Andersen s Algorithm. CS 701 Final Exam (Reminder) Friday, December 12, 4:00 6:00 P.M., 1289 Computer Science.
|
|
- Amos Butler
- 5 years ago
- Views:
Transcription
1 CS 701 Finl Exm (Reminde) Fidy, Deeme 12, 4:00 6:00 P.M., 1289 Comute Siene. Andesen s Algoithm An lgoithm to uild oints-to gh fo C ogm is esented in: Pogm Anlysis nd Seiliztion fo the C ogmming Lnguge, L.O. Andesen, The lgoithm exmines sttements tht ete ointes, one y one, in textul ode (the lgoithm is flowinsensitive). Eh sttement udtes the oints-to gh if it n ete new oints-to eltionshis Six kinds of sttements e onsideed: = &; = q; = *; * = &; * = q; * = *; 2. = q; We dd s fom to eveything q oints to. If new s fom q e lte dded, oesonding s fom must lso e dded (this imlies n itetive o woklist lgoithm). Fo exmle (the dshed is newly dded): We will detil the oints-to gh udtes eh of the sttements indues. 1. = &; We dd n fom to, showing n ossily oint to : q
2 3. = *; Let S e ll the nodes oints to. Let T e ll the nodes memes of S oint to. We dd s fom to ll nodes in T. If lte ointe ssignments inese S o T, new s fom must lso e dded (this gin imlies n itetive o woklist lgoithm). Fo exmle (dshed s e newly dded): d e f 4. * = &; Add n to fom ll nodes oints to. If new s fom e lte dded, new s to must e dded (this imlies n itetive o woklist lgoithm). Fo exmle (dshed s e newly dded): q * = q; Nodes ointed to y must e linked to ll nodes ointed to y q. If lte ointe ssignments dd s fom o q, this ssignment must e evisited (this gin imlies n itetive o woklist lgoithm). Fo exmle (dshed s e newly dded): e 6. * = *; Let S e ll the nodes oints to. Let T e ll the nodes memes of S oint to. We dd s fom ll nodes oints to to ll nodes in T. If lte ointe ssignments inese S o T o link new nodes to, this ssignment must e evisited (this gin imlies n itetive o woklist lgoithm). Fo exmle (dshed s e newly dded): s f q u v d w x
3 Exmle Next: Conside the following ointe mniultions: 1 = &; 2 = &; 1 = 2; = &1; * = &; 3 = *; 2 = &d; We stt with: 1 = &; 2 = &; 1 1 = 2; 1 2 Then: = &1; Next: Finlly: * = &; 2 = &d; Then: 3 = *; d But we en t quite done yet. This lgoithm is flow-insensitive, so we must onside othe exeution odes (nd itetive e-exeution). If we mke nothe ss though the ssignments, we see tht the finl
4 ssignment to 2 n flow to 1, nd then to 3 though : 1 2 d 3 This oints-to gh is the dense, ut it does tue ll the wys ointe vlues might ogte though the vious ointe ssignments. Clls e hndled y teting ointe metes nd ointe etuns s ssignments, done t the oints of ll nd etun. Suogm odies e effetively inlined to tue the oints-to eltions they indue. Given *int eho (*int ) { etun ; } = eho (&); we see the imliit ssignments = &; = ; nd dd the following oints-to infomtion: As n otimiztion, liies n e e-nlyzed to detemine the oints-to eltions they indue. Most my use (ed) ointes ut don t ete ny new oints-to eltions visile outside thei odies. Cll to suh liy outines n e ignoed s f s the lle s oints-to gh is onened. Pefomne of Andesen s Algoithm Exeiene hs shown tht Andesen s Algoithm gives useful oints-to dt nd is f sueio to the nive ddess-tken oh. Inteestingly, exeiments show tht mking the tehnique flow-sensitive o lling ontext-sensitive doesn t imove esults vey muh on tyil enhmks. But exeution time fo modete to lge ogms n e olem. Ceful nlysis shows tht Andesen s Algoithm n equie O(n 3 ) time (whee n is the nume of nodes in the oints-to gh)
5 The eson fo this lge-thnexeted nlysis time is tht sttement like = *q; n foe the lgoithm to visit n 2 nodes (q my oint to n nodes nd eh of these nodes my oint to n nodes). The nume of ointe sttements nlyzed n e O(n), leding to n O(n 3 ) exeution time. Steensgd s Algoithm It would e useful to hve esonly ute oints-to nlysis tht uns in essentilly line time so tht elly lge ogms ould e hndled. This is wht Steensgd s Algoithm offes. (Points-to Anlysis in Almost Line Time, B. Steensgd, 1996 Piniles of Pogmming Lnguges Confeene.) Steensgd s Algoithm is essentilly Andesen s Algoithm, simlified y meging nodes nd if ny ointe n efeene oth Tht is, in Andesen s Algoithm we might hve Steensgd s Algoithm is sometimes less ute thn Andesen s Algoithm. Fo exmle, the following oints-to gh, eted y Andesen s Algoithm, shows tht my oint to o whees q my only oint to : In Steensgd s Algoithm we would insted hve q In effet ny two lotions tht might e ointed to y the sme ointe e led in single equivlene lss. In Steensgd s Algoithm we get q inoetly showing tht if my oint to o then so my q
6 But now sttements like = *q; n t foe the lgoithm to visit n 2 nodes, euse multile nodes efeened y the sme ointe e lwys meged. Using the fst unionfind lgoithm, we n get n exeution time of O(n α(n)) whih is essentilly line in n. Now vey lge ogms n e nlyzed, nd without too muh of loss in eision. Andesen vs. Steensgd in Ptie Howitz nd Shio exmined 61 C ogms, nging in size fom 300 to 24,300 lines. As exeted, Steensgd is less eise: On vege oints-to sets e 4 times igge; t wost 15 times igge. As exeted, Andesen is slowe. On vege 1.5 times slowe: t wost 31 times slowe. Both e muh ette thn the nive ddess tken oh. Bottom line: Use Andesen fo smll ogms, use Steensgd (o something else) fo lge ogms The Howitz-Shio Aoh It would e nie to hve oints-to nlysis tht is meteizle, nging etween the uy of Andesen nd the seed of Steensgd. Howitz nd Shio (Fst nd Aute Flow-Insensitive Points-To Anlysis, 1997 Piniles of Pogmming Lnguges Confeene) esent tehnique intemedite to those oosed y Andesen nd Steensgd. Howitz nd Shio suggest eh node in the oints-to gh e limited to out degee k, whee 1 k n. If k =1 then they hve Steensgd s oh. If k =n (n is nume of nodes in oints to gh), then they hve Andesen s oh. Thei wost se un-time is O(k 2 n), whih is not muh wose thn Steensgd if k is ket esonly smll
7 To use thei oh ssign eh vile tht my e ointed to to one of k tegoies. Now if my oint to x nd my lso oint to y, we mege x nd y only if they oth e in the sme tegoy. If x nd y e in diffeent tegoies, they en t meged, leding to moe ute oints-to estimtes. Exmle 1 = &; 1 = &; 1 = &; 2 = &; Sy we hve k = 2 nd le nd in tegoy 1 nd in tegoy 2. We then uild: 1, 2 This oints-to gh is just s ute s tht uilt y Andesen s oh But... Wht if we hose to le in tegoy 1 nd nd in tegoy 2. We now hve: 1 2, This gh is inext, sine it tells us 2 my oint to, whih is flse. (Steensgd would hve een wose still, inoetly telling us 2 my oint to s well s nd ). Anothe Good Ide Wht if we n Shio nd Howitz s oints-to nlysis twie, eh with diffeent tegoy ssignments? Eh un my odue diffeent oints-to gh. One my sy 2 oints to whees the othe sys it n t. Whih do we elieve? Neithe nlysis misses genuine oints-to eltion. Rthe, meging of nodes sometimes etes flse ointsto infomtion. So we will elieve 2 my oint to only if ll uns sy so. This mens multile uns my filte out mny of the flse oints-to eltions used y meging
8 How Mny Runs e Needed? How e Ctegoies to e Set? We wnt to ssign tegoies so tht duing t lest one un, ny i of ointed-to viles e in diffeent tegoies. This guntees tht if ll the uns tell us my oint to nd, it is not just euse nd lwys hened to e ssigned the sme tegoy. To foe diffeent tegoy ssignments fo eh i of viles, we ssign eh ointed-to vile n index nd wite tht index in se k (the nume of tegoies hosen). Fo exmle, if we hd viles,, nd d, nd hose k = 2, we d use the following iny indies: d 11 Note tht the nume of se k digits needed to eesent indies fom 0 to n-1 is just eiling(log k n). This nume is just the nume of uns we need! Why? In the fist un, we ll use the ight most digit in vile s index s its tegoy. In the next un, we ll use the seond digit fom the ight, then the thid digit fom the ight,... Any two distint viles hve diffeent index vlues, so they must diffe in t lest digit osition. Retuning to ou exmle, d 11 On un #1 we give nd tegoy 0 nd nd d tegoy 1. On un #2, nd get tegoy 0 nd nd d get tegoy 1. So using just 2 uns in this simle se, we eliminte muh of the inuy Steensgd s meging intodues. Run time is now O(log k (n) k 2 n)
9 How Well does this Aoh Wok? On 25 tests, using 3 tegoies, Howitz & Shio oints-to sets on vege e 2.67 lge thn those of Andesen (Steensgd s e 4.75 lge). This oh is slowe thn Steensgd ut on lge ogms it is 7 to 25 times fste thn Andesen. How Well do Points-to Anlyses Wok in Rel Dt Flow Polems? In Whih Pointe Anlysis Should I Use, Hind nd Pioli suvey the effetiveness of nume of ointsto nlyses in tul dt flow nlyses (mod/ef, liveness, ehing defs, inteoedul onstnt ogtion). Thei onlusions e essentilly the sme oss ll these nlyses: Steensgd s nlysis is signifintly moe eise thn ddess-tken nlysis nd not signifintly slowe. Andesen s nlysis odues modest, ut onsistent, imovements ove Steensgd s nlysis Both ontext-sensitive oints-to nlysis nd flow-sensitive oints-to nlysis give little imovement ove Andesen s nlysis. 615
The Area of a Triangle
The e of Tingle tkhlid June 1, 015 1 Intodution In this tile we will e disussing the vious methods used fo detemining the e of tingle. Let [X] denote the e of X. Using se nd Height To stt off, the simplest
More informationChapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:
Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,
More informationWeek 8. Topic 2 Properties of Logarithms
Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationLanguage Processors F29LP2, Lecture 5
Lnguge Pocessos F29LP2, Lectue 5 Jmie Gy Feuy 2, 2014 1 / 1 Nondeteministic Finite Automt (NFA) NFA genelise deteministic finite utomt (DFA). They llow sevel (0, 1, o moe thn 1) outgoing tnsitions with
More information10.3 The Quadratic Formula
. Te Qudti Fomul We mentioned in te lst setion tt ompleting te sque n e used to solve ny qudti eqution. So we n use it to solve 0. We poeed s follows 0 0 Te lst line of tis we ll te qudti fomul. Te Qudti
More information10 Statistical Distributions Solutions
Communictions Engineeing MSc - Peliminy Reding 1 Sttisticl Distiutions Solutions 1) Pove tht the vince of unifom distiution with minimum vlue nd mximum vlue ( is ) 1. The vince is the men of the sques
More information( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x
SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics.
More information1 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 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 informationCS 573 Automata Theory and Formal Languages
Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple
More informationQuality control. Final exam: 2012/1/12 (Thur), 9:00-12:00 Q1 Q2 Q3 Q4 Q5 YOUR NAME
Qulity contol Finl exm: // (Thu), 9:-: Q Q Q3 Q4 Q5 YOUR NAME NOTE: Plese wite down the deivtion of you nswe vey clely fo ll questions. The scoe will be educed when you only wite nswe. Also, the scoe will
More informationAn Analysis of the LRE-Algorithm using Sojourn Times
An Anlysis of the LRE-Algoithm using Sooun Times Nobet Th. Mülle Abteilung Infomtik Univesität Tie D-5486 Tie, Gemny E-mil: muelle@uni-tie.de Tel: ++49-65-0-845 Fx: ++49-65-0-3805 KEYWORDS Disete event
More informationData Compression LZ77. Jens Müller Universität Stuttgart
Dt Compession LZ77 Jens Mülle Univesität Stuttgt 2008-11-25 Outline Intoution Piniple of itiony methos LZ77 Sliing winow Exmples Optimiztion Pefomne ompison Applitions/Ptents Jens Mülle- IPVS Univesität
More informationITI Introduction to Computing II
ITI 1121. Intoduction to Computing II Mcel Tucotte School of Electicl Engineeing nd Compute Science Abstct dt type: Stck Stck-bsed lgoithms Vesion of Febuy 2, 2013 Abstct These lectue notes e ment to be
More informationPreviously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system
436-459 Advnced contol nd utomtion Extensions to bckstepping contolle designs Tcking Obseves (nonline dmping) Peviously Lst lectue we looked t designing nonline contolles using the bckstepping technique
More informationCS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6
CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized
More information1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.
York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech
More informationData Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2.
Element Uniqueness Poblem Dt Stuctues Let x,..., xn < m Detemine whethe thee exist i j such tht x i =x j Sot Algoithm Bucket Sot Dn Shpi Hsh Tbles fo (i=;i
More informationPrerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,
R Pen Towe Rod No Conttos Ae Bistupu Jmshedpu 8 Tel (67)89 www.penlsses.om IIT JEE themtis Ppe II PART III ATHEATICS SECTION I (Totl ks : ) (Single Coet Answe Type) This setion ontins 8 multiple hoie questions.
More informationClass Summary. be functions and f( D) , we define the composition of f with g, denoted g f by
Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g:
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 informationwhere the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b
CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}
More informationMathematical Reflections, Issue 5, INEQUALITIES ON RATIOS OF RADII OF TANGENT CIRCLES. Y.N. Aliyev
themtil efletions, Issue 5, 015 INEQULITIES ON TIOS OF DII OF TNGENT ILES YN liev stt Some inequlities involving tios of dii of intenll tngent iles whih inteset the given line in fied points e studied
More informationLecture 10. Solution of Nonlinear Equations - II
Fied point Poblems Lectue Solution o Nonline Equtions - II Given unction g : R R, vlue such tht gis clled ied point o the unction g, since is unchnged when g is pplied to it. Whees with nonline eqution
More informationTOPIC: LINEAR ALGEBRA MATRICES
Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A -... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED
More informationTopic II.1: Frequent Subgraph Mining
Topi II.1: Fequent Sugph Mining Disete Topis in Dt Mining Univesität des Slndes, Süken Winte Semeste 2012/13 T II.1-1 TII.1: Fequent Sugph Mining 1. Definitions nd Polems 1.1. Gph Isomophism 2. Apioi-Bsed
More informationWeek 10: DTMC Applications Ranking Web Pages & Slotted ALOHA. Network Performance 10-1
Week : DTMC Alictions Rnking Web ges & Slotted ALOHA etwok efonce - Outline Aly the theoy of discete tie Mkov chins: Google s nking of web-ges Wht ge is the use ost likely seching fo? Foulte web-gh s Mkov
More informationSystem Validation (IN4387) November 2, 2012, 14:00-17:00
System Vlidtion (IN4387) Novemer 2, 2012, 14:00-17:00 Importnt Notes. The exmintion omprises 5 question in 4 pges. Give omplete explntion nd do not onfine yourself to giving the finl nswer. Good luk! Exerise
More informationMath 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013
Mth 4318 : Rel Anlysis II Mid-Tem Exm 1 14 Febuy 2013 Nme: Definitions: Tue/Flse: Poofs: 1. 2. 3. 4. 5. 6. Totl: Definitions nd Sttements of Theoems 1. (2 points) Fo function f(x) defined on (, b) nd fo
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More informationCSE 332. Sorting. Data Abstractions. CSE 332: Data Abstractions. QuickSort Cutoff 1. Where We Are 2. Bounding The MAXIMUM Problem 4
Am Blnk Leture 13 Winter 2016 CSE 332 CSE 332: Dt Astrtions Sorting Dt Astrtions QuikSort Cutoff 1 Where We Are 2 For smll n, the reursion is wste. The onstnts on quik/merge sort re higher thn the ones
More informationLinear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
More informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationConvert the NFA into DFA
Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:
More informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationSPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018.
SPA7U/SPA7P: THE GALAXY Solutions fo Cousewok Questions distibuted on: 25 Jnuy 28. Solution. Assessed question] We e told tht this is fint glxy, so essentilly we hve to ty to clssify it bsed on its spectl
More informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationOptimization. x = 22 corresponds to local maximum by second derivative test
Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More informationLogistics. Today s Topics. Overview. Planning. Simplifying Assumptions. CSE 473 Automated Planning. HW1 due in one week (Fri 5/4)
SE 473 utomted Plnning Dn Weld (With slides y UW I fulty & Dn Nu I hve ln - ln tht nnot ossily fil. - Inseto lousseu Logistis HW1 due in one week (Fi 5/4) Pts due in etween: Mondy dft nswe to olem 1 Wed
More informationPart I: Study the theorem statement.
Nme 1 Nme 2 Nme 3 A STUDY OF PYTHAGORAS THEOREM Instrutions: Together in groups of 2 or 3, fill out the following worksheet. You my lift nswers from the reding, or nswer on your own. Turn in one pket for
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
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 informationGeneral Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface
Genel Physics II Chpte 3: Guss w We now wnt to quickly discuss one of the moe useful tools fo clculting the electic field, nmely Guss lw. In ode to undestnd Guss s lw, it seems we need to know the concept
More information10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = =
Chpte 1 nivesl Gvittion 11 *P1. () The un-th distnce is 1.4 nd the th-moon 8 distnce is.84, so the distnce fom the un to the Moon duing sol eclipse is 11 8 11 1.4.84 = 1.4 The mss of the un, th, nd Moon
More informationCSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science
CSCI 340: Computtionl Models Kleene s Theorem Chpter 7 Deprtment of Computer Science Unifiction In 1954, Kleene presented (nd proved) theorem which (in our version) sttes tht if lnguge cn e defined y ny
More informationFactorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
More information22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of:
22: Union Fin CS 473u - Algorithms - Spring 2005 April 14, 2005 1 Union-Fin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x) - rete set tht ontins the single element x. 2. Fin(x)
More informationNon Right Angled Triangles
Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit
More informationLecture 6: Coding theory
Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those
More informationCHAPTER 18: ELECTRIC CHARGE AND ELECTRIC FIELD
ollege Physics Student s Mnul hpte 8 HAPTR 8: LTRI HARG AD LTRI ILD 8. STATI LTRIITY AD HARG: OSRVATIO O HARG. ommon sttic electicity involves chges nging fom nnocoulombs to micocoulombs. () How mny electons
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 informationProbability. b a b. a b 32.
Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,
More informationNON-DETERMINISTIC FSA
Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is
More informationDiscrete Model Parametrization
Poceedings of Intentionl cientific Confeence of FME ession 4: Automtion Contol nd Applied Infomtics Ppe 9 Discete Model Pmetition NOKIEVIČ, Pet Doc,Ing,Cc Deptment of Contol ystems nd Instumenttion, Fculty
More informationEquations from the Millennium Theory of Inertia and Gravity. Copyright 2004 Joseph A. Rybczyk
Equtions fo the illenniu heoy of Ineti nd vity Copyight 004 Joseph A. Rybzyk ollowing is oplete list of ll of the equtions used o deived in the illenniu heoy of Ineti nd vity. o ese of efeene the equtions
More informationMAT 403 NOTES 4. f + f =
MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn
More informationThis immediately suggests an inverse-square law for a "piece" of current along the line.
Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line
More informationAnalysis of Variance for Multiple Factors
Multiple Fto ANOVA Notes Pge wo Fto Anlsis Anlsis of Vine fo Multiple Ftos Conside two ftos (tetments) A nd B with A done t levels nd B done t levels. Within given tetment omintion of A nd B levels, leled
More informationFinite State Automata and Determinisation
Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More information2-Way Finite Automata Radboud University, Nijmegen. Writer: Serena Rietbergen, s Supervisor: Herman Geuvers
2-Wy Finite Automt Rdoud Univesity, Nijmegen Wite: Seen Rietegen, s4182804 Supeviso: Hemn Geuves Acdemic Ye 2017-2018 Contents 1 Intoduction 3 2 One wy utomt, deteministic nd non-deteministic 5 3 Ovehed
More informationThe University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS
The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their
More informationCHAPTER 7 Applications of Integration
CHAPTER 7 Applitions of Integtion Setion 7. Ae of Region Between Two Cuves.......... Setion 7. Volume: The Disk Method................. Setion 7. Volume: The Shell Method................ Setion 7. A Length
More informationModule 4: Moral Hazard - Linear Contracts
Module 4: Mol Hzd - Line Contts Infomtion Eonomis (E 55) Geoge Geogidis A pinipl employs n gent. Timing:. The pinipl o es line ontt of the fom w (q) = + q. is the sly, is the bonus te.. The gent hooses
More informationFluids & Bernoulli s Equation. Group Problems 9
Goup Poblems 9 Fluids & Benoulli s Eqution Nme This is moe tutoil-like thn poblem nd leds you though conceptul development of Benoulli s eqution using the ides of Newton s 2 nd lw nd enegy. You e going
More informationTechnische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution
Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationMark Scheme (Results) January 2008
Mk Scheme (Results) Jnuy 00 GCE GCE Mthemtics (6679/0) Edecel Limited. Registeed in Englnd nd Wles No. 4496750 Registeed Office: One90 High Holbon, London WCV 7BH Jnuy 00 6679 Mechnics M Mk Scheme Question
More informationPart 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
More informationCounting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs
Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationComputing data with spreadsheets. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt
Computing dt with spredsheets Exmple: Computing tringulr numers nd their squre roots. Rell, we showed 1 ` 2 ` `n npn ` 1q{2. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2:
More informationLecture 14. Protocols. Key Distribution Center (KDC) or Trusted Third Party (TTP) KDC generates R1
Lectue 14 Potocols 1 Key Distiution Cente (KDC) o Tusted Thid Pty (TTP) KDC genetes R1 lice otins R1 Msg1: K () Msg2: K (R1 K (R1) ) Msg3: K (R1) o otins R1 nd knows to use s key fo communicting with lice
More information3.15 NMR spectroscopy Different types of NMR There are two main types of NMR 1. C 13 NMR 2. H (proton) NMR
.5 NMR spetrosopy Different types of NMR There re two min types of NMR. NMR. (proton) NMR There is only round % in orgni moleules ut modern NMR mhines re sensitive enough to give full spetr for The spetr
More informationProbabilistic Retrieval
CS 630 Lectue 4: 02/07/2006 Lectue: Lillin Lee Scibes: Pete Bbinski, Dvid Lin Pobbilistic Retievl I. Nïve Beginnings. Motivtions b. Flse Stt : A Pobbilistic Model without Vition? II. Fomultion. Tems nd
More informationπ,π is the angle FROM a! TO b
Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two
More informationReview of Mathematical Concepts
ENEE 322: Signls nd Systems view of Mthemticl Concepts This hndout contins ief eview of mthemticl concepts which e vitlly impotnt to ENEE 322: Signls nd Systems. Since this mteil is coveed in vious couses
More informationQUADRATIC RESIDUES MATH 372. FALL INSTRUCTOR: PROFESSOR AITKEN
QUADRATIC RESIDUES MATH 37 FALL 005 INSTRUCTOR: PROFESSOR AITKEN When is n integer sure modulo? When does udrtic eution hve roots modulo? These re the uestions tht will concern us in this hndout 1 The
More informationType 2: Improper Integrals with Infinite Discontinuities
mth imroer integrls: tye 6 Tye : Imroer Integrls with Infinite Disontinuities A seond wy tht funtion n fil to be integrble in the ordinry sense is tht it my hve n infinite disontinuity (vertil symtote)
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show
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 non-finl.
More informationConnected-components. Summary of lecture 9. Algorithms and Data Structures Disjoint sets. Example: connected components in graphs
Prm University, Mth. Deprtment Summry of lecture 9 Algorithms nd Dt Structures Disjoint sets Summry of this lecture: (CLR.1-3) Dt Structures for Disjoint sets: Union opertion Find opertion Mrco Pellegrini
More informationOn Some Hadamard-Type Inequalıtıes for Convex Functıons
Aville t htt://vuedu/ Al Al Mth ISSN: 93-9466 Vol 9, Issue June 4, 388-4 Alictions nd Alied Mthetics: An Intentionl Jounl AAM On Soe Hdd-Tye Inequlıtıes o, Convex Functıons M Ein Özdei Detent o Mthetics
More informationElectric Potential. and Equipotentials
Electic Potentil nd Euipotentils U Electicl Potentil Review: W wok done y foce in going fom to long pth. l d E dl F W dl F θ Δ l d E W U U U Δ Δ l d E W U U U U potentil enegy electic potentil Potentil
More informationSurds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,
Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationArrow s Impossibility Theorem
Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More information5. Every rational number have either terminating or repeating (recurring) decimal representation.
CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd
More informationNondeterminism and Nodeterministic Automata
Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely
More information1 Error Analysis of Simple Rules for Numerical Integration
cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion
More information