Optimal Experiment Design with Diffuse Prior Information
|
|
- Anastasia Phelps
- 6 years ago
- Views:
Transcription
1 Optiml Experiment Design with Diffuse Prior Informtion Cristin R. Rojs Grhm C. Goodwin Jmes S. Welsh Arie Feuer Abstrct In system identifiction one lwys ims to lern s much s possible bout system from given observtion period. This hs led to on-going interest in the problem of optiml experiment design. Not surprisingly, the more one knows bout system the more focused the experiment cn be. Indeed, mny procedures for optiml experiment design depend, prdoxiclly, on exct knowledge of the system prmeters. This hs motivted recent reserch on, so clled, robust experiment design where one ssumes only prtil prior knowledge of the system. Here we go further nd study the question of optiml experiment design when the -priori informtion bout the system is diffuse. We show tht bndlimited 1/f noise is optiml for prticulr choice of cost function. I. INTRODUCTION In system identifiction, there is lwys strong incentive to lern s much bout system s possible from given observtion period. This hs motivted substntil interest in the topic of optiml experiment design. Indeed, there exists body of work on this topic, both in the sttistics literture [5, 14, 7] nd in the engineering literture [17, 10, 27]. Much of the existing literture is bsed on designing the experiment to optimize some sclr function of the Fisher Informtion Mtrix [10, pg. 6]. However, fundmentl difficulty is tht when the system response depends non-linerly on the prmeters, the Informtion Mtrix depends, interli, on the true system prmeters. Moreover, we note tht models for dynmicl systems (even if liner) typiclly hve the chrcteristic tht their response depends non-linerly on the prmeters. Hence, the informtion mtrix for models of dynmicl systems generlly depends upon the true system prmeters. This mens tht experiment designs which re bsed on the Fisher Informtion Mtrix will, in principle, depend upon knowledge of the true system prmeters. This is prdoxicl since the optiml experiment then depends on the very thing tht the experiment is imed t estimting [13, pg. 427]. The bove resoning hs motivted the study of, so clled, robust optiml experiment designs with respect to uncertinty on priori informtion. In this vein, vrious pproches hve been proposed, e.g. (i) Itertive design where one lterntes between prmeter estimtion nd experiment design bsed on the C. R. Rojs, G. C. Goodwin nd J. S. Welsh re with the School of Electricl Engineering & Computer Science, The University of Newcstle, NSW, Austrli cristin.rojs@studentmil.newcstle.edu.u, jmes.welsh@newcstle.edu.u, grhm.goodwin@newcstle.edu.u A. Feuer is with the Deprtment of Electricl Engineering, Technion, Hif 32000, Isrel. feuer@ee.technion.c.il current estimtes [4, 18, 25]. (ii) Byesin design where one optimizes some function of the expected informtion mtrix, with the expecttion tken over some -priori distribution of the prmeters [1, 3, 6]. (iii) Min-Mx design in which one optimizes the worst cse over bounded set of -priori given prmeter vlues [20, 8, 21]. The ltter designs mentioned bove re closely relted to gme theory. Indeed, gme-theoreticl ides hve been used to chrcterize the optiml robust (in the min-mx sense) experiment. For exmple, severl ppers hve studied different types of one-prmeter robust experiment design problems [21, 11]. It hs been shown for these problems tht the optiml min-mx experiment hs mny interesting properties, e.g. it exists, it is unique, it hs compct support in the frequency domin nd it is chrcterized by line spectrum. For multi-prmeter problems, one usully needs to use gridding strtegies to crry out the robust designs numericlly [21, 25]. A surprising observtion from recent work on min-mx optiml experiment design is tht bnd-limited 1/f noise is ctully quite close to optiml for prticulr problems. Indeed, 1/f noise hs been shown to hve performnce which is within fctor of 2 from the performnce of robust optiml designs for first-order nd resonnt systems [21, 11]. It is importnt to note, however, tht the proof of ner optimlity depends on prticulr property of these systems which llows one to scle the prmeters with respect to frequency. Here we sk more generl question: Sy we re just beginning to experiment on system nd thus hve very little (i.e. diffuse) prior knowledge bout it. Wht would be good initil experiment to use to estimte the system? In this cse we consider s diffuse prior informtion tht the interesting prt of the frequency response of the system lies in n intervl [, b]. This implies tht we re seeking n experiment which is good over very brod clss of possible systems. In this pper, we propose possible solution to this problem, being tht the experiment should consist of bndlimited 1/f noise. The pper is structured s follows. In Section II we discuss the problem of mesuring the goodness of n experiment by using system independent criterion. Section III gives some desirble properties tht such mesure would be expected to possess. In Section IV we consider typicl input constrint generlly used in experiment design. Section V shows preliminry result for choosing suitble cost function which stisfies the properties developed in Section
2 u G(q) Fig. 1. Block digrm describing the reltionship between the input u, the noise n nd the output y of the system G to be identified. III. Sections VI nd VII develop the form of the cost function which stisfies the properties in Section III. In Section VIII we show tht bndlimited 1/f noise is n optiml input signl ccording to this cost function, nd Section IX clerly illustrtes the dvntges of bndlimited 1/f noise by mens of n exmple. We present conclusions in Section X. II. A MEASURE OF THE GOODNESS OF AN EXPERIMENT Our im is to design n experiment which is good for very brod clss of systems. This mens tht we need mesure of goodness of n experiment which is system independent. To construct such mesure, we mke use of the work of Ljung [15], who hs shown tht, for brod clss of liner systems, the vrince of the error in the estimted discrete time frequency response tkes the following symptotic (in both system order nd dt points) form: Vr(Ĝ(ejω )) = K φ n(ω) ; ω [0, 2π], (1) φ u (ω) where φ n is the noise spectrl density nd φ u is the input spectrl density. Here K is function of the number of system prmeters nd the number of observtions. Figure 1 shows how the input u, the noise n nd the output y of the system re relted, i.e. n y(t) = G(q)u(t) + n(t) (2) where G is the trnsfer function of the system, nd q is the forwrd shift opertor. Actully, it hs been rgued in [19] tht better pproximtions exist to tht given in (1) but the simpler expression (1) suffices for our purposes. In fct, the expressions given in [19] for Box-Jenkins nd Output-Error models include fctor which is dependent, for some prticulr specil cses, only upon the poles of G. We note tht this cn be incorported into φ n, thus obtining test signl which is independent on φ n nd the pt. This implies tht the results given here re exct for some clsses of models of finite order. An interesting nd highly desirble property of (1) is tht it is essentilly independent of the system prmeters. This is becuse it depends only on φ n nd φ u. Of course, φ n is somewht problemtic since it would lso be desirble to hve (1) independent of the rel chrcteristics of the noise. This will lso be prt of our considertion. y As rgued in [12, 21, 11, 26], bsolute vrinces re not prticulrly useful when one wnts to crry out n experiment design tht pplies to brod clss of systems. Specificlly, n error stndrd devition of 10 2 in vrible of nominl size 1 would be considered to be insignificnt, wheres the sme error stndrd devition of 10 2 in vrible of nominl size 10 3 would be considered ctstrophic. Hence, it seems preferble to work with reltive errors. Thus, if G(e jω) is the mgnitude of the frequency response of the pt t frequency ω, then eqution (1) suggests tht the reltive vrince t frequency ω is given by φ n (ω) Rel. Vr(Ĝ(ejω )) = K φ u (ω) G(e jω) ; ω [0, 2π]. 2 (3) Finlly, rther thn look t single frequency ω, we will look t n verge mesure over rnge of frequencies. This leds to generl mesure of the goodness of n experiment of the form: J(φ u ) = = F (Vr(Ĝ(ejω ))/ G(e jω ) 2 )W (ω)dω ( F φ u (ω) G(e jω ) 2 ) W (ω)dω, (4) where F nd W re functions to be specified lter, nd 0 < < b < 2π. Here, W is weighting function tht llows the control engineer to define t which frequencies it would be preferble to obtin better model (depending on the control requirements, but not necessrily on the true pt chrcteristics). In the next Section we propose some desirble properties of the functions F nd W. III. DESIRABLE PROPERTIES OF THE COST FUNCTION We consider two sets of criteri. The first reltes principlly to the function F, the second to the function W. In ddition to these properties, we will lso ssume tht F C 1 ([, b], R + 0 ) nd W C1 ([, b], R + ), where C 1 (X, Y ) is the spce of ll functions from X R to Y R hving continuous derivtive. Criteri A It is resonble to consider cost function (4) whose minimum is chieved by function which does not depend on the ctul system chrcteristics. The reson being tht these chrcteristics re typiclly unknown t the time the experiment is pplied, nd in fct it is the purpose of the experiment to revel this informtion. On the other hnd, the cost function (4) should be mesure of the size of the vrince in the estimtion of the pt frequency response. Hence, loosely speking, the cost function should increse ccordingly to n increse of the vrince t ny frequency. The bove rgument implies tht the function F for mesure (4) should be chosen so s to stisfy the following requirements:
3 A.1) The optiml experiment, φ u, which minimizes J in (4), should be independent on the pt G(e jω ) 2 nd the noise vrince φ n. A.2) The integrnd in (4) should increse if the vrince Vr(Ĝ(ejω )) increses t ny frequency. This implies tht F should be monotoniclly incresing function. Criterion B Mny properties of liner systems depend on the rtio of poles nd zeros rther thn on their bsolute loctions in the frequency domin [2, 9, 22]. This implies tht if we scle the frequency ω by constnt, the optiml input must keep its shpe, s the poles nd zeros of the new pt will hve the sme rtios s before. This invrince property must be reflected in the weighting function W, which hs to give equl weight to frequency intervls whose endpoints re in the sme proportion. Thus, the weighting function W should be such tht for every 0 < α < β < 2π nd every k > 0 such tht 0 < kα < kβ < 2π we hve tht β kβ W (ω)dω = W (ω)dω. (5) α kα IV. CONSTRAINTS Our gol will then be to optimize cost function s in (4) where φ u is constrined in some fshion. A typicl constrint used in experiment design is tht the totl input energy should be constrined [10, pg. 125]. Thus, we need to optimize J(φ u ) subject to constrint of the form φ u (ω)dω = 1. (6) Specificlly our gol is to djust F nd W such tht the optiml experiment tht minimizes (4) subject to the constrint (6) stisfies the criteri A.1, A.2 nd B in Section III. V. A PRELIMINARY TECHNICAL RESULT Motivted by the need for mesure to be independent of the system nd such tht criteri A.1, A.2 nd B re met subject to constrint on the input, we hve estblished the following result: Lemm 1: For 0 < < b < 2π, let g, F C 1 ([, b], R + 0 ) nd W C 1 ([, b], R + ). Define, if it exists, f (g) := rg min f C 1 ([,b],r + ) b f(x)dx=1 F ( ) g(x) W (x)dx. (7) f(x) If f (g) does not depend on g, then there re constnts α, β, γ R such tht F (y) = α y + β; g(x) inf x [,b] f(x) y sup g(x) x [,b] f(x), (8) nd f = γw. Proof: Let g, F C 1 ([, b], R + 0 ) nd W C 1 ([, b], R + ) be fixed, nd such tht f (g), s defined in (7), exists. Then, by [16, Section 7.7, Theorem 2], there is constnt λ R for which f (g), is sttionry point of J λ (f) := F ( ) g(x) W (x)dx + λ f(x)dx. (9) f(x) Thus, for ny h C 1 ([, b], R + 0 ) we hve tht δj λ(f ; h) = 0, which mens [16, Section 7.5] tht [ ( ) ( g(x) F f (x) g(x) (f (x)) 2 ) ] W (x) + λ h (x)dx = 0, (10) thus, by [16, Section 7.5, Lemm 1], ( ) g(x) F g(x) f W (x) (x) (f = λ; x [, b]. (11) (x)) 2 Let l(x) := g(x)/f (x), then (11) cn be written s F (l(x))l(x) = λ f (x) ; x [, b]. (12) W (x) The left side of (12) depends on g, but the right does not (becuse of the ssumption on the independence of f upon g). Thus, both sides re equl to constnt, sy, α R, which implies tht F (l(x)) = α ; x [, b]. (13) l(x) Now, by integrting both sides with respect to l between l(x) nd sup l(x), we obtin inf x [,b] x [,b] F (l(x)) = α l(x) + β; x [, b] (14) for some constnt β R. On the other hnd, we hve tht λ f (x) = α, (15) W (x) so if we define γ := α/λ, we conclude tht f = γw. This concludes the proof. VI. CHOICE OF THE FUNCTION F In this Section we use the result of the previous Section to find suitble function F which stisfies Criteri A.1 nd A.2, nd to find the optiml input signl for the resulting cost function. We first exmine the choice of the function F in (4). Now, we my tke, without loss of generlity, α = 1 nd β = 0 for the function F given by Lemm 1. This is becuse, ccording to Lemm 1, every cost function (4) stisfying Criteri A.1 nd A.2 is minimized by the sme f C 1 ([, b], R + ). Thus, such cost function cn be written s J(φ u ) = ( φ u (ω) G(e jω ) 2 ) W (ω)dω. (16)
4 It is then reltively strightforwrd to optimize (16) subject to the constrint given by (6). Indeed, by Lemm 1 the optiml experiment will be essentilly given by scled version of W, i.e. φ 1 u(ω) = W ; ω [, b]. (17) W (x)dx The following Lemm estblishes tht φ u gives not only n extremum, but globl minimum for the cost function (16). Lemm 2: The function φ u defined in (17) gives the globl minimum of the cost function (16). In other words, for 0 < < b < 2π, let W C 1 ([, b], R + ), then, φ u = (18) rg min ( ) φ u (ω) G(e jω ) 2 W (ω)dω. φ u C 1 ([,b],r + ) b φ u(ω)dω=1 Proof: The cost function (16) cn be written s J(φ u ) = C (φ u (ω))w (ω)dω, (19) where C is constnt, independent of φ u, given by Now, if φ u C := ( ) Kφn (ω) G(e jω ) 2 W (ω)dω. (20) is ny function in C 1 ([, b], R + ) such tht φ u (ω)dω = 1, then by (17) we hve tht J(φ u ) = C = C = J(φ u) [φ u(ω) + (φ u (ω) φ u(ω))]w (ω)dω (φ u(ω))w (ω)dω 1 φ u(ω) (φ u(ω) φ u(ω))w (ω)dω h(φ u (ω), φ u(ω))w (ω)dω (21) = J(φ u) h(φ u (ω), φ u(ω))w (ω)dω W (ω)dω (φ u (ω) φ u(ω))dω h(φ u (ω), φ u(ω))w (ω)dω, where h : R + R + R is given by h(x, y) := x y 1 (x y). (22) y Thus, since w > 0, to prove tht φ u gives the globl minimum for the cost function (16), it suffices to show tht h(x, y) < 0 for every x, y R + such tht x y. To this end, notice tht h x (x, y) = 1 x 1 y, (23) thus if x > y, then x h h(x, y) = h(y, y) + ( x, y)d x < 0, (24) x nd similrly for x < y. This proves the Lemm. The reltionship given in (17) highlights the importnce of choosing the correct function W so s to reflect the desired reltive frequency weighting. The choice of W will be explored in the next Section. VII. CHOICE OF THE FUNCTION W A weighting function which is resonble in the sense tht it stisfies Criterion B is described below: Lemm 3: For 0 < < b < 2π, let W C 1 ([, b], R + ). If W stisfies β kβ W (ω)dω = W (ω)dω (25) α for every α < β b nd every k > 0 such tht kα < kβ b, then there is λ > 0 such tht W (x) = λ/x for every x [, b]. Proof: Since W is continuous, we hve from (25) tht +ε W (ω)dω W () = lim ε 0 + for 1 k < b/. Thus, = lim ε 0 + k y = kw (k) kα ε k+kε k W (ω)dω kε (26) W (k) = 1 W (); k < b, (27) k or, by defining x := k nd λ := W (), W (x) = x W () = λ x ; x < b. (28) By the continuity of W, we lso hve tht W (b) = λ/b. This proves the Lemm. With this lst result, nd those of the previous Sections, we cn now proceed to estblish the form of suitble mesure of the goodness of n experiment, nd n optiml input signl ccording to this cost function. This will be done in the next Section.
5 10 Power spectrl density fctor of 2 of the optimum for certin fmilies of oneprmeter problems [12, 21, 11], lthough generl results for multi-prmeter problems re not yet vilble. TABLE I RELATIVE VALUES OF COST FOR DIFFERENT INPUT SIGNALS 1 mx θ Θ [θ2 M(θ, φ u)] 1 Single frequency t ω = Bndlimited white noise Bndlimited 1/f noise 1.43 Robust min-mx optiml input Frequency [rd/s] Fig. 2. Power spectrl density of bndlimited 1/f noise signl for = 1 nd b = 2. VIII. BAND-LIMITED 1/f NOISE If we pply the results of the previous sections to the cost function (16), then we immeditely see tht resonble cost function for mesuring the goodness of n experiment when hving only diffuse prior knowledge bout pt is J(φ u ) = ( φ u (ω) G(e jω ) 2 ) 1 dω. (29) ω Therefore, ccording to (17) nd Lemm 2, the optiml input spectrum is given by φ u(ω) = 1/ω dω ω = 1/ω ; ω [, b]. (30) b Figure 2 shows the spectrl density of this type of signl, known s bndlimited 1/f noise, for = 1 nd b = 2. Thus we see tht, subject to the ssumptions introduced bove, i.e. Criteri A.1, A.2 nd B, 1/f noise is the robust input signl for identifying system when one hs only diffuse prior knowledge. Remrk 1: The fct tht bndlimited 1/f noise is the solution of vritionl problem mens tht it is possible to consider dditionl prior informtion by imposing constrints in the optimistion problem. In this sense, the problem of experiment design resembles the development of the Principle of Mximum Entropy s given in [23, 24]. IX. EXAMPLE We hve seen bove tht bndlimited 1/f noise cn be regrded s robust optiml test signl in the sense described in Section VIII. This result is consistent with erlier findings in the literture, which show tht bndlimited 1/f noise hs ner optiml properties for specific clsses of systems. For exmple, it is known to yield performnce which is within Tble I, reproduced from [21], shows some interesting results. In prticulr, this Tble shows the numericl results for the problem of designing n input signl to identify the prmeter θ of the pt 1 G(s) = s/θ + 1, (31) where it is ssumed priori tht θ lies in the rnge Θ := [0.1, 10]. The cost function used for comprison is the worst cse normlized vrince of n efficient estimtor of θ, J (φ u ) := mx θ Θ 0 ω 2 /θ 2 (ω 2 /θ 2 + 1) 2 φ u(ω)dω 1, (32) where the inputs being compred re (i) A sine wve of frequency 1 (this is the optiml input if the true prmeter is θ = 1). (ii) Bndlimited white noise input, limited to the frequency rnge [0.1, 10]. (iii) Bndlimited 1/f noise input, limited to the frequency rnge [0.1, 10]. (iv) The pproximte discretised robust optiml input generted by Liner Progrmming [21]. Notice tht, for ese of comprison, the costs in Tble I hve been normlized so tht the robust optiml input hs cost Figure 3 shows the performnce of these signls ccording to the normlized vrince obtined s function of the true vlue of θ. Both Tble I nd Figure 3 demonstrte tht bndlimited 1/f noise does indeed yield good performnce t lest in terms of n specific exmple. The results presented in the current pper give theoreticl support to these erlier observtions. X. CONCLUSIONS In this pper, we hve studied the problem of robust experiment design in the fce of diffuse prior informtion. We hve nlysed generl clss of criteri for mesuring how good n experiment is, nd hve found tht there is specific mesure within this clss tht gives system independent optiml experiment design, which is suitble for the cse when one only hs vgue ide bout the pt to be identified. We hve lso shown tht 1/f noise is optiml ccording to this cost function.
6 Cost Fig. 3. Vrition of the normlized vrince [ ] ω 2 /θ (ω 2 /θ 2 +1) 2 φ u (ω)dω s function of θ for vrious input signls: the robust optiml input (solid), sine wve of frequency 1 (dotted), bndlimited white noise (dshed) nd bndlimited 1/f noise (dsh-dotted). θ REFERENCES [1] A. C. Atkinson nd A. N. Doner. Optimum Experiment Design. Clrendon Press, Oxford, [2] H. W. Bode. Network Anlysis nd Feedbck Amplifier Design. D. Vn Nostrnd, [3] K. Chloner nd I. Verdinelli. Byesin experiment design: A review. Sttisticl Science, [4] H. Chernoff. Approches in sequentil design of experiments. In J.N. Srivstv, editor, Survey of Sttisticl Design, pges North Hold, Amsterdm, [5] D. R. Cox. Pning of Experiments. Wiley, New York, [6] M. A. El-Gml nd T. R. Plfrey. Economicl experiments: Byesin efficient experimentl design. Int. J. Gme Theory, 25(4): , [7] V. V. Fedorov. Theory of Optiml Experiments. Acdemic Press, New York nd London, [8] V. V. Fedorov. Convex design theory. Mth. Opertionsforsch. Sttist. Ser. Sttistics, 11(3): , [9] G. C. Goodwin, S. F. Grebe, nd M. E. Slgdo. Control System Design. Prentice Hll, Upper Sddle River, New Jersey, [10] G. C. Goodwin nd R. L. Pyne. Dynmic System Identifiction: Experiment Design nd Dt Anlysis. Acdemic Press, New York, [11] G. C. Goodwin, C. R. Rojs, nd J. S. Welsh. Good, bd nd optiml experiments for identifiction. In T. Gld, editor, Forever Ljung in System Identifiction - Workshop on the occsion of Lennrt Ljung s 60th birthdy. September [12] G. C. Goodwin, J. S. Welsh, A. Feuer, nd M. Derpich. Utilizing prior knowledge in robust optiml experiment design. Proc. of the 14th IFAC SYSID, Newcstle, Austrli, pges , [13] H. Hjlmrsson. From experiment design to closed-loop control. Automtic, 41(3): , Mrch [14] O. Kempthorne. Design nd Anlysis of Experiments. Wiley, New York, [15] L. Ljung. Asymptotic vrince expressions for identified blck-box trnsfer function models. IEEE Trnsctions on Automtic Control, 30: , September [16] D. G. Luenberger. Optimiztion by Vector Spce Methods. John Wiley & Sons, [17] R. K. Mehr. Optiml inputs for system identifiction. IEEE Trnsctions on Automtic Control, AC-19: , [18] W. G. Müller nd B. M. Pötscher. Btch sequentil design for nonliner estimtion problem. In V.V. Fedorov, W.G. Müller, nd I.N. Vuchkov, editors, Model-Orientted Dt Anlysis, Survey of Recent Methods, pges Physic-Verlg, Heidelberg, [19] B. Ninness nd H. Hjlmrsson. Vrince error quntifictions tht re exct for finite-model order. IEEE Trnsctions on Automtic Control, 49(8): , [20] L. Pronzto nd E. Wlter. Robust experiment design vi mxmin optimistion. Mthemticl Biosciences, 89: , [21] C. R. Rojs, G. C. Goodwin, J. S. Welsh, nd A. Feuer. Robust optiml experiment design for system identifiction. Automtic (ccepted for publiction), [22] M. M. Seron, J. H. Brslvsky, nd G. C. Goodwin. Fundmentl Limittions in Filtering nd Control. Springer-Verlg, [23] J. E. Shore nd R. W. Johnson. Axiomtic derivtion of the principle of mximum entropy nd the principle of minimum cross-entropy. IEEE Trnsctions on Informtion Theory, 26(1):26 37, [24] J. Skilling. The xioms of mximum entropy. In G. J. Erickson nd C. R. Smith, editors, Mximum-Entropy nd Byesin Methods in Science nd Engineering (Vol. 1), pges Kluwer Acdemic Publishers, [25] E. Wlter nd L. Pronzto. Identifiction of Prmetric Models from Experimentl Dt. Springer-Verlg, Berlin, Heidelberg, New York, [26] J. S. Welsh, G. C. Goodwin, nd A. Feuer. Evlution nd comprison of robust optiml experiment design criteri. Proceedings of the Americn Control Conference, Minnepolis, USA, pges , [27] M. Zrrop. Optiml Experiment Design for Dynmic System Identifiction, volume 21 of Lecture Notes in Control nd Informtion. Springer, Berlin, New York, 1979.
Review of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
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 informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
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 informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
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 informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
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 information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationCalculus of Variations
Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More information1 Online Learning and Regret Minimization
2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationLecture 1: Introduction to integration theory and bounded variation
Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You
More informationIntroduction to the Calculus of Variations
Introduction to the Clculus of Vritions Jim Fischer Mrch 20, 1999 Abstrct This is self-contined pper which introduces fundmentl problem in the clculus of vritions, the problem of finding extreme vlues
More informationThe Riemann-Lebesgue Lemma
Physics 215 Winter 218 The Riemnn-Lebesgue Lemm The Riemnn Lebesgue Lemm is one of the most importnt results of Fourier nlysis nd symptotic nlysis. It hs mny physics pplictions, especilly in studies of
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
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 informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationPredict Global Earth Temperature using Linier Regression
Predict Globl Erth Temperture using Linier Regression Edwin Swndi Sijbt (23516012) Progrm Studi Mgister Informtik Sekolh Teknik Elektro dn Informtik ITB Jl. Gnesh 10 Bndung 40132, Indonesi 23516012@std.stei.itb.c.id
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 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 informationNon-Linear & Logistic Regression
Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationData Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading
Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method
More informationRecitation 3: Applications of the Derivative. 1 Higher-Order Derivatives and their Applications
Mth 1c TA: Pdric Brtlett Recittion 3: Applictions of the Derivtive Week 3 Cltech 013 1 Higher-Order Derivtives nd their Applictions Another thing we could wnt to do with the derivtive, motivted by wht
More informationEstimation of Binomial Distribution in the Light of Future Data
British Journl of Mthemtics & Computer Science 102: 1-7, 2015, Article no.bjmcs.19191 ISSN: 2231-0851 SCIENCEDOMAIN interntionl www.sciencedomin.org Estimtion of Binomil Distribution in the Light of Future
More informationThe steps of the hypothesis test
ttisticl Methods I (EXT 7005) Pge 78 Mosquito species Time of dy A B C Mid morning 0.0088 5.4900 5.5000 Mid Afternoon.3400 0.0300 0.8700 Dusk 0.600 5.400 3.000 The Chi squre test sttistic is the sum of
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
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 informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationA BRIEF INTRODUCTION TO UNIFORM CONVERGENCE. In the study of Fourier series, several questions arise naturally, such as: c n e int
A BRIEF INTRODUCTION TO UNIFORM CONVERGENCE HANS RINGSTRÖM. Questions nd exmples In the study of Fourier series, severl questions rise nturlly, such s: () (2) re there conditions on c n, n Z, which ensure
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationCS667 Lecture 6: Monte Carlo Integration 02/10/05
CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationDriving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d
Interntionl Industril Informtics nd Computer Engineering Conference (IIICEC 15) Driving Cycle Construction of City Rod for Hybrid Bus Bsed on Mrkov Process Deng Pn1,, Fengchun Sun1,b*, Hongwen He1, c,
More informationStudent Activity 3: Single Factor ANOVA
MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether
More informationReversals of Signal-Posterior Monotonicity for Any Bounded Prior
Reversls of Signl-Posterior Monotonicity for Any Bounded Prior Christopher P. Chmbers Pul J. Hely Abstrct Pul Milgrom (The Bell Journl of Economics, 12(2): 380 391) showed tht if the strict monotone likelihood
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationCredibility Hypothesis Testing of Fuzzy Triangular Distributions
666663 Journl of Uncertin Systems Vol.9, No., pp.6-74, 5 Online t: www.jus.org.uk Credibility Hypothesis Testing of Fuzzy Tringulr Distributions S. Smpth, B. Rmy Received April 3; Revised 4 April 4 Abstrct
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More informationAcceptance Sampling by Attributes
Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire
More information1. Gauss-Jacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. Guss-Jcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationarxiv:math/ v2 [math.ho] 16 Dec 2003
rxiv:mth/0312293v2 [mth.ho] 16 Dec 2003 Clssicl Lebesgue Integrtion Theorems for the Riemnn Integrl Josh Isrlowitz 244 Ridge Rd. Rutherford, NJ 07070 jbi2@njit.edu Februry 1, 2008 Abstrct In this pper,
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationCBE 291b - Computation And Optimization For Engineers
The University of Western Ontrio Fculty of Engineering Science Deprtment of Chemicl nd Biochemicl Engineering CBE 9b - Computtion And Optimiztion For Engineers Mtlb Project Introduction Prof. A. Jutn Jn
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
More informationACM 105: Applied Real and Functional Analysis. Solutions to Homework # 2.
ACM 05: Applied Rel nd Functionl Anlysis. Solutions to Homework # 2. Andy Greenberg, Alexei Novikov Problem. Riemnn-Lebesgue Theorem. Theorem (G.F.B. Riemnn, H.L. Lebesgue). If f is n integrble function
More informationMath 31S. Rumbos Fall Solutions to Assignment #16
Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)
More informationChapter 2 Fundamental Concepts
Chpter 2 Fundmentl Concepts This chpter describes the fundmentl concepts in the theory of time series models In prticulr we introduce the concepts of stochstic process, men nd covrince function, sttionry
More informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More informationFUZZY HOMOTOPY CONTINUATION METHOD FOR SOLVING FUZZY NONLINEAR EQUATIONS
VOL NO 6 AUGUST 6 ISSN 89-668 6-6 Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom FUZZY HOMOTOPY CONTINUATION METHOD FOR SOLVING FUZZY NONLINEAR EQUATIONS Muhmmd Zini Ahmd Nor
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationIN GAUSSIAN INTEGERS X 3 + Y 3 = Z 3 HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A2 IN GAUSSIAN INTEGERS X + Y = Z HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH Elis Lmpkis Lmpropoulou (Term), Kiprissi, T.K: 24500,
More informationCalculus I-II Review Sheet
Clculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties
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 information