Domino Recognizability of Triangular Picture Languages


 George Hodge
 1 years ago
 Views:
Transcription
1 Interntionl Journl of Computer Applictions ( ) Volume 57 No.5 Novemer 0 Domino Recognizility of ringulr icture Lnguges V. Devi Rjselvi Reserch Scholr Sthym University Chenni Klyni Hed of Deprtment of Mthemtics St. Joseph Institute of echnology Chenni D.G. homs Deprtment of Mthemtics Mdrs Christin College Chenni ABSRAC he notion of locl isotringulr picture lnguges nd recognizle isotringulr picture lnguges re introduced. Domino recognizility of isotringulr picture lnguges nd HRLdomino systems re defined. Also the concept tht recognizle isotringulr picture lnguges re chrcterized s projections of hrllocl tringulr picture lnguges is derived. heorems re proved. Keywords Iso tringulr domino system overlpping of iso tringulr pictures.hrl domino systems. INRODUCION A generliztion of forml lnguges to two dimensions is possile to different wys nd severl forml models to recognize or generte two dimensionl ojects hve een proposed in the literture. hese pproches were initilly motivted y prolems rising in the frmework of pttern recognition nd imge processing [3] ut two dimensionl ptterns re lso pper in studied concerning cellulr utomt nd other models of prllel computing [5]. Alredy notion of recognizility of set of pictures in terms of tiling systems is introduced [4]. he underlying ide is to define recognizility y projection of locl properties. Informlly recognition in tiling system is defined in terms of finite set of squre pictures of side two which correspond somehow to utomton trnsitions nd re clled tiles. In picture to e recognized (over the lphet ) ech qudruple of positions form squre to e covered y tile (with symols sy in the lphet ) such tht coherent ssignment of picture positions to lels in is uilt up nd such tht projection from to reestlishes the considered picture. hen the tiles cn e viewed s locl utomton trnsitions nd tiling given picture mens to construct run of the utomton on it []. he locl lnguges re lnguges given y finite set of uthorized tiles of size ( ). he use of locks of size ( ) implies tht in computtionl procedure to recognize given picture the horizontl nd verticl controls re done t the sme time. hen it is nturl to sk wht hppens when the two scnning re done seprtely nd in prticulr wht this cn imply when pply the projections fterwrds. In [] the so clled hvlocl picture lnguges re defined where the squre tiles of side re replced y dominoes tht correspond to two kinds of tiles: horizontl dominoes of size ( ) nd verticl dominoes of size ( ). In this pper the notion of domino systems to recognize isotringulr picture lnguges.. RELIMINARIES In this section some definitions of tiling systems re recollected [6]. Let e finite lphet of symols. A picture over is rectngulr rry of symols over. he set of ll pictures over is denoted y. Given p l (p) nd l (p) denote the numer of rows nd columns respectively of p. he pir (l (p) l (p)) is the size of p; p(i j) denotes the symol t row i nd column j i l (p) nd j l (p). A picture lnguge L is suset of. Let p e picture of size (m n). Let pˆ e the picture of size (m+ n+) otined y ordering p with specil symol. B hk (p) denotes the set of ll supictures of p of size (h k). A tile is picture of size ( ). mn denotes the set of ll pictures of size (m n) over the lphet. Here some sic concepts of isotringulr picture lnguges re given. Definition. An isotringulr picture p over the lphet is n isosceles tringulr rrngement of symols over. he set of ll isotringulr pictures over the lphet is denoted y Σ. An isotringulr picture lnguge over is suset of Σ. Given n isotringulr picture p the numer of rows (counting from the ottom to top) denoted y r(p) is the size of n isotringulr picture. he empty picture is denoted y. Isotringulr pictures cn e clssified into four ctegories.. Upper isotringulr picture. Lower isotringulr picture 3. Right isotringulr picture 4. Left isotringulr picture he upper tringulr isopicture cn e represented in the coordinte system s follows: A lower tringulr isopicture cn e represented in the coordinte system s follows: 6
2 Interntionl Journl of Computer Applictions ( ) Volume 57 No.5 Novemer 0 tiles) over n lphet nd projection : such tht (L) = L. he fmily of recognizle isotringulr picture lnguges will e denoted y IREC. Definition.8 An isotringulr tiling system is 4tuple ( ) where nd re finite set of symols : is projection nd is set of isotringulr tiles over the lphet {}. Definition. If p Σ then pˆ is the isotringulr picture otined y surrounding p with specil oundry symol. Definition.3 Let p Σ is n isotringulr picture. Let nd e two finite lphets nd : e mpping which is clled s rejection. he projection y mpping of isotringulr picture is the picture p such tht (p(i j k)) = p(i j k). Definition.4 Given n isotringulr picture p of size i for k i. Denote B k (p) the set of ll isotringulr supictures of p of size k. B (p) is in fct n isotringulr tile. Definition.5 Let L Σ e n isotringulr picture lnguge. projection of mpping of L is the lnguge (L) = {p / p = (p) p L}. he Definition.6 Let e finite lphet. An isotringulr picture lnguge L Σ is clled locl if there exists finite set of isotringulr tiles over {} such tht L = {p Σ / B ( pˆ ) }. he fmily of locl isotringulr picture lnguges will e denoted y ILOC. Exmple. Let = { } e finite lphet. he isotringulr picture lnguge L Σ is tiling recognizle if there exists tiling system = ( ) such tht L = (L()). It is denoted y L(). he fmily of isotringulr picture lnguges recognizle y isotringulr tiling system is denoted y L(IS). 3. DOMINO RECOGNIZABILIY OF ISORIANGULAR ICURES 3. Overlpping of isotringulr pictures Definition 3. Horizontl Overlpping he horizontl overlpping is etween U isotringulr picture nd D isotringulr picture of equl size nd denoted y the symol over. Exmple 3. Definition 3. Verticl Overlpping he verticl overlpping is defined etween L nd R isotringulr picture of sme size nd it is denoted y the symol over. Exmple 3. U over D = 3 3 Δ L over R = 3 hen L = L() =... he lnguge L() is the set of tringles with size k with lterntive nd in the rows. Clerly L() is locl. Definition.7 Let e finite lphet. An isotringulr picture lnguge L is clled recognizle if there exists isotringulr locl picture lnguge L (given y set of isotringulr Definition 3.3 Right Overlpping he right overlpping is defined etween ny two glule isotringulr pictures of sme size nd is denoted y the symol over. his overlpping includes the following. () D over U () R over U (c) D over L (d) R over L. Exmple 3.3 D over U = 3 3 D U 3 7
3 Interntionl Journl of Computer Applictions ( ) Volume 57 No.5 Novemer 0 Definition 3.4 Left Overlpping he left overlpping is defined etween ny two glule isotringulr pictures of sme size nd it is denoted y the symol over. his overlpping includes the following. () U over R () U over L (c) L over R (d) R over D. Exmple 3.4 R over D = Definition 3.5 he set of ll pictures otined y overlpping n isotringulr pictures of sme size is denoted y O(p). Here dominoes of the following types re considered. (i) Horizontl dominoes (ii) Right nd left dominoes. Horizontl Left dominoes Right Definition 3.6 Let L e n isotringulr picture lnguge. he lnguge L is hrllocl if there exists set of dominoes over the lphet {} such tht L = {p In this cse we write L = L(). Σ / O(B ( pˆ )) }. Definition 3.7 An isotringulr domino system (IDS) is 4tuple ( ) where nd re two finite lphets is finite set of dominoes over the lphet {} such tht : is projection. he isotringulr domino system recognized y n isotringulr picture lnguge L over the lphet nd is defined s L = (L) where L = L() is the hrllocl isotringulr picture lnguge over. he fmily of isotringulr picture lnguges recognized y isotringulr domino system is denoted y L(IDS). roposition 3. If L Γ is hrllocl iso tringulr picture lnguge then L is locl isotringulr picture lnguge. ht is L(IDS) L(IS). roof Let L Γ e hrllocl iso tringulr picture lnguge. hen L = L() where is finite set of dominoes. Here construct finite set of isotringulr tiles of size nd show tht L = L(). Define s follows θ Γ {} / θ Where the symol O denotes overlpping. Let L = L(). Now show tht L = L. Let p L then y definition Hence p L. B (pˆ) O(B (pˆ)) O(θ) Δ. Conversely let p L nd q B (pˆ ). hen O(θ) Δ. his implies tht O(q) O (B(pˆ)) Δ. herefore q nd p L. Hence L = L. Remrk 3. he converse of the roposition. is not true. ht is there re lnguges tht re in ILOC ut not in hrllocl. Lemm 3. Let L e locl isotringulr picture lnguge over n lphet. hen there exists n HRLlocl lnguge L over the lphet nd mpping : such tht L = (L). roof Let L = L() where is finite set of isotringulr tiles of size over {}. By definition contins ll llowed supictures of size of pictures in L. he ide of the proof is to show tht the property of eing n llowed supicture of size of picture in L y mens of domines over n lrger lphet cn e expressed. his is ccomplished y choosing s the set itself nd defining the set of dominoes. Let = ( {}) nd let = 3 3 c c c 3 f 3 3 c 3 d f f 3 d d d 3 c c 3 c c c 3 8
4 Interntionl Journl of Computer Applictions ( ) Volume 57 No.5 Novemer 0 d d d 3 c c c 3 c 3 f c 3 f f f 3 Let L = L(). hen define mpping etween the two lphets nd such tht 3 3 It is esy to verify tht p L nd (p) =. Conversely let p L nd let q B ( pˆ ) e supicture of pˆ of size. o prove tht (p) L. It suffices to show tht (q). Suppose the isotringulr picture q is the following o complete the proof first (L) = L to e proved. Before proving it formlly give n exmple to clrify how picture pl nd picture p L such tht (p) = p re relted. Suppose pˆ hen the corresponding picture pˆ will e the following pˆ 3 In the definition of pˆ severl different order symols re used. More precisely the order symols for re ll isotringulr tiles of size contining. Now L = (L) will e proved. Let p L e of size m. Consider picture pˆ over s follows q where ll the tiles 0 5 nd the four isotringulr tiles of size hen (q) =. Similrly q cn lso e ny lock (Diso tringulr tile) nd in this cse (q). heorem 3. L(IS) = L(IDS)
5 Interntionl Journl of Computer Applictions ( ) Volume 57 No.5 Novemer 0 roof he inclusion L(IDS) L(IS) is n immedite consequence of roposition 3.. he inverse inclusion follows from Lemm 3.. Before concluding n exmple s n ppliction of the theorem s given. Exmple 3. Consider the lnguge L of isotringulr picture over = { }. In the exmple given elow we oserve tht L L(IS). In order to show tht L L(IDS) it suffices to verify tht it cn e otined s projection of the lnguge L over = { } of tringles in which the medin of the tringle crry the symols nd nd the other symol crry the symol nd. It is cler tht L is hrllocl. In fct it is represented y the following set of dominoes. Now L = (L) where : is such tht ( ) = ( ) = nd ( ) = ( ) =. Hence L is recognizle y isotringulr domino system. ht is L L(IDS). 4. CONCLUSION In this pper the overlpping of isotringulr pictures hve een introduced nd the notion of recognizility of isotringulr pictures y new formlism clled domino system hve een investigted. he theorem L(IDS) = L(IS) is proved. ringulr picture lnguges cn generte ll pictures in picture lnguges. he lerning of isotringulr pictures nd unry isotringulr picture lnguges nd their complexity deserve to e studied further. 5. REFERENCES [] M. Ltteux nd D. Simplot Recognizle picture lnguges nd domino tiling Internl Report I Lortoire d Informtique Fond. [] D. Gimmrrsi wo dimensionl nd recognizle functions In roc. Developments in Lnguge heory Finlnd 993. [3] G. Rozenerg nd A. Slom (Eds) World Scientific ulishing Co. Singpore 994 pp [4]. Klyni V.R. Dre nd D.G. homs Locl nd Recognizle iso picture lnguges Lecture notes in Computer science 004 volume 336/004 pp [5] D. Gimmrresi nd A. Restivo Recognizle picture lnguges Interntionl Journl pttern Recognition nd Artificil Intelligence Specil Issue on prllel imge processing M. Nivt nd A. Soudi nd.s.. Wng (Eds) pp Also in roc. First Interntionl Colloquium on rllel imge processing 99 Sme journl vol. 6 No. & 3 pp [6] D. Gimmerresi nd A. Restivo wo dimensionl finite stte recognizility Fundment Informtice Specil Issue: Forml lnguge heory volume 5 no. 3 4 (996) pp
1 Nondeterministic Finite Automata
1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
More informationChapter 2 Finite Automata
Chpter 2 Finite Automt 28 2.1 Introduction Finite utomt: first model of the notion of effective procedure. (They lso hve mny other pplictions). The concept of finite utomton cn e derived y exmining wht
More informationAutomata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ωautomata. Ralf Huuck.
Outline Automt Theory 101 Rlf Huuck Introduction Finite Automt Regulr Expressions ωautomt Session 1 2006 Rlf Huuck 1 Session 1 2006 Rlf Huuck 2 Acknowledgement Some slides re sed on Wolfgng Thoms excellent
More information12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016
CS125 Lecture 12 Fll 2016 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More informationHow to simulate Turing machines by invertible onedimensional cellular automata
How to simulte Turing mchines by invertible onedimensionl cellulr utomt JenChristophe Dubcq Déprtement de Mthémtiques et d Informtique, École Normle Supérieure de Lyon, 46, llée d Itlie, 69364 Lyon Cedex
More information3 Regular expressions
3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll
More informationRegular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene*
Regulr Expressions (RE) Regulr Expressions (RE) Empty set F A RE denotes the empty set Opertion Nottion Lnguge UNIX Empty string A RE denotes the set {} Alterntion R +r L(r ) L(r ) r r Symol Alterntion
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationCS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)
CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts
More informationTypes of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt
More informationState Minimization for DFAs
Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid
More informationTriangles The following examples explore aspects of triangles:
Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x  4 +x xmple 3: ltitude of the
More informationWENJUN LIU AND QUÔ C ANH NGÔ
AN OSTROWSKIGRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of OstrowskiGrüss type on time scles nd thus unify corresponding continuous
More informationChapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)
C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,
More informationGrammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages
5//6 Grmmr Automt nd Lnguges Regulr Grmmr Contextfree Grmmr Contextsensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationFormal Languages and Automata Theory. D. Goswami and K. V. Krishna
Forml Lnguges nd Automt Theory D. Goswmi nd K. V. Krishn Novemer 5, 2010 Contents 1 Mthemticl Preliminries 3 2 Forml Lnguges 4 2.1 Strings............................... 5 2.2 Lnguges.............................
More informationSpeech Recognition Lecture 2: Finite Automata and FiniteState Transducers
Speech Recognition Lecture 2: Finite Automt nd FiniteStte Trnsducers Eugene Weinstein Google, NYU Cournt Institute eugenew@cs.nyu.edu Slide Credit: Mehryr Mohri Preliminries Finite lphet, empty string.
More informationIntuitionistic Fuzzy Lattices and Intuitionistic Fuzzy Boolean Algebras
Intuitionistic Fuzzy Lttices nd Intuitionistic Fuzzy oolen Algebrs.K. Tripthy #1, M.K. Stpthy *2 nd P.K.Choudhury ##3 # School of Computing Science nd Engineering VIT University Vellore632014, TN, Indi
More informationParallel Projection Theorem (Midpoint Connector Theorem):
rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length onehlf the third side. onversely, If line isects
More informationHomework Solution  Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution  et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte nonfinl.
More informationRectangular group congruences on an epigroup
cholrs Journl of Engineering nd Technology (JET) ch J Eng Tech, 015; 3(9):73736 cholrs Acdemic nd cientific Pulisher (An Interntionl Pulisher for Acdemic nd cientific Resources) wwwsspulishercom IN 31435X
More informationLecture 9: LTL and Büchi Automata
Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled
More informationa,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1
CS4 45 Determinisitic Finite Automt : Genertors vs. Checkers Regulr expressions re one wy to specify forml lnguge String Genertor Genertes strings in the lnguge Deterministic Finite Automt (DFA) re nother
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationexpression simply by forming an OR of the ANDs of all input variables for which the output is
2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output
More informationSticker Systems over Permutation Groups
World Applied Sciences Journl 1 (Specil Issue of Applied Mth): 11916, 01 ISSN 1818495; IDOSI Pulictions, 01 DOI: 10.589/idosi.wsj.01.1.m.11 Sticker Systems over Permuttion Groups 1 N.A. Mohd Sery, N.H.
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCKKURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When relvlued
More informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationOn Determinisation of HistoryDeterministic Automata.
On Deterministion of HistoryDeterministic Automt. Denis Kupererg Mich l Skrzypczk University of Wrsw YRICALP 2014 Copenhgen Introduction Deterministic utomt re centrl tool in utomt theory: Polynomil
More informationCS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation
CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes
More informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More informationLearning Goals. Relational Query Languages. Formal Relational Query Languages. Formal Query Languages: Relational Algebra and Relational Calculus
Forml Query Lnguges: Reltionl Alger nd Reltionl Clculus Chpter 4 Lerning Gols Given dtse ( set of tles ) you will e le to express dtse query in Reltionl Alger (RA), involving the sic opertors (selection,
More informationx = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :
Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given
More informationNondeterministic Finite Automata
Nondeterministic Finite Automt From Regulr Expressions to NFA Eliminting nondeterminism Rdoud University Nijmegen Nondeterministic Finite Automt H. Geuvers nd J. Rot Institute for Computing nd Informtion
More informationThe Caucal Hierarchy of Infinite Graphs in Terms of Logic and Higherorder Pushdown Automata
The Cucl Hierrchy of Infinite Grphs in Terms of Logic nd Higherorder Pushdown Automt Arnud Cryol 1 nd Stefn Wöhrle 2 1 IRISA Rennes, Frnce rnud.cryol@iris.fr 2 Lehrstuhl für Informtik 7 RWTH Achen, Germny
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationSTRAND B: NUMBER THEORY
Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,
More informationGeneral Algorithms for Testing the Ambiguity of Finite Automata
Generl Algorithms for Testing the Amiguity of Finite Automt Cyril Alluzen 1,, Mehryr Mohri 2,1, nd Ashish Rstogi 1, 1 Google Reserch, 76 Ninth Avenue, New York, NY 10011. 2 Cournt Institute of Mthemticl
More information[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves
Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.
More informationOn Decentralized Observability of Discrete Event Systems
2011 50th IEEE Conference on Decision nd Control nd Europen Control Conference (CDCECC) Orlndo, FL, USA, Decemer 1215, 2011 On Decentrlized Oservility of Discrete Event Systems M.P. Csino, A. Giu, C.
More informationSeparating Regular Languages with FirstOrder Logic
Seprting Regulr Lnguges with FirstOrder Logic Thoms Plce Mrc Zeitoun LBRI, Bordeux University, Frnce firstnme.lstnme@lri.fr Astrct Given two lnguges, seprtor is third lnguge tht contins the first one
More informationNew Integral Inequalities for ntime Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353362 New Integrl Inequlities for ntime Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More informationHomework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)
More informationA new algorithm for generating Pythagorean triples 1
A new lgorithm for generting Pythgoren triples 1 RH Dye 2 nd RWD Nicklls 3 The Mthemticl Gzette (1998; 82 (Mrch, No. 493, pp. 86 91 http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf 1 Introduction
More informationNONDETERMINISTIC FSA
Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is
More informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4305 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
More informationPreview 11/1/2017. Greedy Algorithms. Coin Change. Coin Change. Coin Change. Coin Change. Greedy algorithms. Greedy Algorithms
Preview Greed Algorithms Greed Algorithms Coin Chnge Huffmn Code Greed lgorithms end to e simple nd strightforwrd. Are often used to solve optimiztion prolems. Alws mke the choice tht looks est t the moment,
More informationComplementing Büchi Automata with a Subsettuple Construction
DEPARTEMENT D INFORMATIQUE DEPARTEMENT FÜR INFORMATIK Bd de Pérolles 90 CH1700 Friourg www.unifr.ch/informtics WORKING PAPER Complementing Büchi Automt with Susettuple Construction J. Allred & U. UltesNitsche
More informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationPrefixFree RegularExpression Matching
PrefixFree RegulrExpression Mthing YoSu Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST PrefixFree RegulrExpression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationGeneral Algorithms for Testing the Ambiguity of Finite Automata and the DoubleTape Ambiguity of FiniteState Transducers
Interntionl Journl of Foundtions of Computer Science c World Scientific Pulishing Compny Generl Algorithms for Testing the Amiguity of Finite Automt nd the DouleTpe Amiguity of FiniteStte Trnsducers
More informationSection: Other Models of Turing Machines. Definition: Two automata are equivalent if they accept the same language.
Section: Other Models of Turing Mchines Definition: Two utomt re equivlent if they ccept the sme lnguge. Turing Mchines with Sty Option Modify δ, Theorem Clss of stndrd TM s is equivlent to clss of TM
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
More informationSet Integral Equations in Metric Spaces
Mthemtic Morvic Vol. 131 2009, 95 102 Set Integrl Equtions in Metric Spces Ion Tişe Abstrct. Let P cp,cvr n be the fmily of ll nonempty compct, convex subsets of R n. We consider the following set integrl
More informationCommunications inmathematicalanalysis Volume 6, Number 2, pp (2009) ISSN
Communictions inmthemticlanlysis Volume 6, Number, pp. 33 41 009) ISSN 19389787 www.communmthnl.org A SHARP GRÜSS TYPE INEQUALITY ON TIME SCALES AND APPLICATION TO THE SHARP OSTROWSKIGRÜSS INEQUALITY
More informationUniversitaireWiskundeCompetitie. Problem 2005/4A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationOverview HC9. Parsing: TopDown & LL(1) ContextFree Grammars (1) Introduction. CFGs (3) ContextFree Grammars (2) Vertalerbouw HC 9: Ch.
Overview H9 Vertlerouw H 9: Prsing: opdown & LL(1) do 3 mei 2001 56 heo Ruys h. 8  Prsing 8.1 ontextfree Grmmrs 8.2 opdown Prsing 8.3 LL(1) Grmmrs See lso [ho, Sethi & Ullmn 1986] for more thorough
More informationCOMPOSITIONALITY AND REACHABILITY WITH CONDITIONS ON PATH LENGTHS
compositionlity Interntionl Journl of Foundtions of Computer Science c World Scientific Pulishing Compny COMPOSITIONALITY AND REACHABILITY WITH CONDITIONS ON PATH LENGTHS INGO FELSCHER Lehrstuhl Informtik
More informationCharacterizations of periods of multidimensional shifts
Chrcteriztions of periods of multidimensionl shifts Emmnuel Jendel, Pscl Vnier To cite this version: Emmnuel Jendel, Pscl Vnier. Chrcteriztions of periods of multidimensionl shifts. 2013.
More informationLIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon
LIP Lortoire de l Informtique du Prllélisme Eole Normle Supérieure de Lyon Institut IMAG Unité de reherhe ssoiée u CNRS n 1398 Onewy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Mrs 1993 Reserh Report N
More informationA Note on Conic Sections and Tangent Circles
Forum Geometricorum Volume 17 017 1 1. FORUM GEOM ISSN 1531178 A Note on Conic Sections nd Tngent Circles Jn Kristin Huglnd Astrct. This rticle presents result on circles tngent to given conic section
More informationLesson Notes: Week 40Vectors
Lesson Notes: Week 40Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples
More informationHow Deterministic are GoodForGames Automata?
How Deterministic re GoodForGmes Automt? Udi Boker 1, Orn Kupfermn 2, nd Mich l Skrzypczk 3 1 Interdisciplinry Center, Herzliy, Isrel 2 The Herew University, Isrel 3 University of Wrsw, Polnd Astrct
More informationQuadratic reciprocity
Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition
More informationAnalytically, vectors will be represented by lowercase boldface Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More information5.1 Definitions and Examples 5.2 Deterministic Pushdown Automata
CSC4510 AUTOMATA 5.1 Definitions nd Exmples 5.2 Deterministic Pushdown Automt Definitions nd Exmples A lnguge cn be generted by CFG if nd only if it cn be ccepted by pushdown utomton. A pushdown utomton
More informationINF1383 Bancos de Dados
3//0 INF383 ncos de Ddos Prof. Sérgio Lifschitz DI PUCRio Eng. Computção, Sistems de Informção e Ciênci d Computção LGER RELCIONL lguns slides sedos ou modificdos dos originis em Elmsri nd Nvthe, Fundmentls
More informationdy ky, dt where proportionality constant k may be positive or negative
Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.
More informationarxiv: v1 [math.ca] 28 Jan 2013
ON NEW APPROACH HADAMARDTYPE INEQUALITIES FOR sgeometrically CONVEX FUNCTIONS rxiv:3.9v [mth.ca 8 Jn 3 MEVLÜT TUNÇ AND İBRAHİM KARABAYIR Astrct. In this pper we chieve some new Hdmrd type ineulities
More informationCHAPTER 1 CENTRES OF MASS
1.1 Introduction, nd some definitions. 1 CHAPTER 1 CENTRES OF MASS This chpter dels with the clcultion of the positions of the centres of mss of vrious odies. We strt with rief eplntion of the mening of
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationResearch Article Determinant Representations of Polynomial Sequences of Riordan Type
Discrete Mthemtics Volume 213, Article ID 734836, 6 pges http://dxdoiorg/11155/213/734836 Reserch Article Determinnt Representtions of Polynomil Sequences of Riordn Type Shengling Yng nd Sinn Zheng Deprtment
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. LmiAthens Lmi 3500 Greece Abstrct Using
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More information8 Automata and formal languages. 8.1 Formal languages
8 Automt nd forml lnguges This exposition ws developed y Clemens Gröpl nd Knut Reinert. It is sed on the following references, ll of which re recommended reding: 1. Uwe Schöning: Theoretische Informtik
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationAutomata for Analyzing and Querying Compressed Documents Barbara FILA, LIFO, Orl eans (Fr.) Siva ANANTHARAMAN, LIFO, Orl eans (Fr.) Rapport No
Automt for Anlyzing nd Querying Compressed Documents Brr FILA, LIFO, Orléns (Fr.) Siv ANANTHARAMAN, LIFO, Orléns (Fr.) Rpport N o 200603 Automt for Anlyzing nd Querying Compressed Documents Brr Fil, Siv
More informationON A CERTAIN PRODUCT OF BANACH ALGEBRAS AND SOME OF ITS PROPERTIES
HE PULISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY Series A OF HE ROMANIAN ACADEMY Volume 5 Number /04 pp. 9 7 ON A CERAIN PRODUC OF ANACH ALGERAS AND SOME OF IS PROPERIES Hossossein JAVANSHIRI Mehdi
More informationThe Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5
The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, BbeşBolyi University Str. Koglnicenu
More informationChapter 1, Part 1. Regular Languages. CSC527, Chapter 1, Part 1 c 2012 Mitsunori Ogihara 1
Chpter 1, Prt 1 Regulr Lnguges CSC527, Chpter 1, Prt 1 c 2012 Mitsunori Ogihr 1 Finite Automt A finite utomton is system for processing ny finite sequence of symols, where the symols re chosen from finite
More informationDiophantine Steiner Triples and PythagoreanType Triangles
Forum Geometricorum Volume 10 (2010) 93 97. FORUM GEOM ISSN 15341178 Diophntine Steiner Triples nd PythgorenType Tringles ojn Hvl bstrct. We present connection between Diophntine Steiner triples (integer
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More informationGoodforGames Automata versus Deterministic Automata.
GoodforGmes Automt versus Deterministic Automt. Denis Kuperberg 1,2 Mich l Skrzypczk 1 1 University of Wrsw 2 IRIT/ONERA (Toulouse) Séminire MoVe 12/02/2015 LIF, Luminy Introduction Deterministic utomt
More informationHarmonic Mean Derivative  Based Closed Newton Cotes Quadrature
IOSR Journl of Mthemtics (IOSRJM) eissn:  pissn: 9X. Volume Issue Ver. IV (My.  Jun. 0) PP  www.iosrjournls.org Hrmonic Men Derivtive  Bsed Closed Newton Cotes Qudrture T. Rmchndrn D.Udykumr nd
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 18, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationLecture V. Introduction to Space Groups Charles H. Lake
Lecture V. Introduction to Spce Groups 2003. Chrles H. Lke Outline:. Introduction B. Trnsltionl symmetry C. Nomenclture nd symols used with spce groups D. The spce groups E. Derivtion nd discussion of
More informationGRADE 4. Division WORKSHEETS
GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.
More informationRefined interfaces for compositional verification
Refined interfces for compositionl verifiction Frédéric Lng INRI Rhônelpes http://www.inrilpes.fr/vsy Motivtion Enumertive verifiction of concurrent systems Prllel composition of synchronous processes
More informationFinite Nondeterministic Automata: Simulation and Minimality
Finite Nondeterministic Automt: Simultion nd Minimlity Cristin S. Clude, Elen Clude, Bkhdyr Khoussinov Abstrct Motivted by recent pplictions of finite utomt to theoreticl physics, we study the minimiztion
More information