DOCSLIB.ORG

LECTURE #08:QUERY EXECUTION Administrivia

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
LECTURE #08:QUERY EXECUTION Administrivia DATA ANALYTICS USING DEEP LEARNING GT 8803 // FALL 2019 // JOY ARULRAJ LECTURE #08:QUERY EXECUTION administrivia • Reminders – Sign up for discussion slots on Thursday – Proposal presentations on next Wednesday – 2 page document + 5 minute presentation – Assignment 1 due on Wednesday GT 8803 // Fall 2019 2 LAST CLASS • Storage models: NSM, DSM, and FSM FLEXIBLE STORAGE MODEL ID University Enrollment City 1 Georgia Tech 15000 Atlanta 2 Wisconsin 30000 Madison 3 Carnegie Mellon 6000 Pittsburgh 4 UC Berkeley 30000 Berkeley GT 8803 // Fall 2019 3 LAST CLASS • Compression – Zone maps – Dictionary encoding Original Data val 100 200 300 400 400 GT 8803 // Fall 2019 4 LAST CLASS • Compression – Zone maps – Dictionary encoding Original Data Zone Map val type val 100 MIN 100 200 MAX 400 300 AVG 280 400 SUM 1400 400 COUNT 5 GT 8803 // Fall 2019 5 LAST CLASS • Compression SELECT * FROM table – Zone maps WHERE val > 600 – Dictionary encoding Original Data Zone Map val type val 100 MIN 100 200 MAX 400 300 AVG 280 400 SUM 1400 400 COUNT 5 GT 8803 // Fall 2019 6 LAST CLASS • Visual Storage Engine – Convolutional Auto Encoder COMPRESSED DATA ENCODER DECODER (CONVOLUTIONS) (DECONVOLUTIONS) GT 8803 // Fall 2019 7 TODAY’s AGENDA • Query Processing Models • Access Methods • Expression Evaluation • Visual Query Execution Engine GT 8803 // Fall 2019 8 QUERY PROCESSING MODELS GT 8803 // Fall 2018 9 ANATOMY OF A DATABASE SYSTEM Connection Manager + Admission Control Process Manager Query Parser Query Optimizer Query Processor Query Executor Query Lock Manager (Concurrency Control) Access Methods (or Indexes) Transactional Storage Manager Buffer Pool Manager Log Manager Memory Manager + Disk Manager Shared Utilities Networking Manager Source: Anatomy of a Database System GT 8803 // Fall 2019 10 QUERY PLAN • The operators are arranged in a tree. – Data flows from the leaves toward the root. – Output of the root node is the result of the query. A.id, B.value SELECT A.id, B.value p FROM A, B WHERE A.id = B.id ⨝ A.id=B.id AND B.value > 100 s value>100 AB GT 8803 // Fall 2019 11 QUERY PROCESSING MODELS • A DBMS's processing model defines how the system executes a query plan. – Different trade-offs for different workloads. – Top-down or Bottom-up execution – Determines what data is sent between operators • Three approaches: – Tuple-at-a-time Model – Operator-at-a-time Model – Vector-at-a-time Model GT 8803 // Fall 2019 12 #1: TUPLE-AT-A-TIME MODEL • Each query plan operator implements a next function. – On each invocation, the operator returns either a single tuple or a null marker if there are no more tuples. – The operator implements a loop that calls next on its children to retrieve their tuples and then process them. • Top-down plan processing. • Also called Iterator / Volcano / Pipeline model. GT 8803 // Fall 2019 13 #1: TUPLE-AT-A-TIME MODEL SELECT A.id, B.value FROM A, B WHERE A.id = B.id AND B.value > 100 p A.id, B.value ⨝ A.id=B.id s value>100 AB GT 8803 // Fall 2018 14 #1: TUPLE-AT-A-TIME MODEL SELECT A.id, B.value for t in child.Next(): emit(projection(t)) FROM A, B WHERE A.id = B.id AND for t1 in left.Next(): B.value > 100 buildHashTable(t1) for t in right.Next(): 2 p A.id, B.value if probe(t2): emit(t1⨝t2) for t in child.Next(): ⨝ A.id=B.id if evalPred(t): emit(t) s value>100 for t in A: for t in B: emit(t) emit(t) AB GT 8803 // Fall 2018 15 #1: TUPLE-AT-A-TIME MODEL SELECT A.id, B.value for t in child.Next(): emit(projection(t)) FROM A, B WHERE A.id = B.id AND for t1 in left.Next(): B.value > 100 buildHashTable(t1) for t in right.Next(): 2 p A.id, B.value if probe(t2): emit(t1⨝t2) for t in child.Next(): ⨝ A.id=B.id if evalPred(t): emit(t) s value>100 for t in A: for t in B: emit(t) emit(t) AB GT 8803 // Fall 2018 16 #1: TUPLE-AT-A-TIME MODEL SELECT A.id, B.value for t in child.Next(): FROM 1 emit(projection(t)) A, B WHERE A.id = B.id AND for t1 in left.Next(): B.value > 100 buildHashTable(t1) for t in right.Next(): 2 p A.id, B.value if probe(t2): emit(t1⨝t2) for t in child.Next(): ⨝ A.id=B.id if evalPred(t): emit(t) s value>100 for t in A: for t in B: emit(t) emit(t) AB GT 8803 // Fall 2018 17 #1: TUPLE-AT-A-TIME MODEL SELECT A.id, B.value for t in child.Next(): FROM 1 emit(projection(t)) A, B WHERE A.id = B.id AND for t1 in left.Next(): B.value > 100 buildHashTable(t1) for t in right.Next(): 2 p A.id, B.value if probe(t2): emit(t1⨝t2) for t in child.Next(): ⨝ A.id=B.id if evalPred(t): emit(t) s value>100 for t in A: for t in B: emit(t) emit(t) AB GT 8803 // Fall 2018 18 #1: TUPLE-AT-A-TIME MODEL SELECT A.id, B.value for t in child.Next(): FROM 1 emit(projection(t)) A, B WHERE A.id = B.id AND for t1 in left.Next(): B.value > 100 buildHashTable(t ) 2 1 for t in right.Next(): 2 p A.id, B.value if probe(t2): emit(t1⨝t2) for t in child.Next(): ⨝ A.id=B.id if evalPred(t): emit(t) s value>100 for t in A: for t in B: emit(t) emit(t) AB GT 8803 // Fall 2018 19 #1: TUPLE-AT-A-TIME MODEL SELECT A.id, B.value for t in child.Next(): FROM 1 emit(projection(t)) A, B WHERE A.id = B.id AND for t1 in left.Next(): B.value > 100 buildHashTable(t ) 2 1 for t in right.Next(): 2 p A.id, B.value if probe(t2): emit(t1⨝t2) for t in child.Next(): ⨝ A.id=B.id if evalPred(t): emit(t) s value>100 for t in A: for t in B: 3 emit(t) emit(t) AB GT 8803 // Fall 2018 20 #1: TUPLE-AT-A-TIME MODEL SELECT A.id, B.value for t in child.Next(): FROM 1 emit(projection(t)) A, B WHERE A.id = B.id AND for t1 in left.Next(): B.value > 100 buildHashTable(t ) 2 1 for t in right.Next(): 2 p A.id, B.value if probe(t2): emit(t1⨝t2) for t in child.Next(): ⨝ A.id=B.id if evalPred(t): 4 emit(t) s value>100 for t in A: for t in B: 3 emit(t) emit(t) 5 AB GT 8803 // Fall 2018 21 #1: TUPLE-AT-A-TIME MODEL • This is used in almost every DBMS. – Allows for tuple pipelining. • Some operators will block until children emit all of their tuples. – Joins, Subqueries, Order By – Known as pipeline breakers • Output control works easily with this approach. – Limit GT 8803 // Fall 2019 22 #2: OPERATOR-AT-A-TIME MODEL • Each operator processes its input all at once and then emits its output all at once. – The operator "materializes" it output as a single result. – The DBMS can push down hints into to avoid scanning too many tuples. • Bottom-up plan processing. GT 8803 // Fall 2019 23 #2: OPERATOR-AT-A-TIME MODEL out = { } SELECT A.id, B.value for t in child.Output(): FROM A, B out.add(projection(t)) WHERE A.id = B.id out = { } AND B.value > 100 for t1 in left.Output(): buildHashTable(t ) 1 p A.id, B.value for t2 in right.Output(): if probe(t2): out.add(t1⨝t2) out = { } ⨝ A.id=B.id for t in child.Output(): if evalPred(t): out.add(t) s value>100 out = { } out = { } for t in A: for t in B: out.add(t) out.add(t) AB GT 8803 // Fall 2018 24 #2: OPERATOR-AT-A-TIME MODEL out = { } SELECT A.id, B.value for t in child.Output(): FROM A, B out.add(projection(t)) WHERE A.id = B.id out = { } AND B.value > 100 for t1 in left.Output(): buildHashTable(t ) 1 p A.id, B.value for t2 in right.Output(): if probe(t2): out.add(t1⨝t2) out = { } ⨝ A.id=B.id for t in child.Output(): if evalPred(t): out.add(t) s value>100 out = { } out = { } 1 for t in A: for t in B: out.add(t) out.add(t) AB GT 8803 // Fall 2018 25 #2: OPERATOR-AT-A-TIME MODEL out = { } SELECT A.id, B.value for t in child.Output(): FROM A, B out.add(projection(t)) WHERE A.id = B.id out = { } AND B.value > 100 for t1 in left.Output(): buildHashTable(t ) 1 p A.id, B.value for t2 in right.Output(): if probe(t2): out.add(t1⨝t2) out = { } ⨝ A.id=B.id for t in child.Output(): if evalPred(t): out.add(t) s value>100 out = { } out = { } 1 for t in A: for t in B: out.add(t) out.add(t) AB GT 8803 // Fall 2018 26 #2: OPERATOR-AT-A-TIME MODEL out = { } SELECT A.id, B.value for t in child.Output(): FROM A, B out.add(projection(t)) WHERE A.id = B.id out = { } AND B.value > 100 for t1 in left.Output(): buildHashTable(t ) 1 p A.id, B.value for t2 in right.Output(): if probe(t2): out.add(t1⨝t2) out = { } ⨝ A.id=B.id for t in child.Output(): if evalPred(t): out.add(t) s value>100 out = { } out = { } 1 for t in A: for t in B: 2 out.add(t) out.add(t) AB GT 8803 // Fall 2018 27 #2: OPERATOR-AT-A-TIME MODEL out = { } SELECT A.id, B.value for t in child.Output(): FROM A, B out.add(projection(t)) WHERE A.id = B.id out = { } AND B.value > 100 for t1 in left.Output(): buildHashTable(t ) 1 p A.id, B.value for t2 in right.Output(): if probe(t2): out.add(t1⨝t2) out = { } ⨝ A.id=B.id for t in child.Output(): 3 if evalPred(t): out.add(t) s value>100 out = { } out = { } 1 for t in A: for t in B: 2 out.add(t) out.add(t) AB GT 8803 // Fall 2018 28 #2: OPERATOR-AT-A-TIME MODEL out = { } SELECT A.id, B.value for t in child.Output(): FROM A, B out.add(projection(t)) WHERE A.id = B.id out = { } AND B.value > 100 for t in left.Output(): 4 1 buildHashTable(t ) 1 p A.id, B.value for t2 in right.Output(): if probe(t2): out.add(t1⨝t2) out = { } ⨝ A.id=B.id for t in child.Output(): 3 if evalPred(t): out.add(t) s value>100 out = { } out = { } 1 for t in A: for t in B: 2 out.add(t) out.add(t) AB GT 8803 // Fall 2018 29 #2: OPERATOR-AT-A-TIME MODEL out = { } SELECT A.id, B.value 5 for t in child.Output(): FROM A, B out.add(projection(t)) WHERE A.id = B.id out = { } AND B.value > 100 for t in left.Output(): 4 1 buildHashTable(t ) 1 p A.id, B.value for t2 in right.Output(): if probe(t2): out.add(t1⨝t2) out = { } ⨝ A.id=B.id for t in child.Output(): 3 if evalPred(t): out.add(t) s value>100 out = { } out = { } 1 for t in A: for t in B: 2 out.add(t) out.add(t) AB GT 8803 // Fall 2018 30 #2: OPERATOR-AT-A-TIME MODEL • Better for OLTP workloads – Transactions typically only access a small number of tuples at a time.
Load more

Recommended publications
  • Mathematics 144 Set Theory Fall 2012 Version
    MATHEMATICS 144 SET THEORY FALL 2012 VERSION Table of Contents I. General considerations.……………………………………………………………………………………………………….1 1. Overview of the course…………………………………………………………………………………………………1 2. Historical background and motivation………………………………………………………….………………4 3. Selected problems………………………………………………………………………………………………………13 I I. Basic concepts. ………………………………………………………………………………………………………………….15 1. Topics from logic…………………………………………………………………………………………………………16 2. Notation and first steps………………………………………………………………………………………………26 3. Simple examples…………………………………………………………………………………………………………30 I I I. Constructions in set theory.………………………………………………………………………………..……….34 1. Boolean algebra operations.……………………………………………………………………………………….34 2. Ordered pairs and Cartesian products……………………………………………………………………… ….40 3. Larger constructions………………………………………………………………………………………………..….42 4. A convenient assumption………………………………………………………………………………………… ….45 I V. Relations and functions ……………………………………………………………………………………………….49 1.Binary relations………………………………………………………………………………………………………… ….49 2. Partial and linear orderings……………………………..………………………………………………… ………… 56 3. Functions…………………………………………………………………………………………………………… ….…….. 61 4. Composite and inverse function.…………………………………………………………………………… …….. 70 5. Constructions involving functions ………………………………………………………………………… ……… 77 6. Order types……………………………………………………………………………………………………… …………… 80 i V. Number systems and set theory …………………………………………………………………………………. 84 1. The Natural Numbers and Integers…………………………………………………………………………….83 2. Finite induction
    [Show full text]
  • Biography Paper – Georg Cantor
    Mike Garkie Math 4010 – History of Math UCD Denver 4/1/08 Biography Paper – Georg Cantor Few mathematicians are house-hold names; perhaps only Newton and Euclid would qualify. But there is a second tier of mathematicians, those whose names might not be familiar, but whose discoveries are part of everyday math. Examples here are Napier with logarithms, Cauchy with limits and Georg Cantor (1845 – 1918) with sets. In fact, those who superficially familier with Georg Cantor probably have two impressions of the man: First, as a consequence of thinking about sets, Cantor developed a theory of the actual infinite. And second, that Cantor was a troubled genius, crippled by Freudian conflict and mental illness. The first impression is fundamentally true. Cantor almost single-handedly overturned the Aristotle’s concept of the potential infinite by developing the concept of transfinite numbers. And, even though Bolzano and Frege made significant contributions, “Set theory … is the creation of one person, Georg Cantor.” [4] The second impression is mostly false. Cantor certainly did suffer from mental illness later in his life, but the other emotional baggage assigned to him is mostly due his early biographers, particularly the infamous E.T. Bell in Men Of Mathematics [7]. In the racially charged atmosphere of 1930’s Europe, the sensational story mathematician who turned the idea of infinity on its head and went crazy in the process, probably make for good reading. The drama of the controversy over Cantor’s ideas only added spice. 1 Fortunately, modern scholars have corrected the errors and biases in older biographies.
    [Show full text]
  • Self-Organizing Tuple Reconstruction in Column-Stores
    Self-organizing Tuple Reconstruction in Column-stores Stratos Idreos Martin L. Kersten Stefan Manegold CWI Amsterdam CWI Amsterdam CWI Amsterdam The Netherlands The Netherlands The Netherlands [email protected] [email protected] [email protected] ABSTRACT 1. INTRODUCTION Column-stores gained popularity as a promising physical de- A prime feature of column-stores is to provide improved sign alternative. Each attribute of a relation is physically performance over row-stores in the case that workloads re- stored as a separate column allowing queries to load only quire only a few attributes of wide tables at a time. Each the required attributes. The overhead incurred is on-the-fly relation R is physically stored as a set of columns; one col- tuple reconstruction for multi-attribute queries. Each tu- umn for each attribute of R. This way, a query needs to load ple reconstruction is a join of two columns based on tuple only the required attributes from each relevant relation. IDs, making it a significant cost component. The ultimate This happens at the expense of requiring explicit (partial) physical design is to have multiple presorted copies of each tuple reconstruction in case multiple attributes are required. base table such that tuples are already appropriately orga- Each tuple reconstruction is a join between two columns nized in multiple different orders across the various columns. based on tuple IDs/positions and becomes a significant cost This requires the ability to predict the workload, idle time component in column-stores especially for multi-attribute to prepare, and infrequent updates. queries [2, 6, 10].
    [Show full text]
  • On Free Products of N-Tuple Semigroups
    n-tuple semigroups Anatolii Zhuchok Luhansk Taras Shevchenko National University Starobilsk, Ukraine E-mail: [email protected] Anatolii Zhuchok Plan 1. Introduction 2. Examples of n-tuple semigroups and the independence of axioms 3. Free n-tuple semigroups 4. Free products of n-tuple semigroups 5. References Anatolii Zhuchok 1. Introduction The notion of an n-tuple algebra of associative type was introduced in [1] in connection with an attempt to obtain an analogue of the Chevalley construction for modular Lie algebras of Cartan type. This notion is based on the notion of an n-tuple semigroup. Recall that a nonempty set G is called an n-tuple semigroup [1], if it is endowed with n binary operations, denoted by 1 ; 2 ; :::; n , which satisfy the following axioms: (x r y) s z = x r (y s z) for any x; y; z 2 G and r; s 2 f1; 2; :::; ng. The class of all n-tuple semigroups is rather wide and contains, in particular, the class of all semigroups, the class of all commutative trioids (see, for example, [2, 3]) and the class of all commutative dimonoids (see, for example, [4, 5]). Anatolii Zhuchok 2-tuple semigroups, causing the greatest interest from the point of view of applications, occupy a special place among n-tuple semigroups. So, 2-tuple semigroups are closely connected with the notion of an interassociative semigroup (see, for example, [6, 7]). Moreover, 2-tuple semigroups, satisfying some additional identities, form so-called restrictive bisemigroups, considered earlier in the works of B. M. Schein (see, for example, [8, 9]).
    [Show full text]
  • SRB MEASURES for ALMOST AXIOM a DIFFEOMORPHISMS José Alves, Renaud Leplaideur
    SRB MEASURES FOR ALMOST AXIOM A DIFFEOMORPHISMS José Alves, Renaud Leplaideur To cite this version: José Alves, Renaud Leplaideur. SRB MEASURES FOR ALMOST AXIOM A DIFFEOMORPHISMS. 2013. hal-00778407 HAL Id: hal-00778407 https://hal.archives-ouvertes.fr/hal-00778407 Preprint submitted on 20 Jan 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. SRB MEASURES FOR ALMOST AXIOM A DIFFEOMORPHISMS JOSE´ F. ALVES AND RENAUD LEPLAIDEUR Abstract. We consider a diffeomorphism f of a compact manifold M which is Almost Axiom A, i.e. f is hyperbolic in a neighborhood of some compact f-invariant set, except in some singular set of neutral points. We prove that if there exists some f-invariant set of hyperbolic points with positive unstable-Lebesgue measure such that for every point in this set the stable and unstable leaves are \long enough", then f admits a probability SRB measure. Contents 1. Introduction 2 1.1. Background 2 1.2. Statement of results 2 1.3. Overview 4 2. Markov rectangles 5 2.1. Neighborhood of critical zone 5 2.2. First generation of rectangles 6 2.3. Second generation rectangles 7 2.4.
    [Show full text]
  • Bounding, Splitting and Almost Disjointness
    Bounding, splitting and almost disjointness Jonathan Schilhan Advisor: Vera Fischer Kurt G¨odelResearch Center Fakult¨atf¨urMathematik, Universit¨atWien Abstract This bachelor thesis provides an introduction to set theory, cardinal character- istics of the continuum and the generalized cardinal characteristics on arbitrary regular cardinals. It covers the well known ZFC relations of the bounding, the dominating, the almost disjointness and the splitting number as well as some more recent results about their generalizations to regular uncountable cardinals. We also present an overview on the currently known consistency results and formulate open questions. 1 CONTENTS Contents Introduction 3 1 Prerequisites 5 1.1 Model Theory/Predicate Logic . .5 1.2 Ordinals/Cardinals . .6 1.2.1 Ordinals . .7 1.2.2 Cardinals . .9 1.2.3 Regular Cardinals . 10 2 Cardinal Characteristics of the continuum 11 2.1 The bounding and the dominating number . 11 2.2 Splitting and mad families . 13 3 Cardinal Characteristics on regular uncountable cardinals 16 3.1 Generalized bounding, splitting and almost disjointness . 16 3.2 An unexpected inequality between the bounding and splitting numbers . 18 3.2.1 Elementary Submodels . 18 3.2.2 Filters and Ideals . 19 3.2.3 Proof of Theorem 3.2.1 (s(κ) ≤ b(κ)) . 20 3.3 A relation between generalized dominating and mad families . 23 References 28 2 Introduction @0 Since it was proven that the continuum hypothesis (2 = @1) is independent of the axioms of ZFC, the natural question arises what value 2@0 could take and it was shown @0 that 2 could be consistently nearly anything.
    [Show full text]
  • Almost-Free E-Rings of Cardinality ℵ1 Rudige¨ R Gob¨ El, Saharon Shelah and Lutz Strung¨ Mann
    Sh:785 Canad. J. Math. Vol. 55 (4), 2003 pp. 750–765 Almost-Free E-Rings of Cardinality @1 Rudige¨ r Gob¨ el, Saharon Shelah and Lutz Strung¨ mann Abstract. An E-ring is a unital ring R such that every endomorphism of the underlying abelian group R+ is multiplication by some ring element. The existence of almost-free E-rings of cardinality greater @ than 2 0 is undecidable in ZFC. While they exist in Godel'¨ s universe, they do not exist in other models @ of set theory. For a regular cardinal @1 ≤ λ ≤ 2 0 we construct E-rings of cardinality λ in ZFC which have @1-free additive structure. For λ = @1 we therefore obtain the existence of almost-free E-rings of cardinality @1 in ZFC. 1 Introduction + Recall that a unital ring R is an E-ring if the evaluation map ": EndZ(R ) ! R given by ' 7! '(1) is a bijection. Thus every endomorphism of the abelian group R+ is multiplication by some element r 2 R. E-rings were introduced by Schultz [20] and easy examples are subrings of the rationals Q or pure subrings of the ring of p-adic integers. Schultz characterized E-rings of finite rank and the books by Feigelstock [9, 10] and an article [18] survey the results obtained in the eighties, see also [8, 19]. In a natural way the notion of E-rings extends to modules by calling a left R-module M an E(R)-module or just E-module if HomZ(R; M) = HomR(R; M) holds, see [1].
    [Show full text]
  • The Axiom of Choice and Its Implications
    THE AXIOM OF CHOICE AND ITS IMPLICATIONS KEVIN BARNUM Abstract. In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial. Contents 1. Introduction 1 2. The Axiom of Choice and Its Equivalents 1 2.1. The Axiom of Choice and its Well-known Equivalents 1 2.2. Some Other Less Well-known Equivalents of the Axiom of Choice 3 3. Applications of the Axiom of Choice 5 3.1. Equivalence Between The Axiom of Choice and the Claim that Every Vector Space has a Basis 5 3.2. Some More Applications of the Axiom of Choice 6 4. Controversial Results 10 Acknowledgments 11 References 11 1. Introduction The Axiom of Choice states that for any family of nonempty disjoint sets, there exists a set that consists of exactly one element from each element of the family. It seems strange at first that such an innocuous sounding idea can be so powerful and controversial, but it certainly is both. To understand why, we will start by looking at some statements that are equivalent to the axiom of choice. Many of these equivalences are very useful, and we devote much time to one, namely, that every vector space has a basis. We go on from there to see a few more applications of the Axiom of Choice and its equivalents, and finish by looking at some of the reasons why the Axiom of Choice is so controversial.
    [Show full text]
  • Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous Krzysztof Ciesielski West Virginia University, [email protected]
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by The Research Repository @ WVU (West Virginia University) Faculty Scholarship 1995 Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous Krzysztof Ciesielski West Virginia University, [email protected] Follow this and additional works at: https://researchrepository.wvu.edu/faculty_publications Part of the Mathematics Commons Digital Commons Citation Ciesielski, Krzysztof, "Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous" (1995). Faculty Scholarship. 822. https://researchrepository.wvu.edu/faculty_publications/822 This Article is brought to you for free and open access by The Research Repository @ WVU. It has been accepted for inclusion in Faculty Scholarship by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Cardinal invariants concerning functions whose sum is almost continuous. Krzysztof Ciesielski1, Department of Mathematics, West Virginia University, Mor- gantown, WV 26506-6310 ([email protected]) Arnold W. Miller1, York University, Department of Mathematics, North York, Ontario M3J 1P3, Canada, Permanent address: University of Wisconsin-Madison, Department of Mathematics, Van Vleck Hall, 480 Lincoln Drive, Madison, Wis- consin 53706-1388, USA ([email protected]) Abstract Let A stand for the class of all almost continuous functions from R to R and let A(A) be the smallest cardinality of a family F ⊆ RR for which there is no g: R → R with the property that f + g ∈ A for all f ∈ F . We define cardinal number A(D) for the class D of all real functions with the Darboux property similarly.
    [Show full text]
  • Georg Cantor English Version
    GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle.
    [Show full text]
  • Almost-At-A-Distance-Accepted
    Almost at-a-distance Andrew McKenzie (Kansas) and Lydia Newkirk (Rutgers) accepted draft in Linguistics & Philosophy [email protected] [email protected] https://doi.org/10.1007/s10988-019-09275-6 Abstract We claim that the meaning of the adverbial almost contains both a scalar proximity measure and a modal that allows it to work sometimes when proximity fails, what we call the at-a-distance reading. Essentially, almost can hold if the proposition follows from the normal uninterrupted out- comes of adding a small enough number of premises to a selection of rel- evant facts. Almost at-a-distance is blocked when the temporal properties of the topic time and Davidsonian event prevent normal outcomes from coming true when they need to. This approach to almost differs from the two general approaches that have emerged in the literature, by replacing the negative polar condition (not p) with a positive antecedent condition that entails not p while avoid- ing the numerous well-documented complications of employing a polar condition. Since this approach to almost involves a circumstantial base with a non-interrupting ordering source, almost behaves in certain ways like the progressive, and shows contextual variability of the same kinds that we see with premise sets. 1 Introduction The adverbial almost has a clear intuitive meaning, but has proven very difficult to reconcile with a formal semantics. Two basic approaches characterize our understanding of almost in the literature. Under scalar alternative accounts, almost p holds when p is false but some close alternative proposition q is true. Modal closeness accounts argue that almost p holds of world w when p is false in w, but true in some close alternative world w0.
    [Show full text]
  • Equivalents to the Axiom of Choice and Their Uses A
    EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper.
    [Show full text]