ITS HISTORY AND APPLICATIONS


 Gregory Lambert
 2 years ago
 Views:
Transcription
1 NEČAS CENTER FOR MATHEMATICAL MODELING, Volume 1 HISTORY OF MATHEMATICS, Volume 29 PRODUCT INTEGRATION, ITS HISTORY AND APPLICATIONS Antonín Slvík (I+ A(x)dx)=I+ b A(x)dx+ b x2 A(x 2 )A(x 1 )dx 1 dx 2 + MATFYZPRESS
2 NEČAS CENTER FOR MATHEMATICAL MODELING, Volume 1 HISTORY OF MATHEMATICS, Volume 29 PRODUCT INTEGRATION, ITS HISTORY AND APPLICATIONS Antonín Slvík PRAGUE 2007
3 NEČAS CENTER FOR MATHEMATICAL MODELING Editoril bord Michl Beneš, Fculty of Nucler Sciences nd Physicl Engineering, Czech Technicl University Edurd Feireisl, Institute of Mthemtics, Acdemy of Sciences of the Czech Republic Miloslv Feistuer, Fculty of Mthemtics nd Physics, Chrles University Josef Málek, Fculty of Mthemtics nd Physics, Chrles University Jn Mlý, Fculty of Mthemtics nd Physics, Chrles University Šárk Nečsová, Institute of Mthemtics, Acdemy of Sciences of the Czech Republic Jiří Neustup, Institute of Mthemtics, Acdemy of Sciences of the Czech Republic Tomáš Roubíček, Fculty of Mthemtics nd Physics, Chrles University HISTORY OF MATHEMATICS Editoril bord Jindřich Bečvář, Fculty of Mthemtics nd Physics, Chrles University Edurd Fuchs, Fculty of Science, Msryk University Ivn Kolář, Fculty of Science, Msryk University Krel Mčák, Fculty of Eduction, Technicl University Liberec Ivn Netuk, Fculty of Mthemtics nd Physics, Chrles University Štefn Schwbik, Institute of Mthemtics, Acdemy of Sciences of the Czech Republic Emilie Těšínská, Institute of Contemporry History, Acdemy of Sciences of the Czech Republic The publiction ws supported by Nečs Center for Mthemticl Modeling nd by grnt MSM of the Czech Ministery of Eduction. Všechn práv vyhrzen. Tto publikce ni žádná její část nesmí být reprodukován nebo šířen v žádné formě, elektronické nebo mechnické, včetně fotokopií, bez písemného souhlsu vydvtele. c Antonín Slvík, 2007 c MATFYZPRESS, vydvtelství Mtemtickofyzikální fkulty Univerzity Krlovy v Prze, 2007 ISBN
4 Tble of contents Prefce 1 Chpter 1. Introduction Ordinry differentil equtions in the 19th century Motivtion to the definition of product integrl Product integrtion in physics Product integrtion in probbility theory Chpter 2. The origins of product integrtion Product integrtion in the work of Vito Volterr Bsic results of mtrix theory Derivtive of mtrix function Product integrl of mtrix function Continuous mtrix functions Multivrible clculus Product integrtion in complex domin Liner differentil equtions t singulr point Chpter 3. Lebesgue product integrtion Riemnn integrble mtrix functions Mtrix exponentil function The indefinite product integrl Product integrl inequlities Lebesgue product integrl Properties of Lebesgue product integrl Double nd contour product integrls Generliztion of Schlesinger s definition Chpter 4. Opertorvlued functions Integrl opertors Product integrl of n opertorvlued function Generl definition of product integrl Chpter 5. Product integrtion in Bnch lgebrs RiemnnGrves integrl Definition of product integrl Useful inequlities Properties of product integrl Integrble nd product integrble functions Additionl properties of product integrl Chpter 6. Kurzweil nd McShne product integrls Kurzweil nd McShne integrls
5 6.2 Product integrls nd their properties Chpter 7. Complements Vrition of constnts Equivlent definitions of product integrl RiemnnStieltjes product integrl Bibliogrphy 145
6 Prefce This publiction is devoted to the theory of product integrl, its history nd pplictions. The text represents n English trnsltion of my disserttion with numerous corrections nd severl complements. The definition of product integrl ppered for the first time in the work of Vito Volterr t the end of the 19th century. Although it is rther elementry concept, it is lmost unknown mong mthemticins. Wheres the ordinry integrl of function A provides solution of the eqution y (x) = f(x), the product integrl helps us to find solutions of the eqution y (x) = A(x)y(x). The function A cn be sclr function, but product integrtion is most useful when A is mtrix function; in the ltter cse, y is vector function nd the bove eqution represents in fct system of liner differentil equtions of the first order. Volterr ws trying (on the whole successfully) to crete nlogy of infinitesiml clculus for the product integrl. However, his first ppers didn t meet with gret response. Only the development of Lebesgue integrl nd the birth of functionl nlysis in the 20th century ws followed by the revivl of interest in product integrtion. The ttempts to generlize the notion of product integrl followed two directions: Product integrtion of mtrix functions whose entries re not Riemnn integrble, nd integrtion of more generl objects thn mtrix functions (e.g. opertorvlued functions). In the 1930 s, the ides of Volterr were tken up by Ludwig Schlesinger, who elborted Volterr s results nd introduced the notion of Lebesgue product integrl. Approximtely t the sme time, Czech mthemticin nd physicist Bohuslv Hostinský proposed definition of product integrl for functions whose vlues re integrl opertors on the spce of continuous functions. New pproches to ordinry integrtion were often followed by similr theories of product integrtion; one of the ims of this work is to document this progress. It cn be lso used s n introductory textbook of product integrtion. Most of the text should be comprehensible to everyone with good knowledge of clculus. Prts of Section 1.1 nd Section 2.8 require bsic knowledge of nlytic functions in complex domin, but both my be skipped. Sections 3.5 to 3.8 ssume tht the reder is fmilir with the bsics of Lebesgue integrtion theory, nd Chpters 4 nd 5 use some elementry fcts from functionl nlysis. Almost every text bout product integrtion contins references to the works of V. Volterr, B. Hostinský, L. Schlesinger nd P. Msni, who hve strongly influenced the present stte of product integrtion theory. The lrgest prt of this 1
7 publiction is devoted to the discussion of their work. There were lso other pioneers of product integrtion such s G. Rsch nd G. Birkhoff, whose works [GR] nd [GB] didn t hve such gret influence nd will not be treted here. The reders with deeper interest in product integrtion should consult the monogrph [DF], which includes n exhusting list of references. All theorems nd proofs tht were tken over from nother work include reference to the originl source. However, especilly the results of V. Volterr were reformulted in the lnguge of modern mthemtics. Some of his proofs contined gps, which I hve either filled, or suggested different proof. Since the work ws originlly written in Czech, it includes references to severl Czech monogrphs nd rticles; I hve lso provided reference to n equivlent work written in English if possible. I m indebted especilly to professor Štefn Schwbik, who supervised my disserttion on product integrtion, suggested vluble dvices nd devoted lot of time to me. I lso thnk to ing. Tomáš Hostinský, who hs kindly provided photogrph of his grndfther Bohuslv Hostinský. 2
8 Chpter 1 Introduction This chpter strts with brief look on the prehistory of product integrtion; the first section summrizes some results concerning ordinry differentil equtions tht were obtined prior the discovery of product integrl. The next prt provides motivtion for the definition of product integrl, nd the lst two sections describe simple pplictions of product integrtion in physics nd in probbility theory. 1.1 Ordinry differentil equtions in the 19th century The notion of product integrl ws introduced by Vito Volterr in connection with the differentil eqution of the nth order y (n) (x) + p 1 (x)y (n 1) (x) + + p n (x)y(x) = q(x). (1.1.1) Such n eqution cn be converted (see Exmple 2.5.5) into system of n liner differentil equtions of the first order y i(x) = n ij (x)y j (x) + b i (x), i = 1,..., n, j=1 which cn be lso written in the vector form y (x) = A(x)y(x) + b(x). (1.1.2) Volterr ws initilly interested in solving this eqution in the rel domin: Given the functions A : [, b] R n n (where R n n denotes the set of ll rel n n mtrices) nd b : [, b] R n, we hve to find ll solutions y : [, b] R n of the system (1.1.2). Lter Volterr considered lso the complex cse, where y : G C n, A : G C n n nd b : G C n, where G C nd C n n denotes the set of ll n n mtrices with complex entries. To be ble to pprecite Volterr s results, let s hve brief look on the theory of Equtions (1.1.1) nd (1.1.2) s developed t the end of the 19th century. A more detiled discussion cn be found e.g. in the book [Kl] (Chpters 21 nd 29). A lrge mount of problems in physics nd in geometry leds to differentil equtions; mthemticins were thus forced to solve differentil equtions lredy since the invention of infinitesiml clculus. The solutions of mny differentil equtions hve been obtined (often in n ingenious wy) in closed form, i.e. expressed s combintions of elementry functions. Leonhrd Euler proposed method for solving Eqution (1.1.1) in cse when the p i re constnts. Substituting y(x) = exp(λx) in the corresponding homogeneous eqution yields the chrcteristic eqution λ n + p 1 λ n p n = 0. 3
9 If the eqution hs n distinct rel roots, then we hve obtined fundmentl system of solutions. Euler knew how to proceed even in the cse of multiple or complex roots nd ws lso ble to solve inhomogeneous equtions. The wellknown method of finding prticulr solution using the vrition of constnts (which works even in the cse of nonconstnt coefficients) ws introduced by Joseph Louis Lgrnge. More complicted equtions of the form (1.1.1) cn be often solved using the power series method: Assuming tht the solution cn be expressed s y(x) = n=0 n(x x 0 ) n nd substituting to the differentil eqution we obtin recurrence reltion for the coefficients n. Of course, this procedure works only in cse when the solution cn be indeed expressed s power series. Consequently, mthemticins begn to be interested in the problems of existence of solutions. The pioneering result ws due to Augustin Louis Cuchy, who proved in 1820 s the existence of solution of the eqution y (x) = f(x, y(x)) y(x 0 ) = y 0 (1.1.3) under the ssumption tht f nd f y re continuous functions. The sttement is lso true for vector functions y, nd thus the liner Eqution (1.1.2) is specil cse of the Eqution (1.1.3). Rudolf Lipschitz lter replced the ssumption of continuity of f y by weker condition (now known s the Lipschitz condition). f(x, y 1 ) f(x, y 2 ) < K y 1 y 2 Tody, the existence nd uniqueness of solution of Eqution (1.1.3) is usully proved using the Bnch fixed point theorem: We put y 1 (x) = y 0 + y n (x) = y 0 + x x 0 f(t, y 0 ) dt, x x 0 f(t, y n 1 (t)) dt, n 2. If f is continuous nd stisfies the Lipschitz condition, then the successive pproximtions {y n } n=1 converge to function y which solves Eqution (1.1.3). The method of successive pproximtions ws lredy known to Joseph Liouville nd ws used by Émile Picrd. Around 1840 Cuchy proved the existence of solution of Eqution (1.1.3) in complex domin using the soclled mjornt method (see [VJ, EH]). We re looking for the solution of Eqution (1.1.3) in the neighbourhood of point x 0 C; the solution is holomorphic function nd thus cn be expressed in the form y(x) = c n (x x 0 ) n (1.1.4) n=0 4
10 in certin neighbourhood of x 0. Suppose tht f is holomorphic for x x 0 nd y y 0 b, i.e. tht f(x, y) = ij (x x 0 ) i (y y 0 ) j. (1.1.5) i,j=0 Substituting the power series (1.1.4) nd (1.1.5) to Eqution (1.1.3) gives n eqution for the unknown coefficients c n ; it is however necessry to prove tht the function (1.1.4) converges in the neighbourhood of x 0. We put nd define F (x, y) = M = sup{ f(x, y), x x 0, y y 0 b} A ij (x x 0 ) i (y y 0 ) j = i,j=0 A ij = M i b j, M (1 (x x 0 )/)(1 (y y 0 )/b). (1.1.6) The coefficients ij cn be expressed using the Cuchy s integrl formul ij = 1 i+j f i!j! x i y j = 1 (2πi) 2 ϕ f(x, y) ϕ b (x x 0 ) i+1 dy dx, (y y 0 ) j+1 where ϕ is circle centered t x 0 with rdius > 0 nd ϕ b is circle centered t y 0 with rdius b > 0. The lst eqution leds to the estimte ij A ij, i.e. the infinite series (1.1.6) is mjornt to the series (1.1.5). Cuchy proved tht there exists solution of the eqution Y (x) = F (x, Y (x)) tht cn be expressed in the form Y (x) = n=0 C n(x x 0 ) n in neighbourhood of x 0 nd such tht c n C n. Consequently the series (1.1.4) is lso convergent in neighbourhood of x 0. In prticulr, for the system of liner equtions (1.1.2) Cuchy rrived t the following result: Theorem Consider functions ij, b j (i, j = 1,..., n) tht re holomorphic in the disk B(x 0, r) = {x C; x x 0 < r}. Then there exists exctly one system of functions y i (x) = yi 0 + c ij (x x 0 ) i (i = 1,..., n) j=1 defined in B(x 0, r) tht stisfies y i(x) = n ij (x)y j (x) + b i (x), j=1 y i (x 0 ) = y 0 i, 5
11 where y 0 1,..., y 0 n C re given numbers. As consequence we obtin the following theorem concerning liner differentil equtions of the nth order: Theorem Consider functions p 1,..., p n, q tht re holomorphic in the disk B(x 0, r) = {x C; x x 0 < r}. Then there exists exctly one holomorphic function y(x) = c k (x x 0 ) k k=0 defined in B(x 0, r) tht stisfies the differentil eqution nd the initil conditions y (n) (x) + p 1 (x)y (n 1) (x) + + p n (x)y(x) = q(x) y(x 0 ) = y 0, y (x 0 ) = y 0,..., y (n 1) (x 0 ) = y (n 1) 0, where y 0, y 0,..., y (n 1) 0 C re given complex numbers. Thus we see tht the solutions of Eqution (1.1.1), whose coefficients p 1,..., p n, q re holomorphic functions, cn be indeed obtined by the power series method. However, it is often necessry to solve Eqution (1.1.1) in cse when the coefficients p 1,..., p n, q hve n isolted singulrity. For exmple, seprtion of vribles in the wve prtil differentil eqution leds to the Bessel eqution y (x) + 1 ) x y (x) + (1 n2 y(x) = 0, whose coefficients hve singulrity t 0. Similrly, seprtion of vribles in the Lplce eqution gives the Legendre differentil eqution y (x) with singulrities t 1 nd 1. x 2 2x n(n + 1) 1 x 2 y (x) + 1 x 2 y(x) = 0 The behviour of solutions in the neighbourhood of singulrity hs been studied by Bernhrd Riemnn nd fter 1865 lso by Lzrus Fuchs. Consider the homogeneous eqution y (n) (x) + p 1 (x)y (n 1) (x) + + p n (x)y(x) = 0 (1.1.7) in the neighbourhood of n isolted singulrity t x 0 C; we ssume tht the functions p i re holomorphic in the ring P (x 0, R) = {x C; 0 < x x 0 < r}. If we choose n rbitrry P (x 0, R), then the functions p i re holomorphic in U(, r) (where r = x 0 ), nd Eqution (1.1.7) hs n linerly independent holomorphic solutions y 1,..., y n in U(, r). We now continue these functions long the circle ϕ(t) = x 0 + ( x 0 ) exp(it), t [0, 2π] 6
12 centered t x 0 nd pssing through. We thus obtin different system of solutions Y 1,..., Y n in U(, r). Since both systems re fundmentl, we must hve Y i = n j=1 M ijy j, or in the mtrix nottion Y = My. By clever choice of the system y 1,..., y n it cn be chieved tht M is Jordn mtrix. Using these fcts, Fuchs ws ble to prove the existence of fundmentl system of solutions of Eqution (1.1.7) in P (x 0, R) tht consists of nlytic functions of the form (x x 0 ) λi ( ϕ i 0(x) + ϕ i 1(x) log(x x 0 ) + + ϕ i n i (x) log ni (x x 0 ) ), i = 1,..., n, where ϕ j k re holomorphic functions in P (x 0, R) nd λ i C is such tht exp(2πiλ i ) is n eigenvlue of M with multiplicity n i. Moreover, if p i hs pole of order t most i t x 0 for i {1,..., n}, then the Fuchs theorem gurntees tht ϕ j k hs pole (i.e. not n essentil singulrity) t x 0. This result implies tht Eqution (1.1.7) hs t lest one solution in the form y(x) = (x x 0 ) r k=0 k(x x 0 ) k ; the numbers r nd k cn be clculted by substituting the solution to Eqution (1.1.7) (this is the Frobenius method). We hve now recpitulted some bsic fcts from the theory of ordinry differentil equtions. In lter chpters we will see tht mny of them cn be lso obtined using the theory of product integrtion. 1.2 Motivtion to the definition of product integrl The theory of product integrl is rther unknown mong mthemticins. following text should provide motivtion for the following chpters. We consider the ordinry differentil eqution The y (t) = f(t, y(t)) (1.2.1) y() = y 0 (1.2.2) where f : [, b] R n R n is given function. Thus, we re seeking solution y : [, b] R n tht stisfies (1.2.1) on [, b] (onesided derivtives re tken t the endpoints of [, b]) s well s the initil condition (1.2.2). An pproximte solution cn be obtined using the Euler method, which is bsed on the observtion tht for smll t, y(t + t). = y(t) + y (t) t = y(t) + f(t, y(t)) t. We choose prtition D : = t 0 < t 1 < < t m = b of intervl [, b] nd put y(t 0 ) = y 0 y(t 1 ) = y(t 0 ) + f(t 0, y(t 0 )) t 1 y(t 2 ) = y(t 1 ) + f(t 1, y(t 1 )) t 2 y(t m ) = y(t m 1 ) + f(t m 1, y(t m 1 )) t m, 7
13 where t i = t i t i 1, i = 1,..., m. We expect tht the finer prtition D we choose, the better pproximtion we get (provided tht f is wellbehved, e.g. continuous, function). We now turn to the specil cse f(t, y(t)) = A(t)y(t), where A(t) R n n is squre mtrix for every t [, b]. The Euler method pplied to the liner eqution yields y(t 0 ) = y 0, y (t) = A(t)y(t) y() = y 0 (1.2.3) y(t 1 ) = (I + A(t 0 ) t 1 )y(t 0 ) = (I + A(t 0 ) t 1 )y 0, y(t 2 ) = (I + A(t 1 ) t 2 )y(t 1 ) = (I + A(t 1 ) t 2 )(I + A(t 0 ) t 1 )y 0, y(t m ) = (I + A(t m 1 ) t m ) (I + A(t 1 ) t 2 )(I + A(t 0 ) t 1 )y 0, where I denotes the identity mtrix. Put P (A, D) = (I + A(t m 1 ) t k ) (I + A(t 1 ) t 2 )(I + A(t 0 ) t 1 ). Provided the entires of A re continuous functions, it is possible to prove (s will be done in the following chpters) tht, if ν(d) 0 (where ν(d) = mx{ t i, i = 1,..., m}), then P (A, D) converges to certin mtrix; this mtrix will be denoted by the symbol (I + A(x) dx) nd will be clled the left product integrl of the mtrix function A over the intervl [, b]. Moreover, the function Y (t) = t (I + A(x) dx) stisfies Y (t) = A(t)Y (t) Y () = I Consequently, the vector function y(t) = t (I + A(x) dx) y 0 is the solution of Eqution (1.2.3). 8
14 1.3 Product integrtion in physics The following exmple shows tht product integrtion lso finds pplictions outside mthemticl nlysis, prticulrly in fluid mechnics ( more generl tretment is given in [DF]). Consider fluid whose motion is described by function S : [t 0, t 1 ] R 3 R 3 ; the vlue S(t, x) corresponds to the position (t the moment t) of the prticle tht ws t position x t the moment t 0. Thus, for every t [t 0, t 1 ], S cn be viewed s n opertor on R 3 ; we emphsize this fct by writing S(t)(x) insted of S(t, x). If x is position of certin prticle t the moment t, it will move to S(t + t) S(t) 1 (x) (where denotes the composition of two opertors) during the intervl [t, t + t]. Consequently, its instntneous velocity t the moment t is given by S(t + t) S(t) 1 ( (x) x V (t)(x) = lim = lim t 0 t t 0 S(t + t) S(t) 1 ) I (x), t where I denotes the identity opertor. The velocity V (t) is n opertor on R 3 for every t [t 0, t 1 ]; in the following chpters it will be clled the left derivtive of the opertor S. Given the velocity opertor V, how to reconstruct the position opertor S? For smll t we hve S(t + t)(x). = (I + V (t) t) S(t)(x). If we choose sufficiently fine prtition D : t 0 = u 0 < u 1 < < u m = t of intervl [t 0, t], we obtin S(t)(x). = (I + V (u m 1 ) u m ) (I + V (u 0 ) u 1 )(x), where u i = u i u i 1, i = 1,..., m. The bove product (or composition) resembles the product encountered in the previous section. Indeed, pssing to the limit ν(d) 0, we see tht S is the left product integrl of opertor V, i.e. S(t) = t t 0 (I + V (u) du), t [t 0, t 1 ]. In certin sense, the left derivtive nd the left product integrl re inverse opertions. 1.4 Product integrtion in probbility theory Some results of probbility theory cn be elegntly expressed in the lnguge of product integrtion. We present two exmples concerning survivl nlysis nd Mrkov processes; both re inspired by [Gil]. Exmple Let T be nonnegtive continuous rndom vrible with distribution function F (t) = P (T t) nd probbility density function f(t) = F (t). 9
15 For exmple, T cn be interpreted s the service life of certin component (or the length of life of person etc.). The probbility of filure in the intervl [t, t + t] is P (t T t + t) = F (t + t) F (t). We remind tht the survivl function is defined s S(t) = 1 F (t) = P (T > t) nd the filure rte (or the hzrd rte) is (t) = f(t) S(t) = f(t) 1 F (t) = S (t) S(t) = d log S(t). (1.4.1) dt The nme filure rte stems from the fct tht P (t T t + t T > t) P (t T t + t) lim = lim = t 0 t t 0 P (T > t) t F (t + t) F (t) = lim = f(t) t 0 S(t) t S(t) = (t), i.e. for smll t, the conditionl probbility of filure during the intervl [t, t + t] is pproximtely (t) t. Given the function, Eqution (1.4.1) tells us how to clculte S: ( t ) S(t) = exp (u) du. (1.4.2) We cn lso proceed in different wy: If we choose n rbitrry prtition then D : 0 = t 0 < t 1 < < t m = t, S(t) = P (T > t) = P (T > t 0 )P (T > t 1 T > t 0 ) P (T > t m T > t m 1 ) = m m = P (T > t i T > t i 1 ) = (1 P (T t i T > t i 1 )). i=1 0 i=1 In cse the prtition is sufficiently fine, the lst product is pproximtely equl to m (1 (t i ) t i ). i=1 This product is similr to the one used in the definition of left product integrl, but the fctors re reversed. Its limit for ν(d) 0 is clled the right product integrl of the function on intervl [0, t] nd will be denoted by the symbol S(t) = (1 (u) du) t. (1.4.3) 0 10
16 Compring Equtions (1.4.2) nd (1.4.3) we obtin the result (1 (u) du) t 0 ( = exp t 0 ) (u) du, which will be proved in Chpter 2 (see Exmple 2.5.6). The product integrl representtion of S hs the dvntge tht it cn be intuitively viewed s the product of probbilities 1 (u) du tht correspond to infinitesiml intervls of length du. The lst exmple corresponds in fct to simple Mrkov process with two sttes s 1 ( the component is operting ) nd s 2 ( the component is broken ). The process strts in the stte s 1 nd goes over to the stte s 2 t time T. We now generlize our clcultion to Mrkov processes with more thn two sttes; before tht we recll the definition of Mrkov process. A stochstic process X on intervl [0, ) is rndom function t X(t), where X(t) is rndom vrible for every t [0, ). We sy tht the process is in the stte X(t) t time t. A Mrkov process is stochstic process such tht the rnge of X is either finite or countbly infinite nd such tht for every choice of numbers n N, n > 1, 0 t 1 < t 2 < < t n, we hve P (X(t n )=x n X(t n 1 )=x n 1,..., X(t 1 )=x 1 ) = P (X(t n )=x n X(t n 1 )=x n 1 ), where x 1,..., x n re rbitrry sttes (i.e. vlues from the rnge of X). The bove condition mens tht the conditionl probbility distribution of the process t time t n depends only on the lst observtion t t n 1 nd not on the whole history. Exmple Let {X(t); t 0} be Mrkov process with finite number of sttes S = {s 1,..., s n }. For exmple, we cn imgine tht X(t) determines the number of ptients in physicin s witing room (whose cpcity is of course finite). Suppose tht the limit ij (t) = P (X(t + t) = s j X(t) = s i ) lim t 0+ t exists for every i, j = 1,..., n, i j nd for every t [0, ). The number ij (t) is clled the trnsition rte from stte i to stte j t time t. For sufficiently smll t we hve P (X(t + t) = s j X(t) = s i ). = ij (t) t, i j, (1.4.4) P (X(t + t) = s i X(t) = s i ). = 1 j i ij (t) t. (1.4.5) We lso define ii (t) = j i ij (t), i = 1,..., n nd denote A(t) = { ij (t)} n i,j=1. Given the mtrix A, we re interested in clculting the probbilities p i (t) = P (X(t) = s i ), t [0, ), i = 1,..., n, 11
17 nd p ij (s, t) = P (X(t) = s j X(s) = s i ), 0 s < t, i, j = 1,..., n. The totl probbility theorem gives p j (t) = n p i (0)p ij (0, t). i=1 The probbilities p i (0), i = 1,..., n re usully given nd it is thus sufficient to clculte the probbilities p ij (0, t), or generlly p ij (s, t). Putting P (s, t) = {p ij (s, t)} n i,j=1 we cn rewrite Equtions (1.4.4) nd (1.4.5) to the mtrix form for sufficiently smll t. P (t, t + t). = I + A(t) t (1.4.6) Using the totl probbility theorem once more we obtin p ij (s, u) = n p ik (s, t)p kj (t, u), (1.4.7) for 0 s < t < u, i, j = 1,..., n. This is equivlent to the mtrix eqution P (s, u) = P (s, t)p (t, u). (1.4.8) If we choose sufficiently fine prtition s = u 0 < u 1 < < u m = t of intervl [s, t], then Equtions (1.4.6) nd (1.4.8) imply m P (s, t) = P (u i 1, u i ) =. m (I + A(u i ) u i ). i=1 Pssing to the limit for ν(d) 0 we obtin mtrix which is clled the right product integrl of the function A over intervl [s, t]: i=1 P (s, t) = (I + A(u) du) t. s The lst result cn be gin intuitively interpreted s the product of mtrices I + A(u) du which correspond to trnsition probbilities in the infinitesiml time intervls of length du. 12
18 Chpter 2 The origins of product integrtion The notion of product integrl hs been introduced by Vito Volterr t the end of the 19th century. We strt with short biogrphy of this eminent Itlin mthemticin nd then proceed to discuss his work on product integrtion. Vito Volterr ws born t Ancon on 3rd My His fther died two yers lter; Vito moved in with his mother Angelic to Alfonso, Angelic s brother, who supported them nd ws like boy s fther. Becuse of their finncil sitution, Angelic nd Alfonso didn t wnt Vito to study his fvourite subject, mthemtics, t university, but eventully Edordo Almgià, Angelic s cousin nd rilrod engineer, helped to persude them. An importnt role ws lso plyed by Volterr s techer Ròiti, who secured him plce of ssistnt in physicl lbortory. Vito Volterr 1 In 1878 Volterr entered the University of Pis; mong his professors ws the fmous Ulisse Dini. In 1879 he pssed the exmintion to Scuol Normle Superiore of Pis. Under the influence of Enrico Betti, his interest shifted towrds mthemticl physics. In 1882 he offered thesis on hydrodynmics, grduted doctor of physics nd becme Betti s ssistent. Shortly fter, in 1883, the young Volterr won the competition for the vcnt chir of rtionl mechnics nd ws promoted 1 Photo from [McT] 13
19 to professor of the University of Pis. After Betti s deth he took over his course in mthemticl physics. In 1893 he moved to the University of Turin, but eventully settled in Rome in The sme yer he mrried Virgini Almgià (the dughter of Edordo Almgià). During the first qurter of the 20th century Volterr not only represented the leding figure of Itlin mthemtics, but lso becme involved in politics nd ws nominted Sentor of the Kingdom in When Itly entered the world wr in 1915, Volterr volunteered the Army Corps of Engineers nd engged himself in perfecting of irships nd firing from them; he lso promoted the collbortion with French nd English scientists. After the end of the wr he returned to scientific work nd teching t the university. Volterr strongly opposed the Mussolini regime which cme to power in As one of the few professors who refused to tke n oth of loylty imposed by the fscists in 1931, he ws forced to leve the University of Rome nd other scientific institutions. After then he spent lot of time brod (giving lectures e.g. in Frnce, Spin, Czechoslovki or Romni) nd lso t his country house in Aricci. Volterr, who ws of Jewish descent, ws lso ffected by the ntisemitic rcil lws of Although he begn to suffer from phlebitis, he still devoted himself ctively to mthemtics. He died in isoltion on 11th October 1940 without greter interest of Itlin scientific community. Despite the fct tht Volterr is best known s mthemticin, he ws mn of universl interests nd devoted himself lso to physics, biology nd economy. His mthemticl reserch often hd origins in physicl problems. Volterr ws lso n enthusistic bibliophile nd his collection, which reched nerly seven thousnd volumes nd is now deposited in the United Sttes, included rre copies of scientific ppers e.g. by Glileo, Brhe, Trtgli, Fermt etc. The monogrph [JG] contins welth of informtion bout Volterr s life nd times. Volterr s nme is closely connected with integrl equtions. He contributed the method of successive pproximtions for solving integrl equtions of the second kind, nd lso noted tht n integrl eqution might be considered s limiting cse of system of lgebric liner equtions; this observtion ws lter utilized by Ivr Fredholm (see lso the introduction to Chpter 4). His investigtions in clculus of vritions led him to the study of functionls (he clled them functions of lines ); in fct he built complete clculus including the definitions of continuity, derivtive nd integrl of functionl. Volterr s pioneering work on integrl equtions nd functionls is often regrded s the dwn of functionl nlysis. An overview of his chievements in this field cn be obtined from the book [VV5]. Volterr ws lso one of the founders of mthemticl biology. The motivtion cme from his soninlw Umberto D Ancon, who ws studying the sttistics of Adritic fishery. He posed to Volterr the problem of explining the reltive increse of predtor fishes, s compred with their prey, during the period of First World Wr (see e.g. [MB]). Volterr interpreted this phenomenon with the help of mthemticl models of struggle between two species; from mthemticl point of 14
20 view, the models were combintions of differentil nd integrl equtions. Volterr s correspondence concerning mthemticl biology ws published in the book [IG]. A more detiled description of Volterr s ctivites (his work on prtil differentil equtions, theory of elsticity) cn be found in the biogrphies [All, JG] nd lso in the books [IG, VV5]. An interesting ccount of Itlin mthemtics nd its intertwining with politics in the first hlf of the 20th century is given in [GN]. 2.1 Product integrtion in the work of Vito Volterr Volterr s first work devoted to product integrtion [VV1] ws published in 1887 nd ws written in Itlin. It introduces the two bsic concepts of the multiplictive clculus, nmely the derivtive of mtrix function nd the product integrl. The topics discussed in [VV1] re essentilly the sme s in Sections 2.3 to 2.6 of the present chpter. The publiction [VV1] ws followed by second prt [VV2] printed in 1902, which is concerned minly with mtrix functions of complex vrible. It includes results which re recpitulted in Sections 2.7 nd 2.8, nd lso tretment of product integrtion on Riemnn surfces. Volterr lso published two short Itlin notes, [VV3] from 1887 nd [VV4] from 1888, which summrize the results of [VV1, VV2] but don t include proofs. Volterr s finl tretment of product integrtion is represented by the book Opértions infinitésimles linéires [VH] written together with Czech mthemticin Bohuslv Hostinský. The publiction ppered in the series Collection de monogrphies sur l théorie des fonctions directed by Émile Borel in More thn two hundred pges of [VH] re divided into eighteen chpters. The first fifteen chpters represent French trnsltion of [VV1, VV2] with only smll chnges nd complements. The remining three chpters, whose uthor is Bohuslv Hostinský, will be discussed in Chpter 4. As Volterr notes in the book s prefce, the publiction of [VH] ws motivted by the results obtined by Bohuslv Hostinský, s well s by n incresed interest in mtrix theory mong mthemticins nd physicists. As the bibliogrphy of [VH] suggests, Volterr ws lredy wre of the ppers [LS1, LS2] by Ludwig Schlesinger, who linked up to Volterr s first works (see Chpter 3). The book [VH] is rther difficult to red for contemporry mthemticins. One of the resons is somewht cumbersome nottion. For exmple, Volterr uses the sme symbol to denote dditive s well s multiplictive integrtion: The sign pplied to mtrix function denotes the product integrl, while the sme sign pplied to sclr function stnds for the ordinry (dditive) integrl. Clcultions with mtrices re usully written out for individul entries, wheres using the mtrix nottion would hve gretly simplified the proofs. Moreover, Volterr didn t hesitte to clculte with infinitesiml quntities, he interchnges the order of summtion nd integrtion or the order of prtil derivtives without ny justifiction etc. The conditions under which individul theorems hold (e.g. continuity or differentibility of the given functions) re often omitted nd must be deduced from the proof. This is certinly surprising, since the rigorous foundtions of mthemticl 15
Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA
CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationGeneralized Inverses: How to Invert a NonInvertible Matrix
Generlized Inverses: How to Invert NonInvertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationExponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.
Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions Section 3 Integrl Equtions Integrl Opertors nd Liner Integrl Equtions As we sw in Section on opertor nottion, we work with functions
More informationDETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.
Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationINTERCHANGING TWO LIMITS. Zoran Kadelburg and Milosav M. Marjanović
THE TEACHING OF MATHEMATICS 2005, Vol. VIII, 1, pp. 15 29 INTERCHANGING TWO LIMITS Zorn Kdelburg nd Milosv M. Mrjnović This pper is dedicted to the memory of our illustrious professor of nlysis Slobodn
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationOstrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias
Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology,
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationEuler Euler Everywhere Using the EulerLagrange Equation to Solve Calculus of Variation Problems
Euler Euler Everywhere Using the EulerLgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch
More informationLecture 2: Matrix Algebra. General
Lecture 2: Mtrix Algebr Generl Definitions Algebric Opertions Vector Spces, Liner Independence nd Rnk of Mtrix Inverse Mtrix Liner Eqution Systems, the Inverse Mtrix nd Crmer s Rule Chrcteristic Roots
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810193 Aveiro, Portugl
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More information19. The FermatEuler Prime Number Theorem
19. The FermtEuler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
More informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More informationCHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
More information6 Energy Methods And The Energy of Waves MATH 22C
6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
More informationNovel Methods of Generating SelfInvertible Matrix for Hill Cipher Algorithm
Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting SelfInvertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationMATLAB Workshop 13  Linear Systems of Equations
MATLAB: Workshop  Liner Systems of Equtions pge MATLAB Workshop  Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB
More informationDerivatives and Rates of Change
Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationThe Riemann Integral. Chapter 1
Chpter The Riemnn Integrl now of some universities in Englnd where the Lebesgue integrl is tught in the first yer of mthemtics degree insted of the Riemnn integrl, but now of no universities in Englnd
More informationPhysics 6010, Fall 2010 Symmetries and Conservation Laws: Energy, Momentum and Angular Momentum Relevant Sections in Text: 2.6, 2.
Physics 6010, Fll 2010 Symmetries nd Conservtion Lws: Energy, Momentum nd Angulr Momentum Relevnt Sections in Text: 2.6, 2.7 Symmetries nd Conservtion Lws By conservtion lw we men quntity constructed from
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More information4 Approximations. 4.1 Background. D. Levy
D. Levy 4 Approximtions 4.1 Bckground In this chpter we re interested in pproximtion problems. Generlly speking, strting from function f(x) we would like to find different function g(x) tht belongs to
More informationNull Similar Curves with Variable Transformations in Minkowski 3space
Null Similr Curves with Vrile Trnsformtions in Minkowski spce Mehmet Önder Cell Byr University, Fculty of Science nd Arts, Deprtment of Mthemtics, Murdiye Cmpus, 45047 Murdiye, Mnis, Turkey. mil: mehmet.onder@yr.edu.tr
More informationCOMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, crossclssified
More informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More informationINVERSE OF A MATRIX AND ITS APPLICATION O Q
Inverse f A tri And Its Appliction DUE  III IVERSE F A ATRIX AD ITS AICATI ET US CSIDER A EXAE: Abhinv spends Rs. in buying pens nd note books wheres Shntnu spends Rs. in buying 4 pens nd note books.we
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More informationGENERALIZED QUATERNIONS SERRETFRENET AND BISHOP FRAMES SERRETFRENET VE BISHOP ÇATILARI
Sy 9, Arlk 0 GENERALIZED QUATERNIONS SERRETFRENET AND BISHOP FRAMES Erhn ATA*, Ysemin KEMER, Ali ATASOY Dumlupnr Uniersity, Fculty of Science nd Arts, Deprtment of Mthemtics, KÜTAHYA, et@dpu.edu.tr ABSTRACT
More informationSolving BAMO Problems
Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only
More informationThe Fundamental Theorem of Calculus for Lebesgue Integral
Divulgciones Mtemátics Vol. 8 No. 1 (2000), pp. 75 85 The Fundmentl Theorem of Clculus for Lebesgue Integrl El Teorem Fundmentl del Cálculo pr l Integrl de Lebesgue Diómedes Bárcens (brcens@ciens.ul.ve)
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationHarvard College. Math 21a: Multivariable Calculus Formula and Theorem Review
Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 tmcwillim@college.hrvrd.edu December 15, 2009 1 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce 5 9.1
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationAA1H Calculus Notes Math1115, Honours 1 1998. John Hutchinson
AA1H Clculus Notes Mth1115, Honours 1 1998 John Hutchinson Author ddress: Deprtment of Mthemtics, School of Mthemticl Sciences, Austrlin Ntionl University Emil ddress: John.Hutchinson@nu.edu.u Contents
More informationKarlstad University. Division for Engineering Science, Physics and Mathematics. Yury V. Shestopalov and Yury G. Smirnov. Integral Equations
Krlstd University Division for Engineering Science, Physics nd Mthemtics Yury V. Shestoplov nd Yury G. Smirnov Integrl Equtions A compendium Krlstd Contents 1 Prefce 4 Notion nd exmples of integrl equtions
More informationModule Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
More informationQUADRATURE METHODS. July 19, 2011. Kenneth L. Judd. Hoover Institution
QUADRATURE METHODS Kenneth L. Judd Hoover Institution July 19, 2011 1 Integrtion Most integrls cnnot be evluted nlyticlly Integrls frequently rise in economics Expected utility Discounted utility nd profits
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationA Note on Complement of Trapezoidal Fuzzy Numbers Using the αcut Method
Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN  Vol.  A Note on Complement of Trpezoidl Fuzzy Numers Using the αcut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment
More informationDlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationwww.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
More informationUNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES
UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES Solution to exm in: FYS30, Quntum mechnics Dy of exm: Nov. 30. 05 Permitted mteril: Approved clcultor, D.J. Griffiths: Introduction to Quntum
More informationMatrix Inverse and Condition
Mtrix Inverse nd Condition Berlin Chen Deprtment of Computer Science & Informtion Engineering Ntionl Tiwn Norml University Reference: 1. Applied Numericl Methods with MATLAB for Engineers, Chpter 11 &
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More information