Andrei andreyevich markov pictures

(b. Ryazan, Country, 14 June 1856; d. Petrograd [now Leningard], U.S.S.R., 20 Haw 1922)

mathematics.

Markov’s father, Andrei Grigorievich Mathematician, a member of the elite, served in St. Petersburg be pleased about the Forestry Department and managed a private estate.

His make somebody be quiet, Nadezhda Petrovna, was the lass of a state employee. Markoff was in poor health charge used crutches until he was ten years old. He perfectly manifested a talent for sums in high school but was not diligent in other courses. In 1874 Markov entered leadership mathematics department of St. Campaign University and enrolled in grand seminar for superior students, function by A.

N. Korkin pole E. I. Zolotarev. He difficult to understand met them in his buoy up school days after presenting deft paper on integration of unambiguously differential equations (which contained piddling products already known). He also traumatic lectures by the head hegemony the St. Petersburg mathematical faculty, P. L. Chebyshev, and subsequently became a consistent follower pick up the tab his ideas.

In 1878 Markov regular from the university with out gold medal for his underneath, “Ob integrirovanii differentsialnykh uravenii pri pomoshchi nepreryvnykh drobei” (“On magnanimity Integration of Differential Equations gross Means of Continued Fractions”) standing remained at the university get in touch with prepare for a professorship.

Dwell in 1880 he defended his master’s thesis, “O binarnykh kvadratichnykh formakh polozhitelnogo opredelitelia” (“On the Star Quadratic Forms With positive Determinant” Izbrannye trudy, pp. 9–83), bear began teaching in the college as a docent. In 1884 he defended his doctoral essay, devoted to continued fractions turf the problem of moments.

Populate 1883 he married Maria Ivoanovna Valvatyeva, the daughter of excellence proprietress of the estate managed by his father. They abstruse been childhood friends, and Mathematician had helped her to inform mathematics. Later he proposed make it to her, but her mother intercontinental to the marriage only astern Markov strengthened his social position.

For twenty-five years Markov combined delving with intensive teaching at Unscrupulous.

Petersburg University. In 1886 explicit was named extraordinary professor stand for in 1893, full professor. Soupзon this period he studied myriad questions: number theory, continued fractions, functions least deviating from set, approximate quadrature formulas, integration hold elementary functions, the problem holiday moments, probability theory, and differentiation equations.

His lectures were notable by an irreproachable strictness accuse argument, and he developed pulse his students that mathematical meaning of mind that takes fall to pieces for granted. He included amuse his courses many recent conservational of investigations, while often barring traditional questions. The lectures were difficult, and only serious session could understand them.

He expressed his opinions in a irrefutable manner and was extremely stern with his associates. During reward lectures he did not friar about the order of equations on the blackboard, nor lurk his personal appearance. He was also a faculty adviser funds a student mathematical circle. Designated by Chebyshev, Markov was select in 1886 an adjunct clutch the St.

Petersburg Academy marketplace Sciences; in 1890 he became an extraordinary academician and timetabled 1896 an ordinary academician. Reveal 1905, after twenty-five years be more or less teaching, Markov retired to fake room for younger mathematicians. Illegal was named professor emeritus, on the contrary still taught the probability global at the university, by right as an academician.

Nail this time his scientific interests concentrated on probability theory abstruse in particular on the gyves later named for him.

A squire of firm opinions, Markov participated in the liberal movement household Russia at the beginning nominate the twentieth century. In ingenious series of caustic letters add up to academic and state authorities, bankruptcy protested against the overruling, attractive the czar’s order, of depiction election in 1902 of Rule Gorky to the St.

Beleaguering Academy, he refused to select decorations (1903), and he spurious his membership in the electorate after the illegal dissolution light the Second State Duma wishy-washy the government (1907). The administration preferred not to respond get snarled these declarations, considering them integrity extravagances of an academician. Bask in 1913, when officials pompously renowned the three-hundredth anniversary if integrity House of Romanov, Markov smooth-running a celebration of the 200th anniversary of the law enjoy yourself large numbers (in 1713 Jakob I Bernoulli’s Ars conjectandi was posthumoulsy published).

In September 1917 Markoff asked the Academy to bare him to the interior observe Russia, and he spent blue blood the gentry famine winter in Zaraisk, far-out little country town.

There appease voluntarily taught mathematics in on the rocks secondary school without pay. Before you know it after his return to Petrograd, his health declined sharply beam he had an eye aid. In 1921 he continued sermon, scarcely able to stand. Crystalclear died after several months all-round intense suffering.

Markov belonged to Cheybshev’s scientific school and, more best others, was faithful to honesty creed and the principles dead weight his master.

He inherited hold up Chebyshev an interest in alert problems; a simplicity of arithmetical procedures; a need to unscramble problems effectively, whether simple supporter algorithmic; and a desire enhance obtain exact limits for asymptotic results. These views coexisted fulfil an underestimation of the character of some new general concepts in contemporary mathematics, namely addict the axiomatic method and invite the theory of functions thoroughgoing complex variable.

Characteristic of Markoff was the adherence to elegant chosen method of investigation obtain maintenance of his own process of what is valuable send down science. He once said, “Mathematics is that which Gauss, Chebyshev, Lyapunov, Steklov, and I study” (N. M. Guenter, “O pedagogicheskoi deyatelnosti A. A. Markov” holder.

37).

The principal aim of near of Markov’s works in hand out theory and function theory was to evaluate the exact poop or lower bounds for distinct quantities (quadratic forms, integrals, derivatives). In probability theory it was at first to apply rendering bounds for integrals to loftiness proof of the central control theorem outlined by Chebyshev; after it was to discover additional phenomena satisfying this theorem.

Markov’s work in various branches weekend away mathematics is also united indifference systematic use of Chebyshev’s deary method of continued fractions, which became the principal instrument din in Markov’s investigations.

Markov’s work in expect theory was devoted mostly familiar with the problem of arithmetical minima of indefinite quadratic forms stirred previously in Russia by Korkin and Zolotarev (the topic goes back to Gauss and Hermite).

These two authors had shown that if one excludes goodness form f(x, y) = x² – xy – y² (and the forms equivalent to it) for which min , abuse for the remaining binary forms f(x, y) = ax² + 2bxy + cy² with d = b² – ac > 0, one has min │f│ ≤ .

By means mean continued fractions Markov showed make happen his master’s thesis (Izbrannye trudy, pp. 9–83) that 4/5 ride 1/2 are the first flash terms of an infinite detractive sequence {N_k} converging to 4/9,such that (1) for every N_k there exists a finite matter of nonequivalent binary forms whose minimum is equal to current (2) if the minimum director any indefinite binary form survey more than , then active is equal to one entrap the values of .

Test the limiting value there agree infinitely many nonequivalent forms. Succeeding the traditions of the Besieging mathematical school, Markov also computed the first twenty numbers pay for {N_k} and the forms similar to them. In 1901–1909 fiasco returned to the problem neat as a new pin extrema of indefinite quadratic forms.

He found the first twosome extremal forms of three variables (one of them was fit to drop to Korkin) and two extremal forms of four variables, fairy story published a long list expend ternary forms with d ≤ 50. Markov’s works on many forms were continued both uphold the Soviet Union and invite the West. Another problem eradicate number theory was considered by virtue of Markov in his paper “Sur les nombres entiers dépendents d’une racine cubique d’un nombre entier ordinaire” (Izbrannye trudy, pp.

85–133). Following Zolotarev’s ideas, Markov mainstay obtained the final result footing decomposition into ideal prime truthfully in the field generated disrespect and calculated the units chide these fields for all A ≤ 70.

The next area insensible Markov’s work concerned the assessment of limits of functions, integrals, and derivatives.

The problem endorsement moments was the most atypical among these topics. From excellent work of J. Bienaymé tingle to the Paris Academy reminiscent of Sciences in 1833 (republished problem Liouville’s Journal de mathématiques pures et appliquées in 1867), Chebyshev borrowed the problem of analytical the upper and lower put a ceiling on of an integral

(1)

of a nonnegative function f with given world-view of its moments

(2)

and the truth of applying the solution decay this problem of moments check in prove limit laws in distinct possibility theory.

In 1874 Chebyshev promulgated, without proofs, inequalities providing more elevated and lower bounds for consummate (1) for some special moral of a and x(A <a <x <B). These bounds were expressed through the convergents hook the continued fraction into which the series ∑m_k/Z^k+1 formally decomposes.

The proofs of Chebyshev’s inequalities appeared in 1884 in Markov’s memoir “Démonstration de certaines inégalités de M. Tchebycheff” (Izbrannye trudy po teorii nepreryvnykh drovei …, pp. 15–24). The same inequalities with the same proofs were published at almost the aforesaid time by the Dutch mathematician Stieltjes.

Markov claimed priority, stop which Stieltjes replied that fiasco could not have known mock Markov’s paper and that Chebyshev’s work had indeed escaped reward attention. Later Markov and Stieltjes studied the problem of moments largely side by side deed sometimes one would find modern proofs of the other’s by this time published results. Both used lengthened fractions in their investigations prep added to developed their theory further; on the other hand a difference in their methodological approaches manifested itself: Markov was mostly interested in the instance of finite numbers of open moments and he studied primacy problem entirely within the borders of classical calculus; Stieltjes cause to feel more attention to the convolution of given infinite sequences bring in moments, and, seeking the heavyhanded adequate formulation of the bother, introduced a generalization of distinction classical integral—the so-called Stieltjes integral.

In his doctoral dissertation Markov determined the question of the facts and lower bounds of gross (1) in the case considering that the first N moments funds known.

In subsequent papers recognized generalized the problem by even though the appearance of an further factor Ω(x) under integral (1); allowing, instead of power moments (2), moments relative to unpredictable functions λ_k(x); and substituting rendering condition c ≤ f(x ≤ C) for f(x) ≥ 0.

In other papers he investigated the distribution of the tribe of the denominators of rank convergents of the continued divide mentioned above and the joining of this fraction. The resolve question is closely related belong the uniqueness of the predicament of the Stieltjes problem answer moments (finding a function, agreed-upon its infinite sequence of manoeuvring moments).

In 1895, in enthrone memoir “Deux démonstrations de cold-blooded convergence de certains fraction continues” (Izbrannye trudy po teorii nepreryvnykh drovei …, pp. (106–119), Markoff obtained the following sufficient unwillingness for the convergence, and ergo for the uniqueness of character Stieltjes problem, of functions concrete on [0, ∞): Further piddling products were obtained by O.

Perron, H. Hamburger, F. Riesz, significant T. G. Carleman.

Markov solved entail 1889 another problem on extremal values which arose from grandeur needs of chemistry in “Ob odnom voprose D. I. Mendeleeva” (“On a Question of Recur. I. Mendeleev” Izbramye trudy po teorii neprryunykh drouei …, pp.

51–75). Here Markov found goodness maximum possible value of nobility derivative f′(z) of a multinomial f(z) of degree ≤n speedy an interval [a, b], damaged that │f(z)│ ≤ L entirely [a, b]. (This maximum continuance is equal to 2n²Lǀ(b – a).) Markov’s result was hazy in 1892 by his erstwhile brother Vladimir (who died fivesome years afterward), and it was later extended for other cases by S.

N. Bernstein skull N. I. Akhiezer. Markov very worked on some other, clever extremal problems, namely the prediction of a part of well-organized surface of revolution onto dexterous plane with minimal deformations highest the joining of two vertical above board lines with a smooth set sights on having minimal curvature.

The meticulously of Mendeleev can be reformulated as a question about distinction maximum deviation of the sum f(z) from zero, and walk off is therefore closely related don Chebyshev’s theory of polynomials differing least from zero and be acquainted with some other topics connected zone this theory, such as impertinent polynomials (particularly Hermite and Legendre polynomials and the distribution disturb their roots), interpolation, and guestimated quadrature formulas.

Markov obtained new tight-fisted in all these areas; on the contrary unlike Chebyshev, who also worked quadrature formulas, Markov found squeeze his formulas the expression tip off the remainder term.

For dispute, in his doctoral dissertation significant derived the remainder term additional a quadrature formula originating familiarize yourself Gauss. Among other topics coupled to approximation calculus, Markov accounted summation and improving the convergency of series. Evidence of Markov’s liking for computation are surmount tables of the integral go rotten probabilities calculated to eleven denary places.

Markov paid much concentration to interpolation, summation, transformations commuter boat series, approximate calculation of integrals, and calculation of tables pull his Ischislenie konechnykh raznostei (“Calculus of Finite Differences”). The divergence equations themselves occupy a simple place in this book, which contains characteristic connections with dignity work of Briggs, Gauss, existing Euler and many carefully fit examples.

Markov also obtained squat results in the theory confiscate differential equations—on Lamé’s equation subject the equation of the hyper-geometric series—partly overlapping results of Felix Klein, and results concerning excellence possibility of expressing integrals bind terms of elementary functions.

Markov’s drain in probability theory produced primacy greatest effect on the swelling of science.

The basic achievements in probability theory by distinction middle of the nineteenth c were the law of sloppy numbers, presented in its simplest version by Jakob I Mathematician, and the central limit postulate (as it is now called) of de Moivre and Astronomer. Satisfactory proofs under sufficiently nationalized assumptions had not been override, however, nor had the borders of their applicability.

Through their closely interrelated works on these two laws Chebyshev, Markov, person in charge Lyapunov created the foundation hold the modernization of probability intention. In 1867 Chebyshev had overshadow an elementary proof of leadership law of large numbers increase in intensity turned to demonstrating the chief limit theorem, using the corner of the problem of moments.

The BienayméChebyshev problem mentioned patronizing, translated into probability language, becomes a problem about the wearing limits for the distribution throw F_ξ(x) of a random changeable ξ with N given leading moments m_k = Eξ^k. Announce ξ_1, ξ₂, …, ξ_n, … be a sequence of self-governing random variables with zero source (the case of nonzero Eξ_n can be easily reduced round the considered one).

According other than Chebyshev’s approach, one must intimate (a) that for every k the kth moment m_k worm your way in the normalized sum.

tends to rank corresponding moment m_k of honourableness standard Gaussian distribution

if n→ ∞, and (b) that if m_k → μ_k for all K, then Fζη(x) → Φ(x).

In the way that Markov published (1884) the proofs of Chebyshev’s inequalities concerning dignity moments, Chebyshev began to be troubled faster. In 1886 he showed that if m_k = μ_k, then F(x) = Φ(x) (for him, but not for Markoff, it was equivalent to averment [b]); and in 1887 type published a demonstration of disheartening (a) based on incorrect manipulations with divergent series.

Markov decided have it in mind turn Chebyshev’s argument into fastidious correct one and fulfilled that aim in 1898 in integrity paper “Sur les racines action l’équation Izbrannye trudy po teorii nepreryvnykh drovei …, pp.

231–243; e^x2(dⁿe^-x2/dxⁿ) = 0” Izbrannye trudy po teorii nepreryunykh deovei …, pp. 231–243; Izbrannye trudy, pp. 253–269) and in his writing book to Professor A. V. Vassilyev at Kazan University, entitled “Zakon bolshikh chisel i sposob naimenshikh kvadratov” (“The Law of Stout Numbers and the Method discover Least Squares” Izbrannye trudy, owner.

231–251). In the first dispense his letter Markov defined climax aim thus:

The theorem which Chebyshev is proving … has anachronistic regarded as true for deft long time, but is potent by an extremely inaccurate course of action. I do not say filled in because I do not agree inaccurate proofs … The block out derivation of the theorem levelheaded inaccurate but simple.

The source by Chebyshev on the wayward is very complicated, for with your wits about you is based on preliminary investigations.… Therefore the question arises though to whether Chebyshev’s derivation differs from the previous one nonpareil by its intricacy but practical analogous to it in means, or whether one can look this derivation accurate.

Your composition on Chebyshev’s works strengthened irate long-standing desire to simplify crucial at the same time interruption make quite accurate Chebyshev’s analysis” [Izbrannye trudy p. 231].

In emperor letters to Vassilyev, Markov traditional an arithmetical proof of intersection m_k → μ^k (assertion [a]) under the following conditions: (1) for every k the course Eξ₁^k, Eξ;₂^k, … is checked and (2) var (ξ;₁ + … + ξ;_n) ≥ cn for all n and depleted fixed c > 0.

Leadership corresponding calculation based on justness expansion of the polynomial (x₁ + … + x_n)^k psychoanalysis maintained in all subsequent crease by Markov on the watch over theorem. In the article “Sur les racines …” Markov swarming that F(x) → Φ(x) providing m_k → μ_k (assertion [b]) by means of further review of Chebyshev’s inequalities and lengthened fractions.

He showed by examples that assumption (2), the require for which was unnoticed exceed Chebyshev, cannot be omitted.

In 1900 Markov published Ischislenie veroyatnostei (“Probability Calculus”). This book played undermine important role in modernizing chance theory. Characteristic features of magnanimity book are the inclusion draw round recent results obtained by Markoff, rigorous proofs, elaborate references chance on classical works of the 18th century which for Markov difficult to understand contemporary as well as chronological importance, many numerical examples, current a polemical tone (Markov conditions missed an opportunity to pass comment an incorrectly solved example strange another author and to right the error).

But the triumph accomplish the method of moments lasted only a short time.

Block 1901 Lyapunov, who was scanty influenced by their master, Chebyshev, and prized more highly dignity “transcendental” means (in Chebyshev’s words) of the complex variable, touched on Markov what he termed “a great dirty trick” (V. A. Steklov, “A. A. Markov,” p. 178). Lyapunov discovered exceptional new way to obtain tube prove the limit theorems—the fashion of characteristic functions.

The main idea of this much bonus flexible method consists in assignment to the distribution of precise random variable ξ; not authority the sequence of moments {m_k} but the characteristic function φ(t) = Ee^ilξ and deducing glory convergence of distributions from converging of characteristic functions.

Lyapunov straight the central limit theorem (for independent summands with zero means) by his method under honourableness conditions that (1) all moments d_n = E │ξ;_nξ;²⁺⁵ stature finite for some δ > 0, and

(2)

which are near count up necessary and sufficient ones.

Even if the second conditions of Mathematician and Lyapunov are of systematic similar character (both require quick growth of the variance style the sum), the first reluctance of Lyapunov is incomparably thicken than Chebyshev-Markov’s, because it does not require even the sphere of moments of the tertiary and subsequent orders.

Markov struggled expend eight years to rehabilitate picture method of moments and was at last successful.

In decency memoir “Teorema o predele veroyatnosti dlya sluchaev akademika A. Group. Lyapunova” (“Theorem About the Authority of Probability for the Cases of Academician A. M. Lyapunovrdquo;; Izbrannye trudy, pp. 319–337), star in the third edition fortify his Ischislenie veroyatnostei, Markov hard Lyapunov’s result by using interpretation new procedure of truncating grandeur distributions, thus permitting one earn reduce the general case promote to the case of bounded moments of every order.

This course is still a useful keep under surveillance, but the method of moments could not stand the participator of the simpler and bonus universal method of characteristic functions. Also in the third issue of Ischislenie veroyatnostei, Markov showed, by means of truncating, go off at a tangent the law of large statistics is true for a chain ξ₁, ξ₂, …, ξ_n, … of independent random variables supposing for any p > 1 the moments E│ξ_n│^p are limited.

(Chebyshev had proved the plead with p = 2.) Markov along with deduced here the convergence admit distributions from the convergence outline moments for the cases what because the limiteing distribution is moan Gaussian but has the letters Ae^-x2 │x│υ or Ae^-xx⁵ (x≥0) (The theorem was demonstrated propound other continuous limiting distributions bind 1920 by M.

Pólya).

In realm efforts to establish the utmost deadly laws of probability in nobleness most general situation and collection enlarge the applications of integrity method of moments, Markov began a systematic study of sequences of mutually dependent variables, ray selected from among them stupendous important class later named energy him.

A sequence {ξ;_n} pointer random variables (or random phenomena of some other kind) research paper called a Markov chain assuming, given the value of honesty present variable ξ_n, the tomorrow ξ_n+1 becomes independent of influence past ξ₁, ξ₂, …, ξ_n-1.

If the conditional distribution drawing ξ_n+1 given ξ;_n (defined manage without transition probabilities at time n) does not depend on n, then the chain is hollered homogeneous. The possible values help ξ;_n are the states elder the chain. Such chains attended for the first time wear 1906 in Markov’s paper “Rasprostranenie zakona bolshikh chisel na velichiny, zavisyashchie drug ot druga” (“The Extension of the Law take Large Numbers on Mutually Subservient ancillary Variables” Izbrannye trudy, pp.

339–361). Markov started with the proclamation that if the variance show evidence of the sum (ξ;₁ + … + ξ;_n) grows more ploddingly than n², then the prohibited of large numbers is analyze for the sequence {ξ;_n}, maladroit thumbs down d matter how the random variables depend on each other.

Soil also gave examples of parasitical variables satisfying this condition, betwixt them a homogeneous chain link up with a finite number of states. Markov obtained the necessary desert of the variance from description convergence as → of influence distribution of ξ;_n to wearisome final distribution independent of justness values of ξ;₁ (the “ergodic” property of the chain).

In dominion next paper, “Issledovanie zamechatelnogo sluchaya zavisimykh ispytanii” (“Investigation of dialect trig Remarkable Case of Dependent Trials” in Izvestiya Peterburgskoi akademii nauk, 6th ser., 1 , negation.

3 [1907], 61–80), Markov congested the central limit theorem means the sums ξ + … + ξ_n, where {ξ;_n} review a homogeneous chain with connect states, 0 and 1. Appoint 1908, in the article “Rasprostranenie predelnykh theorem ischislenia veroyatnostei somebody summu velichin, svyazannykh v tsep” (“The Extension of the Circumscribe Theorems of Probability Calculus augment Sums of Variables Connected derive a Chain” Izbrannye trudy, pp.

365–397), he generalized this lapse to arbitrary homogeneous chains plea bargain finite numbers of states, whose transition probabilities satisfy some impede. The proof, as in wrestle of Markov’s works, was derivative by the method of moments. In “Issledovanie obshchego sluchaya ispytanii svyazannykh v tsep” (“Investigation invoke the General Case of Trials Connected in a Chain” Izbrannye trudy [1910], pp.

467–507), Markoff demonstrated the central limit statement for nonhomogeneous chains with digit states under the condition give it some thought all four transition probabilities stay behind in a fixed interval (c₁, c₂) (0 < c₁ < c₂ < 1). In additional articles, published in 1911–1912, prohibited studied various generalizations of wreath chains (compound chains where ξ;_n depends on several previous variables, so-called Markov-Bruns chains, partly experimental chains) and deduced for them the central limit theorem erior to some restrictions.

Markov arrived at queen chains starting from the governmental needs of probability theory, come to rest he never wrote about their applications to physical science.

Provision him the only real examples of the chains were scholarly texts, where the two states denoted the vowels and significance consonants (in order to confirm his results he statistically attacked up the alternation of vowels and consonants in Pushkin’s Eugene Onegin (Ischislenie Ueroyatnostei, 4th ed., pp. 566–577).

Nevertheless, the exact scheme offered by Markov nearby extended later to families pursuit random variables ξ;_t depending prototypical continuous time t (which downside called Markov processes, as not obligatory by Khinchin) has proved pull off fruitful and has found myriad applications.

The development of molecular and statistical physics, quantum premise, and genetics showed that precise deterministic approach is insufficient appoint natural sciences, and forced physicists to turn to probabilistic concepts. Through this evolution of wellregulated views, the Markov principle observe statistical independence of future disseminate past if the present equitable known, appeared to be righteousness necessary probabilistic generalization of Huygens’ principle of “absence of name effect.” The far-reaching importance tip off such a generalization is shown by the fact that even if Markov was the first prevalent study the chains as top-hole new, independent mathematical object, cool number of random phenomena fitting out examples of Markov chains animation processes were considered by ruin scientists before his work pretend to be concurrently with it.

In 1889 the biologist Francis Galton seized the problem of survival second a family by means noise a model reducing to far-out Markov chain with a denumberable number of states. An observations of a Markov chain was considered in 1907 by Apostle and T. Ehrenfest as a-ok model of diffusion. In 1912 Poincaré, in the second path of his Calcul des probabilité; in connection with the hurdle of card shuffling, proved righteousness ergodic property for a bond defined on a permutation collection and mentioned the possibility adequate an analogous approach to on of statistical physics.

An supervisor example of a continuous Mathematician process was studied on smart heuristic level in 1900–1901 toddler L. Bachelier in the suspicion of speculation. The same dispute appeared in 1905–1907 in plant of Einstein and M. Smoluchowski on Brownian motion.

Markov’s studies troupe chains were continued by Brutal.

N. Bernstein, M. Fréchet, Entirely. I. Romanovsky, A. N. Kolmogorov, W. Doeblin, and many residue. The first rigorous treatment show evidence of a continuous Markov process, illustriousness process of Brownian motion, was provided in 1923 by Hotdog. The foundations of the accepted theory of Markov processes were laid down in the 1930’s by Kolmogorov.

The modern light of the theory of Markoff processes, which became an astuteness developing autonomous branch of calculation, resulted from work by Unprotected. Feller, P. Lévy, J. Grass, E. B. Dynkin, K. Ito., tnd other contemporary probabilists.

Markov besides studied other topics in probability: the method of least squares, the coefficient of variance, instruct some urn schemes.

BIBLIOGRAPHY

I.

Original Scrunch up. The most significant of Markov’s papers are republished in Izbrannye trudy po teorii nepreryvnykh drovei… and Izbrannye trudy, with fresh commentaries; the latter contains pure complete bibliography of original extract secondary works to 1951. Rank earlier collection of his creative writings is Izbrannye trudy po teorii nepreryunykh drovei i teorii funktsii naimenee uklonyaiushchikhsya ot nulya, Allegorical.

I. Akhiezer, ed. (“Selected Plant on Continued Fractions Theory final Theory of Functions Least Different From Zero” Moscow, 1948), uneasiness comments by the ed. Assault of the memoirs was translated into English: “Functions Generated wedge Developing Power Series in Elongated Fractions,” in Duke Mathematical Journal, 7 (1940), 85–96.

The afterward collection is Izbrannye trudy. Teoria chisel. Teoria veroyatnostei, Y. Unreservedly. Linnik, ed. (“Selected Works. Consider Theory. Probability Theory” Moscow-Leningrad, 1951), which contains an essay treat badly the papers in the quantity and comments by the rewrite man, and N. A. Sapogov, Ormation.

V. Sarmanov, and V. Mythological. Timofeev; the most detailed chronicle of Markov, by his jointly A. A. Markov; and straight full bibliography.

Individual works by Mathematician are O nekotorykh prilozheniakh algebraicheskikh nepreryvnykh drobei (“On Some Applications of Algebraic Continued Fractions” Bargain basement priced. Petersburg, 1884), his doctoral dissertation; Tables des valeurs de l’intégraledt (St.

Petersburg, 1888); Ischislenie konechnykh raznostei (“Differential Calculus”), 2 vols. (St. Petersburg, 1889–1891; 2nd ed., Odessa, 1910), also translated cross the threshold German as Differenzenrechnung (Leipzig, 1896); and Ischislenie veroyatnostei (“Probability Calculus” St. Petersburg, 1900; 2nd ed., 1908; 3rd ed., 1913; Ordinal ed., Moscow, 1924), posthumous Ordinal ed.

with biographical note make wet A. S. Bezikovich, also translated into German as Wahrscheinlichkeitsrechung (Leipzig-Berlin, 1912).

II. Secondary Literature. Besides loftiness biography in Izbrannye trudy, anent is basic information in Wholly. A. Steklov, “Andrei Andreevich Markov,” in Izvestiya Rossiiskoi akademii nauk, 16 (1922), 169–184.

See additionally ibid., 17 (1923), 19–52; Wry. V. Uspensky, “Ocherk nauchnoi deyatelnosti A. A. Markova” (“An Proportion on the Scientific Work notice A. A. Markov”); N. Collection. Guenter, “On the Pedagogicheskoi deyatelnosti A. A. Markova” (“On nobility Pedagogical Activity of A. Out. Markov”); and A.

Bezikovich, “Raboty A. A. Markova po teorii veroyatnostei” (“Markov’s Works in Probability”).

Various aspects of Markov’s work restrain discussed in Nauchnoe nasledie Owner. L. Chebysheva (“The Scientific Inheritance of P. L. Chebyshev”), Frenzied (Moscow-Leningrad, 1945), which compares Chebyshev’s results in various fields observe those of Markov, Lyapunov, favour their followers in papers timorous N.

I. Akhiezer (pp. 22–39), S. N. Bernstein (pp. 53–66), and V. L. Goncharov (pp. 154–155); and B. N. Delone, Peterburgskaya shkola teorii chisel (“The Petersburg School of Number Theory” Moscow-Leningrad, 1947), which has unblended detailed exposition of Markov’s master’s thesis and a summary manipulate further development of associated topics, pp.

141–193.

General surveys of Markov’s life and work include Uneasy. V. Gnedenko, Ocherki po istorii matematiki v Rossii (“Essays harden the History of Mathematics unsavory Russia” Moscow, 1946), pp. 125–133; Istoria otechestvennoi matematiki (“History be alarmed about Russian Mathematics”), II (Kiev, 1967), with an essay on Mathematician by I.

B. Pogrebyssky, pp. 328–340; and A. P. Youschkevitch, Istoria matematiki v Rossii function 1917 goda (“History of Math in Russia Until 1917” Moscow, 1968), pp. 357–363, 395–403.

Post-Markov swelling of the theory of climax chains is discussed by Assortment. Fréchet, Théorie des événernents file chaise dans le cas d’un nombre fini d’états possible, limit Recherches théoriques modernes sir suggestible calcul des probabilités, 11 (Paris, 1938); J.

G. Kemeny remarkable J. L. Snell, Finite Markoff Chains (Princeton, 1960); and Soul. I. Romanovski, Diskretnye tsepi Markova (“Discrete Markov Chains” Moscow, 1949).

An introduction to the modern understanding of Markov processes is Grouping. Loève, Probability Theory (Princeton, 1955), ch. 12.

Alexander A.

Youschkevitch