**Preprints**

On the geometry of the cross-cap in the Minkoswki 3-space. (With F.S. Dias) Submitted 2012 pdf file

On the geometry of singular surfaces. (With R. Oset Sinha) Submitted, 2012. Pdf file

**Publications
**

Apparent contours in Minkowski 3-space and first order ordinary differential equations. (With S. Izumiya) To appear in

*Nonlinearity*pdf fileUmbilics of surfaces in the Minkowski 3-space. To appear in

*J. Math. Soc. Japan.*Pdf fileProjections of space curves and duality. To appear in

*Q. J. Math.*doi:10.1093/qmath/har035 (With R. Oset Sinha) pdf fileCurves in the Minkowski plane and their contact with pseudo-circles.

*Geometria Dedicata,*159 (2012), 109--124*.*doi: 10.1007/s10711-011-9649-1. (With A. Saloom) pdf fileA note on binary quintic forms and lines of principal curvature on surfaces in R

^{5}.*Topology and its Applications*, 159 (2012), 562-567. doi:10.1016/j.topol.2011.09.032. (With M. A. S. Ruas) pdf fileCaustics of surfaces in the Minkowski 3-space.

*Q. J. Math.*63 (2012), 189-209. doi: 10.1093/qmath.laq030. Pdf fileFamilies of curve congruence on Lorentzian surfaces and pencils of quadratic forms.

*Proc. Roy. Soc. Edinburgh Sect. A*. 141A (2011), 655--672. (With A. C. Nabarro) pdf fileProjections of timelike surfaces in the de Sitter space.

*Real and Complex Singularities.*Edited by M. Manoel, M. C. Romero Fuster, C. T. C. Wall. London Mathematical Society Lecture Notes Series (No. 380), 2010. (With S. Izumiya) pdf filePairs of foliations on surfaces.

*Real and Complex Singularities.*Edited by M. Manoel, M. C. Romero Fuster, C. T. C. Wall. London Mathematical Society Lecture Notes Series (No. 380), 2010. pdf fileFolding maps on spacelike and timelike surfaces and duality.

*Osaka J. Math.*47 (2010), 839--862. (With S. Izumiya and M. Takahashi) pdf fileFamilies of Gauss indicatrices on smooth surfaces in pseudo-spheres in the Minkowski 4-space.

*Journal of Singularities*, 1 (2010), 116-133. Available from http://www.journalofsing.org/volume1/article8.htmlSelf-adjoint operators on surfaces with singular metrics.

*J. Dyn. Control Syst.*16 (2010), 329--353. (With S. Izumiya) pdf fileProjections of surfaces in the hyperbolic space along horocycles.

*Proc. Roy. Soc. Edinburgh Sect. A*140 (2010), 399-418. (With S. Izumiya) pdf fileFamilies of surfaces and conjugate curve congruences.

*Advances in Geometry,*9 (2009), 279-309. (With A. C. Nabarro) pdf fileSelf-adjoint operators on surfaces in R

^{n}.*Differential Geometry and its Applications*, 27 (2009), 296-306*.*Pdf fileProjections of surfaces in the hyperbolic space to hyperhorospheres and hyperplanes.

*Rev. Mat. Iberoamericana*24 (2008), 895-920. (With S. Izumiya) pdf fileVector fields tangent to foliations.

*Proc. Edinburgh Math. Soc*51 (2008), 765--778*.*(With L. F. Martins) pdf fileTwo-parameter families of binary differential equations.

*Discrete Contin. Dynam. Systems,*22 (2008), 759-789*.*pdf fileOn pairs of regular foliations in R

^{3 }and singularities of map-germs.*Geometria Dedicata*135 (2008), 103-118. (With L. F. Martins and R. D. S. Oliveira) pdf fileAsymptotic curves on surfaces in R

^{5}.*Communications in Contemporary Mathematics,*10 (2008), 309--335. (With M. C. Romero-Fuster and M. A. S. Ruas) pdf fileSurfaces in R

^{4}and their projections to 3-spaces.*Proc. Roy. Soc. Edinburgh Sect. A*137 (2007), 1313--1328. (With J. J. Nuno-Ballesteros) pdf fileOn pairs of geometric foliations on a cross-cap.

*Tohoku Math. J.*59 (2007), 233--258.Geometric properties of the solutions of implicit differential equations.

*Discrete Contin. Dynam. Systems*17 (2007), 349--364.Two-parameter families of implicit differential equations.

*Discrete Contin. Dynam. Systems*13 (2005), 139--262.Dupin indicatrices and families of curve congruences.

*Trans. Amer. Math. Soc.*357 (2005), 267--285. (With J. W. Bruce)On families of square matrices.

*Proc. London Math. Soc.*89 (2004), 738--762. (With J. W. Bruce)Zero curves of families of curve congruences.

*Real and Complex Singularities*(Sao Carlos, 2002), Ed. T. Gaffney and M. Ruas, Contemp. Math., 354, Amer. Math. Soc., Providence, RI, June 2004. (With J. W. Bruce and G. J. Fletcher)On pairs of regular foliations in the plane.

*Hokkaido Math. J.*31 (2002), no. 3, 523--537. (With R. D. S. Oliveira)Families of surfaces in R

^{4}.*Proc. Edinb. Math. Soc.*(2) 45 (2002), 181--203. (With J. W. Bruce)On differential 1-forms in the plane.

*Discrete Contin. Dynam. Systems*6 (2000), no. 3, 519--536. (With R. D. S. Oliveira)Duality and implicit differential equations.

*Nonlinearity*13 (2000), 791--812. (With J. W. Bruce)Bifurcations of binary differential equations.

*Proc. Roy. Soc. Edinburgh Sect. A*130 (2000), 485--506. (With J. W. Bruce and G. J. Fletcher)*Real and Complex Singularities*, J.W. Bruce and F. Tari (Editors), Proceedings of the 5th Workshop on Real and Complex Singularities, 27-31 July 1998, Sao Carlos, Brazil. CRC Research Notes in Mathematics 412, Chapman & Hall/CRC, 2000.Families of surfaces: focal sets, ridges and umbilics.

*Math. Proc. Cambridge Philos. Soc.*125 (1999), 243--268. (With J.W. Bruce and P. J. Giblin)Families of surfaces: height functions and projections to plane.

*Math. Scand.*82 (1998), 165--185. (With J.W. Bruce and P. J. Giblin)On the multiplicity of implicit differential equations.

*J. Differential Equations*148 (1998), 122-147. (With J. W. Bruce)Generic 1-parameter families of binary differential equations.

*Discrete Contin. Dynam. Systems*3 (1997), 79--90. (With J. W. Bruce)Implicit differential equations from singularity theory viewpoint. Singularities and differential equations (Warsaw, 1993), 23-38,

*Banach Center Publ.*, 33, Polish Acad. Sci., Warsaw, 1996. (With J. W. Bruce)Extrema of principal curvature and symmetry.

*Proc. Edinburgh Math. Soc.*39 (1996), 397--402. (With J. W. Bruce)On the geometry of simple germs of corank 1 maps from R

^{3}to R^{3}.*Proc. Camb. Phil. Soc.*119 (1996), 469--481. (With W. L. Marar)Ridges, Crests and Sub-parabolic lines of evolving surfaces.

*Int. J. Computer Vision*18 (1996), 195--210. (With J. W. Bruce and P. J. Giblin)Parabolic curves of evolving surfaces.

*Int. J. Computer Vision*17 (1996), 291--306. (With J. W. Bruce and P. J. Giblin)Families of surfaces: height functions, Gauss maps and duals. In W.L. Marar (Ed.),

*Real and Complex Singularities*, Pitman Research Notes in Mathematics 333 (1995), 148--178. (With J.W. Bruce and P. J. Giblin)On binary differential equations.

*Nonlinearity*8 (1995), 255--271. (With J. W. Bruce)Perpendicular bisectors, duals and local symmetry in the plane.

*Proc. Roy. Soc. Edinburgh Sect. A*125 (1995), 181--194. (With P. J. Giblin)Recognition of

*K*-singularities of functions.*Experimental Mathematics*1 (1992), no. 3, 225--229.Local reflexional and rotational symmetry in the plane.

*Springer Lecture Notes in Mathematics*1462 (1991), 154--171. (With P. J. Giblin)Projections of piecewise-smooth surfaces.

*J. London Math. Soc.*44 (1991), 155--172.

*Last updated: 30 January,
2013.*