# Chapter 3: Ordinal Numbers

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

## Transcript

1 Chapter 3: Ordinal Numbers There are two kinds of number.. Ordinal numbers (0th), st, 2nd, 3rd, 4th, 5th,..., ω, ω +,... ω2, ω2+,... ω 2... answers to the question What position is... in a sequence? What position is... in an ordered set? 2. Cardinal numbers 0,, 2, 3, 4, 5,..., ℵ 0, ℵ, ℵ 2,... answers to the question How may... s are there? How many elements in a set? Finite ordinals correspond to finite cardinals, but the two notions differ in the transfinite. Ordinal numbers this chapter Cardinal numbers chapter 4 2 Ordinal numbers Cantor introduced them as labels for stages in a transfinite iteration. In 882 he became interested in them as numbers in their own right. How to represent them as sets? Finite ordinals = natural numbers 0 = = 0 {0}, 2 = {}, 3 = 2 {2},... n = {0,, 2,... n } Extend this into the transfinite: each ordinal number is the set of all its predecessors. ω = {0,, 2, 3,...} = N ω + = ω {ω} = {0,, 2, 3,..., ω} ω + 2 = (ω + ) {ω + } = {0,, 2, 3,..., ω, ω + } ω + ω (= ω2) = {0,, 2, 3,..., ω, ω +, ω + 2, ω + 3,...} Note: every ordinal is either 0, a successor ordinal, + = {}, a limit ordinal, e.g., ω = lim(0,, 2, 3, 4,...); ω ω = lim(ω, ω 2, ω 3, ω 4,...)

2 Ordering of ordinals: < iff iff. 3 CAUTION: this is not a proper definition of ordinal number. First step: examine the distinctive ordering that occurs in ordinals any non empty subset has a least element ( well ordered ). DEF n. A set A is well ordered by a binary relation iff () x A x x (irreflexive) (2) x, y, z A (x y y z x z) (transitive) (3) S A (S y S ( x S x y y x)) (any non empty subset S has a least element y) We also say (A, ) is a well ordered set (woset), or is a well ordering on A. If (A, ) merely satisfies () and (2), it is called a partially ordered set (poset). Note that () (2) x, y A (x y y x) (anti symmetry). 4 Examples of wosets. (N,<) is a woset. Proof of (3) by induction (Z,<) is a poset but not a woset Likewise, Q and R are partially, but not well, ordered by <. 4. Ordinals are well ordered by < (can t prove it yet). E.g., take ω + 5: ω ω+ ω+2 ω+3 ω+4 Can prove it for ω + 5.

3 5. Order N by m n { (m, n 2 m < n) (m, n > 2 m < n) (m > 2 n 2) 6. Order N by m n iff n < m 7. Order N N by (m, n ) (m 2, n 2 ) iff n < n 2 (n = n 2 m < m 2 )

4 8. Order N N by (m, n ) (m 2, n 2 ) iff m + n < m 2 + n 2 (m + n = m 2 + n 2 m < m 2 ) 9. Order N N by (m, n ) (m 2, n 2 ) iff max{m, n } < max{m 2, n 2 } (m < n = m 2 > n 2 ) (m < m 2 n 2 = n ) (n 2 < n m = m 2 )

5 0. Order N N by 9. Order N by (m, n ) (m 2, n 2 ) iff m < m 2 n < n 2 m n iff { (m, n are even m < n) (m, n are odd m < n) (m is even n is odd) THEOREM 2. A woset (A, ) is linearly ordered, that is, x, y A (x y x = y y x). THEOREM 3. Any subset of a woset is also well ordered. 0 Similar orderings DEF n. Let (A, ) and (B, 2 ) be posets. (A, ) is similar to (B, 2 ), i.e., (A, ) (B, 2 ), iff there exists a function f: A B such that (a) f is a bijection (b) f preserves order: x, y A (x y f(x) 2 f(y)) (f is called a similarity or order isomorphism; we write f: (A, ) (B, 2 ) or just f: A B.) Note that is an equivalence relation on posets.

6 Examples 2. ( π 2, π 2 ) R (usual orderings on both). tan: ( π 2, π 2 ) R 3. ( π 2, π 2 ) (0, ). f: ( π 2, π 2 ) (0, ) defined by x ( π 2, π 2 ) f(x) = x π Consequently, (0, ) R. tan f : (0, ) R. 5. [0, ] R. Assume f: [0, ] R and consider f (f(0) ). 6. Revisiting old examples: Ex 5 ω + 3 Ex 7 ω 2 Ex 8 ω Ex 9 ω Ex ω + ω THEOREM 4. Let (A, ) and (B, 2 ) be similar posets. Then (A, ) is a woset iff (B, 2 ) is. Conjecture. Every woset is similar to a unique ordinal. We need a rigorous definition of ordinal. Burali Forti s Paradox (897) 2 Let W be the set of all ordinals. CLAIM. W is an ordinal. Proof. (Recall that an ordinal is a set that is transitive and well ordered by.) W is transitive since if x W then x W (theorem 5). W is well ordered by since W / (theorem ),, W (since is transitive) any non empty set of ordinals has a least element (theorem 9). Therefore W W, i.e., W < W, contrary to theorem. Contradiction! Burali Forti s explanation: W is only partially ordered, not linearly ordered. Russell s explanation: W is linearly ordered but not well ordered. Jourdain, Russell and Zermelo subsequently decided there was no set W of all ordinals. This resolution is followed in modern set theory.

7 3 Morals () We need a clear definition of ordinal. (2) We need a clear notion of what a set is. We are about to give (). [See separate document.] Do we have (2)? 4 Ordinal Arithmetic What do + and mean? Let and be ordinals. Let S = { (0, x) x } { (, y) y } with order relation defined by (0, x ) (0, x 2 ) iff x < x 2 (, y ) (, y 2 ) iff y < y 2 (0, x) (, y) [Convenient notation: for any set C, C 0 = { (0, x) x C}, C = { (, x) x C}.] Thus S = 0. THEOREM 25. (S, ) is a woset. DEF n. + = the unique ordinal similar to (S, ) (by theorem 23). EXAMPLES. 7. ω + = ω {ω} > ω. Define f: (ω 0, ) (ω {ω},<) by n < ω f(0, n) = n f(, 0) = ω

8 In general, + = {} > ω = ω. Define f: ( 0 ω, ) (ω,<) by f(0, 0) = 0 n < ω f(, n) = n+ Likewise 2+ω = ω: define f: (2 0 ω, ) (ω,<) by f(0, 0) = 0 f(0, ) = n < ω f(, n) = n+2 In general, n < ω n+ω = ω. 9. ω + ω = {0,, 2,..., ω, ω +, ω + 2,...} > ω. Define f: (ω 0 ω, ) ({0,, 2,..., ω, ω +, ω + 2,...},<) by n < ω f(0, n) = n n < ω f(, n) = ω + n Note that + +, in general. 6 THEOREM 26. For any ordinals,,, + ( + ) = ( + )+. Proof. The construction of + ( + ) gives us similarities f: 0 +, g: 0 ( + ) + ( + ) The construction of ( + )+ gives us similarities h: 0 +, i: ( + ) 0 ( + )+ 0 + f f 0 + ( + ) g ( + ) g ( + )+ ( + ) 0 i i + 0 h h

9 Define j: + ( + ) ( + )+ by 7 x j ( g(0, x) y j ( z j ( g(, f(0, y)) g(, f(, z)) ) = i(0, h(0, x)) ) = i(0, h(, y)) ) = i(, z) Then by theorem 20 + ( + ) = ( + )+. What about? Let P = (product of sets), ordered by (x, y ) (x 2, y 2 ) iff (y < y 2 ) (y = y 2 x < x 2 ) THEOREM 27. (P, ) is a woset. DEF n. (Ordinal multiplication) = the unique ordinal similar to (P, ). 2 =, n+ = n for n < ω. 8 EXAMPLES. 20. ω2 = ω + ω. Define f: (ω 2, ) (ω 0 ω, ) by n < ω f(n, 0) = (0, n) n < ω f(n, ) = (, n) Then (ω2,<) (ω 2, ) (ω 0 ω, ) (ω + ω,<) so by theorem 20 ω2 = ω + ω. 2. 2ω = ω. Define f: (2 ω, ) (ω,<) by n < ω f(0, n) = 2n n < ω f(, n) = 2n+ Thus, by definition, 2ω = ω. In general, n < ω nω = ω. Note: in general.

10 9 THEOREM 28. For any ordinals,,, () = (). Proof. The construction of () gives us similarities f: The construction of () gives us similarities h: g: () () i: () () g ( ) X X h X ( ) i ( ) ( ) f X Define j: () () by x y z j ( i(x, h(y, z)) ) = g(f(x, y), z) So by theorem 20 () = (). 20 THEOREM 29. For any ordinals,,, + = ( + ). Proof. The construction of + gives us similarities f:, g:, h: () 0 () +. The construction of ( + ) gives us similarities i: 0 +, j: ( + ) ( + ). X X g f ( ) ( ) 0 h h j X( + ) i + i 0 + ( + )

11 Define a similarity k: + ( + ) by x y k ( ) h(0, f(x, y)) = j(x, i(0, y)) x z k ( ) h(, g(x, z)) = j(x, i(, z)) 2 Then by theorem 20 + = ( + ). BEWARE: ( + ) + in general. THEOREM 30. (ω + )ω = ω 2. Proof. The construction of (ω + )ω gives us similarities f: ω 0 ω + g: (ω + ) ω (ω + )ω The construction of ω 2 gives us a similarity h: ω ω ω 2 22 ω ω ω ( +)X g h ω ω Xω ω ω 0 f ω+ f ( ω +) ω ω 2 ω Define a similarity i: (ω + )ω ω 2 by m i(g(f(0, m), 0)) = h(m, 0) m n i(g(f(0, m), n+)) = h(m+, n+) n i(g(f(, 0), n)) = h(0, n+) So by theorem 20 (ω + )ω = ω 2.

12 23 THEOREM 3. (ω + )ω < ω 2 + ω. Proof. Theorem 30 shows (ω + )ω = ω 2. We must just show ω 2 < ω 2 + ω. The construction of ω 2 + ω gives us a similarity f: (ω 2 ) 0 ω ω 2 + ω We can then define a function g: ω 2 ω 2 + ω by x ω 2 g(x) = f(0, x) This is a similarity from ω 2 to an initial segment of ω 2 + ω (i.e., everything below f(, 0)). So (ω + )ω < ω 2 + ω. THEOREM 32. n < ω (ω + n)ω = ω 2. Proof. As in theorem 30. THEOREM 33. m, n < ω (0 < m (ω + n)m = ωm+n). Proof. Similar to theorem 30. Or, (ω + n)m = (ω + n)(++ + ) 24 = (ω + n)+(ω + n)+ + (ω + n) = ω + (n+ω)+(n+ω)+ +(n+ω)+n = ω + ω + ω + +ω + n by theorem 29 by theorem 26 by example 8 = ω+ω+ω+ +ω+n = ω(+++ + )+n by theorem 29 = ωm+n. EXAMPLE. 22. Reduction to normal form. (Use associativity tacitly.) (ω + )(ω + 2)(ω + 3) = (ω + )(ω + 2)ω + (ω + )(ω + 2)3 = (ω + )ω 2 + (ω + )(ω3+2) = ω 3 + (ω + )ω3+(ω + )2 = ω 3 + ω 2 3+ω2+ by theorem 29 by theorems 32, 33 by theorems 30, 29 by theorems 30, 33

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

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

Διαβάστε περισσότερα

### 2 Composition. Invertible Mappings

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

Διαβάστε περισσότερα

### ORDINAL ARITHMETIC JULIAN J. SCHLÖDER

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

Διαβάστε περισσότερα

### Finite Field Problems: Solutions

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

Διαβάστε περισσότερα

### Chapter 2. Ordinals, well-founded relations.

Chapter 2. Ordinals, well-founded relations. 2.1. Well-founded Relations. We start with some definitions and rapidly reach the notion of a well-ordered set. Definition. For any X and any binary relation

Διαβάστε περισσότερα

### Every set of first-order formulas is equivalent to an independent set

Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

### A Note on Intuitionistic Fuzzy. Equivalence Relation

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

Διαβάστε περισσότερα

### EE512: Error Control Coding

EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3

Διαβάστε περισσότερα

### Example Sheet 3 Solutions

Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note

Διαβάστε περισσότερα

### C.S. 430 Assignment 6, Sample Solutions

C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order

Διαβάστε περισσότερα

### Fractional Colorings and Zykov Products of graphs

Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is

Διαβάστε περισσότερα

### Reminders: linear functions

Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

### Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)

Διαβάστε περισσότερα

### Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that

Διαβάστε περισσότερα

### 5. Choice under Uncertainty

5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation

Διαβάστε περισσότερα

### 3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle

Διαβάστε περισσότερα

### Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

### Congruence Classes of Invertible Matrices of Order 3 over F 2

International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and

Διαβάστε περισσότερα

### Cardinals. Y = n Y n. Define h: X Y by f(x) if x X n X n+1 and n is even h(x) = g

Cardinals 1. Introduction to Cardinals We work in the base theory ZF. The definitions and many (but not all) of the basic theorems do not require AC; we will indicate explicitly when we are using choice.

Διαβάστε περισσότερα

### Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.

Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:

Διαβάστε περισσότερα

### Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all

Διαβάστε περισσότερα

### Section 8.3 Trigonometric Equations

99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

### derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

### SOME PROPERTIES OF FUZZY REAL NUMBERS

Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical

Διαβάστε περισσότερα

### Homework 3 Solutions

Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For

Διαβάστε περισσότερα

### Homomorphism in Intuitionistic Fuzzy Automata

International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic

Διαβάστε περισσότερα

### Abstract Storage Devices

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

Διαβάστε περισσότερα

### Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin

Διαβάστε περισσότερα

### CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter

Διαβάστε περισσότερα

### Sequent Calculi for the Modal µ-calculus over S5. Luca Alberucci, University of Berne. Logic Colloquium Berne, July 4th 2008

Sequent Calculi for the Modal µ-calculus over S5 Luca Alberucci, University of Berne Logic Colloquium Berne, July 4th 2008 Introduction Koz: Axiomatisation for the modal µ-calculus over K Axioms: All classical

Διαβάστε περισσότερα

### k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b

Διαβάστε περισσότερα

### ST5224: Advanced Statistical Theory II

ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known

Διαβάστε περισσότερα

### Matrices and Determinants

Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z

Διαβάστε περισσότερα

### THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS. Daniel A. Romano

235 Kragujevac J. Math. 30 (2007) 235 242. THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS Daniel A. Romano Department of Mathematics and Informatics, Banja Luka University, Mladena Stojanovića

Διαβάστε περισσότερα

### SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)

Διαβάστε περισσότερα

Tutorial 12: Radon-Nikodym Theorem 1 12. Radon-Nikodym Theorem In the following, (Ω, F) is an arbitrary measurable space. Definition 96 Let μ and ν be two (possibly complex) measures on (Ω, F). We say

Διαβάστε περισσότερα

### DIRECT PRODUCT AND WREATH PRODUCT OF TRANSFORMATION SEMIGROUPS

GANIT J. Bangladesh Math. oc. IN 606-694) 0) -7 DIRECT PRODUCT AND WREATH PRODUCT OF TRANFORMATION EMIGROUP ubrata Majumdar, * Kalyan Kumar Dey and Mohd. Altab Hossain Department of Mathematics University

Διαβάστε περισσότερα

### 4.6 Autoregressive Moving Average Model ARMA(1,1)

84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this

Διαβάστε περισσότερα

### Tridiagonal matrices. Gérard MEURANT. October, 2008

Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,

Διαβάστε περισσότερα

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

Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο

Διαβάστε περισσότερα

### Chap. 6 Pushdown Automata

Chap. 6 Pushdown Automata 6.1 Definition of Pushdown Automata Example 6.1 L = {wcw R w (0+1) * } P c 0P0 1P1 1. Start at state q 0, push input symbol onto stack, and stay in q 0. 2. If input symbol is

Διαβάστε περισσότερα

### Commutative Monoids in Intuitionistic Fuzzy Sets

Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,

Διαβάστε περισσότερα

### THE GENERAL INDUCTIVE ARGUMENT FOR MEASURE ANALYSES WITH ADDITIVE ORDINAL ALGEBRAS

THE GENERAL INDUCTIVE ARGUMENT FOR MEASURE ANALYSES WITH ADDITIVE ORDINAL ALGEBRAS STEFAN BOLD, BENEDIKT LÖWE In [BoLö ] we gave a survey of measure analyses under AD, discussed the general theory of order

Διαβάστε περισσότερα

### 6.1. Dirac Equation. Hamiltonian. Dirac Eq.

6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2

Διαβάστε περισσότερα

### Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts

Διαβάστε περισσότερα

### From the finite to the transfinite: Λµ-terms and streams

From the finite to the transfinite: Λµ-terms and streams WIR 2014 Fanny He f.he@bath.ac.uk Alexis Saurin alexis.saurin@pps.univ-paris-diderot.fr 12 July 2014 The Λµ-calculus Syntax of Λµ t ::= x λx.t (t)u

Διαβάστε περισσότερα

### Bounding Nonsplitting Enumeration Degrees

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

Διαβάστε περισσότερα

### MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS

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

Διαβάστε περισσότερα

### Section 9.2 Polar Equations and Graphs

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

Διαβάστε περισσότερα

### Uniform Convergence of Fourier Series Michael Taylor

Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula

Διαβάστε περισσότερα

### ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least

Διαβάστε περισσότερα

### Homomorphism of Intuitionistic Fuzzy Groups

International Mathematical Forum, Vol. 6, 20, no. 64, 369-378 Homomorphism o Intuitionistic Fuzz Groups P. K. Sharma Department o Mathematics, D..V. College Jalandhar Cit, Punjab, India pksharma@davjalandhar.com

Διαβάστε περισσότερα

### Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

### Thus X is nonempty by supposition. By (i), let x be a minimal element of X. Then let

4. Ordinals July 26, 2011 In this chapter we introduce the ordinals, prove a general recursion theorem, and develop some elementary ordinal arithmetic. A set A is transitive iff x A y x(y A); in other

Διαβάστε περισσότερα

### Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα

[ 1 ] Πανεπιστήµιο Κύπρου Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα Νίκος Στυλιανόπουλος, Πανεπιστήµιο Κύπρου Λευκωσία, εκέµβριος 2009 [ 2 ] Πανεπιστήµιο Κύπρου Πόσο σηµαντική είναι η απόδειξη

Διαβάστε περισσότερα

### Completeness Theorem for System AS1

11 The Completeness Theorem for System AS1 1. Introduction...2 2. Previous Definitions...2 3. Deductive Consistency...2 4. Maximal Consistent Sets...5 5. Lindenbaum s Lemma...6 6. Every Maximal Consistent

Διαβάστε περισσότερα

### Overview. Transition Semantics. Configurations and the transition relation. Executions and computation

Overview Transition Semantics Configurations and the transition relation Executions and computation Inference rules for small-step structural operational semantics for the simple imperative language Transition

Διαβάστε περισσότερα

### Solutions for exercises in chapter 1 = 2, 3, 1, 4 ; = 2, 2, 3, 4, 2 1 S 1 1 S 0 = 2, 2, 3, 4, 2, 1, 4,1,3. 1, 1,..., 1, S 0

E1.1 Verify that and Solutions for exercises in chapter 1 S 0 S 1 = 2, 3, 1, 4 (S 0 S 1 ) ( S 1 S 0 ) = 2, 2, 3, 4, 2, 1, 4, 1,3. S 0 S 1 = 2 S 0 S 1 = 2 3 1 S 1 = 2, 3, 1, 4 ; (S 0 S 1 ) ( S 1 S 0 ) =

Διαβάστε περισσότερα

### Statistical Inference I Locally most powerful tests

Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided

Διαβάστε περισσότερα

### Problem Set 3: Solutions

CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C

Διαβάστε περισσότερα

### CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =

Διαβάστε περισσότερα

### TMA4115 Matematikk 3

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

Διαβάστε περισσότερα

### Lecture 13 - Root Space Decomposition II

Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).

Διαβάστε περισσότερα

### Srednicki Chapter 55

Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third

Διαβάστε περισσότερα

### Other Test Constructions: Likelihood Ratio & Bayes Tests

Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :

Διαβάστε περισσότερα

### Solution Series 9. i=1 x i and i=1 x i.

Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x

Διαβάστε περισσότερα

### F19MC2 Solutions 9 Complex Analysis

F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at

Διαβάστε περισσότερα

### Areas and Lengths in Polar Coordinates

Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

### Lecture 15 - Root System Axiomatics

Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the

Διαβάστε περισσότερα

### The Simply Typed Lambda Calculus

Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and

Διαβάστε περισσότερα

### Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the

Διαβάστε περισσότερα

### Cyclic or elementary abelian Covers of K 4

Cyclic or elementary abelian Covers of K 4 Yan-Quan Feng Mathematics, Beijing Jiaotong University Beijing 100044, P.R. China Summer School, Rogla, Slovenian 2011-06 Outline 1 Question 2 Main results 3

Διαβάστε περισσότερα

### Solutions to Selected Homework Problems 1.26 Claim: α : S S which is 1-1 but not onto β : S S which is onto but not 1-1. y z = z y y, z S.

Solutions to Selected Homework Problems 1.26 Claim: α : S S which is 1-1 but not onto β : S S which is onto but not 1-1. Proof. ( ) Since α is 1-1, β : S S such that β α = id S. Since β α = id S is onto,

Διαβάστε περισσότερα

### Jordan Form of a Square Matrix

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

Διαβάστε περισσότερα

### Trigonometric Formula Sheet

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

Διαβάστε περισσότερα

### Knaster-Reichbach Theorem for 2 κ

Knaster-Reichbach Theorem for 2 κ Micha l Korch February 9, 2018 In the recent years the theory of the generalized Cantor and Baire spaces was extensively developed (see, e.g. [1], [2], [6], [4] and many

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

### PARTIAL NOTES for 6.1 Trigonometric Identities

PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot

Διαβάστε περισσότερα

### ( ) 2 and compare to M.

Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8

Διαβάστε περισσότερα

### Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών. Εθνικό Μετσόβιο Πολυτεχνείο. Thales Workshop, 1-3 July 2015.

Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών Εθνικό Μετσόβιο Πολυτεχνείο Thles Worksho, 1-3 July 015 The isomorhism function from S3(L(,1)) to the free module Boštjn Gbrovšek Άδεια Χρήσης Το παρόν

Διαβάστε περισσότερα

### 6.3 Forecasting ARMA processes

122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

### Section 7.6 Double and Half Angle Formulas

09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)

Διαβάστε περισσότερα

### 1. Introduction and Preliminaries.

Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 22:1 (2008), 97 106 ON δ SETS IN γ SPACES V. Renuka Devi and D. Sivaraj Abstract We

Διαβάστε περισσότερα

### On density of old sets in Prikry type extensions.

On density of old sets in Prikry type extensions. Moti Gitik December 31, 2015 Abstract Every set of ordinals of cardinality κ in a Prikry extension with a measure over κ contains an old set of arbitrary

Διαβάστε περισσότερα

### Second Order Partial Differential Equations

Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y

Διαβάστε περισσότερα

### Areas and Lengths in Polar Coordinates

Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

### Concrete Mathematics Exercises from 30 September 2016

Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)

Διαβάστε περισσότερα

### ( y) Partial Differential Equations

Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate

Διαβάστε περισσότερα

### The challenges of non-stable predicates

The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates

Διαβάστε περισσότερα

### Math 6 SL Probability Distributions Practice Test Mark Scheme

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

Διαβάστε περισσότερα

### About these lecture notes. Simply Typed λ-calculus. Types

About these lecture notes Simply Typed λ-calculus Akim Demaille akim@lrde.epita.fr EPITA École Pour l Informatique et les Techniques Avancées Many of these slides are largely inspired from Andrew D. Ker

Διαβάστε περισσότερα

### Pg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is

Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =

Διαβάστε περισσότερα

### Math221: HW# 1 solutions

Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin

Διαβάστε περισσότερα

### Numerical Analysis FMN011

Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =

Διαβάστε περισσότερα

### New bounds for spherical two-distance sets and equiangular lines

New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a

Διαβάστε περισσότερα

### Intuitionistic Fuzzy Ideals of Near Rings

International Mathematical Forum, Vol. 7, 202, no. 6, 769-776 Intuitionistic Fuzzy Ideals of Near Rings P. K. Sharma P.G. Department of Mathematics D.A.V. College Jalandhar city, Punjab, India pksharma@davjalandhar.com

Διαβάστε περισσότερα