Regular Language Membership Constraint

Size: px
Start display at page:

Download "Regular Language Membership Constraint"

Transcription

1 Regulr Lnguge Memership Constrint Niko Pltzer Universität des Srlndes, Deutschlnd, Advisor: Lutz Strßurger Astrct. This pper minly dels with the work of Gilles Pesnt on the stretch constrint nd its reformultion s regulr lnguge memership constrint. Some definitions nd exmples should introduce nd explin the notion of stretch, the stretch constrint nd the regulr constrint. The consistency lgorithm for the regulr constrint is explined nd illustrted with n dditionl exmple. Some comprtive remrks on the stretch constrint nd the regulr constrint re included s well. 1 Introduction The first question I sked myself when I went in for the regulr lnguge memership constrint ws the question for the ppliction re. It seemed to e something rtificil tht does not relly mtter in usul constrint prolems s for exmple scheduling or logicl prolems. Therefore it ws necessry to go little step ck in history to discover how Gilles Pesnt hit on the introduction of the regulr constrint in [1]. It turns out tht the regulr constrint is just generliztion of the stretch constrint tht Pesnt introduced in [3]. And the min ppliction field of this stretch constrint re in fct scheduling especilly rostering prolems. Therefore I first wnt to give n exmple for such rostering prolem nmely the construction of rotting schedule. Then this prolem ist formulted with the stretch constrint. Finlly, from the resulting exmple, deterministic finite utomton (DFA) is deduced. Then we consider the forml definition of the regulr constrint. Afterwrds, I shortly recll the definition of hyper rc consistency since tht is wht the susequent descried consistency lgorithm for the regulr constrint chieves. A smll exmple should mke cler, how this lgorithm works. Some specilized enchmrks concerning the ppliction to stretch instnces re given. Finlly, nother exmple shows different possile (ut unfortuntely inefficient) ppliction of the regulr constrint, nmely the lldifferent constrint.

2 2 Rotting Schedules In todys usiness, it is norml for lot of compnies to e ville 24 hours round the clock. Therefore it is necessry to hve time tle for the workers tht on the one hnd ensures tht worker ( tem of workers) is present the whole dy nd on the other hnd gurntees tht ech worker hs enough dys off per week to regenerte. For this prolem, so clled rotting schedules were introduced. These schedules re orgnised s follows: shift tle for single worker or tem of workers for severl weeks is creted the different workers (resp. tems) hve ll the sme shift tle ut ech strts with different offset of dys Therefore the shift tle is done in prllel. While the first worker (or tem) strts for exmple with the shifts of the first week in the tle, the second one performs the shifts of the second week nd so on. Tle 1 shows n exmple where we hve three diffent shift types, nmely dy shifts (D), night shifts (N) nd dys off (-). Tle 1. smple rotting schedule feturing dy shifts, night shifts nd dys off mo tu we th fr s su D D D - - N N N - - D D D D - N N N N - - week 1 week 2 week 3 tem 1 tem 3 tem 2 tem 2 tem 1 tem 3 tem 3 tem 2 tem 1 3 Stretch Constrint Given sequence of vlues, stretch is consecutive susequence of identicl vlues. Additionlly, stretch is lwys s long s possile i.e. the preceding nd succeeding vlues re different from. A stretch S consisting of n vlues is clled n -stretch of length n (length(s) = n). The types of two consecutive stretches (considered s n ordered pir) re clled pttern. E.g. the sequence cc consists of n -stretch of length 3 nd c-stretch of length 2 nd these two stretches form the pttern (, c).

3 The forml definition of the stretch constrint requires set of shift types T = {t 1,..., t m }, two vectors min nd mx of length m, sequence of FD vriles s = < s 0,..., s n 1 >, finite domins D si T, set of ptterns Π T T, oolen vlue cyclic. The vectors min nd mx restrict the length of the occuring stretches. For ech shift type t T, they contin n integer vlue. In oder to stte consistent constrint, min t mx t must hold for ll t. The set Π contins ll vlid ptterns i.e. only the ptterns in Π re llowed to occur in sequence of vlues ssigned to s. The oolen flg cyclic denotes if the sequence s should e considered s cycle. For exmple if cyclic = true then the sequence cc lso forms the pttern (c, ). And the sequence cc only consists of two stretches (one c- stretch of length 2 nd one -stretch of length 3) insted of three stretches in the non-cyclic cse (two c-stretches of length 1 nd one -stretch of length 3). The stretch constrint is then stted with nd it ensures tht stretch (s, min, mx, Π, cyclic) stretches S of type t : min t length(s) mx t, consecutive stretches S nd S of type t nd t : (t, t ) Π. 3.1 Exmple: Rotting Schedule Suppose now tht we wnt to model stretch constrint to crete rotting schedule with the follwing properties: dy shifts, night shifts nd dys off lternting etween dy nd night work work stretches of length 3 or 4 fter (night) work, (1+) 1 or 2 dys off exctly one dy shift nd one night shift per dy The lst requirement cn not e formulted within the stretch constrint, therefore we ssume tht it is ensured y some other glol constrints. Since we hve two shifts dy (dy shift nd night shift), our rotting schedule hs to hve t lest 14 dys. But we wnt our workers to hve some time for regenertion, we choose length 21 dys to gurntee enough spce for dys off. Therfore, we hve 21 vriles s 0,..., s 20.

4 Becuse we wnt the workers to lternte etween dy nd night work, we hve to introduce two different shift types for dys off, nmely O D for dys off fter dy work nd O N fter night work. The respective ptterns re (D, O D ) nd (N, O N ). Now we only hve to introduce two more ptterns (O D, N) nd (O N, D) tht gurntee night shift fter dy work nd vice vers. The full model of the stretch constrint looks s follows: T = {D, N, O D, O N } s =< s 0,..., s 20 > D si = T Π = {(D, O D ), (O D, N), (N, O N ), (O N, D)} min D,N,OD,O N = (3, 3, 1, 2) mx D,N,OD,O N = (4, 4, 2, 3) cyclic = true stretch (s, min, mx, Π, cyclic) Tle 2 shows n exmple shift tle fulfilling this constrint. It is equivlent to the one shown in Tle 1 except for the two different types (O D nd O N ) for dys off. Tle 2. Smple shift tle fulfilling the stretch constrint D D D O D O D N N N O N O N D D D D O D N N N N O N O N 4 Regulr Constrint To illustrte the trnsformtion of the stretch constrint into regulr lnguge memership constrint we drw the DFA in Fig. 1. The DFA restricts the length of the stretches in the sme wy the stretch constrint does, since for exmple only the nodes where three or four work shifts hve een completed re finl sttes. Only the cyclic cse is modelled here since the cyclic one cuses some overhed tht will e discussed lter. Since the clss of lnguges ccepted y DFA is in fct equivlent to the clss of lnguges descried y regulr expression, it is indeed regulr lnguge memership prolem.

5 D D D D 1 d 2 d 3 d 4 d O N D D O D O D O N O N O D O N N N O N O D N N N N 4 n 3 n 2 n 1 n Fig. 1. the DFA from stretch of the exmple given in Section 3.1 The forml definition of the regulr constrint is quite smller thn the one for the stretch constrint, since the informtion of min, mx nd Π is included in the DFA: sequence of FD vriles s = < s 1, s 2,..., s n > DFA M = (Q, Σ, δ, q 0, F ) finite domins D si Σ The DFA s usul consists of the finite set of sttes Q, the finite lphet Σ, the trnsition function δ, the initil stte q 0 nd the set of finl sttes F. Stting the regulr constrint regulr (s, M) ensures tht vlue(s 1 ) vlue(s 2 )... vlue(s n ) L(M) i.e. every sequence of vlues tken y the vriles of s hve to e memer of the regulr lnguge recognised y M. 5 Hyper Arc Consistency The consistency lgorithm for the regulr constrint discussed in Section 6 chieves hyper rc consistency so let me shortly recll the definition from [2]. It is often lso clled generlized rc consistency or domin consistency. Rememer tht rc consistency itself is only defined for inry constrints i.e. constrints over exctly two vriles. Since hyper rc consistency is the generlistion of rc consistency, it is defined for constrints over ny (finite) numer of vriles.

6 The forml definition looks s follows: A constrint C D 1... D k is clled hyper rc consistent iff D i D i : ( 1,..., i 1, i+1,..., k ) D 1... D i 1 D i+1... D k : ( 1,..., i 1,, i+1,..., k ) C Intuitively expressed, constrint C is hyper rc consistent iff single ritrry vrile constrined y C cn e determined to n ritrry vlue from its domin nd C is still fesile. So ech vlue from ny domin tken y the corresponding vrile is prt of t lest one solution to C. The dvntge of CSP, consisting of only one hyper rc consistent constrint is the fct tht no cktrcking is needed when distriuting over the domins. Since the distriution methods only split domins with more thn one vrile, it follows from the definition of hyper rc consistency tht such CSP hs t lest two solutions. After the distriution step, ech creted suprolem hs t lest one of these solutions. Apt defines in [2] hyper rc consistency even for CSPs: A CSP is hyper rc consistent iff ll its constrints re hyper rc consistent. In my opinion, this is somehow missleding terminology ecuse the ove descried feture is not gurnteed in such CSP! In fct, - per definition - hyper rc consistent CSP does not need to hve solution even if ll domins re nonempty. Therefore I would prefer to cll CSP hyper rc consistent iff it consists of only one hyper rc consistent constrint. 6 Consistency Algorithm Now I wnt to give n informl description of the consistency lgorithm introduced y Gilles Pesnt. The interested reder cn find the forml definition nd the pseudo code in [1]. The lgorithm consists of the initil phse where the necessry dt structures re initilized nd mintennce phse where these dt structures re modified ccording to chnges of the ffected domins. The min dt structure is directed cyclic grph (DAG). The nodes of the grph re grouped into levels. Ech node (q, i) is lelled with the corresponding stte q of M nd the level i. Therefore ny stte of M occurs t most once on ech level. Level 0 only contins the root of the DAG i.e. the initil stte of M. Then level i corresponds to vrile s i. So the grph consists of n + 1 levels. The rcs in the grph re lelled with vlues. Arcs pointing to node (q, i) cn only e lelled with vlue v D si. Such n rc is clled v-supporting rc. Vice vers, fter the initil phse, vlue v is removed from D si if there is no v-supporting rc on level i. All nodes (q, n) on level n hve to e finl sttes of M i.e. q F. The correltion etween this grph nd the constrint is the following: For ech solution of the constrint there exists pth from the root node on level 0 to node on level n in the grph.

7 6.1 Initil Phse The initil phse gin consists of three suphses, forwrd phse, ckwrd phse nd clen-up phse. Ech phse is performed with n steps. forwrd phse At first, the initil node of M is inserted into level 0. Then we iterte from i = 1 to n. In itertion i, for ll nodes k = (q, i 1) nd ll vlues v D si, the node k = (q, i) with q = δ(q, v) nd the rc (k, k ) re inserted into the DAG. 1 ckwrd phse This phse strts with removing ll nodes (q, n) nd their corresponding incoming rcs with q / F. Then we iterte from i = n 1 to 0. In itertion i, we remove ll nodes (q, i) nd their corresponding incoming rcs tht hve no outgoing rcs. clen-up phse Now we hve the desired DAG, where ech pth from level 0 to level n corresponds to solution to the constrint. For i = 1 to n we remove ll vlues v from D si tht hve no supporting rc pointing to node on level i. After tht, we hve reched hyper rc consistency since ll vlues tht do not occur in t lest one pth (resp. one solution) hve een removed. 6.2 Mintennce Phse If other glol constrints remove vlue from domin of the regulr constrint, n updte on the DAG needs to e performed. For ech vlue tht is removed from D si, the corresponding supporting rcs pointing to level i hve to e removed from the DAG. For ech rc pointing to level i we remove, the following steps re executed recursively: remove unrechle nodes on level i (nodes tht hve no incoming rcs nymore) nd their outgoing rcs pointing to level i + 1 (recursion!) remove invlid nodes on level i 1 (nodes tht hve no outgoing rcs nymore) nd their incoming rcs (recursion!) remove those vlues from D si 1 nd D si+1 tht hve no supporting rcs nymore 1 rememer tht δ is the trnsition function of M

8 6.3 Exmple: Regulr Constrint Now we wnt to consider smll exmple to visulise how the lgorithm works. Therfore we introduce sequence of four vriles s =< s 1, s 2, s 3, s 4 > with domins D si = {, }. The DFA M is given y Fig. 2. The lphet Σ only consists of the two letters nd. The words of the lnguge ccepted y M re restricted to lternting -stretches of length 1 nd -stretches of length 2 or 3. Fig. 3 to Fig. 6 show the initilistion nd the mintennce of the grph Fig. 2. smple DFA M tht ccepts lternting -stretches of length 1 nd -stretches of length 2 or Properties of the Algorithm As lredy mentioned, this lgorithm chieves hyper rc consistency ecuse vlue v only remins in domin iff there exists pth in the grph tht includes v-supporting rc. And since pth in the grph is equivlent to solution to the constrint, vlue only remins in domin iff it is prt of t lest one solution to the constrint. The running time nd spce is in O(n Σ Q ), detils cn e found in [1] s well s short description of slightly modified lgorithm. This lgorithm does not mintin the whole grph ut only one v-supporting edge per vlue nd level. Clerly, if one of these edges is removed, it hs to e checked, if we cn find nother v-supporting edge tht we omitted so fr. But the given enchmrks do not show n significntly improved running time.

9 D s1 D s2 D s3 D s4 {, } {, } {, } {, } Lvl 1 Lvl 2 Lvl 3 Lvl Fig. 3. smple DAG fter the forwrd phse: finl sttes of M re lso mrked s finl nodes on level 4 D s1 D s2 D s3 D s4 {, } {} {} {, } Lvl 1 Lvl 2 Lvl 3 Lvl Fig. 4. smple DAG fter the ckwrd phse nd the clen-up phse: the nonfinl node on level 4 nd the invlid node on level 3 hve een removed; vlue hs een removed from D s2 nd D s3

10 D s1 D s2 D s3 D s4 {, } {} {} {} Lvl 1 Lvl 2 Lvl 3 Lvl Fig. 5. smple DAG t the eginning of the mintennce phse: vlue hs een removed from D s4 y nother glol constrint, therefore the corresponding supporting rcs hve een removed D s1 D s2 D s3 D s4 {} {} {} {} Lvl 1 Lvl 2 Lvl 3 Lvl Fig. 6. smple DAG fter the mintennce phse: ll unrechle nd invlid nodes nd their corresponding rcs hve een removed; vlue hs een removed from D s1 since the only -supporting rc hs een removed

11 Pesnt lso gives the hint tht, depending on the wy the DFA is creted, it might e preferle to construct minimum stte DFA in O( Q log Q ) efore pssing it to the regulr constrint. 6.5 Remrks on Stretch In the enchmrk from [1], the consistency lgorithm for the regulr constrint is fster thn stretch filtering implementtion from [3] for instnces with n 50 nd Σ 5. Since the filering lgorithm does not chieve hyper rc consistency, it hs to perform lot of cktrcking on hrder instnces. Together with Lrs Hellsten nd Peter vn Beek, Gilles Pesnt presented in [4] new lgorithm for the stretch constrint tht chieves hyper rc consistency nd runs in O(n m 2 ). Unfortuntely no enchmrk is given tht compres this new version with one for the regulr constrint. As lredy mentioned, it is not stright forwrd to model circulr stretch prolems s regulr constrint. For this purpose we hve to duplicte l vriles where l is the mximum of the llowed stretch-lengths. Furthermore we hve to introduce nother 2 l 2 dummy vriles (for more detils see [1]). Depending on the prolem this cn cuse significnt mount of overhed. 6.6 Exmple: Alldifferent In the end I wnt to give lst exmple not feturing stretches ut the well known lldifferent constrint. The regulr constrint cn e intuitively used to intuitively model this importnt glol constrint. The sttes of the DFA simply represent the set of the lredy tken vlues. All sttes re mrked finl, since from the construction, no invlid lloction of the vriles is permitted. Unfortuntely this is very inefficient, since the numer of sttes is exponentil in the numer of letters in the lphet ( Q = 2 Σ ). If the numer of vriles n is strictly smller thn the numer of different vlues, the size of the DFA cn e reduced y removing ll nodes tht re more thn n steps wy from the initil node. The resulting DFA cn gin e minimized with the resp. lgorithm. Fig. 7 shows n exmple DFA for the lphet Σ = {,, c}.

12 {} {, } c c {} {} {, c} {,, c} c c {c} {, c} Fig. 7. smple DFA to formulte the lldifferent constrint unsing the regulr constrint References 1. Gilles Pesnt: A Regulr Lnguge Memership Constrint for Finite Sequences of Vriles. Springer-Verlg Berlin Heidelerg (2004) 2. Krzysztof R. Apt: Principles of Constrint Progrmming: Cmridge University Press (2003) 3. Gilles Pesnt: A Filtering Algorithm for the Stretch Constrint. Springer-Verlg Berlin Heidelerg (2001) 4. Lrs Hellsten, Gilles Pesnt nd Peter vn Beek: A Domin Consistency Algorithm for the Stretch Constrint. Springer-Verlg Berlin Heidelerg (2004)

Homework 3 Solutions

Homework 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 information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

Finite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh

Finite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

One Minute To Learn Programming: Finite Automata

One Minute To Learn Programming: Finite Automata Gret Theoreticl Ides In Computer Science Steven Rudich CS 15-251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge

More information

Formal Languages and Automata Exam

Formal Languages and Automata Exam Forml Lnguges nd Automt Exm Fculty of Computers & Informtion Deprtment: Computer Science Grde: Third Course code: CSC 34 Totl Mrk: 8 Dte: 23//2 Time: 3 hours Answer the following questions: ) Consider

More information

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

Variable Dry Run (for Python)

Variable Dry Run (for Python) Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 20-50 minutes

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

1 Numerical Solution to Quadratic Equations

1 Numerical Solution to Quadratic Equations cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

More information

3 Determinization of Büchi-Automata

3 Determinization of Büchi-Automata 3 Determiniztion of Büchi-Automt Mrkus Roggenbch Bremen Institute for Sfe Systems Bremen University For Bene Introduction To determinize Büchi utomt it is necessry to switch to nother clss of ω-utomt,

More information

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929  Math Learning Center Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of

More information

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

More information

Solution to Problem Set 1

Solution to Problem Set 1 CSE 5: Introduction to the Theory o Computtion, Winter A. Hevi nd J. Mo Solution to Prolem Set Jnury, Solution to Prolem Set.4 ). L = {w w egin with nd end with }. q q q q, d). L = {w w h length t let

More information

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

1.2 The Integers and Rational Numbers

1.2 The Integers and Rational Numbers .2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl

More information

FORMAL LANGUAGES, AUTOMATA AND THEORY OF COMPUTATION EXERCISES ON REGULAR LANGUAGES

FORMAL LANGUAGES, AUTOMATA AND THEORY OF COMPUTATION EXERCISES ON REGULAR LANGUAGES FORMAL LANGUAGES, AUTOMATA AND THEORY OF COMPUTATION EXERCISES ON REGULAR LANGUAGES Introduction This compendium contins exercises out regulr lnguges for the course Forml Lnguges, Automt nd Theory of Computtion

More information

Calculus of variations with fractional derivatives and fractional integrals

Calculus of variations with fractional derivatives and fractional integrals Anis do CNMAC v.2 ISSN 1984-820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810-193 Aveiro, Portugl

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Modular Generic Verification of LTL Properties for Aspects

Modular Generic Verification of LTL Properties for Aspects Modulr Generic Verifiction of LTL Properties for Aspects Mx Goldmn Shmuel Ktz Computer Science Deprtment Technion Isrel Institute of Technology {mgoldmn, ktz}@cs.technion.c.il ABSTRACT Aspects re seprte

More information

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility

More information

A Note on Complement of Trapezoidal Fuzzy Numbers Using the α-cut Method

A Note on Complement of Trapezoidal Fuzzy Numbers Using the α-cut Method Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN - Vol. - A Note on Complement of Trpezoidl Fuzzy Numers Using the α-cut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment

More information

Curve Sketching. 96 Chapter 5 Curve Sketching

Curve Sketching. 96 Chapter 5 Curve Sketching 96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

More information

Small Businesses Decisions to Offer Health Insurance to Employees

Small Businesses Decisions to Offer Health Insurance to Employees Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employer-sponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

5 a LAN 6 a gateway 7 a modem

5 a LAN 6 a gateway 7 a modem STARTER With the help of this digrm, try to descrie the function of these components of typicl network system: 1 file server 2 ridge 3 router 4 ckone 5 LAN 6 gtewy 7 modem Another Novell LAN Router Internet

More information

0.1 Basic Set Theory and Interval Notation

0.1 Basic Set Theory and Interval Notation 0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined

More information

A new algorithm for generating Pythagorean triples

A new algorithm for generating Pythagorean triples A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.

Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule. Text questions, Chpter 5, problems 1-5: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed

More information

AntiSpyware Enterprise Module 8.5

AntiSpyware Enterprise Module 8.5 AntiSpywre Enterprise Module 8.5 Product Guide Aout the AntiSpywre Enterprise Module The McAfee AntiSpywre Enterprise Module 8.5 is n dd-on to the VirusScn Enterprise 8.5i product tht extends its ility

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

Windows 7/8. Windows 7/8. Installation Guide Sawgrass SG400/SG800. v

Windows 7/8. Windows 7/8. Installation Guide Sawgrass SG400/SG800. v Windows 7/8 Windows 7/8 Instlltion Guide Swgrss SG400/SG800 v20150521 Contents Virtuoso SG400/SG800 Initil Setup...2 Browser Instlltion...3 Internet Connection Speeds...3 CS Print nd Color Mnger Downlod...

More information

Network Configuration Independence Mechanism

Network Configuration Independence Mechanism 3GPP TSG SA WG3 Security S3#19 S3-010323 3-6 July, 2001 Newbury, UK Source: Title: Document for: AT&T Wireless Network Configurtion Independence Mechnism Approvl 1 Introduction During the lst S3 meeting

More information

Concept Formation Using Graph Grammars

Concept Formation Using Graph Grammars Concept Formtion Using Grph Grmmrs Istvn Jonyer, Lwrence B. Holder nd Dine J. Cook Deprtment of Computer Science nd Engineering University of Texs t Arlington Box 19015 (416 Ytes St.), Arlington, TX 76019-0015

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

Regular Languages and Finite Automata

Regular Languages and Finite Automata N Lecture Notes on Regulr Lnguges nd Finite Automt for Prt IA of the Computer Science Tripos Mrcelo Fiore Cmbridge University Computer Lbortory First Edition 1998. Revised 1999, 2000, 2001, 2002, 2003,

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

New Internet Radio Feature

New Internet Radio Feature XXXXX XXXXX XXXXX /XW-SMA3/XW-SMA4 New Internet Rdio Feture EN This wireless speker hs een designed to llow you to enjoy Pndor*/Internet Rdio. In order to ply Pndor/Internet Rdio, however, it my e necessry

More information

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

Protocol Analysis. 17-654/17-764 Analysis of Software Artifacts Kevin Bierhoff

Protocol Analysis. 17-654/17-764 Analysis of Software Artifacts Kevin Bierhoff Protocol Anlysis 17-654/17-764 Anlysis of Softwre Artifcts Kevin Bierhoff Tke-Awys Protocols define temporl ordering of events Cn often be cptured with stte mchines Protocol nlysis needs to py ttention

More information

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS 4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem

More information

1.00/1.001 Introduction to Computers and Engineering Problem Solving Fall 2011 - Final Exam

1.00/1.001 Introduction to Computers and Engineering Problem Solving Fall 2011 - Final Exam 1./1.1 Introduction to Computers nd Engineering Problem Solving Fll 211 - Finl Exm Nme: MIT Emil: TA: Section: You hve 3 hours to complete this exm. In ll questions, you should ssume tht ll necessry pckges

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Plotting and Graphing

Plotting and Graphing Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured

More information

Rational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and

Rational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

RTL Power Optimization with Gate-level Accuracy

RTL Power Optimization with Gate-level Accuracy RTL Power Optimiztion with Gte-level Accurcy Qi Wng Cdence Design Systems, Inc Sumit Roy Clypto Design Systems, Inc 555 River Oks Prkwy, Sn Jose 95125 2903 Bunker Hill Lne, Suite 208, SntClr 95054 qwng@cdence.com

More information

Version 001 CIRCUITS holland (1290) 1

Version 001 CIRCUITS holland (1290) 1 Version CRCUTS hollnd (9) This print-out should hve questions Multiple-choice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire

More information

Outline of the Lecture. Software Testing. Unit & Integration Testing. Components. Lecture Notes 3 (of 4)

Outline of the Lecture. Software Testing. Unit & Integration Testing. Components. Lecture Notes 3 (of 4) Outline of the Lecture Softwre Testing Lecture Notes 3 (of 4) Integrtion Testing Top-down ottom-up ig-ng Sndwich System Testing cceptnce Testing istriution of ults in lrge Industril Softwre System (ISST

More information

Strong acids and bases

Strong acids and bases Monoprotic Acid-Bse Equiliri (CH ) ϒ Chpter monoprotic cids A monoprotic cid cn donte one proton. This chpter includes uffers; wy to fi the ph. ϒ Chpter 11 polyprotic cids A polyprotic cid cn donte multiple

More information

Mathematics Higher Level

Mathematics Higher Level Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

More information

Mechanics Cycle 1 Chapter 5. Chapter 5

Mechanics Cycle 1 Chapter 5. Chapter 5 Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies

More information

Quantity Oriented Resource Allocation Strategy on Multiple Resources Projects under Stochastic Conditions

Quantity Oriented Resource Allocation Strategy on Multiple Resources Projects under Stochastic Conditions Interntionl Conference on Industril Engineering nd Systems Mngement IESM 2009 My 13-15 MONTRÉAL - CANADA Quntity Oriented Resource Alloction Strtegy on Multiple Resources Projects under Stochstic Conditions

More information

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00 COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided

More information

DATABASDESIGN FÖR INGENJÖRER - 1056F

DATABASDESIGN FÖR INGENJÖRER - 1056F DATABASDESIGN FÖR INGENJÖRER - 06F Sommr 00 En introuktionskurs i tssystem http://user.it.uu.se/~ul/t-sommr0/ lt. http://www.it.uu.se/eu/course/homepge/esign/st0/ Kjell Orsorn (Rusln Fomkin) Uppsl Dtse

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Words Symbols Diagram. abcde. a + b + c + d + e

Words Symbols Diagram. abcde. a + b + c + d + e Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

More information

Generating In-Line Monitors For Rabin Automata

Generating In-Line Monitors For Rabin Automata Generting In-Line Monitors For Rin Automt Hugues Chot, Rphel Khoury, nd Ndi Twi Lvl University, Deprtment of Computer Science nd Softwre Engineering, Pvillon Adrien-Pouliot, 1065, venue de l Medecine Queec

More information

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

More information

Basic Research in Computer Science BRICS RS-02-13 Brodal et al.: Solving the String Statistics Problem in Time O(n log n)

Basic Research in Computer Science BRICS RS-02-13 Brodal et al.: Solving the String Statistics Problem in Time O(n log n) BRICS Bsic Reserch in Computer Science BRICS RS-02-13 Brodl et l.: Solving the String Sttistics Prolem in Time O(n log n) Solving the String Sttistics Prolem in Time O(n log n) Gerth Stølting Brodl Rune

More information

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent! MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

More information

MODULE 3. 0, y = 0 for all y

MODULE 3. 0, y = 0 for all y Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

Gene Expression Programming: A New Adaptive Algorithm for Solving Problems

Gene Expression Programming: A New Adaptive Algorithm for Solving Problems Gene Expression Progrmming: A New Adptive Algorithm for Solving Prolems Cândid Ferreir Deprtmento de Ciêncis Agráris Universidde dos Açores 9701-851 Terr-Chã Angr do Heroísmo, Portugl Complex Systems,

More information

JaERM Software-as-a-Solution Package

JaERM Software-as-a-Solution Package JERM Softwre-s--Solution Pckge Enterprise Risk Mngement ( ERM ) Public listed compnies nd orgnistions providing finncil services re required by Monetry Authority of Singpore ( MAS ) nd/or Singpore Stock

More information

A Network Management System for Power-Line Communications and its Verification by Simulation

A Network Management System for Power-Line Communications and its Verification by Simulation A Network Mngement System for Power-Line Communictions nd its Verifiction y Simultion Mrkus Seeck, Gerd Bumiller GmH Unterschluerscher-Huptstr. 10, D-90613 Großhersdorf, Germny Phone: +49 9105 9960-51,

More information

Section A-4 Rational Expressions: Basic Operations

Section A-4 Rational Expressions: Basic Operations A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

More information

All pay auctions with certain and uncertain prizes a comment

All pay auctions with certain and uncertain prizes a comment CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

More information

A Visual and Interactive Input abb Automata. Theory Course with JFLAP 4.0

A Visual and Interactive Input abb Automata. Theory Course with JFLAP 4.0 Strt Puse Step Noninverted Tree A Visul nd Interctive Input Automt String ccepted! 5 nodes generted. Theory Course with JFLAP 4.0 q0 even 's, even 's q2 even 's, odd 's q1 odd 's, even 's q3 odd 's, odd

More information

Regular Repair of Specifications

Regular Repair of Specifications Regulr Repir of Specifictions Michel Benedikt Oxford University michel.enedikt@coml.ox.c.uk Griele Puppis Oxford University griele.puppis@coml.ox.c.uk Cristin Riveros Oxford University cristin.riveros@coml.ox.c.uk

More information

Automated Grading of DFA Constructions

Automated Grading of DFA Constructions Automted Grding of DFA Constructions Rjeev Alur nd Loris D Antoni Sumit Gulwni Dileep Kini nd Mhesh Viswnthn Deprtment of Computer Science Microsoft Reserch Deprtment of Computer Science University of

More information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

More information

Bypassing Space Explosion in Regular Expression Matching for Network Intrusion Detection and Prevention Systems

Bypassing Space Explosion in Regular Expression Matching for Network Intrusion Detection and Prevention Systems Bypssing Spce Explosion in Regulr Expression Mtching for Network Intrusion Detection n Prevention Systems Jignesh Ptel, Alex Liu n Eric Torng Dept. of Computer Science n Engineering Michign Stte University

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting

More information