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J. Laurie Snell

October 21, 2023

James Laurie Snell (January 15, 1925 in Wheaton, Illinois – March 19, 2011 in Hanover, New Hampshire) was an American mathematician and educator.

James Laurie Snell

Biography Edit

J. Laurie Snell was the son of Roy Snell, an adventure author, and Lucille, a concert pianist. Lucille taught the three sons (Jud, John and Laurie) to play piano, cello, and violin. The family had a life-lease on a cabin in Isle Royale National Park where they would go for summer holidays.[1]

Graduate work Edit

Snell studied mathematics at the University of Illinois with Joseph L. Doob from 1948 through 1951; Doob introduced him to martingales, an aspect of probability theory.[a] Doob assigned such topics by having students attempt to solve a series of problems that he kept on file cards.[b][2] Snell earned his Ph.D. in 1951 ("Applications of Martingale System Theorems"), with Doob as his supervisor.

Dartmouth College Edit

At Dartmouth College Snell became involved in a mathematics department project to develop a course on modern mathematics used in biological and social sciences. He worked with John G. Kemeny and Gerald L. Thompson to write Introduction to Finite Mathematics (1957) which described probability theory, linear algebra, and applications in sociology, genetics, psychology, anthropology, and economics. They found "the basic ideas of finite mathematics were easier to state and theorems about them considerably easier to prove than their infinite counterparts." A French translation was made by M. C. Loyau and published in 1960 by Donod.[3]

Another colleague at Dartmouth, Hazleton Mirkil, joined the team to write Finite Mathematical Structures (1959) for sophomores at Dartmouth studying science. Infinite problems are considered after their finite counterparts are fully developed in the text. In 1962 the publisher Prentice-Hall issued a third book from a Dartmouth team: Kemeny, Snell, Thompson, and Arthur Schleifer Jr. wrote Finite Mathematics with Business Applications which included applications: computer circuits, critical path analysis, flow diagrams for computing and accounting procedures, Monte Carlo simulation of decision processes, reliability, decision theory, waiting line theory, a simple approach to mathematics of finance, matrix games, and the simplex method for solving linear programming problems. A second edition of the first text came out in 1966.

Writings Edit

In 1959 Snell published a survey article on Markov chains.[4] He worked the material into a book Finite Markov Chains with Kemeny. As the "first self-contained account in English",[5] it attracted wide interest. While one reviewer said "the exposition is of high quality",[6] other reviewers found fault: Too little attention paid to assumptions inherent in a model.[7] "Interest builds steadily as one peruses the book." But "little attention to historical development."[8] "From the point of view of an undergraduate ... the opening chapter on mathematical prerequisites is rather frightening."[9] "Does not supersede the corresponding chapters in Feller's classic Introduction to Probability; "No index and not even the sketchiest bibliography."[10]

Snell started Chance News in 1992 to "review news and journal articles pertaining to probability and statistics in the real world." One feature is Forsooth for statistical gaffes in media reports, a column originally found in the newsletter of the Royal Statistical Society. In 2005 Chance News was moved to Chance Wiki where there is an archive of Forsooths and previous News. Out of collaborations in Chance News with Charles M. Grinstead and William P. Peterson, a book Probability Tales (2011) was published by American Mathematical Society in the Student Mathematical Library. The book covers four topics: streaks in sports as streaks of successful Bernoulli trials (like hitting streaks), constructing stock market models, estimating expected value of a lottery ticket, and reliability of fingerprint identification.

Legacy Edit

Snell retired in 1995 and was elected as a fellow of the American Statistical Association in 1996.

The Snell envelope, used in stochastics and mathematical finance, is the smallest supermartingale dominating the price process. The Snell envelope refers to results in a 1952 paper Applications of martingale system theorems.[11]

Books Edit

  • 1957: (with John G. Kemeny and Gerald L. Thompson) Introduction to Finite Mathematics Prentice Hall Online
  • 1959: (with Kemeny, Thompson & Hazleton Mirkil) Finite Mathematical Structures
  • 1960: (with John G. Kemeny) Finite Markov Chains, D. van Nostrand Company ISBN 0-442-04328-7
  • 1962: (with Kemeny, Thompson & Arthur Schleifer Jr.) Finite Mathematics with Business Applications
  • 1962: (with John G. Kemeny) Mathematical Models in the Social Sciences, Ginn and Company
  • 1966: (with J.G. Kemeny & A.W. Knapp) Denumerable Markov Chains, second edition 1976, Springer-Verlag
  • 1980: (with Ross Kindermann) Markov Random Fields and Their Applications, American Mathematical Society ISBN 0-8218-5001-6, ISBN 978-0-8218-5001-5
  • 1980: (with Ross P. Kindermann) "On the relation between Markov random fields and social networks", Journal of Mathematical Sociology 7(1): 1–13.
  • 1984: (with Peter G. Doyle) Random Walks and Electrical Networks, Mathematical Association of America ISBN 0-88385-024-9
  • 1988: Introduction To Probability, Random House ISBN 0-394-34485-5
  • 1997: (with Charles Grinsted) Introduction to Probability second edition, American Mathematical Society, ISBN 0-8218-0749-8, ISBN 978-0-8218-0749-1 (online 2011-07-27 at the Wayback Machine)
  • 2011: (with C.M. Grinstead & W.P. Peterson) Probability Tales, American Mathematical Society ISBN 978-0-8218-5261-3

Notes Edit

  1. ^ Quoted from Snell's Obituary of Joseph L. Doob: A discrete-time martingale is a sequence of random variables with finite expectation such that the expected value of any of the random variables, given the previous outcomes, is equal to the last outcome. Thus, if we interpret the outcomes as our fortune in a game, at each stage the game seems fair. So, we can think of a martingale as representing a fair game. If the expected value is less than or equal to the last outcome, then we say the process is a supermartingale and, if it is greater than or equal to the last value, then it is called a submartingale. Thus, a supermartingale represents an unfavorable game and a submartingale a favorable game. These names are suggested by probabilistic potential theory, where martingales correspond to harmonic functions, supermartingales to superharmonic functions, and submartingales to subharmonic functions.[2]
  2. ^ Quoted from Snell's Obituary of Joseph L. Doob: Doob kept a card file of ideas for theses. When he got a new graduate student he would pull out a card and suggest the problem on the card. If the student could not solve it, Doob put it back in the file and chose the next card...I succeeded on the third card, which proposed extending to submartingales an inequality called the "upcrossing inequality" that Doob proved for martingales and used to prove his martingale convergence theorem. This inequality for a submartingale   would, for a < b, give an upper bound in terms of   for the expected number of times the sample path can go from below a to above b, up to time n. This bound implied that if   for some constant k, then the sample paths cannot oscillate infinitely often between a and b with positive probability, which implies that the submartingale converges with probability 1.[2]

References Edit

  1. ^ Breen/Snell Camp from Isle Royale Institute at Michigan Technological University
  2. ^ a b c J.L. Snell (2005) "Obituary: Joseph L. Doob", Journal of Applied Probability 42(1): 247–56 doi:10.1017/S002190020000019X
  3. ^ Algèbre Moderne et Activités Humaines
  4. ^ J.L. Snell (1959) "Finite Markov Chains and Their Applications", American Mathematical Monthly 66: 99–104
  5. ^ Harrison White (1961) American Journal of Sociology 66(1): 427
  6. ^ D.J. Thompson Quarterly Review of Biology 37(1) doi:10.1086/403629
  7. ^ Glen E. Baxter (1961) Journal of the American Statistical Association 56: 182,3 doi:10.2307/2282356
  8. ^ K. A. Bush (1960) American Mathematical Monthly 67(10): 1039
  9. ^ S. D. Silvey (1960) Proceedings of the Edinburgh Mathematical Society 12(1)
  10. ^ Benoit Mandelbrot (1960) Information and Control
  11. ^ J. L. Snell (1952) "Applications of martingale system theorems", Transactions of the American Mathematical Society 73: 293–312

External links Edit

October 21, 2023
laurie, snell, james, laurie, snell, january, 1925, wheaton, illinois, march, 2011, hanover, hampshire, american, mathematician, educator, james, laurie, snell, contents, biography, graduate, work, dartmouth, college, writings, legacy, books, notes, references. James Laurie Snell January 15 1925 in Wheaton Illinois March 19 2011 in Hanover New Hampshire was an American mathematician and educator James Laurie Snell Contents 1 Biography 2 Graduate work 3 Dartmouth College 4 Writings 5 Legacy 6 Books 7 Notes 8 References 9 External linksBiography EditJ Laurie Snell was the son of Roy Snell an adventure author and Lucille a concert pianist Lucille taught the three sons Jud John and Laurie to play piano cello and violin The family had a life lease on a cabin in Isle Royale National Park where they would go for summer holidays 1 Graduate work EditSnell studied mathematics at the University of Illinois with Joseph L Doob from 1948 through 1951 Doob introduced him to martingales an aspect of probability theory a Doob assigned such topics by having students attempt to solve a series of problems that he kept on file cards b 2 Snell earned his Ph D in 1951 Applications of Martingale System Theorems with Doob as his supervisor Dartmouth College EditAt Dartmouth College Snell became involved in a mathematics department project to develop a course on modern mathematics used in biological and social sciences He worked with John G Kemeny and Gerald L Thompson to write Introduction to Finite Mathematics 1957 which described probability theory linear algebra and applications in sociology genetics psychology anthropology and economics They found the basic ideas of finite mathematics were easier to state and theorems about them considerably easier to prove than their infinite counterparts A French translation was made by M C Loyau and published in 1960 by Donod 3 Another colleague at Dartmouth Hazleton Mirkil joined the team to write Finite Mathematical Structures 1959 for sophomores at Dartmouth studying science Infinite problems are considered after their finite counterparts are fully developed in the text In 1962 the publisher Prentice Hall issued a third book from a Dartmouth team Kemeny Snell Thompson and Arthur Schleifer Jr wrote Finite Mathematics with Business Applications which included applications computer circuits critical path analysis flow diagrams for computing and accounting procedures Monte Carlo simulation of decision processes reliability decision theory waiting line theory a simple approach to mathematics of finance matrix games and the simplex method for solving linear programming problems A second edition of the first text came out in 1966 Writings EditIn 1959 Snell published a survey article on Markov chains 4 He worked the material into a book Finite Markov Chains with Kemeny As the first self contained account in English 5 it attracted wide interest While one reviewer said the exposition is of high quality 6 other reviewers found fault Too little attention paid to assumptions inherent in a model 7 Interest builds steadily as one peruses the book But little attention to historical development 8 From the point of view of an undergraduate the opening chapter on mathematical prerequisites is rather frightening 9 Does not supersede the corresponding chapters in Feller s classic Introduction to Probability No index and not even the sketchiest bibliography 10 Snell started Chance News in 1992 to review news and journal articles pertaining to probability and statistics in the real world One feature is Forsooth for statistical gaffes in media reports a column originally found in the newsletter of the Royal Statistical Society In 2005 Chance News was moved to Chance Wiki where there is an archive of Forsooths and previous News Out of collaborations in Chance News with Charles M Grinstead and William P Peterson a book Probability Tales 2011 was published by American Mathematical Society in the Student Mathematical Library The book covers four topics streaks in sports as streaks of successful Bernoulli trials like hitting streaks constructing stock market models estimating expected value of a lottery ticket and reliability of fingerprint identification Legacy EditSnell retired in 1995 and was elected as a fellow of the American Statistical Association in 1996 The Snell envelope used in stochastics and mathematical finance is the smallest supermartingale dominating the price process The Snell envelope refers to results in a 1952 paper Applications of martingale system theorems 11 Books Edit1957 with John G Kemeny and Gerald L Thompson Introduction to Finite Mathematics Prentice Hall Online 1959 with Kemeny Thompson amp Hazleton Mirkil Finite Mathematical Structures 1960 with John G Kemeny Finite Markov Chains D van Nostrand Company ISBN 0 442 04328 7 1962 with Kemeny Thompson amp Arthur Schleifer Jr Finite Mathematics with Business Applications 1962 with John G Kemeny Mathematical Models in the Social Sciences Ginn and Company 1966 with J G Kemeny amp A W Knapp Denumerable Markov Chains second edition 1976 Springer Verlag 1980 with Ross Kindermann Markov Random Fields and Their Applications American Mathematical Society ISBN 0 8218 5001 6 ISBN 978 0 8218 5001 5 1980 with Ross P Kindermann On the relation between Markov random fields and social networks Journal of Mathematical Sociology 7 1 1 13 1984 with Peter G Doyle Random Walks and Electrical Networks Mathematical Association of America ISBN 0 88385 024 9 1988 Introduction To Probability Random House ISBN 0 394 34485 5 1997 with Charles Grinsted Introduction to Probability second edition American Mathematical Society ISBN 0 8218 0749 8 ISBN 978 0 8218 0749 1 online Archived 2011 07 27 at the Wayback Machine 2011 with C M Grinstead amp W P Peterson Probability Tales American Mathematical Society ISBN 978 0 8218 5261 3Notes Edit Quoted from Snell s Obituary of Joseph L Doob A discrete time martingale is a sequence of random variables with finite expectation such that the expected value of any of the random variables given the previous outcomes is equal to the last outcome Thus if we interpret the outcomes as our fortune in a game at each stage the game seems fair So we can think of a martingale as representing a fair game If the expected value is less than or equal to the last outcome then we say the process is a supermartingale and if it is greater than or equal to the last value then it is called a submartingale Thus a supermartingale represents an unfavorable game and a submartingale a favorable game These names are suggested by probabilistic potential theory where martingales correspond to harmonic functions supermartingales to superharmonic functions and submartingales to subharmonic functions 2 Quoted from Snell s Obituary of Joseph L Doob Doob kept a card file of ideas for theses When he got a new graduate student he would pull out a card and suggest the problem on the card If the student could not solve it Doob put it back in the file and chose the next card I succeeded on the third card which proposed extending to submartingales an inequality called the upcrossing inequality that Doob proved for martingales and used to prove his martingale convergence theorem This inequality for a submartingale Y 1 Y 2 displaystyle Y 1 Y 2 nbsp would for a lt b give an upper bound in terms of E Y n displaystyle E Y n nbsp for the expected number of times the sample path can go from below a to above b up to time n This bound implied that if E Y n lt k displaystyle E Y n lt k nbsp for some constant k then the sample paths cannot oscillate infinitely often between a and b with positive probability which implies that the submartingale converges with probability 1 2 References Edit Breen Snell Camp from Isle Royale Institute at Michigan Technological University a b c J L Snell 2005 Obituary Joseph L Doob Journal of Applied Probability 42 1 247 56 doi 10 1017 S002190020000019X Algebre Moderne et Activites Humaines J L Snell 1959 Finite Markov Chains and Their Applications American Mathematical Monthly 66 99 104 Harrison White 1961 American Journal of Sociology 66 1 427 D J Thompson Quarterly Review of Biology 37 1 doi 10 1086 403629 Glen E Baxter 1961 Journal of the American Statistical Association 56 182 3 doi 10 2307 2282356 K A Bush 1960 American Mathematical Monthly 67 10 1039 S D Silvey 1960 Proceedings of the Edinburgh Mathematical Society 12 1 Benoit Mandelbrot 1960 Information and Control J L Snell 1952 Applications of martingale system theorems Transactions of the American Mathematical Society 73 293 312External links EditLiterature by and about J Laurie Snell in the German National Library catalogue Website at Dartmouth College J Laurie Snell at the Mathematics Genealogy Project Retrieved from https en wikipedia org w index php title J Laurie Snell amp oldid 1154871617, wikipedia, wiki, book, books, library,

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