Defining Sets
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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]. -
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]). -
Enumeration of Finite Automata 1 Z(A) = 1
INFOI~MATION AND CONTROL 10, 499-508 (1967) Enumeration of Finite Automata 1 FRANK HARARY AND ED PALMER Department of Mathematics, University of Michigan, Ann Arbor, Michigan Harary ( 1960, 1964), in a survey of 27 unsolved problems in graphical enumeration, asked for the number of different finite automata. Re- cently, Harrison (1965) solved this problem, but without considering automata with initial and final states. With the aid of the Power Group Enumeration Theorem (Harary and Palmer, 1965, 1966) the entire problem can be handled routinely. The method involves a confrontation of several different operations on permutation groups. To set the stage, we enumerate ordered pairs of functions with respect to the product of two power groups. Finite automata are then concisely defined as certain ordered pah's of functions. We review the enumeration of automata in the natural setting of the power group, and then extend this result to enumerate automata with initial and terminal states. I. ENUMERATION THEOREM For completeness we require a number of definitions, which are now given. Let A be a permutation group of order m = ]A I and degree d acting on the set X = Ix1, x~, -.. , xa}. The cycle index of A, denoted Z(A), is defined as follows. Let jk(a) be the number of cycles of length k in the disjoint cycle decomposition of any permutation a in A. Let al, a2, ... , aa be variables. Then the cycle index, which is a poly- nomial in the variables a~, is given by d Z(A) = 1_ ~ H ~,~(°~ . (1) ~$ a EA k=l We sometimes write Z(A; al, as, .. -
Formal Construction of a Set Theory in Coq
Saarland University Faculty of Natural Sciences and Technology I Department of Computer Science Masters Thesis Formal Construction of a Set Theory in Coq submitted by Jonas Kaiser on November 23, 2012 Supervisor Prof. Dr. Gert Smolka Advisor Dr. Chad E. Brown Reviewers Prof. Dr. Gert Smolka Dr. Chad E. Brown Eidesstattliche Erklarung¨ Ich erklare¨ hiermit an Eides Statt, dass ich die vorliegende Arbeit selbststandig¨ verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Statement in Lieu of an Oath I hereby confirm that I have written this thesis on my own and that I have not used any other media or materials than the ones referred to in this thesis. Einverstandniserkl¨ arung¨ Ich bin damit einverstanden, dass meine (bestandene) Arbeit in beiden Versionen in die Bibliothek der Informatik aufgenommen und damit vero¨ffentlicht wird. Declaration of Consent I agree to make both versions of my thesis (with a passing grade) accessible to the public by having them added to the library of the Computer Science Department. Saarbrucken,¨ (Datum/Date) (Unterschrift/Signature) iii Acknowledgements First of all I would like to express my sincerest gratitude towards my advisor, Chad Brown, who supported me throughout this work. His extensive knowledge and insights opened my eyes to the beauty of axiomatic set theory and foundational mathematics. We spent many hours discussing the minute details of the various constructions and he taught me the importance of mathematical rigour. Equally important was the support of my supervisor, Prof. Smolka, who first introduced me to the topic and was there whenever a question arose. -
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. -
Python Mock Test
PPYYTTHHOONN MMOOCCKK TTEESSTT http://www.tutorialspoint.com Copyright © tutorialspoint.com This section presents you various set of Mock Tests related to Python. You can download these sample mock tests at your local machine and solve offline at your convenience. Every mock test is supplied with a mock test key to let you verify the final score and grade yourself. PPYYTTHHOONN MMOOCCKK TTEESSTT IIII Q 1 - What is the output of print tuple[2:] if tuple = ′abcd′, 786, 2.23, ′john′, 70.2? A - ′abcd′, 786, 2.23, ′john′, 70.2 B - abcd C - 786, 2.23 D - 2.23, ′john′, 70.2 Q 2 - What is the output of print tinytuple * 2 if tinytuple = 123, ′john′? A - 123, ′john′, 123, ′john′ B - 123, ′john′ * 2 C - Error D - None of the above. Q 3 - What is the output of print tinytuple * 2 if tinytuple = 123, ′john′? A - 123, ′john′, 123, ′john′ B - 123, ′john′ * 2 C - Error D - None of the above. Q 4 - Which of the following is correct about dictionaries in python? A - Python's dictionaries are kind of hash table type. B - They work like associative arrays or hashes found in Perl and consist of key-value pairs. C - A dictionary key can be almost any Python type, but are usually numbers or strings. Values, on the other hand, can be any arbitrary Python object. D - All of the above. Q 5 - Which of the following function of dictionary gets all the keys from the dictionary? A - getkeys B - key C - keys D - None of the above. -
Efficient Skyline Computation Over Low-Cardinality Domains
Efficient Skyline Computation over Low-Cardinality Domains MichaelMorse JigneshM.Patel H.V.Jagadish University of Michigan 2260 Hayward Street Ann Arbor, Michigan, USA {mmorse, jignesh, jag}@eecs.umich.edu ABSTRACT Hotel Parking Swim. Workout Star Name Available Pool Center Rating Price Current skyline evaluation techniques follow a common paradigm Slumber Well F F F ⋆ 80 that eliminates data elements from skyline consideration by find- Soporific Inn F T F ⋆⋆ 65 ing other elements in the dataset that dominate them. The perfor- Drowsy Hotel F F T ⋆⋆ 110 mance of such techniques is heavily influenced by the underlying Celestial Sleep T T F ⋆ ⋆ ⋆ 101 Nap Motel F T F ⋆⋆ 101 data distribution (i.e. whether the dataset attributes are correlated, independent, or anti-correlated). Table 1: A sample hotels dataset. In this paper, we propose the Lattice Skyline Algorithm (LS) that is built around a new paradigm for skyline evaluation on datasets the Soporific Inn. The Nap Motel is not in the skyline because the with attributes that are drawn from low-cardinality domains. LS Soporific Inn also contains a swimming pool, has the same number continues to apply even if one attribute has high cardinality. Many of stars as the Nap Motel, and costs less. skyline applications naturally have such data characteristics, and In this example, the skyline is being computed over a number of previous skyline methods have not exploited this property. We domains that have low cardinalities, and only one domain that is un- show that for typical dimensionalities, the complexity of LS is lin- constrained (the Price attribute in Table 1). -
Canonical Models for Fragments of the Axiom of Choice∗
Canonical models for fragments of the axiom of choice∗ Paul Larson y Jindˇrich Zapletal z Miami University University of Florida June 23, 2015 Abstract We develop a technology for investigation of natural forcing extensions of the model L(R) which satisfy such statements as \there is an ultrafil- ter" or \there is a total selector for the Vitali equivalence relation". The technology reduces many questions about ZF implications between con- sequences of the axiom of choice to natural ZFC forcing problems. 1 Introduction In this paper, we develop a technology for obtaining certain type of consistency results in choiceless set theory, showing that various consequences of the axiom of choice are independent of each other. We will consider the consequences of a certain syntactical form. 2 ! Definition 1.1. AΣ1 sentence Φ is tame if it is of the form 9A ⊂ ! (8~x 2 !! 9~y 2 A φ(~x;~y))^(8~x 2 A (~x)), where φ, are formulas which contain only numerical quantifiers and do not refer to A anymore, and may refer to a fixed analytic subset of 2! as a predicate. The formula are called the resolvent of the sentence. This is a syntactical class familiar from the general treatment of cardinal invari- ants in [11, Section 6.1]. It is clear that many consequences of Axiom of Choice are of this form: Example 1.2. The following statements are tame consequences of the axiom of choice: 1. there is a nonprincipal ultrafilter on !. The resolvent formula is \T rng(x) is infinite”; ∗2000 AMS subject classification 03E17, 03E40. -
Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 5-2014 Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected] Follow this and additional works at: http://scholarworks.boisestate.edu/ math_undergraduate_theses Part of the Set Theory Commons Recommended Citation Rahim, Farighon Abdul, "Axioms of Set Theory and Equivalents of Axiom of Choice" (2014). Mathematics Undergraduate Theses. Paper 1. Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Advisor: Samuel Coskey Boise State University May 2014 1 Introduction Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set. In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite. -
Session 5 – Main Theme
Database Systems Session 5 – Main Theme Relational Algebra, Relational Calculus, and SQL Dr. Jean-Claude Franchitti New York University Computer Science Department Courant Institute of Mathematical Sciences Presentation material partially based on textbook slides Fundamentals of Database Systems (6th Edition) by Ramez Elmasri and Shamkant Navathe Slides copyright © 2011 and on slides produced by Zvi Kedem copyight © 2014 1 Agenda 1 Session Overview 2 Relational Algebra and Relational Calculus 3 Relational Algebra Using SQL Syntax 5 Summary and Conclusion 2 Session Agenda . Session Overview . Relational Algebra and Relational Calculus . Relational Algebra Using SQL Syntax . Summary & Conclusion 3 What is the class about? . Course description and syllabus: » http://www.nyu.edu/classes/jcf/CSCI-GA.2433-001 » http://cs.nyu.edu/courses/fall11/CSCI-GA.2433-001/ . Textbooks: » Fundamentals of Database Systems (6th Edition) Ramez Elmasri and Shamkant Navathe Addition Wesley ISBN-10: 0-1360-8620-9, ISBN-13: 978-0136086208 6th Edition (04/10) 4 Icons / Metaphors Information Common Realization Knowledge/Competency Pattern Governance Alignment Solution Approach 55 Agenda 1 Session Overview 2 Relational Algebra and Relational Calculus 3 Relational Algebra Using SQL Syntax 5 Summary and Conclusion 6 Agenda . Unary Relational Operations: SELECT and PROJECT . Relational Algebra Operations from Set Theory . Binary Relational Operations: JOIN and DIVISION . Additional Relational Operations . Examples of Queries in Relational Algebra . The Tuple Relational Calculus . The Domain Relational Calculus 7 The Relational Algebra and Relational Calculus . Relational algebra . Basic set of operations for the relational model . Relational algebra expression . Sequence of relational algebra operations . Relational calculus . Higher-level declarative language for specifying relational queries 8 Unary Relational Operations: SELECT and PROJECT (1/3) . -
Notes on Sets, Relations, and Functions
Sets, Relations, and Functions S. F. Ellermeyer May 15, 2003 Abstract We give definitions of the concepts of Set, Relation, and Function, andlookatsomeexamples. 1Sets A set is a well—defined collection of objects. An example of a set is the set, A,defined by A = 1, 2, 5, 10 . { } The set A has four members (also called elements). The members of A are the numbers 1, 2, 5, and 10. Another example of a set is the set, B,defined by B = Arkansas, Hawaii, Michigan . { } The set B hasthreemembers—thestatesArkansas,Hawaii,andMichi- gan. In this course, we will restrict our attention to sets whose members are real numbers or ordered pairs of real numbers. An example of a set whose members are ordered pairs of real numbers is the set, C,defined by C = (6, 8) , ( 4, 7) , (5, 1) , (10, 10) . { − − } Note that the set C has four members. When describing a set, we never list any of its members more than once. Thus, the set 1, 2, 5, 5, 10 isthesameastheset 1, 2, 5, 10 . Actually, it is { } { } 1 not even correct to write this set as 1, 2, 5, 5, 10 because, in doing so, we are listing one of the members more than{ once. } If an object, x, is a member of a set, A,thenwewrite x A. ∈ This notation is read as “x is a member of A”, or as “x is an element of A” or as “x belongs to A”. If the object, x, is not a member of the set A,then we write x/A. -
ALGEBRA II 6.1 Mathematical Functions
Page 1 of 7 ALGEBRA II 6.1 mathematical functions: In mathematics, a function[1] is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. The input variable(s) are sometimes referred to as the argument(s) of the function. Functions of various kinds are "the central objects of investigation"[2] in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation. The input and output of a function can be expressed as an ordered pair, ordered so that the first element is the input (or tuple of inputs, if the function takes more than one input), and the second is the output. In the example above, f(x) = x2, we have the ordered pair (−3, 9).