Synchronization of different 3D chaotic systems by generalized active control


 Everett Lindsey
 1 years ago
 Views:
Transcription
1 ISSN , Englnd, UK Journl of Informtion nd Computing Siene Vol. 7, No. 4, 0, pp. 78 Synhroniztion of different D hoti systems y generlized tive ontrol Mohmmd Ali Khn Deprtment of Mthemtis, Grhet Rmsundr Vidyhn, Grhet, Pshim Medinipur, West Bengl, Indi (Reeived My 9, 0, epted Septemer, 0) Astrt. This pper designs sheme for ontrolling hoti system to period system using tive ontrol tehnique. We disuss this tking hoti Genesio system s exmple. We hve lso disussed the synhroniztion sheme etween two different oupled hoti systems (Genesio nd Nuler spin genertor(nsg)system) s well s two identil oupled hoti systems (Foursroll ttrtor) vi tive ontrol. Numeril Simultion results re presented to show the effetiveness of the proposed sheme. Keywords: Choti system, Chos ontrol, Chos synhroniztion, Ative ontrol, Genesio system, NSG system, Foursroll ttrtor.. Introdution Chos synhroniztion is n importnt topi in the nonliner siene. In 990, Peorr nd Crroll [] introdued the ide of hos synhroniztion. Two or more oupled hoti systems re lled synhronized if their ehviors re losely relted. Chos synhroniztion hs reeived muh ttention due to its pplitions in mny re suh seure ommunition, informtion proessing, iologil systems nd hemil retions. Usully two dynmil systems re lled synhronized if the distne etween their orresponding sttes onverges to zero s time goes to infinity. This type of synhroniztion is lled identil synhroniztion. A generliztion, of this onept for unidiretionlly oupled dynmil systems ws proposed y Rulkov et.l.(995) [] where two oupled systems re lled synhronized if stti funtionl reltionship exists etween the sttes of the systems. This kind of synhroniztion is lled generlized synhroniztion(gs). In 009 synhroniztion in unidiretionlly oupled Rossler system ws proposed y Khn nd Mndl [] nd lso synhroniztion in idiretionlly oupled systems reported y Tri et.l. [4,5]. In 0 Khn et.l. [6] hve disussed vrious type of ontrol for ontrolling hos in unified hoti system nd reently in 0 generlized ntisynhroniztion of different hoti systems ws proposed y Khn et.l. [7]. In 996, Korev nd Prlitz [8] formulted ondition for the ourrene of GS for the mster nd slve system. Using tehnique from tive ontrol theory, ERWeii nd Krl.E.Lonngren(997) [9] demonstrte tht oupled Lorenz systems n synhronize. Further ERWeii nd Krl.E.Lonngren [0] hve investigted sequentil synhroniztion of two Lorenz system using tive ontrol. In 00, synhroniztion of Rossler nd Chen hotil dynmil sytems using tive ontrol ws studied y Agiz nd Yssen []. In 00 MingChung Ho, YoChen Hung [] generlized the tehnique of tive ontrol theory nd pplied them to synhronize two different systems. Sinh et.l. [] proposed generl pproh in the design of tive ontrollers for nonliner systems exhiiting hos in 00. Synhroniztion of two hoti fourdimensionl systems using tive ontrol tehnique ws proposed y Youming nd Wenxin in 007 [4]. In 009 Sudheer nd Sir [5] ws investigted hyrid synhroniztion of hyperhoti Lu system vi tive ontrol. Reently in 0 Shhzd [6] ws investigted hos synhroniztion of n ellipsoidl stellite vi tive ontrol. Synhroniztion sheme for mny oupled hoti systems re not explored till now. Synhroniztion strtegy for oupled hoti Genesio system is very importnt from the theoretil point of view nd this strtegy is reported y us in this Pulished y World Ademi Press, World Ademi Union
2 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp pper. The synhroniztion strtegy etween two identil oupled hoti systems s well s two different hoti systems re lso very importnt from the pplition point of view. This motivtes us to study the synhroniztion etween Genesio system nd Nuler spin genertor systems nd synhroniztion etween Foursroll ttrtor systems in this pper.. Genesio system The Genesio system, proposed y Genesio et.l [7], it ptures mny fetures of hoti systems. It inludes simple squre prt nd three simple ordinry differentil equtions tht depend on three positive rel prmeters. The dynmil equtions re desried y = y = z () = x y z+ x where x, y nd z re stte vriles nd, nd re positive onstnts, stisfying <. The Genesio system exhiits hoti ttrtor t the prmeter vlues =., =.9 nd =6 shown in Fig.... Control of hoti Genesio system to period system In order to oserve the ontrol ehvior we tke three dimensionl period systems s mster system with three stte vriles denoted y the susript nd the slve systems denoted y the susript. The initil onditions for the mster system re different from tht of the slve system. The mster nd slve systems re defined s follows Mster system = y = z () = y Slve system = y + u = z + u () = x y z + x + u where U = [ u,, ] T u u is the ontroller funtion introdued in the slve system. Using tive ontrol tehniques, we sustrt eqution () from eqution () nd get JIC emil for susription:
3 74 Mohmmd Ali Khn: Synhroniztion of different D hoti systems y generlized tive ontrol = y + u = z + u = x y z + x + y + u where x = x x; y = y y nd z = z z. We define the tive ontrol funtions u, u nd u s u = V u = V (5) u = x + y + z x y + V This leds to = y+ V = z+ V (6) = V Eqution (6) desries the error dynmis nd n e onsidered in terms of ontrol prolem where the system to e ontrolled is liner system with ontrol input V, Vnd V s funtions of x, y nd z. Our im is to stilize the systems x, y nd z t origin. This implies tht two systems re synhronized with feedk ontrol. There re mny possile hoies for the ontrol V, V nd V. We hoose V x V = A y (7) V z where A is onstnt mtrix. For proper hoie of elements of the mtrix A, the feedk system must hve ll of the eigenvlues with negtive rel prts. In this se, the losed loop system will e stle. Let us hoose prtiulr form of the mtrix A tht is given y 0 A = 0 (8) 0 0 For this prtiulr hoie, the losed loop system hs eigenvlues tht re found to e , nd . This hoie will led to stle systems nd s we will oserve in numeril investigtion, led to the synhroniztion of Genesio system to e period system. (4) Fig.. Solution of Genesio system to e period system with the tive ontrol detivted JIC emil for ontriution:
4 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp Fig.. Solution of Genesio system to e period system with the tive ontrol tivted.. Numeril Simultion We selet the prmeters =., =.9 nd = 6 to ensure the existene of the hoti ehvior(fig.). The initil vlues re tken s x (0) = 0.6, y (0) =.6 nd z (0) =.5. Simultion results for unoupled system re presented in Fig.. nd tht of ontrolled system re shown in Fig.. The synhronizing of Genesio system to period system n lso e oserved y monitoring the differene of the two signls x, y nd z. For numeril simultion the initil vlues of differene of two signls re tken s x(0) = 4.0, y(0) = 0.0 nd z (0) =.0. Fig.4. displys the differene signls with the tive ontroller disonneted into the iruit. Fig.5. displys the sme signls with the tive ontroller onneted into the iruit. Fig.4. Disply the differene signls when tive ontroller detivted JIC emil for susription:
5 76 Mohmmd Ali Khn: Synhroniztion of different D hoti systems y generlized tive ontrol Fig.5. Disply the differene signls when tive ontroller tivted.. Nuler spin genertor (NSG) system The nuler spin genertor prolem ws studied y Shdev nd Srthy [8] nd Hegzi et.l. [9]. They showed tht the system displys rih nd typil ifurtion nd hoti phenomen for some vlues of the ontrol prmeters. The system onsists suitle simple of mtter ontining proper nulei in reltively strong mgneti field defining Zdiretion, n exiting oil with xis in the Xdiretion, perpendiulr to Z, pikup oil with xis in the Xdiretion, perpendiulr to oth X nd Z nd highgin mplifier feeding the voltge indued in the pikup oil k to the exiting oil. We first reformulte its eqution in the following form = β x+ y = x βy( κz) (9) = βα ( ( z)) κy ) suh tht α, β nd κ re prmeters, where βα 0 nd β 0 re liner dmping terms, the nonlinerity prmeters βκ is proportionl to the mplifier gin in the voltge feedk. Physil onsidertion limits the prmeters α to the rnge 0< α. Fig.6. Choti ttrtor of NSG system t β =.75, α =.5 nd κ =.0.. Synhroniztion of two different hoti systems JIC emil for ontriution:
6 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp Now, we wnt to synhronize two different oupled hoti systems. Tke Genesio nd Nuler spin generting systems into onsidertion. Mster system(genesio system) = y = z (0) Slve system(nsg system) = x y z + x βα κ = β x + y + u = x βy ( κz ) + u = ( ( z ) y ) + u () Sustrt (0) from () we get = β x + y + u = x βy ( κz ) z + u = βα ( ( z ) κy ) + x + y + z x + u () We define the tive ontrol funtions u, und u s u = β x + V u = x + βy( κz) + z+ V u = βα ( ( z ) + κy ) x y z + x + V () This leds to = y + V = V = V (4) Eqution (4) desrie the error dynmil system whih is liner system with ontrol input V, V nd V s funtions of x, y nd z. Our gol is hieved when these feedks stilize the system x, y nd z t zero. There re mny possile hoies for the ontrol V, V nd V. We hoose V x V = A y (5) V z Where A is onstnt mtrix. Let us hoose prtiulr form of the mtrix A tht is given y β 0 A = β 0 (6) 0 0 αβ The eqution (6) n e written s X & = BX (7) where JIC emil for susription:
7 78 Mohmmd Ali Khn: Synhroniztion of different D hoti systems y generlized tive ontrol β 0 B = β αβ Finlly, we hve to produe mtrix B suh tht three eigen vlues hs negtive rel prts. (8) Fig.7. Solution of Genesio system nd NSG systems with the tive ontrol detivted. Fig.8. Solution of Genesio system nd NSG systems with the tive ontrol tivted... Numeril Simultion We selet the prmeters for NSG systems s β = 0.75, α = 0.5 nd κ = to ensure the hoti ehvior shown in Fig.6. The initil vlues of Genesio system re tken s x (0) = 0.0, y (0) =.0, z (0) =.0 nd for Nuler spin genertor system x (0) =.0, y (0) =.6, z (0) =.5. Simultion results for unontrolled system re presented in Fig.7. nd tht of ontrolled system re shown in Fig.8. Fig.9. displys the differene signls with the tive ontroller into the iruit. JIC emil for ontriution:
8 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp Fig.9. Disply the differene signl of Genesio nd NSG system when tive ontrol tivted 4. Foursroll ttrtor Consider the following Foursroll ttrtor = x yz = y+ xz (9) = z+ xy where, nd re positive ontrol prmeters. This system hs hoti ttrtor s shown in Fig.0. t the prmeter vlues = 0.4, = nd = Synhroniztion of two identil hoti systems To oserve the synhroniztion ehvior for Foursroll ttrtor, we hve two Foursroll ttrtor systems where the mster system with three stte vriles denoted y the susript nd slve system hving identil equtions denoted y the susript. The mster nd slve systems re defined s follows JIC emil for susription:
9 80 Mohmmd Ali Khn: Synhroniztion of different D hoti systems y generlized tive ontrol nd = x y z = y + x z = z + x y (0) Sustrting (0) from () we get = x y z + u = y + x z + u = z + x y + u = x + y z y z + u = y + x z x z + u = z + x y x y + u By suitle hoie of u, u nd u the system () eomes = x + V where = y + V = z + V u = y z + y z + V u = x z x z + V u = x y x y + V There re mny possile hoies for ontroller V, V nd V. We hoose V x V = A y V z Where A is onstnt mtrix. Let us hoose prtiulr form of the mtrix A tht is given y ( + ) 0 0 A = Therefore differene signl system eomes = x = y = z () () () (4) (5) (6) JIC emil for ontriution:
10 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp Fig.. Disply the solution of the oupled Foursroll ttrtor system of equtions with the tive ontrol detivted. Fig.. Disply the solution of the oupled Foursroll ttrtor system of equtions with the tive ontrol tivted. 4.. Numeril Simultion We selet the prmeters for Foursroll ttrtor systems s = 0.4, = nd = 5 to ensure the hoti ehvior shown in Fig.0. The initil vlues of oupled Foursroll ttrtor system re tken s x(0) = 0.0, y(0) =.0, z(0) =.0 nd x(0) = 5.6, y(0) =.6, z(0) =.5. The results of the simultion of oupled Foursroll ttrtor system with the tive ontroller detivted re shown in Fig.. Fig.. displys the sme sequene of signls when tive ontrol tivted nd differene signls re shown in Fig.. with the tive ontroller onneted into the iruit. JIC emil for susription:
11 8 Mohmmd Ali Khn: Synhroniztion of different D hoti systems y generlized tive ontrol 5. Conlusion Fig.. Disply the differene signl of x, y nd z when tive ontrol tivted. We hve disussed synhroniztion of Genesio system to period system using tive ontrol tehnique. This strtegy n e used to synhronize ny hoti system to periodi system with desired period. This tehnique my e useful for ontrolling hoti osilltions in eletroni systems, iologil ells et. We hve lso disussed the synhroniztion etween two different hoti systems onsidering Genesio nd NSG systems s well s two identil Foursroll ttrtor systems vi tive ontrol. We elieve these synhroniztion sheme my e useful for sending serete messge. We show tht hoti system n e synhronized to periodi system or ny other system. Aknowledgement I would like to express my sinere thnks to Dr. Swrup pori of the Dept. Of Applied Mthemtis, Clutt University for motivting me y vlule suggestions for preprtion of this work. 6. Referenes [] [] [] [4] [5] [6] [7] Peorr nd Crroll, Synhroniztion in hoti systems, Phys. Rev. Lett. 64 (990), pp.8. Rulkov N.F, Sushik M.M nd T. Simring L.S, Generlized synhroniztion of hos diretionlly oupled hoti system, Phys. Rev. E. 5 (995), pp M.A.Khn nd A.K.Mndl, Generlized hos synhroniztion of oupled Rossler systems,bull. Cl. Mth. So. 0 (009), pp A.Tri, S.Pori nd P.Chtterjee, Synhroniztion of idiretionlly oupled hoti Chen s System with dely, Chos Solitons nd Frtls, doi: 0.06/j.hos A.Tri, S.Pori nd P.Chtterjee, Synhroniztion of generlized linerly idiretionlly oupled unified hoti system, Chos Solitons nd Frtls. 40 (009), pp M.A.Khn, A.K.Mondl nd S.Pori, Three ontrol strtegies for unified hoti system, Int. J. of Applied Mehnis nd Engineering. 6 (0), pp M.A.Khn, S.N.Pl nd S.Pori, Generlized ntisynhroniztion of different hoti systems, Int. J. of Applied Mehnis nd Engineering. 7 (0), pp JIC emil for ontriution:
12 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp [8] [9] Korev L nd Prlitz U, Generlized synhroniztion, preditility nd equivlene of unidiretionlly oupled dynmil systems, Phys. Rev. Lett. 76 (996), pp.86. ERWeii nd Krl E. Lonngren, Synhroniztion of two Lorenz systems using tive ontrol, Chos, Solitons nd Frtls. 8 (997), pp [0] [] [] [] [4] [5] [6] [7] ERWeii nd Krl E. Lonngren, Sequentil synhroniztion of two Lorenz systems using tive ontrol, Chos,Solitons nd Frtls. (997), pp.04. Agiz nd Yssen, Synhroniztion of Rossler nd Chen hotil dynmil systems using tive ontrol, Physis Letters A. 78 (00), pp.9 MingChung Ho nd YC Hung, Synhroniztion of different systems using generlized tive ontrol, Physis Letters A. 0 (00), pp Sinh SC, Henrihs JT nd Rvindr BA, A generl pproh in the design of tive ontrollers for nonliner systems exhiiting hos, Int. J. Bifurt. Chos (00) pp L.Youming, X. Wei nd X. Wenxin, Synhroniztion of two hoti fourdimensionl systems using tive ontrol, Chos, Solitons nd Frtls. (007), pp K. Sestin Sudheer nd M. Sir, Hyrid synhroniztion of hyperhoti Lu system, Prmnjournl of physis. 7 (009) pp Genesio R nd Tesi A, A hrmoni lne methods for the nlysis of hoti dynmis In nonliner systems, Automti, 8 (99), pp Mohmmd Shhzd, Chos synhroniztion of n ellipsoidl stellite vi tive ontrol, Progress in Applied Mthemtis. (0), pp.6. [8] Shdev nd Srthy, Periodi nd hoti solutions for nonliner system rising from nuler spin genertor, Chos,solitons nd Frtls. 4 (994), pp [9] Hegzi,Agiz nd Eledessoky, Adptive ontrol nd synhroniztion of modified Chu s iruit system, Chos,solitons nd Frtls. (00), pp JIC emil for susription:
1.3 SCALARS AND VECTORS
Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd
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 informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd AmGm inequlity 2. Elementry inequlities......................
More informationA Mathematical Model for UnemploymentTaking an Action without Delay
Advnes in Dynmil Systems nd Applitions. ISSN 97353 Volume Number (7) pp. 8 Reserh Indi Publitions http://www.ripublition.om A Mthemtil Model for UnemploymentTking n Ation without Dely Gulbnu Pthn Diretorte
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 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 informationSymmetrical Components 1
Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedystte nlysis of high voltge trnsmission systems, we mke use of the perphse equivlent
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
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 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 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 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 informationChapter Gauss Quadrature Rule of Integration
Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to
More informationLecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and
Leture 7: iffusion of Ions: Prt : oupled diffusion of tions nd nions s desried y NernstPlnk Eqution Tody s topis Continue to understnd the fundmentl kinetis prmeters of diffusion of ions within n eletrilly
More informationIf only one fertilizer x is used, the dependence of yield z(x) on x first was given by Mitscherlich (1909) in form of the differential equation
Mitsherlih s Lw: Generliztion with severl Fertilizers Hns Shneeerger Institute of Sttistis, University of ErlngenNürnerg, Germny 00, 5 th August Astrt: It is shown, tht the ropyield z in dependene on
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 informationModeling and Simulation of Permanent Magnet Brushless Motor Drives using Simulink
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 72102, DECEMBER 2729, 2002 25 Modeling nd Simultion of Permnent Mgnet Brushless Motor Dries using Simulink Mukesh Kumr, Bhim Singh nd B.P.Singh Astrt: Permnent
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 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 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 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 informationMATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.
MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].
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 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 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 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 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 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 informationGreedoid polynomial, chipfiring, and Gparking function for directed graphs. Connections in Discrete Mathematics
Greedoid polynomil, hipfiring, nd Gprking funtion for direted grphs Swee Hong Chn Cornell University Connetions in Disrete Mthemtis June 15, 2015 Motivtion Tutte polynomil [Tut54] is polynomil defined
More informationA new algorithm for generating Pythagorean triples 1
A new lgorithm for generting Pythgoren triples 1 RH Dye 2 nd RWD Nicklls 3 The Mthemticl Gzette (1998; 82 (Mrch, No. 493, pp. 86 91 http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf 1 Introduction
More informationA Comparison of Dynamic Tyre Models for Vehicle Shimmy Stability Analysis
A Comprison of Dynmi Tyre Models for Vehile Shimmy Stility Anlysis J.W.L.H. Ms DCT 009.101 MS Thesis Supervisors: Prof. Dr. H. Nijmeijer (TU/e) Dr. Ir. I.J.M. Besselink (TU/e) Ir. S.G.J. de Cok (DAF Truks)
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More informationON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs
ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,
More informationILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS
ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu
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 informationWave Equation on a Two Dimensional Rectangle
Wve Eqution on Two Dimensionl Rectngle In these notes we re concerned with ppliction of the method of seprtion of vriles pplied to the wve eqution in two dimensionl rectngle. Thus we consider u tt = c
More informationH 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3.
. Spetrosopy Mss spetrosopy igh resolution mss spetrometry n e used to determine the moleulr formul of ompound from the urte mss of the moleulr ion For exmple, the following moleulr formuls ll hve rough
More informationA Nonparametric Approach in Testing Higher Order Interactions
A Nonprmetri 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 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 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 informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationAvailable online at ScienceDirect. Procedia Engineering 120 (2015 ) EUROSENSORS 2015
Aville online t www.sienediret.om SieneDiret Proedi Engineering 10 (015 ) 887 891 EUROSENSORS 015 A Fesiility Study for SelfOsillting Loop for Three DegreeofFreedom Coupled MEMS Resontor Fore Sensor
More informationHARMONIC BALANCE SOLUTION OF COUPLED NONLINEAR NONCONSERVATIVE DIFFERENTIAL EQUATION
GNIT J. nglesh Mth. So. ISSN  HRMONIC LNCE SOLUTION OF COUPLED NONLINER NONCONSERVTIVE DIFFERENTIL EQUTION M. Sifur Rhmn*, M. Mjeur Rhmn M. Sjeur Rhmn n M. Shmsul lm Dertment of Mthemtis Rjshhi University
More informationGénération aléatoire uniforme pour les réseaux d automates
Génértion létoire uniforme pour les réseux d utomtes Niols Bsset (Trvil ommun ve Mihèle Sori et Jen Miresse) Université lire de Bruxelles Journées Alé 2017 1/25 Motivtions (1/2) p q Automt re omnipresent
More informationPreLie algebras, rooted trees and related algebraic structures
PreLie lgers, rooted trees nd relted lgeri strutures Mrh 23, 2004 Definition 1 A prelie lger is vetor spe W with mp : W W W suh tht (x y) z x (y z) = (x z) y x (z y). (1) Exmple 2 All ssoitive lgers
More informationCalculus Cheat Sheet. Integrals Definitions. where F( x ) is an antiderivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous AntiDerivtive : An ntiderivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
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 informationThe EmissionAbsorption of Energy analyzed by QuantumRelativity. Abstract
The missionabsorption of nergy nlyzed by QuntumReltivity Alfred Bennun* & Néstor Ledesm** Abstrt The uslity horizon llows progressive quntifition, from n initil nk prtile, whih yields its energy s blk
More informationReflection Property of a Hyperbola
Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the
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 informationz TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability
TRASFORS Trnsform Bsics Trnsfer Functions Bck to the Time Domin Trnsfer Function nd Stility DSPG 6. Trnsform Bsics The definition of the trnsform for digitl signl is: n X x[ n is complex vrile The trnsform
More informationdx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.
Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd
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 informationThe Modified Heinz s Inequality
Journl of Applied Mthemtics nd Physics, 03,, 6570 Pulished Online Novemer 03 (http://wwwscirporg/journl/jmp) http://dxdoiorg/0436/jmp03500 The Modified Heinz s Inequlity Tkshi Yoshino Mthemticl Institute,
More informationLecture 7 notes Nodal Analysis
Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions
More informationTransition systems (motivation)
Trnsition systems (motivtion) Course Modelling of Conurrent Systems ( Modellierung neenläufiger Systeme ) Winter Semester 2009/0 University of DuisurgEssen Brr König Tehing ssistnt: Christoph Blume In
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 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 informationLine Integrals and Entire Functions
Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series
More informationON LEFT(RIGHT) SEMIREGULAR AND greguar posemigroups. Sang Keun Lee
KngweonKyungki Mth. Jour. 10 (2002), No. 2, pp. 117 122 ON LEFT(RIGHT) SEMIREGULAR AND greguar posemigroups Sng Keun Lee Astrt. In this pper, we give some properties of left(right) semiregulr nd gregulr
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 informationLIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon
LIP Lortoire de l Informtique du Prllélisme Eole Normle Supérieure de Lyon Institut IMAG Unité de reherhe ssoiée u CNRS n 1398 Onewy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Mrs 1993 Reserh Report N
More informationm m m m m m m m P m P m ( ) m m P( ) ( ). The oordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
COORDINTE GEOMETR II I Qudrnt Qudrnt (.+) (++) X X    0  III IV Qudrnt  Qudrnt ()  (+) Region CRTESIN COORDINTE SSTEM : Retngulr Coordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
More informationSECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS
SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS INSITU PROBING OF DOMAIN POLING IN Bi 4 Ti 3 O 12 THIN FILMS BY OPTICAL SECOND HARMONIC GENERATION YANIV BARAD, VENKATRAMAN GOPALAN Mterils Reserh Lortory
More informationA Functorial Query Language
A Funtoril Query Lnguge Ryn Wisnesky, Dvid Spivk Deprtment of Mthemtis Msshusetts Institute of Tehnology {wisnesky, dspivk}@mth.mit.edu Presented t Boston Hskell April 16, 2014 Outline Introdution to FQL.
More informationa) Read over steps (1) (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.
Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You
More informationA NEW COMBINED BRACKETING METHOD FOR SOLVING NONLINEAR EQUATIONS
Aville online t htt://ik.org J. Mth. Comut. Si. 3 (013), No. 1, 8793 ISSN: 1975307 A NEW COMBINED BRACKETING METHOD FOR SOLVING NONLINEAR EQUATIONS M.A. HAFIZ Dertment of mthemti, Fulty of Siene nd rt,
More informationSemantics of RTL and Validation of Synthesized RTL Designs using Formal Verification in Reconfigurable Computing Systems
emntis of TL nd Vlidtion of ynthesized TL Designs using Forml Verifition in eonfigurle Computing ystems Phn C. Vinh nd Jonthn P. Bowen London outh Bnk University Centre for Applied Forml Methods, Institute
More informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4305 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
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 informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 218, pp 4448): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 218, pp 4448): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview IntroductionModelling Bsic concepts to understnd n ODE. Description nd properties
More informationThis enables us to also express rational numbers other than natural numbers, for example:
Overview Study Mteril Business Mthemtis 0506 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers
More informationVariational Techniques for SturmLiouville Eigenvalue Problems
Vritionl Techniques for SturmLiouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationSIDESWAY MAGNIFICATION FACTORS FOR STEEL MOMENT FRAMES WITH VARIOUS TYPES OF COLUMN BASES
Advned Steel Constrution Vol., No., pp. 788 () 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 informationI 3 2 = I I 4 = 2A
ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents
More information( ) as a fraction. Determine location of the highest
AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to
More informationCS241 Week 6 Tutorial Solutions
241 Week 6 Tutoril olutions Lnguges: nning & ontextfree Grmmrs Winter 2018 1 nning Exerises 1. 0x0x0xd HEXINT 0x0 I x0xd 2. 0xend HEXINT 0xe I nd ER  MINU  3. 1234120x INT 1234 INT 120 I x 4.
More informationDoes the electromotive force (always) represent work?
rxiv.org > physis > rxiv:1405.7474 Does the eletromotive fore (lwys) represent work?. J. Pphristou 1, A. N. Mgouls 1 Deprtment of Physil Sienes, Nvl Ademy of Greee, Pireus, Greee Emil: pphristou@snd.edu.gr
More information= state, a = reading and q j
4 Finite Automt CHAPTER 2 Finite Automt (FA) (i) Derterministi Finite Automt (DFA) A DFA, M Q, q,, F, Where, Q = set of sttes (finite) q Q = the strt/initil stte = input lphet (finite) (use only those
More informationThis chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2
1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion
More informationThe Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5
The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle
More informationGrammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages
5//6 Grmmr Automt nd Lnguges Regulr Grmmr Contextfree Grmmr Contextsensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive
More informationSEQUENTIAL MOVEMENT IN THE GREAT FISH WAR: A PRELIMINARY ANALYSIS
Mjlh Bdn Pengkjin dn Penerpn Teknologi No. 79, Feruri 997 ISSN 066569 SEQUENTIAL MOVEMENT IN THE GREAT FISH WAR: A PRELIMINARY ANALYSIS Budy P. Resosudrmo Direktort Pengkjin Sistem Sosil, Ekonomi dn Pengemngn
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More informationH (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.
Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining
More informationMA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES
MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES CHARLIE COLLIER UNIVERSITY OF BATH These notes hve been typeset by Chrlie Collier nd re bsed on the leture notes by Adrin Hill nd Thoms Cottrell. These
More informationThe Ellipse. is larger than the other.
The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)
More informationarxiv: v4 [condmat.statmech] 18 May 2017
Quntum Vertex Model for Reversile Clssil Computing rxiv:1604.05354v4 [ondmt.sttmeh] 18 My 2017 C. Chmon, 1,. R. Muiolo, 2 A.. Rukenstein, 1 nd Z.C. Yng 1 1 Physis Deprtment, Boston University, 590 Commonwelth
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 informationPhysics 505 Homework No. 11 Solutions S111
Physis 55 Homework No 11 s S111 1 This problem is from the My, 24 Prelims Hydrogen moleule Consider the neutrl hydrogen moleule, H 2 Write down the Hmiltonin keeping only the kineti energy terms nd the
More informationOn the CoOrdinated Convex Functions
Appl. Mth. In. Si. 8, No. 3, 0850 0 085 Applied Mthemtis & Inormtion Sienes An Interntionl Journl http://.doi.org/0.785/mis/08038 On the CoOrdinted Convex Funtions M. Emin Özdemir, Çetin Yıldız, nd Ahmet
More informationThe solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr
Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the evectors nd evlues
More informationNondeterministic Finite Automata
Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)
More informationThe RiemannStieltjes Integral
Chpter 6 The RiemnnStieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0
More informationSTURM LIOUVILLE OPERATORS WITH MEASUREVALUED COEFFICIENTS. 1. Introduction. + q(x)y(x) = zr(x)y(x)
STUM LIOUVILLE OPEATOS WITH MEASUEVALUED COEFFICIENTS JONATHAN ECKHADT AND GEALD TESCHL Abstrt. We give omprehensive tretment of Sturm Liouville opertors whose oeffiients re mesures inluding full disussion
More informationCS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation
CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes
More informationInvestigations on Power Quality Disturbances Using Discrete Wavelet Transform
I J E E E Interntionl Journl of Eletril, Eletronis ISSN No. (Online): 7766 nd omputer Engineering (): 4753(13) Investigtions on Power Qulity Disturnes Using Disrete Wvelet Trnsform hvn Jin, Shilendr
More information8.3 THE HYPERBOLA OBJECTIVES
8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS
More informationDesigning Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.
EECS 16A Designing Informtion Devices nd Systems I Fll 2016 Bk Ayzifr, Vldimir Stojnovic Homework 6 This homework is due Octoer 11, 2016, t Noon. 1. Homework process nd study group Who else did you work
More information