CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions

Size: px
Start display at page:

Download "CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions"

Transcription

1 CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags: Prl, Java, Python, tc - grat for pattrn matching oprations). Exampls: a * b ( ) * = (+)* Dfinition. R is a rgular xprssion, if R is on of th following:. ε 2. a, for som a Σ (R R 2 ), whr R and R 2 ar rgular xprssions 5. R R 2, whr R and R 2 ar rgular xprssions 6. (R ) *, whr R is a rgular xprssion Not: it is a inductiv dfinition - it is dfind basd on itslf. Not: th + symbol will at tims b usd for union (+)* R R + + = RR! " = * R L((+)*(+)*) = {w w contains a in th middl} (Not: othr rgular xprssion xampls on p.65)

2 Equivalnc of Rgular Exprssions and Finit Automata Thorm. A languag is rgular if and only if som rgular xprssion dscribs it. W nd to prov two dirctions:. If a languag is dscribd by a rgular xprssion, thn it is rgular. 2. If a languag is rgular, thn thr is a rgular xprssion that dscribs it. Part. This is th asy part - "If a languag is dscribd by a rgular xprssion, thn it is rgular." Say that a rgular xprssion R dscribs som languag A. W will convrt R to an NFA N that rcognizs A. Thn, A has to b rgular. R has on of six possibl forms:. R = ε. Thn, 2. R = a, for som a Σ. Thn, a 3. R =. Thn, 4. R = R R R = R R R = (R ) * In th last thr cass, th constructions givn in th proofs that th class of rgular languags ar closd undr th rgular oprations can b usd hr as wll. That is, w assum R and R 2 ar rcognizd by NFAs N and N 2, and us th sam constructions to crat N from N and N 2. Convrt th following rgular xprssion to a NFA: (+)*(+)*

3 Not: using th lttr "" to rprsnt ε Not: For th abov rsulting NFA - major limination of stats and transistion can tak plac. Doubl 's can b rippd out. What ls can b liminatd? Part 2: W nd to show that if a languag is rgular, thn it is dscribd by a rgular xprssion. "If a languag is rgular, thn thr is a rgular xprssion that dscribs it." If a languag A is rgular, thn it is rcognizd by a DFA M. W will show how to convrt an arbitrary DFA M into an quivalnt rgular xprssion. Th main ida is that w will gradually liminat th stats of M. Stratgy: DFA --> GNFA --> rgular xprssions GNFA - gnralizd nondtrministic finit automaton Dfinition (informal dtails in th book). A GNFA is a Finit Automaton, xcpt:. Thr is only on start stat, on final stat, and th two ar distinct. 2. Th start stat has arrows going to vry othr stat. 3. Thr ar no arrows going to th start stat. 4. Th final stat has arrows coming from vry othr stat. 5. Thr ar no arrows laving th final stat. 6. Evry stat (xcpt start & final) has arrows going to vry othr stat. 7. Th labls of th arrows ar rgular xprssions.

4 To convrt a DFA into a GNFA:. Crat a nw start stat, with ε-transitions to th prvious start stat and -transitions to all othr stats. 2. Crat a nw final stat, with ε-transitions from th prvious final stat and - transitions from all othr stats. 3. Add -transitions from vry stat to vry stat (othr than start & final). Exampl: (Not: a vn mor profound xampl would b a DFA with multipl accpt stats) Th rsulting GNFA always has 2 stats. (It actually always has 3 stats, but now w will start rmoving stats until it has xactly 2 stats.) To convrt from GNFA to RE: W will rmov stats incrmntally, and at th sam tim build up a rgular xprssion. Whnvr th GNFA has 2 stats, w liminat a stat as follows: Th stat q blow is any stat othr than th (uniqu) start and (uniqu) final stats. That stat q, has possibly a numbr of transitions to it, including a slf-transition, and a numbr of transitions from it. q

5 For vry pair of stats p and r (including start & final), q is btwn p and r: p always has a transition to q, and r always has a transition from q. W rmov q, and in its plac put a rgular xprssion dscribing how to gt from p to r through q: R2 p R q R3 r p RR2*R3 r Th abov 2 GNFA sgmnts ar quivalnt. This ida of quivalncy mans: to gt from p to r, ithr gt through th arrow from p to r, or gt through th old q-path, in which cas you nd to go through th old p-to-q transition, thn possibly loop in q an arbitrary numbr of tims, and thn go through th old q-to-r path. Continu this procss of liminating stats, until w ar lft with just th start and final stats. Thn, E is th rgular xprssion w nd! E will accpt xactly th strings that th old DFA was accpting. Multipl dgs can b rmovd: a+b a a b b a+b Practic.. Find a DFA accpting th languag + ( + )*

6 Stp. Find NFA Stp2. Transform NFA to DFA Q $ " Q # = { q} Q Q = { q2} = { q, q4} = { q3} { q2} { q2} { q3}! { q, q4} { q3} { q2}! 2,,4 2 3, 2. Find th rgular xprssion accptd by this NFA.

7 , Answr: (+)*(+)+ = (+)* Simplifying formulas for REs (prov thm as an xrcis!): Not: E rprsnts th alphabt. + E = E εe = Eε = E E = E = (E * ) * = E * * = ε ε * = ε E + E = E E(F + G) = EF + EG (F + G)E = FE + GE Th UNIX-styl + -oprator, matchs a string onc or mor tims (not zro tims). W could dfin it hr as E + = EE * = E * E. Thn, chck as an xrcis that E * = E + ε. Rfrncs: Introduction to th Thory of Computation (2nd d.) Michal Sipsr Problm Solving in Automata, Languags, and Complxity Ding-Zhu Du and Kr-I Ko

8

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) 92.222 - Linar Algbra II - Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial

More information

Lecture 3: Diffusion: Fick s first law

Lecture 3: Diffusion: Fick s first law Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th

More information

New Basis Functions. Section 8. Complex Fourier Series

New Basis Functions. Section 8. Complex Fourier Series Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ral-valud Fourir sris is xplaind and formula ar givn for convrting

More information

14.3 Area Between Curves

14.3 Area Between Curves 14. Ara Btwn Curvs Qustion 1: How is th ara btwn two functions calculatd? Qustion : What ar consumrs and producrs surplus? Earlir in this chaptr, w usd dfinit intgrals to find th ara undr a function and

More information

Genetic Drift and Gene Flow Illustration

Genetic Drift and Gene Flow Illustration Gntic Drift and Gn Flow Illustration This is a mor dtaild dscription of Activity Ida 4, Chaptr 3, If Not Rac, How do W Explain Biological Diffrncs? in: How Ral is Rac? A Sourcbook on Rac, Cultur, and Biology.

More information

A Derivation of Bill James Pythagorean Won-Loss Formula

A Derivation of Bill James Pythagorean Won-Loss Formula A Drivation of Bill Jams Pythagoran Won-Loss Formula Ths nots wr compild by John Paul Cook from a papr by Dr. Stphn J. Millr, an Assistant Profssor of Mathmatics at Williams Collg, for a talk givn to th

More information

Question 3: How do you find the relative extrema of a function?

Question 3: How do you find the relative extrema of a function? ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating

More information

Non-Homogeneous Systems, Euler s Method, and Exponential Matrix

Non-Homogeneous Systems, Euler s Method, and Exponential Matrix Non-Homognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous first-ordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach

More information

Exponential Growth and Decay; Modeling Data

Exponential Growth and Decay; Modeling Data Exponntial Growth and Dcay; Modling Data In this sction, w will study som of th applications of xponntial and logarithmic functions. Logarithms wr invntd by John Napir. Originally, thy wr usd to liminat

More information

5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power

5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim

More information

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST: .4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This

More information

The example is taken from Sect. 1.2 of Vol. 1 of the CPN book.

The example is taken from Sect. 1.2 of Vol. 1 of the CPN book. Rsourc Allocation Abstract This is a small toy xampl which is wll-suitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of C-nts. Hnc, it can b rad by popl

More information

Statistical Machine Translation

Statistical Machine Translation Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm

More information

Lecture 20: Emitter Follower and Differential Amplifiers

Lecture 20: Emitter Follower and Differential Amplifiers Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.

More information

Singleton Theorem Using Models

Singleton Theorem Using Models Singlton Thorm Using Modls Srivathsan B, Igor Walukiwicz LaBRI Paris, March 2010 Srivathsan B, Igor Walukiwicz (LaBRI) Singlton Thorm Using Modls Paris, March 2010 1 / 17 Introduction Singlton Thorm [Statman

More information

Planar Graphs. More precisely: there is a 1-1 function f : V R 2 and for each e E there exists a 1-1 continuous g e : [0, 1] R 2 such that

Planar Graphs. More precisely: there is a 1-1 function f : V R 2 and for each e E there exists a 1-1 continuous g e : [0, 1] R 2 such that Planar Graphs A graph G = (V, E) is planar i it can b drawn on th plan without dgs crossing xcpt at ndpoints a planar mbdding or plan graph. Mor prcisly: thr is a - unction : V R 2 and or ach E thr xists

More information

Factorials! Stirling s formula

Factorials! Stirling s formula Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical

More information

Econ 371: Answer Key for Problem Set 1 (Chapter 12-13)

Econ 371: Answer Key for Problem Set 1 (Chapter 12-13) con 37: Answr Ky for Problm St (Chaptr 2-3) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc

More information

The Normal Distribution: A derivation from basic principles

The Normal Distribution: A derivation from basic principles Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn

More information

A Note on Approximating. the Normal Distribution Function

A Note on Approximating. the Normal Distribution Function Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and

More information

Graph Theory. 1 Graphs and Subgraphs

Graph Theory. 1 Graphs and Subgraphs 1 Graphs and Subgraphs Graph Thory Dfinition 1.1. A multigraph or just graph is an ordrd pair G = (V, E) consisting of a nonmpty vrtx st V of vrtics and an dg st E of dgs such that ach dg E is assignd

More information

Simulated Radioactive Decay Using Dice Nuclei

Simulated Radioactive Decay Using Dice Nuclei Purpos: In a radioactiv sourc containing a vry larg numbr of radioactiv nucli, it is not possibl to prdict whn any on of th nucli will dcay. Although th dcay tim for any on particular nuclus cannot b prdictd,

More information

SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* Rostov-on-Don. Russia

SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* Rostov-on-Don. Russia SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* Rostov-on-Don. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs

More information

II. Equipment. Magnetic compass, magnetic dip compass, Helmholtz coils, HP 6212 A power supply, Keithley model 169 multimeter

II. Equipment. Magnetic compass, magnetic dip compass, Helmholtz coils, HP 6212 A power supply, Keithley model 169 multimeter Magntic fild of th arth I. Objctiv: Masur th magntic fild of th arth II. Equipmnt. Magntic compass, magntic dip compass, Hlmholtz s, HP 6212 A powr supply, Kithly modl 169 multimtr III Introduction. IIIa.

More information

Adverse Selection and Moral Hazard in a Model With 2 States of the World

Adverse Selection and Moral Hazard in a Model With 2 States of the World Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,

More information

Category 7: Employee Commuting

Category 7: Employee Commuting 7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil

More information

QUANTITATIVE METHODS CLASSES WEEK SEVEN

QUANTITATIVE METHODS CLASSES WEEK SEVEN QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.

More information

Extended Multi Bottom-up Tree Transducers

Extended Multi Bottom-up Tree Transducers Motivation Extndd multi ottom-up tr transducr Thortical rsults Extndd Multi Bottom-up Tr Transducrs Joost Englfrit 1, Eric Lilin 2, and Andras Maltti 3 1 LIACS, Lidn, Th Nthrlands 2 Univrsité d Lill, Franc

More information

ME 612 Metal Forming and Theory of Plasticity. 6. Strain

ME 612 Metal Forming and Theory of Plasticity. 6. Strain Mtal Forming and Thory of Plasticity -mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.

More information

In the previous two chapters, we clarified what it means for a problem to be decidable or undecidable.

In the previous two chapters, we clarified what it means for a problem to be decidable or undecidable. Chaptr 7 Computational Complxity 7.1 Th Class P In th prvious two chaptrs, w clarifid what it mans for a problm to b dcidabl or undcidabl. In principl, if a problm is dcidabl, thn thr is an algorithm (i..,

More information

Traffic Flow Analysis (2)

Traffic Flow Analysis (2) Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,

More information

10/06/08 1. Aside: The following is an on-line analytical system that portrays the thermodynamic properties of water vapor and many other gases.

10/06/08 1. Aside: The following is an on-line analytical system that portrays the thermodynamic properties of water vapor and many other gases. 10/06/08 1 5. Th watr-air htrognous systm Asid: Th following is an on-lin analytical systm that portrays th thrmodynamic proprtis of watr vapor and many othr gass. http://wbbook.nist.gov/chmistry/fluid/

More information

SPECIAL VOWEL SOUNDS

SPECIAL VOWEL SOUNDS SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)

More information

Module 7: Discrete State Space Models Lecture Note 3

Module 7: Discrete State Space Models Lecture Note 3 Modul 7: Discrt Stat Spac Modls Lctur Not 3 1 Charactristic Equation, ignvalus and ign vctors For a discrt stat spac modl, th charactristic quation is dfind as zi A 0 Th roots of th charactristic quation

More information

Some Useful Integrals of Exponential Functions

Some Useful Integrals of Exponential Functions prvious indx nxt Som Usful Intgrls of Exponntil Functions Michl Fowlr W v shown tht diffrntiting th xponntil function just multiplis it by th constnt in th xponnt, tht is to sy, d x x Intgrting th xponntil

More information

by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia

by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs

More information

Principles of Humidity Dalton s law

Principles of Humidity Dalton s law Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

More information

Inference by Variable Elimination

Inference by Variable Elimination Chaptr 5 Infrnc by Variabl Elimination Our purpos in this chaptr is to prsnt on of th simplst mthods for gnral infrnc in Baysian ntworks, known as th mthod of Variabl Elimination. 5.1 Introduction Considr

More information

Section 5-5 Inverse of a Square Matrix

Section 5-5 Inverse of a Square Matrix - Invrs of a Squar Matrix 9 (D) Rank th playrs from strongst to wakst. Explain th rasoning hind your ranking. 68. Dominan Rlation. Eah mmr of a hss tam plays on math with vry othr playr. Th rsults ar givn

More information

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000 hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails

More information

Sigmoid Functions and Their Usage in Artificial Neural Networks

Sigmoid Functions and Their Usage in Artificial Neural Networks Sigmoid Functions and Thir Usag in Artificial Nural Ntworks Taskin Kocak School of Elctrical Enginring and Computr Scinc Applications of Calculus II: Invrs Functions Eampl problm Calculus Topic: Invrs

More information

Gas Radiation. MEL 725 Power-Plant Steam Generators (3-0-0) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi

Gas Radiation. MEL 725 Power-Plant Steam Generators (3-0-0) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi Gas Radiation ME 725 Powr-Plant Stam Gnrators (3-0-0) Dr. Prabal Talukdar Assistant Profssor Dpartmnt of Mchanical Enginring T Dlhi Radiation in absorbing-mitting mdia Whn a mdium is transparnt to radiation,

More information

http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force

http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd

More information

Examples. Epipoles. Epipolar geometry and the fundamental matrix

Examples. Epipoles. Epipolar geometry and the fundamental matrix Epipoar gomtry and th fundamnta matrix Epipoar ins Lt b a point in P 3. Lt x and x b its mapping in two imags through th camra cntrs C and C. Th point, th camra cntrs C and C and th (3D points corrspon

More information

Basis risk. When speaking about forward or futures contracts, basis risk is the market

Basis risk. When speaking about forward or futures contracts, basis risk is the market Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also

More information

e = C / electron Q = Ne

e = C / electron Q = Ne Physics 0 Modul 01 Homwork 1. A glass rod that has bn chargd to +15.0 nc touchs a mtal sphr. Aftrword, th rod's charg is +8.00 nc. What kind of chargd particl was transfrrd btwn th rod and th sphr, and

More information

CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.

CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List. Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by

More information

Deer: Predation or Starvation

Deer: Predation or Starvation : Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,

More information

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s1660-015-0741-1 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans Wi-Mao Qian

More information

Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.

Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final. Vrsion.0 Gnral Crtificat of Education (A-lvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,

More information

Cloud and Big Data Summer School, Stockholm, Aug., 2015 Jeffrey D. Ullman

Cloud and Big Data Summer School, Stockholm, Aug., 2015 Jeffrey D. Ullman Cloud and Big Data Summr Scool, Stockolm, Aug., 2015 Jffry D. Ullman Givn a st of points, wit a notion of distanc btwn points, group t points into som numbr of clustrs, so tat mmbrs of a clustr ar clos

More information

Modelling and Solving Two-Step Equations: ax + b = c

Modelling and Solving Two-Step Equations: ax + b = c Modlling and Solving To-Stp Equations: a + b c Focus on Aftr this lsson, you ill b abl to φ modl problms φ ith to-stp linar quations solv to-stp linar quations and sho ho you ord out th ansr Cali borrod

More information

(Analytic Formula for the European Normal Black Scholes Formula)

(Analytic Formula for the European Normal Black Scholes Formula) (Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually

More information

Dept. of Materials Science and Engineering. Problem Set 8 Solutions

Dept. of Materials Science and Engineering. Problem Set 8 Solutions MSE 30/ECE 30 Elctrical Prortis Of Matrials Dt. of Matrials Scinc and Enginring Fall 0/Bill Knowlton Problm St 8 Solutions. Using th rlationshi of n i n i i that is a function of E g, rcrat th lot shown

More information

Solutions to Homework 8 chem 344 Sp 2014

Solutions to Homework 8 chem 344 Sp 2014 1. Solutions to Homwork 8 chm 44 Sp 14 .. 4. All diffrnt orbitals mans thy could all b paralll spins 5. Sinc lctrons ar in diffrnt orbitals any combination is possibl paird or unpaird spins 6. Equivalnt

More information

Constraint-Based Analysis of Gene Deletion in a Metabolic Network

Constraint-Based Analysis of Gene Deletion in a Metabolic Network Constraint-Basd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFG-Rsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, 14195 Brlin, Grmany

More information

Making and Using the Hertzsprung - Russell Diagram

Making and Using the Hertzsprung - Russell Diagram Making and Using th Hrtzsprung - Russll Diagram Nam In astronomy th Hrtzsprung-Russll Diagram is on of th main ways that w organiz data dscribing how stars volv, ags of star clustrs, masss of stars tc.

More information

Graph Theory Definitions

Graph Theory Definitions Grph Thory Dfinitions A grph is pir of sts (V, E) whr V is finit st ll th st of vrtis n E is st of 2-lmnt susts of V, ll th st of gs. W viw th gs s st of onntions twn th nos. Hr is n xmpl of grph G: G

More information

Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY

Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl

More information

Ground Fault Current Distribution on Overhead Transmission Lines

Ground Fault Current Distribution on Overhead Transmission Lines FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 19, April 2006, 71-84 Ground Fault Currnt Distribution on Ovrhad Transmission Lins Maria Vintan and Adrian Buta Abstract: Whn a ground fault occurs on

More information

7 Timetable test 1 The Combing Chart

7 Timetable test 1 The Combing Chart 7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing

More information

CIRCUITS AND ELECTRONICS. Basic Circuit Analysis Method (KVL and KCL method)

CIRCUITS AND ELECTRONICS. Basic Circuit Analysis Method (KVL and KCL method) 6. CIRCUITS AND ELECTRONICS Basic Circuit Analysis Mthod (KVL and KCL mthod) Cit as: Anant Agarwal and Jffry Lang, cours matrials for 6. Circuits and Elctronics, Spring 7. MIT 6. Fall Lctur Rviw Lumpd

More information

Creating Your Own Coffeehouse At-Home. An Interactive Guide Brought to you by New Folgers Coffeehouse Blend

Creating Your Own Coffeehouse At-Home. An Interactive Guide Brought to you by New Folgers Coffeehouse Blend Crating Yr Own ffhs t-hom n Intractiv Guid Brght to y by Nw Folgrs ffhs Blnd ro -Drip ff HOW DO YOU LIKE YOUR COFFEE? vr ff ld Th Folgr ff mpany rss ff ch n Icd ff B r w C o f f How to Mak ff with a Traditional

More information

LABORATORY 1 IDENTIFICATION OF CIRCUIT IN A BLACK-BOX

LABORATORY 1 IDENTIFICATION OF CIRCUIT IN A BLACK-BOX LABOATOY IDENTIFICATION OF CICUIT IN A BLACK-BOX OBJECTIES. To idntify th configuration of an lctrical circuit nclosd in a two-trminal black box.. To dtrmin th valus of ach componnt in th black box circuit.

More information

Who uses our services? We have a growing customer base. with institutions all around the globe.

Who uses our services? We have a growing customer base. with institutions all around the globe. not taking xpr Srvic Guid 2013 / 2014 NTE i an affordabl option for audio to txt convrion. Our rvic includ not or dirct trancription rvic from prviouly rcordd audio fil. Our rvic appal pcially to tudnt

More information

A Project Management framework for Software Implementation Planning and Management

A Project Management framework for Software Implementation Planning and Management PPM02 A Projct Managmnt framwork for Softwar Implmntation Planning and Managmnt Kith Lancastr Lancastr Stratgis Kith.Lancastr@LancastrStratgis.com Th goal of introducing nw tchnologis into your company

More information

Noble gas configuration. Atoms of other elements seek to attain a noble gas electron configuration. Electron configuration of ions

Noble gas configuration. Atoms of other elements seek to attain a noble gas electron configuration. Electron configuration of ions Valnc lctron configuration dtrmins th charactristics of lmnts in a group Nobl gas configuration Th nobl gass (last column in th priodic tabl) ar charactrizd by compltly filld s and p orbitals this is a

More information

Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective

Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective Ovrviw Lctur 4 Projctions - 3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl - Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog

More information

Differential Equations (MTH401) Lecture That a non-homogeneous linear differential equation of order n is an equation of the form n

Differential Equations (MTH401) Lecture That a non-homogeneous linear differential equation of order n is an equation of the form n Diffrntial Equations (MTH40) Ltur 7 Mthod of Undtrmind Coffiints-Surosition Aroah Rall. That a non-homognous linar diffrntial quation of ordr n is an quation of th form n n d d d an + a a a0 g( ) n n +

More information

intro Imagine that someone asked you to describe church using only the bible. What would you say to them?

intro Imagine that someone asked you to describe church using only the bible. What would you say to them? intro Imagin that somon askd you to dscrib church using only th bibl. What would you say to thm? So many of th things w'v mad church to b arn't ssntial in scriptur. W'r on a journy of r-imagining what

More information

CHAPTER 4c. ROOTS OF EQUATIONS

CHAPTER 4c. ROOTS OF EQUATIONS CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03 - Computation Mthod in Civil Enginring II Dpartmnt o Civil

More information

Regular Languages and Finite State Machines

Regular Languages and Finite State Machines Regular Languages and Finite State Machines Plan for the Day: Mathematical preliminaries - some review One application formal definition of finite automata Examples 1 Sets A set is an unordered collection

More information

Foreign Exchange Markets and Exchange Rates

Foreign Exchange Markets and Exchange Rates Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls

More information

6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010. Class 4 Nancy Lynch

6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010. Class 4 Nancy Lynch 6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 4 Nancy Lynch Today Two more models of computation: Nondeterministic Finite Automata (NFAs)

More information

Magic Square of Squares

Magic Square of Squares Intrnatinal Jurnal f Enginring and Tchnical Rsarch (IJETR) ISSN: 232-869, Vlum-, Issu-8, Octbr 23 Magic Squar f Squars Shubhankar Paul Abstract A n n array f intgrs is calld a magic squar whn all th rws,

More information

SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY

SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY 1 SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY ALEXA Vasil ABSTRACT Th prsnt papr has as targt to crat a programm in th Matlab ara, in ordr to solv, didactically

More information

Optical Modulation Amplitude (OMA) and Extinction Ratio

Optical Modulation Amplitude (OMA) and Extinction Ratio Application Not: HFAN-.. Rv; 4/8 Optical Modulation Amplitud (OMA) and Extinction Ratio AVAILABLE Optical Modulation Amplitud (OMA) and Extinction Ratio Introduction Th optical modulation amplitud (OMA)

More information

AP Calculus AB 2008 Scoring Guidelines

AP Calculus AB 2008 Scoring Guidelines AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity.

More information

HANDOUT E.19 - EXAMPLES ON FEEDBACK CONTROL SYSTEMS

HANDOUT E.19 - EXAMPLES ON FEEDBACK CONTROL SYSTEMS MEEN 64 Parauram Lctur 9, Augut 5, HANDOUT E9 - EXAMPLES ON FEEDBAC CONTOL SSTEMS Exampl Conidr th ytm hown blow Th opn loop tranfr function i givn by Th clod loop tranfr function i Exampl Conidr th ytm

More information

Architecture of the proposed standard

Architecture of the proposed standard Architctur of th proposd standard Introduction Th goal of th nw standardisation projct is th dvlopmnt of a standard dscribing building srvics (.g.hvac) product catalogus basd on th xprincs mad with th

More information

Budget Optimization in Search-Based Advertising Auctions

Budget Optimization in Search-Based Advertising Auctions Budgt Optimization in Sarch-Basd Advrtising Auctions ABSTRACT Jon Fldman Googl, Inc. Nw York, NY jonfld@googl.com Martin Pál Googl, Inc. Nw York, NY mpal@googl.com Intrnt sarch companis sll advrtismnt

More information

Gateway 125,126,130 Fall 2006 Exam 1 KEY p1

Gateway 125,126,130 Fall 2006 Exam 1 KEY p1 Gatway 125,126,130 Fall 2006 Exam 1 KEY p1 Q16 (1 point ach) Plas plac th corrct lttr/s in th box. 1) How many lctrons can th third principal quantum lvl (n = 3) hold? a. 2 b. 8 c. 16 d. 18. 32 2) Arrang

More information

3. Yes. You can put 20 of the 6-V lights in series, or you can put several of the 6-V lights in series with a large resistance.

3. Yes. You can put 20 of the 6-V lights in series, or you can put several of the 6-V lights in series with a large resistance. CHAPTE 6: DC Circuits sponss to Qustions. Evn though th bird s ft ar at high potntial with rspct to th ground, thr is vry littl potntial diffrnc btwn thm, bcaus thy ar clos togthr on th wir. Th rsistanc

More information

Step 1: Checking the shaft

Step 1: Checking the shaft Stp 1: Chcking th shaft Th shaft diamtr at th baring sat should b within tolranc as shown in our Product Catalog. As a gnral guid, +0.000", 0.002" is adquat. Any tapr or out of roundnss in th shaft at

More information

HSBC Bank International Expat Explorer Survey 08

HSBC Bank International Expat Explorer Survey 08 HSBC Bank Intrnational Expat Explorr Survy 08 Rport On: Expat Existnc Th Survy Th Expat Explorr survy qustiond 2,155 xpatriats across four continnts about th opportunitis and challngs thy fac. Th survy

More information

Use a high-level conceptual data model (ER Model). Identify objects of interest (entities) and relationships between these objects

Use a high-level conceptual data model (ER Model). Identify objects of interest (entities) and relationships between these objects Chaptr 3: Entity Rlationship Modl Databas Dsign Procss Us a high-lvl concptual data modl (ER Modl). Idntify objcts of intrst (ntitis) and rlationships btwn ths objcts Idntify constraints (conditions) End

More information

Fundamentals of Tensor Analysis

Fundamentals of Tensor Analysis MCEN 503/ASEN 50 Chptr Fundmntls of Tnsor Anlysis Fll, 006 Fundmntls of Tnsor Anlysis Concpts of Sclr, Vctor, nd Tnsor Sclr α Vctor A physicl quntity tht cn compltly dscrid y rl numr. Exmpl: Tmprtur; Mss;

More information

Today: Electron waves?!

Today: Electron waves?! Today: Elctron wavs?! 1. Exampl of wav diffraction: Light. 2. Prdicting th diffraction of lctrons. 3. Th Davisson, Grmr xprimnt as an xampl. W can't dfin anything prcisly. If w attmpt to, w gt into that

More information

MAXIMAL CHAINS IN THE TURING DEGREES

MAXIMAL CHAINS IN THE TURING DEGREES MAXIMAL CHAINS IN THE TURING DEGREES C. T. CHONG AND LIANG YU Abstract. W study th problm of xistnc of maximal chains in th Turing dgrs. W show that:. ZF + DC+ Thr xists no maximal chain in th Turing dgrs

More information

Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental

Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal Zvi Lotkr zvilo@cs.bgu.ac.il Dpt. of Comm. Systms Enginring Bn Gurion Univrsity Br Shva Isral Boaz Patt-Shamir Dror Rawitz {boaz, rawitz}@ng.tau.ac.il

More information

Parallel and Distributed Programming. Performance Metrics

Parallel and Distributed Programming. Performance Metrics Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:

More information

Job shop scheduling with unit processing times

Job shop scheduling with unit processing times Job shop schduling with unit procssing tims Nikhil Bansal Tracy Kimbrl Maxim Sviridnko Abstract W considr randomizd algorithms for th prmptiv job shop problm, or quivalntly, th cas in which all oprations

More information

Nondeterministic Finite Automata

Nondeterministic Finite Automata Chapter, Part 2 Nondeterministic Finite Automata CSC527, Chapter, Part 2 c 202 Mitsunori Ogihara Fundamental set operations Let A and B be two languages. In addition to union and intersection, we consider

More information

Section 7.4: Exponential Growth and Decay

Section 7.4: Exponential Growth and Decay 1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart

More information

Automata and Languages

Automata and Languages Automata and Languages Computational Models: An idealized mathematical model of a computer. A computational model may be accurate in some ways but not in others. We shall be defining increasingly powerful

More information

Finite Element Vibration Analysis

Finite Element Vibration Analysis Finit Elmnt Vibration Analysis Introduction In prvious topics w larnd how to modl th dynamic bhavior of multi-dof systms, as wll as systms possssing infinit numbrs of DOF. As th radr may raliz, our discussion

More information

Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D

Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D 24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd

More information

Hardware Modules of the RSA Algorithm

Hardware Modules of the RSA Algorithm SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 11, No. 1, Fbruary 2014, 121-131 UDC: 004.3`142:621.394.14 DOI: 10.2298/SJEE140114011S Hardwar Moduls of th RSA Algorithm Vlibor Škobić 1, Branko Dokić 1,

More information

Quantum Graphs I. Some Basic Structures

Quantum Graphs I. Some Basic Structures Quantum Graphs I. Som Basic Structurs Ptr Kuchmnt Dpartmnt of Mathmatics Txas A& M Univrsity Collg Station, TX, USA 1 Introduction W us th nam quantum graph for a graph considrd as a on-dimnsional singular

More information

Systems of Equations and Inequalities

Systems of Equations and Inequalities CHAPTER CONNECTIONS Systms of Equations and Inqualitis CHAPTER OUTLINE.1 Linar Systms in Two Variabls with Applications 04. Linar Systms in Thr Variabls with Applications 16.3 Nonlinar Systms of Equations

More information