MATH 590: Meshfree Methods
|
|
|
- Bryce Jesse Norman
- 9 years ago
- Views:
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 [email protected] 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 [email protected] 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. [email protected] 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. [email protected] MATH 590 Chapter 7 5
![.., 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.](/docs-images/43/18897974/images/page_5.jpg "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.")
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. [email protected] 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. [email protected] 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. [email protected] 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. [email protected] 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 [email protected] 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. [email protected] 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. [email protected] 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. [email protected] 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. [email protected] MATH 590 Chapter 7 10
![Assume [c, d] T is a solution of the homogeneous linear system, i.e., with y = 0.](/docs-images/43/18897974/images/page_16.jpg "We show that [c, d] T = 0 is the only possible solution. fasshauer@iit.")
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. [email protected] MATH 590 Chapter 7 10
![Assume [c, d] T is a solution of the homogeneous linear system, i.e., with y = 0.](/docs-images/43/18897974/images/page_17.jpg "We show that [c, d] T = 0 is the only possible solution.")
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) [email protected] MATH 590 Chapter 7 10
![We show that [c, d] T = 0 is the only possible solution.](/docs-images/43/18897974/images/page_18.jpg "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.")
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. [email protected] 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. [email protected] 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). [email protected] 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 [email protected] 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. [email protected] 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)]. [email protected] 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. [email protected] 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). [email protected] 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 [email protected] 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. [email protected] 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. [email protected] 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 ). [email protected] 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. [email protected] 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 }. [email protected] 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 }. [email protected] 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. [email protected] 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)]). [email protected] 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. [email protected] 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. [email protected] 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. [email protected] 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. [email protected] 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. [email protected] 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. [email protected] 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)]. [email protected] 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. [email protected] 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). [email protected] 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 [email protected] MATH 590 Chapter 7 22
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 [email protected].
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 [email protected] This paper contains a collection of 31 theorems, lemmas,
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
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
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,
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
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
[1] Diagonal factorization
![[1] Diagonal factorization](/thumbs/40/21671034.jpg "[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:
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
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
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
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
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
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
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
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
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) =
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
SECRET sharing schemes were introduced by Blakley [5]
![SECRET sharing schemes were introduced by Blakley [5]](/thumbs/24/4404980.jpg "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
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
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
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
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
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
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
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
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
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
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.
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
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
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
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.
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.
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
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
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 =
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
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
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:
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.
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:
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
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
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.
( ) 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
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
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
HETEROGENEOUS AGENTS AND AGGREGATE UNCERTAINTY. Daniel Harenberg [email protected]. University of Mannheim. Econ 714, 28.11.
COMPUTING EQUILIBRIUM WITH HETEROGENEOUS AGENTS AND AGGREGATE UNCERTAINTY (BASED ON KRUEGER AND KUBLER, 2004) Daniel Harenberg [email protected] University of Mannheim Econ 714, 28.11.06 Daniel Harenberg
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
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
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
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
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
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
![Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.](/thumbs/26/8276997.jpg "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
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
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
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
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).