The wave equation in elastodynamic

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "The wave equation in elastodynamic"

Transcript

1 The wave equation in elastodynamic Wave propagation in a non-homogeneous anisotropic elastic medium occupying a bounded domain R d, d = 2, 3, with boundary Γ, is described by the linear wave equation: ρ 2 v τ 2 = f, in (, T ), (1) τ = C ɛ, (2) v = v, v =, (3) where v(x, t) R d, is the displacement, τ is the stress tensor, ρ(x) is the density of the material depending on x, t is the time variable, T is a final time, and f(x, t) R d, is a given source function. 1

2 Further, ɛ is the strain tensor with components coupled to τ by Hooke s law ɛ ij = ɛ ij (v) = 1 2 ( v i x j + v j x i ) (4) τ ij = d k=1 d C ijkl ɛ kl, (5) l=1 where C is a cyclic symmetric tensor, satisfying C ijkl = C klij = C jkli. (6) If the constants C ijkl (x) do not depend on x, the material of the body is said to be homogeneous. If the constants C ijkl (x) do not depend on the choice of the coordinate system, the material of the body is said to be isotropic at the point x. Otherwise, the material is anisotropic at the point x. 2

3 In the isotropic case C can be written as C ijkl = λδ ij δ kl + µ(δ ij δ kl + δ il δ jk ), (7) where δ ij is Kronecker symbol, in which case (5) takes the form of Hooke s law d τ ij = λδ ij ɛ kk + 2µɛ ij, (8) k=1 where λ and µ are the Lame s coefficients, depending on x, given by µ = E 2(1 + ν), λ = Eν (1 + ν)(1 2ν), (9) where E is the modulus of elasticity (Young modulus) and ν is the Poisson s ratio of the elastic material. We have that λ >, µ > E >, < ν < 1/2. (1) 3

4 4

5 ρ 2 v 1 2 ρ 2 v 2 2 ρ 2 v 3 2 (( ) v 1 λ + 2µ + λ v 2 + λ v 3 ) x 1 x 1 x 2 x 3 ( ( v 1 µ + v 2 )) x 2 x 2 x 1 ( ( v 1 µ + v 3 )) = f1, x 3 x 3 x 1 ( v 2 (λ + 2µ) + λ v 1 + λ v 3 ) x 2 x 2 x 1 x 3 ( ( v 1 µ + v 2 )) x 1 x 2 x 1 ( ( v 2 µ + v 3 )) = f2, x 3 x 3 x 2 ( v 3 (λ + 2µ) + λ v 2 + λ v 1 ) x 3 x 3 x 2 x 1 ( ( v 3 µ + v 2 )) x 2 x 2 x 3 ( ( v 1 µ + v 3 ) ) = f 3, x 1 x 3 5 x 1

6 or in more compact form ρ 2 v (µ v) ((λ + µ) v) = f. (11) 2 6

7 The inverse scattering problem We consider the elastics system in a non-homogeneous isotropic medium in a bounded domain R d, d = 2, 3, with boundary Γ: ρ 2 v (µ v) ((λ + µ) v) = f in (, T ), (12) 2 v(x, t) v(x, ) = v (t), =, v Γ =. (13) where v = (v 1, v 2, v 3 ) T, v = ( v 1, v 2, v 3 ) T, and v is the divergence of the vector field v. 7

8 The inverse problem for (12-13) can be formulated as follows: find a controls ρ, µ, λ which belongs to the set of admissible controls U = {ρ, µ, λ L 2 (); < ρ min < ρ(x) < ρ max, and minimizes the quantity E(v, ρ, λ, µ) = 1 T 2 < λ min < λ(x) < λ max, < µ min < µ(x) < µ max } (14) (v ṽ) 2 δ obs dxdt γ γ 2 µ 2 dx γ 3 ρ 2 dx λ 2 dx, (15) where ṽ is observed data at a finite set of observation points x obs, v satisfies (12) and thus depends on ρ, µ, λ, δ obs = δ(x obs ) is a sum of delta-functions corresponding to the observation points, and γ i, i = 1, 2, 3 are a regularization parameters. 8

9 To approach this minimization problem, we introduce the Lagrangian L(α, v, ρ, µ, λ) = E(v, ρ, λ, µ) + T ( α v ρ + µ α v + (λ + µ) v α fα) dxdt, and search for a stationary point with respect to (α, v, ρ, µ, λ) satisfying for all (ᾱ, v, ρ, λ, µ) L (α, v, ρ, µ, λ)(ᾱ, v, ρ, µ, λ) =, (16) where L is the gradient of L and we assume that α(, T ) = ᾱ(, T ) =, α = and v(, ) = v(, ) =. 9

10 = L (α, v, ρ, µ, λ)(ᾱ) = α = L v T ( ρ ᾱ v + µ ᾱ v (17) + (λ + µ) v ᾱ fᾱ ) dxdt, T (α, v, ρ, µ, λ)( v) = (v ṽ) v δ obs dxdt (18) + T ρ α v + µ α v + (λ + µ) v α dxdt, 1

11 = L (α, v, ρ, µ, λ)( ρ) = ρ = L (α, v, ρ, µ, λ)( µ) = µ = L λ (α, v, ρ, µ, λ)( λ) = T + γ 1 T + γ 2 T + γ 3 α(x, t) ρ ρ dx, x. v(x, t) ρ dxdt (19) ( α v + v α) µ dxdt µ µ dx, x. v α λ dxdt (2) λ λ dx, x. 11

12 The equation (17) is a weak form of the state equation (12-13), the equation (18) is a weak form of the adjoint state equation ρ 2 α 2 (µ α) ((λ + µ) α) = (v ṽ)δ obs, x, < t < T, α(t ) = α(t ) =, (21) α = on Γ (, T ), and (19) - (2) expresses stationarity with respect to ρ(x), µ(x), λ(x). 12

13 To solve the minimization problem we shall use a discrete form of the following steepest descent or gradient method starting from an initial guess ρ, µ, λ and computing a sequence ρ n, µ n, λ n in the following steps: 1. Compute the solution v n = (v n 1, v n 2, v n 3 ) of the forward problem (12-13) with ρ = ρ n, µ = µ n, λ = λ n. 2. Compute the solution α n = (α n 1, α n 2, α n 3 ) of the adjoint problem (22). 3. Update the ρ, µ, λ according to ρ n+1 (x) = ρ n (x) β n( T µ n+1 (x) = µ n (x) β n( T λ n+1 (x) = λ n (x) β n( T α n (x, t) v n (x, t) dt + γ 1 ρ n (x) ), α n v n + v n α n + γ 2 µ n (x) ), v n α n + γ 3 λ n (x) ), 13

14 Finite element discretization. We now formulate a finite element method for (16) based on using continuous piecewise linear functions in space and time. We discretize (, T ) in the usual way denoting by K h = {K} a partition of the domain into elements K (triangles in R 2 and tetrahedra in R 3 with h = h(x) being a mesh function representing the local diameter of the elements), and we let J k = {J} be a partition of the time interval I = (, T ) into time intervals J = (t k 1, t k ] of uniform length τ = t k t k 1. In fully discrete form the resulting method corresponds to a centered finite difference approximation for the second order time derivative and a usual finite element approximation of the Laplacian. To formulate the finite element method for (16) we introduce the finite 14

15 element spaces V h, W v h and W α h defined by : V h := {v L 2 () : v P (K), K K h }, W v := {v [ H 1 ( I) ] 3 : v(, ) =, v Γ = }, W α := {α [ H 1 ( I) ] 3 : α(, T ) = α Γ = }, W v h := {v W v : v K J [P 1 (K) P 1 (J)] 3, K K h, J J k }, W α h := {v W α : v K J [P 1 (K) P 1 (J)] 3, K K h, J J k }, where P 1 (K) and P 1 (J) are the set of linear functions on K and J, respectively. 15

16 The finite element method now reads: Find ρ h V h, µ h V h, λ h V h, α h Wh α, v h Wh v, such that L (α h, v h, ρ h, µ h, λ h )(ᾱ, v, ρ, µ, λ) = ρ V h, µ V h, λ V h, ᾱ W α h, v W v h. 16

17 Fully discrete scheme Expanding v, α in terms of the standard continuous piecewise linear functions ϕ i (x) in space and ψ i (t) in time and substituting this into (17-18), the following system of linear equations is obtained: M(v k+1 2v k + v k 1 ) = τ 2 τ 2 M(α k+1 2α k + α k 1 ) = τ 2 τ 2 ρ F k τ 2 ρ µk(1 6 vk vk vk+1 ) ρ (λ + µ)dvk, (22) ρ Sk τ 2 ρ µk(1 6 αk αk αk+1 ) ρ (λ + µ)dαk, (23) 17

18 with initial conditions : v() =, v(), (24) α(t ) =, α(t ). (25) Here, M is the mass matrix in space, K is the stiffness matrix, D is the divergence matrice, k = 1, 2, 3... denotes the time level, F k, S k are the load vectors, v is the unknown discrete field values of v, α is the unknown discrete field values of α and τ is the time step. 18

19 The explicit formulas for the entries in system (22-23) at the element level can be given as: M e i,j = (ϕ i, ϕ j ) e, (26) K e i,j = ( ϕ i, ϕ j ) e, (27) D e i,j = ( ϕ i, ϕ j ) e, (28) F e j,m = (f, ϕ j ψ m ) e J, (29) S e j,m = (v ṽ, ϕ j ψ m ) e J, (3) where (.,.) e denotes the L 2 (e) scalar product. The matrix M e is the contribution from element e to the global assembled matrix in space M, K e is the contribution from element e to the global assembled matrix K,D e is the contribution from element e to the global assembled matrix D, F e and S e are the contributions from element e to the assembled source vectors F and vector of the right hand side of (22), correspondingly. 19

20 To obtain an explicit scheme we approximate M with the lumped mass matrix M L, the diagonal approximation obtained by taking the row sum of M. By multiplying (22) - (23) with (M L ) 1 and replacing the terms 1 6 vk vk vk+1 and 1 6 αk αk αk+1 by v k and α k, respectively, we obtain an efficient explicit formulation: v k+1 = τ 2 τ 2 α k 1 = τ 2 τ 2 ρ (M L ) 1 F k + 2v k τ 2 ρ (λ + µ)(m L ) 1 Dv k v k 1, ρ (M L ) 1 S k + 2α k τ 2 ρ (λ + µ)(m L ) 1 Dα k α k+1. ρ µ(m L ) 1 Kv k (31) ρ µ(m L ) 1 Kα k (32) 2

21 The discrete version of gradients takes the form: = L (α, v, ρ(x), µ(x), λ(x))( ρ) (33) ρ T α h v h = ρ dxdt + γ 1 ρ h ρ dx, ρ V h. = L (α, v, ρ(x), µ(x), λ(x))( µ) (34) µ T = ( α h v h + v h α h ) µ dxdt + γ 2 µ h µ dx, x. = L λ (α, v, ρ(x), µ(x), λ(x))( λ) (35) T = v h α h λ dxdt + γ3 λ h λ dx, x. 21

22 An a posteriori error estimate for the Lagrangian We start by writing an equation for the error e in the Lagrangian as e = L(α, v, ρ, µ, λ) L(α h, v h, ρ h, µ h, λ h ) = 1 2 L (α h, v h, ρ h, µ h, λ h )((α, v, ρ, µ, λ) (α h, v h, ρ h, µ h, λ h )) + R = 1 2 L (α h, v h, ρ h, µ h, λ h ))(α α h, v v h, ρ ρ h, µ µ h, λ λ h ) + R, where R denotes (a small) second order term. Using the Galerkin orthogonality and the splitting α α h = (α αh I ) + (αi h α h), v v h = (v vh I ) + (vi h v h), ρ ρ h = (ρ ρ I h ) + (ρi h ρ h), µ µ h = (µ µ I h ) + (µi h µ h), λ λ h = (λ λ I h ) + (λi h λ h), where (αh I, vi h, ρi h, µi h, λi h ) denotes an interpolant of (α, v, ρ, µ, λ) Wh α W h v V h V h V h, and neglecting the term R, 22

23 we get: e 1 2 L (α h, v h, ρ h, µ h, λ h ))(α α h, v v h, ρ ρ h, µ µ h, λ λ h ) = 1 2 (I 1 + I 2 + I 3 + I 4 + I 5 ), (36) 23

24 where I 1 = T ρ h (α α I h ) v h + µ h (α α I h) v h + (λ h + µ h )( v h (α αh)) I f(α αh)) I dxdt, T I 2 = (v h ṽ)(v vh) I δ obs dxdt + T T ρ h α h (v v I h ) + (λ h + µ h ) (v vh) I α h dxdt, T α h (x, t) I 3 = I 4 = I 5 = T + µ h α h (v v I h) v h (x, t) (ρ ρ I h) dxdt + ( α h v h + v h α h ) (µ µ I h) dxdt + ( v h α h )(λ λ I h) dxdt + ρ h (ρ ρ I h) dx, λ h (λ λ I h) dx. µ h (µ µ I h) dx, 24

25 Defining the residuals R v1 = f, R v2 = µ h max 2 S K h 1 [ ] k s v h, R v3 = ρ h 2 τ 1 [ ] v ht, R α1 = v h ṽ, R α2 = µ h max 2 S K h 1 [ ] k s α h, R α3 = ρ h 2 τ 1 [ ] α ht, R ρ1 = α h v h, R ρ2 = ρ h, R µ1 = α h v h + α h v h, R µ2 = µ h, R λ1 = α h v h, R λ2 = λ h, (37) 25

26 and interpolation errors in the form [ ] [ ] σ α = Cτ αh + Ch αh, (38) n [ ] [ ] σ v = Cτ vh + Ch vh, (39) n σ ρ = C [ρ h ], σ µ = C [µ h ], σ λ = C [λ h ] (4) 26

27 we obtain the following a posteriori estimate e 1 ( T T R v1 σ α dxdt + R v2 σ α dxdt + 2 T T + R α1 σ v dxdt + R α2 σ v dxdt T R ρ1 σ ρ dxdt + R µ2 σ µ dx + T R ρ2 σ ρ dx + R λ1 σ λ dxdt + T T T R µ1 σ µ R λ2 σ λ dx ). R v3 σ α R α3 σ v dxdt dxdt dxdt 27

28 Adaptive algorithm In the computations below we use the following variant of the gradient method with adaptive mesh selection: 1. Choose an initial mesh K h and an initial time partition J k of the time interval (, T ). 2. Compute the solution v n = (v n 1, v n 2, v n 3 ) on K h and J k of the forward problem (12-13) with ρ = ρ n, µ = µ n, λ = λ n. 3. Compute the solution α n = (α n 1, α n 2, α n 3 ) of the adjoint problem ρ 2 α 2 µ α (λ+µ) ( α) = (v ṽ)δ obs, x, < t < T on K h and J k. 28

29 4. Update the ρ, µ, λ according to ρ n+1 (x) = ρ n (x) β n( T µ n+1 (x) = µ n (x) β n( T λ n+1 (x) = λ n (x) β n( T α n (x, t) v n (x, t) dt + γ 1 ρ n (x) ), α n v n + v n α n + γ 2 µ n (x) ), v n α n + γ 3 λ n (x) ). Make steps 1 4 as long the gradient quickly decreases. 5. Refine all elements, where (R ρ1 + R ρ2 )σ ρ + (R µ1 + R µ2 )σ µ + (R λ1 + R λ2 )σ λ > tol and construct a new mesh K h and a new time partition J k. Here tol is a tolerance chosen by the user. Return to 1. 29

Partial Differential Equations in Biology The boundary element method. March 26, 2013

Partial Differential Equations in Biology The boundary element method. March 26, 2013 The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet

Διαβάστε περισσότερα

Concrete Mathematics Exercises from 30 September 2016

Concrete Mathematics Exercises from 30 September 2016 Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)

Διαβάστε περισσότερα

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1. Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ. Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action

Διαβάστε περισσότερα

ES440/ES911: CFD. Chapter 5. Solution of Linear Equation Systems

ES440/ES911: CFD. Chapter 5. Solution of Linear Equation Systems ES440/ES911: CFD Chapter 5. Solution of Linear Equation Systems Dr Yongmann M. Chung http://www.eng.warwick.ac.uk/staff/ymc/es440.html Y.M.Chung@warwick.ac.uk School of Engineering & Centre for Scientific

Διαβάστε περισσότερα

6.3 Forecasting ARMA processes

6.3 Forecasting ARMA processes 122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear

Διαβάστε περισσότερα

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013 Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering

Διαβάστε περισσότερα

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all

Διαβάστε περισσότερα

w o = R 1 p. (1) R = p =. = 1

w o = R 1 p. (1) R = p =. = 1 Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:

Διαβάστε περισσότερα

D Alembert s Solution to the Wave Equation

D Alembert s Solution to the Wave Equation D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

Finite difference method for 2-D heat equation

Finite difference method for 2-D heat equation Finite difference method for 2-D heat equation Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen

Διαβάστε περισσότερα

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

4.6 Autoregressive Moving Average Model ARMA(1,1)

4.6 Autoregressive Moving Average Model ARMA(1,1) 84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this

Διαβάστε περισσότερα

Homework 8 Model Solution Section

Homework 8 Model Solution Section MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx

Διαβάστε περισσότερα

Dr. D. Dinev, Department of Structural Mechanics, UACEG

Dr. D. Dinev, Department of Structural Mechanics, UACEG Lecture 4 Material behavior: Constitutive equations Field of the game Print version Lecture on Theory of lasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACG 4.1 Contents

Διαβάστε περισσότερα

Homework 3 Solutions

Homework 3 Solutions Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For

Διαβάστε περισσότερα

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

6.1. Dirac Equation. Hamiltonian. Dirac Eq. 6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2

Διαβάστε περισσότερα

Numerical Analysis FMN011

Numerical Analysis FMN011 Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =

Διαβάστε περισσότερα

2 Composition. Invertible Mappings

2 Composition. Invertible Mappings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

Διαβάστε περισσότερα

Local Approximation with Kernels

Local Approximation with Kernels Local Approximation with Kernels Thomas Hangelbroek University of Hawaii at Manoa 5th International Conference Approximation Theory, 26 work supported by: NSF DMS-43726 A cubic spline example Consider

Διαβάστε περισσότερα

Example Sheet 3 Solutions

Example Sheet 3 Solutions Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note

Διαβάστε περισσότερα

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)

Διαβάστε περισσότερα

1 String with massive end-points

1 String with massive end-points 1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε

Διαβάστε περισσότερα

Tridiagonal matrices. Gérard MEURANT. October, 2008

Tridiagonal matrices. Gérard MEURANT. October, 2008 Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,

Διαβάστε περισσότερα

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

The kinetic and potential energies as T = 1 2. (m i η2 i k(η i+1 η i ) 2 ). (3) The Hooke s law F = Y ξ, (6) with a discrete analog

The kinetic and potential energies as T = 1 2. (m i η2 i k(η i+1 η i ) 2 ). (3) The Hooke s law F = Y ξ, (6) with a discrete analog Lecture 12: Introduction to Analytical Mechanics of Continuous Systems Lagrangian Density for Continuous Systems The kinetic and potential energies as T = 1 2 i η2 i (1 and V = 1 2 i+1 η i 2, i (2 where

Διαβάστε περισσότερα

C.S. 430 Assignment 6, Sample Solutions

C.S. 430 Assignment 6, Sample Solutions C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order

Διαβάστε περισσότερα

Other Test Constructions: Likelihood Ratio & Bayes Tests

Other Test Constructions: Likelihood Ratio & Bayes Tests Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :

Διαβάστε περισσότερα

Uniform Convergence of Fourier Series Michael Taylor

Uniform Convergence of Fourier Series Michael Taylor Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula

Διαβάστε περισσότερα

Approximation of distance between locations on earth given by latitude and longitude

Approximation of distance between locations on earth given by latitude and longitude Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth

Διαβάστε περισσότερα

Lecture 13 - Root Space Decomposition II

Lecture 13 - Root Space Decomposition II Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).

Διαβάστε περισσότερα

Appendix A. Curvilinear coordinates. A.1 Lamé coefficients. Consider set of equations. ξ i = ξ i (x 1,x 2,x 3 ), i = 1,2,3

Appendix A. Curvilinear coordinates. A.1 Lamé coefficients. Consider set of equations. ξ i = ξ i (x 1,x 2,x 3 ), i = 1,2,3 Appendix A Curvilinear coordinates A. Lamé coefficients Consider set of equations ξ i = ξ i x,x 2,x 3, i =,2,3 where ξ,ξ 2,ξ 3 independent, single-valued and continuous x,x 2,x 3 : coordinates of point

Διαβάστε περισσότερα

Solution Series 9. i=1 x i and i=1 x i.

Solution Series 9. i=1 x i and i=1 x i. Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x

Διαβάστε περισσότερα

ECE Spring Prof. David R. Jackson ECE Dept. Notes 2

ECE Spring Prof. David R. Jackson ECE Dept. Notes 2 ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =

Διαβάστε περισσότερα

EE512: Error Control Coding

EE512: Error Control Coding EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3

Διαβάστε περισσότερα

g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King

g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i

Διαβάστε περισσότερα

derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

Second Order Partial Differential Equations

Second Order Partial Differential Equations Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y

Διαβάστε περισσότερα

Math221: HW# 1 solutions

Math221: HW# 1 solutions Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin

Διαβάστε περισσότερα

Problem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.

Problem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ. Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +

Διαβάστε περισσότερα

HW 3 Solutions 1. a) I use the auto.arima R function to search over models using AIC and decide on an ARMA(3,1)

HW 3 Solutions 1. a) I use the auto.arima R function to search over models using AIC and decide on an ARMA(3,1) HW 3 Solutions a) I use the autoarima R function to search over models using AIC and decide on an ARMA3,) b) I compare the ARMA3,) to ARMA,0) ARMA3,) does better in all three criteria c) The plot of the

Διαβάστε περισσότερα

Figure A.2: MPC and MPCP Age Profiles (estimating ρ, ρ = 2, φ = 0.03)..

Figure A.2: MPC and MPCP Age Profiles (estimating ρ, ρ = 2, φ = 0.03).. Supplemental Material (not for publication) Persistent vs. Permanent Income Shocks in the Buffer-Stock Model Jeppe Druedahl Thomas H. Jørgensen May, A Additional Figures and Tables Figure A.: Wealth and

Διαβάστε περισσότερα

Solutions to Exercise Sheet 5

Solutions to Exercise Sheet 5 Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X

Διαβάστε περισσότερα

Matrices and Determinants

Matrices and Determinants Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z

Διαβάστε περισσότερα

Section 8.3 Trigonometric Equations

Section 8.3 Trigonometric Equations 99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.

Διαβάστε περισσότερα

Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices

Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices Chi-Kwong Li Department of Mathematics The College of William and Mary Williamsburg, Virginia 23187-8795

Διαβάστε περισσότερα

Statistical Inference I Locally most powerful tests

Statistical Inference I Locally most powerful tests Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided

Διαβάστε περισσότερα

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch: HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying

Διαβάστε περισσότερα

Second Order RLC Filters

Second Order RLC Filters ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor

Διαβάστε περισσότερα

Space-Time Symmetries

Space-Time Symmetries Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a

Διαβάστε περισσότερα

CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD

CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.

Διαβάστε περισσότερα

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)

Διαβάστε περισσότερα

Exercises to Statistics of Material Fatigue No. 5

Exercises to Statistics of Material Fatigue No. 5 Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can

Διαβάστε περισσότερα

Eulerian Simulation of Large Deformations

Eulerian Simulation of Large Deformations Eulerian Simulation of Large Deformations Shayan Hoshyari April, 2018 Some Applications 1 Biomechanical Engineering 2 / 11 Some Applications 1 Biomechanical Engineering 2 Muscle Animation 2 / 11 Some Applications

Διαβάστε περισσότερα

Forced Pendulum Numerical approach

Forced Pendulum Numerical approach Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.

Διαβάστε περισσότερα

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =

Διαβάστε περισσότερα

Problem Set 3: Solutions

Problem Set 3: Solutions CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C

Διαβάστε περισσότερα

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required) Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts

Διαβάστε περισσότερα

ST5224: Advanced Statistical Theory II

ST5224: Advanced Statistical Theory II ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known

Διαβάστε περισσότερα

Introduction to Theory of. Elasticity. Kengo Nakajima Summer

Introduction to Theory of. Elasticity. Kengo Nakajima Summer Introduction to Theor of lasticit Summer Kengo Nakajima Technical & Scientific Computing I (48-7) Seminar on Computer Science (48-4) elast Theor of lasticit Target Stress Governing quations elast 3 Theor

Διαβάστε περισσότερα

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We

Διαβάστε περισσότερα

3+1 Splitting of the Generalized Harmonic Equations

3+1 Splitting of the Generalized Harmonic Equations 3+1 Splitting of the Generalized Harmonic Equations David Brown North Carolina State University EGM June 2011 Numerical Relativity Interpret general relativity as an initial value problem: Split spacetime

Διαβάστε περισσότερα

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- ----------------- Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin

Διαβάστε περισσότερα

Higher Derivative Gravity Theories

Higher Derivative Gravity Theories Higher Derivative Gravity Theories Black Holes in AdS space-times James Mashiyane Supervisor: Prof Kevin Goldstein University of the Witwatersrand Second Mandelstam, 20 January 2018 James Mashiyane WITS)

Διαβάστε περισσότερα

The Simply Typed Lambda Calculus

The Simply Typed Lambda Calculus Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and

Διαβάστε περισσότερα

Written Examination. Antennas and Propagation (AA ) April 26, 2017.

Written Examination. Antennas and Propagation (AA ) April 26, 2017. Written Examination Antennas and Propagation (AA. 6-7) April 6, 7. Problem ( points) Let us consider a wire antenna as in Fig. characterized by a z-oriented linear filamentary current I(z) = I cos(kz)ẑ

Διαβάστε περισσότερα

Econ Spring 2004 Instructor: Prof. Kiefer Solution to Problem set # 5. γ (0)

Econ Spring 2004 Instructor: Prof. Kiefer Solution to Problem set # 5. γ (0) Cornell University Department of Economics Econ 60 - Spring 004 Instructor: Prof. Kiefer Solution to Problem set # 5. Autocorrelation function is defined as ρ h = γ h γ 0 where γ h =Cov X t,x t h =E[X

Διαβάστε περισσότερα

Bayesian statistics. DS GA 1002 Probability and Statistics for Data Science.

Bayesian statistics. DS GA 1002 Probability and Statistics for Data Science. Bayesian statistics DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Frequentist vs Bayesian statistics In frequentist

Διαβάστε περισσότερα

CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS

CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3

Διαβάστε περισσότερα

Tutorial on Multinomial Logistic Regression

Tutorial on Multinomial Logistic Regression Tutorial on Multinomial Logistic Regression Javier R Movellan June 19, 2013 1 1 General Model The inputs are n-dimensional vectors the outputs are c-dimensional vectors The training sample consist of m

Διαβάστε περισσότερα

Parametrized Surfaces

Parametrized Surfaces Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some

Διαβάστε περισσότερα

Srednicki Chapter 55

Srednicki Chapter 55 Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third

Διαβάστε περισσότερα

Pg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is

Pg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =

Διαβάστε περισσότερα

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter

Διαβάστε περισσότερα

Lecture 10 - Representation Theory III: Theory of Weights

Lecture 10 - Representation Theory III: Theory of Weights Lecture 10 - Representation Theory III: Theory of Weights February 18, 2012 1 Terminology One assumes a base = {α i } i has been chosen. Then a weight Λ with non-negative integral Dynkin coefficients Λ

Διαβάστε περισσότερα

Every set of first-order formulas is equivalent to an independent set

Every set of first-order formulas is equivalent to an independent set Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent

Διαβάστε περισσότερα

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β 3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle

Διαβάστε περισσότερα

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1 Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the

Διαβάστε περισσότερα

Differential equations

Differential equations Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential

Διαβάστε περισσότερα

Strain gauge and rosettes

Strain gauge and rosettes Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified

Διαβάστε περισσότερα

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset

Διαβάστε περισσότερα

( ) 2 and compare to M.

( ) 2 and compare to M. Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8

Διαβάστε περισσότερα

DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C

DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C By Tom Irvine Email: tomirvine@aol.com August 6, 8 Introduction The obective is to derive a Miles equation which gives the overall response

Διαβάστε περισσότερα

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max

Διαβάστε περισσότερα

Macromechanics of a Laminate. Textbook: Mechanics of Composite Materials Author: Autar Kaw

Macromechanics of a Laminate. Textbook: Mechanics of Composite Materials Author: Autar Kaw Macromechanics of a Laminate Tetboo: Mechanics of Composite Materials Author: Autar Kaw Figure 4.1 Fiber Direction θ z CHAPTER OJECTIVES Understand the code for laminate stacing sequence Develop relationships

Διαβάστε περισσότερα

Reminders: linear functions

Reminders: linear functions Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U

Διαβάστε περισσότερα

9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr

9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr 9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values

Διαβάστε περισσότερα

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max

Διαβάστε περισσότερα

Review Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Review Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 1) sin - 11π 1 1) + - + - - ) sin 11π 1 ) ( -

Διαβάστε περισσότερα

Durbin-Levinson recursive method

Durbin-Levinson recursive method Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again

Διαβάστε περισσότερα

1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r(t) = 3cost, 4t, 3sint

1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r(t) = 3cost, 4t, 3sint 1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P, π, rt) cost, t, sint ). b) 5 points) Find curvature of the curve at the point P. Solution: a) r t) sint,,

Διαβάστε περισσότερα

6. MAXIMUM LIKELIHOOD ESTIMATION

6. MAXIMUM LIKELIHOOD ESTIMATION 6 MAXIMUM LIKELIHOOD ESIMAION [1] Maximum Likelihood Estimator (1) Cases in which θ (unknown parameter) is scalar Notational Clarification: From now on, we denote the true value of θ as θ o hen, view θ

Διαβάστε περισσότερα

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0. DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec

Διαβάστε περισσότερα

A Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics

A Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics A Bonus-Malus System as a Markov Set-Chain Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics Contents 1. Markov set-chain 2. Model of bonus-malus system 3. Example 4. Conclusions

Διαβάστε περισσότερα

MA 342N Assignment 1 Due 24 February 2016

MA 342N Assignment 1 Due 24 February 2016 M 342N ssignment Due 24 February 206 Id: 342N-s206-.m4,v. 206/02/5 2:25:36 john Exp john. Suppose that q, in addition to satisfying the assumptions from lecture, is an even function. Prove that η(λ = 0,

Διαβάστε περισσότερα

Lecture 26: Circular domains

Lecture 26: Circular domains Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains

Διαβάστε περισσότερα

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R + Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b

Διαβάστε περισσότερα

Empirical best prediction under area-level Poisson mixed models

Empirical best prediction under area-level Poisson mixed models Noname manuscript No. (will be inserted by the editor Empirical best prediction under area-level Poisson mixed models Miguel Boubeta María José Lombardía Domingo Morales eceived: date / Accepted: date

Διαβάστε περισσότερα