MATH 590: Meshfree Methods

Size: px
Start display at page:

Download "MATH 590: Meshfree Methods"

Transcription

1 MATH 590: Meshfree Methods Chapter 7: Conditionally Positive Definite Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 MATH 590 Chapter 7 1

2 Outline 1 Conditionally Positive Definite Functions Defined 2 CPD Functions and Generalized Fourier Transforms fasshauer@iit.edu MATH 590 Chapter 7 2

3 Outline Conditionally Positive Definite Functions Defined 1 Conditionally Positive Definite Functions Defined 2 CPD Functions and Generalized Fourier Transforms fasshauer@iit.edu MATH 590 Chapter 7 3

4 Conditionally Positive Definite Functions Defined In this chapter we generalize positive definite functions to conditionally positive definite and strictly conditionally positive definite functions of order m. These functions provide a natural generalization of RBF interpolation with polynomial reproduction. Examples of strictly conditionally positive definite (radial) functions are given in the next chapter. fasshauer@iit.edu MATH 590 Chapter 7 4

5 Conditionally Positive Definite Functions Defined Definition A complex-valued continuous function Φ is called conditionally positive definite of order m on R s if N j=1 k=1 N c j c k Φ(x j x k ) 0 (1) for any N pairwise distinct points x 1,..., x N R s, and c = [c 1,..., c N ] T C N satisfying N c j p(x j ) = 0, j=1 for any complex-valued polynomial p of degree at most m 1. The function Φ is called strictly conditionally positive definite of order m on R s if the quadratic form (1) is zero only for c 0. fasshauer@iit.edu MATH 590 Chapter 7 5

6 Conditionally Positive Definite Functions Defined An immediate observation is Lemma A function that is (strictly) conditionally positive definite of order m on R s is also (strictly) conditionally positive definite of any higher order. In particular, a (strictly) positive definite function is always (strictly) conditionally positive definite of any order. fasshauer@iit.edu MATH 590 Chapter 7 6

7 Conditionally Positive Definite Functions Defined An immediate observation is Lemma A function that is (strictly) conditionally positive definite of order m on R s is also (strictly) conditionally positive definite of any higher order. In particular, a (strictly) positive definite function is always (strictly) conditionally positive definite of any order. Proof. The first statement follows immediately from the definition. fasshauer@iit.edu MATH 590 Chapter 7 6

8 Conditionally Positive Definite Functions Defined An immediate observation is Lemma A function that is (strictly) conditionally positive definite of order m on R s is also (strictly) conditionally positive definite of any higher order. In particular, a (strictly) positive definite function is always (strictly) conditionally positive definite of any order. Proof. The first statement follows immediately from the definition. The second statement is true since (by convention) the case m = 0 yields the class of (strictly) positive definite functions, i.e., (strictly) conditionally positive definite functions of order zero are (strictly) positive definite. fasshauer@iit.edu MATH 590 Chapter 7 6

9 Conditionally Positive Definite Functions Defined As for positive definite functions we also have (see [Wendland (2005a)] for more details) Theorem A real-valued continuous even function Φ is called conditionally positive definite of order m on R s if N j=1 k=1 N c j c k Φ(x j x k ) 0 (2) for any N pairwise distinct points x 1,..., x N R s, and c = [c 1,..., c N ] T R N satisfying N c j p(x j ) = 0, j=1 for any real-valued polynomial p of degree at most m 1. The function Φ is called strictly conditionally positive definite of order m on R s is zero only for c 0. fasshauer@iit.edu MATH 590 Chapter 7 7

10 Conditionally Positive Definite Functions Defined Remark If the function Φ is strictly conditionally positive definite of order m, then the matrix A with entries A jk = Φ(x j x k ) can be interpreted as being positive definite on the space of vectors c such that N c j p(x j ) = 0, p Π s m 1. j=1 fasshauer@iit.edu MATH 590 Chapter 7 8

11 Conditionally Positive Definite Functions Defined Remark If the function Φ is strictly conditionally positive definite of order m, then the matrix A with entries A jk = Φ(x j x k ) can be interpreted as being positive definite on the space of vectors c such that N c j p(x j ) = 0, p Π s m 1. j=1 In this sense A is positive definite on the space of vectors c perpendicular to s-variate polynomials of degree at most m 1. fasshauer@iit.edu MATH 590 Chapter 7 8

12 Conditionally Positive Definite Functions Defined We can now generalize the theorem we had in the previous chapter for constant precision interpolation to the case of general polynomial reproduction: MATH 590 Chapter 7 9

13 Conditionally Positive Definite Functions Defined We can now generalize the theorem we had in the previous chapter for constant precision interpolation to the case of general polynomial reproduction: Theorem If the real-valued even function Φ is strictly conditionally positive definite of order m on R s and the points x 1,..., x N form an (m 1)-unisolvent set, then the system of linear equations [ A P P T O ] [ c d ] = [ y 0 ] (3) is uniquely solvable. fasshauer@iit.edu MATH 590 Chapter 7 9

14 Proof Conditionally Positive Definite Functions Defined The proof is almost identical to the proof of the earlier theorem for constant reproduction. MATH 590 Chapter 7 10

15 Proof Conditionally Positive Definite Functions Defined The proof is almost identical to the proof of the earlier theorem for constant reproduction. Assume [c, d] T is a solution of the homogeneous linear system, i.e., with y = 0. fasshauer@iit.edu MATH 590 Chapter 7 10

16 Proof Conditionally Positive Definite Functions Defined The proof is almost identical to the proof of the earlier theorem for constant reproduction. Assume [c, d] T is a solution of the homogeneous linear system, i.e., with y = 0. We show that [c, d] T = 0 is the only possible solution. fasshauer@iit.edu MATH 590 Chapter 7 10

17 Proof Conditionally Positive Definite Functions Defined The proof is almost identical to the proof of the earlier theorem for constant reproduction. Assume [c, d] T is a solution of the homogeneous linear system, i.e., with y = 0. We show that [c, d] T = 0 is the only possible solution. Multiplication of the top block of (3) by c T yields c T Ac + c T Pd = 0. fasshauer@iit.edu MATH 590 Chapter 7 10

18 Proof Conditionally Positive Definite Functions Defined The proof is almost identical to the proof of the earlier theorem for constant reproduction. Assume [c, d] T is a solution of the homogeneous linear system, i.e., with y = 0. We show that [c, d] T = 0 is the only possible solution. Multiplication of the top block of (3) by c T yields c T Ac + c T Pd = 0. From the bottom block of the system we know P T c = 0. This implies c T P = 0 T, and therefore c T Ac = 0. (4) fasshauer@iit.edu MATH 590 Chapter 7 10

19 Conditionally Positive Definite Functions Defined Since the function Φ is strictly conditionally positive definite of order m by assumption we know that the above quadratic form of A (with coefficients such that P T c = 0) is zero only for c = 0. fasshauer@iit.edu MATH 590 Chapter 7 11

20 Conditionally Positive Definite Functions Defined Since the function Φ is strictly conditionally positive definite of order m by assumption we know that the above quadratic form of A (with coefficients such that P T c = 0) is zero only for c = 0. Therefore (4) tells us that c = 0. fasshauer@iit.edu MATH 590 Chapter 7 11

21 Conditionally Positive Definite Functions Defined Since the function Φ is strictly conditionally positive definite of order m by assumption we know that the above quadratic form of A (with coefficients such that P T c = 0) is zero only for c = 0. Therefore (4) tells us that c = 0. The unisolvency of the data sites, i.e., the linear independence of the columns of P (c.f. one of our earlier remarks), and the fact that c = 0 guarantee d = 0 from the top block Ac + Pd = 0 of the homogeneous version of (3). fasshauer@iit.edu MATH 590 Chapter 7 11

22 Outline CPD Functions and Generalized Fourier Transforms 1 Conditionally Positive Definite Functions Defined 2 CPD Functions and Generalized Fourier Transforms fasshauer@iit.edu MATH 590 Chapter 7 12

23 CPD Functions and Generalized Fourier Transforms As before, integral characterizations help us identify functions that are strictly conditionally positive definite of order m on R s. fasshauer@iit.edu MATH 590 Chapter 7 13

24 CPD Functions and Generalized Fourier Transforms As before, integral characterizations help us identify functions that are strictly conditionally positive definite of order m on R s. An integral characterization of conditionally positive definite functions of order m, i.e., a generalization of Bochner s theorem, can be found in the paper [Sun (1993b)]. fasshauer@iit.edu MATH 590 Chapter 7 13

25 CPD Functions and Generalized Fourier Transforms As before, integral characterizations help us identify functions that are strictly conditionally positive definite of order m on R s. An integral characterization of conditionally positive definite functions of order m, i.e., a generalization of Bochner s theorem, can be found in the paper [Sun (1993b)]. However, since the subject matter is rather complicated, and since it does not really help us solve the scattered data interpolation problem, we do not mention any details here. fasshauer@iit.edu MATH 590 Chapter 7 13

26 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform In order to formulate the Fourier transform characterization of strictly conditionally positive definite functions of order m on R s we require some advanced tools from analysis (see Appendix B). fasshauer@iit.edu MATH 590 Chapter 7 14

27 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform In order to formulate the Fourier transform characterization of strictly conditionally positive definite functions of order m on R s we require some advanced tools from analysis (see Appendix B). First we define the Schwartz space of rapidly decreasing test functions S = {γ C (R s ) : lim x α (D β γ)(x) = 0, α, β N s 0 }, x where we use the multi-index notation D β = x β 1 1 β x βs s, β = s β i. i=1 fasshauer@iit.edu MATH 590 Chapter 7 14

28 CPD Functions and Generalized Fourier Transforms Some properties of the Schwartz space The Schwartz Space and the Generalized Fourier Transform S consists of all those functions γ C (R s ) which, together with all their derivatives, decay faster than any power of 1/ x. fasshauer@iit.edu MATH 590 Chapter 7 15

29 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform Some properties of the Schwartz space S consists of all those functions γ C (R s ) which, together with all their derivatives, decay faster than any power of 1/ x. S contains the space C 0 (Rs ), the space of all infinitely differentiable functions on R s with compact support. fasshauer@iit.edu MATH 590 Chapter 7 15

30 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform Some properties of the Schwartz space S consists of all those functions γ C (R s ) which, together with all their derivatives, decay faster than any power of 1/ x. S contains the space C 0 (Rs ), the space of all infinitely differentiable functions on R s with compact support. C 0 (Rs ) is a true subspace of S since, e.g., the function γ(x) = e x 2 belongs to S but not to C 0 (Rs ). fasshauer@iit.edu MATH 590 Chapter 7 15

31 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform Some properties of the Schwartz space S consists of all those functions γ C (R s ) which, together with all their derivatives, decay faster than any power of 1/ x. S contains the space C 0 (Rs ), the space of all infinitely differentiable functions on R s with compact support. C 0 (Rs ) is a true subspace of S since, e.g., the function γ(x) = e x 2 belongs to S but not to C 0 (Rs ). γ S has a classical Fourier transform ˆγ which is also in S. fasshauer@iit.edu MATH 590 Chapter 7 15

32 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform Of particular importance are the following subspaces S m of S S m = {γ S : γ(x) = O( x m ) for x 0, m N 0 }. fasshauer@iit.edu MATH 590 Chapter 7 16

33 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform Of particular importance are the following subspaces S m of S S m = {γ S : γ(x) = O( x m ) for x 0, m N 0 }. Furthermore, the set V of slowly increasing functions is given by V = {f C(R s ) : f (x) p(x) for some polynomial p Π s }. fasshauer@iit.edu MATH 590 Chapter 7 16

34 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform The generalized Fourier transform is now given by Definition Let f V be complex-valued. A continuous function ˆf : R s \ {0} C is called the generalized Fourier transform of f if there exists an integer m N 0 such that f (x)ˆγ(x)dx = ˆf (x)γ(x)dx R s R s is satisfied for all γ S 2m. The smallest such integer m is called the order of ˆf. fasshauer@iit.edu MATH 590 Chapter 7 17

35 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform The generalized Fourier transform is now given by Definition Let f V be complex-valued. A continuous function ˆf : R s \ {0} C is called the generalized Fourier transform of f if there exists an integer m N 0 such that f (x)ˆγ(x)dx = ˆf (x)γ(x)dx R s R s is satisfied for all γ S 2m. The smallest such integer m is called the order of ˆf. Remark Various definitions of the generalized Fourier transform exist in the literature (see, e.g., [Gel fand and Vilenkin (1964)]). fasshauer@iit.edu MATH 590 Chapter 7 17

36 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform Since one can show that the generalized Fourier transform of an s-variate polynomial of degree at most 2m is zero, it follows that the inverse generalized Fourier transform is only unique up to addition of such a polynomial. fasshauer@iit.edu MATH 590 Chapter 7 18

37 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform Since one can show that the generalized Fourier transform of an s-variate polynomial of degree at most 2m is zero, it follows that the inverse generalized Fourier transform is only unique up to addition of such a polynomial. The order of the generalized Fourier transform is nothing but the order of the singularity at the origin of the generalized Fourier transform. fasshauer@iit.edu MATH 590 Chapter 7 18

38 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform Since one can show that the generalized Fourier transform of an s-variate polynomial of degree at most 2m is zero, it follows that the inverse generalized Fourier transform is only unique up to addition of such a polynomial. The order of the generalized Fourier transform is nothing but the order of the singularity at the origin of the generalized Fourier transform. For functions in L 1 (R s ) the generalized Fourier transform coincides with the classical Fourier transform. fasshauer@iit.edu MATH 590 Chapter 7 18

39 CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform Since one can show that the generalized Fourier transform of an s-variate polynomial of degree at most 2m is zero, it follows that the inverse generalized Fourier transform is only unique up to addition of such a polynomial. The order of the generalized Fourier transform is nothing but the order of the singularity at the origin of the generalized Fourier transform. For functions in L 1 (R s ) the generalized Fourier transform coincides with the classical Fourier transform. For functions in L 2 (R s ) it coincides with the distributional Fourier transform. fasshauer@iit.edu MATH 590 Chapter 7 18

40 CPD Functions and Generalized Fourier Transforms Fourier Transform Characterization This general approach originated in the manuscript [Madych and Nelson (1983)]. Many more details can be found in the original literature as well as in [Wendland (2005a)]. The following result is due to [Iske (1994)]. Theorem Suppose the complex-valued function Φ V possesses a generalized Fourier transform ˆΦ of order m which is continuous on R s \ {0}. Then Φ is strictly conditionally positive definite of order m if and only if ˆΦ is non-negative and non-vanishing. fasshauer@iit.edu MATH 590 Chapter 7 19

41 CPD Functions and Generalized Fourier Transforms Fourier Transform Characterization This general approach originated in the manuscript [Madych and Nelson (1983)]. Many more details can be found in the original literature as well as in [Wendland (2005a)]. The following result is due to [Iske (1994)]. Theorem Suppose the complex-valued function Φ V possesses a generalized Fourier transform ˆΦ of order m which is continuous on R s \ {0}. Then Φ is strictly conditionally positive definite of order m if and only if ˆΦ is non-negative and non-vanishing. Remark This theorem states that strictly conditionally positive definite functions on R s are characterized by the order of the singularity of their generalized Fourier transform at the origin, provided that this generalized Fourier transform is non-negative and non-zero. fasshauer@iit.edu MATH 590 Chapter 7 19

42 CPD Functions and Generalized Fourier Transforms Fourier Transform Characterization Since integral characterizations similar to our earlier theorems of Schoenberg for positive definite radial functions are so complicated in the conditionally positive definite case we do not pursue the concept of a conditionally positive definite radial function here. Such theorems can be found in [Guo et al. (1993a)]. fasshauer@iit.edu MATH 590 Chapter 7 20

43 CPD Functions and Generalized Fourier Transforms Fourier Transform Characterization Since integral characterizations similar to our earlier theorems of Schoenberg for positive definite radial functions are so complicated in the conditionally positive definite case we do not pursue the concept of a conditionally positive definite radial function here. Such theorems can be found in [Guo et al. (1993a)]. Examples of radial functions via the Fourier transform approach are given in the next chapter. In Chapter 9 we will explore the connection between completely and multiply monotone functions and conditionally positive definite radial functions. fasshauer@iit.edu MATH 590 Chapter 7 20

44 References I Appendix References Buhmann, M. D. (2003). Radial Basis Functions: Theory and Implementations. Cambridge University Press. Fasshauer, G. E. (2007). Meshfree Approximation Methods with MATLAB. World Scientific Publishers. Gel fand, I. M. and Vilenkin, N. Ya. (1964). Generalized Functions Vol. 4. Academic Press (New York). Iske, A. (2004). Multiresolution Methods in Scattered Data Modelling. Lecture Notes in Computational Science and Engineering 37, Springer Verlag (Berlin). Wendland, H. (2005a). Scattered Data Approximation. Cambridge University Press (Cambridge). fasshauer@iit.edu MATH 590 Chapter 7 21

45 References II Appendix References Guo, K., Hu, S. and Sun, X. (1993a). Conditionally positive definite functions and Laplace-Stieltjes integrals. J. Approx. Theory 74, pp Iske, A. (1994). Charakterisierung bedingt positiv definiter Funktionen für multivariate Interpolationsmethoden mit radial Basisfunktionen. Ph.D. Dissertation, Universität Göttingen. Madych, W. R. and Nelson, S. A. (1983). Multivariate interpolation: a variational theory. manuscript. Sun, X. (1993b). Conditionally positive definite functions and their application to multivariate interpolation. J. Approx. Theory 74, pp fasshauer@iit.edu MATH 590 Chapter 7 22

These axioms must hold for all vectors ū, v, and w in V and all scalars c and d.

These axioms must hold for all vectors ū, v, and w in V and all scalars c and d. DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms

More information

Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I. Ronald van Luijk, 2012 Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

by the matrix A results in a vector which is a reflection of the given

by the matrix A results in a vector which is a reflection of the given Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

Section 4.4 Inner Product Spaces

Section 4.4 Inner Product Spaces Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer

More information

Moving Least Squares Approximation

Moving Least Squares Approximation Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the so-called moving least squares method. As we will see below, in this method the

More information

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

More information

4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces Geometrical Characterization of RN-operators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg

More information

Factoring Cubic Polynomials

Factoring Cubic Polynomials Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than

More information

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

1 Review of Least Squares Solutions to Overdetermined Systems

1 Review of Least Squares Solutions to Overdetermined Systems cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares

More information

minimal polyonomial Example

minimal polyonomial Example Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

More information

BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBERT SPACE REVIEW BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

More information

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

More information

Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

More information

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein) Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

More information

The Characteristic Polynomial

The Characteristic Polynomial Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

More information

The Factor Theorem and a corollary of the Fundamental Theorem of Algebra

The Factor Theorem and a corollary of the Fundamental Theorem of Algebra Math 421 Fall 2010 The Factor Theorem and a corollary of the Fundamental Theorem of Algebra 27 August 2010 Copyright 2006 2010 by Murray Eisenberg. All rights reserved. Prerequisites Mathematica Aside

More information

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued). MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

More information

Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom. Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

More information

Inner product. Definition of inner product

Inner product. Definition of inner product Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

More information

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A = MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

More information

Methods for Finding Bases

Methods for Finding Bases Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

[1] Diagonal factorization

[1] Diagonal factorization 8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES 1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

More information

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

More information

8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

More information

ASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1

ASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1 19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point

More information

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively. Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

5. Factoring by the QF method

5. Factoring by the QF method 5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the

More information

Math 181 Handout 16. Rich Schwartz. March 9, 2010

Math 181 Handout 16. Rich Schwartz. March 9, 2010 Math 8 Handout 6 Rich Schwartz March 9, 200 The purpose of this handout is to describe continued fractions and their connection to hyperbolic geometry. The Gauss Map Given any x (0, ) we define γ(x) =

More information

5 Homogeneous systems

5 Homogeneous systems 5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m

More information

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

More information

8.1 Examples, definitions, and basic properties

8.1 Examples, definitions, and basic properties 8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

Lecture L3 - Vectors, Matrices and Coordinate Transformations S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

More information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN TRIPLES KEITH CONRAD PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall that two vectors in are perpendicular or orthogonal provided that their dot Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

More information

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

More information

Separation Properties for Locally Convex Cones

Separation Properties for Locally Convex Cones Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

More information

How To Prove The Dirichlet Unit Theorem

How To Prove The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

Orthogonal Projections

Orthogonal Projections Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

More information

LEARNING OBJECTIVES FOR THIS CHAPTER

LEARNING OBJECTIVES FOR THIS CHAPTER CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

More information

SECRET sharing schemes were introduced by Blakley [5]

SECRET sharing schemes were introduced by Blakley [5] 206 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 1, JANUARY 2006 Secret Sharing Schemes From Three Classes of Linear Codes Jin Yuan Cunsheng Ding, Senior Member, IEEE Abstract Secret sharing has

More information

LS.6 Solution Matrices

LS.6 Solution Matrices LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

Maximum Likelihood Estimation

Maximum Likelihood Estimation Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

More information

The Notebook Series. The solution of cubic and quartic equations. R.S. Johnson. Professor of Applied Mathematics

The Notebook Series. The solution of cubic and quartic equations. R.S. Johnson. Professor of Applied Mathematics The Notebook Series The solution of cubic and quartic equations by R.S. Johnson Professor of Applied Mathematics School of Mathematics & Statistics University of Newcastle upon Tyne R.S.Johnson 006 CONTENTS

More information

Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 51 First Exam January 29, 2015 Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

More information

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

More information

Notes on Factoring. MA 206 Kurt Bryan

Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

Differentiation and Integration

Differentiation and Integration This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

More information

1 Inner Products and Norms on Real Vector Spaces

1 Inner Products and Norms on Real Vector Spaces Math 373: Principles Techniques of Applied Mathematics Spring 29 The 2 Inner Product 1 Inner Products Norms on Real Vector Spaces Recall that an inner product on a real vector space V is a function from

More information

Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels

Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels Tanya M. Morton 1) and Marian Neamtu ) Abstract. The problem of solving pseudodifferential equations on

More information

Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

More information

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m) Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

More information

Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

More information

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

More information

1 Sufficient statistics

1 Sufficient statistics 1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

More information

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

More information

Homogeneous systems of algebraic equations. A homogeneous (ho-mo-geen -ius) system of linear algebraic equations is one in which

Homogeneous systems of algebraic equations. A homogeneous (ho-mo-geen -ius) system of linear algebraic equations is one in which Homogeneous systems of algebraic equations A homogeneous (ho-mo-geen -ius) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x + + a n x n = a

More information

Fuzzy Probability Distributions in Bayesian Analysis

Fuzzy Probability Distributions in Bayesian Analysis Fuzzy Probability Distributions in Bayesian Analysis Reinhard Viertl and Owat Sunanta Department of Statistics and Probability Theory Vienna University of Technology, Vienna, Austria Corresponding author:

More information

3. INNER PRODUCT SPACES

3. INNER PRODUCT SPACES . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

More information

Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12 CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

More information

( ) which must be a vector

( ) which must be a vector MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

More information

Duality of linear conic problems

Duality of linear conic problems Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

More information

WHEN DOES A CROSS PRODUCT ON R n EXIST?

WHEN DOES A CROSS PRODUCT ON R n EXIST? WHEN DOES A CROSS PRODUCT ON R n EXIST? PETER F. MCLOUGHLIN It is probably safe to say that just about everyone reading this article is familiar with the cross product and the dot product. However, what

More information

HETEROGENEOUS AGENTS AND AGGREGATE UNCERTAINTY. Daniel Harenberg daniel.harenberg@gmx.de. University of Mannheim. Econ 714, 28.11.

HETEROGENEOUS AGENTS AND AGGREGATE UNCERTAINTY. Daniel Harenberg daniel.harenberg@gmx.de. University of Mannheim. Econ 714, 28.11. COMPUTING EQUILIBRIUM WITH HETEROGENEOUS AGENTS AND AGGREGATE UNCERTAINTY (BASED ON KRUEGER AND KUBLER, 2004) Daniel Harenberg daniel.harenberg@gmx.de University of Mannheim Econ 714, 28.11.06 Daniel Harenberg

More information

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

it is easy to see that α = a

it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

More information

A characterization of trace zero symmetric nonnegative 5x5 matrices

A characterization of trace zero symmetric nonnegative 5x5 matrices A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

More information

MATH 551 - APPLIED MATRIX THEORY

MATH 551 - APPLIED MATRIX THEORY MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points

More information

Notes from February 11

Notes from February 11 Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The

More information

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)! Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

More information

Markov random fields and Gibbs measures

Markov random fields and Gibbs measures Chapter Markov random fields and Gibbs measures 1. Conditional independence Suppose X i is a random element of (X i, B i ), for i = 1, 2, 3, with all X i defined on the same probability space (.F, P).

More information