Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T19:16:45.132Z Has data issue: false hasContentIssue false

Bibliography

Published online by Cambridge University Press:  24 March 2017

Marian B. Pour-El
Affiliation:
University of Minnesota
J. Ian Richards
Affiliation:
University of Minnesota
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aberth, O. 1971 The failure in computable analysis of a classical existence theorem for differential equations, Proc. Amer. Math. Soc. 30, 151–156 Google Scholar
Aberth, O. 1980 Computable analysis, McGraw-Hill:New York
Ahlfors, L. V. 1953 Complex Analysis, McGraw-Hill: New York
Baez, J.C. 1983 Recursivity in quantum mechanics, Trans. Amer. Math. Soc. 280, 339–350 Google Scholar
Beeson, M.J. 1985 Foundations of constructive mathematics, Springer: Heidelberg, New York
Bishop, E.A. 1967 Foundations of constructive analysis, McGraw-Hill: New York
Bishop, E.A. 1970 Mathematics as a numerical language, Intuitionism and Proof Th.; 1968 Buffalo 53–71; ed. by Kino, A., Myhill, J., Vesley, R.E.. North Holland: Amsterdam
Bjshop, E.A., Bridges, D.S. 1985 Constructive analysis, Springer: Heidelberg, New York
Blum, L., Shub, M., Smale, S. On a theory of computation and complexity over the real numbers: NP completeness, récursive functions and universal machines (to appear in Bull. Amer. Math. Soc.)
Bridges, D.S. 1979 Constructive functional analysis, Pitman Publ.: Belmont, London v + 203pp
Brouwer, L.E.J. 1908 The unreliability of the logical principles (Dutch), Tijdsch. Wijsbegeerte 2, 152–158; transl. in: Coll. Works I, 107–111 (1975) (English)
Brouwer, L.E.J. 1975 Collected works I, North Holland: Amsterdam
Brouwer, L.E.J. 1981 Brouwer's Cambridge lectures on intuitionism, Cambridge Univ. Pr.: Cambridge, GB
Caldwell, J., Pour-El, B.M. 1975 On a simple definition of computable function of a real variable—with applications to functions of a complex variable, Z. Math. Logik Grundlagen Math. 21, 1–19 Google Scholar
Cleave, J.P. 1969 The primitive recursive analysis of ordinary differential equations and the complexity of their solutions, J. Comp. Syst. Sci. 3, 447–455 Google Scholar
Cook, S.A. 1971 The complexity of theorem proving procedures, ACM Symp. Th. of Comput. (3); 1971 Shaker Heights 151–158 Google Scholar
Cutland, N.J. 1980 Computability. An introduction to recursive function theory, Cambridge Univ. Pr.: Cambridge, GB
Dalen van, D. 1973 Lectures on intuitionism, Cambridge Summer School Math. Log.; 1971 Cambridge, GB 1–94; ed. by Rogers, H., Mathias, A.R.D.. Lect. Notes Math. 337, Springer: Heidelberg, New York
Davis, M.D. 1958 Computability and unsolvability, McGraw-Hill: New York
Denef, J., Lipshitz, L. 1984 Power series solutions of algebraic differential equations, Math. Ann. 267, 213–238 Google Scholar
Dieudonné, J.A. 1969 Foundations of modern analysis, Academic Pr.: New York
Dunford, N., Schwartz, J.T. 1958 Linear Operators, Part I, Intersci. Publ.: New York
Feferman, S. 1977 Theories of finite type related to mathematical practice, Handb. of Math. Logic 913–971; ed. by Barwise, J.. Stud. Logic Found. Math. 90, North Holland: Amsterdam
Feferman, S. 1979 Constructive theories of functions and classes, Logic Colloq.; 1978 Mons 159–224; ed. By Boffa, M., Dalen van, D., McAloon, K.. Stud. Logic Found. Math. 97, North Holland: Amsterdam
Feferman, S. 1984 Between constructive and classical mathematics, Computation and Proof Theory, Logic Colloq.; 1983 Aachen 2, 143–162; ed. by Börger, E., Oberschelp, W., Richter, M.M., Schinzel, B., Thomas, W.. Lect. Notes Math. 1104, Springer: Heidelberg, New York
Friedman, H.M., Ker-I, Ko, 1982 Computational complexity of real functions, Theor. Comput. Sci. 20, 323–352 Google Scholar
Friedman, H.M. 1984 The computational complexity of maximization and integration, Adv. Math. 53, 80–98 Google Scholar
Goodstein, R.L. 1961 Recursive analysis, North Holland: Amsterdam
Grzegorczyk, A. 1955 Computable functionals, Fund. Math. 42, 168–202 Google Scholar
Grzegorczyk, A. 1957 On the definitions of computable real continuous functions, Fund. Math. 44, 61–71 Google Scholar
Grzegorczyk, A. 1959 Some approaches to constructive analysis, Constructivity in Math.; 1957 Amsterdam 43–61; ed. by Heyting, A.. Stud. Logic Found. Math., North Holland: Amsterdam
Halmos, P.R. 1950 Measure theory, Van Nostrand: New York
Halmos, P.R. 1951 Introduction to Hilbert space, Chelsea: New York
Hauck, J. 1981 Berechenbarkeit in topologischen Räumen mit rekursiver Basis, Z. Math. Logik Grundlagen Math. 27, 473–480 Google Scholar
Hauck, J. 1984 Zur Wellengleichung mit konstruktiven Randbedingungen, Z. Math. Logik Grundlagen Math. 30, 561–566 Google Scholar
Hellwig, G. 1964 Partial differential equations, Blaisdell: New York
Heyting, A. 1956 Intuitionism. An introduction, North Holland: Amsterdam
Huang, W., Nerode, A. 1985 Application of pure recursion theory in recursive analysis (Chinese), Shuxue Xuebao 28/5, 625–635 Google Scholar
Hyland, J.M.E. 1982 Applications of constructivity, Int. Congr. Log., Meth. and Phil, of Sci. (6, Proa); 1979 Hannover 145–152; ed. by Cohen, L.J., Łos, J., Pfeiffer, H., Podewski, K.-P.. Stud. Logic Found. Math. 104, North Holland: Amsterdam
Kalantari, L, Weitkamp, G. 1985 Effective topological spaces I: A definability theory. II: A hierarchy, Ann. Pure Appl. Logic 29, 1–27,207–224 Google Scholar
Karp, R.M. 1972 Reducibility among combinatorial problems, Compl. of Computer Computation; 1972 York-town Heights 85–103; edited by Miller, R.E., Thatcher, J.W., Bohlinger, J.D.. Plenum Publ.: New York
Klaua, D. 1961 Konstruktive Analysis, Dt. Verlag Wiss.: Berlin
Kleene, S.C. 1952 Introduction to metamathematics, North Holland: Amsterdam
Kleene, S.C. 1956 A note on computable functions, Indag. Math. 18, 275–280 Google Scholar
Ker-I, Ko 1983 On the computational complexity of ordinary differential equations, Inform, and Control 58, 157–194 Google Scholar
Kreisel, G., Lacombe, D. 1957 Ensembles récursivement mesurables et ensembles récursivement ouverts ou fermés, C. R. Acad. Sci., Paris 245,1106–1109
Kreisel, G. 1958 Review of “Meschkowski, H.: ‘Zur rekursiven Funktionentheorie’, Acta Math. 95(1956), 9–23”, MR 19, 238Google Scholar
Kreisel, G. 1974 A notion of mechanistic theory, Synthese 29, 11–16 Google Scholar
Kreisel, G. 1982 Brouwer's Cambridge lectures on intuitionism, Canad. Phil. Rev. 2, 249–251 Google Scholar
Kreitz, C, Weihrauch, K. 1984 A unified approach to constructive and recursive analysis, Computation and Proof Theory, Logic Colloq.; 1983 Aachen 2, 259–278; edited by Boerger, E., Oberschelp, W., Richter, M.M., Schinzel, B. Thomas, W., Lect. Notes Math. 1104, Springer: Heidelberg, New York
Kushner, B.A. 1973 Lectures on constructive mathematical analysis (Russian), Nauka: Moskva 447pp; transl.: Amer. Math. Soc: Providence (1984) (English)Google Scholar
Lachlan, A.H. 1963 Recursive real numbers, J. Symb. Logic 28, 1–16 Google Scholar
Lacombe, D. 1955a Extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles. I, C. R. Acad. Sci., Paris 240, 2478–2480
Lacombe, D. 1955b Extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables reelles II, III, C. R. Acad. Sci., Paris 241, 13–14,151–153
Lacombe, D. 1955c Remarque sur les opérateurs récursifs et sur les fonctions récursives d'une variable réelle, C. R. Acad. Sci., Paris 241, 1250–1252
Lacombe, D. 1957a Quelques propriétés d'analyse récursive, C. R. Acad. Sci., Paris 244, 838–840, 996–997
Lacombe, D. 1957b Les ensembles récursivement ouverts ou fermés, et leurs applications à l'analyse récursive, C. R. Acad. Sci., Paris 245, 1040–1043
Lacombe, D. 1958 Sur les possibilites d'extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles, Raisonn. en Math, et Sci. Expér.; 1955 Paris 67–71, Colloq. Int. CNRS. 70, CNRS Inst. B. Pascal: Paris 140pp
Lacombe, D. 1959 Quelques procédés de definition en topologie récursive, Constructivity in Math.; 1957 Amsterdam 129–158 ed. by Heyting, A.. Stud. Logic Found. Math., North Holland: Amsterdam
Loomis, L.H. 1953 An introduction to abstract harmonic analysis, Van Nostrand: New York
Mazur, S. 1963 Computable analysis, PWN: Warsaw
Metakides, G., Nerode, A. 1982 The introduction of non-recursive methods into mathematics, Brouwer Centenary Symp.; 1981 Noordwijkerhout 319–335; in Troelstra, A.S., Dalen van, D. (eds.) (1982)Google Scholar
Metakides, G., Nerode, A., Shore, R.A. 1985 Recursive limits on the Hahn-Banach theorem, E. Bishop—Reflection on Him & Research; 1983 San Diego 85–91; edited by Rosenblatt, M.. Contemp. Math. 39, Amer. Math. Soc: Providence
Moschovakis, Y.N. 1964 Recursive metric spaces, Fund. Math. 55, 215–238 Google Scholar
Moschovakis, Y.N. 1966 Notation systems and récursive ordered fields, Compos. Math. 17, 40–71 Google Scholar
Mostowski, A. 1957 On computable sequences, Fund. Math. 44, 37–51 Google Scholar
Mycielski, J. 1981 Analysis without actual infinity, J. Symb. Logic 46, 625–633 Google Scholar
Myhill, J.R. 1953 Criteria of constructibility for real numbers, J. Symb. Logic 18, 7–10 Google Scholar
Myhill, J.R. 1971 A recursive function, defined on a compact interval and having a continuous derivative that is not recursive, Michigan Math. J. 18, 97–98 Google Scholar
Orevkov, V.P. 1963 A constructive mapping of the square onto itself displacing every constructive point (Russian), Dokl. Akad. Nauk SSSR 152, 55–58; transl. in: Sov. Math., Dokl. 4, 1253–1256 (1963) (English)Google Scholar
Petrovskij, I.G. 1967 Partial differential equations, Saunders: Philadelphia
Pour-El, M.B. 1974 Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers), Trans. Amer. Math. Soc. 199, 1–28 Google Scholar
Pour-El, M.B., Richards, I. 1979 A computable ordinary differential equation which possesses no computable solution, Ann. Math. Logic 17, 61–90 Google Scholar
Pour-El, M.B., Richards, I. 1981 The wave equation with computable initial data such that its unique solution is not computable, Adv. Math. 39, 215–239 Google Scholar
Pour-El, M.B., Richards, I. 1983a Computability and noncomputability in classical analysis, Trans. Amer. Math. Soc. 275, 539–560 Google Scholar
Pour-El, M.B., Richards, I. 1983b Noncomputability in analysis and physics: a complete determination of the class of non-computable linear operators, Adv. Math. 48, 44–74 Google Scholar
Pour-El, M.B., Richards, I. 1984 Lp-computability in récursive analysis, Proc. Amer. Math. Soc. 92, 93–97 Google Scholar
Pour-El, M.B., Richards, I. 1987 The eigenvalues of an effectively determined self-adjoint operator are computable, but the sequence of eigenvalues is not, Adv. Math. 63, 1–41 Computability on a Banach space—the eigenvector theorem, (to appear)Google Scholar
Remmel, J.B. 1978 Recursively enumerable boolean algebras, Ann. Math. Logic 15, 75–107 Google Scholar
Rice, H.G. 1954 Recursive real numbers, Proc. Amer. Math. Soc. 5, 784–791 Google Scholar
Richardson, D.B. 1968 Some undecidable problems involving elementary functions of a real variable, J. Symb. Logic 33, 514–520 Google Scholar
Riesz, F., Sz.-Nagy, B. 1955 Functional analysis, Ungar: New York and London
Robinson, R.M. 1951 Review of “Peter, R.: ‘Rekursive Funktionen’,Akad. Kiado, Budapest, 1951”, J. Symb. Logic 16, 280 Google Scholar
RogersJr., H. 1967 Theory of recursive functions and effective computability, McGraw-Hill: New York
Rosenbloom, P.C. 1945 An elementary constructive proof of the fundamental theorem of algebra, Amer. Math. Mon. 52, 562–570 Google Scholar
Rudin, W. 1973 Real and complex analysis, McGraw-Hill: New York
Ščedrov, A. 1984 Differential equations in constructive analysis and in the récursive realizability topos, J. Pure Appl. Algebra 33, 69–80 Google Scholar
Schütte, K. 1977 Proof theory, Springer: Heidelberg, New York
Shanin, N.A. 1962 Constructive real numbers and constructive function spaces (Russian), Tr. Mat. Inst. Steklov 67, 15–294; Akad. Nauk SSSR: Moskva; transl.: Amer. Math. Soc: Providence (1968) (English)Google Scholar
Shepherdson, J.C. 1976 On the definition of computable function of a real variable, Z. Math. Logik Grundlagen Math. 22, 391–402 Google Scholar
Shohat, J.A., Tamarkin, J.D. 1943 The problem of moments, Amer. Math. Soc: Providence Google Scholar
Simpson, S.G. 1984 Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations?, J. Symb. Logic 49, 783–802 Google Scholar
Soare, R.I. 1969 Recursion theory and Dedekind cuts, Trans. Amer. Math. Soc. 140, 271–294 Google Scholar
Soare, R.I. 1987 Recursively enumerable sets and degrees, Springer: Heidelberg, New York
Specker, E. 1949 Nicht konstruktiv beweisbare Sätze der Analysis, J. Symb. Logic 14, 145–158 Google Scholar
Specker, E. 1959 Der Satz vom Maximum in der rekursiven Analysis, Constructivity in Math.; 1957 Amsterdam 254–265; edited by Heyting, A. Stud. Logic Found. Math., North Holland: Amsterdam
Specker, E. 1969 The fundamental theorem of algebra in recursive analysis, Constr. Aspects Fund. Thm. Algeb.; 1967 Zürich 321–329; edited by Dejon, B., Henrici, P.. Wiley and Sons: New York
Traub, J.F., Wasilkowski, G.W., Wozniakowski, H. 1988 Information-based complexity, Academic Press Inc., Boston, New York
Troelstra, A.S. 1969 Principles of intuitionism, Springer: Heidelberg, New York
Troelstra, A.S., Dalen van, D. (eds.) 1982 The L.E.J. Brouwer centenary symposium, Stud. Logic Found. Math. 110, North Holland: Amsterdam
Tsejtin, G.S. 1959 Algorithmic operators in constructive complete separable metric spaces (Russian), Dokl. Akad. Nauk SSSR 128, 49–52 Google Scholar
Tsejtin, G.S. 1962 Mean value theorems in constructive analysis (Russian), Tr. Mat. Inst. Steklov 67, 362–384; Akad. Nauk SSSR: Moskva; transl. in: Amer. Math. Soc, Transl., Ser. 2 98, 11–40 (1971) (English)Google Scholar
Tsejtin, G.S., Zaslavskij, I.D. 1962 On singular coverings and properties of constructive functions connected with them (Russian), Tr. Mat. Inst. Steklov 67, 458–502; Akad. Nauk SSSR: Moskva; transl. in: Amer. Math. Soc, Transl., Ser. 2 98, 41–89 (1971) (English)Google Scholar
Turing, A.M. 1936 On computable numbers, with an application to the “Entscheidungsproblem”, Proc London Math. Soc, Ser. 2 42, 230–265; corr. ibid. 43, 544–546 (1937)Google Scholar
Weyl, H. 1918 Das Kontinuum, Veit: Leipzig
Wolfram, S. 1985 Undecidability and intractability in theoretical physics, Phys. Rev. Lett. 54, 735–738 Google Scholar
Zaslavskij, I.D. 1955 Disproof of some theorems of classical analysis in constructive analysis (Russian), Usp. Mat. Nauk 10/4, 209–210 Google Scholar
Zaslavskij, I.D. 1962 Some properties of constructive real numbers and constructive functions (Russian), Tr. Mat. Inst. Steklov 67, 385–457 Akad. Nauk SSSR: Moskva; transl. in: Amer. Math. Soc, Transl., Ser. 2 57, 1–84 (1966) (English)Google Scholar
Zygmund, A. 1959 Trigonometric series. Vol. I, II, Cambridge Univ. Pr.: Cambridge, GB

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Bibliography
  • Marian B. Pour-El, University of Minnesota, J. Ian Richards, University of Minnesota
  • Book: Computability in Analysis and Physics
  • Online publication: 24 March 2017
  • Chapter DOI: https://doi.org/10.1017/9781316717325.013
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Bibliography
  • Marian B. Pour-El, University of Minnesota, J. Ian Richards, University of Minnesota
  • Book: Computability in Analysis and Physics
  • Online publication: 24 March 2017
  • Chapter DOI: https://doi.org/10.1017/9781316717325.013
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Bibliography
  • Marian B. Pour-El, University of Minnesota, J. Ian Richards, University of Minnesota
  • Book: Computability in Analysis and Physics
  • Online publication: 24 March 2017
  • Chapter DOI: https://doi.org/10.1017/9781316717325.013
Available formats
×