{"title":"List of Figures","authors":"","doi":"10.2307/j.ctvqsdxqd.3","DOIUrl":"https://doi.org/10.2307/j.ctvqsdxqd.3","url":null,"abstract":"","PeriodicalId":119327,"journal":{"name":"Statistical Inference via Convex Optimization","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128342242","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}
Day 2: Basic of Algorithms Analysis "For me, great algorithms are the poetry of computation. Just like verse, they can be terse, allusive, dense, and even mysterious. But once unlocked, they cast a brilliant new light on some aspect of computing."-Francis Sullivan
{"title":"On Computational Tractability","authors":"Neil Rhodes, F. Sullivan, C. Babbage, Jon Bentley","doi":"10.2307/j.ctvqsdxqd.8","DOIUrl":"https://doi.org/10.2307/j.ctvqsdxqd.8","url":null,"abstract":"Day 2: Basic of Algorithms Analysis \"For me, great algorithms are the poetry of computation. Just like verse, they can be terse, allusive, dense, and even mysterious. But once unlocked, they cast a brilliant new light on some aspect of computing.\"-Francis Sullivan","PeriodicalId":119327,"journal":{"name":"Statistical Inference via Convex Optimization","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122952279","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}
This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.
{"title":"Sparse Recovery via ℓ1 Minimization","authors":"J. Stillwell","doi":"10.2307/j.ctvqsdxqd.9","DOIUrl":"https://doi.org/10.2307/j.ctvqsdxqd.9","url":null,"abstract":"This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.","PeriodicalId":119327,"journal":{"name":"Statistical Inference via Convex Optimization","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126647023","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}
This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.
{"title":"From Hypothesis Testing to Estimating Functionals","authors":"J. Stillwell","doi":"10.2307/j.ctvqsdxqd.11","DOIUrl":"https://doi.org/10.2307/j.ctvqsdxqd.11","url":null,"abstract":"This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.","PeriodicalId":119327,"journal":{"name":"Statistical Inference via Convex Optimization","volume":"1037 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116270493","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}
This chapter develops the basic results of computability theory, many of which are about noncomputable sequences and sets, with the goal of revealing the limits of computable analysis. Two of the key examples are a bounded computable sequence of rational numbers whose limit is not computable, and a computable tree with no computable infinite path. Computability is an unusual mathematical concept, because it is most easily used in an informal way. One often talks about it in terms of human activities, such as making lists, rather than by applying a precise definition. Nevertheless, there is a precise definition of computability, so this informal description of computations can be formalized.
{"title":"Signal Recovery by Linear Estimation","authors":"J. Stillwell","doi":"10.2307/j.ctvqsdxqd.12","DOIUrl":"https://doi.org/10.2307/j.ctvqsdxqd.12","url":null,"abstract":"This chapter develops the basic results of computability theory, many of which are about noncomputable sequences and sets, with the goal of revealing the limits of computable analysis. Two of the key examples are a bounded computable sequence of rational numbers whose limit is not computable, and a computable tree with no computable infinite path. Computability is an unusual mathematical concept, because it is most easily used in an informal way. One often talks about it in terms of human activities, such as making lists, rather than by applying a precise definition. Nevertheless, there is a precise definition of computability, so this informal description of computations can be formalized.","PeriodicalId":119327,"journal":{"name":"Statistical Inference via Convex Optimization","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121353833","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 : 2019-09-24DOI: 10.23943/princeton/9780691197296.003.0006
J. Stillwell
This chapter develops the basic results of computability theory, many of which are about noncomputable sequences and sets, with the goal of revealing the limits of computable analysis. Two of the key examples are a bounded computable sequence of rational numbers whose limit is not computable, and a computable tree with no computable infinite path. Computability is an unusual mathematical concept, because it is most easily used in an informal way. One often talks about it in terms of human activities, such as making lists, rather than by applying a precise definition. Nevertheless, there is a precise definition of computability, so this informal description of computations can be formalized.
{"title":"Solutions to Selected Exercises","authors":"J. Stillwell","doi":"10.23943/princeton/9780691197296.003.0006","DOIUrl":"https://doi.org/10.23943/princeton/9780691197296.003.0006","url":null,"abstract":"This chapter develops the basic results of computability theory, many of which are about noncomputable sequences and sets, with the goal of revealing the limits of computable analysis. Two of the key examples are a bounded computable sequence of rational numbers whose limit is not computable, and a computable tree with no computable infinite path. Computability is an unusual mathematical concept, because it is most easily used in an informal way. One often talks about it in terms of human activities, such as making lists, rather than by applying a precise definition. Nevertheless, there is a precise definition of computability, so this informal description of computations can be formalized.","PeriodicalId":119327,"journal":{"name":"Statistical Inference via Convex Optimization","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121115182","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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