UC SPARK - University of Canterbury - New Zealand

Professor Douglas Sutherland Bridges

Mathematics and Statistics



Fields of Research

  • Constructive reverse mathematics
  • Computability and complexity
  • Constructive analysis
  • Mathematical economics
  • Mathematical foundations of computer science
  • Mathematical foundations of quantum mechanics
  • Topology

Researcher Summary

Research interests include: constructive foundations of mathematics: in particular, constructive foundations of analysis (real and functional) and topology (the axiomatic theory of apartness), and ring theory; constructive reverse mathematics: classifying constructive proofs by the principles they require, and nonconstructive proofs by the nonconstructive principles they depend on; computability and abstract complexity theory; mathematical economics: the numerical representation of preference relations; demand functions; equilibrium theory; and quantum logic and operator-theoretic foundations of physics.

Subject Area: Disciplines

Research Groups

Affiliations

Research/Scholarly/Creative Works

(Displaying research/scholarly/creative work for 1998 - 2008)
Journal Articles
  • Bridges DS. and Diener H. (2008) The pseudocompactness of [0,1] is equivalent to the uniform continuity theorem. Journal of Symbolic Logic 72(4): 1379-1384. http://dx.doi.org/10.2178/jsl/1203350793.
  • Bridges DS. and Vîta LS. (2008) Proximal connectedness. Fundamenta Informaticae 83(1-2): 25-34.
  • Bridges D., Ishihara H., Mines R., Richman F., Schuster P. and Vita L. (2007) Almost locatedness in uniform spaces. Czechoslovak Mathematical Journal 57(1): 1-12.
  • Bridges D. and Schuster P. (2006) A simple constructive proof of Kronecker's density theorem. Elemente der Mathematik 61(4): 152-154.
Authored Books
  • Bridges DS. and Vîta LS. (2006) Techniques of Constructive Analysis. New York: Springer. 216pp.
Journal Articles
  • Bridges DS., Ishihara H., Schuster P. and Vita LS. (2005) Products in the category of apartness spaces. Cahiers de Topologie et Geometrie Differentielle Categoriques 46: 139-153.
  • Bridges D., Ishihara H. and Vita L. (2004) Computing infima on convex sets, with applications in Hilbert spaces. Proceedings of the American Mathematics Society 132: 2723-2732.
  • Bridges DS. (2004) Constructive complements of unions of two closed sets. Mathematical Logic Quarterly 50(3): 293-294.
  • Bridges DS. (2004) Constructivity in mathematics. Cubo 6(1): 209-258.
  • Bridges DS. (2004) First steps in constructive game theory. Mathematical Logic Quarterly 50(4/5): 501-506.
  • Bridges DS. and Vita LS. (2004) More on Cauchy nets in apartness spaces. Scientiae Mathematicae Japonicae 9: 375-380.
  • Bridges DS., Mines R., Richman F. and Schuster PM. (2004) The polydisk Nullstellensatz. Proceedings of the American Mathematics Society 132(7): 2133-2140.
  • Bridges D. and Vita LS. (2003) Separatedness in constructive topology. Documenta Mathematica 8: 567-576.
  • Bridges D. and Vita LS. (2003) Strong and uniform continuity - the uniform space case. London Mathematical Society Journal of Computing and Mathematics 6: 326-334.
  • Bridges DS. (2003) Apartness spaces - a progress report. Bulletin of the European Association for Theoretical Computer Science 81
  • Bridges DS. and Popa G. (2003) Exact, Continuous Boundary Crossings Out of Convex Sets in R^N. Quarterly Journal of Mathematics 54(4): 391-398.
  • Bridges DS. and Vita LS. (2003) A constructive theory of point-set nearness. Theoretical Computer Science 305(1-3): 473-489.
  • Bridges DS. and Vita LS. (2003) A proof-technique in uniform space theory. Journal of Symbolic Logic 68(3): 795-802.
  • Bridges DS. and Vita LS. (2003) Apartness spaces as a framework for constructive topology. Annals of Pure and Applied Logic 119(1-3): 61-83.
  • Bridges DS. and Vita LS. (2003) Cauchy nets in uniform apartness spaces. Revue Roumaine des Math'ematiques Pures et Appliqu'ees 48(5-6): 467-474.
  • Bridges DS., Ishihara H. and Spitters B. (2003) Locating the range of an operator with an adjoint. Proc. Koningklijke Nederlandse Akad. Wetenschappen (Indagationes Mathematicae) 11(3): 1-8.
  • Bridges DS., van Dalen D. and Ishihara H. (2003) Ishihara's proof technique in constructive analysis. Proc. Koninklijke Nederlandse Akad. Wetenschappen (Indagationes Mathematicae) 14(2): 163-168.
  • Schuster P., Vita LS. and Bridges D. (2003) Strong versus uniform continuity: A constructive round. Quaestiones Mathematicae 26(2): 171-190.
  • Bridges D. and Vita LS. (2002) Cauchy nets in the constructive theory of apartness spaces. Scientiâe Mathimaticae Japonicae 56(1): 123-132.
  • Bridges D., Richman F. and Schuster P. (2002) Trace-class operators. Houston Journal of Mathematics 28: 565-583.
  • Bridges D., Schuster P. and Vita LS. (2002) Apartness, topology, and uniformity: a constructive view. Mathematical Logic Quarterly 48(1): 16-28.
  • Bridges D. (2001) Prime and maximal ideals in constructive ring theory. Comm. Algebra 29(7): 2787-2803.
  • Bridges D. and Havea R. (2001) A constructive version of the spectral mapping theorem. Math. Logic Quarterly 47(3): 299-304.
  • Bridges D. and Ishihara H. (2001) Constructive closed range and open mapping theorems. Proc. Koninklijke Nederlandse Akad. Wetenschappen (Indag. Math) 11(4): 509-516.
  • Bridges D. and Mahalanobis A. (2001) Bounded variation implies regulated: a constructive proof. Journal of Symbolic Logic 66(4): 1695-1700.
  • Bridges D. and Mahalanobis A. (2001) Increasing, nondecreasing, and virtually continuous functions. J Automata, Languages and Combinatorics 6(2): 139-143.
  • Bridges D. and Vita LS. (2001) Characterising near continuity constructively. Math. Logic Quarterly 27(4): 535-538.
  • Bridges D. (2000) A constructive look at functions of bounded variation. Bull. London Math. Soc 32(3): 316-324.
  • Bridges D. (2000) Constructive methods in Banach algebra theory. Math. Japonica 52(1): 145-161.
  • Bridges D. (2000) Converses of Bishop's lemma on located sets. Proc. Koninklijke Nederlandse Akad. Wetenschappen (Indag. Math) 11(1): 31-38.
  • Bridges D. and Havea R. (2000) A constructive analysis of a proof that the numerical range is convex. London Math. Soc. Journal of Mathematics and Computation 3: 191-206.
  • Bridges D. and Mahalanobis A. (2000) Sequential continuity of functions in recursive analysis. Mathematical Logic Quarterly 46(1): 139-143.
  • Bridges D. and Svozil K. (2000) Constructive mathematics and quantum physics. International Journal of Theoretical Physics 39: 501-513.
  • Bridges D. and Vita LS. (2000) Constructing extensions of ultraweakly continuous linear functionals. Journal of Functional Analysis 178(2): 421-434.
  • Bridges D. and Vita LS. (2000) The weak-operator continuity of left multiplication. Proc. Royal Netherlands Academy (Indag. Math) 11(1): 39-42.
  • Bridges D. and Vita LS. (2000) The weak-operator sequential continuity of left multiplication. Proc. Koninklijke Nederlandse Akad. Wetenschappen (Indag. Math) 11(1): 39-42.
  • Bridges D., Ishihara H. and Schuster P. (2000) Sequential compactness in constructive analysis. Österr. Akad. Wiss. Math. -Natur. Kl. Sitzungsberichte II 208: 159-163.
  • Bridges D., Richman F. and Schuster P. (2000) A weak countable choice principle. Proc. Amer. Math. Soc 128(9): 2749-2752.
  • Bridges D., Richman F. and Schuster P. (2000) Adjoints, absolute values and polar decompositions. Journal Operator Theory 44: 243-254.
  • Bridges D. (1999) Can constructive mathematics be applied in physics? Journal of Philosophical Logic 28(5): 439-453.
  • Bridges D. (1999) Constructive mathematics: a foundation for computable analysis. Theoretical Computer Science 219(1-2): 95-109.
  • Bridges D. (1999) Constructive methods in mathematical economics. Zeitschrift für Nationalökonomie Supplementum 8(Mathematical Utitlity Theory): 1-21.
  • Bridges D. and Reeves S. (1999) Constructive mathematics, in theory and programming practice. Philosophia Mathematica 7(1): 65-104.
  • Bridges D. and Richman F. (1999) A constructive proof of Gleason's theorem. Journal of Functional Analysis 162: 287-312.
  • Bridges D. and Vita LS. (1999) Weak-continuity properties in constructive analysis. Logic Journal of the IGPL 7(3): 277-281.
  • Bridges D. and Vita LS. (1999) Weak-operator continuity and the existence of adjoints. Math. Logic Quarterly 45: 203-206.
  • Bridges D., Calude C., Pavlov B. and Stefanescu D. (1999) The constructive implicit function theorem and applications in physics. Chaos, Solitons & Fractals 10: 927-934.
  • Bridges D., Richman F. and Schuster P. (1999) Linear independence and choice. Annals of Pure and Applied Logic 101(1): 95-102.
  • Bridges D. and Dudley Ward N. (1998) Constructing ultraweakly continuous linear functionals on B(H). Proc. Amer. Math. Soc 126(11): 3347-3353.
  • Bridges D. and Ishihara H. (1998) A definitive constructive open mapping theorem? Math. Logic Quarterly 44: 545-552.
  • Bridges D. and Mines R. (1998) Sequentially continuous linear mappings in constructive analysis. Journal of Symbolic Logic 63(2): 579-583.
  • Bridges D. and Yuchuan W. (1998) Constructing best approximations on a Jordan curve. Journal of Approximation Theory 94(2): 222-234.
  • Bridges D. and Yuchuan W. (1998) Constructive weak solutions of the Dirichlet problem. Journal of the London Mathematical Society 57(3): 655-667.
  • Bridges D. and Yuchuan W. (1998) Weak solutions of the Dirichlet Problem and the Locatedness of Hv(0)¹. New Zealand Journal of Mathematics 28(1): 1-5.
Chapters
  • Bridges D. (2008) Philosophy of Mathematics: Five Questions. Chapter 5. In Hendricks VF; Leitgeb H (Ed.), Philosophy of Mathematics: Five Questions: 45-49. New York: Automatic Press/VIP.
  • Bridges DS. (2008) A reverse look at Brouwer's Fan Theorem. In van Atten M; Boldini P; Bourdeau M; Heinzmann G (Ed.), One Hundred Years of Intuitionism (1907-2007): 316-325. Basel: Birkhaeuser.
  • Bridges DS. and Vita LS. (2008) Journey into Apartness Space. In Lindström S; Palmgren E; Segerberg K; Stoltenberg-Hansen V (Ed.), Logicism, Intuitionism, and Formalism: 167-188. Berlin: Springer-Verlag.
  • Bridges DS. (2005) Errett Bishop. In Crosilla L; Schuster P (Ed.), From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics Oxford: Clarendon Press.
Conference Contributions - Other
  • Bridges DS., Ishihara H. and Schuster PM. (2003) Compactness and continuity, constructively revisited. Edinburgh, UK: Computer Science Logic, 22 Sep 2002
  • Bridges D. (2001) Constructive comments on Dini's Theorem. Constanta, Romania: Combinatorics, Computability and Logic (Proceedings of DMTCS'01)., 2 Jul 2001
  • Bridges D. (2001) Reality and virtual reality in Mathematics. Bucharest, Romania: Millennium III, Black Sea University and Romanian Academy.
  • Bridges D. and Ishihara H. (2001) Constructive unbounded operators. San Servolo/Venice, Italy: Reuniting the Antipodes - Constructive and Nonstandard Views of the Continuum, Proceedings of the Symposium, May 17-22, 1999,, 17 May 2001
  • Bridges S. and Vita LS. (2001) Constructive apartness spaces. Anogia, Crete: Proceedings of the 3rd Panhellenic Logic Conference, University of Crete and National Technical University of Athens, 1 Jul 2001
  • Schuster P., Vita LS. and Bridges D. (2001) Apartness as a relation between subsets. Constanta, Romania,: Proceedings of The Third International Conference on Combinatorics, Computability and Logic, DMTCS Monographs, 2 Jul 2001
  • Vita LS. and Bridges D. (1999) Constructive notes on uniform and locally convex spaces. Iasi, Romania: Proceedings of the 12th International Symposium, FCT'99, 26 Jun 1999
  • Bridges D. (1998) Constructive truth in practice. Mussomeli, Sicily: Truth in Mathematics (Proceedings of Conference, 13-21 September 1995), 1 Jan 1998
Additional Publications
  • Bridges DS. (2008) Utility Maximization, Choice and Preference. SIAM Review 50(3): 597-599. [Book Review].
  • Bridges DS. (2007) Colocatedness and Lebesgue Integrability. Siena, Italy: Computability in Europe (CiE) 2007, 18-23 Jun 2007. In Lecture Notes in Computer Science (LNCS) 4497(Computation and Logic in the Real World): 98-104. http://dx.doi.org/10.1007/978-3-540-73001-9.
  • Bridges DS. (2007) Romanian Connections. : 260-261. Bucureşti: Editura Academiei Române. [Collection of invited articles].