving the rendering equation computer graphics path tracing 2009 fabio pellacini 2 solving rendering
More informationMatrix Differentiation
1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have
More informationA characterization of trace zero symmetric nonnegative 5x5 matrices
A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationIntroduction to time series analysis
Introduction to time series analysis Margherita Gerolimetto November 3, 2010 1 What is a time series? A time series is a collection of observations ordered following a parameter that for us is time. Examples
More informationExample: Boats and Manatees
Figure 9-6 Example: Boats and Manatees Slide 1 Given the sample data in Table 9-1, find the value of the linear correlation coefficient r, then refer to Table A-6 to determine whether there is a significant
More informationCorrelation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs
Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in
More informationTraffic Behavior Analysis with Poisson Sampling on High-speed Network 1
Traffic Behavior Analysis with Poisson Sampling on High-speed etwork Guang Cheng Jian Gong (Computer Department of Southeast University anjing 0096, P.R.China) Abstract: With the subsequent increasing
More informationMarkov Chains. Table of Contents. Schedules
Markov Chains These notes contain material prepared by colleagues who have also presented this course at Cambridge, especially James Norris. The material mainly comes from books of Norris, Grimmett & Stirzaker,
More informationGrade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %
Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationFactoring by Quantum Computers
Factoring by Quantum Computers Ragesh Jaiswal University of California, San Diego A Quantum computer is a device that uses uantum phenomenon to perform a computation. A classical system follows a single
More informationFACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project
9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module
More informationStochastic Gene Expression in Prokaryotes: A Point Process Approach
Stochastic Gene Expression in Prokaryotes: A Point Process Approach Emanuele LEONCINI INRIA Rocquencourt - INRA Jouy-en-Josas ASMDA Mataró June 28 th 2013 Emanuele LEONCINI (INRIA) Stochastic Gene Expression
More informationAnyone know these guys?
Anyone know these guys? Gavin Brown and Miles Reid We observe that some of our diptych varieties have a beautiful description in terms of key 5-folds V (k) A k+5 that are almost homogeneous spaces. By
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More information5 Homogeneous systems
5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationStochastic Gene Expression in Prokaryotes: A Point Process Approach
Stochastic Gene Expression in Prokaryotes: A Point Process Approach Emanuele LEONCINI INRIA Rocquencourt - INRA Jouy-en-Josas Mathematical Modeling in Cell Biology March 27 th 2013 Emanuele LEONCINI (INRIA)
More informationLights and Darks of the Star-Free Star
Lights and Darks of the Star-Free Star Edward Ochmański & Krystyna Stawikowska Nicolaus Copernicus University, Toruń, Poland Introduction: star may destroy recognizability In (finitely generated) trace
More informationA stochastic individual-based model for immunotherapy of cancer
A stochastic individual-based model for immunotherapy of cancer Loren Coquille - Joint work with Martina Baar, Anton Bovier, Hannah Mayer (IAM Bonn) Michael Hölzel, Meri Rogava, Thomas Tüting (UniKlinik
More informationRank one SVD: un algorithm pour la visualisation d une matrice non négative
Rank one SVD: un algorithm pour la visualisation d une matrice non négative L. Labiod and M. Nadif LIPADE - Universite ParisDescartes, France ECAIS 2013 November 7, 2013 Outline Outline 1 Data visualization
More information15th European Union Contest for Young Scientists
15th written projects conception to conclusion European Union Contest for Young Scientists Budapest, Hungary 20-26 September, 2003 Millenary Park Organizers: originality and creativity reasoning and clarity
More informationSolutions to Linear First Order ODE s
First Order Linear Equations In the previous session we learned that a first order linear inhomogeneous ODE for the unknown function x = x(t), has the standard form x + p(t)x = q(t) () (To be precise we
More informationDisorder-induced rounding of the phase transition. in the large-q-state Potts model. F. Iglói SZFKI - Budapest
Disorder-induced rounding of the phase transition in the large-q-state Potts model M.T. Mercaldo J-C. Anglès d Auriac Università di Salerno CNRS - Grenoble F. Iglói SZFKI - Budapest Motivations 2. CRITICAL
More informationAnalysis/resynthesis with the short time Fourier transform
Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TU-Berlin IRCAM Analysis/Synthesis
More informationFactoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
More informationLogarithmic and Exponential Equations
11.5 Logarithmic and Exponential Equations 11.5 OBJECTIVES 1. Solve a logarithmic equation 2. Solve an exponential equation 3. Solve an application involving an exponential equation Much of the importance
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column
More information2.1 The Present Value of an Annuity
2.1 The Present Value of an Annuity One example of a fixed annuity is an agreement to pay someone a fixed amount x for N periods (commonly months or years), e.g. a fixed pension It is assumed that the
More informationLecture notes on linear algebra
Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra
More information6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks
6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In
More informationStock price fluctuations and the mimetic behaviors of traders
Physica A 382 (2007) 172 178 www.elsevier.com/locate/physa Stock price fluctuations and the mimetic behaviors of traders Jun-ichi Maskawa Department of Management Information, Fukuyama Heisei University,
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver. Finite Difference Methods for Partial Differential Equations As you are well aware, most differential equations are much too complicated to be solved by
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationRobust Staff Level Optimisation in Call Centres
Robust Staff Level Optimisation in Call Centres Sam Clarke Jesus College University of Oxford A thesis submitted for the degree of M.Sc. Mathematical Modelling and Scientific Computing Trinity 2007 Abstract
More informationOpen Problems in Quantum Information Processing. John Watrous Department of Computer Science University of Calgary
Open Problems in Quantum Information Processing John Watrous Department of Computer Science University of Calgary #1 Open Problem Find new quantum algorithms. Existing algorithms: Shor s Algorithm (+ extensions)
More informationQuantum Computers. And How Does Nature Compute? Kenneth W. Regan 1 University at Buffalo (SUNY) 21 May, 2015. Quantum Computers
Quantum Computers And How Does Nature Compute? Kenneth W. Regan 1 University at Buffalo (SUNY) 21 May, 2015 1 Includes joint work with Amlan Chakrabarti, U. Calcutta If you were designing Nature, how would
More informationQuantum Monte Carlo and the negative sign problem
Quantum Monte Carlo and the negative sign problem or how to earn one million dollar Matthias Troyer, ETH Zürich Uwe-Jens Wiese, Universität Bern Complexity of many particle problems Classical 1 particle:
More informationPart 1: Link Analysis & Page Rank
Chapter 8: Graph Data Part 1: Link Analysis & Page Rank Based on Leskovec, Rajaraman, Ullman 214: Mining of Massive Datasets 1 Exam on the 5th of February, 216, 14. to 16. If you wish to attend, please
More informationWhat does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
More information1 Introduction. JS'12, Cnam Paris, 3-4 Avril 2012. Ourouk Jawad 1, David Lautru 2, Jean Michel Dricot, François Horlin, Philippe De Doncker 3
Statistical Study of SAR under Wireless Channel Exposure in Indoor Environment Etude Statistique du DAS sous Exposition d un Canal sans Fils en Environnement Intérieur Ourouk Jawad 1, David Lautru 2, Jean
More informationNumerology - A Case Study in Network Marketing Fractions
Vers l analyse statique de programmes numériques Sylvie Putot Laboratoire de Modélisation et Analyse de Systèmes en Interaction, CEA LIST Journées du GDR et réseau Calcul, 9-10 novembre 2010 Sylvie Putot
More informationAuger width of metastable states in antiprotonic helium II
«Избранные вопросы теоретической физики и астрофизики». Дубна: ОИЯИ, 2003. С. 153 158. Auger width of metastable states in antiprotonic helium II J. Révai a and A. T. Kruppa b a Research Institute for
More informationHow To Find The Optimal Control Function On A Unitary Operation
Quantum Computation as Geometry arxiv:quant-ph/0603161v2 21 Mar 2006 Michael A. Nielsen, Mark R. Dowling, Mile Gu, and Andrew C. Doherty School of Physical Sciences, The University of Queensland, Queensland
More informationLargest Fixed-Aspect, Axis-Aligned Rectangle
Largest Fixed-Aspect, Axis-Aligned Rectangle David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created: February 21, 2004 Last Modified: February
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationLecture 11: 0-1 Quadratic Program and Lower Bounds
Lecture : - Quadratic Program and Lower Bounds (3 units) Outline Problem formulations Reformulation: Linearization & continuous relaxation Branch & Bound Method framework Simple bounds, LP bound and semidefinite
More informationIntroduction to Group Theory with Applications in Molecular and Solid State Physics
Introduction to Group Theory with Applications in Molecular and Solid State Physics Karsten Horn Fritz-Haber-Institut der Max-Planck-Gesellschaft 3 84 3, e-mail horn@fhi-berlin.mpg.de Symmetry - old concept,
More informationTime Ordered Perturbation Theory
Michael Dine Department of Physics University of California, Santa Cruz October 2013 Quantization of the Free Electromagnetic Field We have so far quantized the free scalar field and the free Dirac field.
More informationOscillations of the Sending Window in Compound TCP
Oscillations of the Sending Window in Compound TCP Alberto Blanc 1, Denis Collange 1, and Konstantin Avrachenkov 2 1 Orange Labs, 905 rue Albert Einstein, 06921 Sophia Antipolis, France 2 I.N.R.I.A. 2004
More informationThe two dimensional heat equation
The two dimensional heat equation Ryan C. Trinity University Partial Differential Equations March 6, 2012 Physical motivation Consider a thin rectangular plate made of some thermally conductive material.
More informationFinancial Mathematics and Simulation MATH 6740 1 Spring 2011 Homework 2
Financial Mathematics and Simulation MATH 6740 1 Spring 2011 Homework 2 Due Date: Friday, March 11 at 5:00 PM This homework has 170 points plus 20 bonus points available but, as always, homeworks are graded
More information1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs
More informationCollatz Sequence. Fibbonacci Sequence. n is even; Recurrence Relation: a n+1 = a n + a n 1.
Fibonacci Roulette In this game you will be constructing a recurrence relation, that is, a sequence of numbers where you find the next number by looking at the previous numbers in the sequence. Your job
More informationUsing the ac Method to Factor
4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trial-and-error
More informationLet s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure 2. R. Figure 1.
Examples of Transient and RL Circuits. The Series RLC Circuit Impulse response of Circuit. Let s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure.
More information