The ε-pseudospectrum of a Matrix

Save this PDF as:
 WORD  PNG  TXT  JPG

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "The ε-pseudospectrum of a Matrix"

Transcript

1 The ε-pseudospectrum of a Matrix Feb 16, 2015 () The ε-pseudospectrum of a Matrix Feb 16, / 18

2 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of Problems () The ε-pseudospectrum of a Matrix Feb 16, / 18

3 Let A M n (C) matrix. () The ε-pseudospectrum of a Matrix Feb 16, / 18

4 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} () The ε-pseudospectrum of a Matrix Feb 16, / 18

5 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + () The ε-pseudospectrum of a Matrix Feb 16, / 18

6 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + M = AA () The ε-pseudospectrum of a Matrix Feb 16, / 18

7 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + M = AA Singular values of A { s s σ(aa )} () The ε-pseudospectrum of a Matrix Feb 16, / 18

8 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + M = AA Singular values of A { s s σ(aa )} s i (A) i th largest singular value of A () The ε-pseudospectrum of a Matrix Feb 16, / 18

9 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + M = AA Singular values of A { s s σ(aa )} s i (A) i th largest singular value of A Spectral Norm of A A = max x =1 Ax 2 = s 1 (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

10 Let A M n (C) matrix. Spectrum of A σ(a) = {λ C det(λi A) = 0} Positive Semi-definite matrix M = M and σ(m) R + M = AA Singular values of A { s s σ(aa )} s i (A) i th largest singular value of A Spectral Norm of A A = max x =1 Ax 2 = s 1 (A) Note: The spectrum of A is an unstable. Invertibility is affected quickly even by small perturbations. Some generalizations of the spectrum are 1 numerical range 2 polynomial numerical hull 3 pseudospectrum () The ε-pseudospectrum of a Matrix Feb 16, / 18

11 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of Problems () The ε-pseudospectrum of a Matrix Feb 16, / 18

12 The ε Pseudospectrum Let A M n (C), a norm on M n (C), and ε > 0 () The ε-pseudospectrum of a Matrix Feb 16, / 18

13 The ε Pseudospectrum Let A M n (C), a norm on M n (C), and ε > 0 σ ε (A) = {z C z σ(a + E) for some E with E < ε} σ ε (A) = {z C (z A)v < ε for some v with v = 1} σ ε (A) = {z C (zi A) 1 1 ε } () The ε-pseudospectrum of a Matrix Feb 16, / 18

14 The ε Pseudospectrum Let A M n (C), a norm on M n (C), and ε > 0 σ ε (A) = {z C z σ(a + E) for some E with E < ε} σ ε (A) = {z C (z A)v < ε for some v with v = 1} σ ε (A) = {z C (zi A) 1 1 ε } If is the spectral norm, then σ ε (A) = {z C s n (zi A) ε} () The ε-pseudospectrum of a Matrix Feb 16, / 18

15 Pseudospectral Radius and Abscissa Pseudospectral Radius ρ ε (A) = max z σ ε z The radius of the smallest circle centered at the origin containing σ ε (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

16 Pseudospectral Radius and Abscissa Pseudospectral Radius ρ ε (A) = max z σ ε z The radius of the smallest circle centered at the origin containing σ ε (A) Pseudospectral Abscissa ρ ε (A) = max z σ ε Re(z) The rightmost vertical support line of σ ε (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

17 Goal To completely describe the geometry of the pseudospectrum of a matrix. () The ε-pseudospectrum of a Matrix Feb 16, / 18

18 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of Problems () The ε-pseudospectrum of a Matrix Feb 16, / 18

19 Reducing Properties () The ε-pseudospectrum of a Matrix Feb 16, / 18

20 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) () The ε-pseudospectrum of a Matrix Feb 16, / 18

21 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 () The ε-pseudospectrum of a Matrix Feb 16, / 18

22 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 2 σ ε (UAU ) = σ ε (A) for any unitary U () The ε-pseudospectrum of a Matrix Feb 16, / 18

23 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 2 σ ε (UAU ) = σ ε (A) for any unitary U 3 If A is normal, i.e. AA = A A, then σ ε (A) = z σ(a) D(z, ε) () The ε-pseudospectrum of a Matrix Feb 16, / 18

24 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 2 σ ε (UAU ) = σ ε (A) for any unitary U 3 If A is normal, i.e. AA = A A, then σ ε (A) = 4 σ ε (A 1 A 2 ) = σ ε (A 1 ) σ ε (A 2 ) z σ(a) D(z, ε) () The ε-pseudospectrum of a Matrix Feb 16, / 18

25 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 2 σ ε (UAU ) = σ ε (A) for any unitary U 3 If A is normal, i.e. AA = A A, then σ ε (A) = 4 σ ε (A 1 A 2 ) = σ ε (A 1 ) σ ε (A 2 ) z σ(a) D(z, ε) 5 σ ε (αi + βa) = α + βσ ε β (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

26 Reducing Properties ) 1 σ ε (diag(d 1,, d n ) = n D(d i, ε) i=1 2 σ ε (UAU ) = σ ε (A) for any unitary U 3 If A is normal, i.e. AA = A A, then σ ε (A) = 4 σ ε (A 1 A 2 ) = σ ε (A 1 ) σ ε (A 2 ) z σ(a) D(z, ε) 5 σ ε (αi + βa) = α + βσ ε β (A) 6 σ ε (A ) = σ ε (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

27 Containment Properties 1 σ(a) σ ε (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

28 Containment Properties 1 σ(a) σ ε (A) 2 If ε 1 ε 2, then σ ε1 (A) σ ε2 (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

29 Containment Properties 1 σ(a) σ ε (A) 2 If ε 1 ε 2, then σ ε1 (A) σ ε2 (A) 3 σ ε E (A) σ ε (A + E) σ ε+ E (A) () The ε-pseudospectrum of a Matrix Feb 16, / 18

30 Containment Properties 1 σ(a) σ ε (A) 2 If ε 1 ε 2, then σ ε1 (A) σ ε2 (A) 3 σ ε E (A) σ ε (A + E) σ ε+ E (A) 4 If κ(x ) = X X 1 = s1(x ) s n(x ), then σ ε κ(s) (SAS 1 ) σ ε (A) σ εκ(s) (SAS 1 ) () The ε-pseudospectrum of a Matrix Feb 16, / 18

31 Containment Properties 1 σ(a) σ ε (A) 2 If ε 1 ε 2, then σ ε1 (A) σ ε2 (A) 3 σ ε E (A) σ ε (A + E) σ ε+ E (A) 4 If κ(x ) = X X 1 = s1(x ) s n(x ), then σ ε κ(s) (SAS 1 ) σ ε (A) σ εκ(s) (SAS 1 ) 5 If u, v are unit vectors such that X v = λv and u X = λu, then κ(λ) = 1/ u v. If A has n distinct eigenvalues, then σ ε (A) D(λ, εκ(λ)) λ σ(a) () The ε-pseudospectrum of a Matrix Feb 16, / 18

32 Containment Properties 1 σ ε (A) λ W (A) D(λ, ε) () The ε-pseudospectrum of a Matrix Feb 16, / 18

33 Containment Properties 1 σ ε (A) D(λ, ε) λ W (A) 2 σ ε (A) D(λ, ε + dep(a)) λ σ(a) () The ε-pseudospectrum of a Matrix Feb 16, / 18

34 Containment Properties 1 σ ε (A) D(λ, ε) λ W (A) 2 σ ε (A) D(λ, ε + dep(a)) λ σ(a) 3 If A = [a ij ] and r j = n a jk, then k=1 j n σ ε (A) D(a jj, r j + ε n) j=1 () The ε-pseudospectrum of a Matrix Feb 16, / 18

35 Geometric Properties 1 σ ε (A) has at most n connected components () The ε-pseudospectrum of a Matrix Feb 16, / 18

36 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A () The ε-pseudospectrum of a Matrix Feb 16, / 18

37 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi () The ε-pseudospectrum of a Matrix Feb 16, / 18

38 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi 4 σ ε (A) has no flat portions () The ε-pseudospectrum of a Matrix Feb 16, / 18

39 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi 4 σ ε (A) has no flat portions 5 σ ε (A) generally not convex, nor simply-connected () The ε-pseudospectrum of a Matrix Feb 16, / 18

40 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi 4 σ ε (A) has no flat portions 5 σ ε (A) generally not convex, nor simply-connected 6 If σ ε (A) = n D(z i, ε) such that D(z j, ε) D(z i, ε) for all j. Then A is normal. i=1 i=1 j () The ε-pseudospectrum of a Matrix Feb 16, / 18

41 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi 4 σ ε (A) has no flat portions 5 σ ε (A) generally not convex, nor simply-connected 6 If σ ε (A) = n D(z i, ε) such that D(z j, ε) D(z i, ε) for all j. Then A is normal. i=1 i=1 j 7 If σ ε (A) = σ ε (B) for any ε, then σ ε (A) and σ ε (B) have the same minimal polynomial. () The ε-pseudospectrum of a Matrix Feb 16, / 18

42 Geometric Properties 1 σ ε (A) has at most n connected components 2 Each connected component contains at least one eigenvalue of A [ ] xi A ɛi 3 z = x + iy σ ε (A) if and only if iy is an eigenvalue of ɛi A xi 4 σ ε (A) has no flat portions 5 σ ε (A) generally not convex, nor simply-connected 6 If σ ε (A) = n D(z i, ε) such that D(z j, ε) D(z i, ε) for all j. Then A is normal. i=1 i=1 j 7 If σ ε (A) = σ ε (B) for any ε, then σ ε (A) and σ ε (B) have the same minimal polynomial. Converse is not true. () The ε-pseudospectrum of a Matrix Feb 16, / 18

43 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of Problems () The ε-pseudospectrum of a Matrix Feb 16, / 18

44 2 2 Matrix Reduction () The ε-pseudospectrum of a Matrix Feb 16, / 18

45 2 2 Matrix Reduction [ ] a b Let A = with σ(a) = {λ 1, λ 2 } c d () The ε-pseudospectrum of a Matrix Feb 16, / 18

46 2 2 Matrix Reduction [ ] a b Let A = with σ(a) = {λ 1, λ 2 } c d = σ ε (A) = (a+d) 2 + σ ε ([ a d 2 b c d a 2 ]) () The ε-pseudospectrum of a Matrix Feb 16, / 18

47 2 2 Matrix Reduction [ ] a b Let A = with σ(a) = {λ 1, λ 2 } c d = σ ε (A) = (a+d) 2 + σ ε ([ a d 2 b c d a 2 ]) ([ã ]) b = α + e iβ σ ε c ã () The ε-pseudospectrum of a Matrix Feb 16, / 18

48 2 2 Matrix Reduction [ ] a b Let A = with σ(a) = {λ 1, λ 2 } c d = σ ε (A) = (a+d) 2 + σ ε ([ a d 2 b c d a 2 ]) = σ ε (A) = σ ε (UAU ) = α + e iβ σ ε ([ r s ([ã ]) b = α + e iβ σ ε c ã 0 r ]), where () The ε-pseudospectrum of a Matrix Feb 16, / 18

49 2 2 Matrix Reduction [ ] a b Let A = with σ(a) = {λ 1, λ 2 } c d = σ ε (A) = (a+d) 2 + σ ε ([ a d 2 b c d a 2 ]) = σ ε (A) = σ ε (UAU ) = α + e iβ σ ε ([ r s ([ã ]) b = α + e iβ σ ε c ã 0 r ]), where α = λ1+λ2 2 and re 2iβ = (λ 1 λ 2 ) 2 and s = tr((a αi )(A αi ) ) 2r 2 () The ε-pseudospectrum of a Matrix Feb 16, / 18

50 2 2 Matrices [ ] r s Let A = 0 r () The ε-pseudospectrum of a Matrix Feb 16, / 18

51 2 2 Matrices [ ] r s Let A = 0 r If r = 0, then () The ε-pseudospectrum of a Matrix Feb 16, / 18

52 2 2 Matrices [ ] r s Let A = 0 r If r = 0, then σ ε (A) = D(0, ε 2 + εs) () The ε-pseudospectrum of a Matrix Feb 16, / 18

53 2 2 Matrices [ ] r s Let A = 0 r If r = 0, then σ ε (A) = D(0, ε 2 + εs) If r 0, then any z = x + iy σ ε (A) satisfies () The ε-pseudospectrum of a Matrix Feb 16, / 18

54 2 2 Matrices [ ] r s Let A = 0 r If r = 0, then σ ε (A) = D(0, ε 2 + εs) If r 0, then any z = x + iy σ ε (A) satisfies y 2 + x 2 + ε2 s 2 4r 2 2 ( ε 2 + r 1 + s2 4r 2 ) y 2 + x 2 + ε2 s 2 4r 2 r 2 ( ε s2 4r 2 ) 0 () The ε-pseudospectrum of a Matrix Feb 16, / 18

55 2 2 Matrices 1 ρ ε (A) = e iθ ρ ε (A 1 ) () The ε-pseudospectrum of a Matrix Feb 16, / 18

56 2 2 Matrices 1 ρ ε (A) = e iθ ρ ε (A 1 ) 2 α ε (A) = tr(a) 2 + α ε (A 0 ) () The ε-pseudospectrum of a Matrix Feb 16, / 18

57 2 2 Matrices 1 ρ ε (A) = e iθ ρ ε (A 1 ) 2 α ε (A) = tr(a) 2 + α ε (A 0 ) Method of Lagrange Multipliers maximize f (x, y) subject to g(x, y) = c Λ(x, y, λ) = f (x, y) + λ(g(x, y) c) = Λ(x, y, λ) = 0 () The ε-pseudospectrum of a Matrix Feb 16, / 18

58 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of Problems () The ε-pseudospectrum of a Matrix Feb 16, / 18

59 Problems 1 A generalized quadratic matrix is of the form [ ] ai bt A = ct di What can we say about the pseudospectral radius/abscissa of A? 2 Consider real structured ε pseudospectrum of A M n (R) σ R ε (A) = {z σ(a + E) E M n (R), E < ε} 3 Consider Special types of nonnormal Matrices. () The ε-pseudospectrum of a Matrix Feb 16, / 18

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

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

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

Finite Field Problems: Solutions

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

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

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

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

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

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

Jordan Form of a Square Matrix

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

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

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

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

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

Αλγόριθμοι και πολυπλοκότητα NP-Completeness (2)

Αλγόριθμοι και πολυπλοκότητα NP-Completeness (2) ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Αλγόριθμοι και πολυπλοκότητα NP-Completeness (2) Ιωάννης Τόλλης Τμήμα Επιστήμης Υπολογιστών NP-Completeness (2) x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 12 22 32 11 13 21

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

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

Πρόβλημα 1: Αναζήτηση Ελάχιστης/Μέγιστης Τιμής

Πρόβλημα 1: Αναζήτηση Ελάχιστης/Μέγιστης Τιμής Πρόβλημα 1: Αναζήτηση Ελάχιστης/Μέγιστης Τιμής Να γραφεί πρόγραμμα το οποίο δέχεται ως είσοδο μια ακολουθία S από n (n 40) ακέραιους αριθμούς και επιστρέφει ως έξοδο δύο ακολουθίες από θετικούς ακέραιους

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

Ενότητα 3: Ακρότατα συναρτήσεων μίας ή πολλών μεταβλητών. Νίκος Καραμπετάκης Τμήμα Μαθηματικών

Ενότητα 3: Ακρότατα συναρτήσεων μίας ή πολλών μεταβλητών. Νίκος Καραμπετάκης Τμήμα Μαθηματικών ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΝΟΙΧΤΑ ΑΚΑΔΗΜΑΙΚΑ ΜΑΘΗΜΑΤΑ Ενότητα 3: Ακρότατα συναρτήσεων μίας ή πολλών μεταβλητών Νίκος Καραμπετάκης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative

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

Trigonometric Formula Sheet

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

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

x j (t) = e λ jt v j, 1 j n

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

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

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

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

The Jordan Form of Complex Tridiagonal Matrices

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

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

( ) 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

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

ENGR 691/692 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework 1: Bayesian Decision Theory (solutions) Due: September 13

ENGR 691/692 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework 1: Bayesian Decision Theory (solutions) Due: September 13 ENGR 69/69 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework : Bayesian Decision Theory (solutions) Due: Septemer 3 Prolem : ( pts) Let the conditional densities for a two-category one-dimensional

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο

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 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

On a four-dimensional hyperbolic manifold with finite volume

On a four-dimensional hyperbolic manifold with finite volume BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In

Διαβάστε περισσότερα

Section 9.2 Polar Equations and Graphs

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

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Τέλος Ενότητας Χρηματοδότηση Το παρόν εκπαιδευτικό υλικό έχει αναπτυχθεί

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

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 24/3/2007

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 24/3/2007 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Όλοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα μικρότεροι του 10000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Αν κάπου κάνετε κάποιες υποθέσεις

Διαβάστε περισσότερα

Math 6 SL Probability Distributions Practice Test Mark Scheme

Math 6 SL Probability Distributions Practice Test Mark Scheme Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry

Διαβάστε περισσότερα

TMA4115 Matematikk 3

TMA4115 Matematikk 3 TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet

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

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) 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

Introduction to Time Series Analysis. Lecture 16.

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

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

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

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

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

ΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011

ΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011 Διάρκεια Διαγωνισμού: 3 ώρες Απαντήστε όλες τις ερωτήσεις Μέγιστο Βάρος (20 Μονάδες) Δίνεται ένα σύνολο από N σφαιρίδια τα οποία δεν έχουν όλα το ίδιο βάρος μεταξύ τους και ένα κουτί που αντέχει μέχρι

Διαβάστε περισσότερα

Section 8.2 Graphs of Polar Equations

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

( )( ) ( ) ( )( ) ( )( ) β = Chapter 5 Exercise Problems EX α So 49 β 199 EX EX EX5.4 EX5.5. (a)

( )( ) ( ) ( )( ) ( )( ) β = Chapter 5 Exercise Problems EX α So 49 β 199 EX EX EX5.4 EX5.5. (a) hapter 5 xercise Problems X5. α β α 0.980 For α 0.980, β 49 0.980 0.995 For α 0.995, β 99 0.995 So 49 β 99 X5. O 00 O or n 3 O 40.5 β 0 X5.3 6.5 μ A 00 β ( 0)( 6.5 μa) 8 ma 5 ( 8)( 4 ) or.88 P on + 0.0065

Διαβάστε περισσότερα

Lecture 2. Soundness and completeness of propositional logic

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

A Note on Intuitionistic Fuzzy. Equivalence Relation

A Note on Intuitionistic Fuzzy. Equivalence Relation International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com

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

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

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 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

Space Physics (I) [AP-3044] Lecture 1 by Ling-Hsiao Lyu Oct Lecture 1. Dipole Magnetic Field and Equations of Magnetic Field Lines

Space Physics (I) [AP-3044] Lecture 1 by Ling-Hsiao Lyu Oct Lecture 1. Dipole Magnetic Field and Equations of Magnetic Field Lines Space Physics (I) [AP-344] Lectue by Ling-Hsiao Lyu Oct. 2 Lectue. Dipole Magnetic Field and Equations of Magnetic Field Lines.. Dipole Magnetic Field Since = we can define = A (.) whee A is called the

Διαβάστε περισσότερα

Strukturalna poprawność argumentu.

Strukturalna poprawność argumentu. Strukturalna poprawność argumentu. Marcin Selinger Uniwersytet Wrocławski Katedra Logiki i Metodologii Nauk marcisel@uni.wroc.pl Table of contents: 1. Definition of argument and further notions. 2. Operations

Διαβάστε περισσότερα

MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS

MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS FUMIE NAKAOKA AND NOBUYUKI ODA Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006 Some properties of minimal closed sets and maximal closed

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

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 10η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 10η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 0η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Best Response Curves Used to solve for equilibria in games

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή

Διαβάστε περισσότερα

Correction Table for an Alcoholometer Calibrated at 20 o C

Correction Table for an Alcoholometer Calibrated at 20 o C An alcoholometer is a device that measures the concentration of ethanol in a water-ethanol mixture (often in units of %abv percent alcohol by volume). The depth to which an alcoholometer sinks in a water-ethanol

Διαβάστε περισσότερα

Answers - Worksheet A ALGEBRA PMT. 1 a = 7 b = 11 c = 1 3. e = 0.1 f = 0.3 g = 2 h = 10 i = 3 j = d = k = 3 1. = 1 or 0.5 l =

Answers - Worksheet A ALGEBRA PMT. 1 a = 7 b = 11 c = 1 3. e = 0.1 f = 0.3 g = 2 h = 10 i = 3 j = d = k = 3 1. = 1 or 0.5 l = C ALGEBRA Answers - Worksheet A a 7 b c d e 0. f 0. g h 0 i j k 6 8 or 0. l or 8 a 7 b 0 c 7 d 6 e f g 6 h 8 8 i 6 j k 6 l a 9 b c d 9 7 e 00 0 f 8 9 a b 7 7 c 6 d 9 e 6 6 f 6 8 g 9 h 0 0 i j 6 7 7 k 9

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

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

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 Lambda Model Characterizing Computational Behaviours of Terms

A Lambda Model Characterizing Computational Behaviours of Terms A Lambda Model Characterizing Computational Behaviours of Terms joint paper with Silvia Ghilezan RPC 01, Sendai, October 26, 2001 1 Plan of the talk normalization properties inverse limit model Stone dualities

Διαβάστε περισσότερα

Lecture Notes Introduction to Cluster Algebra

Lecture Notes Introduction to Cluster Algebra Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: May 29, 2017 7.2 Properties of Exchangeable roots The notion of exchangeable is explained as follows Proposition 7.20. If C and C = C

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

Abstract Storage Devices

Abstract Storage Devices Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD

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

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

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

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

MATRICES

MATRICES MARICES 1. Matrix: he arrangement of numbers or letters in the horizontal and vertical lines so that each horizontal line contains same number of elements and each vertical row contains the same numbers

Διαβάστε περισσότερα

Ανάκτηση Πληροφορίας

Ανάκτηση Πληροφορίας ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΪΚΑ ΜΑΘΗΜΑΤΑ Ενότητα 8: Λανθάνουσα Σημασιολογική Ανάλυση (Latent Semantic Analysis) Απόστολος Παπαδόπουλος Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό

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 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

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

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

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 11/3/2006

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 11/3/2006 ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 11/3/26 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα μικρότεροι το 1 εκτός αν ορίζεται διαφορετικά στη διατύπωση

Διαβάστε περισσότερα

ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ. ΕΠΛ342: Βάσεις Δεδομένων. Χειμερινό Εξάμηνο Φροντιστήριο 10 ΛΥΣΕΙΣ. Επερωτήσεις SQL

ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ. ΕΠΛ342: Βάσεις Δεδομένων. Χειμερινό Εξάμηνο Φροντιστήριο 10 ΛΥΣΕΙΣ. Επερωτήσεις SQL ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΕΠΛ342: Βάσεις Δεδομένων Χειμερινό Εξάμηνο 2013 Φροντιστήριο 10 ΛΥΣΕΙΣ Επερωτήσεις SQL Άσκηση 1 Για το ακόλουθο σχήμα Suppliers(sid, sname, address) Parts(pid, pname,

Διαβάστε περισσότερα

Properties of Some Generalizations of Kac-Murdock-Szegö Matrices

Properties of Some Generalizations of Kac-Murdock-Szegö Matrices Appeared In Structured Matrices in Mathematics, Computer Science and Engineering II AMS Contemporary Mathematics Series 281 (2001) 233 245 Properties of Some Generalizations of Kac-Murdock-Szegö Matrices

Διαβάστε περισσότερα

Μηχανική Μάθηση Hypothesis Testing

Μηχανική Μάθηση Hypothesis Testing ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΕΤΑΙΡΕΙΑ IΔ ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΟΛΥΜΠΙΑΔΑ 2013 21 ΑΠΡΙΛΙΟΥ 2013 Β & Γ ΛΥΚΕΙΟΥ. www.cms.org.cy

ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΕΤΑΙΡΕΙΑ IΔ ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΟΛΥΜΠΙΑΔΑ 2013 21 ΑΠΡΙΛΙΟΥ 2013 Β & Γ ΛΥΚΕΙΟΥ. www.cms.org.cy ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΕΤΑΙΡΕΙΑ IΔ ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΟΛΥΜΠΙΑΔΑ 2013 21 ΑΠΡΙΛΙΟΥ 2013 Β & Γ ΛΥΚΕΙΟΥ www.cms.org.cy ΘΕΜΑΤΑ ΣΤΑ ΕΛΛΗΝΙΚΑ ΚΑΙ ΑΓΓΛΙΚΑ PAPERS IN BOTH GREEK AND ENGLISH ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΟΛΥΜΠΙΑΔΑ

Διαβάστε περισσότερα

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

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

ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ

ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ ΕΛΕΝΑ ΦΛΟΚΑ Επίκουρος Καθηγήτρια Τµήµα Φυσικής, Τοµέας Φυσικής Περιβάλλοντος- Μετεωρολογίας ΓΕΝΙΚΟΙ ΟΡΙΣΜΟΙ Πληθυσµός Σύνολο ατόµων ή αντικειµένων στα οποία αναφέρονται

Διαβάστε περισσότερα

DETERMINANT AND PFAFFIAN OF SUM OF SKEW SYMMETRIC MATRICES. 1. Introduction

DETERMINANT AND PFAFFIAN OF SUM OF SKEW SYMMETRIC MATRICES. 1. Introduction Unspecified Journal Volume 00, Number 0, Pages 000 000 S????-????(XX)0000-0 DETERMINANT AND PFAFFIAN OF SUM OF SKEW SYMMETRIC MATRICES TIN-YAU TAM AND MARY CLAIR THOMPSON Abstract. We completely describe

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. 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

1. For each of the following power series, find the interval of convergence and the radius of convergence:

1. For each of the following power series, find the interval of convergence and the radius of convergence: Math 6 Practice Problems Solutios Power Series ad Taylor Series 1. For each of the followig power series, fid the iterval of covergece ad the radius of covergece: (a ( 1 x Notice that = ( 1 +1 ( x +1.

Διαβάστε περισσότερα

Bounding Nonsplitting Enumeration Degrees

Bounding Nonsplitting Enumeration Degrees Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,

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

ORDINAL ARITHMETIC JULIAN J. SCHLÖDER

ORDINAL ARITHMETIC JULIAN J. SCHLÖDER ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.

Διαβάστε περισσότερα

arxiv: v1 [math.ra] 19 Dec 2017

arxiv: v1 [math.ra] 19 Dec 2017 TWO-DIMENSIONAL LEFT RIGHT UNITAL ALGEBRAS OVER ALGEBRAICALLY CLOSED FIELDS AND R HAHMED UBEKBAEV IRAKHIMOV 3 arxiv:7673v [mathra] 9 Dec 7 Department of Math Faculty of Science UPM Selangor Malaysia &

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

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 Spiral of Theodorus, Numerical Analysis, and Special Functions

The Spiral of Theodorus, Numerical Analysis, and Special Functions Theo p. / The Spiral of Theodorus, Numerical Analysis, and Special Functions Walter Gautschi wxg@cs.purdue.edu Purdue University Theo p. 2/ Theodorus of ca. 46 399 B.C. Theo p. 3/ spiral of Theodorus 6

Διαβάστε περισσότερα

MATRIX INVERSE EIGENVALUE PROBLEM

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

SOLVING CUBICS AND QUARTICS BY RADICALS

SOLVING CUBICS AND QUARTICS BY RADICALS SOLVING CUBICS AND QUARTICS BY RADICALS The purpose of this handout is to record the classical formulas expressing the roots of degree three and degree four polynomials in terms of radicals. We begin with

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

Probability and Random Processes (Part II)

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

wave energy Superposition of linear plane progressive waves Marine Hydrodynamics Lecture Oblique Plane Waves:

wave energy Superposition of linear plane progressive waves Marine Hydrodynamics Lecture Oblique Plane Waves: 3.0 Marine Hydrodynamics, Fall 004 Lecture 0 Copyriht c 004 MIT - Department of Ocean Enineerin, All rihts reserved. 3.0 - Marine Hydrodynamics Lecture 0 Free-surface waves: wave enery linear superposition,

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 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

Cable Systems - Postive/Negative Seq Impedance

Cable Systems - Postive/Negative Seq Impedance Cable Systems - Postive/Negative Seq Impedance Nomenclature: GMD GMR - geometrical mead distance between conductors; depends on construction of the T-line or cable feeder - geometric mean raduius of conductor

Διαβάστε περισσότερα

Multi-dimensional Central Limit Theorem

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

Solution to Review Problems for Midterm III

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

(1) Describe the process by which mercury atoms become excited in a fluorescent tube (3)

(1) Describe the process by which mercury atoms become excited in a fluorescent tube (3) Q1. (a) A fluorescent tube is filled with mercury vapour at low pressure. In order to emit electromagnetic radiation the mercury atoms must first be excited. (i) What is meant by an excited atom? (1) (ii)

Διαβάστε περισσότερα