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
  • Berger, J. and Bridges, D. (2008) The Fan Theorem and Positive-Valued Uniformly Continuous Functions on Compact Intervals. New Zealand Journal of Mathematics 38: 129-135.
  • Bridges, DS. and Vî, LS. (2008) Proximal connectedness. Fundamenta Informaticae 83(1-2): 25-34.
  • Bridges, D. and Diener, H. (2007) 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, D. and Havea, R. (2007) Powers of a Hermitian Element in a Banach Algebra. New Zealand Journal of Mathematics 36: 1-10.
  • 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.
Journal articles
  • Bridges, DS., Havea, R. and Schuster, P. (2006) Finitely generated Banach algebras and local Nullstellensatze. Publicationes Mathematicae Debrecen 69(1-2)
  • Bridges, DS., Ishihara, H. and Vita, L. (2006) A new constructive version of Baire's Theorem. Hokkaido Mathematics Journal 35(1): 107-118.
  • Bridges, DS. and Havea, R. (2005) Approximating the numerical range in a Banach algebra. Oxford Logic Guides 48(From Sets and Types to Topology and Analysis): 293-303.
  • Bridges, DS. and Vita, L. (2005) The constructive uniqueness of the locally convex topology on Rn. Oxford Logic Guides 48(From Sets and Types to Topology and Analysis): 304-315.
  • Bridges, DS., Ishihara, H., Schuster, P. and Vita, LS. (2005) Products in the category of apartness spaces. Cahiers de Topologie et Géometrie Différentielle 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) 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 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. 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) Constructive methods in Banach algebra theory. Math. Japonica 52(1): 145-161.
  • 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, DS. (2000) A constructive look at functions of bounded variation. Bulletin of the London Mathematical Society 32(3): 316-324.
  • Bridges, DS. (2000) Converses of Bishop's Lemma on located sets. Indagationes Mathematicae 11(1): 31-38.
  • 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, DS. (1999) Can constructive mathematics be applied in physics? Journal of Philosophical Logic 28(5): 439-453.
  • Bridges, DS. (1999) Constructive mathematics: A foundation for computable analysis. Theoretical Computer Science 219(1-2): 95-109.
  • Bridges, DS. (1999) Constructive methods in mathematical economics. JOURNAL OF ECONOMICS-ZEITSCHRIFT FUR NATIONALOKONOMIE : 1-21.
  • 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. (2006) Church's thesis and Bishop's constructivism. In Olszewski, A, Wolenski, J, Janusz, R (Ed.), Church's Thesis After 70 Years: 58-61. Frankfurt: Ontos 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 - Other
  • Bridges, DS., Ishihara, H. and Schuster, PM. (2003) Compactness and continuity, constructively revisited. Edinburgh, UK: Computer Science Logic, 22-9-2002. 89-102.
  • Bridges, D. (2001) Constructive comments on Dini's Theorem. Constanta, Romania: Combinatorics, Computability and Logic (Proceedings of DMTCS'01)., 2-7-2001. 69-80.
  • 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,, 17-5-2001. 25-37.
  • 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, 7-2001.
  • 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, 2-7-2001. 203-214.
  • Vita, LS. and Bridges, D. (1999) Constructive notes on uniform and locally convex spaces. Iasi, Romania: Proceedings of the 12th International Symposium, FCT'99, 26-6-1999. 195-203.
  • Bridges, D. (1998) Constructive truth in practice. Mussomeli, Sicily: Truth in Mathematics (Proceedings of Conference, 13-21 September 1995), 1998. 53-69.
Additional Publications
  • Bridges, DS. (2008) Utility Maximization, Choice and Preference. SIAM Review 50(3): 597-599. Book Review.
  • Bridges, Douglas S. (2007) Colocatedness and Lebesgue integrability. Siena, ITALY: 3rd Conference on Computability in Europe (CiE 2007), 18 Jun 2007 - 23 Jun 2007. In Computation and Logic in the Real World, Proceedings 4497: 98-104.
  • Bridges, DS. (2007) Romanian Connections.: 260-261. Bucureşti: Editura Academiei Române. Collection of invited articles.