Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "{apolin},{mcampos}@ieee.org"

Transcription

1

2

3

4

5

6 x( ) 2 x( ) x( ) = ( ) x = [ ( ) x ı x + ( ) y ( ) y ( ) z ı y + ( ) z ] T ı z 2 x ( ) = 2 ( ) x + 2 ( ) 2 y + 2 ( ) 2 z 2

7 2 E = 1 2 E c 2 t 2 s(x,t) 2 s x + 2 s 2 y + 2 s 2 z = 1 2 s 2 c 2 t 2 c E x = [x y z] T

8 s(x,t) s(x,t) = Ae j(ωt k xx kyy kzz) A kx ky kz ω 0

9 s(x,t) k 2 xs(x,t)+k 2 ys(x,t)+k 2 zs(x,t) = 1 c 2ω2 s(x,t) s(x,t) k x 2 +k y 2 +k z 2 = 1 c 2ω2

10 s(x,t) = Ae j(ωt k xx kyy kzz) x = [0 0 0] T s(0,t) = Ae jωt

11 s(x,t) = Ae j(ωt k xx kyy kzz) s(x,t) kxx+kyy +kzz = C C

12 k k = [kx ky kz] T s(x,t) = Ae j(ωt kt x) s(x,t) k

13 δx δt s(x,t) = s(x+δx,t+δt) Ae j(ωt kt x) = Ae j [ω(t+δt) k T (x+δx)] = ωδt k T δx = 0

14 k δx k T δx = k δx = ωδt = k δx k 2 = ω 2 /c 2 δx δt = ω k

15 T = 2π/ω λ δx = λ δt = T = 2π ω T = λ k ω 2π = λ = k k ω

16 s(x,t) = Ae j(ωt kt x) = Ae jω(t αt x) α = k/ω c = ω/ k α c

17 s(x,t) = s(t α T x) ω0 s(x,t) = s(t α T x) = n= Sne jnω 0(t α T x) Sn = 1 T T 0 s(u)e jnω 0u du

18 s(x,t) ω = nω0 k α = k/ω

19 s(x,t) = s(t α T x) = 1 2π S(ω)e jω(t αt x) dω S(ω) = s(u)e jωu du

it = α + β i + γ 1 t + γ 2 t + γ 3 t +λ 1 ( i ) + λ 2 ( i ) + λ 3 ( i ) +δx i + ϵ it, it i i t t t i λ 1 λ 3 t t t i = α + β i + δx i + ϵ i i i i 12 Harrison Cleveland McKinley Roosevelt 10 8 6

More information

α α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =

More information

Fraternity & Sorority Academic Report Spring 2016

Fraternity & Sorority Academic Report Spring 2016 Fraternity & Sorority Academic Report Organization Overall GPA Triangle 17-17 1 Delta Chi 88 12 100 2 Alpha Epsilon Pi 77 3 80 3 Alpha Delta Chi 28 4 32 4 Alpha Delta Pi 190-190 4 Phi Gamma Delta 85 3

More information

Fraternity & Sorority Academic Report Fall 2015

Fraternity & Sorority Academic Report Fall 2015 Fraternity & Sorority Academic Report Organization Lambda Upsilon Lambda 1-1 1 Delta Chi 77 19 96 2 Alpha Delta Chi 30 1 31 3 Alpha Delta Pi 134 62 196 4 Alpha Sigma Phi 37 13 50 5 Sigma Alpha Epsilon

More information

Tips, tricks and formulae on H.C.F and L.C.M. Follow the steps below to find H.C.F of given numbers by prime factorization method.

Tips, tricks and formulae on H.C.F and L.C.M. Follow the steps below to find H.C.F of given numbers by prime factorization method. Highest Common Factor (H.C.F) Tips, tricks and formulae on H.C.F and L.C.M H.C.F is the highest common factor or also known as greatest common divisor, the greatest number which exactly divides all the

More information

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

More information

EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION

EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7-669. UL: http://ejde.math.txstate.edu

More information

Fourier Analysis and its applications

Fourier Analysis and its applications Fourier Analysis and its applications Fourier analysis originated from the study of heat conduction: Jean Baptiste Joseph Fourier (1768-1830) Fourier analysis enables a function (signal) to be decomposed

More information

Pacific Journal of Mathematics

Pacific Journal of Mathematics Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000

More information

x o R n a π(a, x o ) A R n π(a, x o ) π(a, x o ) A R n a a x o x o x n X R n δ(x n, x o ) d(a, x n ) d(, ) δ(, ) R n x n X d(a, x n ) δ(x n, x o ) a = a A π(a, xo ) a a A = X = R π(a, x o ) = (x o + ρ)

More information

Acoustics. Lecture 2: EE E6820: Speech & Audio Processing & Recognition. Spherical waves & room acoustics. Oscillations & musical acoustics

Acoustics. Lecture 2: EE E6820: Speech & Audio Processing & Recognition. Spherical waves & room acoustics. Oscillations & musical acoustics EE E6820: Speech & Audio Processing & Recognition Lecture 2: Acoustics 1 The wave equation 2 Acoustic tubes: reflections & resonance 3 Oscillations & musical acoustics 4 Spherical waves & room acoustics

More information

Lecture 2: Acoustics

Lecture 2: Acoustics EE E6820: Speech & Audio Processing & Recognition Lecture 2: Acoustics 1 The wave equation Dan Ellis & Mike Mandel Columbia University Dept. of Electrical Engineering http://www.ee.columbia.edu/ dpwe/e6820

More information

14: FM Radio Receiver

14: FM Radio Receiver (1) (2) (3) DSP and Digital Filters (2015-7310) FM Radio: 14 1 / 12 (1) (2) (3) FM spectrum: 87.5 to 108 MHz Each channel: ±100 khz Baseband signal: Mono (L + R): ±15kHz Pilot tone: 19 khz Stereo (L R):

More information

Lecture 4: BK inequality 27th August and 6th September, 2007

Lecture 4: BK inequality 27th August and 6th September, 2007 CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

More information

Frequency Domain and Fourier Transforms

Frequency Domain and Fourier Transforms Chapter 4 Frequency Domain and Fourier Transforms Frequency domain analysis and Fourier transforms are a cornerstone of signal and system analysis. These ideas are also one of the conceptual pillars within

More information

Towards a Structuralist Interpretation of Saving, Investment and Current Account in Turkey

Towards a Structuralist Interpretation of Saving, Investment and Current Account in Turkey Towards a Structuralist Interpretation of Saving, Investment and Current Account in Turkey MURAT ÜNGÖR Central Bank of the Republic of Turkey http://www.muratungor.com/ April 2012 We live in the age of

More information

University of Maryland Fraternity & Sorority Life Spring 2015 Academic Report

University of Maryland Fraternity & Sorority Life Spring 2015 Academic Report University of Maryland Fraternity & Sorority Life Academic Report Academic and Population Statistics Population: # of Students: # of New Members: Avg. Size: Avg. GPA: % of the Undergraduate Population

More information

20.1 Revisiting Maxwell s equations

20.1 Revisiting Maxwell s equations Scott Hughes 28 April 2005 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 20: Wave equation & electromagnetic radiation 20.1 Revisiting Maxwell s equations In our

More information

Introduction to Frequency Domain Processing 1

Introduction to Frequency Domain Processing 1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.151 Advanced System Dynamics and Control Introduction to Frequency Domain Processing 1 1 Introduction - Superposition In this

More information

12.4 UNDRIVEN, PARALLEL RLC CIRCUIT*

12.4 UNDRIVEN, PARALLEL RLC CIRCUIT* + v C C R L - v i L FIGURE 12.24 The parallel second-order RLC circuit shown in Figure 2.14a. 12.4 UNDRIVEN, PARALLEL RLC CIRCUIT* We will now analyze the undriven parallel RLC circuit shown in Figure

More information

Spring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations

Spring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring

More information

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.685 Electric Machines Class Notes 3: Eddy Currents, Surface Impedances and Loss Mechanisms c 2005 James

More information

Chapter 3. Integral Transforms

Chapter 3. Integral Transforms Chapter 3 Integral Transforms This part of the course introduces two extremely powerful methods to solving differential equations: the Fourier and the Laplace transforms. Beside its practical use, the

More information

CHAPTER IV - BROWNIAN MOTION

CHAPTER IV - BROWNIAN MOTION CHAPTER IV - BROWNIAN MOTION JOSEPH G. CONLON 1. Construction of Brownian Motion There are two ways in which the idea of a Markov chain on a discrete state space can be generalized: (1) The discrete time

More information

Lecture 9. Poles, Zeros & Filters (Lathi 4.10) Effects of Poles & Zeros on Frequency Response (1) Effects of Poles & Zeros on Frequency Response (3)

Lecture 9. Poles, Zeros & Filters (Lathi 4.10) Effects of Poles & Zeros on Frequency Response (1) Effects of Poles & Zeros on Frequency Response (3) Effects of Poles & Zeros on Frequency Response (1) Consider a general system transfer function: zeros at z1, z2,..., zn Lecture 9 Poles, Zeros & Filters (Lathi 4.10) The value of the transfer function

More information

TTT4110 Information and Signal Theory Solution to exam

TTT4110 Information and Signal Theory Solution to exam Norwegian University of Science and Technology Department of Electronics and Telecommunications TTT4 Information and Signal Theory Solution to exam Problem I (a The frequency response is found by taking

More information

Scalar Field Theory. Chapter 3. 3.1 Canonical Formulation. The dispersion relation for a particle of mass m is. E 2 = p 2 + m 2, p 2 = p p, (3.

Scalar Field Theory. Chapter 3. 3.1 Canonical Formulation. The dispersion relation for a particle of mass m is. E 2 = p 2 + m 2, p 2 = p p, (3. Chapter 3 Scalar Field Theory 3. Canonical Formulation The dispersion relation for a particle of mass m is or, in relativistic notation, with p 0 = E, A wave equation corresponding to this relation is

More information

Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of this system is: k m therefore, k

Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of this system is: k m therefore, k Physics 1C Midterm 1 Summer Session II, 2011 Solutions 1. If F = kx, then k m is (a) A (b) ω (c) ω 2 (d) Aω (e) A 2 ω Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of

More information

ACTS 4301 FORMULA SUMMARY Lesson 1: Probability Review. Name f(x) F (x) E[X] Var(X) Name f(x) E[X] Var(X) p x (1 p) m x mp mp(1 p)

ACTS 4301 FORMULA SUMMARY Lesson 1: Probability Review. Name f(x) F (x) E[X] Var(X) Name f(x) E[X] Var(X) p x (1 p) m x mp mp(1 p) ACTS 431 FORMULA SUMMARY Lesson 1: Probability Review 1. VarX)= E[X 2 ]- E[X] 2 2. V arax + by ) = a 2 V arx) + 2abCovX, Y ) + b 2 V ary ) 3. V ar X) = V arx) n 4. E X [X] = E Y [E X [X Y ]] Double expectation

More information

Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Dr Glenn Vinnicombe HANDOUT 3. Stability and pole locations.

Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Dr Glenn Vinnicombe HANDOUT 3. Stability and pole locations. Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Dr Glenn Vinnicombe HANDOUT 3 Stability and pole locations asymptotically stable marginally stable unstable Imag(s) repeated poles +

More information

Pure Math 450/650, Winter 2012

Pure Math 450/650, Winter 2012 Compact course notes Pure Math 450/650, Winter 202 Lebesgue integration and Fourier analysis Professor: N. Spronk transcribed by: J. Lazovskis University of Waterloo pril, 202 Contents Toward the Lebesgue

More information

Techniques of Mathematical Modelling. Warning: these are rather longer than actual fhs questions would be. In parts they are also somewhat harder.

Techniques of Mathematical Modelling. Warning: these are rather longer than actual fhs questions would be. In parts they are also somewhat harder. Specimen fhs questions. Techniques of Mathematical Modelling Warning: these are rather longer than actual fhs questions would be. In parts they are also somewhat harder. 1. Explain what is meant by a conservation

More information

The 1-D Wave Equation

The 1-D Wave Equation The -D Wave Equation 8.303 Linear Partial Differential Equations Matthew J. Hancock Fall 006 -D Wave Equation : Physical derivation Reference: Guenther & Lee., Myint-U & Debnath.-.4 [Oct. 3, 006] We consider

More information

Laboratory #6: Dipole and Monopole Antenna Design

Laboratory #6: Dipole and Monopole Antenna Design EEE 171 Lab #6 1 Laboratory #6: Dipole and Monopole Antenna Design I. OBJECTIVES Design several lengths of dipole antennas. Design appropriate impedance matching networks for those antennas. The antennas

More information

Electromagnetism - Lecture 2. Electric Fields

Electromagnetism - Lecture 2. Electric Fields Electromagnetism - Lecture 2 Electric Fields Review of Vector Calculus Differential form of Gauss s Law Poisson s and Laplace s Equations Solutions of Poisson s Equation Methods of Calculating Electric

More information

Differential Equations and Linear Algebra Lecture Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University

Differential Equations and Linear Algebra Lecture Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University Differential Equations and Linear Algebra Lecture Notes Simon J.A. Malham Department of Mathematics, Heriot-Watt University Contents Chapter. Linear second order ODEs 5.. Newton s second law 5.2. Springs

More information

F en = mω 0 2 x. We should regard this as a model of the response of an atom, rather than a classical model of the atom itself.

F en = mω 0 2 x. We should regard this as a model of the response of an atom, rather than a classical model of the atom itself. The Electron Oscillator/Lorentz Atom Consider a simple model of a classical atom, in which the electron is harmonically bound to the nucleus n x e F en = mω 0 2 x origin resonance frequency Note: We should

More information

IU Fraternity & Sorority Spring 2012 Grade Report

IU Fraternity & Sorority Spring 2012 Grade Report SORORITY CHAPTER RANKINGS 1 Chi Delta Phi 3.550 2 Alpha Omicron Pi 3.470 3 Kappa Delta 3.447 4 Alpha Gamma Delta 3.440 5 Delta Gamma 3.431 6 Alpha Chi Omega 3.427 7 Phi Mu 3.391 8 Chi Omega 3.372 8 Kappa

More information

Theory and applications of the relativistic Boltzmann equation. I. Special relativity

Theory and applications of the relativistic Boltzmann equation. I. Special relativity Theory and applications of the. I. Special relativity Gilberto Medeiros Departamento de Física Universidade Federal do Paraná kremer@fisica.ufpr.br 49th Winter School of Theoretical Physics La dek-zdrój,

More information

Lecture 14 More on Real Business Cycles. Noah Williams

Lecture 14 More on Real Business Cycles. Noah Williams Lecture 14 More on Real Business Cycles Noah Williams University of Wisconsin - Madison Economics 312 Optimality Conditions Euler equation under uncertainty: u C (C t, 1 N t) = βe t [u C (C t+1, 1 N t+1)

More information

SPRING 2011 FRATERNITY/SORORITY GRADE REPORT SORORITY CHAPTER RANKINGS

SPRING 2011 FRATERNITY/SORORITY GRADE REPORT SORORITY CHAPTER RANKINGS SPRING 2011 FRATERNITY/SORORITY GRADE REPORT SORORITY CHAPTER RANKINGS 1 Kappa Alpha Theta 3.4643 2 Phi Mu 3.4273 3 Kappa Delta 3.4260 4 Alpha Omicron Pi 3.4072 5 Delta Gamma 3.4072 6 Alpha Chi Omega 3.3989

More information

Interaction of Atoms and Electromagnetic Waves

Interaction of Atoms and Electromagnetic Waves Interaction of Atoms and Electromagnetic Waves Outline - Review: Polarization and Dipoles - Lorentz Oscillator Model of an Atom - Dielectric constant and Refractive index 1 True or False? 1. The dipole

More information

6.025J Medical Device Design Lecture 3: Analog-to-Digital Conversion Prof. Joel L. Dawson

6.025J Medical Device Design Lecture 3: Analog-to-Digital Conversion Prof. Joel L. Dawson Let s go back briefly to lecture 1, and look at where ADC s and DAC s fit into our overall picture. I m going in a little extra detail now since this is our eighth lecture on electronics and we are more

More information

Lecture 7 ELE 301: Signals and Systems

Lecture 7 ELE 301: Signals and Systems Lecture 7 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 22 Introduction to Fourier Transforms Fourier transform as a limit

More information

CLASS TEST GRADE 11. PHYSICAL SCIENCES: PHYSICS Test 3: Electricity and magnetism

CLASS TEST GRADE 11. PHYSICAL SCIENCES: PHYSICS Test 3: Electricity and magnetism CLASS TEST GRADE 11 PHYSICAL SCIENCES: PHYSICS Test 3: Electricity and magnetism MARKS: 45 TIME: 1 hour INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. 2. You may use non-programmable calculators.

More information

Two-Stage Stochastic Linear Programs

Two-Stage Stochastic Linear Programs Two-Stage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 Two-Stage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic

More information

14: FM Radio Receiver

14: FM Radio Receiver (1) (2) DSP and Digital Filters (2015-7310) FM Radio: 14 1 / 12 FM Radio Block Diagram (1) (2) FM spectrum: 87.5 to108mhz [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and

More information

5.4 The Heat Equation and Convection-Diffusion

5.4 The Heat Equation and Convection-Diffusion 5.4. THE HEAT EQUATION AND CONVECTION-DIFFUSION c 6 Gilbert Strang 5.4 The Heat Equation and Convection-Diffusion The wave equation conserves energy. The heat equation u t = u xx dissipates energy. The

More information

Objectives: 1. Be able to list the assumptions and methods associated with the Virial Equation of state.

Objectives: 1. Be able to list the assumptions and methods associated with the Virial Equation of state. Lecture #19 1 Lecture 19 Objectives: 1. Be able to list the assumptions and methods associated with the Virial Equation of state. 2. Be able to compute the second virial coefficient from any given pair

More information

ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES

ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES by Xiaofeng Qian Doctor of Philosophy, Boston University, 27 Bachelor of Science, Peking University, 2 a Project

More information

MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 425, PRACTICE FINAL EXAM SOLUTIONS. MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

More information

Use bandpass filters to discriminate against wide ranges of frequencies outside the passband.

Use bandpass filters to discriminate against wide ranges of frequencies outside the passband. Microwave Filter Design Chp6. Bandstop Filters Prof. Tzong-Lin Wu Department of Electrical Engineering National Taiwan University Bandstop Filters Bandstop filter V.S. Bandpass filter Use bandpass filters

More information

MATH 4552 Cubic equations and Cardano s formulae

MATH 4552 Cubic equations and Cardano s formulae MATH 455 Cubic equations and Cardano s formulae Consider a cubic equation with the unknown z and fixed complex coefficients a, b, c, d (where a 0): (1) az 3 + bz + cz + d = 0. To solve (1), it is convenient

More information

Manual for SOA Exam MLC.

Manual for SOA Exam MLC. Chapter 5 Life annuities Extract from: Arcones Manual for the SOA Exam MLC Fall 2009 Edition available at http://wwwactexmadrivercom/ 1/70 Due n year deferred annuity Definition 1 A due n year deferred

More information

Detailed Product Information for Greek Stationery and Sorority Stationery

Detailed Product Information for Greek Stationery and Sorority Stationery Sorority Stationery Sorority Stationary Sorority Letterhead Many Layouts Sorority Envelopes Self Seal Envelopes Mailing Envelopes Security Envelopes Window Envelopes Large Envelopes Sorority Notecards

More information

Using the Impedance Method

Using the Impedance Method Using the Impedance Method The impedance method allows us to completely eliminate the differential equation approach for the determination of the response of circuits. In fact the impedance method even

More information

About the Ingham type proof for the boundary observability of a square and its generalization in N-d

About the Ingham type proof for the boundary observability of a square and its generalization in N-d About the Ingham type proof for the boundary observability of a square and its generalization in N-d M. Mehrenberger University of Strasbourg (France), INRIA TONUS team Rome 015 Rome, September 30, 015

More information

SOLUTIONS TO PROBLEM SET 3

SOLUTIONS TO PROBLEM SET 3 SOLUTIONS TO PROBLEM SET 3 MATTI ÅSTRAND The General Cubic Extension Denote L = k(α 1, α 2, α 3 ), F = k(a 1, a 2, a 3 ) and K = F (α 1 ). The polynomial f(x) = x 3 a 1 x 2 + a 2 x a 3 = (x α 1 )(x α 2

More information

221A Lecture Notes Path Integral

221A Lecture Notes Path Integral 1A Lecture Notes Path Integral 1 Feynman s Path Integral Formulation Feynman s formulation of quantum mechanics using the so-called path integral is arguably the most elegant. It can be stated in a single

More information

Statistical Energy Analysis

Statistical Energy Analysis 1 2 1 Long. 2 Long. 3 Long. 3 1 Bend. 2 Bend. 3 Bend. [E] = [C] -1 [W] Lectures Time: Wednesday 10-12 Place: TA 201 Lecturer: Prof. Björn Petersson Tutorials Time: Wednesday 12-14 Place: TA K001 Tutor:

More information

American Criminal Justice Association Lambda Alpha Epsilon Psi Omega. Judicial Board Guidelines

American Criminal Justice Association Lambda Alpha Epsilon Psi Omega. Judicial Board Guidelines American Criminal Justice Association Lambda Alpha Epsilon Psi Omega Judicial Board Guidelines Created Submitted by: Sergeant- At- Arms Anthony Bouchard 2 The Sergeant-at-Arms shall be the chairperson

More information

How do we obtain the solution, if we are given F (t)? First we note that suppose someone did give us one solution of this equation

How do we obtain the solution, if we are given F (t)? First we note that suppose someone did give us one solution of this equation 1 Green s functions The harmonic oscillator equation is This has the solution mẍ + kx = 0 (1) x = A sin(ωt) + B cos(ωt), ω = k m where A, B are arbitrary constants reflecting the fact that we have two

More information

Decomposing total risk of a portfolio into the contributions of individual assets

Decomposing total risk of a portfolio into the contributions of individual assets Decomposing total risk of a portfolio into the contributions of individual assets Abstract. Yukio Muromachi Graduate School of Social Sciences, Tokyo Metropolitan University 1-1 Minami-Ohsawa, Hachiohji,

More information

SOLID MECHANICS DYNAMICS TUTORIAL MOMENT OF INERTIA. This work covers elements of the following syllabi.

SOLID MECHANICS DYNAMICS TUTORIAL MOMENT OF INERTIA. This work covers elements of the following syllabi. SOLID MECHANICS DYNAMICS TUTOIAL MOMENT OF INETIA This work covers elements of the following syllabi. Parts of the Engineering Council Graduate Diploma Exam D5 Dynamics of Mechanical Systems Parts of the

More information

The Ergodic Theorem and randomness

The Ergodic Theorem and randomness The Ergodic Theorem and randomness Peter Gács Department of Computer Science Boston University March 19, 2008 Peter Gács (Boston University) Ergodic theorem March 19, 2008 1 / 27 Introduction Introduction

More information

Probability for Estimation (review)

Probability for Estimation (review) Probability for Estimation (review) In general, we want to develop an estimator for systems of the form: x = f x, u + η(t); y = h x + ω(t); ggggg y, ffff x We will primarily focus on discrete time linear

More information

PHYS 2425 Engineering Physics I EXPERIMENT 9 SIMPLE HARMONIC MOTION

PHYS 2425 Engineering Physics I EXPERIMENT 9 SIMPLE HARMONIC MOTION PHYS 2425 Engineering Physics I EXPERIMENT 9 SIMPLE HARMONIC MOTION I. INTRODUCTION The objective of this experiment is the study of oscillatory motion. In particular the springmass system and the simple

More information

Manual for SOA Exam MLC.

Manual for SOA Exam MLC. Chapter 5. Life annuities. Extract from: Arcones Manual for the SOA Exam MLC. Spring 2010 Edition. available at http://www.actexmadriver.com/ 1/114 Whole life annuity A whole life annuity is a series of

More information

14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style

14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of

More information

1 Introduction. 2 Energy Conversion Process: J.L. Kirtley Jr.

1 Introduction. 2 Energy Conversion Process: J.L. Kirtley Jr. Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.061 Introduction to Power Systems Class Notes Chapter 8 Electromagnetic Forces and Loss Mechanisms J.L.

More information

Exam 3: Equation Summary

Exam 3: Equation Summary MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

More information

č é é č Á Ě Č Á š Á Ó Á Á ď ú ď Š ň Ý ú ď Ó č ď Ě ů ň Č Š š ď Ň ď ď Č ý Ž Ý Ý Ý ČÚ Ž é úč ž ý ž ý ý ý č ů ý é ý č ý ý čů ý ž ž ý č č ž ž ú é ž š é é é č Ž ý ú é ý š é Ž č Ž ů Ů Ť ý ý ý Á ý ý Č Ť É Ď ň

More information

Lecture 12: DC Analysis of BJT Circuits.

Lecture 12: DC Analysis of BJT Circuits. Whites, 320 Lecture 12 Page 1 of 9 Lecture 12: D Analysis of JT ircuits. n this lecture we will consider a number of JT circuits and perform the D circuit analysis. For those circuits with an active mode

More information

Massachusetts Institute of Technology Department of Mechanical Engineering Cambridge, MA 02139

Massachusetts Institute of Technology Department of Mechanical Engineering Cambridge, MA 02139 Massachusetts Institute of Technology Department of Mechanical Engineering Cambridge, MA 02139 2.002 Mechanics and Materials II Spring 2004 Laboratory Module No. 1 Elastic behavior in tension, bending,

More information

Nonlinear Stochastic Integrals for Hyperfinite Lévy Processes

Nonlinear Stochastic Integrals for Hyperfinite Lévy Processes Dept of Math University of Oslo Pure Mathematics No 25 ISSN 86 2439 September 25 Nonlinear Stochastic Integrals for Hyperfinite Lévy Processes Tom Lindstrøm Abstract We develop a notion of nonlinear stochastic

More information

RANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis

RANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied

More information

Generalized BIC for Singular Models Factoring through Regular Models

Generalized BIC for Singular Models Factoring through Regular Models Generalized BIC for Singular Models Factoring through Regular Models Shaowei Lin http://math.berkeley.edu/ shaowei/ Department of Mathematics, University of California, Berkeley PhD student (Advisor: Bernd

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

Ordinary Differential Equations

Ordinary Differential Equations Ordinary Differential Equations Dan B. Marghitu and S.C. Sinha 1 Introduction An ordinary differential equation is a relation involving one or several derivatives of a function y(x) with respect to x.

More information

11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Rotational Motion: 11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

More information

MATH 31B: MIDTERM 1 REVIEW. 1. Inverses. yx 3y = 1. x = 1 + 3y y 4( 1) + 32 = 1

MATH 31B: MIDTERM 1 REVIEW. 1. Inverses. yx 3y = 1. x = 1 + 3y y 4( 1) + 32 = 1 MATH 3B: MIDTERM REVIEW JOE HUGHES. Inverses. Let f() = 3. Find the inverse g() for f. Solution: Setting y = ( 3) and solving for gives and g() = +3. y 3y = = + 3y y. Let f() = 4 + 3. Find a domain on

More information

2. Dynamo models. Chris Jones Department of Applied Mathematics. University of Leeds, UK. Nordita Winter School January 12th Stockholm

2. Dynamo models. Chris Jones Department of Applied Mathematics. University of Leeds, UK. Nordita Winter School January 12th Stockholm 2. Dynamo models Chris Jones Department of Applied Mathematics University of Leeds, UK Nordita Winter School January 12th Stockholm Dynamo simulations 1 Solve induction equation and the Navier-Stokes equations

More information

Private Equity Fund Valuation and Systematic Risk

Private Equity Fund Valuation and Systematic Risk An Equilibrium Approach and Empirical Evidence Axel Buchner 1, Christoph Kaserer 2, Niklas Wagner 3 Santa Clara University, March 3th 29 1 Munich University of Technology 2 Munich University of Technology

More information

2.016 Hydrodynamics Prof. A.H. Techet Fall 2005

2.016 Hydrodynamics Prof. A.H. Techet Fall 2005 .016 Hydrodynamics Reading #7.016 Hydrodynamics Prof. A.H. Techet Fall 005 Free Surface Water Waves I. Problem setu 1. Free surface water wave roblem. In order to determine an exact equation for the roblem

More information

Frequency-domain and stochastic model for an articulated wave power device

Frequency-domain and stochastic model for an articulated wave power device Frequency-domain stochastic model for an articulated wave power device J. Cândido P.A.P. Justino Department of Renewable Energies, Instituto Nacional de Engenharia, Tecnologia e Inovação Estrada do Paço

More information

OHIO REGION PHI THETA KAPPA 2012-13

OHIO REGION PHI THETA KAPPA 2012-13 OHIO REGION PHI THETA KAPPA REGION HALLMARK AWARDS HONORS IN ACTION HALLMARK WINNER Alpha Rho Epsilon Columbus State Community College HONORS IN ACTION HALLMARK FIRST RUNNER-UP Washington State Community

More information

POINT OF INTERSECTION OF TWO STRAIGHT LINES

POINT OF INTERSECTION OF TWO STRAIGHT LINES POINT OF INTERSECTION OF TWO STRAIGHT LINES THEOREM The point of intersection of the two non parallel lines bc bc ca ca a x + b y + c = 0, a x + b y + c = 0 is,. ab ab ab ab Proof: The lines are not parallel

More information

A First Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A First Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved A First Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction

More information

Chapter 9 Rigid Body Motion in 3D

Chapter 9 Rigid Body Motion in 3D Chapter 9 Rigid Body Motion in 3D Rigid body rotation in 3D is a complicated problem requiring the introduction of tensors. Upon completion of this chapter we will be able to describe such things as the

More information

The Kelly criterion for spread bets

The Kelly criterion for spread bets IMA Journal of Applied Mathematics 2007 72,43 51 doi:10.1093/imamat/hxl027 Advance Access publication on December 5, 2006 The Kelly criterion for spread bets S. J. CHAPMAN Oxford Centre for Industrial

More information

Antenna A mean for radiating and receiving radio waves Transitional structure between free-space and a guiding device. Application: Radiation

Antenna A mean for radiating and receiving radio waves Transitional structure between free-space and a guiding device. Application: Radiation Antenna A mean for radiating and receiving radio waves Transitional structure between free-space and a guiding device Application: adiation Introduction An antenna is designed to radiate or receive electromagnetic

More information

Gabriele Bianchi Dipartimento di Matematica, Universita di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy

Gabriele Bianchi Dipartimento di Matematica, Universita di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy THE SCALAR CURVATURE EQUATION ON R n AND S n Gabriele Bianchi Dipartimento di Matematica, Universita di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy Abstract. We study the existence of positive solutions

More information

Signal detection and goodness-of-fit: the Berk-Jones statistics revisited

Signal detection and goodness-of-fit: the Berk-Jones statistics revisited Signal detection and goodness-of-fit: the Berk-Jones statistics revisited Jon A. Wellner (Seattle) INET Big Data Conference INET Big Data Conference, Cambridge September 29-30, 2015 Based on joint work

More information

Math 267 - Practice exam 2 - solutions

Math 267 - Practice exam 2 - solutions C Roettger, Fall 13 Math 267 - Practice exam 2 - solutions Problem 1 A solution of 10% perchlorate in water flows at a rate of 8 L/min into a tank holding 200L pure water. The solution is kept well stirred

More information

Energy conservation. = h(ν- ν 0 ) Recombination: electron gives up energy = ½ mv f. Net energy that goes into heating: ½ mv i.

Energy conservation. = h(ν- ν 0 ) Recombination: electron gives up energy = ½ mv f. Net energy that goes into heating: ½ mv i. Thermal Equilibrium Energy conservation equation Heating by photoionization Cooling by recombination Cooling by brehmsstralung Cooling by collisionally excited lines Collisional de-excitation Detailed

More information

FINITE ELEMENT : MATRIX FORMULATION. Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 7633

FINITE ELEMENT : MATRIX FORMULATION. Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 7633 FINITE ELEMENT : MATRIX FORMULATION Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 76 FINITE ELEMENT : MATRIX FORMULATION Discrete vs continuous Element type Polynomial approximation

More information