Research

Papers:
1. A Note on harmonic maps of semi-Riemannian manifold, Proceedings of the Workshop on Global Analysis, Differential Geometry and Lie Algebras (Thessaloniki, 1995), 33-37, BSG Proc.,1. (with Ianus, S.)--Abstract
2. Geodesics on a certain step 2 sub-Riemannian manifold, Annals of Global Analysis and Geometry, Vol.22, No.4, pp. 317-339, (2002).-Abstract
3. Variational calculus on sub-Riemannian manifolds, Balcan Journal of Geometry and Applications, vol. 8, 2003 (with Mangione, V.)--Pdf.file
4. Geodesics with constraints on Heisenberg manifolds, Results in Mathematics, 44, 2003, p. 44-53 (with Mangione, V.).--Pdf file.
5. Some PDE's from the geometric point of view, Bulletin Mathematique de la Societé des Sciences Mathematiques de Roumanie, 47 (95), no. 3-4, (2004), pp. 159-165. (with Mangione, V.).--Abstract
6. Some results on Heisenberg manifolds, published in Quaderni del Dipartimento di Matematica, Univ. Parma, 282, 2002. (with Mangione, V.).
7. Invariant formulas for the Euler-Lagrange equation associated to sub-Riemannian geodesics, Quaderni del Dipartimento di Matematica, Univ. Parma, 290, 2002. (with Mangione, V.).
8. On sub-Riemannian geodesics, Analysis and Its Applications, vol 1,issue 3, 2003, pp. 289-350 (with Greiner, P). -Abstract
9. The Missing Direction and Differential geometry on Heisenberg Group, Ph.D. thesis, University of Toronto, 2000.--Pdf.file
10. Real and complex Hamiltonian mechanics on some SubRiemannian manifolds, Asian Journal of Mathematics, vol 18, no.1, pp. 137-160 (with Chang, D.C., Greiner, P.C.--Abstract, Pdf.file
11. On a Step 2(k+1) Sub-Riemannian manifold, Journal of Geometric Analysis, vol 14, no.1, 2004, pp.1-18. (with Chang, D.C., Greiner, P.C.) --Pdf.file
12. Geometric Mechanics on a step 4 subRiemannian manifold, Taiwanese Journal of Mathematics, June 2005, vol. 9, No. 2, pp. 261-280. (with Chang, D.C.)--Pdf.file
13. Geometric Analysis on SubRiemannian Manifolds, Advances in Analysis: Proceedings of the 4th ISAAC Congress, World Scientific, (ed. by M.W. Wong), 17-38, 2005 (with Chang, D.C., Greiner, P.C.).-book abstract
14. The Geometry on a Step 3 Grusin Model, Arbeitsgruppe Partielle Differentialgleichungen und komplexe Analysis, Universitat Postdam, Institut fur Mathematik, April 2004 (with Chang, D.C.)
15. Geometric mechanics on the Heisenberg group, Bulletin of the Institute of Mathematics, Academia Sinica, vol.33, no.3, Sep., 2005 (with Chang, D.C.)--Pdf.file
16. The geometry on a step 3 Grushin model, Applicable Analysis,Vol. 84, Issue 2, February 2005, pages 111-129(with Chang, D.C.).--Pdf.file
17. The Geometry of a step two Grushin operator, Proceedings of International Conference on Complex Analysis and Dynamical Systems, (ed. by L. Karp and L. Zalcman), Contemporary Math., Amer. Math. Soc., vol. 382, pp. 89-111, 2005(with Chang, D-C, and Greiner, P., Kannai, Y.).--Abstract
18. Hermite operator on the Heisenberg group, Harmonic Analysis, Signal Proceedings and Complexity, (ed. by D. Walnut and D. Struppa), Birkhauser, Boston, Massachusetts, pp. 37-54, 2005 (with D.C. Chang and J. Tie). --Pdf.file
19. Solving Asset Pricing Models when the Price-Dividend Function is Anaytic, Econometrica, vol. 73, pp. 961-982, 2005 (with Yu Chen, T. F. Cosimano and A. Himonas)--Abstract,Pdf.file and supplement
20. Geometric Analysis on a Family of Pseudoconvex Hypersurfaces, Complex Variables, Theory and Applications, vol. 50, no 7-11, pp. 803-835, 2005 (with D.C. Chang).--Pdf.file
21. Fundamental solutions for Hermite and subelliptic operators, J. d'Analyse Math., Vol. 100, 2006 (with D.C. Chang and J. Tie). --Abstract, Pdf.file
22. Fundamental solutions for a family of sub-elliptic PDEs, Pure and Applied Mathematics Quarterly, vol. 3, no. 2, pp. 1-23, 2007 (with D.C. Chang and S. Fricke).--Abstract
23. Periodic solutions for a family of Euler-Lagrange systems, Asian J. Math., vol. 11, no. 1, pp. 69-88, March 2007 (with D.C. Chang and Yau, S. T). --Pdf.file
24. SubRiemannian geometry on S^3, Canadian J. of Math., Vol. 61, no.4, pp. 721-739, 2009 (with D.C. Chang and Markina, I). --Pdf.file
25. Isometric Embeddings of SubRiemannian Manifolds in R^n, Advanced Studies in Contemporary Mathematics, Vol. 15, No.1, pp. 21-36, July 2007 (with D.C. Chang). --Pdf.file
26. Geometric Analysis on H-type groups related to division algebras, Math. Nachr. 282, nr. 1, 2009 (with D.C. Chang, Markina, I.).--Pdf.file
27. SubRiemannian geometry, a variational approach, Journal of Differential Geometry, vol. 80, #1, pp. 23-43 (with D.C. Chang).--Abstract
28. Generalized Hamilton-Jacobi Equation and Heat Kernel on Step Two Nilpotent Lie Groups, Analysis and Mathematical Physics. New Trends in Mathematics, Birkhauser, 2009, pp. 49-76 (with D.C. Chang, I. Markina)-Pdf.file
29. Heat kernels for Kolmogorov-type operators and its applications, Journal of Fourier Analysis and Applications, Vol. 15, Issue: 6, Publisher: Birhauser, Boston,INC, pp. 816-838, 2009 (with D.C. Chang).--Abstract, Pdf.file
30. Heat kernels for operators with gravitational potential operators in one and two variables, Pure and Applied Mathematics Quarterly, vol. 6., no.3, 2010, pp.677-692 (with D.C. Chang).--Pdf.file
31. Non-Connectivity Example in SubRiemannian Geometry, Complex Variables and Elliptic Equations,Vol. 56 Issue 1-4, p.35-47, 2011 (with D.C. Chang).-- Pdf.file
32. Generalizations of conjugate connections, Trends in Diff. geometry, complex analysis, and mathematical physics, 26-34, World Sci. Publ., Hackensack, NJ, 2009 (with Matsuzoe, H., Zhang, J.).--Pdf. File
33. Stochastic regression in terms of Brownian motion, (with D.C. Chang and J. Hu), Applicable Analysis, Volume 90, Issue 6, pp. 899-919, June 2011--Pdf file
34. Stochastic Trigonometry and Stochastic Invariants (with D.C. Chang), Journal of Nonlinear and Convex Analysis, Vol. 11, No.1, 2010 --Pdf file
35. Non-holonomic system and sub-Riemannian geometry, (with D.C. Chen and S.T. Yau), Comm. Informations and Systems, vol. 10, No.4, pp. 293-316, 2010.Abstract
36. On a Class of Nilpotent Distributions (with DC Chang), Taiwanese Journal of Math., Vol. 15, No. 2, pp. 875-881, April 2011-- Pdf.file
37. Heat Kernels for differential operators with radical function coefficients, (with D.C. Chang),Taiwanesse Journal of Math., Vol. 15, No.4, pp. 1629-1636, August 2011--Pdf.file
38. Asian Options with Stochastic Strike Price, (with D.C. Chen and B. Alshamary), Applicable Analysis,Vol. 91, Issue 1, 2012, pp.91-104.--Pdf. file
39. Geometric Optimization in Prey-Predator Problems (with B. Alshamary), Journal of Mathematical Biology, Vol: 63, issue 5, pp.881-900, 2011 -Pdf file1, Pdf.file2
40. Heat kernels for a class of degenerate operators using the stochastic method (with D.C. Chang, Y. L and J. Hui) Complex Variables and Elliptic Equations, 57, no.2-4, pp.155-168, 2012--Pdf. file, Pdf.file
41.On Heat Kernels of a Class of Degenerate Elliptic Operators (with D.C. Chang, Y. L and J. Hui), Journal of Nonlinear and Convex Analysis, 12, no.2, pp.309-340, 2011--Pdf file
42.On the Heat Kernel of a Left Invariant Elliptic Operator, (with D.C. Chang, Y. Li), Excursions in Harmonic Analysis, vol.2, pp.197-209, Appl. Numer. Harmon. Anal., Birkhauser/Springer, New York, 2013--Pdf.file
43. Stochastic optimization approach of car value depreciation, (with B. Alshamary), Applied Stochastic Models in Business and Industry, Vol. 29, Issue 3, May/June 208–223, 2013.- Pdf.file
44.Global connectivity and optimization on an infinite step distribution (with B. AlShamary and D.C. Chen), Adv. Appl. Math. 50, no. 4, pp.634-644, 2013- PDF.file.
45. Integrability conditions for the Grushin and Martinet distribution (with D.C. Chang and M. Eastwood), Bulletin of the Institute of Mathematics, Academia Sinica (New series), Vol. 8, No.2, June 2013
.-PDF.file
46. Pricing a stochastic car value depreciation deal, (with B.Alshamary), Applied Stochastic Models in Business and Industry, 30, no.4, pp. 509-516, 2014 -Pdf.file
47. Transience of diffusions on Heisenberg group, Journal of Geometry and Physics, vol. 77, March 2014, pp. 131-142.-Pdf. file
48. Integrability conditions for Heisenberg and Grushin-type distributions (with D.C. Chang and M. Eastwood), Analysis and Mathematical Physics, pp. 99-114, vol. 4, issue 1-2, 2014.- Pdf. file
49. A stochastic variant of Chow-Rashewski Theorem on the Grushin Distribution, (with C. Udriste and I. Tevy) Balkan Journal of Geometry and Applications, 19, no.1, pp.1-12, 2014. -Pdf
50. Stochastic sub-Riemannian geodesics on the Grushin distribution, Balkan Journal of Geometry and Applications (with C. Udriste and I. Tevy), 19, no.2, pp.37-49, 2014. -Pdf.file
51. Isometric elastic deformations (with F. AlAzemi), pp. 1-8, BGS Proceedings, 2014
.-Pdf. file
52. Poincare’s lemma on the Heisenberg group (with J. Hu and D.C. Chen), Adv. in Appl. Math. 60, pp. 90-102, 2014.-Pdf.file
53. Integrability conditions on Engel-type manifolds, Anal. Math. Physics, vol.5, issue 3, pp. 217-231, 2015.-Pdf.file
54. Asian Options with Harmonic Average (with F. AlAzemi), Applied Mathematics and Information Sciences, vol.9, no. 6, 2015-Pdf.file
55. An Optimality Model for Rust Formation and Evolution (with B. AlShamary), accepted to
Applied Stochastic Models in Business and Industry, 2015 - Pdf.file
56. Integrability Conditions on a Sub-Riemannian structure on S^3, submitted.
57. Integrability Conditions on a Sub-Riemannian structure on H^3, in preparation.
58. Bond Valuation for term structure of interest with jumps, in preparation.

Books/Monographs:
1. Geometric Mechanics on Riemannian Manifolds, with Applications to Partial Differential Equations, published by Birkhauser (Boston Basel Berlin), Fall 2004 (with Chang, D-C)
2. Euclidean and Non-Euclidean Geometry: A metric approach, Wiley (Custom Services), 2004.
3. An Introduction to Vectorial Geometry, McGraw Hills Custom Publishing, 2005.
4. Wp-Calculus, An Introduction to Formal Languages, Wiley Custom Services, 1st ed. 2006, 2nd ed. 2013 (with Zeiger, M.).
5. Geometric Analysis on Heisenberg Manifolds and Generalizations (with Chang, D-C, and Greiner, P.) Studies in Adv. Math. No. 40, AMS/IP, 2007.
6. SubRiemannian Geometry -- Theory and examples (with Chang, D-C), Encyclopedia of Math and Its Appl., 126 Cambridge University Press, 2009.
7. Heat Kernels-- Methods and Techniques, Applied and Numerical Harmonic Analysis, Birkhauser (Boston Basel Berlin), Fall 2010 (with Chang, D.C., Furutani, K. and Iwasaki, C).

8. Geometric Modeling in Probability and Statistics (with C. Udriste), Springer, 2014.
9. An Informal Introduction to Stochastic Calculus with Applications, World Scientific, 2015.