Investigation of Determinacy for Games of Variable Length
|
|
- Ὀλυμπιόδωρος Ζαΐμης
- 5 χρόνια πριν
- Προβολές:
Transcript
1 UNLV Theses, Dissertations, Professional Papers, and Capstones May 217 Investigation of Determinacy for Games of Variable Length Emi Ikeda University of Nevada, Las Vegas, Follow this and additional works at: Part of the Mathematics Commons Repository Citation Ikeda, Emi, "Investigation of Determinacy for Games of Variable Length" 217). UNLV Theses, Dissertations, Professional Papers, and Capstones This Dissertation is brought to you for free and open access by Digital It has been accepted for inclusion in UNLV Theses, Dissertations, Professional Papers, and Capstones by an authorized administrator of Digital For more information, please contact
2 INVESTIGATION OF DETERMINACY FOR GAMES OF VARIABLE LENGTH Emi Ikeda Bachelor of Science - Mathematical Sciences Yamaguchi University, Japan 24 Master of Science - Mathematical Sciences Yamaguchi University, Japan 26 A dissertation submitted in partial fulfillment of the requirements for the Doctor of Philosophy - Mathematical Sciences Department of Mathematical Sciences College of Sciences The Graduate College University of Nevada, Las Vegas May 217
3 Copyright 217 by Emi Ikeda All Rights Reserved
4 Dissertation Approval The Graduate College The University of Nevada, Las Vegas March 31, 217 This dissertation prepared by Emi Ikeda entitled Investigation of Determinacy for Games of Variable Length is approved in partial fulfillment of the requirements for the degree of Doctor of Philosophy - Mathematical Sciences Department of Mathematical Sciences Derrick DuBose, Ph.D. Examination Committee Chair Kathryn Hausbeck Korgan, Ph.D. Graduate College Interim Dean Douglas Burke, Ph.D. Examination Committee Member Peter Shiue, Ph.D. Examination Committee Member Pushkin Kachroo, Ph.D. Graduate College Faculty Representative ii
5 Abstract Many well-known determinacy results calibrate determinacy strength in terms of large cardinals e.g., a measurable cardinal) or a large cardinal type property e.g., zero sharp exists). Some of the other results are of the form that subsets of reals of a certain complexity will satisfy a well-known property when a certain amount of determinacy holds. The standard game tree considered in the study of determinacy involves games in which all moves are from omega and all plays have length omega i.e. the game tree is ω <ω and the body of the game tree is ω ω ). There are also many well-known results on the game trees ω <α for α countable all moves from omega and all paths are of fixed length α). However, one can easily construct a nondetermined open game on a game tree T, in which all moves are from ω, but some paths of T have length omega while the others of length ω + 1. Many determinacy results consider games on a fixed game tree with each path having the same length. In this dissertation, we investigate the determinacy of games on game trees with variable length paths. Especially, we investigate two types of such game trees, which we named Type 1 and Type 2. The length of each path in a Type 1 tree is determined by its first ω moves. A Type 2 tree is generalization of a Type 1 tree. In other words, a Type 1 tree is a special case of a Type 2 tree. We shall consider collections C of such game trees, iii
6 that will be defined from particular parameters ranging over certain sets. A T ree 1 collection will be a collection of Type 1 trees. A T ree 2 collection will be a collection of Type 2 trees. Given a T ree 1 respectively, T ree 2 ) collection C and a fixed complexity e.g., open, Borel, Σ 1 1), we calibrate the strength of the determinacy of games with that complexity on all trees in the collection C in terms of well-known determinacy. iv
7 Acknowledgments I would like to take this time to thank the many people that have helped me achieving my goal of acquiring my Ph.D. First, I would like to thank my advisor Dr. Derrick DuBose for his help and guidance throughout my time here at UNLV and my dissertation process. He has taken many of days and long nights to help me gain a better understanding of Set Theory. I would also like to especially thank Dr. Douglas Burke for taking time out of his schedule to help me through my whole time at UNLV. He has been like a co-advisor to me. He has taken time out of his day to meet with me every week to help me study and present me with a lot of ideas to help me in understanding Set Theory. He has helped me solve many of the problems I had concerning my studies. He guided me to the kind of game trees to look for in my dissertation. He has been a great influence for me. I would remise if I did not thank him for his guidance and patience. Without these two I do not know if I could had made it to the position I am in today. I would like to thank my other committee members Dr. Peter Shiue and Graduate College Representative Dr. Pushkin Kachroo. They have generously taken time out of their schedules to listen to my comprehensive exam and dissertation defense. I would also like to thank the Mathematical Sciences Department at UNLV for affording me the opportunity to study and achieve my dream of obtaining my v
8 Ph.D. My original area of focus was topology while I was working on my Master s. I was planning on continuing to study Topology when I came to UNLV. Sadly, I was not able to find a professor to work with me in topology. When I was taking MAT 71 and MAT 72, Dr. DuBose suggested to me to study Set Theory. I learned Set Theory from scratch. He gave me the opportunity to pursue a new field of study and he invited me to the Set Theory Seminars. For that I am grateful for the opportunity to learn this new and exciting field of which I have been studying for the last few years. I would also like to thank a Ph.D. student Josh Reagan and a Master s student Katherine Yost for studying with me in the Set Theory Student Workshop here on campus. Finally, I would like to give a special acknowledgment to my family and friends for their loving support throughout the years of my studies as a Ph.D. student. vi
9 Table of Contents Abstract iii Acknowledgments v Table of Contents xii List of Figures xiii Chapter 1 Preliminaries and Introduction General notations for a product space Definition of a game Definitions related to a game tree Definitions related to a game on a tree Definition of complexities Definitions related to topologies Open sets Borel hierarchy Projective hierarchy Difference hierarchy vii
10 1.4 Well-known determinacy results Determinacy results from ZFC Determinacy results from large cardinals Introduction to this dissertation Motivation to study long trees Difference from usual determinacy results Notations for this dissertation Chapter 2 Type 1 Tree : T Ψ,B X,Y Definition of a Type 1 tree Definition of a T ree 1 collection and a collection of games on a T ree 1 collection with complexity Ξ Equivalence between Σ α and Π α determinacy on a T ree 1 collection and equivalence between Σ 1 n and Π 1 n determinacy on a T ree 1 collection Using the determinacy of games on a T ree 1 collection to obtain the determinacy of games on X <ω ZF-P) Using 1 determinacy on a T ree 1 collection to obtain finite Borel determinacy on X <ω Using Σ 1 determinacy on a T ree 1 collection to obtain the determinacy of games on X <ω Using α-π 1 1 determinacy on T ree 1 collection to obtain α + 1-Π 1 1 determinacy on X ω for even α ω viii
11 2.5 Getting the determinacy of the games on a T ree 1 collection from the determinacy of the games on X <ω Reversed direction of section 2.4) Getting the determinacy of games on a T ree 1 collection with countable Y from the determinacy of games on X <ω Obtaining the determinacy of open games on a T ree 1 collection with countable Y from the determinacy of games on X <ω Obtaining the determinacy of Borel games on a T ree 1 collection with countable Y from the determinacy of Borel games on X <ω Obtaining the determinacy of projective games on a T ree 1 collection with countable Y from the determinacy of projective games on X <ω Well-known results about uncountable Y = N Determinacy equivalences between games on X <ω and games on T ree 1 collections Determinacy equivalence between Borel games on X <ω and games on T ree 1 collections Determinacy equivalence between projective games on X <ω and games on T ree 1 collections Generalization of a Type 1 tree Chapter 3 Type 2 Tree : T Ψ,B Definition of a Type 2 tree Definition of a T ree 2 collection and a collection of games on a T ree 2 collection with complexity Ξ ix
12 3.3 Equivalence between Σ α and Π α determinacy on Type 2 trees and equivalence between Σ 1 1 and Π 1 1 determinacy on Type 2 trees Using the determinacy of games on a T ree 2 collection to obtain the determinacy of games on X <ω ZF-P) Using 1 determinacy on a T ree 2 collection to obtain Borel determinacy on X <ω Using Σ 1 determinacy on a T ree 2 collection to obtain the determinacy of games on X <ω Using α-π 1 1 determinacy on T ree 2 collection to obtain α + 1-Π 1 1 determinacy on X <ω for even α ω Using the determinacy of open games on a T ree 2 collection to obtain the determinacy of α-π 1 1 games on X <ω Getting the determinacy of games on a T ree 2 collection from the determinacy of games on X <ω Reversed direction of section 3.4) Getting the determinacy of games on a T ree 2 collection with F W F and CW F from the determinacy of games on X <ω Obtaining the open determinacy on T ree 2 collection with F W F and CW F from the determinacy of games on X <ω Obtaining the determinacy of Borel games on a T ree 2 collection with F W F and CW F from the determinacy of Borel games on X <ω Obtaining the determinacy of projective games on a T ree 2 collection with CW F from the determinacy of projective games on X <ω x
13 3.5.5 Comment about a Type 2 tree T Ψ,B for T sq with not well-founded trees or moves over an uncountable set Determinacy equivalence between games on X <ω and games on T ree 2 collections Determinacy equivalence between Borel games on X <ω and games on T ree 2 collections Determinacy equivalence between projective games on X <ω and games on T ree 2 collections Generalization of a Type 2 tree Chapter 4 Definitions of Type 3, Type 4, Type 5 trees and future questions Definition of a Type 3 tree Definition of a Type 4 tree Definition of a Type 5 tree Appendix A Big picture Appendix B List of Symbols B.1 Letters with special meanings B.2 Symbols related to Type 1 and Type 2 trees B.2.1 Symbols related to Type 1 trees B.2.2 Symbols related to Type 2 trees B.3 Other notations Appendix C Definition of Trees C.1 Type 1 tree : T Ψ,B X,Y xi
14 C.2 Type 2 tree : T Ψ,B C.3 Type 3 tree : T Φ,π C.4 Type 4 tree : T Φ,π C.5 Type 5 tree : T f α α 2 ω T α α 2 ω C.6 Yost tree T α g.t Appendix D Well-known Determinacy Results D.1 Determinacy results from ZFC D.2 Results related to the existence of measurable cardinals D.3 Projective determinacy D.4 Lightface results related to the existence of # Appendix E Definitions and Notations E.1 Chapter E.2 Chapter E.3 Chapter Appendix F Determinacy Results F.1 Chapter F.2 Chapter References Curriculum Vitae xii
15 List of Figures Illustration of p T and x [T ] Illustration of a play x [T ] for lhx) > ω Illustration of winning strategies Illustration of q and Oq) Diagram of Borel hierarchy for X ω Illustration of a projection along Y Illustration of difference kernel Illustration of well-known boldface determinacy results. The Det before each class is suppressed/excluded Illustration of paths h [T ] for a Type 1 tree T = T Ψ,B X,Y for B Illustration of h [SftT )] with lh h) > ω Illustration of p T, lhp) > ω according to II s strategy s Illustration of p T, lhp) > ω according to I s strategy s Illustration of x X ω Y l+1 l is even) according to I s strategy s corresponding to the ) direction of the equivalence 2.4) on page 85) Illustration of x X ω Y l+1 l is even) according to II s strategy s corresponding to the ) direction of the equivalence 2.4) on page 85) xiii
16 2.4.3 Illustration of x X ω Y l+1 l is odd) according to I s strategy s corresponding to the ) direction of the equivalence 2.6) on page 85) Illustration of x X ω Y l+1 l is odd) according to II s strategy s corresponding to the ) direction of the equivalence 2.6) on page 85) Illustration of x X ω Y l l is even) according to I s strategy s corresponding to the ) direction of the equivalence 2.4) on page 85) Illustration of x X ω Y l l is even) according to II s strategy s corresponding to the ) direction of the equivalence 2.4) on page 85) Illustration of x X ω Y l l is odd) according to I s strategy s corresponding to the ) direction of the equivalence 2.6) on page 85) Illustration of x X ω Y l l is odd) according to II s strategy s corresponding to the ) direction of the equivalence 2.6) on page 85) Illustration of paths h [T ] for a Type 2 tree T = T Ψ,B for B Illustration of h [Sft 2 T )] with lh h) > ω Illustration of p T, lhp) > ω according to II s strategy s Illustration of p T, lhp) > ω according to I s strategy s A..1Illustration of the determinacy equivalences between well-known results and some of the results in this dissertation xiv
17 Chapter 1 Preliminaries and Introduction The determinacy of games has been an active area of study in set theory. In this dissertation, we will focus on two-player perfect information games on a certain type of long trees, all of which have heights greater than or equal to) ω. Our goal to this dissertation will be the classification of certain long games. Before we start discussing games, we will review standard definitions and well-known theorems. In this chapter, we will review the basic concepts of games and set of notations for this dissertation. In section 1.1, we will review some notations for a product space and sequences. In section 1.2, we will define trees and games. In section 1.3, we will define complexities. We will use the product topology taking each set as a discrete space. Thus defining open sets, we will use finiteness. Then we will define the Borel, projective, and difference hierarchies. In section 1.4, we will review several well-known determinacy results for games on trees ω <ω and X <ω for any nonempty set X. Then in section 1.5, we will start the introduction to this dissertation and introduce some new concepts and notations, particular to this dissertation. 1
18 We will use the following notation 1..1 throughout the paper. Notation We use to signify that this is the end of the statement of definition, theorem, proposition, lemma, corollary, observation and notation. By using notation 1..1, it is easier to distinguish the end of a statement. While we use symbol to identify the end of a theorem, we will use this symbol to identify the end of a proof. For the material in this dissertation, the following books and publication are standard references: Martin 217 draft). Borel and Projective Games unpublished). dam/booketc/thebook.pdf. The main reference for this dissertation is Martin s unpublished book. The 217 draft does not include Chapter 5. The cited page numbers and theorems for Chapter 5 are from an older draft. Jech 23). Descriptive Set Theory, the Third Millennium Edition, Revised and Expanded. Springer, 23. Kechris 21). Classical Descriptive Set Theory, Graduate Texts in Mathematics: vol Springer-Verlag. 2
19 Moschovakis 29). Descriptive Set Theory, Second Edition. American Mathematical Society. Neeman 24). The Determinacy of Long Games: de Gruyter Series in Logic and Its Applications, vol. 7 Berlin, Germany: de Gruyter GmbH, Walter. Steel 1988). Long Games. In: Kechris A.S., Martin D.A., Steel J.R. eds) Cabal Seminar Lecture Notes in Mathematics, vol Springer, Berlin, Heidelberg These and the additional references are listed under the References on page 418. General notations for a product space 3
20 1.1 General notations for a product space In this section, we will review some standard notations for sequences. Definition Definition of countable, denumerable and uncountable sets) A set X is finite if there is a bijection between X and some finite subset of the set of natural numbers. A set X is denumerable if there is a bijection between X and the set of all natural numbers. A set X is countable if it is either finite or denumerable. A set X is uncountable if it is not countable, i.e., it is infinite and not denumerable. Definition ω is the least countable ordinal and ω 1 is the least uncountable ordinal. Suppose X and Y are nonempty sets. X Y is a set of functions from Y into X and thus it is called a function space. In particular, we will consider the case that Y is an ordinal number α. Then X α = {f f : α X }. Since the domain of each function in X α is an ordinal α, by letting x β = f β) for each β α, each function f can be identified with a sequence of length α. Thus each element of X α is a sequence x, x 1,..., x β,... where β α and each x β X. Each x β is called the β-th entry of the sequence. Hence X α is the α Cartesian product of X, i.e., X X multiplied α times. Recall {, 1} ω = 2 ω is called the Cantor space and ω ω is called the Baire space. We also use N to represent the Baire space. We define X <α and X α by X <α = β<α Xβ and X α = β α Xβ. Notation The length of a sequence p is the domain of p and is denoted by lhp). Note that for any sequence p, there exists a unique ordinal α such that p = p, p 1,..., p i,..., i α. 4
21 Thus p can be identified with the function { i, p i i α}. Hence the domain of a sequence p is the domain of the corresponding function, i.e., lhp) = α. Definition Definition of a concatenation) Suppose f = f), f1),... and g = g), g1),... are sequences. Then a concatenation of f and g, denoted by f g is defined to mean f), f1),..., g), g1),.... i.e., if α = domf) and β = domg), then domf g) = α + β and f γ) if γ < α, f g γ) = g δ) if α γ < β, where γ = α + δ. Notation Definition of for a sequence) Suppose x is a sequence of length α. Then for any β α, define x β to be the sequence of length β such that x β)γ) = xγ) for γ β, i.e., x β and x have the same γth component for any γ β. If β > α, then we define x β to be x. For a function f and a set A, the restriction of f to A, f A, f A = f A domf)). Since any sequence x can be identified as a function, we can obtain x β = x β domx)). Since the domain of x is the length of x, if β > lhx), then β domx) = β lhx) = lhx). Thus we obtain x β = x as in notation x β = x β lhx)) = x lhx) = x. Definition Definition of an initial segment and an extension of a sequence) If s = t α for some ordinal α, then we say s is an initial segment of t and t is an extension 5
22 of s possibly s = t). If s = t α for some ordinal α and s t, then we say s is a proper initial segment of t and t is a proper extension of s. Definition of a game 6
23 1.2 Definition of a game In this section, we will give standard definitions related to game trees. We regularly refer to a game tree as a tree. Then we will give standard definitions related to a game on a tree. By a game, we mean a two-player perfect information game Definitions related to a game tree Definition Definition of a game tree) T is a game tree if T satisfies the following 4 properties. 1. T is a set of sequences. 2. T is closed under initial segments, i.e., if t T then t α T for all α lht). 3. If s T and lhs) is a limit ordinal, then there exists t T such that s t. Property 3 is a convention that we need to fix to avoid confusion. This assumption implies that there is no path of limit length which is a position. A path and a position are defined in definition below.) When we say a tree, we mean a game tree. Note that every tree contains the empty sequence. Note that ω ω is not a tree since it is not closed under initial segments. The typical example of a game tree is ω <ω. If the game tree is not specified, we assume this tree. Definition Definition of the height of a tree) 7
24 Suppose T is a tree. The height of T, denoted by htt ) is defined by ht T ) = sup lh p)) p T Definition Definition of a position, a move, a play and a path) Suppose T is a tree. Each p T is called a position. For any p T, define M p = {m p m T }. Then a move at p in T is an a such that a M p, i.e., p a T. a is called a move if there exists a position p T such that a M p. A play is a sequence x in which every proper initial segment of x is in T and for any move a, x a is not in T. Each play is also called a branch or a path through the tree T. Note that property 3 in definition affects of a definition of a play. Suppose every proper initial segment of x is in T and no proper extension of x is in T. If the length of x is a successor ordinal and x T, then x is a play in T. If the length of x is a limit ordinal, then x is a play in T but x / T. Definition Definition of the body of a tree) Suppose T is a tree. The body of a tree is the set of all plays in T and is denoted by [T ]. If x [T ]\T, then the length of x is a limit ordinal. If x T [T ], then the length of x is a successor ordinal. 8
25 Figure 1.2.1: Illustration of p T and x [T ]. Definition Definition of a well-founded tree) Suppose T is a tree. If T has no infinite branches, T is called well-founded. Otherwise, T is called ill-founded. Definition Definition of the rank of a well-founded tree) Suppose T is a well-founded tree. Then [T ] T. Define the rank of T recursively. rank T : T ω if p [T ], p sup {rank T p k ) + 1 p k T } if p T \ [T ] Definitions related to a game on a tree Definition Definition of a two-player perfect information game GA; T )) Define a two-player perfect information game GA; T ) as follows see Figure ): 9
26 1. There are two players, usually called player I and player II. 2. Player I and player II alternatively play moves as follows: Suppose p T. a) If lhp) is even e.g., lhp) = ), then I plays a move a such that a M p, i.e., p a T. b) If lhp) is odd, then player II plays a move a such that a M p. 3. Each player has complete knowledge of the previous moves of the way that has been played, i.e., when a player makes a move a p at a position p T, then the player knows p, therefore knows all moves previous to a p. 4. A play of the game is exactly a play on the tree, i.e., f [T ]. 5. f is a win for player I if and only if f A, respectively, f is a win for player II if and only if f / A. A is called the payoff set for player I. [T ]\A is called the payoff set for player II. We denote such a game by GA; T ). We will also use GA, [T ]) for the notation sometimes it is easier to use [T ] rather than T for notational issues with cross products). I x x 2 x ω x ω+2 x II x 1 x 3 x ω+1 x = x, x 1,..., x ω, x ω+1,... [T ]. stops when x [T ] Figure 1.2.2: Illustration of a play x [T ] for lhx) > ω. The notation GA; T ) is not universal. In Jech 23, p. 627), he uses G A with fixed tree ω <ω. In Moschovakis 29, p. 218), he uses G X A) for a fixed tree X <ω. In Kechris 21, 1
27 p. 137), he uses GX, A) or GA) for a fixed tree X <ω. Since we will be considering games on different trees, we will use Martin s notation GA; T ) by Martin 217 draft, p. 5). We will sometimes use GA; [T ]) since we can uniquely obtain T from [T ]. From now on, we will only consider two-player perfect information games. Definition Definition of a strategy) Suppose T is a tree. Recall M p = {m p m T } for each p T. A strategy s for player I is a function such that s : {p T \[T ] lh p) is even} p T M p and sp) M p. Similarly, a strategy s for player II is a function such that s : {p T \[T ] lh p) is odd} p T M p and sp) M p. s is a strategy on the tree T if s is a strategy for player I or player II. Definition Definition of being according to a strategy) Suppose T is a tree and s is a strategy on T. For any f T [T ], f is according to s if and only if for any β such that f β doms), fβ) = sf β). Note that each strategy s gives rise to the following tree T s = T T, s). Notation Suppose T is a tree and s is a strategy on T. Define T s = {p T p is according to s}. 11
28 Definition Definition of a winning strategy for a game GA; T )) Suppose T is a tree and A [T ] is a payoff set for player I. A strategy s is a winning strategy for player I for GA; T ) if for any f [T ] according to s, f A, i.e., [T s ] A. Similarly, a strategy s is a winning strategy for player II for GA, T ) if for any f [T ] according to s, f / A, i.e., [T s ] [T ]\A. Illustration of a winning strategy σ for I Illustration of a winning strategy τ for II σ restricts I's moves τ restricts II's moves I x played by I x I II I II I x1 played by II x 1 II I II I II [ T] All plays in Tσ are in A All plays in T τ are in [ T]\ A Figure 1.2.3: Illustration of winning strategies. Definition Definition of a game being determined) Suppose T is a tree and A [T ] is a payoff set for player I. We say the game GA; T ) is determined if and only if player I or player II has a winning strategy, i.e., there exists a strategy s on T such that [T s ] A and s is a strategy of player I, or [T s ] [T ]\A and s is a strategy for player II. Notice that determinacy corresponds to the existence of a subtree T σ or T τ of T as illustrated in Figure
29 1.3 Definition of complexities In this section, we will review standard complexities on subsets of [T ]. We will first define open sets in a space, from which we will define Borel hierarchy, projective hierarchy, and the difference hierarchy on Π 1 1 sets. Notation Definition of a complexity) In this dissertation, whenever we mention a complexity in chapters 2 and 3, we mean the complexities defined in this section, i.e., Borel, projective and difference hierarchy, unless specified. More precisely, the definition of a complexity in this dissertation is the following: Suppose we have Ξ such that for each tree T, Ξ [T ] [T ]) is defined e.g., Σ α, Π α, Σ 1 n, Π 1 n). Then we say Ξ is a complexity Definitions related to topologies First, we will review the definition of topologies. Definition Definition of a topology) Suppose X is a set. A topology on a set X is a collection τ of subsets of X such that: 1., X τ, 2. Any union of elements in τ is in τ, 3. The finite intersection of elements of τ is in τ. A set X with a topology τ, X, τ) is called a topological space. The elements of τ are called open sets in X. 13
30 A basis for a topology X, τ) is B τ in which every open set A τ can be written as unions of elements of B. Definition Definition of a basis) Suppose X is a set. A basis of a topology X, τ) is a collection B of subsets in X such that 1. For every x X, there exists B B such that x B. 2. If there exist B 1, B 2 B such that x B 1 B 2, then there exists B 3 B such that x B 3 B 1 B 2. If B satisfies both of the conditions 1 and 2, then there is a unique topology on X for which B is a basis. It is called the topology generated by B. We will consider the product topology i I X i with each X i discrete. We will review the product topology and the discrete topology. Definition Definition of the product topology) Suppose X i are sets and τ i is a topology for X i for i I. Consider the Cartesian product i I X i. The basic open sets of i I X i are sets of the form i I U i where each U i is an open set in X i and U i X i for finitely many i I that is, finiteness ). Definition Definition of the discrete topology) Suppose X is a set. The discrete topology on X is defined by setting every subset of X to be an open set in X. 14
31 1.3.2 Open sets We will review open sets in product topology i I X i with each X i discrete. Then we will define open sets over a tree T by using finiteness. Observation Suppose X i for i I are nonempty sets with the discrete topology. Then for every x i X i, {x i } is an open set in X i. Consider the product topology for i I X i. Note that every x i I X i is a sequence x = x i i I with x i X i. The basic open sets are of the form { O { i, x i i E }) = f } X i f { i, x i i E } i I for some finite E I and some x i X i for i I. For a tree T, we will define open sets over T in a way similar to our definition of open sets in the product topology, by using finiteness. Once we define open sets over [T ], we can naturally define the Borel and projective sets on [T ]. Definition Suppose T is a tree. Define F initet ) by F initet ) = {q q is finite p T q p)} The basic open sets in [T ] are the Oq) s for q F initet ) where Oq) = {h [T ] h q}. Figure 1.3.1: Illustration of q and Oq). 15
32 Proposition Suppose T is a tree. The set of open sets defined in definition form a basis for a topology on [T ] Borel hierarchy Sets are classified in hierarchies according to the complexity. The collection of Borel sets on a set [T ] are the smallest collection containing all open sets and closed under complements and countable unions. We will denote the class of Borel sets over [T ] by B [T ]. Borel sets are defined by the smallest σ-algebra containing all open sets. We will review the definitions of algebra and σ-algebra. Definition Definition of an algebra and σ-algebra) An algebra of sets is a collection S of subsets of a given set S such that 1. S S, 2. if X S and Y S then X Y S, 3. if X S then S\X S. Note that S is also closed under finite intersections. A σ-algebra is additionally closed under countable unions and countable intersections): 4. If X n S for all n ω, then n ω X n S. Now, we define the Borel sets over a tree [T ]. First, we will define the restriction notation over classes. 16
33 Notation Moschovakis, 29, p. 27) Suppose X is a space and A is an arbitrary collection sets. Then define A X by A X = {A X A A}. If the space is clear from the context, we will omit it. Definition Definition of the Borel sets over [T ]) Suppose T is a tree. A set B [T ] is Borel if it belongs to the smallest σ-algebra of subsets of [T ] that contains all open sets of [T ]. We will use B [T ] to represent the collection of Borel sets over [T ]. We will review the definition of the Borel Hierarchy. The notation of the Σ s, Π s and s were introduced by Addison 1959). For more details, see Moschovakis 29, p. 48) and Jech 23, p. 153). Definition Hierarchy of Borel sets for [T ])Notation by Addison, 1959) Suppose T is a tree. For any 1 α ω 1, Σ 1 [T ]= the collection of all open sets on [T ], Π 1 [T ]= the collection of all closed sets on [T ], Σ α [T ]= the collection of all sets A= n ω A n,where each A n Π β n [T ] for some β n α, Π α [T ]= the collection of all complements of sets in Σ α [T ], Π α [T ]= the collection of all sets A= n ω A n,where each A n Σ β n [T ] for some β n α, α [T ]=Σ α [T ] Π α [T ]. Note that B = α ω 1 Σ α [T ] = α ON Σ α [T ] = α ON Π α [T ] = α ω 1 Π α [T ]. 17
34 where ON represents the class of all ordinal numbers. Proposition Martin, 217 draft, p. 7, Lemma for X ω ) Suppose each collection in Figure is defined over X ω. For any α ω 1, we have the following inclusions. Σ 1 Σ 2 Σ α α α+1 Π 1 Π 2 Π α Figure 1.3.2: Diagram of Borel hierarchy for X ω. To show the diagram above, one needs to show Σ 1 X ω Σ 2 X ω. For a countable X, one can use separability to get this. However, for even uncountable X, Σ 1 X ω Σ 2 X ω holds as shown in Martin 217 draft). In Martin 217 draft), for T = X <ω, to show Σ 1 [T ] Σ 2 [T ], he uses that O n = {[T p ] p T lh p) = n [T p ] A} is clopen where T p = {q T q p p q }. It is routine to adjust the above argument to get Σ 1 [T ] Σ 2 [T ] for game trees of countable height. Thus, the diagram in figure is true for any tree with countable height. In general, one must be careful whether the above diagram holds for other game trees T. Dr. Burke communicated that Σ 1 2 ω 1 Σ 2 2 ω 1 so that the above diagram is false when T = 2 <ω 1. 18
35 Assume that T = 2 <ω 1. Let O = {f [T ] β ω 1 such that fβ) = }. Then O Σ 1 [T ]. Notice that [T ]\O = {f 1 } where f 1 : ω 1 {, 1} is the constant function fα) = 1 for any α ω 1. We show that O / Σ 2 [T ]. Suppose, for a contradiction, O Σ 2 [T ]. Then there exists C n n ω such that each C n Π 1 [T ] and O = n ω C n. Then [T ]\O = n ω O n where each O n is a complement of C n. Thus each O n is open. Hence each O n = m ω Bm n where each B m n a basic open neighborhood. Then for each B m n, there is q m n F initet ) such that B m n = Oq m n ) see notations for definition 1.3.7). Hence {f 1 } = [T ]\O = n ω O n = n ω m ω O qm n ). Thus, for any n ω, there exists m n such that f 1 O q m n n is a sequence of countable length such that every entry is 1. Define ), i.e., f 1 q m n n. Hence each q m n n π : ω ω n µi ω f 1 O q i n)) 1 Then we have f 1 O ) qn πn) O n ω n ω m ω qm n ) = O n = [T ]\O. n ω Let r = sup domqn πn) ). Then r ω 1 since ω 1 is regular. 2 Let f 2 ω 1 such that f r is a n ω sequence with every entry 1 and fr) =. Then for every n ω, f qn πn). Thus f O ) qn πn) [T ]\O = {f1 } n ω 1 µ represents the least. 2 An infinite cardinal α is regular if cofinality of α is α. 19
36 Since f f 1, this is a contradiction. Hence O / Σ 2 [T ] Projective hierarchy The collection of Borel sets of reals is closed under countable unions and intersections and closed under complements, but it is not closed under continuous images. The image of a Borel set by a continuous function need not be a Borel set Jech, 23, p. 142). Beyond the Borel hierarchy, we have the projective hierarchy. The Σ 1 1 [T ] sets are obtained from taking projections of a closed subset of [T ] N along the Baire space. The Π 1 1 [T ] sets are the complement of Σ 1 1 [T ] sets. In general, for 1 n < ω, the Σ 1 n+1 [T ] N n ) sets are obtained from taking projections of Π 1 n [T ] N n+1 ) sets along the Baire space. In this section, we will review basic definitions associated with projective hierarchy. We will denote the class of projective sets over [T ] by P [T ]. Definition Definition of the projection of S along Y )Moschovakis, 29, p. 19) The projection of a set S X Y along Y into X) is the set P S = {x X y Y x, y S)}. Figure 1.3.3: Illustration of a projection along Y. 2
37 Suslin first discovered that there are Σ 1 1 sets which are not Borel. Together with Lusin, they established most of the basic properties of analytic sets as cited in Moschovakis, 29, p. 2). Projective sets were introduced by Lusin in 1925 and independently by Sierpinski in as cited in Moschovakis, 29, p. 47). See more historic details in Moschovakis 29, p. 2, p. 47). Definition Hierarchy of projective sets over [T ])Lusin, ) Suppose T is a tree. Define Σ 1 [T ] = Σ 1 [T ] and Π 1 = Π 1 [T ]. For each n ω and i ω, inductively define Σ 1 n+1 [T ] N i )= the collection of the projections along N of the Π 1 n [T ] N i+1 ) sets, Π 1 n+1 [T ] N i )= the collection of complements of the Σ 1 n+1 [T ] N i ) sets, 1 n+1 [T ] N i )=Σ 1 n+1 [T ] N i ) Π 1 n+1 [T ] N i ). Denote that the collection of projective sets over [T ] by P [T ]. Thus, for example, for any A [T ], A is Σ 1 1 [T ] if and only if A is the projection of a closed set of [T ] N along N and the collection of projective sets P [T ] is P [T ] = n ω Σ1 n [T ] = n ω Π1 n [T ]. B [T ] 1 1 [T ] is obtained from the following well-known proposition. Proposition Sierpinski, ) Σ 1 n [T ] and Π 1 n [T ] are closed under countable unions and countable intersections. There is a proof for the cases Σ 1 1 and Π 1 1 in Jech 23, pp ). See lemma and lemma for proofs of proposition as cited in Moschovakis 29, p. 29). 4 as cited in Moschovakis 29, p. 47). 21
38 Theorem Suslin 5 ) Suppose T is a countable tree. Every Σ 1 1 [T ] whose complement is also Σ 1 1 [T ] is a Borel set. Thus 1 1 [T ] = B [T ]. Definition Definition of an open-separated union)martin, 199; Martin, 217 draft, p.8) Suppose T is a tree. A [T ] is the open separated union of {B j [T ] j J} where each B j [T ], if 1. A = j J B j 2. there are disjoint open sets D j, j J such that B j D j for each j J Definition Definition of a quasi-borel set)martin, 199; Martin, 217 draft, p.8) Suppose T is a tree. The quasi-borel subsets of [T ] form the smallest class of subsets of [T ] containing all open sets and closed under the operations: 1. complementation 2. countable union 3. open-separated union We will denote the collection of quasi-borel sets on [T ] by qb [T ]. By closure under complementation 1) and countable union 2) of quasi-borel sets, B [T ] qb [T ] for any tree T. 5 as cited in Jech 23, p. 145, Theorem 11.1). 22
39 Theorem Martin, 199, p281 Remarks 1)) Suppose T is tree. If T is countable, the quasi-borel subsets of [T ] are Borel subsets of [T ]. Thus qb [T ] = B [T ] for a countable tree T. If T is uncountable, not all the quasi-borel subsets of [T ] are Borel. For example, let T = { a p p ω <ω α ω 1 }. For each α ω 1, fix B α Π α ω ω ) \Σ α ω ω ). Define A = { a y y B α }. Then A is quasi-borel but not Borel Martin, 217 draft, p. 83, Remarka)). Suslin s theorem generalizes: Theorem Hansell, ) For any tree T, 1 1 [T ] = qb [T ]. This is shown in Martin 199, p. 281, Theorem 1) Difference hierarchy The difference kernel was discussed by Hausdorff as cited in Welch, 1996, p. 1). Definition Definition of the difference kernel)hausdorff, ) Denote the difference kernel of A = A β β α by dk A) and define dk A) = {x [T ] µβ x / A β β = α) is odd}. Definition Suppose Λ is a class of subsets of [T ] and Λ is closed under countable intersections. Suppose α ω 1. Define { α-λ [T ] = A [T ] A = A β β α each A β Λ and A = dk A) )}. 6 as cited in Martin 199); Martin 217 draft, p. 84, Theorem 2.2.3). 7 as cited in Welch 1996, p. 1). 23
40 Since Λ is closed under countable intersections, without loss of generality, we can assume each A β A γ for any β < γ. Note that 1-Λ = Λ In general, for any finite n, 2-Λ = {A [T ] A, A 1 ΛA = A \ A }{{} 1 )} 1-Λ 3-Λ = {A [T ] A, A 1, A 2 ΛA = A \ A 1 \A 2 ))} }{{} 2-Λ 4-Λ = {A [T ] A, A 1, A 2, A 3 ΛA = A \ A 1 \ A 2 \A 3 )))} }{{} 3-Λ. n-λ = {A [T ] A, A 1,..., A n 1 ΛA = A \ A 1 \ A 2 \ A 3 \ A n 2 \A n 1 ))))))} }{{} n 1)-Λ Figure 1.3.4: Illustration of difference kernel. Consider 2-Λ. Then A }{{} Λ [T ] \ A 1 }{{} Λ [T ] = A }{{} Λ [T ] X ω \A 1 ) }{{} co-λ [T ] = X ω \A 1 ) \ X ω \A }{{} ). }{{} co-λ [T ] co-λ [T ] 24
41 Thus 2-Λ [T ] = Λ co-λ) [T ] = 2-co-Λ) [T ] where the notation is defined in notation on page 43. In particular, 2-Π 1 1 X ω = Σ 1 1 Π 1 1) X ω = 2-Σ 1 1 X ω. This gives us co-2-π 1 1 X ω = { A X } ) ω X ω \A 2-Π 1 1 = Σ 1 1 Π 1 1 X ω where notation is defined in notation on page 43. We also have Σ 1 1 X ω 2-Π 1 1 X ω since for any E Σ 1 1 X ω, E = }{{} X ω \ X ω 1 \E) 2-Π1 X ω. }{{} Π 1 1 Xω Π 1 1 Xω The following classes are also well-known and are presented in Martin 217 draft). Note that Martin 217 draft) does not include Chapter 5. The page numbers listed below under Chapter 5 are from an older draft. Definition Martin, 217 draft, p. 24, Chapter 5, p. 23) Diff Π 1 1 [T ] ) = α ω 1 α-π 1 1 [T ] Definition Martin, 217 draft, p.275, Chapter 5 Section 5.4) Define Σ 1 Π 1 1) to be the collection of all countable unions of Boolean combinations of sets belonging to Π 1 1 sets. Lemma Martin, 217 draft, p. 276, Chapter 5 Lemma 5.4.1) 25
42 Suppose T is a tree and let A [T ]. Then A Σ 1 Π 1 1) [T ] if and only if A is a countable union of differences of Π 1 1 sets. Thus for any γ β, we have γ Π 1 1 [T ] β Π 1 1 [T ] Diff Π 1 1 [T ] ) Σ 1 Π 1 1 ) [T ]. Well-known determinacy results 26
43 1.4 Well-known determinacy results In this section, we will list some well-known determinacy results. In section 1.4.1, we will list some well-known determinacy results from ZFC. In section 1.4.2, we will list some well-known determinacy results from large cardinal properties. The list of well-known determinacy results are also on page 367 Appendix D. Definition Axioms of Zermelo-Fraenkel ZF) and ZFC)Jech, 23) 1. Axiom of Extensionality. If X and Y have the same elements, then X = Y. 2. Axiom of Pairing. For any a and b, there exists a set {a, b} that contains exactly a and b. 3. Axiom Schema of Separation Comprehension). If P is a property with parameter p), then for any X and p, there exists a set Y = {u X P u, p)} that contains all those u X that have property P. 4. Axiom of Union. For any X, there exists a set Y = X, the union of all elements of X. 5. Axiom of Power Set. For any X, there exists a set Y = X), the set of all subsets of X. 6. Axiom of Infinity. There exists an infinite set. 27
44 7. Axiom Schema of Replacement. If a class F is a function, then for any X there exists a set Y = F X) = {F x) x X}. 8. Axiom of Regularity Foundation). Every nonempty set has an -minimal element. 9. Axiom of Choice. Every family of nonempty sets has a choice function. The theory with axioms 1-8 is the Zermelo-Fraenkel axiomatic set theory ZF; ZFC denotes the theory ZF with the Axiom of choice; ZF-P denotes the theory with ZF without the Power Set Axiom Determinacy results from ZFC Theorem through theorem are theorems of ZFC. Theorem Gale and Stewart, 1953) Suppose T is a tree. If T is well-founded, then for any A [T ], GA; T ) is determined. Theorem AC)Gale and Stewart, 1953)as cited in Moschovakis, 29, p. 222, 6A.6) There exists A ω ω such that GA; ω <ω ) is not determined. Definition Definition of an open game) Suppose T is a tree. Suppose A [T ]. If A is an open set, we call GA; T ) is called an open game. Similarly for the other complexities. 28
45 Notation Suppose T is a tree. We denote all open games on T are determined by DetΣ 1 [T ]). In this case, we say Σ 1 determinacy on T holds. Similarly for the other complexities. Theorem Gale and Stewart, 1953) Suppose T = X <ω for some nonempty X. Then DetΣ 1 [T ]) and DetΠ 1 [T ]). Theorem Wolfe, 1955) Suppose T = X <ω for some nonempty X. Then DetΣ 2 [T ]). Theorem Martin, 1975; Martin, 199) Suppose T = X <ω for some nonempty X. Then DetB [T ]). Theorem Martin, 199) Suppose T = X <ω for some nonempty X. Then DetqB [T ]) Determinacy results from large cardinals An uncountable cardinal κ is a measurable cardinal if there is a κ-complete nonprincipal ultrafilter on κ. We will review definitions of filters related to the definition of a measurable cardinal. Definition Definitions of a filter, a principal filter, an ultrafilter and a κ-complete filter) A filter on a nonempty set S is a collection F of subsets of S such that 1. S F and F, 29
46 2. if X F and Y F, then X Y F, 3. if X, Y S, X F and X Y, then Y F. Let X be a nonempty subset of S. The filter F = {X S X X } is a principal filter. A filter U on S is an ultrafilter if for every X S, either X U or S\X U. If κ is a regular uncountable cardinal and F is a filter on S, then F is called κ-complete if F is closed under intersection of less than κ sets, i.e., for any {X α F α γ} with γ κ, α γ X α F. Definition Definition of a measurable cardinal) An uncountable cardinal κ is measurable if there is a κ-complete nonprincipal ultrafilter U on κ List of results related to the existence of measurable cardinals The following are results obtained from the existence of a measurable cardinal. Theorem Martin, 197) If there is a measurable cardinal, then DetΠ 1 1 ω ω ). Theorem Martin, 197)as cited in Martin, 217 draft, p.187, Theorem 4.1.6) Let T be a game tree. Assume there is a measurable cardinal larger than T. Then DetΠ 1 1 [T ]). Theorem Martin, 199, p. 287, Theorem 3) If there is a measurable cardinal, then Detω 2 -Π 1 1 ω ω ). 3
47 Martin proved the above result in 197 s. In the 198 s he proved the following generalization which uses quasi-borel determinacy. Theorem Martin, 199, p. 292, Theorem 4) If there is a measurable cardinal, then Det ω 2 + 1)-Π 1 1 ω ω ). Theorem Martin, 217 draft, p.241, Chapter 5 Theorem ) Let α be a countable ordinal and T = X <ω. If the class of measurable cardinals greater than T has order type α, then Det ω 2 α + 1)-Π 1 1 [T ]). Martin s student John Simms proved the following in his dissertation. Theorem Simms ) Let T = X <ω. If there is a measurable limit of measurable cardinals that is larger than T, then DetΣ 1Π 1 1) [T ]) Projective Determinacy In general, to obtain each level of projective determinacy DetΠ 1 n+1 ω ω ), we will need the existence of n Woodin cardinals. We will review the definition of an elementary embedding and define a Woodin cardinal. Definition Definition of an elementary embedding and a critical point) Suppose M = M, E) and N = N, F ) are models of set theory. An elementary embedding of M into N is a function j : M N such that for any formula φ v 1,..., v n ) of the language of set theory and for any a 1,..., a n M, 8 as cited in Martin 217 draft, p. 281, Chapter 5 Theorem 5.4.5). 31
48 M φ [a 1,..., a n ] N φ [j a 1 ),..., j a n )] Suppose M and N are both transitive and j : M N is an elementary embedding. Then an ordinal κ M is the critical point if κ is the least such that jκ) κ. If α is an ordinal, then jα) is an ordinal and if α < β, then jα) < jβ) so that α jα). Thus we can replace jκ) κ by jκ) > κ. Also, for any n ω, jn) = n and thus jω) = ω. Hence κ > ω. Theorem Jech, 23, p. 287) If there exists a measurable cardinal, then there exists a nontrivial elementary embedding of the universe. Conversely, if j : V M is a nontrivial elementary embedding, then there exists a measurable cardinal. Definition Definition of the cumulative hierarchy V α of sets ) Inductively, for each ordinal α, define a set V α by : 1. V = ; 2. V α+1 = V α ); 3. V λ = α<λ V α if λ is a limit ordinal. Define the class V = α ON V α. Definition Definition of a Woodin cardinal) A cardinal δ is a Woodin cardinal if for all A V δ there are arbitrary large κ < δ such that 32
49 for all λ < δ there exists an elementary embedding j : V M with critical point κ such that jκ) > λ, V λ M and A V λ = ja) V λ. Each level of projective determinacy DetΠ 1 n+1 ω ω ) is obtained from the existence of a measurable cardinal above n Woodin cardinals. Theorem Martin and Steel, 1985) For n ω, if there exist n Woodin cardinals with a measurable cardinal above them, then DetΠ 1 n+1 ω ω ). Projective determinacy DetP ω ω ) is obtained from the existence of infinitely many Woodin cardinals. Theorem Martin and Steel, 1985) Suppose there are infinitely many Woodin cardinals. Then DetP ω ω ) Lightface results related to the existence of # We will observe theorems of difference hierarchy of lightface version. Recall that definition of Π 1 1 is obtained from a recursive relation. We will review the definition of #. The theory of # is provided in Jech 23, p. 313, chapter 18). First, we will review the definition of Gödel s constructible universe L. Definition Definition of the Gödel s constructible universe L)Martin, 217 draft) Gödel s constructible universe L and hierarchy of constructible sets are defined as follows: 1. L = 33
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 περισσότεραEvery 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 περισσότεραOrdinal 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 περισσότεραω 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 περισσότεραExample 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 περισσότεραChapter 3: Ordinal Numbers
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
Διαβάστε περισσότεραCongruence 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 περισσότεραMINIMAL 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 περισσότεραEE512: 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 περισσότεραA 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 περισσότεραLecture 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 περισσότερα4.6 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 περισσότεραCommutative 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 περισσότεραSOME 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 περισσότεραST5224: 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 περισσότεραBounding 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 περισσότεραORDINAL 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 περισσότεραC.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 περισσότεραHomework 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 περισσότεραFractional 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 περισσότεραHomomorphism 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 περισσότερα3.4 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 περισσότεραConcrete 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 περισσότεραCHAPTER 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 περισσότεραKnaster-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 περισσότεραMatrices 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 περισσότερα1. 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 περισσότεραAbstract 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 περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραPhys460.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 περισσότεραGenerating Set of the Complete Semigroups of Binary Relations
Applied Mathematics 06 7 98-07 Published Online January 06 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/036/am067009 Generating Set of the Complete Semigroups of Binary Relations Yasha iasamidze
Διαβάστε περισσότεραOther 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 περισσότεραSCHOOL 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 περισσότεραSecond 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 περισσότεραUniform 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 περισσότεραStatistical 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 περισσότεραLecture 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 περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότερα2. 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 περισσότεραMain 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 περισσότεραSection 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 περισσότεραSome new generalized topologies via hereditary classes. Key Words:hereditary generalized topological space, A κ(h,µ)-sets, κµ -topology.
Bol. Soc. Paran. Mat. (3s.) v. 30 2 (2012): 71 77. c SPM ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v30i2.13793 Some new generalized topologies via hereditary
Διαβάστε περισσότεραHomomorphism 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 περισσότεραderivation 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 περισσότεραFinite 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 περισσότερα6.3 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 περισσότεραSplitting stationary sets from weak forms of Choice
Splitting stationary sets from weak forms of Choice Paul Larson and Saharon Shelah February 16, 2008 Abstract Working in the context of restricted forms of the Axiom of Choice, we consider the problem
Διαβάστε περισσότερα5. 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 περισσότεραOn a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
Διαβάστε περισσότεραCardinals. 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 περισσότεραNowhere-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 περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότεραLecture 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 περισσότεραTridiagonal 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 περισσότεραReminders: 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 περισσότεραThe 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 περισσότεραChapter 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 περισσότεραF A S C I C U L I M A T H E M A T I C I
F A S C I C U L I M A T H E M A T I C I Nr 46 2011 C. Carpintero, N. Rajesh and E. Rosas ON A CLASS OF (γ, γ )-PREOPEN SETS IN A TOPOLOGICAL SPACE Abstract. In this paper we have introduced the concept
Διαβάστε περισσότεραStrain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Διαβάστε περισσότεραMath 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 περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότεραOn 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 περισσότεραk 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 περισσότεραInstruction Execution Times
1 C Execution Times InThisAppendix... Introduction DL330 Execution Times DL330P Execution Times DL340 Execution Times C-2 Execution Times Introduction Data Registers This appendix contains several tables
Διαβάστε περισσότερα6.1. 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 περισσότεραMath221: 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 περισσότεραDIRECT 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 περισσότεραPractice 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 περισσότερα5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
Διαβάστε περισσότεραTHE 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( 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 περισσότεραFourier 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 περισσότεραInverse 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 περισσότεραEcon 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 περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραSection 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 περισσότεραChapter 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 περισσότεραSrednicki 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 περισσότεραSequent 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 περισσότεραANSWERSHEET (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 περισσότεραPotential Dividers. 46 minutes. 46 marks. Page 1 of 11
Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and
Διαβάστε περισσότεραΜηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
Διαβάστε περισσότεραDémographie spatiale/spatial Demography
ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΙΑΣ Démographie spatiale/spatial Demography Session 1: Introduction to spatial demography Basic concepts Michail Agorastakis Department of Planning & Regional Development Άδειες Χρήσης
Διαβάστε περισσότεραSection 1: Listening and responding. Presenter: Niki Farfara MGTAV VCE Seminar 7 August 2016
Section 1: Listening and responding Presenter: Niki Farfara MGTAV VCE Seminar 7 August 2016 Section 1: Listening and responding Section 1: Listening and Responding/ Aκουστική εξέταση Στο πρώτο μέρος της
Διαβάστε περισσότεραProblem 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 περισσότεραPARTIAL 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 περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραAreas 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 περισσότεραThe 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 περισσότεραAreas 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 περισσότεραModels for Probabilistic Programs with an Adversary
Models for Probabilistic Programs with an Adversary Robert Rand, Steve Zdancewic University of Pennsylvania Probabilistic Programming Semantics 2016 Interactive Proofs 2/47 Interactive Proofs 2/47 Interactive
Διαβάστε περισσότεραCRASH 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 περισσότεραSection 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 περισσότεραHomomorphism and Cartesian Product on Fuzzy Translation and Fuzzy Multiplication of PS-algebras
Annals of Pure and Applied athematics Vol. 8, No. 1, 2014, 93-104 ISSN: 2279-087X (P), 2279-0888(online) Published on 11 November 2014 www.researchmathsci.org Annals of Homomorphism and Cartesian Product
Διαβάστε περισσότεραGÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)
IFSCOM016 1 Proceeding Book No. 1 pp. 155-161 (016) ISBN: 978-975-6900-54-3 SOME RESULTS ON S α,β AND T α,β INTUITIONISTIC FUZZY MODAL OPERATORS GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)
Διαβάστε περισσότεραλρ-calculus 1. each λ-variable is a λρ-term, called an atom or atomic term; 2. if M and N are λρ-term then (MN) is a λρ-term called an application;
λρ-calculus Yuichi Komori komori@math.s.chiba-u.ac.jp Department of Mathematics, Faculty of Sciences, Chiba University Arato Cho aratoc@g.math.s.chiba-u.ac.jp Department of Mathematics, Faculty of Sciences,
Διαβάστε περισσότεραthe total number of electrons passing through the lamp.
1. A 12 V 36 W lamp is lit to normal brightness using a 12 V car battery of negligible internal resistance. The lamp is switched on for one hour (3600 s). For the time of 1 hour, calculate (i) the energy
Διαβάστε περισσότεραSolution 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