∗Most of the material is from [1] When this assumption is violated, a degenerate basic variable (with zero value) occurs in a basic feasible solution. Often it can be handled as a nondegenerate basic feasible solution. However, it is possible that at pivoting the new variable will come in at zero value. This implies that the zero-valued basic variable is the one to go out. The objective will not decrease and the new basic feasible solution will also be degenerate. The result is a cycle that could be repeated indefinitely. Methods have been developed to avoid such cycles [1, pp. 78].
{"title":"The simplex method","authors":"Marcos Singer","doi":"10.4324/9781351032148-6","DOIUrl":"https://doi.org/10.4324/9781351032148-6","url":null,"abstract":"∗Most of the material is from [1] When this assumption is violated, a degenerate basic variable (with zero value) occurs in a basic feasible solution. Often it can be handled as a nondegenerate basic feasible solution. However, it is possible that at pivoting the new variable will come in at zero value. This implies that the zero-valued basic variable is the one to go out. The objective will not decrease and the new basic feasible solution will also be degenerate. The result is a cycle that could be repeated indefinitely. Methods have been developed to avoid such cycles [1, pp. 78].","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129094659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-07-01DOI: 10.1142/9789814759786_0014
J. Bernstein
{"title":"The Schrödinger equation","authors":"J. Bernstein","doi":"10.1142/9789814759786_0014","DOIUrl":"https://doi.org/10.1142/9789814759786_0014","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129105606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Jones polynomial","authors":"A. R. Nizami","doi":"10.1090/mbk/121/73","DOIUrl":"https://doi.org/10.1090/mbk/121/73","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133656958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Smale’s paradox","authors":"Geometría","doi":"10.1090/mbk/121/46","DOIUrl":"https://doi.org/10.1090/mbk/121/46","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"1198 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133269759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Society for American Baseball\u0000 Research","authors":"Спорт","doi":"10.1090/mbk/121/59","DOIUrl":"https://doi.org/10.1090/mbk/121/59","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"520 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116571242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-03-01DOI: 10.1215/00182702-38-1-189
Robert Leonard
{"title":"Theory of games and economic behavior","authors":"Robert Leonard","doi":"10.1215/00182702-38-1-189","DOIUrl":"https://doi.org/10.1215/00182702-38-1-189","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131776706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1973-02-01DOI: 10.14492/HOKMJ/1381759014
Hiroshi Tanaka
Let R, R’ be hyperbolic Riemann surfaces and phi be an analytic mapping of R into R’. Let K_{0} be a closed disk in R and let R_{0}=R-K_{0} . Let acute{C} be the Kuramochi capacity on R_{0}cupDelta_{N} and Delta_{1} be the set of all minimal Kuramochi boundary points of R. For a metrizable compactification R^{prime*} of R’, we denote by mathscr{F}(phi) the set of all points in Delta_{1} at which phi has a fine limit in R^{prime*} . There are two typical extensions of Beurling’s theorem [1] to analytic mappings of a Riemann surface to another one, i . e. , Z. Kuramochi’s [5, 6, 7] and C. Constantinescu and A. Cornea’s theorems [3, 4] . The former result states that if phi is an almost finitely sheeted mapping and R^{prime*} is H. D. separative, then tilde{C}(Delta_{1}-mathscr{F}(phi))=0 . The latter one states that if phi is a Dirichlet mapping and R^{prime*} is a quotient space of the Royden compactification of R’, then overline{C}(Delta_{1}-^{Gamma j}(phi))=0 . The present author [9] proved that these two results are independent. In this paper we shall give an another extension of Beurling’s theorem such that it contains the above two results: If phi is a Dirichlet mapping and R^{prime*} is H. D. separative, then Beurling’s theorem is valid. Notation and terminology Let R be a hyperbolic Riemann surface. For a subset A of R, we denote by partial A and A^{i} the (relative) boundary and the interior of A respectively. We call a closed or open subset A of R is regular if partial A is nonempty and consists of at most a countable number of analytic arcs clustering nowhere in R. We fix a closed disk K_{0} in R once for all and let R_{0}=
{"title":"Beurling’s theorem","authors":"Hiroshi Tanaka","doi":"10.14492/HOKMJ/1381759014","DOIUrl":"https://doi.org/10.14492/HOKMJ/1381759014","url":null,"abstract":"Let R, R’ be hyperbolic Riemann surfaces and phi be an analytic mapping of R into R’. Let K_{0} be a closed disk in R and let R_{0}=R-K_{0} . Let acute{C} be the Kuramochi capacity on R_{0}cupDelta_{N} and Delta_{1} be the set of all minimal Kuramochi boundary points of R. For a metrizable compactification R^{prime*} of R’, we denote by mathscr{F}(phi) the set of all points in Delta_{1} at which phi has a fine limit in R^{prime*} . There are two typical extensions of Beurling’s theorem [1] to analytic mappings of a Riemann surface to another one, i . e. , Z. Kuramochi’s [5, 6, 7] and C. Constantinescu and A. Cornea’s theorems [3, 4] . The former result states that if phi is an almost finitely sheeted mapping and R^{prime*} is H. D. separative, then tilde{C}(Delta_{1}-mathscr{F}(phi))=0 . The latter one states that if phi is a Dirichlet mapping and R^{prime*} is a quotient space of the Royden compactification of R’, then overline{C}(Delta_{1}-^{Gamma j}(phi))=0 . The present author [9] proved that these two results are independent. In this paper we shall give an another extension of Beurling’s theorem such that it contains the above two results: If phi is a Dirichlet mapping and R^{prime*} is H. D. separative, then Beurling’s theorem is valid. Notation and terminology Let R be a hyperbolic Riemann surface. For a subset A of R, we denote by partial A and A^{i} the (relative) boundary and the interior of A respectively. We call a closed or open subset A of R is regular if partial A is nonempty and consists of at most a countable number of analytic arcs clustering nowhere in R. We fix a closed disk K_{0} in R once for all and let R_{0}=","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"187 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1973-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124359475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract : The purpose of this note is to indicate how a simple lemma due originally to Fekete, and in generalized form to Polya-Szego, permits a simple derivation of some interesting ergodic theorems. We have already indicated some applications to dynamic programming.
{"title":"The ergodic theorem","authors":"R. Bellman","doi":"10.1090/mbk/107/29","DOIUrl":"https://doi.org/10.1090/mbk/107/29","url":null,"abstract":"Abstract : The purpose of this note is to indicate how a simple lemma due originally to Fekete, and in generalized form to Polya-Szego, permits a simple derivation of some interesting ergodic theorems. We have already indicated some applications to dynamic programming.","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1960-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133964423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1938-02-18DOI: 10.1080/00029890.1948.11999317
G. Mackey
{"title":"William Lowell Putnam Mathematical\u0000 Competition","authors":"G. Mackey","doi":"10.1080/00029890.1948.11999317","DOIUrl":"https://doi.org/10.1080/00029890.1948.11999317","url":null,"abstract":"","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1938-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114440632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Anna Karenina principle is named after the opening sentence in the eponymous novel: Happy families are all alike; every unhappy family is unhappy in its own way. The Two Envelopes Problem (TEP) is a much-studied paradox in probability theory, mathematical economics, logic, and philosophy. Time and again a new analysis is published in which an author claims finally to explain what actually goes wrong in this paradox. Each author (the present author included) emphasizes what is new in their approach and concludes that earlier approaches did not get to the root of the matter. We observe that though a logical argument is only correct if every step is correct, an apparently logical argument which goes astray can be thought of as going astray at different places. This leads to a comparison between the literature on TEP and a successful movie franchise: it generates a succession of sequels, and even prequels, each with a different director who approaches the same basic premise in a personal way. We survey resolutions in the literature with a view to synthesis, correct common errors, and give a new theorem on order properties of an exchangeable pair of random variables, at the heart of most TEP variants and interpretations. A theorem on asymptotic independence between the amount in your envelope and the question whether it is smaller or larger shows that the pathological situation of improper priors or infinite expectation values has consequences as we merely approach such a situation.
{"title":"Two envelopes problem","authors":"R. Gill","doi":"10.1090/mbk/121/70","DOIUrl":"https://doi.org/10.1090/mbk/121/70","url":null,"abstract":"The Anna Karenina principle is named after the opening sentence in the eponymous novel: Happy families are all alike; every unhappy family is unhappy in its own way. The Two Envelopes Problem (TEP) is a much-studied paradox in probability theory, mathematical economics, logic, and philosophy. Time and again a new analysis is published in which an author claims finally to explain what actually goes wrong in this paradox. Each author (the present author included) emphasizes what is new in their approach and concludes that earlier approaches did not get to the root of the matter. We observe that though a logical argument is only correct if every step is correct, an apparently logical argument which goes astray can be thought of as going astray at different places. This leads to a comparison between the literature on TEP and a successful movie franchise: it generates a succession of sequels, and even prequels, each with a different director who approaches the same basic premise in a personal way. We survey resolutions in the literature with a view to synthesis, correct common errors, and give a new theorem on order properties of an exchangeable pair of random variables, at the heart of most TEP variants and interpretations. A theorem on asymptotic independence between the amount in your envelope and the question whether it is smaller or larger shows that the pathological situation of improper priors or infinite expectation values has consequences as we merely approach such a situation.","PeriodicalId":423691,"journal":{"name":"100 Years of Math Milestones","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131409121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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