N 1. (q k+1 q k ) 2 + α 3. k=0


 MargaretMargaret Betty Grant
 2 years ago
 Views:
Transcription
1 Teoretisk Fysik Handin problem B, SI1142, Spring 2010 In 1955 Fermi, Pasta and Ulam 1 numerically studied a simple model for a one dimensional chain of nonlinear oscillators to see how the energy distribution is randomized in a system with many degrees of freedom. Instead of the rapid transition to a uniform energy distribution one would naively expect, they observed an extremely slow randomization. In subsequent work by Zabusky and Kruskal 2 the relation of the FermiPastaUlam model to the Kortewegde Vries (KdV) equation was noted, and in a numeric solution of the KdV equation the appearance of solitary wave pulses was observed which they named solitons. This work motivated the search for a analytic solution of the KdV equation which was achieved two years later. 3 This latter work started a period of very intensive research on a class of nonlinear partial differential equations which today are known as soliton equations. The FermiPastaUlam model describes a weakly nonlinear, fixedend onedimensional chain of (N 1) moving mass points and is given by the Hamiltonian H = k=1 1 2m p2 k + K 2 k=0 (q k+1 q k ) 2 + α 3 k=0 (q k+1 q k ) 3 (1) where q 0 q N 0 and q k and p k are the coordinate and momentum for the kth particle, and the parameter α is a small nonlinear coupling; the positive parameters m and K are the particle mass and spring constant, respectively. As was realized by Zabusky and Kruskal (ZK), the equations of motion following from this Hamiltonian (the dots mean differentiation with respect to time t), m q k = K(q k+1 2q k + q k 1 ) + α[(q k+1 q k ) 2 (q k q k 1 ) 2 ] (2) in the lowestorder continuum limit take the following form, q tt = (c ɛq x )q xx (3) with some constants c 0 and ɛ. Eq. (3) can be viewed as ordinary wave equation whose wave speed c depends on the spatial derivative q x, c 2 = (c ɛq x). Typical solutions of this equations for positive ɛ describe pulses moving to the left or right while changing shape and steepening until their leading edge develops a vertical shock front (like water waves in the sea before they break close to the beach), at which point Eq. (3) looses validity. Nonetheless, prior to the formation of the shock, it was found that Eq. (3) provides a quite reasonable description of the Fermi PastaUlam model behavior. ZK found that, to avoid shock wave behavior, one has to take into account dissipation, which can be done by including the nextorder spatial derivative term, q tt = c 2 0q xx + ɛq x q xx + βq xxxx (4) for some constant β. For both convenience and simplicity, ZK now insist on periodic boundary conditions, restrict their attention to waves traveling on one direction only, and elect a moving frame. After replacing x by x = x c 1 t, t by t = c 2 t, q x (x, t) by c 3 u( x, t) in Eq. (4) with particular constants c j and neglecting terms proportional to ɛ 2 they obtained the KdV equation in the following form, u t + uu x + δ 2 u x x x = 0, with δ a (small) constant. In the following we will write this equation as u t + uu x + δ 2 u xxx = 0, u = u(x, t) (5) 1 Los Alamos Scientific Report, unpublished 2 Zabusky and Kruskal, Phys. Rev. Lett. 15, 240 (1965) 3 Gardner, Greene, Kruskal and Miura, Phys. Rev. Lett. 19, 1095 (1967) 1
2 with u = u(x, t), to simplify notation. In their short paper referred to above, ZK first recalled the welllocalized traveling wave solutions of the the KdV equation, with U(x) = u(x, t) = U(x ct) (6) 3c cosh 2 ( c(x x 0 )/2δ), (7) which they describe as solitary wave pulses. The amplitude of such a pulse is proportional to its velocity c > 0, and the constant x 0 equal to the location of its centerofmass at t = 0. ZK then discuss the equation following from (3) without dissipative term, together with the initial condition Using the fact that u t + uu x = 0, (8) u(0, x) = cos πx. (9) u(t, x) = f(x u(t, x)t) (10) is an implicit solution of (8) for any differentiable function f, they find that the solution of Eqs. (8) and (9) tends to become discontinuous at x = 1/2 and t = T B = 1/π. They then describe their results of a numeric solution of the KdV equation with small but nonzero dissipative term, δ = (11) They find that for times smaller than the breakdown time T B, the solution is nearly the same as for δ = 0, but then the dissipative term becomes important. For nonzero δ there is no breakdown any more, but instead solitary wave pulses appear: after a while the solution looks very much like a superposition of special solutions in Eqs. (6) and (7), with different values of c and x 0, 4 and the localized pulses are very stable, preserving shape and speed always except when two of them meet, in which case there is a certain interaction period in which the pulses merge. After such interactions the individual pulses reappear which the same shapes and velocities as before the interaction. ZK also observe that, during the interaction period, the joint amplitude of the interacting solitons decreases, in contradistinction to what would happen if the pulses overlapped linearly. In this problem you are asked to fill in the details of the derivations outlined above (part A). You are then asked to solve the KdV equation numerically and, in particular, reproduce the ZK result (part B). More specific instructions together with further hints will be given below. 4 Note that this is very remarkable since the KdV equation is nonlinear, and the superposition principle therefore does not hold. 2
3 Part A: To do: Derive Eq. (2) from Eq. (1). Hint: Use the following variational principle (= Hamiltonian principle), ( t2 N 1 ) δ dt p k (t) q k (t) H(p 1 (t),..., p N 1 (t), q 1 (t),..., q N 1 (t)) t 1 k=1 = 0. (12) From this obtain 2(N 1) first order differential equations of the form q k =..., ṗ k =... (= Hamilton equations), which, after eliminating half of the variables, yield (2). Derive in detail the continuum limit of Eq. (2) as described above. Hint: To obtain the continuum limit in Eq. (3), introduce x = ka where a > 0 is the lattice constant (i.e. the distance of adjacent mass points when the chain is not moving), and write q(t, x) = q k (t) where x = ka. Approximate q k±1 (t) = q(t, x ± a) by a Taylor series, keeping only certain lower order derivative terms. Do not forget to give the relation between the parameters (i.e. write down the formulas for the parameters c 0 and ɛ in Eq. (3) in terms of m, K, a, α, etc.). Generalize the argument to get Eq. (4). Give then all the details leading to Eq. (5). Verify in detail the shock wave behavior of Eq. (8), as described above. Hint: Verify that Eq. (10) is an implicit solution of Eq. (8). Use (10) to sketch the solution of Eqs. (8) and (9) for a few different times between 0 and T B = 1/π, to show the steepening of the wave. Verify that the wave breaks at the time T B. (If you cannot show this analytically, you can also hand in numeric verifications of these facts produced with the help of MATLAB.) Derive the solutions in Eqs. (6) and (7) from the KdV equation (5). Hint: This is easy if you read Chapter 9 in FMM, of course. Part (B): To do: Solve Eq. (5) for δ = numerically by using the finite difference method (FDM; see Section 8.1 in FMM), for 0 x 2 and periodic boundary conditions, u(x + 2, t) = u(x, t). Test your program by comparing the numeric solution with the analytical solution given in Eqs. (6) and (7) for x 0 = 0.4 and some value of c for times 0 t 0.6/c. Hand in a plot showing your result together with a printout of your MATLAB program. Hint: We suggest you use the following discretization of the KdV equation, with u i,j+1 = u i,j k 6h (u i+1,j + u i,j + u i 1,j ) (u i+1,j u i 1,j ) δ 2 k 2h 3 (u i+2,j 2u i+1,j + 2u i 1,j u i 2,j ) (13) x i = ih, h = 2, i = 0, 1, 2... N 1 N t j = jk, k = t max, j = 0, 1, 2... M (14) M 3
4 and u i+n,j = u i,j (periodic boundary conditions). The initial condition is u i,0 = u(x i, 0), of course. Show that Eq. (13) indeed is a discretization of Eq. (5). To solve these equations on a computer, we suggest you use MATLAB (you can also use other program tools if you prefer). A similar MATLAB program which you can use as a starting point can be found on the course homepage. You will find that not all values of c, h and k are numerically stable, and you need to experiment to find good values. Values that are numerically stable are e.g. c = 0.1, h = 0.06, k = 0.005, (15) but you can get better results with smaller values for h and k. 5 program by either decreasing the ratio k/h 3 or/and decreasing c. We could stabilize our Find a simpler FDM discretization of the KdV equation. Implement your discretization in MATLAB (e.g.) and test your program. Hand in your program and one representative plot showing your test results. 6 Hand in plots showing your numerical solution of the KdV equation with the following initial conditions, (i) u(x, 0) = A cos(πx) 0 t 2/(Aπ) (ii) u(x, 0) = 3U(x) for x 0 = 0.4, 0 t 0.6/c (iii) u(x, 0) = U(x) for x 0 = 0.4, 0 t 0.6/c (16) with U(x) in Eq. (7), and some value of c > 0 and A > 0 which you choose, showing the solution in a six equidistant times (including initial and final time) in the given time interval. Discuss your results. Hint: ZK give the result for (i) with A = 1 which therefore would be the preferred choice, but to get reasonable results for that you need much smaller h and k values than the ones given above. If you computer cannot handle that choose a smaller values for A like A = 0.1. The initial condition u(x, 0) = 1 2n(n + 1)U(x) for n = 1, 2,... is known to be a nsoliton solution at the moment when all solitons are at the same spot (nsoliton collision). For t > 0 the solitons move to the right and separate. For negative initial condition one does not have any solitons but only dispersive waves moving to the left. Try our different initial conditions and different values for k and h in your program. Hand in one plot (iv) with an initial condition of your own choice which you find interesting. Hint: It is fine if you choose u(x, 0) = AU(x) for some other parameters of c, x 0 and A > 0, but perhaps you find something more interesting. Results from an unstable program are not interesting, of course. 5 If your computer is slow and cannot handle such small values of h and k it is OK to have larger values. 6 A negative result (program based on your discretization cannot be made stable) is OK if described properly. 4
5 GENERAL INSTRUCTIONS: A MAIN GOAL IN THIS PROJECT IS TO WRITE A GOOD SCIENTIFIC REPORT. Your solution should be selfcontained and preferably hand written (except for the program code, of course). Give a clear description of your reasoning. Your plots should have a figure caption explaining what they display, and the axes should be labeled. (The plot produced by the MATLAB program provided by us is not OK as it comes out of the program! It is OK to write the labels and figure captions by hand.) Please put your papers together (no loose pages)! Collaborations are allowed but only up to two students in one team, and each of the students should hand in his/her own report. If you collaborate and/or hand in a report in LaTex or the like you should be prepared to be asked to explain your solution in an oral presentation. Do not forget to write name, ADDRESS and personal number on your papers. 5
Finite Difference Approach to Option Pricing
Finite Difference Approach to Option Pricing February 998 CS5 Lab Note. Ordinary differential equation An ordinary differential equation, or ODE, is an equation of the form du = fut ( (), t) (.) dt where
More informationIntroduction to Engineering System Dynamics
CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are
More informationDynamics. Basilio Bona. DAUINPolitecnico di Torino. Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30
Dynamics Basilio Bona DAUINPolitecnico di Torino 2009 Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30 Dynamics  Introduction In order to determine the dynamics of a manipulator, it is
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationPlate waves in phononic crystals slabs
Acoustics 8 Paris Plate waves in phononic crystals slabs J.J. Chen and B. Bonello CNRS and Paris VI University, INSP  14 rue de Lourmel, 7515 Paris, France chen99nju@gmail.com 41 Acoustics 8 Paris We
More informationThe integrating factor method (Sect. 2.1).
The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable
More informationPSTricks. pstode. A PSTricks package for solving initial value problems for sets of Ordinary Differential Equations (ODE), v0.7.
PSTricks pstode A PSTricks package for solving initial value problems for sets of Ordinary Differential Equations (ODE), v0.7 27th March 2014 Package author(s): Alexander Grahn Contents 2 Contents 1 Introduction
More informationHomotopy Perturbation Method for Solving Partial Differential Equations with Variable Coefficients
Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 28, 13951407 Homotopy Perturbation Method for Solving Partial Differential Equations with Variable Coefficients Lin Jin Modern Educational Technology
More informationApplications to Data Smoothing and Image Processing I
Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is
More information2.5 Physicallybased Animation
2.5 Physicallybased Animation 320491: Advanced Graphics  Chapter 2 74 Physicallybased animation Morphing allowed us to animate between two known states. Typically, only one state of an object is known.
More information1.3.1 Position, Distance and Displacement
In the previous section, you have come across many examples of motion. You have learnt that to describe the motion of an object we must know its position at different points of time. The position of an
More informationThe dynamic equation for the angular motion of the wheel is R w F t R w F w ]/ J w
Chapter 4 Vehicle Dynamics 4.. Introduction In order to design a controller, a good representative model of the system is needed. A vehicle mathematical model, which is appropriate for both acceleration
More informationIntroduction to SME and Scattering Theory. Don Colladay. New College of Florida Sarasota, FL, 34243, U.S.A.
June 2012 Introduction to SME and Scattering Theory Don Colladay New College of Florida Sarasota, FL, 34243, U.S.A. This lecture was given at the IUCSS summer school during June of 2012. It contains a
More informationTCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS
TCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS 1. Bandwidth: The bandwidth of a communication link, or in general any system, was loosely defined as the width of
More informationPhysics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE
1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object
More informationLecture 11. 3.3.2 Cost functional
Lecture 11 3.3.2 Cost functional Consider functions L(t, x, u) and K(t, x). Here L : R R n R m R, K : R R n R sufficiently regular. (To be consistent with calculus of variations we would have to take L(t,
More informationA First Course in Elementary Differential Equations. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A First Course in Elementary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction Field of y = f(t, y)
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationPHYS 1624 University Physics I. PHYS 2644 University Physics II
PHYS 1624 Physics I An introduction to mechanics, heat, and wave motion. This is a calculus based course for Scientists and Engineers. 4 hours (3 lecture/3 lab) Prerequisites: Credit for MATH 2413 (Calculus
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.
More informationOptimization of Supply Chain Networks
Optimization of Supply Chain Networks M. Herty TU Kaiserslautern September 2006 (2006) 1 / 41 Contents 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationCFD modelling of floating body response to regular waves
CFD modelling of floating body response to regular waves Dr Yann Delauré School of Mechanical and Manufacturing Engineering Dublin City University Ocean Energy Workshop NUI Maynooth, October 21, 2010 Table
More informationIncorporating Internal Gradient and Restricted Diffusion Effects in Nuclear Magnetic Resonance Log Interpretation
The OpenAccess Journal for the Basic Principles of Diffusion Theory, Experiment and Application Incorporating Internal Gradient and Restricted Diffusion Effects in Nuclear Magnetic Resonance Log Interpretation
More informationRobert Collins CSE598G. More on Meanshift. R.Collins, CSE, PSU CSE598G Spring 2006
More on Meanshift R.Collins, CSE, PSU Spring 2006 Recall: Kernel Density Estimation Given a set of data samples x i ; i=1...n Convolve with a kernel function H to generate a smooth function f(x) Equivalent
More informationHigh Speed Aerodynamics Prof. K. P. Sinhamahapatra Department of Aerospace Engineering Indian Institute of Technology, Kharagpur
High Speed Aerodynamics Prof. K. P. Sinhamahapatra Department of Aerospace Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 06 Onedimensional Gas Dynamics (Contd.) We
More informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationAP Physics 1 and 2 Lab Investigations
AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks
More informationKinetic effects in the turbulent solar wind: capturing ion physics with a Vlasov code
Kinetic effects in the turbulent solar wind: capturing ion physics with a Vlasov code Francesco Valentini francesco.valentini@fis.unical.it S. Servidio, D. Perrone, O. Pezzi, B. Maruca, F. Califano, W.
More informationNetwork Traffic Modelling
University of York Dissertation submitted for the MSc in Mathematics with Modern Applications, Department of Mathematics, University of York, UK. August 009 Network Traffic Modelling Author: David Slade
More information9. Sampling Distributions
9. Sampling Distributions Prerequisites none A. Introduction B. Sampling Distribution of the Mean C. Sampling Distribution of Difference Between Means D. Sampling Distribution of Pearson's r E. Sampling
More informationMaximum likelihood estimation of mean reverting processes
Maximum likelihood estimation of mean reverting processes José Carlos García Franco Onward, Inc. jcpollo@onwardinc.com Abstract Mean reverting processes are frequently used models in real options. For
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationNumerical Wave Generation In OpenFOAM R
Numerical Wave Generation In OpenFOAM R Master of Science Thesis MOSTAFA AMINI AFSHAR Department of Shipping and Marine Technology CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden, 2010 Report No. X10/252
More informationVehicleBridge Interaction Dynamics
VehicleBridge Interaction Dynamics With Applications to HighSpeed Railways Y. B. Yang National Taiwan University, Taiwan J. D. Yau Tamkang University, Taiwan Y. S. Wu Sinotech Engineering Consultants,
More informationτ θ What is the proper price at time t =0of this option?
Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min
More informationLecture 8 ELE 301: Signals and Systems
Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 22 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 22 / 37 Properties of the Fourier Transform Properties of the Fourier
More informationCS 29473 Software Engineering for Scientific Computing. http://www.cs.berkeley.edu/~colella/cs294fall2013. Lecture 16: Particle Methods; Homework #4
CS 29473 Software Engineering for Scientific Computing http://www.cs.berkeley.edu/~colella/cs294fall2013 Lecture 16: Particle Methods; Homework #4 Discretizing TimeDependent Problems From here on in,
More informationVARIANCE REDUCTION TECHNIQUES FOR IMPLICIT MONTE CARLO SIMULATIONS
VARIANCE REDUCTION TECHNIQUES FOR IMPLICIT MONTE CARLO SIMULATIONS An Undergraduate Research Scholars Thesis by JACOB TAYLOR LANDMAN Submitted to Honors and Undergraduate Research Texas A&M University
More informationA) F = k x B) F = k C) F = x k D) F = x + k E) None of these.
CT161 Which of the following is necessary to make an object oscillate? i. a stable equilibrium ii. little or no friction iii. a disturbance A: i only B: ii only C: iii only D: i and iii E: All three Answer:
More informationNotes for AA214, Chapter 7. T. H. Pulliam Stanford University
Notes for AA214, Chapter 7 T. H. Pulliam Stanford University 1 Stability of Linear Systems Stability will be defined in terms of ODE s and O E s ODE: Couples System O E : Matrix form from applying Eq.
More informationPresentation of problem T1 (9 points): The Maribo Meteorite
Presentation of problem T1 (9 points): The Maribo Meteorite Definitions Meteoroid. A small particle (typically smaller than 1 m) from a comet or an asteroid. Meteorite: A meteoroid that impacts the ground
More information(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7)
Chapter 4. Lagrangian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7 4.1 Important Notes on Notation In this chapter, unless otherwise stated, the following
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More informationNCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )
Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates
More informationBINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract
BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple
More information2DI36 Statistics. 2DI36 Part II (Chapter 7 of MR)
2DI36 Statistics 2DI36 Part II (Chapter 7 of MR) What Have we Done so Far? Last time we introduced the concept of a dataset and seen how we can represent it in various ways But, how did this dataset came
More information2After completing this chapter you should be able to
After completing this chapter you should be able to solve problems involving motion in a straight line with constant acceleration model an object moving vertically under gravity understand distance time
More informationOrbital Mechanics. Angular Momentum
Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely
More informationBayesian Adaptive Trading with a Daily Cycle
Bayesian Adaptive Trading with a Daily Cycle Robert Almgren and Julian Lorenz July 28, 26 Abstract Standard models of algorithmic trading neglect the presence of a daily cycle. We construct a model in
More informationNonlinear Modal Analysis of Mechanical Systems with Frictionless Contact Interfaces
Nonlinear Modal Analysis of Mechanical Systems with Frictionless Contact Interfaces Denis Laxalde, Mathias Legrand and Christophe Pierre Structural Dynamics and Vibration Laboratory Department of Mechanical
More informationStocks paying discrete dividends: modelling and option pricing
Stocks paying discrete dividends: modelling and option pricing Ralf Korn 1 and L. C. G. Rogers 2 Abstract In the BlackScholes model, any dividends on stocks are paid continuously, but in reality dividends
More informationMath 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Practice Test, 1/28/2008 (with solutions)
Math 370, Actuarial Problemsolving Spring 008 A.J. Hildebrand Practice Test, 1/8/008 (with solutions) About this test. This is a practice test made up of a random collection of 0 problems from past Course
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 informationValuation of American Options
Valuation of American Options Among the seminal contributions to the mathematics of finance is the paper F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationNonlinear physics (solitons, chaos, discrete breathers)
Nonlinear physics (solitons, chaos, discrete breathers) N. Theodorakopoulos Konstanz, June 2006 Contents Foreword vi 1 Background: Hamiltonian mechanics 1 1.1 Lagrangian formulation of dynamics.......................
More informationAssignment 2: Option Pricing and the BlackScholes formula The University of British Columbia Science One CS 20152016 Instructor: Michael Gelbart
Assignment 2: Option Pricing and the BlackScholes formula The University of British Columbia Science One CS 20152016 Instructor: Michael Gelbart Overview Due Thursday, November 12th at 11:59pm Last updated
More informationComputational Statistics and Data Analysis
Computational Statistics and Data Analysis 53 (2008) 17 26 Contents lists available at ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Coverage probability
More informationi=1 In practice, the natural logarithm of the likelihood function, called the loglikelihood function and denoted by
Statistics 580 Maximum Likelihood Estimation Introduction Let y (y 1, y 2,..., y n be a vector of iid, random variables from one of a family of distributions on R n and indexed by a pdimensional parameter
More informationThe Backpropagation Algorithm
7 The Backpropagation Algorithm 7. Learning as gradient descent We saw in the last chapter that multilayered networks are capable of computing a wider range of Boolean functions than networks with a single
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationDYNAMIC RESPONSE OF VEHICLETRACK COUPLING SYSTEM WITH AN INSULATED RAIL JOINT
11 th International Conference on Vibration Problems Z. Dimitrovová et al. (eds.) Lisbon, Portugal, 912 September 2013 DYNAMIC RESPONSE OF VEHICLETRACK COUPLING SYSTEM WITH AN INSULATED RAIL JOINT Ilaria
More informationRock Bolt Condition Monitoring Using Ultrasonic Guided Waves
Rock Bolt Condition Monitoring Using Ultrasonic Guided Waves Bennie Buys Department of Mechanical and Aeronautical Engineering University of Pretoria Introduction Rock Bolts and their associated problems
More informationState Newton's second law of motion for a particle, defining carefully each term used.
5 Question 1. [Marks 28] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding
More informationOptimal Control of Switched Networks for Nonlinear Hyperbolic Conservation Laws
Optimal Control of Switched Networks for Nonlinear Hyperbolic Conservation Laws Stefan Ulbrich TU Darmstadt Günter Leugering Universität ErlangenNürnberg SPP 1253 Optimization with Partial Differential
More informationThe Basics of Interest Theory
Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest
More informationFLOODING AND DRYING IN DISCONTINUOUS GALERKIN DISCRETIZATIONS OF SHALLOW WATER EQUATIONS
European Conference on Computational Fluid Dynamics ECCOMAS CFD 26 P. Wesseling, E. Oñate and J. Périaux (Eds) c TU Delft, The Netherlands, 26 FLOODING AND DRING IN DISCONTINUOUS GALERKIN DISCRETIZATIONS
More informationECE 533 Project Report Ashish Dhawan Aditi R. Ganesan
Handwritten Signature Verification ECE 533 Project Report by Ashish Dhawan Aditi R. Ganesan Contents 1. Abstract 3. 2. Introduction 4. 3. Approach 6. 4. Preprocessing 8. 5. Feature Extraction 9. 6. Verification
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 informationIntroduction to CFD Basics
Introduction to CFD Basics Rajesh Bhaskaran Lance Collins This is a quickanddirty introduction to the basic concepts underlying CFD. The concepts are illustrated by applying them to simple 1D model problems.
More informationIs there chaos in Copenhagen problem?
Monografías de la Real Academia de Ciencias de Zaragoza 30, 43 50, (2006). Is there chaos in Copenhagen problem? Roberto Barrio, Fernando Blesa and Sergio Serrano GME, Universidad de Zaragoza Abstract
More informationNotes on Elastic and Inelastic Collisions
Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just
More informationMODELING, SIMULATION AND DESIGN OF CONTROL CIRCUIT FOR FLEXIBLE ENERGY SYSTEM IN MATLAB&SIMULINK
MODELING, SIMULATION AND DESIGN OF CONTROL CIRCUIT FOR FLEXIBLE ENERGY SYSTEM IN MATLAB&SIMULINK M. Pies, S. Ozana VSBTechnical University of Ostrava Faculty of Electrotechnical Engineering and Computer
More informationUSING MS EXCEL FOR DATA ANALYSIS AND SIMULATION
USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION Ian Cooper School of Physics The University of Sydney i.cooper@physics.usyd.edu.au Introduction The numerical calculations performed by scientists and engineers
More informationQuasistatic evolution and congested transport
Quasistatic evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed
More information3. Experimental Results
Experimental study of the wind effect on the focusing of transient wave groups J.P. Giovanangeli 1), C. Kharif 1) and E. Pelinovsky 1,) 1) Institut de Recherche sur les Phénomènes Hors Equilibre, Laboratoire
More informationMathematical Modeling and Engineering Problem Solving
Mathematical Modeling and Engineering Problem Solving Berlin Chen Department of Computer Science & Information Engineering National Taiwan Normal University Reference: 1. Applied Numerical Methods with
More informationCHAPTER 1 Splines and Bsplines an Introduction
CHAPTER 1 Splines and Bsplines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the
More informationComputer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science  Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: WireFrame Representation Object is represented as as a set of points
More informationThe mhr model is described by 30 ordinary differential equations (ODEs): one. ion concentrations and 23 equations describing channel gating.
Online Supplement: Computer Modeling Chris Clausen, PhD and Ira S. Cohen, MD, PhD Computer models of canine ventricular action potentials The mhr model is described by 30 ordinary differential equations
More informationDynamic Process Modeling. Process Dynamics and Control
Dynamic Process Modeling Process Dynamics and Control 1 Description of process dynamics Classes of models What do we need for control? Modeling for control Mechanical Systems Modeling Electrical circuits
More informationThe finite element immersed boundary method: model, stability, and numerical results
Te finite element immersed boundary metod: model, stability, and numerical results Lucia Gastaldi Università di Brescia ttp://dm.ing.unibs.it/gastaldi/ INdAM Worksop, Cortona, September 18, 2006 Joint
More informationHello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation.
Hello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation. 1 Without any doubts human capital is a key factor of economic growth because
More informationPhysics 2A, Sec B00: Mechanics  Winter 2011 Instructor: B. Grinstein Final Exam
Physics 2A, Sec B00: Mechanics  Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry
More informationOPTIMAL CONTROL IN DISCRETE PEST CONTROL MODELS
OPTIMAL CONTROL IN DISCRETE PEST CONTROL MODELS Author: Kathryn Dabbs University of Tennessee (865)4697 katdabbs@gmail.com Supervisor: Dr. Suzanne Lenhart University of Tennessee (865)974427 lenhart@math.utk.edu.
More informationParameter Estimation for Bingham Models
Dr. Volker Schulz, Dmitriy Logashenko Parameter Estimation for Bingham Models supported by BMBF Parameter Estimation for Bingham Models Industrial application of ceramic pastes Material laws for Bingham
More informationCross section, Flux, Luminosity, Scattering Rates
Cross section, Flux, Luminosity, Scattering Rates Table of Contents Paul Avery (Andrey Korytov) Sep. 9, 013 1 Introduction... 1 Cross section, flux and scattering... 1 3 Scattering length λ and λ ρ...
More informationTutorial Creating a regular grid for point sampling
This tutorial describes how to use the fishnet, clip, and optionally the buffer tools in ArcGIS 10 to generate a regularlyspaced grid of sampling points inside a polygon layer. The steps below should
More informationNumerical modeling of damreservoir interaction seismic response using the Hybrid Frequency Time Domain (HFTD) method
Numerical modeling of damreservoir interaction seismic response using the Hybrid Frequency Time Domain (HFTD) method Graduation Committee: Prof. Dr. A.V. Metrikine Dr. Ir. M.A.N. Hriks Dr. Ir. E.M. Lourens
More informationAn OnLine Algorithm for Checkpoint Placement
An OnLine Algorithm for Checkpoint Placement Avi Ziv IBM Israel, Science and Technology Center MATAM  Advanced Technology Center Haifa 3905, Israel avi@haifa.vnat.ibm.com Jehoshua Bruck California Institute
More informationDynamic Analysis. Mass Matrices and External Forces
4 Dynamic Analysis. Mass Matrices and External Forces The formulation of the inertia and external forces appearing at any of the elements of a multibody system, in terms of the dependent coordinates that
More informationAdvanced CFD Methods 1
Advanced CFD Methods 1 Prof. Patrick Jenny, FS 2014 Date: 15.08.14, Time: 13:00, Student: Federico Danieli Summary The exam took place in Prof. Jenny s office, with his assistant taking notes on the answers.
More informationSystem of First Order Differential Equations
CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions
More informationThe Dispersal of an Initial Concentration of Motile Bacteria
Journal of General Microbiology (I976), 9,253 I Printed in Great Britain The Dispersal of an Initial Concentration of Motile Bacteria By P. C. THONEMANN AND C. J. EVANS Department of Physics, University
More informationStatistical Study of Magnetic Reconnection in the Solar Wind
WDS'13 Proceedings of Contributed Papers, Part II, 7 12, 2013. ISBN 9788073782511 MATFYZPRESS Statistical Study of Magnetic Reconnection in the Solar Wind J. Enžl, L. Přech, J. Šafránková, and Z. Němeček
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More information14.11. Geodesic Lines, Local GaussBonnet Theorem
14.11. Geodesic Lines, Local GaussBonnet Theorem Geodesics play a very important role in surface theory and in dynamics. One of the main reasons why geodesics are so important is that they generalize
More information