ITS HISTORY AND APPLICATIONS

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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 Mtemticko-fyzikální 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. Opertor-vlued functions Integrl opertors Product integrl of n opertor-vlued function Generl definition of product integrl Chpter 5. Product integrtion in Bnch lgebrs Riemnn-Grves 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 Riemnn-Stieltjes 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. opertor-vlued 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 n-th 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 well-known method of finding prticulr solution using the vrition of constnts (which works even in the cse of non-constnt 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 so-clled 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 n-th 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] (one-sided 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 well-behved, 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 non-negtive 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 son-in-lw 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

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