# The Spiral of Theodorus, Numerical Analysis, and Special Functions

Save this PDF as:

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

## Transcript

1 Theo p. / The Spiral of Theodorus, Numerical Analysis, and Special Functions Walter Gautschi Purdue University

2 Theo p. 2/ Theodorus of ca B.C.

3 Theo p. 3/ spiral of Theodorus

4 Theo p. 3/ spiral of Theodorus numerical analysis k 3/2 + k /2

5 Theo p. 3/ spiral of Theodorus numerical analysis k 3/2 + k /2 =

6 Theo p. 3/ spiral of Theodorus numerical analysis k 3/2 + k /2 = special functions F (x) = e x2 x e t2 dt Dawson s integral

7 Theo p. 4/ discrete spiral of Theodorus (also known as Quadratwurzelschnecke ; Hlawka, 98) 2 T 5 T 4 T 3 T T T 2 T 8 T 7 T n 2 Tn

8 Theo p. 4/ discrete spiral of Theodorus (also known as Quadratwurzelschnecke ; Hlawka, 98) 2 T 5 T 4 T 3 T T T 2 T 8 T 7 T n 2 Tn parametric representation T (α) C, α

9 Theo p. 4/ discrete spiral of Theodorus (also known as Quadratwurzelschnecke ; Hlawka, 98) 2 T 5 T 4 T 3 T T T 2 T 8 T 7 T n 2 Tn parametric representation T (α) C, α defining properties T (n) = T n T n = n T n+ T n = n =,, 2,...

10 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral

11 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral solution (P.J. Davis, 993; inspired by Euler s work on the gamma function)

12 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral solution (P.J. Davis, 993; inspired by Euler s work on the gamma function) T (α) = + i/ k + i/ k + α, α (i = )

13 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral solution (P.J. Davis, 993; inspired by Euler s work on the gamma function) T (α) = + i/ k + i/ k + α, α (i = ) properties T (α) = α ( ) T (α + ) = + i α T (α)

14 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral solution (P.J. Davis, 993; inspired by Euler s work on the gamma function) T (α) = + i/ k + i/ k + α, α (i = ) properties T (α) = α ( ) T (α + ) = + i α T (α) heart of the spiral: T (α), α 2

15 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral solution (P.J. Davis, 993; inspired by Euler s work on the gamma function) T (α) = + i/ k + i/ k + α, α (i = ) properties T (α) = α ( ) T (α + ) = + i α T (α) heart of the spiral: T (α), α 2 (Gronau, 24) Davis s function is the unique solution of the above difference equation with T (α) and arg T (α) monotonically increasing, and T () =.

16 a little bit of number theory Theo p. 6/

17 Theo p. 6/ a little bit of number theory distribution of the angles ϕ n = T T T n+ = n sin k +, n =, 2, 3,...

18 Theo p. 6/ a little bit of number theory distribution of the angles ϕ n = T T T n+ = n sin k +, n =, 2, 3,... equidistribution (Hlawka, 979) The sequence {ϕ n } n= is equidistributed mod 2π

19 Theo p. 6/ a little bit of number theory distribution of the angles ϕ n = T T T n+ = n sin k +, n =, 2, 3,... equidistribution (Hlawka, 979) The sequence {ϕ n } n= is equidistributed mod 2π more generally ϕ n (α) = T (α)t T (α + n) = n sin k + α < α < 2, n =, 2, 3,...

20 Theo p. 6/ a little bit of number theory distribution of the angles ϕ n = T T T n+ = n sin k +, n =, 2, 3,... equidistribution (Hlawka, 979) The sequence {ϕ n } n= is equidistributed mod 2π more generally ϕ n (α) = T (α)t T (α + n) = n sin k + α < α < 2, n =, 2, 3,... The sequence {ϕ n (α)} n= is equidistributed mod 2π for any α with < α < 2 (Niederreiter, Feb. 3, 29)

21 Theo p. 7/ logarithmic derivative of T (α) T (α) T (α) = + i/ k + α + i/ k d dα ( + i/ ) k + i/ k + α

22 Theo p. 7/ logarithmic derivative of T (α) T (α) T (α) = + i/ k + α + i/ k d dα ( + i/ ) k + i/ k + α = ( + i/ k + α ) i 2 (k + α ) 3/2 ( + i/ k + α ) 2

23 Theo p. 7/ logarithmic derivative of T (α) T (α) T (α) = + i/ k + α + i/ k d dα ( + i/ ) k + i/ k + α = ( + i/ k + α ) i 2 (k + α ) 3/2 ( + i/ k + α ) 2 = i 2 (k + α )( k + α + i)

24 Theo p. 7/ logarithmic derivative of T (α) T (α) T (α) = + i/ k + α + i/ k d dα ( + i/ ) k + i/ k + α = ( + i/ k + α ) i 2 (k + α ) 3/2 ( + i/ k + α ) 2 = i 2 = i 2 (k + α )( k + α + i) k + α i (k + α )(k + α)

25 Theo p. 8/ logarithmic derivative of T (α) (cont ) = 2 (k + α )(k + α) + i 2 (k + α ) 3/2 + (k + α ) /2

26 Theo p. 8/ logarithmic derivative of T (α) (cont ) = 2 (k + α )(k + α) + i 2 = 2 (k + α ) 3/2 + (k + α ) /2 ( k + α ) + i k + α 2 U(α)

27 Theo p. 8/ logarithmic derivative of T (α) (cont ) = 2 (k + α )(k + α) + i 2 where = 2 (k + α ) 3/2 + (k + α ) /2 ( k + α ) + i k + α 2 U(α) = 2α + i 2 U(α) U(α) = (k + α ) 3/2 + (k + α ) /2

28 T (α) T (α) = 2α + i 2 U(α) Theo p. 9/

29 Theo p. 9/ integrate from to α> T (α) T (α) = 2α + i 2 U(α) ln T (α) = ln(α /2 ) + i 2 α U(α)dα

30 Theo p. 9/ integrate from to α> T (α) T (α) = 2α + i 2 U(α) ln T (α) = ln(α /2 ) + i 2 polar representation of T (α) T (α) = ( i α exp 2 α α U(α)dα ) U(α)dα, α >

31 Theo p. 9/ integrate from to α> T (α) T (α) = 2α + i 2 U(α) ln T (α) = ln(α /2 ) + i 2 polar representation of T (α) T (α) = ( i α exp 2 α α U(α)dα ) U(α)dα, α > since T () = + i U(), the slope of the tangent vector to the 2 2 spiral at α = is U() = (Theodorus constant) k 3/2 + k /2

32 Theo p. 9/ integrate from to α> T (α) T (α) = 2α + i 2 U(α) ln T (α) = ln(α /2 ) + i 2 polar representation of T (α) T (α) = ( i α exp 2 α α U(α)dα ) U(α)dα, α > since T () = + i U(), the slope of the tangent vector to the 2 2 spiral at α = is U() = (Theodorus constant) k 3/2 + k /2 numerical analysis and special functions: compute and identify U(α) = U(α)dα for < α < 2, (k+α ) 3/2 +(k+α ) /2 α

33 Theo p. / first digression summation by integration (G. & Milovanović, 985) s = a k, a k = (Lf)(k)

34 Theo p. / first digression summation by integration (G. & Milovanović, 985) s = a k, a k = (Lf)(k) s = (Lf)(k) = e kt f(t)dt

35 Theo p. / first digression summation by integration (G. & Milovanović, 985) s = a k, a k = (Lf)(k) = s = (Lf)(k) = ( ) e kt f(t)dt = e kt f(t)dt t e t f(t) t dt

36 Theo p. / first digression summation by integration (G. & Milovanović, 985) s = a k, a k = (Lf)(k) Thus = s = (Lf)(k) = ( ) e kt f(t)dt = a k = ε(t) = t e t f(t) t ε(t)dt, e kt f(t)dt t e t f(t) t f = L a Bose Einstein distribution dt

37 Theo p. / Theodorus: a k = k /2 = k 3/2 + k/2 k + convolution theorem for Laplace transform Lg Lh = Lg h, (g h)(t) = t g(τ)h(t τ)dτ

38 Theo p. / Theodorus: a k = k /2 = k 3/2 + k/2 k + convolution theorem for Laplace transform Lg Lh = Lg h, (g h)(t) = t g(τ)h(t τ)dτ application to a k k /2 = k + ( L t /2 π ) (k) = ( L e t) (k)

39 Theo p. / Theodorus: a k = k /2 = k 3/2 + k/2 k + convolution theorem for Laplace transform Lg Lh = Lg h, (g h)(t) = t g(τ)h(t τ)dτ application to a k k /2 = k + ( L t /2 π ) (k) = ( L e t) (k) a k = ( L t /2 π ) (k) (L e t ) (k) = t (L π ) τ /2 e (t τ) dτ (k)

40 Theo p. 2/ Theodorus (cont ) f(t) = π e t t = 2 π e t t τ /2 e τ dτ e x2 dx = 2 π F ( t)

41 Theo p. 2/ Theodorus (cont ) f(t) = π e t t = 2 π e t t τ /2 e τ dτ e x2 dx = 2 π F ( t) k 3/2 + k /2 = f(t) t ε(t)dt = 2 π F ( t) t w(t)dt

42 Theo p. 2/ Theodorus (cont ) f(t) = π e t t = 2 π e t t τ /2 e τ dτ e x2 dx = 2 π F ( t) k 3/2 + k /2 = f(t) t ε(t)dt = 2 π F ( t) t w(t)dt w(t) = t /2 ε(t) = t/2 e t

43 Theo p. 3/ second digression Gaussian quadrature n-point quadrature formula g(t)w(t)dt = n λ (n) k g(τ (n) k ), g P 2n

44 Theo p. 3/ second digression Gaussian quadrature n-point quadrature formula g(t)w(t)dt = n λ (n) k g(τ (n) k ), g P 2n orthogonal polynomials (π k, π l ) =, k l, where (u, v) = u(t)v(t)w(t)dt

45 Theo p. 3/ second digression Gaussian quadrature n-point quadrature formula g(t)w(t)dt = n λ (n) k g(τ (n) k ), g P 2n orthogonal polynomials (π k, π l ) =, k l, where (u, v) = three-term recurrence relation u(t)v(t)w(t)dt π k+ (t) = (t α k )π k (t) β k π k (t), k =,, 2,... π (t) =, π (t) = where α k = α k (w) R, β k = β k (w) >, β = w(t)dt

46 Theo p. 4/ Jacobi matrix J n (w) = α β β α β 2 β 2 α β n β n α n

47 Theo p. 4/ Jacobi matrix J n (w) = α β β α β 2 β 2 α β n β n α n Gaussian nodes and weights (Golub & Welsch, 969) τ (n) k = eigenvalues of J n, λ (n) k = β v 2 k, v k, = first component of (normalized) eigenvector v k

48 Theo p. 4/ Jacobi matrix J n (w) = α β β α β 2 β 2 α β n β n α n Gaussian nodes and weights (Golub & Welsch, 969) τ (n) k = eigenvalues of J n, λ (n) k = β v 2 k, v k, = first component of (normalized) eigenvector v k moments of w µ k = t k w(t)dt, k =,,..., 2n

49 Theo p. 4/ Jacobi matrix J n (w) = α β β α β 2 β 2 α β n β n α n Gaussian nodes and weights (Golub & Welsch, 969) τ (n) k = eigenvalues of J n, λ (n) k = β v 2 k, v k, = first component of (normalized) eigenvector v k moments of w µ k = Chebyshev algorithm t k w(t)dt, k =,,..., 2n {µ k } 2n k= {α k, β k } n k=

50 Theo p. 5/ numerical results for Gaussian quadrature (in 5D-arithmetic) µ k = t k w(t)dt = t k+/2 e t dt = Γ(k + 3/2)ζ(k + 3/2)

51 Theo p. 5/ numerical results for Gaussian quadrature (in 5D-arithmetic) µ k = t k w(t)dt = t k+/2 e t dt = Γ(k + 3/2)ζ(k + 3/2) k 3/2 + k = 2 [F ( t)/ t ] w(t) /2 π s n = 2 n ( ) λ (n) π k F τ (n) k / τ (n) k

52 Theo p. 5/ numerical results for Gaussian quadrature (in 5D-arithmetic) µ k = t k w(t)dt = t k+/2 e t dt = Γ(k + 3/2)ζ(k + 3/2) k 3/2 + k = 2 [F ( t)/ t ] w(t) /2 π s n = 2 n ( ) λ (n) π k F τ (n) k / τ (n) k n s n

53 Theo p. 6/ computation and identification of U(α) and α U(α)dα (k + α ) /2 (k + α ) + = t ) (L τ /2 e (t τ) dτ (k + α ) π

54 Theo p. 6/ computation and identification of U(α) and α U(α)dα (k + α ) /2 (k + α ) + = t ) (L τ /2 e (t τ) dτ (k + α ) π applying the shift property for Laplace transform yields (Lg) (s + b) = ( L e bt g(t) ) (s) (k + α ) /2 (k + α ) + = t L (e αt π ) τ /2 e τ dτ (k)

55 Theo p. 6/ computation and identification of U(α) and α U(α)dα (k + α ) /2 (k + α ) + = t ) (L τ /2 e (t τ) dτ (k + α ) π applying the shift property for Laplace transform yields hence (Lg) (s + b) = ( L e bt g(t) ) (s) (k + α ) /2 (k + α ) + = t L (e αt π f(t) = π e αt t ) τ /2 e τ dτ (k) τ /2 e τ dτ = 2 π e (α )t F ( t)

56 Theo p. 7/ U(α) = f(t) t ε(t)dt = 2 e (α )t F ( t) π t w(t)dt w(t) = t/2 e t, α < 2

57 Theo p. 7/ U(α) = f(t) t ε(t)dt = 2 e (α )t F ( t) π t w(t)dt w(t) = t/2 e t, α < 2 identification of U(α) as a Laplace transform U(α) = (Lu) (α ), u(t) = 2 π F ( t) t w(t)

58 Theo p. 7/ U(α) = f(t) t ε(t)dt = 2 e (α )t F ( t) π t w(t)dt w(t) = t/2 e t, α < 2 identification of U(α) as a Laplace transform U(α) = (Lu) (α ), u(t) = 2 π F ( t) t w(t) the integral of U(α) α U(α)dα = 2(α ) π e (α )t (α )t F ( t) t w(t)dt

59 Theo p. 8/ Epilogue summation theory can be generalized to series s + = k ν r(k), s = ( ) k k ν r(k), < ν < where r(k) is a rational function

60 Theo p. 8/ Epilogue summation theory can be generalized to series s + = k ν r(k), s = ( ) k k ν r(k), < ν < where r(k) is a rational function regularizing transformation (J. Waldvogel, in preparation) α = r 2, r R

61 Theo p. 8/ Epilogue summation theory can be generalized to series s + = k ν r(k), s = ( ) k k ν r(k), < ν < where r(k) is a rational function regularizing transformation (J. Waldvogel, in preparation) then T (α) = T (r 2 ) = α = r 2, + i + i/r k=2 r R + i/ k + i/ r 2 + k is analytic for r C \ [i, i ] [ i, i ]

62 Theo p. 8/ Epilogue summation theory can be generalized to series s + = k ν r(k), s = ( ) k k ν r(k), < ν < where r(k) is a rational function regularizing transformation (J. Waldvogel, in preparation) then T (α) = T (r 2 ) = α = r 2, + i + i/r k=2 r R + i/ k + i/ r 2 + k is analytic for r C \ [i, i ] [ i, i ] r > : spiral of Theodorus

63 Theo p. 8/ Epilogue summation theory can be generalized to series s + = k ν r(k), s = ( ) k k ν r(k), < ν < where r(k) is a rational function regularizing transformation (J. Waldvogel, in preparation) then T (α) = T (r 2 ) = α = r 2, + i + i/r k=2 r R + i/ k + i/ r 2 + k is analytic for r C \ [i, i ] [ i, i ] r > : spiral of Theodorus r < : analytic continuation of the spiral

64 Theo p. 9/ twin-spiral of Theodorus

### 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

### Section 9.2 Polar Equations and Graphs

180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify

Διαβάστε περισσότερα

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Trigonometric Formula Sheet

Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ

Διαβάστε περισσότερα

### 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

### Integrals in cylindrical, spherical coordinates (Sect. 15.7)

Integrals in clindrical, spherical coordinates (Sect. 5.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.

Διαβάστε περισσότερα

### Bessel functions. ν + 1 ; 1 = 0 for k = 0, 1, 2,..., n 1. Γ( n + k + 1) = ( 1) n J n (z). Γ(n + k + 1) k!

Bessel functions The Bessel function J ν (z of the first kind of order ν is defined by J ν (z ( (z/ν ν Γ(ν + F ν + ; z 4 ( k k ( Γ(ν + k + k! For ν this is a solution of the Bessel differential equation

Διαβάστε περισσότερα

### 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

### Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal

Διαβάστε περισσότερα

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Mock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =

Mock Eam 7 Mock Eam 7 Section A. Reference: HKDSE Math M 0 Q (a) ( + k) n nn ( )( k) + nk ( ) + + nn ( ) k + nk + + + A nk... () nn ( ) k... () From (), k...() n Substituting () into (), nn ( ) n 76n 76n

Διαβάστε περισσότερα

### High order interpolation function for surface contact problem

3 016 5 Journal of East China Normal University Natural Science No 3 May 016 : 1000-564101603-0009-1 1 1 1 00444; E- 00030 : Lagrange Lobatto Matlab : ; Lagrange; : O41 : A DOI: 103969/jissn1000-56410160300

Διαβάστε περισσότερα

### MATRIX INVERSE EIGENVALUE PROBLEM

English NUMERICAL MATHEMATICS Vol.14, No.2 Series A Journal of Chinese Universities May 2005 A STABILITY ANALYSIS OF THE (k) JACOBI MATRIX INVERSE EIGENVALUE PROBLEM Hou Wenyuan ( ΛΠ) Jiang Erxiong( Ξ)

Διαβάστε περισσότερα

### A Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering

Electronic Companion A Two-Sie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix

Διαβάστε περισσότερα

### Foundations of fractional calculus

Foundations of fractional calculus Sanja Konjik Department of Mathematics and Informatics, University of Novi Sad, Serbia Winter School on Non-Standard Forms of Teaching Mathematics and Physics: Experimental

Διαβάστε περισσότερα

### ( ) 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

### 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

### Finite Field Problems: Solutions

Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The

Διαβάστε περισσότερα

### Variational Wavefunction for the Helium Atom

Technische Universität Graz Institut für Festkörperphysik Student project Variational Wavefunction for the Helium Atom Molecular and Solid State Physics 53. submitted on: 3. November 9 by: Markus Krammer

Διαβάστε περισσότερα

### Nachrichtentechnik I WS 2005/2006

Nachrichtentechnik I WS 2005/2006 1 Signals & Systems wt 10/2005 1 Overview (Signals & Systems) Signals: definition & classification properties basic signals Signal transformations Fourier transformation

Διαβάστε περισσότερα

### Probability and Random Processes (Part II)

Probability and Random Processes (Part II) 1. If the variance σ x of d(n) = x(n) x(n 1) is one-tenth the variance σ x of a stationary zero-mean discrete-time signal x(n), then the normalized autocorrelation

Διαβάστε περισσότερα

### SPECIAL FUNCTIONS and POLYNOMIALS

SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 8.195

Διαβάστε περισσότερα

### 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

### 2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

Διαβάστε περισσότερα

### Numerical Methods for Civil Engineers. Lecture 10 Ordinary Differential Equations. Ordinary Differential Equations. d x dx.

Numerical Metods for Civil Engineers Lecture Ordinar Differential Equations -Basic Ideas -Euler s Metod -Higer Order One-step Metods -Predictor-Corrector Approac -Runge-Kutta Metods -Adaptive Stepsize

Διαβάστε περισσότερα

### Section 8.2 Graphs of Polar Equations

Section 8. Graphs of Polar Equations Graphing Polar Equations The graph of a polar equation r = f(θ), or more generally F(r,θ) = 0, consists of all points P that have at least one polar representation

Διαβάστε περισσότερα

### ΔΙΑΚΡΙΤΟΣ ΜΕΤΑΣΧΗΜΑΤΙΣΜΟΣ FOURIER - Discrete Fourier Transform - DFT -

ΔΙΑΚΡΙΤΟΣ ΜΕΤΑΣΧΗΜΑΤΙΣΜΟΣ FOURIER - Discrete Fourier Transform - DFT - Α. ΣΚΟΔΡΑΣ ΣΗΜΑΤΑ ΚΑΙ ΣΥΣΤΗΜΑΤΑ ΙΙ (22Y603) ΕΝΟΤΗΤΑ 4 ΔΙΑΛΕΞΗ 1 ΔΙΑΦΑΝΕΙΑ 1 Διαφορετικοί Τύποι Μετασχηµατισµού Fourier Α. ΣΚΟΔΡΑΣ

Διαβάστε περισσότερα

### ON NEGATIVE MOMENTS OF CERTAIN DISCRETE DISTRIBUTIONS

Pa J Statist 2009 Vol 25(2), 135-140 ON NEGTIVE MOMENTS OF CERTIN DISCRETE DISTRIBUTIONS Masood nwar 1 and Munir hmad 2 1 Department of Maematics, COMSTS Institute of Information Technology, Islamabad,

Διαβάστε περισσότερα

### 1. If log x 2 y 2 = a, then dy / dx = x 2 + y 2 1] xy 2] y / x. 3] x / y 4] none of these

1. If log x 2 y 2 = a, then dy / dx = x 2 + y 2 1] xy 2] y / x 3] x / y 4] none of these 1. If log x 2 y 2 = a, then x 2 + y 2 Solution : Take y /x = k y = k x dy/dx = k dy/dx = y / x Answer : 2] y / x

Διαβάστε περισσότερα

### 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

### Ψηφιακή Επεξεργασία Φωνής

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Επεξεργασία Φωνής Ενότητα 1η: Ψηφιακή Επεξεργασία Σήματος Στυλιανού Ιωάννης Τμήμα Επιστήμης Υπολογιστών CS578- Speech Signal Processing Lecture 1: Discrete-Time

Διαβάστε περισσότερα

### Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1

Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a

Διαβάστε περισσότερα

### Introduction to Time Series Analysis. Lecture 16.

Introduction to Time Series Analysis. Lecture 16. 1. Review: Spectral density 2. Examples 3. Spectral distribution function. 4. Autocovariance generating function and spectral density. 1 Review: Spectral

Διαβάστε περισσότερα

### Risk! " #\$%&'() *!'+,'''## -. / # \$

Risk! " #\$%&'(!'+,'''## -. / 0! " # \$ +/ #%&''&(+(( &'',\$ #-&''&\$ #(./0&'',\$( ( (! #( &''/\$ #\$ 3 #4&'',\$ #- &'',\$ #5&''6(&''&7&'',\$ / ( /8 9 :&' " 4; < # \$ 3 " ( #\$ = = #\$ #\$ ( 3 - > # \$ 3 = = " 3 3, 6?3

Διαβάστε περισσότερα

### The Jordan Form of Complex Tridiagonal Matrices

The Jordan Form of Complex Tridiagonal Matrices Ilse Ipsen North Carolina State University ILAS p.1 Goal Complex tridiagonal matrix α 1 β 1. γ T = 1 α 2........ β n 1 γ n 1 α n Jordan decomposition T =

Διαβάστε περισσότερα

### Testing for Indeterminacy: An Application to U.S. Monetary Policy. Technical Appendix

Testing for Indeterminacy: An Application to U.S. Monetary Policy Technical Appendix Thomas A. Lubik Department of Economics Johns Hopkins University Frank Schorfheide Department of Economics University

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 Contents 1. Markov set-chain 2. Model of bonus-malus system 3. Example 4. Conclusions

Διαβάστε περισσότερα

### Math 248 Homework 1. Edward Burkard. Exercise 1. Prove the following Fourier Transforms where a > 0 and c R: f (x) = b. f(x c) = e.

Math 48 Homework Ewar Burkar Exercise. Prove the following Fourier Transforms where a > an c : a. f(x) f(ξ) b. f(x c) e πicξ f(ξ) c. eπixc f(x) f(ξ c). f(ax) f(ξ) a e. f (x) πiξ f(ξ) f. xf(x) f(ξ) πi ξ

Διαβάστε περισσότερα

Graded Refractive-Index Common Devices Methodologies for Graded Refractive Index Methodologies: Ray Optics WKB Multilayer Modelling Solution requires: some knowledge of index profile n 2 x Ray Optics for

Διαβάστε περισσότερα

### CBC MATHEMATICS DIVISION MATH 2412-PreCalculus Exam Formula Sheets

System of Equations and Matrices 3 Matrix Row Operations: MATH 41-PreCalculus Switch any two rows. Multiply any row by a nonzero constant. Add any constant-multiple row to another Even and Odd functions

Διαβάστε περισσότερα

### If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2

Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

### UNIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet

UNIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet Solution for take home exam: FYS3, Oct. 4, 3. Problem. Ĥ ɛ K K + ɛ K K + β K K + α K K For Ĥ Ĥ : ɛ ɛ, β α. The operator ˆT can be written

Διαβάστε περισσότερα

### Elements of Information Theory

Elements of Information Theory Model of Digital Communications System A Logarithmic Measure for Information Mutual Information Units of Information Self-Information News... Example Information Measure

Διαβάστε περισσότερα

### If we restrict the domain of y = sin x to [ π 2, π 2

Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

### 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

### ΑΝΩΤΑΤΟ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΠΕΙΡΑΙΑ ΣΧΟΛΗ ΤΕΧΝΟΛΟΓΙΚΩΝ ΕΦΑΡΜΟΓΩΝ ΤΜΗΜΑ ΠΟΛΙΤΙΚΩΝ ΔΟΜΙΚΩΝ ΕΡΓΩΝ

ΑΝΩΤΑΤΟ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΠΕΙΡΑΙΑ ΣΧΟΛΗ ΤΕΧΝΟΛΟΓΙΚΩΝ ΕΦΑΡΜΟΓΩΝ ΤΜΗΜΑ ΠΟΛΙΤΙΚΩΝ ΔΟΜΙΚΩΝ ΕΡΓΩΝ Δυναμική Ανάλυση Κατασκευών με Κατανεμημένη Μάζα και Ακαμψία Πτυχιακή Εργασία Φουκάκη Βαρβάρα

Διαβάστε περισσότερα

X = [ 1 2 4 6 12 15 25 45 68 67 65 98 ] X X double[] X = { 1, 2, 4, 6, 12, 15, 25, 45, 68, 67, 65, 98 }; double X.Length double double[] x1 = { 0, 8, 12, 20 }; double[] x2 = { 8, 9, 11, 12 }; double mean1

Διαβάστε περισσότερα

### Μονοβάθμια Συστήματα: Εξίσωση Κίνησης, Διατύπωση του Προβλήματος και Μέθοδοι Επίλυσης. Απόστολος Σ. Παπαγεωργίου

Μονοβάθμια Συστήματα: Εξίσωση Κίνησης, Διατύπωση του Προβλήματος και Μέθοδοι Επίλυσης VISCOUSLY DAMPED 1-DOF SYSTEM Μονοβάθμια Συστήματα με Ιξώδη Απόσβεση Equation of Motion (Εξίσωση Κίνησης): Complete

Διαβάστε περισσότερα

### 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

### Solution to Review Problems for Midterm III

Solution to Review Problems for Mierm III Mierm III: Friday, November 19 in class Topics:.8-.11, 4.1,4. 1. Find the derivative of the following functions and simplify your answers. (a) x(ln(4x)) +ln(5

Διαβάστε περισσότερα

### 2. Μηχανικό Μαύρο Κουτί: κύλινδρος με μια μπάλα μέσα σε αυτόν.

Experiental Copetition: 14 July 011 Proble Page 1 of. Μηχανικό Μαύρο Κουτί: κύλινδρος με μια μπάλα μέσα σε αυτόν. Ένα μικρό σωματίδιο μάζας (μπάλα) βρίσκεται σε σταθερή απόσταση z από το πάνω μέρος ενός

Διαβάστε περισσότερα

### A General Note on δ-quasi Monotone and Increasing Sequence

International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in

Διαβάστε περισσότερα

### Bayesian modeling of inseparable space-time variation in disease risk

Bayesian modeling of inseparable space-time variation in disease risk Leonhard Knorr-Held Laina Mercer Department of Statistics UW May, 013 Motivation Ohio Lung Cancer Example Lung Cancer Mortality Rates

Διαβάστε περισσότερα

### 2x 2 y x 4 +y 2 J (x, y) (0, 0) 0 J (x, y) = (0, 0) I ϕ(t) = (t, at), ψ(t) = (t, t 2 ), a ÑL<ÝÉ b, ½-? A? 2t 2 at t 4 +a 2 t 2 = lim

9çB\$ø`çü5 (-ç ) Ch.Ch4 b. è. [a] #8ƒb f(x, y) = { x y x 4 +y J (x, y) (, ) J (x, y) = (, ) I ϕ(t) = (t, at), ψ(t) = (t, t ), a ÑL

Διαβάστε περισσότερα

### 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

### Wavelet based matrix compression for boundary integral equations on complex geometries

1 Wavelet based matrix compression for boundary integral equations on complex geometries Ulf Kähler Chemnitz University of Technology Workshop on Fast Boundary Element Methods in Industrial Applications

Διαβάστε περισσότερα

### Jordan Form of a Square Matrix

Jordan Form of a Square Matrix Josh Engwer Texas Tech University josh.engwer@ttu.edu June 3 KEY CONCEPTS & DEFINITIONS: R Set of all real numbers C Set of all complex numbers = {a + bi : a b R and i =

Διαβάστε περισσότερα

### Lecture 2. Soundness and completeness of propositional logic

Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness

Διαβάστε περισσότερα

### DuPont Suva. DuPont. Thermodynamic Properties of. Refrigerant (R-410A) Technical Information. refrigerants T-410A ENG

Technical Information T-410A ENG DuPont Suva refrigerants Thermodynamic Properties of DuPont Suva 410A Refrigerant (R-410A) The DuPont Oval Logo, The miracles of science, and Suva, are trademarks or registered

Διαβάστε περισσότερα

### ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω

0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +

Διαβάστε περισσότερα

### Multi-dimensional Central Limit Theorem

Mult-dmensonal Central Lmt heorem Outlne () () () t as () + () + + () () () Consder a sequence of ndependent random proceses t, t, dentcal to some ( t). Assume t 0. Defne the sum process t t t t () t ();

Διαβάστε περισσότερα

### Problem 3.16 Given B = ˆx(z 3y) +ŷ(2x 3z) ẑ(x+y), find a unit vector parallel. Solution: At P = (1,0, 1), ˆb = B

Problem 3.6 Given B = ˆxz 3y) +ŷx 3z) ẑx+y), find a unit vector parallel to B at point P =,0, ). Solution: At P =,0, ), B = ˆx )+ŷ+3) ẑ) = ˆx+ŷ5 ẑ, ˆb = B B = ˆx+ŷ5 ẑ = ˆx+ŷ5 ẑ. +5+ 7 Problem 3.4 Convert

Διαβάστε περισσότερα

### On the summability of divergent power series solutions for certain first-order linear PDEs Masaki HIBINO (Meijo University)

On the summability of divergent power series solutions for certain first-order linear PDEs Masaki HIBINO (Meijo University) 1 1 Introduction (E) {1+x 2 +β(x,y)}y u x (x,y)+{x+b(x,y)}y2 u y (x,y) +u(x,y)=f(x,y)

Διαβάστε περισσότερα

### The k-fractional Hilfer Derivative

Int. Journal of Math. Analysis, Vol. 7, 213, no. 11, 543-55 The -Fractional Hilfer Derivative Gustavo Abel Dorrego and Rubén A. Cerutti Faculty of Exact Sciences National University of Nordeste. Av. Libertad

Διαβάστε περισσότερα

### 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

### Computable error bounds for asymptotic expansions formulas of distributions related to gamma functions

Computable error bounds for asymptotic expansions formulas of distributions related to gamma functions Hirofumi Wakaki (Math. of Department, Hiroshima Univ.) 20.7. Hiroshima Statistical Group Meeting at

Διαβάστε περισσότερα

### Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions

International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for

Διαβάστε περισσότερα

### Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών. ΗΥ-570: Στατιστική Επεξεργασία Σήµατος. ιδάσκων : Α. Μουχτάρης. εύτερη Σειρά Ασκήσεων.

Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 2015 ιδάσκων : Α. Μουχτάρης εύτερη Σειρά Ασκήσεων Λύσεις Ασκηση 1. 1. Consder the gven expresson for R 1/2 : R 1/2

Διαβάστε περισσότερα

### 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

### Physics 554: HW#1 Solutions

Physics 554: HW#1 Solutions Katrin Schenk 7 Feb 2001 Problem 1.Properties of linearized Riemann tensor: Part a: We want to show that the expression for the linearized Riemann tensor, given by R αβγδ =

Διαβάστε περισσότερα

### 2.1

181 8588 2 21 1 e-mail: sekig@th.nao.ac.jp 1. G ab kt ab, (1) k 8pGc 4, G c 2. 1 2.1 308 2009 5 3 1 2) ( ab ) (g ab ) (K ab ) 1 2.2 3 1 (g ab, K ab ) 1 t a S n a a b a 2.3 a b i (t a ) 2 1 2.4 1 g ab ab

Διαβάστε περισσότερα

### Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test

Διαβάστε περισσότερα

### MATHEMATICS. 1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81

1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81 We know that KA = A If A is n th Order 3AB =3 3 A. B = 27 1 3 = 81 3 2. If A= 2 1 0 0 2 1 then

Διαβάστε περισσότερα

### CYLINDRICAL & SPHERICAL COORDINATES

CYLINDRICAL & SPHERICAL COORDINATES Here we eamine two of the more popular alternative -dimensional coordinate sstems to the rectangular coordinate sstem. First recall the basis of the Rectangular Coordinate

Διαβάστε περισσότερα

### Orbital angular momentum and the spherical harmonics

Orbital angular momentum and the spherical harmonics March 8, 03 Orbital angular momentum We compare our result on representations of rotations with our previous experience of angular momentum, defined

Διαβάστε περισσότερα

### CORDIC Background (4A)

CORDIC Background (4A Copyright (c 20-202 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later

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 Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth

Διαβάστε περισσότερα

### x j (t) = e λ jt v j, 1 j n

9.5: Fundamental Sets of Eigenvector Solutions Homogenous system: x 8 5 10 = Ax, A : n n Ex.: A = Characteristic Polynomial: (degree n) p(λ) = det(a λi) Def.: The multiplicity of a root λ i of p(λ) is

Διαβάστε περισσότερα

### ΚΕΦΑΛΑΙΟ 2. Περιγραφή της Κίνησης. 2.1 Κίνηση στο Επίπεδο

ΚΕΦΑΛΑΙΟ 2 Περιγραφή της Κίνησης Στο κεφάλαιο αυτό θα δείξουμε πώς να προγραμματίσουμε απλές εξισώσεις τροχιάς ενός σωματιδίου και πώς να κάνουμε βασική ανάλυση των αριθμητικών αποτελεσμάτων. Χρησιμοποιούμε

Διαβάστε περισσότερα

### arxiv: v1 [math.ca] 2 Apr 2016

Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions Banu Yılmaz YAŞAR and Mehmet Ali ÖZARSLAN Eastern Mediterranean University, Department of Mathematics Gazimagusa, TRNC, Mersin,

Διαβάστε περισσότερα

### EL 625 Lecture 2. State equations of finite dimensional linear systems

EL 625 Lecture 2 EL 625 Lecture 2 State equations of finite dimensional linear systems Continuous-time: ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) + D(t)u(t) Discrete-time: x(t k+ ) = A(t k )x(t k ) +

Διαβάστε περισσότερα

### Product Identities for Theta Functions

Product Identities for Theta Functions Zhu Cao April 7, 008 Defining Products of Theta Functions n 1 (a; q) 0 = 1, (a; q) n = (1 aq k ), n 1, k=0 (a; q) = lim (a; q) n, q < 1. n Jacobi s triple product

Διαβάστε περισσότερα

### Solutions Ph 236a Week 8

Solutions Ph 236a Week 8 Page 1 of 9 Solutions Ph 236a Week 8 Kevin Barkett, Jonas Lippuner, and Mark Scheel December 1, 2015 Contents Problem 1................................... 2...................................

Διαβάστε περισσότερα

### Estimation for ARMA Processes with Stable Noise. Matt Calder & Richard A. Davis Colorado State University

Estimation for ARMA Processes with Stable Noise Matt Calder & Richard A. Davis Colorado State University rdavis@stat.colostate.edu 1 ARMA processes with stable noise Review of M-estimation Examples of

Διαβάστε περισσότερα

### Derivations of Useful Trigonometric Identities

Derivations of Useful Trigonometric Identities Pythagorean Identity This is a basic and very useful relationship which comes directly from the definition of the trigonometric ratios of sine and cosine

Διαβάστε περισσότερα

### Boundary-Layer Flow over a Flat Plate Approximate Method

Bounar-aer lo oer a lat Plate Approimate Metho Transition Turbulent aminar The momentum balance on a control olume o the bounar laer leas to the olloing equation: + () The approimate metho o bounar laer

Διαβάστε περισσότερα