DIGRAPHS in TERMS of SET THEORY 1. Sets and Ordered Pairs
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Rank of Weighted Digraphs with Blocks
Rank of weighted digraphs with blocks∗ Ranveer Singh† Swarup Kumar Panda ‡ Naomi Shaked-Monderer§ Abraham Berman¶ October 10, 2018 Abstract Let G be a digraph and r(G) be its rank. Many interesting results on the rank of an undirected graph appear in the literature, but not much information about the rank of a digraph is available. In this article, we study the rank of a digraph using the ranks of its blocks. In particular, we define classes of digraphs, namely r2-digraph, and r0-digraph, for which the rank can be exactly determined in terms of the ranks of subdigraphs of the blocks. Furthermore, the rank of directed trees, simple biblock graphs, and some simple block graphs are studied. Keywords: r2-digraphs, r0-digraphs, tree digraphs, block graphs, biblock graphs AMS Subject Classifications. 15A15, 15A18, 05C50 1 Introduction A well-known problem, proposed in 1957 by Collatz and Sinogowitz, is to characterize graphs with positive nullity [19]. Nullity of graphs is applicable in various branches of science, in particular, quantum chemistry, H¨uckel molecular orbital theory [10], [13] and social network theory [14]. For more detail on the applications see [10]. Many significant results on the nullity of undirected graphs are available in the literature, see [4, 7, 11, 12, 15, 12, 2, 3, 20]. Recently in [9, 8, 6] nullity of undirected graphs was studied using cut-vertices and connected components. The results are used to calculate the nullity of line graphs of undirected graphs. Apart from the connected components, it turns out that the blocks are also interesting subgraphs of a graph, which can be utilized to know its determinant [17, 16]. -
31 Vowel Digraph Oo
Sort Vowel Digraph oo 31 Objectives • To identify spelling patterns of vowel digraph oo Words • To read, sort, and write words with vowel digraph oo o˘o oˉo = uˉ Oddball brook nook fool spool could Materials for Within Word Pattern crook soot groom spoon should Big Book of Rhymes, “The Puppet Show,” page 55 foot stood hoop stool would hood wood noon tool Whiteboard Activities DVD-ROM, Sort 31 hook wool root troop Teacher Resource CD-ROM, Sort 31 and Follow the Dragon Game Student Book, pages 121–124 Words Their Way Library, The House That Stood on Booker Hill Introduce/Model Small Groups • Read a Rhyme Read “The Puppet Show,” and Extend the Sort emphasize words with the vowel digraph oo. (wood, book, took; soon, noon) Ask students to locate these Alternative Sort: Brainstorming words, and help them write the words in two columns, Ask students to think of other words that contain according to vowel sound. Help students hear the oo. Write their responses on index cards. When different pronunciations of the vowel digraph oo. students have completed brainstorming, ask them to identify and sort all the words they named • Model Use the whiteboard DVD or the CD word according to the vowel sound of oo. cards. Define in context any that may be unfamiliar to students. Demonstrate how to sort the words ELL English Language Learners according to the sound of the digraph oo. Point out Explain that a nook is “a hidden place,” and that that would, could, and should do not contain the groom can have several meanings, including “a digraph oo, so they belong in the oddball category. -
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. -
Locutour Guide to Letters, Sounds, and Symbols
LOCUTOUROUR® Guide to Letters, Sounds, and Symbols LOCUTOUR ABEL LASSIFICATION LACE OR RTICULATION PELLED AS XAMPLES S PELLINGP E L L I N G L C P A IPA S E Consonants p bilabial plosive voiceless lips /p/ p, pp pit, puppy b bilabial plosive voiced lips /b/ b, bb bat, ebb t lingua-alveolar plosive voiceless tongue tip + upper gum ridge /t/ t, ed, gth, th, tt, tw tan, tipped, tight, thyme, attic, two d lingua-alveolar plosive voiced tongue tip + upper gum ridge /d/ d, dd, ed dad, ladder, bagged k lingua-velar plosive voiceless back of tongue and soft palate /k/ c, cc, ch, ck cab, occur, school, duck que, k mosque, kit g lingua-velar plosive voiced back of tongue and soft palate /g/ g, gg, gh, gu, gue hug, bagged, Ghana, guy, morgue f labiodental fricative voiceless lower lip + upper teeth /f/ f, ff , gh, ph feet, fl u uff, enough, phone v labiodental fricative voiced lower lip + upper teeth /v/ v, vv, f, ph vet, savvy, of, Stephen th linguadental fricative voiceless tongue + teeth /T/ th thin th linguadental fricative voiced tongue + teeth /D/ th, the them, bathe s lingua-alveolar fricative voiceless tongue tip + upper gum ridge or /s/ s, ss, sc, ce, ci, sit, hiss, scenic, ace, city tongue tip + lower gum ridge cy, ps, z cycle, psychology, pizza z lingua-alveolar fricative voiced tongue tip + upper gum ridge or /z/ z, zz, s, ss, x, cz Zen, buzz, is, scissors, tongue tip + lower gum ridge Xerox, czar sh linguapalatal fricative voiceless tongue blade and hard palate /S/ sh, ce, ch, ci, sch, si, sheep, ocean, chef, glacier, kirsch ss, su, -
How Do Speakers of a Language with a Transparent Orthographic System
languages Article How Do Speakers of a Language with a Transparent Orthographic System Perceive the L2 Vowels of a Language with an Opaque Orthographic System? An Analysis through a Battery of Behavioral Tests Georgios P. Georgiou Department of Languages and Literature, University of Nicosia, Nicosia 2417, Cyprus; [email protected] Abstract: Background: The present study aims to investigate the effect of the first language (L1) orthography on the perception of the second language (L2) vowel contrasts and whether orthographic effects occur at the sublexical level. Methods: Fourteen adult Greek learners of English participated in two AXB discrimination tests: one auditory and one orthography test. In the auditory test, participants listened to triads of auditory stimuli that targeted specific English vowel contrasts embedded in nonsense words and were asked to decide if the middle vowel was the same as the first or the third vowel by clicking on the corresponding labels. The orthography test followed the same procedure as the auditory test, but instead, the two labels contained grapheme representations of the target vowel contrasts. Results: All but one vowel contrast could be more accurately discriminated in the auditory than in the orthography test. The use of nonsense words in the elicitation task Citation: Georgiou, Georgios P. 2021. eradicated the possibility of a lexical effect of orthography on auditory processing, leaving space for How Do Speakers of a Language with the interpretation of this effect on a sublexical basis, primarily prelexical and secondarily postlexical. a Transparent Orthographic System Conclusions: L2 auditory processing is subject to L1 orthography influence. Speakers of languages Perceive the L2 Vowels of a Language with transparent orthographies such as Greek may rely on the grapheme–phoneme correspondence to with an Opaque Orthographic System? An Analysis through a decode orthographic representations of sounds coming from languages with an opaque orthographic Battery of Behavioral Tests. -
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. -
Expansion of the Extipa and Voqs Combining Diacritics
Expansion of the extIPA and VoQS Kirk Miller, [email protected] Martin Ball, [email protected] 2020 Apr 14 This is a request for additions to Unicode to support recent expansion of the extIPA (Extensions to the IPA for Disordered Speech), including superscript modifier letters for unusual releases of plosives, and the VoQS (Voice Quality Symbols). The JIPA articles on the 2015 revision of the extIPA and the 2016 expansion of the VoQS are publicly available online from Cambridge University Press. (See references.) Thanks to Deborah Anderson of the Universal Scripts Project for her assistance. As summarized in the JIPA article, the motivation behind the original 1990 version of the extIPA symbols, and subsequent revisions to them, has been to supply transcriptional resources to those needing to describe disordered speech of all types. Symbols have been chosen following requests from speech-language pathologists and clinical phoneticians. As for the recent update, at the Oslo conference of the International Clinical Phonetics and Linguistics Association (ICPLA) in 2010, a special panel on the transcription of disordered speech reviewed the extIPA symbols and the chart that displays them, with input also from the audience at the panel presentation (Ball et al. 2010). The results of this review, together with informal consultations with colleagues, led to a set of suggested changes. These were presented in a poster in June 2016 at the ICPLA conference in Halifax, Nova Scotia (Ball et al. 2016), and approved by the membership. The changes were presented as a motion at the Business Meeting (all ICPLA members present at conference) and approved nem. -
Orthographies in Early Modern Europe
Orthographies in Early Modern Europe Orthographies in Early Modern Europe Edited by Susan Baddeley Anja Voeste De Gruyter Mouton An electronic version of this book is freely available, thanks to the support of libra- ries working with Knowledge Unlatched. KU is a collaborative initiative designed to make high quality books Open Access. More information about the initiative can be found at www.knowledgeunlatched.org An electronic version of this book is freely available, thanks to the support of libra- ries working with Knowledge Unlatched. KU is a collaborative initiative designed to make high quality books Open Access. More information about the initiative can be found at www.knowledgeunlatched.org ISBN 978-3-11-021808-4 e-ISBN (PDF) 978-3-11-021809-1 e-ISBN (EPUB) 978-3-11-021806-2 ISSN 0179-0986 e-ISSN 0179-3256 ThisISBN work 978-3-11-021808-4 is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License, ase-ISBN of February (PDF) 978-3-11-021809-1 23, 2017. For details go to http://creativecommons.org/licenses/by-nc-nd/3.0/. e-ISBN (EPUB) 978-3-11-021806-2 LibraryISSN 0179-0986 of Congress Cataloging-in-Publication Data Ae-ISSN CIP catalog 0179-3256 record for this book has been applied for at the Library of Congress. ISBN 978-3-11-028812-4 e-ISBNBibliografische 978-3-11-028817-9 Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliogra- fie;This detaillierte work is licensed bibliografische under the DatenCreative sind Commons im Internet Attribution-NonCommercial-NoDerivs über 3.0 License, Libraryhttp://dnb.dnb.deas of February of Congress 23, 2017.abrufbar. -
First : Arabic Transliteration Alphabet
E/CONF.105/137/CRP.137 13 July 2017 Original: English and Arabic Eleventh United Nations Conference on the Standardization of Geographical Names New York, 8-17 August 2017 Item 14 a) of the provisional agenda* Writing systems and pronunciation: Romanization Romanization System from Arabic letters to Latinized letters 2007 Submitted by the Arabic Division ** * E/CONF.105/1 ** Prepared by the Arabic Division Standard Arabic System for Transliteration of Geographical Names From Arabic Alphabet to Latin Alphabet (Arabic Romanization System) 2007 1 ARABIC TRANSLITERATION ALPHABET Arabic Romanization Romanization Arabic Character Character ٛ GH ؽٔيح ء > ف F ا } م Q ة B ى K د T ٍ L س TH ّ M ط J ٕ ػ N % ٛـ KH ؿ H ٝاُزبء أُوثٛٞخ ك٢ ٜٗب٣خ أٌُِخ W, Ū ٝ ك D ١ Y, Ī م DH a Short Opener ه R ā Long Opener ى Z S ً ā Maddah SH ُ ☺ Alif Maqsourah u Short Closer ٓ & ū Long Closer ٗ { ٛ i Short Breaker # ī Long Breaker ظ ! ّ ّلح Doubling the letter ع < - 1 - DESCRIPTION OF THE NEW ALPHABET How to describe the transliteration Alphabet: a. The new alphabet has neglected the following Latin letters: C, E, O, P, V, X in addition to the letter G unless it is coupled with the letter H to form a digraph GH .(اُـ٤ٖ Ghayn) b. This Alphabet contains: 1. Latin letters which have similar phonetic letters in Arabic : B,T,J,D,R,Z,S,Q,K,L,M,N,H,W,Y. ة، ،د، ط، ك، ه، ى، ً، م، ى، ٍ، ّ، ٕ، ٛـ، ٝ، ١ 2. -
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.