Review of CFGs and Parsing I Context-free Languages and Grammars. Winter 2014 Costas Busch - RPI 1

Size: px
Start display at page:

Download "Review of CFGs and Parsing I Context-free Languages and Grammars. Winter 2014 Costas Busch - RPI 1"

Transcription

1 Review of CFGs d Prsig I Cotext-free Lguges d Grmmrs Witer 2014 Costs Busch - RPI 1

2 Cotext-Free Lguges { b : { ww } 0} R Regulr Lguges *b* ( b) * Witer 2014 Costs Busch - RPI 2

3 Cotext-Free Lguges Cotext-Free Grmmrs Pushdow Automt stck utomto Witer 2014 Costs Busch - RPI 3

4 Cotext-Free Grmmrs Witer 2014 Costs Busch - RPI 4

5 Forml Defiitios Grmmr: G = ( V, T,, P) et of Vribles/ No-termil et of termil trt ymbol et of productios symbols Witer 2014 Costs Busch - RPI 5

6 Cotext-Free Grmmr: G = ( V, T,, P) All productios i P re of the form A s No-termil trig of No-termil d termils Witer 2014 Costs Busch - RPI 6

7 xmple of Cotext-Free Grmmr b λ productios P = { b, λ} G = ( V, T,, P) V = {} vribles T = {, b} termils strt vrible Witer 2014 Costs Busch - RPI 7

8 Lguge of Grmmr: For grmmr G with strt vrible * L( G) = { w : w, w T*} trig of termils or λ Witer 2014 Costs Busch - RPI 8

9 xmple equece of termils d o-termils Grmmr: b λ No-termils The right side my be λ Witer 2014 Costs Busch - RPI 9

10 Grmmr: b λ Derivtio of strig : b b b b λ Witer 2014 Costs Busch - RPI 10

11 Grmmr: b λ bb Derivtio of strig : b bb bb b λ Witer 2014 Costs Busch - RPI 11

12 Grmmr: b λ Lguge of the grmmr: L = { b : 0} Witer 2014 Costs Busch - RPI 12

13 A Coveiet Nottio We write: * bbb for zero or more derivtio steps Isted of: b bb bbb bbb Witer 2014 Costs Busch - RPI 13

14 I geerl we write: w * 1 w If: w 1 w 2 w 3 w i zero or more derivtio steps Trivilly: w * w Witer 2014 Costs Busch - RPI 14

15 Cotext-Free Lguge: A lguge is cotext-free if there is cotext-free grmmr with L L = L(G) G Witer 2014 Costs Busch - RPI 15

16 xmple: L = b { : 0} is cotext-free lguge sice cotext-free grmmr : G b λ geertes L ( G) = L Witer 2014 Costs Busch - RPI 16

17 Aother xmple Cotext-free grmmr G : bb λ xmple derivtios: bb bb bb bb bb L(G) = { ww R : w {, b}*} Plidromes of eve legth Witer 2014 Costs Busch - RPI 17

18 Aother xmple Cotext-free grmmr G : b λ xmple derivtios: b b b b b bb bb L(G) = Describes mtched pretheses: { w : d ( w) = i y ( v) ( w), prefix ( v) v} Witer 2014 Costs Busch - RPI 18 b b () ((( ))) (( )) = (, b = )

19 Derivtio Order d Derivtio Trees Witer 2014 Costs Busch - RPI 19

20 Derivtio Order Cosider the followig exmple grmmr with 5 productios: 1. AB 2. A A 4. B Bb 3. A λ 5. B λ Witer 2014 Costs Busch - RPI 20

21 1. AB 2. A A 4. B Bb 3. A λ 5. B λ Leftmost derivtio order of strig : b ABABBBbb 4 5 At ech step, we substitute the leftmost vrible Witer 2014 Costs Busch - RPI 21

22 1. AB 2. A A 4. B Bb 3. A λ 5. B λ Rightmost derivtio order of strig : b AB ABb AbAbb At ech step, we substitute the rightmost vrible Witer 2014 Costs Busch - RPI 22

23 1. AB 2. A A 4. B Bb 3. A λ 5. B λ Leftmost derivtio of : b ABABBBbb 4 5 Rightmost derivtio of b : AB ABb 3 AbAbb Witer 2014 Costs Busch - RPI 23

24 Derivtio Trees Cosider the sme exmple grmmr: AB A A λ B Bb λ Ad derivtio of : b AB AB ABb Bb b Witer 2014 Costs Busch - RPI 24

25 AB A A λ B Bb λ AB A B yield AB Witer 2014 Costs Busch - RPI 25

26 AB A A λ B Bb λ AB AB A B A yield AB Witer 2014 Costs Busch - RPI 26

27 AB A A λ B Bb λ AB AB ABb A B A B b yield ABb Witer 2014 Costs Busch - RPI 27

28 AB A A λ B Bb λ AB AB ABb Bb A B A B b λ yield λbb = Bb Witer 2014 Costs Busch - RPI 28

29 AB A A λ B Bb λ AB AB ABb Bb b Derivtio Tree (prse tree) A B A B b λ yield λλb = b Witer 2014 Costs Busch - RPI 29 λ

30 ometimes, derivtio order does t mtter Leftmost derivtio: AB AB B Bb b Rightmost derivtio: AB ABb Ab Ab b Give sme derivtio tree A B A B b Witer 2014 Costs Busch - RPI 30 λ λ

31 Ambiguity Witer 2014 Costs Busch - RPI 31

32 Grmmr for mthemticl expressios ( ) xmple strigs: ( ) ( ( )) Deotes y umber Witer 2014 Costs Busch - RPI 32

33 Witer 2014 Costs Busch - RPI 33 A leftmost derivtio for * ) (

34 Witer 2014 Costs Busch - RPI 34 Aother leftmost derivtio for ) (

35 Witer 2014 Costs Busch - RPI 35 Two derivtio trees for ) (

36 tke = 2 = Witer 2014 Costs Busch - RPI 36

37 Good Tree Bd Tree = = Compute expressio result usig the tree Witer 2014 Costs Busch - RPI 37

38 Two differet derivtio trees my cuse problems i pplictios which use the derivtio trees: vlutig expressios I geerl, i compilers for progrmmig lguges Witer 2014 Costs Busch - RPI 38

39 Ambiguous Grmmr: A cotext-free grmmr if there is strig G w L(G) is mbiguous which hs: two differet derivtio trees or two leftmost derivtios (Two differet derivtio trees give two differet leftmost derivtios d vice-vers) Witer 2014 Costs Busch - RPI 39

40 Witer 2014 Costs Busch - RPI 40 strig hs two derivtio trees ) ( this grmmr is mbiguous sice xmple:

41 Witer 2014 Costs Busch - RPI 41 strig hs two leftmost derivtios * ) ( this grmmr is mbiguous lso becuse

42 Aother mbiguous grmmr: IF_TMT if XPR the TMT if XPR the TMT else TMT Vribles Termils Very commo piece of grmmr i progrmmig lguges Witer 2014 Costs Busch - RPI 42

43 If expr1 the if expr2 the stmt1 else stmt2 IF_TMT if expr1 the TMT if expr2 the stmt1 else stmt2 IF_TMT Two derivtio trees if expr1 the TMT else stmt2 if expr2 the stmt1 Witer 2014 Costs Busch - RPI 43

44 I geerl, mbiguity is bd d we wt to remove it ometimes it is possible to fid o-mbiguous grmmr for lguge But, i geerl we cot do so Witer 2014 Costs Busch - RPI 44

45 A successful exmple: Ambiguous Grmmr ( ) quivlet No-Ambiguous Grmmr T T T T F F F ( ) geertes the sme lguge Witer 2014 Costs Busch - RPI 45

46 Witer 2014 Costs Busch - RPI 46 F F F F T T T F T T T T T F F T F F F F T T T T ) ( Uique derivtio tree for

47 A u-successful exmple: L = { b c m } { b m c m }, m 0 L is iheretly mbiguous: every grmmr tht geertes this lguge is mbiguous Witer 2014 Costs Busch - RPI 47

48 Witer 2014 Costs Busch - RPI 48 } { } { m m m c b c b L = λ 1 1 Ab A A c λ 2 2 bbc B B 1 2 xmple (mbiguous) grmmr for : L

49 Delig with Ambiguity There re severl wys to hdle mbiguity Most direct method is to rewrite grmmr umbiguously ' ' id ʹ id () ʹ () forces precedece of * over ' Professor Alex Aike Lecture #5 49

50 Ambiguity No geerl techiques for hdlig mbiguity Impossible to covert utomticlly mbiguous grmmr to umbiguous oe Used with cre, mbiguity c simplify the grmmr. ometimes llows more turl defiitios. However, we still eed to evetully elimite mbiguity. How? 50

51 Precedece d Associtivity Declrtios Isted of rewritig the grmmr Use the more turl (mbiguous) grmmr Alog with dismbigutig declrtios Most prser-geertors llow precedece d ssocitivity declrtios to dismbigute grmmrs 51

52 Associtivity Declrtios Ambiguous two prse trees of it it it Left ssocitivity declrtio: %left it it it it it it 52

53 Precedece Declrtios Cosider the strig it it * it Precedece declrtios: %left %left * * it it * it it it it 53

54 Properties of Cotext-Free lguges Witer 2014 Costs Busch - RPI 54

55 Uio Cotext-free lguges re closed uder: Uio L1 is cotext free L2 is cotext free L1 L 2 is cotext-free Witer 2014 Costs Busch - RPI 55

56 xmple Lguge Grmmr L = 1 { b } 1 1b λ L = 2 R { ww } 2 2 b2b λ Uio L = { b } { ww R } 1 2 Witer 2014 Costs Busch - RPI 56

57 I geerl: For cotext-free lguges with cotext-free grmmrs d strt vribles L 1, L 2 G 1, G 2 1, 2 The grmmr of the uio hs ew strt vrible 1 L 2 L d dditiol productio 1 2 Witer 2014 Costs Busch - RPI 57

58 Coctetio Cotext-free lguges re closed uder: Coctetio L1 is cotext free L2 is cotext free L 1 L 2 is cotext-free Witer 2014 Costs Busch - RPI 58

59 xmple Lguge Grmmr L = 1 { b } 1 1b λ L = 2 R { ww } 2 2 b2b λ L = { b Coctetio }{ ww R } 1 2 Witer 2014 Costs Busch - RPI 59

60 I geerl: For cotext-free lguges with cotext-free grmmrs d strt vribles L 1, L 2 G 1, G 2 1, 2 The grmmr of the coctetio hs ew strt vrible L 1 L 2 d dditiol productio 1 2 Witer 2014 Costs Busch - RPI 60

61 tr Opertio Cotext-free lguges re closed uder: tr-opertio L is cotext free * L is cotext-free Witer 2014 Costs Busch - RPI 61

62 xmple Lguge Grmmr L = { b } b λ tr Opertio L = { b }* 1 1 λ Witer 2014 Costs Busch - RPI 62

63 I geerl: For cotext-free lguge with cotext-free grmmr d strt vrible L G The grmmr of the str opertio hs ew strt vrible 1 d dditiol productio 1 1 L* λ Witer 2014 Costs Busch - RPI 63

64 Cotext-free Lguges do t hve the followig Properties Witer 2014 Costs Busch - RPI 64

65 Itersectio Cotext-free lguges re ot closed uder: itersectio L1 L2 is cotext free is cotext free L1 L 2 ot ecessrily cotext-free Witer 2014 Costs Busch - RPI 65

66 L = { 1 b c m Cotext-free: } xmple L = { 2 b m c m Cotext-free: } AC AB A Ab λ A A λ C cc λ B bbc λ Itersectio L L = 1 2 { b c } NOT cotext-free Witer 2014 Costs Busch - RPI 66

67 Complemet Cotext-free lguges re ot closed uder: complemet L is cotext free L ot ecessrily cotext-free. I.e., the complemet of cotext-free lguge my or my ot be cotext-free. Witer 2014 Costs Busch - RPI 67

68 L = { 1 b c m Cotext-free: } xmple L = { 2 b m c m Cotext-free: } AC AB A Ab λ A A λ C cc λ B bbc λ 1 Complemet L2 L1 L2 { NOT cotext-free L = = Witer 2014 Costs Busch - RPI 68 b c }

69 Itersectio of Cotext-free lguges d Regulr Lguges Witer 2014 Costs Busch - RPI 69

70 The itersectio of cotext-free lguge d regulr lguge is cotext-free lguge L1 cotext free L2 regulr L1 L 2 cotext-free Witer 2014 Costs Busch - RPI 70

71 Applictios of Regulr Closure Witer 2014 Costs Busch - RPI 71

72 The itersectio of cotext-free lguge d regulr lguge is cotext-free lguge L1 L2 cotext free regulr Regulr Closure L1 L 2 cotext-free Witer 2014 Costs Busch - RPI 72

73 A Applictio of Regulr Closure Prove tht: L = { b : 100, 0} is cotext-free Witer 2014 Costs Busch - RPI 73

74 We kow: { b : 0} is cotext-free Witer 2014 Costs Busch - RPI 74

75 We lso kow: 100 L 1 = { b 100 } is regulr * L1 = {( b) } { b } is regulr Witer 2014 Costs Busch - RPI 75

76 { b } cotext-free 100 L1 = {( b) } { b * regulr 100 } (regulr closure) { b } L 1 cotext-free { b } L1 = { b : 100, 0} = L is cotext-free Witer 2014 Costs Busch - RPI 76

77 Aother Applictio of Regulr Closure Prove tht: L = { w: = b = c } is ot cotext-free Witer 2014 Costs Busch - RPI 77

78 If L = { w: = b = c } is cotext-free The (regulr closure) L { * b* c*} = { b c } cotext-free regulr cotext-free Impossible!!! Therefore, L is ot cotext free Witer 2014 Costs Busch - RPI 78

Applications of Regular Closure

Applications of Regular Closure Applictios of Regulr Closure 1 The itersectio of cotext-free lguge d regulr lguge is cotext-free lguge L1 L2 cotext free regulr Regulr Closure L1 L 2 cotext-free 2 Liz 6 th, sectio 8.2, exple 8.7, pge

More information

Formal Languages The Pumping Lemma for CFLs

Formal Languages The Pumping Lemma for CFLs Forl Lguges The Pupig Le for CFLs Review: pupig le for regulr lguges Tke ifiite cotext-free lguge Geertes ifiite uer of differet strigs Exple: 3 I derivtio of log strig, vriles re repeted derivtio: 4 Derivtio

More information

Content. Languages, Alphabets and Strings. Operations on Strings. a ab abba baba. aaabbbaaba b 5. Languages. A language is a set of strings

Content. Languages, Alphabets and Strings. Operations on Strings. a ab abba baba. aaabbbaaba b 5. Languages. A language is a set of strings CD5560 FABER Forml guges, Automt d Models of Computtio ecture Mälrdle Uiversity 006 Cotet guges, Alphets d Strigs Strigs & Strig Opertios guges & guge Opertios Regulr Expressios Fiite Automt, FA Determiistic

More information

Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:

Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right: Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the

More information

Geometric Sequences. Geometric Sequence. Geometric sequences have a common ratio.

Geometric Sequences. Geometric Sequence. Geometric sequences have a common ratio. s A geometric sequece is sequece such tht ech successive term is obtied from the previous term by multiplyig by fixed umber clled commo rtio. Exmples, 6, 8,, 6,..., 0, 0, 0, 80,... Geometric sequeces hve

More information

8.3 Sequences & Series: Convergence & Divergence

8.3 Sequences & Series: Convergence & Divergence 8.3 Sequeces & Series: Covergece & Divergece A sequece is simply list of thigs geerted by rule More formlly, sequece is fuctio whose domi is the set of positive itegers, or turl umbers,,,3,. The rge of

More information

Formal Languages Simplifications of CFGs

Formal Languages Simplifications of CFGs Forml Lnguges implifictions of CFGs ubstitution Rule Equivlent grmmr b bc ubstitute b bc bbc b 2 ubstitution Rule b bc bbc ubstitute b bc bbc bc Equivlent grmmr 3 In generl: xz y 1 ubstitute y 1 xz xy1z

More information

1.4 Nonregular Languages

1.4 Nonregular Languages 74 1.4 Nonregulr Lnguges The number of forml lnguges over ny lphbet (= decision/recognition problems) is uncountble On the other hnd, the number of regulr expressions (= strings) is countble Hence, ll

More information

CD5080 AUBER RECAPITULATION. Context-Free Languages. Models of Computation, Languages and Automata. S asb. S bsb

CD5080 AUBER RECAPITULATION. Context-Free Languages. Models of Computation, Languages and Automata. S asb. S bsb CD5080 AUBR RCAPIULAION Models of Coputtio, Lguges d Autot Leture 1 Cotext-ree Lguges, CL Pushdow Autot, PDA Pupig Le for CL eleted CL Proles Cotext-ree Lguges Mälrdle Uiversity 00 1 3 Cotext-ree Lguges

More information

Some Properties of Brzozowski Derivatives of Regular Expressions

Some Properties of Brzozowski Derivatives of Regular Expressions Itertiol Jourl of Computer Treds d Techology (IJCTT) volume 13 umber 1 Jul 014 Some Properties of Brzozoski erivtives of Regulr Expressios NMuruges #1, OVShmug Sudrm * #1 Assistt Professor, ept of Mthemtics,

More information

Course Material. CS Lecture 1 Deterministic Finite Automata. Grading and Policies. Workload. Website:http://www.cs.colostate.

Course Material. CS Lecture 1 Deterministic Finite Automata. Grading and Policies. Workload. Website:http://www.cs.colostate. Course Mteril CS 301 - Lecture 1 Determiistic Fiite Automt Fll 2008 Wesite:http://www.cs.colostte.edu/~cs301 Syllus, Outlie, Grdig Policies Homework d Slides Istructor: D Mssey Office hours: 2-3pm Tues

More information

SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11

SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11 UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum

More information

MATH 118 HW 7 KELLY DOUGAN, ANDREW KOMAR, MARIA SIMBIRSKY, BRANDEN LASKE

MATH 118 HW 7 KELLY DOUGAN, ANDREW KOMAR, MARIA SIMBIRSKY, BRANDEN LASKE MATH 118 HW 7 KELLY DOUGAN, ANDREW KOMAR, MARIA SIMBIRSKY, BRANDEN LASKE Prt 1. Let be odd rime d let Z such tht gcd(, 1. Show tht if is qudrtic residue mod, the is qudrtic residue mod for y ositive iteger.

More information

Introduction to Computational Molecular Biology. Suffix Trees

Introduction to Computational Molecular Biology. Suffix Trees 18.417 Itroductio to Computtiol Moleculr Biology Lecture 11: October 14, 004 Scribe: Athich Muthitchroe Lecturer: Ross Lippert Editor: Toy Scelfo Suffix Trees This is oe of the most complicted dt structures

More information

Section 6.3: Geometric Sequences

Section 6.3: Geometric Sequences 40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.

More information

SM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory

SM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.

More information

1.3 Regular Expressions

1.3 Regular Expressions 56 1.3 Regulr xpressions These hve n importnt role in describing ptterns in serching for strings in mny pplictions (e.g. wk, grep, Perl,...) All regulr expressions of lphbet re 1.Ønd re regulr expressions,

More information

Discrete Mathematics I Tutorial 12

Discrete Mathematics I Tutorial 12 Discrete Mthemtics I Tutoril Refer to Chpter 4., 4., 4.4. For ech of these sequeces fid recurrece reltio stisfied by this sequece. (The swers re ot uique becuse there re ifiitely my differet recurrece

More information

RULES FOR MANIPULATING SURDS b. This is the addition law of surds with the same radicals. (ii)

RULES FOR MANIPULATING SURDS b. This is the addition law of surds with the same radicals. (ii) SURDS Defiitio : Ay umer which c e expressed s quotiet m of two itegers ( 0 ), is clled rtiol umer. Ay rel umer which is ot rtiol is clled irrtiol. Irrtiol umers which re i the form of roots re clled surds.

More information

Chapter 7 Infinite Series

Chapter 7 Infinite Series MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2

More information

Convergence rates of approximate sums of Riemann integrals

Convergence rates of approximate sums of Riemann integrals Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki

More information

CS 275 Automata and Formal Language Theory

CS 275 Automata and Formal Language Theory CS 275 Automt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Problem (II) Chpter II.5.: Properties of Context Free Grmmrs (14) Anton Setzer (Bsed on book drft by J. V. Tucker nd K. Stephenson)

More information

1.3 Continuous Functions and Riemann Sums

1.3 Continuous Functions and Riemann Sums mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be

More information

B. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i

B. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio

More information

INFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1

INFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1 Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series

More information

Project 3: Using Identities to Rewrite Expressions

Project 3: Using Identities to Rewrite Expressions MAT 5 Projet 3: Usig Idetities to Rewrite Expressios Wldis I lger, equtios tht desrie properties or ptters re ofte lled idetities. Idetities desrie expressio e repled with equl or equivlet expressio tht

More information

Surds, Indices, and Logarithms Radical

Surds, Indices, and Logarithms Radical MAT 6 Surds, Idices, d Logrithms Rdicl Defiitio of the Rdicl For ll rel, y > 0, d ll itegers > 0, y if d oly if y where is the ide is the rdicl is the rdicd. Surds A umber which c be epressed s frctio

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

More information

For convenience, we rewrite m2 s m2 = m m m ; where m is repeted m times. Since xyz = m m m nd jxyj»m, we hve tht the string y is substring of the fir

For convenience, we rewrite m2 s m2 = m m m ; where m is repeted m times. Since xyz = m m m nd jxyj»m, we hve tht the string y is substring of the fir CSCI 2400 Models of Computtion, Section 3 Solutions to Homework 4 Problem 1. ll the solutions below refer to the Pumping Lemm of Theorem 4.8, pge 119. () L = f n b l k : k n + lg Let's ssume for contrdiction

More information

( a n ) converges or diverges.

( a n ) converges or diverges. Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite

More information

Lecture 6 Regular Grammars

Lecture 6 Regular Grammars Lecture 6 Regulr Grmmrs COT 4420 Theory of Computtion Section 3.3 Grmmr A grmmr G is defined s qudruple G = (V, T, S, P) V is finite set of vribles T is finite set of terminl symbols S V is specil vrible

More information

Compact Representation of Ambiguities. Generating Sentences by Grammar Specialization. Compact Representation of Ambiguities

Compact Representation of Ambiguities. Generating Sentences by Grammar Specialization. Compact Representation of Ambiguities Compct Represettio of Ambiguities Geertig Seteces by Grmmr Speciliztio Jürge Wedekid Prsig Chrt: represets ltertive sytctic lyses pcked f-structures (cotexted costrit stisfctio, Mxwell & Kpl 91) Geertio

More information

Unit 1. Extending the Number System. 2 Jordan School District

Unit 1. Extending the Number System. 2 Jordan School District Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets

More information

FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),

FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx), FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the

More information

Review of Sections

Review of Sections Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,

More information

Linear Programming. Preliminaries

Linear Programming. Preliminaries Lier Progrmmig Prelimiries Optimiztio ethods: 3L Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio

More information

Context-free grammars and. Basics of string generation methods

Context-free grammars and. Basics of string generation methods Cotext-free grammars ad laguages Basics of strig geeratio methods What s so great about regular expressios? A regular expressio is a strig represetatio of a regular laguage This allows the storig a whole

More information

Automata and Languages

Automata and Languages Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive

More information

Similar idea to multiplication in N, C. Divide and conquer approach provides unexpected improvements. Naïve matrix multiplication

Similar idea to multiplication in N, C. Divide and conquer approach provides unexpected improvements. Naïve matrix multiplication Next. Covered bsics of simple desig techique (Divided-coquer) Ch. of the text.. Next, Strsse s lgorithm. Lter: more desig d coquer lgorithms: MergeSort. Solvig recurreces d the Mster Theorem. Similr ide

More information

Error-free compression

Error-free compression Error-free compressio Useful i pplictio where o loss of iformtio is tolerble. This mybe due to ccurcy requiremets, legl requiremets, or less th perfect qulity of origil imge. Compressio c be chieved by

More information

1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2

1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2 Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit

More information

10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form

10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form 0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully

More information

In an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case

In an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio

More information

The total number of permutations of S is n!. We denote the set of all permutations of S by

The total number of permutations of S is n!. We denote the set of all permutations of S by DETERMINNTS. DEFINITIONS Def: Let S {,,, } e the set of itegers from to, rrged i scedig order. rerrgemet jjj j of the elemets of S is clled permuttio of S. S. The totl umer of permuttios of S is!. We deote

More information

Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best

Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =

More information

MTH 146 Class 16 Notes

MTH 146 Class 16 Notes MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si

More information

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k. . Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric

More information

PROGRESSIONS AND SERIES

PROGRESSIONS AND SERIES PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.

More information

Name: Period: Date: 2.1 Rules of Exponents

Name: Period: Date: 2.1 Rules of Exponents SM NOTES Ne: Period: Dte:.1 Rules of Epoets The followig properties re true for ll rel ubers d b d ll itegers d, provided tht o deoitors re 0 d tht 0 0 is ot cosidered. 1 s epoet: 1 1 1 = e.g.) 7 = 7,

More information

CS 275 Automata and Formal Language Theory

CS 275 Automata and Formal Language Theory CS 275 Automt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Problem (II) Chpter II.6.: Push Down Automt Remrk: This mteril is no longer tught nd not directly exm relevnt Anton Setzer (Bsed

More information

Definite Integral. The Left and Right Sums

Definite Integral. The Left and Right Sums Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give

More information

Non-Deterministic Finite Automata. Fall 2018 Costas Busch - RPI 1

Non-Deterministic Finite Automata. Fall 2018 Costas Busch - RPI 1 Non-Deterministic Finite Automt Fll 2018 Costs Busch - RPI 1 Nondeterministic Finite Automton (NFA) Alphbet ={} q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 2 Nondeterministic Finite Automton (NFA) Alphbet

More information

General properties of definite integrals

General properties of definite integrals Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties

More information

Review of the Riemann Integral

Review of the Riemann Integral Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of

More information

Introduction to mathematical Statistics

Introduction to mathematical Statistics More ttistics tutoril t wwwdumblittledoctorcom Itroductio to mthemticl ttistics Chpter 7 ypothesis Testig Exmple (ull hypothesis) : the verge height is 5 8 or more ( : μ 5'8" ) (ltertive hypothesis) :

More information

MATRIX ALGEBRA, Systems Linear Equations

MATRIX ALGEBRA, Systems Linear Equations MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,

More information

RADICALS. Upon completion, you should be able to. define the principal root of numbers. simplify radicals

RADICALS. Upon completion, you should be able to. define the principal root of numbers. simplify radicals RADICALS m 1 RADICALS Upo completio, you should be ble to defie the pricipl root of umbers simplify rdicls perform dditio, subtrctio, multiplictio, d divisio of rdicls Mthemtics Divisio, IMSP, UPLB Defiitio:

More information

POWER SERIES R. E. SHOWALTER

POWER SERIES R. E. SHOWALTER POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise

More information

PRACTICE FINAL SOLUTIONS

PRACTICE FINAL SOLUTIONS CSE 303 PRACTICE FINAL SOLUTIONS FOR FINAL stud Practice Fial (mius PUMPING LEMMA ad Turig Machies) ad Problems from Q1 Q4, Practice Q1 Q4, ad Midterm ad Practice midterm. I will choose some of these problems

More information

Sequence and Series of Functions

Sequence and Series of Functions 6 Sequece d Series of Fuctios 6. Sequece of Fuctios 6.. Poitwise Covergece d Uiform Covergece Let J be itervl i R. Defiitio 6. For ech N, suppose fuctio f : J R is give. The we sy tht sequece (f ) of fuctios

More information

CS 314 Principles of Programming Languages

CS 314 Principles of Programming Languages C 314 Principles of Progrmming Lnguges Lecture 6: LL(1) Prsing Zheng (Eddy) Zhng Rutgers University Ferury 5, 2018 Clss Informtion Homework 2 due tomorrow. Homework 3 will e posted erly next week. 2 Top

More information

5. Solving recurrences

5. Solving recurrences 5. Solvig recurreces Time Complexity Alysis of Merge Sort T( ) 0 if 1 2T ( / 2) otherwise sortig oth hlves mergig Q. How to prove tht the ru-time of merge sort is O( )? A. 2 Time Complexity Alysis of Merge

More information

Convergence rates of approximate sums of Riemann integrals

Convergence rates of approximate sums of Riemann integrals Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem

More information

lecture 16: Introduction to Least Squares Approximation

lecture 16: Introduction to Least Squares Approximation 97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d

More information

MA123, Chapter 9: Computing some integrals (pp )

MA123, Chapter 9: Computing some integrals (pp ) MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how

More information

Section IV.6: The Master Method and Applications

Section IV.6: The Master Method and Applications Sectio IV.6: The Mster Method d Applictios Defiitio IV.6.1: A fuctio f is symptoticlly positive if d oly if there exists rel umer such tht f(x) > for ll x >. A cosequece of this defiitio is tht fuctio

More information

Summer MA Lesson 4 Section P.3. such that =, denoted by =, is the principal square root

Summer MA Lesson 4 Section P.3. such that =, denoted by =, is the principal square root Suer MA 00 Lesso Sectio P. I Squre Roots If b, the b is squre root of. If is oegtive rel uber, the oegtive uber b b b such tht, deoted by, is the pricipl squre root of. rdicl sig rdicl expressio rdicd

More information

Approximations of Definite Integrals

Approximations of Definite Integrals Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl

More information

Limit of a function:

Limit of a function: - Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive

More information

Approximate Integration

Approximate Integration Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:

More information

SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11

SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11 SUTCLIFFE S NOTES: CALCULUS SWOKOWSKI S CHAPTER Ifiite Series.5 Altertig Series d Absolute Covergece Next, let us cosider series with both positive d egtive terms. The simplest d most useful is ltertig

More information

Non Deterministic Automata. Linz: Nondeterministic Finite Accepters, page 51

Non Deterministic Automata. Linz: Nondeterministic Finite Accepters, page 51 Non Deterministic Automt Linz: Nondeterministic Finite Accepters, pge 51 1 Nondeterministic Finite Accepter (NFA) Alphbet ={} q 1 q2 q 0 q 3 2 Nondeterministic Finite Accepter (NFA) Alphbet ={} Two choices

More information

Positive Properties of Context-Free languages

Positive Properties of Context-Free languages CS 30 - Leture 8 Properties of Cotext Free Graars Fall 008 Review Laguages ad Graars Alphabets, strigs, laguages Regular Laguages Deteriisti Fiite ad Nodeteriisti Autoata Equivalee of NFA ad DFA Regular

More information

Overview HC9. Parsing: Top-Down & LL(1) Context-Free Grammars (1) Introduction. CFGs (3) Context-Free Grammars (2) Vertalerbouw HC 9: Ch.

Overview HC9. Parsing: Top-Down & LL(1) Context-Free Grammars (1) Introduction. CFGs (3) Context-Free Grammars (2) Vertalerbouw HC 9: Ch. Overview H9 Vertlerouw H 9: Prsing: op-down & LL(1) do 3 mei 2001 56 heo Ruys h. 8 - Prsing 8.1 ontext-free Grmmrs 8.2 op-down Prsing 8.3 LL(1) Grmmrs See lso [ho, Sethi & Ullmn 1986] for more thorough

More information

Parse trees, ambiguity, and Chomsky normal form

Parse trees, ambiguity, and Chomsky normal form Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs

More information

Chapter System of Equations

Chapter System of Equations hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee

More information

Algebra II, Chapter 7. Homework 12/5/2016. Harding Charter Prep Dr. Michael T. Lewchuk. Section 7.1 nth roots and Rational Exponents

Algebra II, Chapter 7. Homework 12/5/2016. Harding Charter Prep Dr. Michael T. Lewchuk. Section 7.1 nth roots and Rational Exponents Algebr II, Chpter 7 Hrdig Chrter Prep 06-07 Dr. Michel T. Lewchuk Test scores re vilble olie. I will ot discuss the test. st retke opportuit Sturd Dec. If ou hve ot tke the test, it is our resposibilit

More information

Chapter Real Numbers

Chapter Real Numbers Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:

More information

Pikachu, Domosaur, and other Monolexical Languages

Pikachu, Domosaur, and other Monolexical Languages Pikchu, Domosur, d other Moolexicl Lguges Srh Alle, Jesse Dodge, Domosur Mrch 2014 Abstrct M complicted techiques hve bee itroduced to id i computer processig of turl lguges. While this is geerll cosidered

More information

Vectors. Vectors in Plane ( 2

Vectors. Vectors in Plane ( 2 Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector

More information

Non-deterministic Finite Automata

Non-deterministic Finite Automata Non-deterministic Finite Automt Eliminting non-determinism Rdoud University Nijmegen Non-deterministic Finite Automt H. Geuvers nd T. vn Lrhoven Institute for Computing nd Informtion Sciences Intelligent

More information

Chapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures

Chapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)

More information

A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD

A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method

More information

 n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2!

 n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2! mth power series, prt ii 7 A Very Iterestig Emple Oe of the first power series we emied ws! + +! + + +!! + I Emple 58 we used the rtio test to show tht the itervl of covergece ws (, ) Sice the series coverges

More information

Properties of Regular Languages. Reading: Chapter 4

Properties of Regular Languages. Reading: Chapter 4 Properties of Regular Laguages Readig: Chapter 4 Topics ) How to prove whether a give laguage is regular or ot? 2) Closure properties of regular laguages 3) Miimizatio of DFAs 2 Some laguages are ot regular

More information

The Reimann Integral is a formal limit definition of a definite integral

The Reimann Integral is a formal limit definition of a definite integral MATH 136 The Reim Itegrl The Reim Itegrl is forml limit defiitio of defiite itegrl cotiuous fuctio f. The costructio is s follows: f ( x) dx for Reim Itegrl: Prtitio [, ] ito suitervls ech hvig the equl

More information

5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9.

5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9. Regulr Expressions, Pumping Lemm, Right Liner Grmmrs Ling 106 Mrch 25, 2002 1 Regulr Expressions A regulr expression descries or genertes lnguge: it is kind of shorthnd for listing the memers of lnguge.

More information

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More

More information

5.1 - Areas and Distances

5.1 - Areas and Distances Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 63u Office Hours: R 9:5 - :5m The midterm will cover the sectios for which you hve received homework d feedbck Sectios.9-6.5 i your book.

More information

DETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1

DETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1 NOD6 (\Dt\04\Kot\J-Advced\SMP\Mths\Uit#0\NG\Prt-\0.Determits\0.Theory.p65. INTRODUCTION : If the equtios x + b 0, x + b 0 re stisfied by the sme vlue of x, the b b 0. The expressio b b is clled determit

More information

MAS221 Analysis, Semester 2 Exercises

MAS221 Analysis, Semester 2 Exercises MAS22 Alysis, Semester 2 Exercises Srh Whitehouse (Exercises lbelled * my be more demdig.) Chpter Problems: Revisio Questio () Stte the defiitio of covergece of sequece of rel umbers, ( ), to limit. (b)

More information

FABER Formal Languages, Automata and Models of Computation

FABER Formal Languages, Automata and Models of Computation DVA337 FABER Forml Lnguges, Automt nd Models of Computtion Lecture 5 chool of Innovtion, Design nd Engineering Mälrdlen University 2015 1 Recp of lecture 4 y definition suset construction DFA NFA stte

More information

M3P14 EXAMPLE SHEET 1 SOLUTIONS

M3P14 EXAMPLE SHEET 1 SOLUTIONS M3P14 EXAMPLE SHEET 1 SOLUTIONS 1. Show tht for, b, d itegers, we hve (d, db) = d(, b). Sice (, b) divides both d b, d(, b) divides both d d db, d hece divides (d, db). O the other hd, there exist m d

More information

If a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?

If a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero? 2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a

More information

Eigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x):

Eigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x): Eigefuctio Epsio: For give fuctio o the iterl the eigefuctio epsio of f(): f ( ) cmm( ) m 1 Eigefuctio Epsio (Geerlized Fourier Series) To determie c s we multiply oth sides y Φ ()r() d itegrte: f ( )

More information

18.01 Calculus Jason Starr Fall 2005

18.01 Calculus Jason Starr Fall 2005 18.01 Clculus Jso Strr Lecture 14. October 14, 005 Homework. Problem Set 4 Prt II: Problem. Prctice Problems. Course Reder: 3B 1, 3B 3, 3B 4, 3B 5. 1. The problem of res. The ciet Greeks computed the res

More information

Schrödinger Equation Via Laplace-Beltrami Operator

Schrödinger Equation Via Laplace-Beltrami Operator IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,

More information

Chapter 2 Infinite Series Page 1 of 9

Chapter 2 Infinite Series Page 1 of 9 Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric

More information

Lecture 11 Context-Free Languages

Lecture 11 Context-Free Languages Lecture 11 Context-Free Languages COT 4420 Theory of Computation Chapter 5 Context-Free Languages n { a b : n n { ww } 0} R Regular Languages a *b* ( a + b) * Example 1 G = ({S}, {a, b}, S, P) Derivations:

More information

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I ICS4: Discrete Mathematics for Computer Sciece I Dept. Iformatio & Computer Sci., Ja Stelovsky based o slides by Dr. aek ad Dr. Still Origials by Dr. M. P. Frak ad Dr. J.L. Gross Provided by McGraw-Hill

More information