First mean return time in decoherent quantum walks


 Tyler Arnold
 3 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 unistochastic 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 /
Chapter 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 informationLECTURE 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 informationPerformance Analysis, Autumn 2010
Performance Analysis, Autumn 2010 Bengt Jonsson November 16, 2010 Kendall Notation Queueing process described by A/B/X /Y /Z, where Example A is the arrival distribution B is the service pattern X the
More informationgeometric transforms
geometric transforms 1 linear algebra review 2 matrices matrix and vector notation use column for vectors m 11 =[ ] M = [ m ij ] m 21 m 12 m 22 =[ ] v v 1 v = [ ] T v 1 v 2 2 3 matrix operations addition
More informationLecture 2: Essential quantum mechanics
Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/ kvanttilaskenta/ p. 1/46 Quantum information and computing Lecture 2: Essential quantum mechanics JaniPetri Martikainen
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
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 informationUnit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12
Unit 3: Algebra Date Topic Page (s) Algebra Terminology Variables and Algebra Tiles 3 5 Like Terms 6 8 Adding/Subtracting Polynomials 9 1 Expanding Polynomials 13 15 Introduction to Equations 16 17 One
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 informationSolving Linear Recurrence Relations. Niloufar Shafiei
Solving Linear Recurrence Relations Niloufar Shafiei Review A recursive definition of a sequence specifies Initial conditions Recurrence relation Example: a 0 =0 and a 1 =3 Initial conditions a n = 2a
More informationH = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the
1 INTRODUCTION 1 1. Introduction In the discussion of random matrix theory and information theory, we basically explained the statistical aspect of ensembles of random matrices, The minimal information
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 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 informationFor the case of an Ndimensional spinor the vector α is associated to the onedimensional . N
1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. onhermitian
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 informationTons of Free Math Worksheets at:
Topic : Equations of Circles  Worksheet 1 form which has a center at (6, 4) and a radius of 5. x 28x+y 28y12=0 circle with a center at (4,4) and passes through the point (7, 3). center located at
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 twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
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 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 informationDO WE REALLY UNDERSTAND QUANTUM MECHANICS?
DO WE REALLY UNDERSTAND QUANTUM MECHANICS? COMPRENONSNOUS VRAIMENT LA MECANIQUE QUANTIQUE? VARIOUS INTERPRETATIONS OF QUANTUM MECHANICS IHES, 29 janvier 2015 Franck Laloë, LKB, ENS Paris 1 INTRODUCTION
More informationarxiv: v1 [physics.genph] 10 Jul 2015
Infinite circuits are easy. How about long ones? Mikhail Kagan, Xinzhe Wang Penn State Abington arxiv:507.08v [physics.genph] 0 Jul 05 Abstract We consider a long but finite ladder) circuit with alternating
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 13:22:3705:00 Copyright 2003 Dan
More informationSystems of Linear Equations
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Systems of Linear Equations Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION AttributionNonCommercialShareAlike (CC
More informationNumber Theory Homework.
Number Theory Homework. 1. Pythagorean triples and rational points on quadratics and cubics. 1.1. Pythagorean triples. Recall the Pythagorean theorem which is that in a right triangle with legs of length
More informationMarkov Chains, Stochastic Processes, and Advanced Matrix Decomposition
Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition Jack Gilbert Copyright (c) 2014 Jack Gilbert. Permission is granted to copy, distribute and/or modify this document under the terms
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 informationSolving Systems by Elimination 35
10/21/13 Solving Systems by Elimination 35 EXAMPLE: 5x + 2y = 1 x 3y = 7 1.Multiply the Top equation by the coefficient of the x on the bottom equation and write that equation next to the first equation
More informationheterogeneous diusion equation is a superposition of our previous solutions. The
Chapter 6 Absorption and Scattering When we have both variations in both absorption and scattering, the solution to the heterogeneous diusion equation is a superposition of our previous solutions The Born
More informationMth 95 Module 2 Spring 2014
Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression
More informationThe Essentials of Quantum Mechanics
The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008Oct22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum
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 informationSimplifying Radical Expressions
9.2 Simplifying Radical Expressions 9.2 OBJECTIVES. Simplify expressions involving numeric radicals 2. Simplify expressions involving algebraic radicals In Section 9., we introduced the radical notation.
More informationNotes: Tensor Operators
Notes: Tensor Operators Ben Baragiola I. VECTORS We are already familiar with the concept of a vector, but let s review vector properties to refresh ourselves. A component Cartesian vector is v = v x
More informationSYSTEMS OF EQUATIONS
SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which
More informationSPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BIDIAGONAL REPRESENTATIONS FOR PHASE TYPE DISTRIBUTIONS AND MATRIXEXPONENTIAL DISTRIBUTIONS
Stochastic Models, 22:289 317, 2006 Copyright Taylor & Francis Group, LLC ISSN: 15326349 print/15324214 online DOI: 10.1080/15326340600649045 SPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BIDIAGONAL
More information05 Systems of Linear Equations and Inequalities
1. Solve each system of equations by graphing. Graph each equation in the system. The lines intersect at the point (1, 3). This ordered pair is the solution of the system. CHECK esolutions Manual  Powered
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 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 informationGENERALIZED PYTHAGOREAN TRIPLES AND PYTHAGOREAN TRIPLE PRESERVING MATRICES. Mohan Tikoo and Haohao Wang
VOLUME 1, NUMBER 1, 009 3 GENERALIZED PYTHAGOREAN TRIPLES AND PYTHAGOREAN TRIPLE PRESERVING MATRICES Mohan Tikoo and Haohao Wang Abstract. Traditionally, Pythagorean triples (PT) consist of three positive
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 information56 The Remainder and Factor Theorems
Use synthetic substitution to find f (4) and f ( 2) for each function. 1. f (x) = 2x 3 5x 2 x + 14 Divide the function by x 4. The remainder is 58. Therefore, f (4) = 58. Divide the function by x + 2.
More informationLecture 18: Quantum Mechanics. Reading: Zumdahl 12.5, 12.6 Outline. Problems (Chapter 12 Zumdahl 5 th Ed.)
Lecture 18: Quantum Mechanics Reading: Zumdahl 1.5, 1.6 Outline Basic concepts of quantum mechanics and molecular structure A model system: particle in a box. Demos how Q.M. actually obtains a wave function.
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 informationCharacterizations are given for the positive and completely positive maps on n n
Special Classes of Positive and Completely Positive Maps ChiKwong Li and Hugo J. Woerdemany Department of Mathematics The College of William and Mary Williamsburg, Virginia 387 Email: ckli@cs.wm.edu
More informationOn the general equation of the second degree
On the general equation of the second degree S Kesavan The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai  600 113 email:kesh@imscresin Abstract We give a unified treatment of the
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 information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationWatch Your Step You May Collide!
Mike Blakely Mathematics Academy Lesson Plan: Solving Systems of Equations Graphically Cover Page Watch Your Step You May Collide! Introduction: In Algebra systems of equations are taught graphically,
More informationYoung s Two Slit Interference
4/7/ Young s Two Slit Interference Equipment Needed Flashlight Jack, Table Laser, HeNe Meter stick Multiple Slit Set Pasco Optics Bench Pasco Viewing Screen, Optics Bench Power Supply, Laser HeNe Ruler,
More informationThis page must be the first page of your solutions!
ENERGY TRANSFER Submission deadline of December Total number of points = 9 +10+ 34 + 13+30+23 = 119 Name: FRET Max Real 5,1 A 3 B 3 C 3 5,2 A 3 B 3 C 4 5,3 A 15 B 5 C 8 D 6 5,4 A 3 B 5 C 5 5,5 30 5,6 A
More informationSolution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute
Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) 1/2 x, x 1, 2, 3,..., zero elsewhere, be the pmf of the random variable X. Find the mgf, the mean, and the variance of X. Solution 1.9.2. Using the
More informationThe counterpart to a DAC is the ADC, which is generally a more complicated circuit. One of the most popular ADC circuit is the successive
The counterpart to a DAC is the ADC, which is generally a more complicated circuit. One of the most popular ADC circuit is the successive approximation converter. 1 2 The idea of sampling is fully covered
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 informationDifference of Squares and Perfect Square Trinomials
4.4 Difference of Squares and Perfect Square Trinomials 4.4 OBJECTIVES 1. Factor a binomial that is the difference of two squares 2. Factor a perfect square trinomial In Section 3.5, we introduced some
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 informationarxiv: v1 [physics.genph] 29 Aug 2016
arxiv:160907984v1 [physicsgenph] 9 Aug 016 No quantum process can explain the existence of the preferred basis: decoherence is not universal Hitoshi Inamori Société Générale Boulevard Franck Kupka, 9800
More information5.1 The Unit Circle. Copyright Cengage Learning. All rights reserved.
5.1 The Unit Circle Copyright Cengage Learning. All rights reserved. Objectives The Unit Circle Terminal Points on the Unit Circle The Reference Number 2 The Unit Circle In this section we explore some
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 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 informationGroups, Rings, and Fields. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, S S = {(x, y) x, y S}.
Groups, Rings, and Fields I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ : S S S. A binary
More informationINTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28
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 informationWe consider a hydrogen atom in the ground state in the uniform electric field
Lecture 13 Page 1 Lectures 1314 Hydrogen atom in electric field. Quadratic Stark effect. Atomic polarizability. Emission and Absorption of Electromagnetic Radiation by Atoms Transition probabilities and
More informationThe Australian Journal of Mathematical Analysis and Applications
The Australian Journal of Mathematical Analysis and Applications Volume 7, Issue, Article 11, pp. 114, 011 SOME HOMOGENEOUS CYCLIC INEQUALITIES OF THREE VARIABLES OF DEGREE THREE AND FOUR TETSUYA ANDO
More informationAnswer on Question #48173 Math Algebra
Answer on Question #48173 Math Algebra On graph paper, draw the axes, and the lines y = 12 and x = 6. The rectangle bounded by the axes and these two lines is a pool table with pockets in the four corners.
More informationTangent line of a circle can be determined once the tangent point or the slope of the line is known.
Worksheet 7: Tangent Line of a Circle Name: Date: Tangent line of a circle can be determined once the tangent point or the slope of the line is known. Straight line: an overview General form : Ax + By
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 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 informationit s refraction, it s diffraction, it s a kinoform lens  new concepts in focusing xrays detlef smilgies chess
it s refraction, it s diffraction, it s a kinoform lens  new concepts in focusing xrays detlef smilgies chess what is a lens? incoherent source > geometric optics > refraction Snell s law > lensmaker
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 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 information4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved.
4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions. Sketch
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 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 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 informationCSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo
Propositional Logic CSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 37 Discrete Mathematics What is Discrete
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 informationBellType Quantum Field Theory
BellType Quantum Field Theory Work with Detlef Dürr, Shelly Goldstein, & Roderich Tumulka Nino Zanghì Università di Genova 3rd International Summer School in Philosophy of Physics: The Ontology of Physics,
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 information10 Quadratic Equations and
www.ck1.org Chapter 10. Quadratic Equations and Quadratic Functions CHAPTER 10 Quadratic Equations and Quadratic Functions Chapter Outline 10.1 GRAPHS OF QUADRATIC FUNCTIONS 10. QUADRATIC EQUATIONS BY
More informationCorinne: I m thinking of a number between 220 and 20. What s my number? Benjamin: Is it 25?
Walk the Line Adding Integers, Part I Learning Goals In this lesson, you will: Model the addition of integers on a number line. Develop a rule for adding integers. Corinne: I m thinking of a number between
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 informationImpurities in the Heisenberg magnet and the general recipe of Weyl
Impurities in the Heisenberg magnet and the general recipe of Weyl B. Lulek Division of Mathematical Physics, The Institute of Physics A. Mickiewicz University, Poznań, Poland Abstract The general recipe
More informationModule 3: SecondOrder Partial Differential Equations
Module 3: SecondOrder Partial Differential Equations In Module 3, we shall discuss some general concepts associated with secondorder linear PDEs. These types of PDEs arise in connection with various
More informationThe Ramsey Discounting Formula for a HiddenState Stochastic Growth Process / 8
The Ramsey Discounting Formula for a HiddenState Stochastic Growth Process Martin L. Weitzman May 2012 Bergen Conference LongTerm Social Discount Rates What is Approach of This Paper? Increasing fuzziness
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 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 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 informationChapter 8 Graphs and Functions:
Chapter 8 Graphs and Functions: Cartesian axes, coordinates and points 8.1 Pictorially we plot points and graphs in a plane (flat space) using a set of Cartesian axes traditionally called the x and y axes
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 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 informationLINEAR ALGEBRA OF PASCAL MATRICES
LINEAR ALGEBRA OF PASCAL MATRICES LINDSAY YATES Abstract. The famous Pascal s triangle appears in many areas of mathematics, such as number theory, combinatorics and algebra. Pascal matrices are derived
More informationJoint distributions Math 217 Probability and Statistics Prof. D. Joyce, Fall 2014
Joint distributions Math 17 Probability and Statistics Prof. D. Joyce, Fall 14 Today we ll look at joint random variables and joint distributions in detail. Product distributions. If Ω 1 and Ω are sample
More informationFrom Random Matrices to Geometry: the "topological recursion"
From Random Matrices to Geometry: the "topological recursion" Bertrand Eynard, IPHT CEA Saclay, CERN Brunel workshop on random matri theory december 17th 1 Contents 1. Random Matrices, statistical properties
More informationMathematics Notes for Class 12 chapter 3. Matrices
1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form
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 information