UC Research Profile - University of Canterbury

Professor Douglas Sutherland Bridges

Contact

Department: Mathematics and Statistics

Email: douglas.bridges@canterbury.ac.nz

Office:

Languages: English, French, German, Italian, Spanish

About

Research / Creative works

Networks

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

Mathematics: Pure Mathematics

Resources

Staff webpage

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Research/Scholarly/Creative Works

(Displaying research/scholarly/creative work from 1998 to 2008)

Journal Articles

Baroni MA. and Bridges DS. (2008) Continuity Properties of Preference Relations. Electronic Notes in Theoretical Computer Science 202: 19-25. http://dx.doi.org/10.1016/j.entcs.2008.03.004.

Baroni MA. and Bridges DS. (2008) Continuity properties of preference relations. Mathematical Logic Quarterly 54(5): 454-459. http://dx.doi.org/10.1002/malq.200710059.

Berger J. and Bridges D. (2008) The anti-Specker property, a Heine-Borel property, and uniform continuity. Archive for Mathematical Logic 46(7-8): 583-592. http://dx.doi.org/10.1007/s00153-007-0063-1.

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 D., Ishihara H., Schuster PM. and Vita L. (2008) Apartness, compactness and nearness. Theoretical Computer Science 405(1-2): 3-10. http://dx.doi.org/10.1016/j.tcs.2008.06.019.

Bridges DS. (2008) Product a-frames and proximity. Mathematical Logic Quarterly 54(1): 12-25. http://dx.doi.org/10.1002/malq.200710033.

Bridges DS. (2008) Uniform Continuity Properties of Preference Relations. Notre Dame Journal of Formal Logic 49(1): 97-106.

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 Havea RS. (2008) Constructive notions of maximality for ideals. Journal of Universal Computer Science 14(22): 3648-3657.

Bridges DS. and Vîta LS. (2008) Proximal connectedness. Fundamenta Informaticae 83(1-2): 25-34.

Berger J. and Bridges D. (2007) A fan-theoretic equivalent of the antithesis of Specker's theorem. Indagationes Mathematicae 18(2): 195-202. http://dx.doi.org/10.1016/S0019-3577(07)00012-2.

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. and Vîţǎ L. (2007) Corrigendum to "Constructing extensions of ultraweakly continuous linear functionals" [J. Funct. Anal. 178 (2000) 421-434] (DOI:10.1006/jfan.2000.3657). Journal of Functional Analysis 244(2): 691-692. http://dx.doi.org/10.1016/j.jfa.2006.12.002.

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 DS. (2007) Constructing local optima on a compact interval. Archive for Mathematical Logic 46(2): 149-154. http://dx.doi.org/10.1007/s00153-006-0032-0.

Berger J. and Bridges DS. (2006) A bizarre property equivalent to the П01-fan theorem. Logic Journal of IGPL 14(6): 867-871. http://dx.doi.org/10.1093/jigpal/jzl026.

Bridges D. and Diener H. (2006) A constructive treatment of Urysohn's lemma in an apartness space. Mathematical Logic Quarterly 52(5): 464-469. http://dx.doi.org/10.1002/malq.200610006.

Bridges D. and Schuster P. (2006) A simple constructive proof of Kronecker's density theorem. Elemente der Mathematik 61(4): 152-154.

Bridges D. and Vita LS. (2006) Pre-apartness structures on spaces of functions. Journal of Complexity 22(6): 881-893. http://dx.doi.org/10.1016/j.jco.2006.04.008.

Bridges D., Havea R. and Schuster P. (2006) Finitely generated Banach algebras and local Nullstellensätze. Publicationes Mathematicae 69(1-2): 171-184.

Authored Books

Bridges DS. and Vîta LS. (2006) Techniques of Constructive Analysis. New York: Springer. 216pp.

Journal Articles

Bridges DS., Berger J. and Schuster P. (2006) The fan theorem and unique existence of maxima. Journal of Symbolic Logic 71(2): 713-720. http://dx.doi.org/10.2178/jsl/1146620167.

Bridges DS., Havea R. and Schuster P. (2006) Finitely generated Banach algebras and local Nullstellensatze. Publicationes Mathematicae Debrecen 69(1-2): 10pp.

Bridges DS., Havea R. and Schuster P. (2006) Ideals in constructive Banach algebra theory. Journal of Complexity 22(6): 729-737. http://dx.doi.org/10.1016/j.jco.2006.03.004.

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. (2005) A weak constructive sequential compactness property and the fan theorem. Logic Journal of the IGPL 13(2): 151-158. http://dx.doi.org/10.1093/jigpal/jzi011.

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. and Vita LS. (2005) A general constructive proof technique. Electronic Notes in Theoretical Computer Science 120: 31-43. http://dx.doi.org/10.1016/j.entcs.2004.06.032.

Bridges DS., Ishihara H., Schuster P. and Vita L. (2005) Strong continuity implies uniform sequential continuity. Archive for Mathematical Logic 44(7): 887-895. http://dx.doi.org/10.1007/s00153-005-0291-1.

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. and Vîţă L. (2004) Corrigendum to “A proof-technique in uniform space theory”. Journal of Symbolic Logic 69(1): 328. http://dx.doi.org/10.2178/jsl/1080938843.

Bridges D. and Vîţǎ L. (2004) Erratum: A proof-technique in uniform space theory (The Journal of Symbolic Logic). Journal of Symbolic Logic 69(1): 328.

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. (2004) First steps in constructive game theory. Mathematical Logic Quarterly 50(4-5): 501-506. http://dx.doi.org/10.1002/malq.200310115.

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 Vîţă L. (2003) A proof-technique in uniform space theory. Journal of Symbolic Logic 68(3): 765-802. http://dx.doi.org/10.2178/jsl/1058448439.

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., Ishihara H. and Spitters B. (2002) Locating the range of an operator with an adjoint. Indagationes Mathematicae 13(4): 433-440. http://dx.doi.org/10.1016/S0019-3577(02)80024-6.

Bridges D., Richman F. and Schuster P. (2002) Trace-class operators. Houston Journal of Mathematics 28: 565-583.

Bridges D., Schuster P. and Vîţǎ L. (2002) Apartness, topology, and uniformity: A constructive view. Mathematical Logic Quarterly 48(SUPPL. 1): 16-28. http://dx.doi.org/10.1002/1521-3870(200210)48:1+<16::AID-MALQ16>3.0.CO;2-7.

Bridges D., Schuster P. and Vita LS. (2002) Apartness, topology, and uniformity: a constructive view. Mathematical Logic Quarterly 48(1): 16-28.

Bridges DS. and Ward ND. (2002) Kernels of seminorms in constructive analysis. Theoretical Computer Science 284(2): 259-267. http://dx.doi.org/10.1016/S0304-3975(01)00067-6.

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 Vîţǎ L. (2001) Characterising near continuity constructively. Mathematical Logic Quarterly 47(4): 535-538. http://dx.doi.org/10.1002/1521-3870(200111)47:4<535::AID-MALQ535>3.0.CO;2-3.

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(3): 503-515. http://dx.doi.org/10.1023/A:1003613131948.

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., Richman F. and Schuster P. (1999) Linear independence without choice. Annals of Pure and Applied Logic 101(1): 95-102. http://dx.doi.org/10.1016/s0168-0072(99)00030-5.

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 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) Romanian Connections. : 260-261. Bucureşti: Editura Academiei Române. [Collection of invited articles].

Bridges DS. (2007) Colocatedness and Lebesgue integrability. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 4497 LNCS: 98-104. http://dx.doi.org/10.1007/978-3-540-73001-9_10.

Bridges DS. (2007) Colocatedness and Lebesgue Integrability. 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.

Affiliations

Royal Society of Edinburgh (RSE) (Professional Organisation): Corresponding Fellow
Royal Society of New Zealand (RSNZ) (Professional Organisation): Fellow (FRSNZ)

Research Groups

Centre for Computable and Constructive Mathematics (CCCM)