The Spiral of Theodorus, Numerical Analysis, and Special Functions
|
|
|
- Γάδ Αλεξίου
- 8 χρόνια πριν
- Προβολές:
Transcript
1 Theo p. / The Spiral of Theodorus, Numerical Analysis, and Special Functions Walter Gautschi wxg@cs.purdue.edu 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
Computing the Macdonald function for complex orders
Macdonald p. 1/1 Computing the Macdonald function for complex orders Walter Gautschi wxg@cs.purdue.edu Purdue University Macdonald p. 2/1 Integral representation K ν (x) = complex order ν = α + iβ e x
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
The circle theorem and related theorems for Gauss-type quadrature rules
OP.circle p. / The circle theorem and related theorems for Gauss-type quadrature rules Walter Gautschi wxg@cs.purdue.edu Purdue University OP.circle p. 2/ Web Site http : //www.cs.purdue.edu/ archives/22/wxg/codes
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 = =
DiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
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
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
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)
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
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 +
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
Arithmetical applications of lagrangian interpolation. Tanguy Rivoal. Institut Fourier CNRS and Université de Grenoble 1
Arithmetical applications of lagrangian interpolation Tanguy Rivoal Institut Fourier CNRS and Université de Grenoble Conference Diophantine and Analytic Problems in Number Theory, The 00th anniversary
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)
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
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
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
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
An Inventory of Continuous Distributions
Appendi A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by Also, define Γ(α; ) = 1 with = G(α; ) = Z 0 Z 0 Z t α 1 e t dt, α > 0, >0 t α 1 e t dt, α >
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.
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
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.
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
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
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,
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
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
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:
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.
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
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
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
ECE 468: Digital Image Processing. Lecture 8
ECE 468: Digital Image Processing Lecture 8 Prof. Sinisa Todorovic sinisa@eecs.oregonstate.edu 1 Image Reconstruction from Projections X-ray computed tomography: X-raying an object from different directions
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
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
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
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
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
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
Mean-Variance Analysis
Mean-Variance Analysis Jan Schneider McCombs School of Business University of Texas at Austin Jan Schneider Mean-Variance Analysis Beta Representation of the Risk Premium risk premium E t [Rt t+τ ] R1
University of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing
University of Illinois at Urbana-Champaign ECE : Digital Signal Processing Chandra Radhakrishnan PROBLEM SET : SOLUTIONS Peter Kairouz Problem Solution:. ( 5 ) + (5 6 ) + ( ) cos(5 ) + 5cos( 6 ) + cos(
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
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
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) =
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
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
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
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
Lecture 34 Bootstrap confidence intervals
Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α
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
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
A summation formula ramified with hypergeometric function and involving recurrence relation
South Asian Journal of Mathematics 017, Vol. 7 ( 1): 1 4 www.sajm-online.com ISSN 51-151 RESEARCH ARTICLE A summation formula ramified with hypergeometric function and involving recurrence relation Salahuddin
b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
DETERMINATION OF DYNAMIC CHARACTERISTICS OF A 2DOF SYSTEM. by Zoran VARGA, Ms.C.E.
DETERMINATION OF DYNAMIC CHARACTERISTICS OF A 2DOF SYSTEM by Zoran VARGA, Ms.C.E. Euro-Apex B.V. 1990-2012 All Rights Reserved. The 2 DOF System Symbols m 1 =3m [kg] m 2 =8m m=10 [kg] l=2 [m] E=210000
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,...,
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
Discontinuous Hermite Collocation and Diagonally Implicit RK3 for a Brain Tumour Invasion Model
1 Discontinuous Hermite Collocation and Diagonally Implicit RK3 for a Brain Tumour Invasion Model John E. Athanasakis Applied Mathematics & Computers Laboratory Technical University of Crete Chania 73100,
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
( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
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
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)
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
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
The k-α-exponential Function
Int Journal of Math Analysis, Vol 7, 213, no 11, 535-542 The --Exponential Function Luciano L Luque and Rubén A Cerutti Faculty of Exact Sciences National University of Nordeste Av Libertad 554 34 Corrientes,
Overview. Transition Semantics. Configurations and the transition relation. Executions and computation
Overview Transition Semantics Configurations and the transition relation Executions and computation Inference rules for small-step structural operational semantics for the simple imperative language Transition
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
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
ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Survival Analysis: One-Sample Problem /Two-Sample Problem/Regression. Lu Tian and Richard Olshen Stanford University
Survival Analysis: One-Sample Problem /Two-Sample Problem/Regression Lu Tian and Richard Olshen Stanford University 1 One sample problem T 1,, T n 1 S( ), C 1,, C n G( ) and T i C i Observations: (U i,
( ) 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
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
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
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
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
Answer sheet: Third Midterm for Math 2339
Answer sheet: Third Midterm for Math 339 November 3, Problem. Calculate the iterated integrals (Simplify as much as possible) (a) e sin(x) dydx y e sin(x) dydx y sin(x) ln y ( cos(x)) ye y dx sin(x)(lne
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
Section 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
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 =
Additional Results for the Pareto/NBD Model
Additional Results for the Pareto/NBD Model Peter S. Fader www.petefader.com Bruce G. S. Hardie www.brucehardie.com January 24 Abstract This note derives expressions for i) the raw moments of the posterior
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
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
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
Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type
Noname manuscript No. will be inserted by the editor Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type Victor Nijimbere Received: date / Accepted: date Abstract
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) =
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
SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
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
P AND P. P : actual probability. P : risk neutral probability. Realtionship: mutual absolute continuity P P. For example:
(B t, S (t) t P AND P,..., S (p) t ): securities P : actual probability P : risk neutral probability Realtionship: mutual absolute continuity P P For example: P : ds t = µ t S t dt + σ t S t dw t P : ds
Fundamentals of Signals, Systems and Filtering
Fundamentals of Signals, Systems and Filtering Brett Ninness c 2000-2005, Brett Ninness, School of Electrical Engineering and Computer Science The University of Newcastle, Australia. 2 c Brett Ninness
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
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
The Pohozaev identity for the fractional Laplacian
The Pohozaev identity for the fractional Laplacian Xavier Ros-Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya (joint work with Joaquim Serra) Xavier Ros-Oton (UPC) The Pohozaev
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
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
Laplace Expansion. Peter McCullagh. WHOA-PSI, St Louis August, Department of Statistics University of Chicago
Laplace Expansion Peter McCullagh Department of Statistics University of Chicago WHOA-PSI, St Louis August, 2017 Outline Laplace approximation in 1D Laplace expansion in 1D Laplace expansion in R p Formal
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
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)
![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)](/thumbs/40/21364883.jpg "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
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
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
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
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,