DETERMINING SIGNIFICANT FACTORS AND THEIR EFFECTS ON SOFTWARE ENGINEERING PROCESS QUALITY
|
|
- Marcus Underwood
- 6 years ago
- Views:
Transcription
1 DETERMINING SIGNIFINT FTORS ND THEIR EFFETS ON SOFTWRE ENGINEERING PROESS QULITY R. Rdhrmnn Jeng-Nn Jung Mil to: Shool of Engineering, Merer Universit, Mon, G 37 US strt This pper nlzes the qulit of n ongoing softwre mintenne projet using defet densit dt from prior nd urrent relese hnges. The ojetive is to test the signifine of ftors suh s developer experiene, the omplexit of the hnge, the size of the hnge (mesured in lines of ode), nd the vrious intertions ginst the defet densit of prtiulr hnge. The two phses of the hnge tht del diretl with ode nd impt qulit the most re ode design nd ode development, so the hve oth een nlzed to see if there re n signifint ftors. For the signifint ftors, regreion equtions hve een developed followed residul nlses. In the ode Design (D) phse of the softwre development, developer experiene nd ode hnge omplexit were found to e ftors tht n impt the ost nd shedule of the overll projet. For the ode Development Wlkthrough (W) phse, no ftors were found to e signifint. It is due to high vriilit in softwre development nd this m hnge with the ddition of more input dt. Sine defet densit is the item of importne in this stud, it would e helpful to rete upper nd lower ontrol limits for different tretment omintions. This would llow the projet mnger to monitor the proe performne nd t ordingl to n normlities. Kewords: Softwre engineering, ode design, ode development, proe qulit, performne mesures, signifint ftors. Introdution The use of sttistis hs long een regrded s w to mesure the performne nd qulit for ll tpes of engineering prolems [, ]. lthough most ommonl used in the mnufturing world, where there re unlimited supplies of dt, the use of sttistis hs slowl moved into the softwre engineering environment. Mn still question the use of sttistis in softwre engineering simpl euse of the nture of the work. In mn ses, proees re not repetle or one hnge is just muh smller or lrger thn the previous. The question then eomes: How do ou ompre lrge softwre hnge to smll softwre hnge nd feel omfortle with the results? The introdution of the Softwre Engineering Institute nd the MMI proe for developing softwre hs helped pioneer the use of sttistis in softwre engineering [3]. The MMI model fouses on using sttistis to mnge the performne of the projet with respet to meeting time nd udget onstrints s well s mintining the qulit of the softwre. Mnging time nd udget onstrints is not new prolem n streth of the imgintion nd is firl ommon ro ll tpes of engineering projets [4]. The ide of mnging qulit in softwre is somewht hrder to understnd for most people. For strters, how do ou define qulit? Is it the
2 numer of softwre defets relesed to the user, or does it inlude ll of the defets found during testing? If it is the first, how n ou ount defets if ou don t know the exist? Mn defets go undisovered for ers fter the softwre is relesed while others surfe lmost immeditel. One ommon w to mesure the qulit of the softwre is to tke qulit mesurements t predetermined milestones nd ompre the dt to onfidene intervls to p judgment on the qulit of the softwre [5]. In this pper, the qulit of n ongoing softwre mintenne projet is nlzed using defet densit dt from prior nd urrent relese hnges [6]. The ojetive is to test the signifine of ftors suh s developer experiene, the omplexit of the hnge, the size of the hnge (mesured in lines of ode), nd the vrious intertions ginst the defet densit of prtiulr hnge. The two phses of the hnge tht del diretl with ode nd impt qulit the most re ode design nd ode development, so the will oth e nlzed to see if there re n signifint ftors [7, 8]. For the ftors tht were found to e signifint, regreion equtions were developed followed residul nlses. Methodolog The lrgest prolem tht projet mnger fes when deling with softwre is how to mnge qulit [9]. Poor qulit often leds to shedule nd ost overruns tht n jeoprdize future worklod. Of the mn ftors tht go into softwre development, the most prominent ftors re experiene of the emploee (), omplexit of the hnge (), nd the lines of ode in the hnge (). sed on these three ftors, the projet mnger should e le to predit whether prtiulr hnge will enounter qulit iues in the future. prediting when there re going to e qulit iues, the projet mnger n e protive putting more experiened developer on the hnge or even split up the hnge to redue the size nd omplexit []. Sine there re three ftors, the est w to ddre the signifine or lk thereof is to ondut three ftor fixed effet experiment []. lthough more replites re etter, the dt set does meet the minimum requirement of t lest two replites without whih the error sum of squres, whih is n importnt prt of the nlsis, ould not e omputed. Using the populr dot nottion, the totl sum of squres nd the sum of squres for ftors,, nd re omputed from the following equtions: T i j k n l ijkl, () i j k i..., () n. j.., (3) n.. k.. (4) n
3 To ompute the two-ftor intertion sum of squres, the totls for the x, x, nd x re needed; the sum of squres for the two-ftor intertions re: i j ij.. n, (5) i k i. k. n, (6) j k. jk. n, (7) The three-ftor intertion is omputed from the three-w ell totls using: i j k ijk. n, (8) The error sum of squres is simpl the totl sum of squres minus the sum of squres for eh effet nd intertion. E SS. (9) T sutotls( ) fter omputing the sum of squres, the entire nlsis of vrine (NOV) tle n e ompleted. uming n lph vlue of, the signifine of ftor is verified using the f-test []. Further nlsis is needed on the dt set to ensure tht there re no violtions of si umptions tht ould invlidte the results. So the residul vlues of the experiment need to e lulted. In order to lulte the residuls, the min effets s well s the two-ftor nd the three-ftor intertion effets need to e estimted. The min effets re estimted using: [ ()], () [ ()], () [ ()]. () The two-ftor intertion effets re estimted using: [ + () + + ], (3) [ + () + + ], (4)
4 [ + () + + ]. (5) The three-ftor intertion effet is estimted tking the verge differene etween the intertion t the two levels of or: [ ()]. (6) The effet estimtes n then e used to develop regreion model to lulte the residuls of the experiment. The fitted regreion model is: where: 3 β + β x + β x + β x +... (7)... β ; β ; β ; β 3. (8) The remining oeffiients n e found in similr fshion. The tul regreion model will onl onsist of the oeffiients tht orrespond to the ftors deemed signifint. The residuls n then e lulted using the regreion model nd evluting the oserved vlues t eh tretment omintion. The results n then e plotted on norml proilit plot to illustrte n normlities []. Results nd Disuions The smples tken over period of three ers from softwre development projet t the 58 th Softwre Mintenne Squdron t Roins ir Fore se, Georgi, during ode Design (D) nd ode Development Wlkthrough (W) phses re shown in Tles nd respetivel. The tles outline different ode hnges ordered the experiene level of the developer who mde the hnge, the omplexit level of the hnge (tehnil diffiult), nd the size of the hnge (mesured in lines of ode). For the dt nlsis, the Minit sttistil nlsis softwre pkge ws used to perform NOV, regreion nlsis, nd omputtion of norml plots nd residuls []. Tle 3 shows the NOV for the D phse. The P-vlues indite Developer Experiene nd ode omplexit re signifint ftors (ounting P-vlue of le thn s signifint). The D phse R vlue is 7.43%. The norml proilit plot is shown in Figure. Figure shows the residuls plot. Tle. D phse defet densit 3
5 Tle. W phse defet densit Tle 3. D phse NOV Norml Proilit Plot (response is D) Versus Fits (response is D) Perent Fitted Vlue Figure. Norml proilit plot Figure. s plot Tle 4 shows the NOV for the W phse nd it does not indite n signifint ftors. In ddition, the W phse R vlue is poor (39.9%). The norml proilit plot nd the residuls plot re shown in Figures 3 nd 4 respetivel. Tle 4. W phse NOV
6 Norml Proilit Plot (response is W) Versus Fits (response is W) Perent Fitted Vlue.5 Figure 3. Norml proilit plot Figure 4. s plot Tle 4 shows the NOV for the regreion nlsis during D phse onsidering developer experiene, R omplexit, nd R size s ftors. Tle 4. D phse NOV - Regreion nlsis The fitted regreion eqution is: D Developer Exp. + R omplexit - 49 R Size (9) The Minit regreion nlsis of the D phse shows firl poor R vlue of 54.6%. This is due to Minit s ehvior of ounting ll ftors, even non-signifint ones, in the regreion nlsis. When the non-signifint ftor, R Size, is exluded from the nlsis, muh etter R vlue of 9% is found. Figure 5 shows the norml proilit plot. The residuls plot is shown in Figure 6. Norml Proilit Plot (response is D) Versus Fits (response is D) Perent Fitted Vlue Figure 5. Norml proilit plot Figure 6. s plot
7 When onl signifint ftors were inluded to fit stright line to the D phse dt, the R vlue of the signifint ftors (developer experiene nd R omplexit) is found to e 9.5% (Figure 7). This mens tht these two ftors must e n importnt prt of the predition model for ode Design. When plotted s n exponentil model, the R vlue is even higher, 95.44%, with one outlier (Figure 8). The residuls plot lso shows different pttern when onl signifint ftor vlues re used (Figure 9). Norml Sore Norml Proilit Plot 3.449x +.5 R s Figure 7. Norml proilit plot Norml Sore Norml Proilit Plot.3759e.95x R s Figure 8. Norml proilit plot s vs Fitted s Fitted Figure 9. s plot Tle 5 shows the NOV for the regreion nlsis during W phse onsidering developer experiene, R omplexit, nd R size s ftors. Tle 5. W phse NOV - Regreion nlsis The fitted regreion eqution is: W Developer Exp R omplexit R Size () The Minit regreion nlsis of the W phse shows poor R vlue of 8.6% nd P-vlue of.4. From the regreion nlsis, it is ler tht n useful predition model from the
8 olleted dt nnot e derived. The norml proilit plot is shown in Figure. Figure shows the residuls plot. Norml Proilit Plot (response is W) Versus Fits (response is W) Perent Fitted Vlue.5 Figure. Norml proilit plot Figure. s plot onlusions This stud hs shown tht in the ode Design phse of the softwre development projet, developer experiene nd ode hnge omplexit should e onsidered s ftors tht n impt the ost nd shedule of the overll projet. For the ode Wlkthrough phse, no ftors s reorded n e onsidered signifint. lthough this m hnge with the ddition of more input dt, this result is not unexpeted due to the high vriilit of softwre development. The results of this stud n e used in the rel world in the res of Quntittive Projet Mngement nd Orgniztionl Proe Performne, whih involve the use of sttistis to mesure performne nd mke deisions [9]. From these results, the softwre tem will e le to etter predit res of onern in future development les nd determine w to est hndle them. This will help the tem mintin the gol of produing qulit produt while sting within time nd udget onstrints. This stud lso leves plent of room for future stud in determining onfidene nd predition intervls. Sine defet densit is the item of importne in this stud, it would e helpful to rete n upper nd lower ontrol limits for the different tretment omintions. This would llow the projet mnger to monitor the proe performne nd t ordingl to n normlities. Referenes [] Hines, W. W., Montgomer, D.., Goldsmn, D. M., nd orror. M., Proilit nd Sttistis in Engineering, 4 th Edition, John Wile & Sons, 3. [] Lewis, E.E.; Introdution to Reliilit Engineering, [3] rnegie Mellon, MMI Model V., Softwre Engineering Institute, 6. nd Edition, Wile & Sons, 996. [4] Jeffre, R.., Proilit nd the rt of Judgment, mridge Universit Pre, 99. [5] 58 SMXS Tehnil Stff, 58 th SMXS Squdron Defined Softwre Proe, 58 SMXS, ugust 4. [6] 58 SMXS Tehnil Stff, Softwre Qulit urne Pln V., 58 SMXS, Ferur 7. [7] 58 SMXS Flight D Tehnil Stff, M-3H Softwre Proedure Mnul V., 58 SMXS, Jnur 7.
9 [8] 58 SMXS Flight D Tehnil Stff, Softwre Development Pln for the M-3H omt Tlon II Opertionl Flight Progrm V3., 58 SMXS, Jnur 7. [9] 58 SMXS Flight D Tehnil Stff, Quntittive Projet Mngement Metri Inditors for the omt Tlon II, 58 SMXS, Deemer 6. [] 58 SMXS Flight D Tehnil Stff, M-3H omt Tlon II hnge Request omplexit Rnking Proedure V., 58 SMXS, pril 6.
Project 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 informationIowa Training Systems Trial Snus Hill Winery Madrid, IA
Iow Trining Systems Tril Snus Hill Winery Mdrid, IA Din R. Cohrn nd Gil R. Nonneke Deprtment of Hortiulture, Iow Stte University Bkground nd Rtionle: Over the lst severl yers, five sttes hve een evluting
More informationReview Topic 14: Relationships between two numerical variables
Review Topi 14: Reltionships etween two numeril vriles Multiple hoie 1. Whih of the following stterplots est demonstrtes line of est fit? A B C D E 2. The regression line eqution for the following grph
More informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationA Non-parametric Approach in Testing Higher Order Interactions
A Non-prmetri Approh in Testing igher Order Intertions G. Bkeerthn Deprtment of Mthemtis, Fulty of Siene Estern University, Chenkldy, Sri Lnk nd S. Smit Deprtment of Crop Siene, Fulty of Agriulture University
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 information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationANALYSIS AND MODELLING OF RAINFALL EVENTS
Proeedings of the 14 th Interntionl Conferene on Environmentl Siene nd Tehnology Athens, Greee, 3-5 Septemer 215 ANALYSIS AND MODELLING OF RAINFALL EVENTS IOANNIDIS K., KARAGRIGORIOU A. nd LEKKAS D.F.
More informationTHE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL
THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL P3.1 Kot Iwmur*, Hiroto Kitgw Jpn Meteorologil Ageny 1. INTRODUCTION Jpn Meteorologil Ageny
More informationTable of Content. c 1 / 5
Tehnil Informtion - t nd t Temperture for Controlger 03-2018 en Tble of Content Introdution....................................................................... 2 Definitions for t nd t..............................................................
More informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
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 informationIntermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths
Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t
More informationSomething found at a salad bar
Nme PP Something found t sld r 4.7 Notes RIGHT TRINGLE hs extly one right ngle. To solve right tringle, you n use things like SOH-H-TO nd the Pythgoren Theorem. n OLIQUE TRINGLE hs no right ngles. To solve
More informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More informationComparing the Pre-image and Image of a Dilation
hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity
More informationChapter 8 Roots and Radicals
Chpter 8 Roots nd Rdils 7 ROOTS AND RADICALS 8 Figure 8. Grphene is n inredily strong nd flexile mteril mde from ron. It n lso ondut eletriity. Notie the hexgonl grid pttern. (redit: AlexnderAIUS / Wikimedi
More informationCore 2 Logarithms and exponentials. Section 1: Introduction to logarithms
Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................
More informationGeneralization of 2-Corner Frequency Source Models Used in SMSIM
Generliztion o 2-Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville
More information] dx (3) = [15x] 2 0
Leture 6. Double Integrls nd Volume on etngle Welome to Cl IV!!!! These notes re designed to be redble nd desribe the w I will eplin the mteril in lss. Hopefull the re thorough, but it s good ide to hve
More informationVIBRATION ANALYSIS OF AN ISOLATED MASS WITH SIX DEGREES OF FREEDOM Revision G
B Tom Irvine Emil: tom@virtiondt.om Jnur 8, 3 VIBRATION ANALYSIS OF AN ISOLATED MASS WITH SIX DEGREES OF FREEDOM Revision G Introdution An vionis omponent m e mounted with isoltor grommets, whih t s soft
More informationare coplanar. ˆ ˆ ˆ and iˆ
SML QUSTION Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor ii The question pper onsists of 6 questions divided into three Setions, B nd C iii Question No
More informationData Structures LECTURE 10. Huffman coding. Example. Coding: problem definition
Dt Strutures, Spring 24 L. Joskowiz Dt Strutures LEURE Humn oing Motivtion Uniquel eipherle oes Prei oes Humn oe onstrution Etensions n pplitions hpter 6.3 pp 385 392 in tetook Motivtion Suppose we wnt
More informationBehavior Composition in the Presence of Failure
Behvior Composition in the Presene of Filure Sestin Srdin RMIT University, Melourne, Austrli Fio Ptrizi & Giuseppe De Giomo Spienz Univ. Rom, Itly KR 08, Sept. 2008, Sydney Austrli Introdution There re
More information= x x 2 = 25 2
9.1 Wrm Up Solve the eqution. 1. 4 2 + 3 2 = x 2 2. 13 2 + x 2 = 25 2 3. 3 2 2 + x 2 = 5 2 2 4. 5 2 + x 2 = 12 2 Mrh 7, 2016 Geometry 9.1 The Pythgoren Theorem 1 Geometry 9.1 The Pythgoren Theorem 9.1
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More informationHS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem
HS Pre-Alger Notes Unit 9: Roots, Rel Numers nd The Pythgoren Theorem Roots nd Cue Roots Syllus Ojetive 5.4: The student will find or pproximte squre roots of numers to 4. CCSS 8.EE.-: Evlute squre roots
More informationAVL Trees. D Oisín Kidney. August 2, 2018
AVL Trees D Oisín Kidne August 2, 2018 Astrt This is verified implementtion of AVL trees in Agd, tking ides primril from Conor MBride s pper How to Keep Your Neighours in Order [2] nd the Agd stndrd lirr
More information6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR
6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht
More information6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR
6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht
More information22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of:
22: Union Fin CS 473u - Algorithms - Spring 2005 April 14, 2005 1 Union-Fin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x) - rete set tht ontins the single element x. 2. Fin(x)
More informationAutomatic Synthesis of New Behaviors from a Library of Available Behaviors
Automti Synthesis of New Behviors from Lirry of Aville Behviors Giuseppe De Giomo Università di Rom L Spienz, Rom, Itly degiomo@dis.unirom1.it Sestin Srdin RMIT University, Melourne, Austrli ssrdin@s.rmit.edu.u
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 single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More information5. Every rational number have either terminating or repeating (recurring) decimal representation.
CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd
More 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 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 informationElectromagnetism Notes, NYU Spring 2018
Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More informationMathematics SKE: STRAND F. F1.1 Using Formulae. F1.2 Construct and Use Simple Formulae. F1.3 Revision of Negative Numbers
Mthemtis SKE: STRAND F UNIT F1 Formule: Tet STRAND F: Alger F1 Formule Tet Contents Setion F1.1 Using Formule F1. Construt nd Use Simple Formule F1.3 Revision of Negtive Numers F1.4 Sustitution into Formule
More information16z z q. q( B) Max{2 z z z z B} r z r z r z r z B. John Riley 19 October Econ 401A: Microeconomic Theory. Homework 2 Answers
John Riley 9 Otober 6 Eon 4A: Miroeonomi Theory Homework Answers Constnt returns to sle prodution funtion () If (,, q) S then 6 q () 4 We need to show tht (,, q) S 6( ) ( ) ( q) q [ q ] 4 4 4 4 4 4 Appeling
More informationNON-DETERMINISTIC FSA
Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is
More informationChapter 4 State-Space Planning
Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different
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 informationMaintaining Mathematical Proficiency
Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More information, g. Exercise 1. Generator polynomials of a convolutional code, given in binary form, are g. Solution 1.
Exerise Genertor polynomils of onvolutionl ode, given in binry form, re g, g j g. ) Sketh the enoding iruit. b) Sketh the stte digrm. ) Find the trnsfer funtion T. d) Wht is the minimum free distne of
More informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or
More informationExercise 3 Logic Control
Exerise 3 Logi Control OBJECTIVE The ojetive of this exerise is giving n introdution to pplition of Logi Control System (LCS). Tody, LCS is implemented through Progrmmle Logi Controller (PLC) whih is lled
More information21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae
MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Contents STRAND G: Alger Unit 1 Formule Student Tet Contents Setion 1.1 Using Formule 1. Construt nd Use Simple Formule 1.3 Revision of Negtive Numers 1.4
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
More informationBEGINNING ALGEBRA (ALGEBRA I)
/0 BEGINNING ALGEBRA (ALGEBRA I) SAMPLE TEST PLACEMENT EXAMINATION Downlod the omplete Study Pket: http://www.glendle.edu/studypkets Students who hve tken yer of high shool lger or its equivlent with grdes
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationTIME AND STATE IN DISTRIBUTED SYSTEMS
Distriuted Systems Fö 5-1 Distriuted Systems Fö 5-2 TIME ND STTE IN DISTRIUTED SYSTEMS 1. Time in Distriuted Systems Time in Distriuted Systems euse eh mhine in distriuted system hs its own lok there is
More information9.1 Day 1 Warm Up. Solve the equation = x x 2 = March 1, 2017 Geometry 9.1 The Pythagorean Theorem 1
9.1 Dy 1 Wrm Up Solve the eqution. 1. 4 2 + 3 2 = x 2 2. 13 2 + x 2 = 25 2 3. 3 2 2 + x 2 = 5 2 2 4. 5 2 + x 2 = 12 2 Mrh 1, 2017 Geometry 9.1 The Pythgoren Theorem 1 9.1 Dy 2 Wrm Up Use the Pythgoren
More informationNondeterministic Automata vs Deterministic Automata
Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n
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 minimum-ost
More informationAlpha Algorithm: Limitations
Proess Mining: Dt Siene in Ation Alph Algorithm: Limittions prof.dr.ir. Wil vn der Alst www.proessmining.org Let L e n event log over T. α(l) is defined s follows. 1. T L = { t T σ L t σ}, 2. T I = { t
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationEffects of Applying Accumulator Straw in Soil on Nutrient Uptake and Soil Enzyme Activity of Capsella bursa-pastoris under Cadmium Stress
Interntionl Conferene on Mnufturing Siene nd Engineering (ICMSE 2015) Effets of Applying Aumultor Strw in Soil on Nutrient Uptke nd Soil Enzyme Ativity of Cpsell burs-pstoris under Cdmium Stress Jin Wng1,,
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationTrigonometry Revision Sheet Q5 of Paper 2
Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.
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 informationFinite State Automata and Determinisation
Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions
More informationEE 330/330L Energy Systems (Spring 2012) Laboratory 1 Three-Phase Loads
ee330_spring2012_l_01_3phse_lods.do 1/5 EE 330/330L Energy Systems (Spring 2012) Lortory 1 ThreePhse Lods Introdution/Bkground In this lortory, you will mesure nd study the voltges, urrents, impednes,
More informationAlpha Algorithm: A Process Discovery Algorithm
Proess Mining: Dt Siene in Ation Alph Algorithm: A Proess Disovery Algorithm prof.dr.ir. Wil vn der Alst www.proessmining.org Proess disovery = Ply-In Ply-In event log proess model Ply-Out Reply proess
More informationReference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.
I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the
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 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 informationCS 491G Combinatorial Optimization Lecture Notes
CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,
More informationLearning Objectives of Module 2 (Algebra and Calculus) Notes:
67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under
More informationGlobal alignment. Genome Rearrangements Finding preserved genes. Lecture 18
Computt onl Biology Leture 18 Genome Rerrngements Finding preserved genes We hve seen before how to rerrnge genome to obtin nother one bsed on: Reversls Knowledge of preserved bloks (or genes) Now we re
More informationAppendix C Partial discharges. 1. Relationship Between Measured and Actual Discharge Quantities
Appendi Prtil dishrges. Reltionship Between Mesured nd Atul Dishrge Quntities A dishrging smple my e simply represented y the euilent iruit in Figure. The pplied lternting oltge V is inresed until the
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More information8 THREE PHASE A.C. CIRCUITS
8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the so-lled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),
More informationModeling of Catastrophic Failures in Power Systems
Modeling of tstrophi Filures in Power Systems hnn Singh nd lex Sprintson Deprtment of Eletril nd omputer Engineering Texs &M hnn Singh nd lex Sprintson Modeling of tstrophi Filures Motivtion Reent events
More information1 Find the volume of each solid, correct to one decimal place where necessary. 12 cm 14 m. 25 mm. p c 5 ffiffiffi
1 Find the volume of eh solid, orret to one deiml le where neessry. 8 m 6 m m 14 m 65 m 2 2 m d 7.6 mm 2 m 4 m 4 m 7 m 25 mm Stge 5.3 See Chter 1 See Chter 7 See Chter 9 See Chter 9 See Chter 13 2 Simlify
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 informationCS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014
S 224 DIGITAL LOGI & STATE MAHINE DESIGN SPRING 214 DUE : Mrh 27, 214 HOMEWORK III READ : Relte portions of hpters VII n VIII ASSIGNMENT : There re three questions. Solve ll homework n exm prolems s shown
More informationCS 347 Parallel and Distributed Data Processing
CS 347 Prllel nd Distriuted Dt Proessing Spring 06 Network Prtitions Susets of nodes m e isolted or nodes m e slow in responding Notes 8: Network Prtitions CS 347 Notes 8 Network Prtitions Cuses ired network
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 informationIntegration. antidifferentiation
9 Integrtion 9A Antidifferentition 9B Integrtion of e, sin ( ) nd os ( ) 9C Integrtion reognition 9D Approimting res enlosed funtions 9E The fundmentl theorem of integrl lulus 9F Signed res 9G Further
More informationThe Trapezoidal Rule
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
More informationEstimation of Global Solar Radiation in Onitsha and Calabar Using Empirical Models
Communitions in Applied Sienes ISS 0-77 Volume, umer, 0, 5-7 Estimtion of Glol Solr dition in Onitsh nd Clr Using Empiril Models M.. nuhi, J. E. Ekpe nd G. F Ieh Deprtment of Industril Physis, Eonyi Stte
More informationEffects of Drought on the Performance of Two Hybrid Bluegrasses, Kentucky Bluegrass and Tall Fescue
TITLE: OBJECTIVE: AUTHOR: SPONSORS: Effets of Drought on the Performne of Two Hyrid Bluegrsses, Kentuky Bluegrss nd Tll Fesue Evlute the effets of drought on the visul qulity nd photosynthesis in two hyrid
More informationArrow s Impossibility Theorem
Rep Fun Gme Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties
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 information1 This question is about mean bond enthalpies and their use in the calculation of enthalpy changes.
1 This question is out men ond enthlpies nd their use in the lultion of enthlpy hnges. Define men ond enthlpy s pplied to hlorine. Explin why the enthlpy of tomistion of hlorine is extly hlf the men ond
More informationSIDESWAY MAGNIFICATION FACTORS FOR STEEL MOMENT FRAMES WITH VARIOUS TYPES OF COLUMN BASES
Advned Steel Constrution Vol., No., pp. 7-88 () 7 SIDESWAY MAGNIFICATION FACTORS FOR STEEL MOMENT FRAMES WIT VARIOUS TYPES OF COLUMN BASES J. ent sio Assoite Professor, Deprtment of Civil nd Environmentl
More informationHOMEWORK FOR CLASS XII ( )
HOMEWORK FOR CLASS XII 8-9 Show tht the reltion R on the set Z of ll integers defined R,, Z,, is, divisile,, is n equivlene reltion on Z Let f: R R e defined if f if Is f one-one nd onto if If f, g : R
More informationAlgorithm Design and Analysis
Algorithm Design nd Anlysis LECTURE 5 Supplement Greedy Algorithms Cont d Minimizing lteness Ching (NOT overed in leture) Adm Smith 9/8/10 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov,
More informationCounting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs
Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if
More informationApplying Hyperaccumulator Straw in Cd-Contaminated Soil Enhances Nutrient Uptake and Soil Enzyme Activity of Capsella bursa-pastoris
Interntionl Conferene on Mnufturing Siene nd Engineering (ICMSE 2015) Applying Hyperumultor Strw in Cd-Contminted Soil Enhnes Nutrient Uptke nd Soil Enzyme Ativity of Cpsell burs-pstoris Jin Wng1,, Keng
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 informationVectors. a Write down the vector AB as a column vector ( x y ). A (3, 2) x point C such that BC = 3. . Go to a OA = a
Streth lesson: Vetors Streth ojetives efore you strt this hpter, mrk how onfident you feel out eh of the sttements elow: I n lulte using olumn vetors nd represent the sum nd differene of two vetors grphilly.
More informationPYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles
More informationSpecial Numbers, Factors and Multiples
Specil s, nd Student Book - Series H- + 3 + 5 = 9 = 3 Mthletics Instnt Workooks Copyright Student Book - Series H Contents Topics Topic - Odd, even, prime nd composite numers Topic - Divisiility tests
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 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 information