Logic Synthesis and Verification


 Morgan Ray
 1 years ago
 Views:
Transcription
1 Logi Synthesis nd Verifition SOPs nd Inompletely Speified Funtions JieHong Rolnd Jing 江介宏 Deprtment of Eletril Engineering Ntionl Tiwn University Fll 2010 Reding: Logi Synthesis in Nutshell Setion 2 most of the following slides re y ourtesy of Andres Kuehlmnn 1 2 Boolen Funtion Representtion Sum of Produts A funtion n e represented y sum of ues (produts): E.g., f = + + Sine eh ue is produt of literls, this is sum of produts (SOP) representtion An SOP n e thought of s set of ues F E.g., F = {,, } A set of ues tht represents f is lled over of f E.g., F 1 ={,, } nd F 2 ={,,, } re overs of f = + +. List of Cues (Cover Mtrix) We often use mtrix nottion to represent over: Exmple F = + d = d d or d Eh row represents ue 1 mens tht the positive literl ppers in the ue 0 mens tht the negtive literl ppers in the ue 2 (or ) mens tht the vrile does not pper in the ue. It impliitly represents oth 0 nd 1 vlues. 3 4
2 PLA A PLA is (multipleoutput) funtion f : B n B m represented in SOP form n=3, m=3 over mtrix PLA Eh distint ue ppers just one in the ANDplne, nd n e shred y (multiple) outputs in the ORplne, e.g., ue () f 1 f 2 f Extensions from singleoutput to multipleoutput minimiztion theory re strightforwrd f 1 f 2 f SOP Irredundnt Cue The over (set of SOPs) n effiiently represent mny prtil logi funtions (i.e., for mny prtil funtions, there exist smll overs) Twolevel minimiztion seeks the over of minimum size (lest numer of ues) = onset minterm Note tht eh onset minterm is overed y t lest one of the ues! None of the offset minterms is overed 7 Let F = { 1, 2,, k } e over for f, i.e., f = i k =1 i A ue i F is irredundnt if F\{ i } f Exmple f = + + F\{} f Not overed 8
3 Prime Cue A literl x ( vrile or its negtion) of ue F (over of f) is prime if (F \ {}) { x } f, where x (oftor w.r.t. x) is with literl x of deleted A ue of F is prime if ll its literls re prime Exmple f = xy + xz + yz = xy; y = x (literl y deleted) F \ {} { y } = x + xz + yz z yz xz x Prime nd Irredundnt Cover Definition 1. A over is prime (resp. irredundnt) if ll its ues re prime (resp. irredundnt) Definition 2. A prime (ue) of f is essentil (essentil prime) if there is onset minterm (essentil vertex) in tht prime ut not in ny other prime. Definition 3. Two ues re orthogonl if they do not hve ny minterm in ommon E.g. 1 = x y 2 = y z re orthogonl 1 = x y 2 = y z re not orthogonl inequivlent to f sine offset vertex is overed x y 9 10 Prime nd Irredundnt Cover Exmple f = + d + d is prime nd irredundnt. is essentil sine d, ut not in d or d or d Why is d not n essentil vertex of? Wht is n essentil vertex of? Wht other ue is essentil? Wht prime is not essentil? d d d Inompletely Speified Funtion Let F = (f, d, r) : B n {0, 1, *}, where * represents don t re. f = onset funtion f(x)=1 F(x)=1 r = offset funtion r(x)=1 F(x)=0 d = don t re funtion d(x)=1 F(x)=* (f,d,r) forms prtition of B n, i.e, f + d + r = B n (f d) = (f r) = (d r) = (pirwise disjoint) (Here we don t distinguish hrteristi funtions nd the sets they represent) 11 12
4 Inompletely Speified Funtion A ompletely speified funtion g is over for F = (f,d,r) if f g f+d g r = if x d (i.e. d(x)=1), then g(x) n e 0 or 1; if x f, then g(x) = 1; if x r, then g(x) = 0 We don t re whih vlue g hs t x d Prime of Inompletely Speified Funtion Definition. A ue is prime of F = (f,d,r) if f+d (n implint of f+d), nd no other implint (of f+d) ontins (i.e., it is simply prime of f+d) Definition. Cue j of over G = { i } of F = (f,d,r) is redundnt if f G\{ j }; otherwise it is irredundnt Note tht f+d r = Prime of Inompletely Speified Funtion Exmple Consider logi minimiztion of F(,,)=(f,d,r) with f= + + nd d = + on off don t re F 1 ={,, } Expnd F 2 ={,, } is redundnt is prime F 3 = {, } Expnd Cheking of Prime nd Irredundny Let G e over of F = (f,d,r). Let D e over for d i G is redundnt iff i (G\{ i }) D (1) (Let G i G\{ i } D. Sine i G i nd f G f+d, then i i f+ i dnd i f G\{ i }. Thus f G\{ i }.) A literl l i is prime if ( i \{ l }) ( = ( i ) l ) is not n implint of F A ue i is prime of F iff ll literls l i re prime Literl l i is not prime ( i ) l f+d (2) Note: Both tests (1) nd (2) n e heked y tutology (to e explined): (G i ) i 1 (implies i redundnt) (f d) (i)l 1 (implies l not prime) The ove two oftors re with respet to ues insted of literls F 4 = {, } 15 16
5 (Literl) Coftor Let f : B n B e Boolen funtion, nd x= (x 1, x 2,, x n ) the vriles in the support of f; the oftor f of f y literl = x i or = x i is f xi (x 1, x 2,, x n ) = f (x 1,, x i1, 1, x i+1,, x n ) f xi (x 1, x 2,, x n ) = f (x 1,, x i1, 0, x i+1,, x n ) The omputtion of the oftor is fundmentl opertion in Boolen resoning! Exmple f = + f = 17 (Literl) Coftor The oftor Cx j of ue C (representing some Boolen funtion) with respet to literl x j is C if x j nd x j do not pper in C C\{x j } if x j ppers positively in C, i.e., x j C if x j ppers negtively in C, i.e., x j C Exmple C = x 1 x 4 x 6, Cx 2 = C (x 2 nd x 2 do not pper in C ) Cx 1 = x 4 x 6 (x 1 ppers positively in C) Cx 4 = (x 4 ppers negtively in C) 18 (Literl) Coftor Exmple F = + d + d F = + d (Just drop everywhere nd throw wy ues ontining literl ) Coftor nd disjuntion ommute! Shnnon Expnsion Let f : B n B Shnnon Expnsion: f = x i f xi + x i f xi Theorem: F is over of f. Then F = x i F xi + x i F xi We sy tht f nd F re expnded out x i, nd x i is lled the splitting vrile 19 20
6 Shnnon Expnsion Exmple F = + + F = F + F = (++)+ () = +++ Cue got split into two ues (Cue) Coftor The oftor f C of f y ue C is f with the fixed vlues indited y the literls of C E.g., if C = x i x j, then x i = 1 nd x j = 0 For C = x 1 x 4 x 6, f C is just the funtion f restrited to the suspe where x 1 = x 6 = 1 nd x 4 = 0 Note tht f C does not depend on x 1,x 4 or x 6 nymore (However, we still onsider f C s funtion of ll n vriles, it just hppens to e independent of x 1,x 4 nd x 6 ) x 1 f f x1 E.g., for f = +, f = f = nd f = (Cue) Coftor The oftor of the over F of some funtion f is the sum of the oftors of eh of the ues of F If F={ 1, 2,, k } is over of f, then F = {( 1 ), ( 2 ),, ( k ) } is over of f Continment vs. Tutology A fundmentl theorem tht onnets funtionl ontinment nd tutology: Theorem. Let e ue nd f funtion. Then f f 1. Proof. We use the ft tht xf x = xf, nd f x is independent of x. ( ) Suppose f 1. Then f = f =. Thus, f. ( ) Suppose f. Then f+=f. In ddition, (f+) = f +1=1. Thus, f =1. f 23 24
7 Cheking of Prime nd Irredundny (Revisited) Let G e over of F = (f,d,r). Let D e over for d i G is redundnt iff i (G\{ i }) D (1) (Let G i G\{ i } D. Sine i G i nd f G f+d, then i i f+ i dnd i f G\{ i }. Thus f G\{ i }.) A literl l i is prime if ( i \{ l }) ( = ( i ) l ) is not n implint of F A ue i is prime of F iff ll literls l i re prime Literl l i is not prime ( i ) l f+d (2) Note: Both tests (1) nd (2) n e heked y tutology (explined): (G i ) i 1 (implies i redundnt) (f d) (i)l 1 (implies l not prime) The ove two oftors re with respet to ues insted of literls Generlized Coftor Definition. Let f, g e ompletely speified funtions. The generlized oftor of f with respet to g is the inompletely speified funtion: o( f,g) ( f g,g, f g) Definition. Let = (f, d, r) nd g e given. Then o(,g) ( f g,d g,r g) Shnnon vs. Generlized Coftor Shnnon vs. Generlized Coftor Let g = x i. Shnnon oftor is f xi (x 1, x 2,, x n ) = f (x 1,, x i1, 1, x i+1,, x n ) Generlized oftor with respet to g=x i is Note tht o( f, x i ) ( f x i, x i, f x i ) f on off Don t t re f x i f xi f x i x i f x i In ft f xi is the unique over of o(f, x i ) independent of the vrile x i. o( f,) ( f,, f ) f 27 28
8 Shnnon vs. Generlized Coftor Shnnon vs. Generlized Coftor o( f,) ( f,, f ) Shnnon Coftor Generlized Coftor f So f f f f f f x f x x f x f f x y f xy f g y f y g y f x f x f g o( f,g) g o( f,g) o(o( f, g),h) o( f,gh) o( f g,h) o( f,h) o(g,h) o( f,g) o( f,g) We will get k to the use of generlized oftor lter Dt Struture for SOP Mnipultion most of the following slides re y ourtesy of Andres Kuehlmnn Opertion on Cue Lists AND opertion: tke two lists of ues ompute pirwise AND etween individul ues nd put result on new list represent ues in omputer words implement set opertions s itvetor opertions Algorithm AND(List_of_Cues C1,List_of_Cues C2) { C = foreh 1 C1 { foreh 2 C2 { = 1 2 C = C } } return C } 31 32
Logic Synthesis and Verification
Logi Synthesis nd Verifition SOPs nd Inompletely Speified Funtions JieHong Rolnd Jing 江介宏 Deprtment of Eletril Engineering Ntionl Tiwn University Fll 22 Reding: Logi Synthesis in Nutshell Setion 2 most
More informationEngr354: Digital Logic Circuits
Engr354: Digitl Logi Ciruits Chpter 4: Logi Optimiztion Curtis Nelson Logi Optimiztion In hpter 4 you will lern out: Synthesis of logi funtions; Anlysis of logi iruits; Tehniques for deriving minimumost
More informationPropositional models. Historical models of computation. Application: binary addition. Boolean functions. Implementation using switches.
Propositionl models Historil models of omputtion Steven Lindell Hverford College USA 1/22/2010 ISLA 2010 1 Strt with fixed numer of oolen vriles lled the voulry: e.g.,,. Eh oolen vrile represents proposition,
More informationSection 4.4. Green s Theorem
The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higherdimensionl nlogues) with the definite integrls
More informationUnit 4. Combinational Circuits
Unit 4. Comintionl Ciruits Digitl Eletroni Ciruits (Ciruitos Eletrónios Digitles) E.T.S.I. Informáti Universidd de Sevill 5/10/2012 Jorge Jun 2010, 2011, 2012 You re free to opy, distriute
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationLecture 1  Introduction and Basic Facts about PDEs
* 18.15  Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1  Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
More informationf (x)dx = f(b) f(a). a b f (x)dx is the limit of sums
Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx
More informationDiscrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α
Disrete Strutures, Test 2 Mondy, Mrh 28, 2016 SOLUTIONS, VERSION α α 1. (18 pts) Short nswer. Put your nswer in the ox. No prtil redit. () Consider the reltion R on {,,, d with mtrix digrph of R.. Drw
More informationMatrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix
tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri
More information( ) { } [ ] { } [ ) { } ( ] { }
Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationQUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 004 Exerise  0509 Exerise  093 Exerise  3 45 Exerise  4 6 Answer Key 78 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
More informationCS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6
CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized
More informationarxiv: v1 [math.ca] 21 Aug 2018
rxiv:1808.07159v1 [mth.ca] 1 Aug 018 Clulus on Dul Rel Numbers Keqin Liu Deprtment of Mthemtis The University of British Columbi Vnouver, BC Cnd, V6T 1Z Augest, 018 Abstrt We present the bsi theory of
More informationTechnische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution
Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:
More informationUniversity of Sioux Falls. MAT204/205 Calculus I/II
University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques
More informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationBoolean Algebra. Boolean Algebra
Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with:  two inry opertions, commonly denoted y + nd,  unry opertion, usully denoted y ˉ or ~ or,  two elements usully clled zero nd
More informationFunctions. mjarrar Watch this lecture and download the slides
9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. OnetoOne Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides
More information7.2 Riemann Integrable Functions
7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous
More informationNecessary and sucient conditions for some two. Abstract. Further we show that the necessary conditions for the existence of an OD(44 s 1 s 2 )
Neessry n suient onitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos, M. Mitrouli y, n Jennifer Seerry z Deite to Professor Anne Penfol Street Astrt We give new lgorithm whih llows us
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationApril 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm
More informationBoolean algebra.
http://en.wikipedi.org/wiki/elementry_boolen_lger Boolen lger www.tudorgir.com Computer science is not out computers, it is out computtion nd informtion. computtion informtion computer informtion Turing
More informationPart 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationLecture Notes No. 10
2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite
More informationSolutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!
Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:
More informationNONDETERMINISTIC FSA
Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is
More informationLogic Synthesis and Verification
Logic Synthesis and Verification JieHong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 24 TwoLevel Logic Minimization (/2) Reading: Logic Synthesis in a Nutshell
More informationOverview of Today s Lecture:
CPS 4 Computer Orgniztion nd Progrmming Lecture : Boolen Alger & gtes. Roert Wgner CPS4 BA. RW Fll 2 Overview of Tody s Lecture: Truth tles, Boolen functions, Gtes nd Circuits Krnugh mps for simplifying
More informationChapter 4 StateSpace Planning
Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 StteSpe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different
More informationwhere the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b
CS 2942 9/11/04 Quntum Ciruit Model, SolovyKitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits  Universl Gte Sets A lssil iruit implements multioutput oolen funtion f : {0,1}
More information1. Logic verification
. Logi verifition Bsi priniples of OBDD s Vrile ordering Network of gtes => OBDD s FDD s nd OKFDD s Resoning out iruits Struturl methods Stisfiility heker Logi verifition The si prolem: prove tht two iruits
More informationTOPIC: LINEAR ALGEBRA MATRICES
Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A ... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More information18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2106
8. Problem Set Due Wenesy, Ot., t : p.m. in  Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the oneelement, linerly epenent sets forme from these. (b) Wht re the twoelement, linerly
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationLinear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
More informationDiscrete Structures Lecture 11
Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.
More informationT b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.
Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene
More informationEvaluating Definite Integrals. There are a few properties that you should remember in order to assist you in evaluating definite integrals.
Evluting Definite Integrls There re few properties tht you should rememer in order to ssist you in evluting definite integrls. f x dx= ; where k is ny rel constnt k f x dx= k f x dx ± = ± f x g x dx f
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationGreen s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e
Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let
More informationy1 y2 DEMUX a b x1 x2 x3 x4 NETWORK s1 s2 z1 z2
BOOLEAN METHODS Giovnni De Miheli Stnford University Boolen methods Exploit Boolen properties. { Don't re onditions. Minimiztion of the lol funtions. Slower lgorithms, etter qulity results. Externl don't
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATHGA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1. 1 [(y ) 2 + yy + y 2 ] dx,
MATH3403: Green s Funtions, Integrl Equtions nd the Clulus of Vritions 1 Exmples 5 Qu.1 Show tht the extreml funtion of the funtionl I[y] = 1 0 [(y ) + yy + y ] dx, where y(0) = 0 nd y(1) = 1, is y(x)
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationfor all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx
Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion
More informationKENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS CLASS  XII MATHEMATICS (Relations and Functions & Binary Operations)
KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS 67 CLASS  XII MATHEMATICS (Reltions nd Funtions & Binry Opertions) For Slow Lerners:  A Reltion is sid to e Reflexive if.. every A
More informationF / x everywhere in some domain containing R. Then, + ). (10.4.1)
0.4 Green's theorem in the plne Double integrls over plne region my be trnsforme into line integrls over the bounry of the region n onversely. This is of prtil interest beuse it my simplify the evlution
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255  Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationMath 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)
Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte singlevrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion
More informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationpadic Egyptian Fractions
padic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Setup 3 4 pgreedy Algorithm 5 5 pegyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationArrow s Impossibility Theorem
Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationFor a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then
Slrs7.2ADV.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities:
More informationELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE378: Computer Hardware Design Winter Notes  Unit 1
ELETRIL ND OMPUTER ENGINEERING DEPRTMENT, OKLND UNIVERSIT EE78: omputer Hrdwre Design Winter 016 INTRODUTION TO LOGI IRUITS Notes  Unit 1 OOLEN LGER This is the oundtion or designing nd nlyzing digitl
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationThe Area of a Triangle
The e of Tingle tkhlid June 1, 015 1 Intodution In this tile we will e disussing the vious methods used fo detemining the e of tingle. Let [X] denote the e of X. Using se nd Height To stt off, the simplest
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationCh. 2.3 Counting Sample Points. Cardinality of a Set
Ch..3 Counting Smple Points CH 8 Crdinlity of Set Let S e set. If there re extly n distint elements in S, where n is nonnegtive integer, we sy S is finite set nd n is the rdinlity of S. The rdinlity of
More informationChapter 3. Vector Spaces. 3.1 Images and Image Arithmetic
Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner
More informationis equal to  (A) abc (B) 2abc (C) 0 (D) 4abc (sinx) + a 2 (sin 2 x) a n (A) 1 (B) 1 (C) 0 (D) 2 is equal to 
JMthemtics XRCIS  0 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR). The vlue of determinnt c c c c c c (A) c (B) c (C) 0 (D) 4c. If sin x cos x cos 4x cos x cos x sin x 4 cos x sin x
More informationComputing data with spreadsheets. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt
Computing dt with spredsheets Exmple: Computing tringulr numers nd their squre roots. Rell, we showed 1 ` 2 ` `n npn ` 1q{2. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2:
More informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
More informationAT100  Introductory Algebra. Section 2.7: Inequalities. x a. x a. x < a
Section 2.7: Inequlities In this section, we will Determine if given vlue is solution to n inequlity Solve given inequlity or compound inequlity; give the solution in intervl nottion nd the solution 2.7
More informationDATABASE DESIGN I  1DL300
DATABASE DESIGN I  DL300 Fll 00 An introductory course on dtse systems http://www.it.uu.se/edu/course/homepge/dstekn/ht0/ Mnivskn Sesn Uppsl Dtse Lortory Deprtment of Informtion Technology, Uppsl University,
More informationMetodologie di progetto HW Technology Mapping. Last update: 19/03/09
Metodologie di progetto HW Tehnology Mpping Lst updte: 19/03/09 Tehnology Mpping 2 Tehnology Mpping Exmple: t 1 = + b; t 2 = d + e; t 3 = b + d; t 4 = t 1 t 2 + fg; t 5 = t 4 h + t 2 t 3 ; F = t 5 ; t
More informationPart I: Study the theorem statement.
Nme 1 Nme 2 Nme 3 A STUDY OF PYTHAGORAS THEOREM Instrutions: Together in groups of 2 or 3, fill out the following worksheet. You my lift nswers from the reding, or nswer on your own. Turn in one pket for
More informationCombinational Logic. Precedence. Quick Quiz 25/9/12. Schematics à Boolean Expression. 3 Representations of Logic Functions. Dr. Hayden So.
5/9/ Comintionl Logic ENGG05 st Semester, 0 Dr. Hyden So Representtions of Logic Functions Recll tht ny complex logic function cn e expressed in wys: Truth Tle, Boolen Expression, Schemtics Only Truth
More informationReview: The Riemann Integral Review: The definition of R b
eview: The iemnn Integrl eview: The definition of b f (x)dx. For ontinuous funtion f on the intervl [, b], Z b f (x) dx lim mx x i!0 nx i1 f (x i ) x i. This limit omputes the net (signed) re under the
More informationSolutions to Assignment 1
MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Clculus of One Vrile II Fll 2017 Textook: Single Vrile Clculus: Erly Trnscendentls, 7e, y Jmes Stewrt Unit 2 Skill Set Importnt: Students should expect test questions tht require synthesis of
More information8 THREE PHASE A.C. CIRCUITS
8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the solled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationAP Calculus AB Unit 4 Assessment
Clss: Dte: 004 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationProbability. b a b. a b 32.
Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationHardware Verification 2IMF20
Hrdwre Verifition 2IMF20 Julien Shmltz Leture 02: Boolen Funtions, ST, CEC Course ontent  Forml tools Temporl Logis (LTL, CTL) Domin Properties System Verilog ssertions demi & Industrils Proessors Networks
More informationAP CALCULUS Test #6: Unit #6 Basic Integration and Applications
AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret
More informationFinite State Automata and Determinisation
Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Nondeterministi Finite Stte Automt (nf) 4 Regulr Expressions
More informationEigenvectors and Eigenvalues
MTB 050 1 ORIGIN 1 Eigenvets n Eigenvlues This wksheet esries the lger use to lulte "prinipl" "hrteristi" iretions lle Eigenvets n the "prinipl" "hrteristi" vlues lle Eigenvlues ssoite with these iretions.
More informationBoolean Algebra. Boolean Algebras
Boolen Algebr Boolen Algebrs A Boolen lgebr is set B of vlues together with:  two binry opertions, commonly denoted by + nd,  unry opertion, usully denoted by or ~ or,  two elements usully clled zero
More informationCS 310 (sec 20)  Winter Final Exam (solutions) SOLUTIONS
CS 310 (sec 20)  Winter 2003  Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)
More informationInstructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting.
ID: B CSE 2021 Computer Orgniztion Midterm Test (Fll 2009) Instrutions This is losed ook, 80 minutes exm. The MIPS referene sheet my e used s n id for this test. An 8.5 x 11 Chet Sheet my lso e used s
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationTutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.
Mth 5 Tutoril Week 1  Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationLecture 11 Binary Decision Diagrams (BDDs)
C 474A/57A ComputerAie Logi Design Leture Binry Deision Digrms (BDDs) C 474/575 Susn Lyseky o 3 Boolen Logi untions Representtions untion n e represente in ierent wys ruth tle, eqution, Kmp, iruit, et
More information