• Celestial Mechanics – perturbed Kepler problem.
• Probability and Statistics – sampling distributions, Normal approximation, parameter estimation, autoregressive models.
• Monte Carlo techniques – applied to Quantum Chemistry
• Numerical solution of integral equations – applied to Mechanical Engineering problems

Representative Publications

• J. Vrbik,
  “Solving Lunar problem via perturbed K-S equation”
  New Astronomy 11 #5(2006), 366-373
• J. Vrbik,
  “Population moments of sampling distributions”
  Computational Statistics 20 #4(2005), 611-621
• J. Vrbik,
  “Finite-n corrections to Normal approximation”
  Communications in Statistics (Simulation and Computation) 34 #4(2005), 827-837
• J. Vrbik,
  “Moments of AR(1)-model estimators”
  Communications in Statistics (Simulation and Computation) 34 #3(2005), 595-600
• J. Vrbik,
  “Zonal-harmonics perturbations”
  Celestial Mechanics & Dynamical Astronomy91 (2005), 217-237
• Rokne J, Singh B M, Dhaliwal R S and J. Vrbik,
  “The axisymmetric Boussinesq-type problem for half-space under optimal heating of arbitrary profile”
  Int. J. of Math. and Math. Sciences 40 (2004), 2123-2131
• Vrbik J, Singh BM, Rokne J, et al.,
  “On the expansion of a penny-shaped crack by a rigid circular disc inclusion in a thick plate”
  ZAMM 84 #8(2004), 538-550
• Rokne J, Singh BM, Dhaliwal RS and Vrbik J,
  “The Reissner-Sagoci type problem for a nonhomogeneous elastic cylinder embedded in and elastic non-homogeneous half-space”
  IMA J. Appl. Math.69 #2(2004), 159-173
• Rokne J, Singh B M, Dhaliwal RS and Vrbik J,
  “Mixed boundary value problem in a nonhomogeneous medium under steady distribution of temperature”
  Meccanica39 #2(2004), 113-123
• J. Vrbik,
  “A novel solution to Kepler’s problem”
  European J. of Phys.24 #6 (2003), 575-583
• Singh BM, Danyluk HT, Vrbik J, et al.,
  “The Reisner-Sagoci problem for a non-homogeneous half-space with a surface constraint”
  Meccanica38 #4 (2003), 453-465
• Singh B M, Rokne, J, Dhaliwal R S and J. Vrbik,
  “Contact problem for bonded nonhomogeneous materials under shear loading”
  Int. J. of Math. and Math. Sci.2003 #29 (2003), 1821-1832
• Vrbik J, Singh B M, Rokne J, at al.,
  “The breaking of a non-homogeneous fiber embedded in an infinite non-homogeneous medium”
  Z. Angew Math. Phys.54 #2 (2003), 212-223
• Vrbik J, Singh B M, Rokne J, et al.,
  “The problem of a penny-shaped crack in a nonhomogeneous medium under shear”
  Eur. J. Mech. A-Solid21 #5 (2002), 773-777
• Vrbik J,
  “Erratum: oblateness perturbations to forth order”
  Monthly Notices of the Royal Astronomical Society331 #4 (2002), 1072
• Vrbik J, Singh B M, Rokne J and Dhaliwal R S,
  “Plastic deformation at the tip of an edge crack”
  ZAMM81 #9 (2001), 642-647
• Singh B M, Danyluk H T, Selvadurai A P S, Dhaliwal R S, Rokne J and Vrbik J, et al.,
  “Effects of inclusion in ductile cracks”
  ZAMP52 (2001), 680-695
• J. Vrbik, Singh B M, Rokne J, Dhaliwal R S,
  “External crack in torsion in an infinite nonhomogeneous medium with a non-homogeneous cylindrical inclusion”
  ZAMM81 #7 (2001), 489-497
• J. Vrbik,
  “Quaternionic processo”
  Celestial Mechanics and Dynamical Astronomy80 #2 (2001), 111-118
• J. Vrbik,
  “Perturbative solution of an asteroid’s motion in resonance with Jupiter”
  Monthly Notices of the Royal Astronomical Society (2000)
• J. Vrbik,
  “Simple simulation of solar system”
  Astrophysics and Space Science 266 #4 (1999), 557-567.
• J. Vrbik,
  “Iterative solution to perturbed Kepler problem via Kustaanheimo-Stiefel equation”
  Celestial Mechanics and Dynamical Astronomy 71 (1999), 273-287.
• J. Vrbik,
  “Novel analysis of tadpole and horseshoe orbits”
  Celestial Mechanics and Dynamical Astronomy 69 (1998), 283-291.
• P. Langfelder, S.M. Rothstein, and J. Vrbik,
  “Diffusion quantum Monte Carlo calculations of nondifferential properties for atomic ground states”
  J. Chem. Phys. 107 (1997), 8525-35.
• J. Vrbik,
  “Oblateness perturbations to fourth order”
  Mon. Not. R. Astron. Soc. 291 (1997), 65-70.
• J. Vrbik,
  “Resonance formation of Kirkwood gaps and asteroid clusters”
  J. Phys. A 29 #12 (1996), 3311-16.
• H.T. Danyluk, B.M. Singh, and J. Vrbik,
  “A Dugdale-type estimation of the plastic zone for a penny-shaped crack in a thick transversely isotropic layer due to radial shear”
  Engineering Fracture Mechanics 51 (1995), 735-740.
• J. Vrbik,
  “Perturbed Kepler problem in quaternionic form”
  J. Phys. A 28 (1995), 193-198.
• J. Vrbik,
  “Dirac equation and Clifford algebra”
  J. Math. Phys. 35 #5 (May 1994), 2309-14.
• J. Vrbik,
  “Celestial mechanics via quaternions”
  Can. J. Phys. 72 (1994), 141-146.
• P. Belohorec, S.M. Rothstein, and J. Vrbik,
  “Infinitesimal diffusion quantum Monte Carlo study of CuH spectroscopic constants”
  J. Chem. Phys. 98 #8 (1993), 6401-05.
• J. Vrbik,
  “A chain sliding off a table”
  Am. J. Phys. 61 #3 (March 1993), 258-261.
• H.T. Danyluk, B.M. Singh, and J. Vrbik,
  “Ductile penny-shaped crack in a transversely isotropic cylinder”
  Int. J. Fract. 51 #4 (1991), 331-342.
• J. Vrbik,
  “A simple approximation to the pseudoskin factor resulting from restricted entry”
  SPE Formation Evaluation (Dec 1991), 444-446.
• A.L.L. East, S.M. Rothstein, and J. Vrbik,
  “Reply to comment on: ‘Sampling the exact electron distribution by diffusion quantum Monte Carlo”
  J. Chem. Phys. 92 #3 (1990), 2120.
• D.A. Lagare, S.M. Rothstein, and J. Vrbik,
  “Infinitesimal differential diffusion quantum Monte Carlo: Diatomic molecular properties”
  J. Chem. Phys. 92 (Jan 1990), 1221-27.
• M.C. Au, A. Cardou, B.M. Singh, A.P.S. Selvadurai, and J. Vrbik,
  “Closed from solution for two finite-length cracks moving in an orthotropic layer under anti-plane loading”
  Theor. Appl. Fract. Mech. 10 #3 (Dec 1988), 197-212.
• S.M. Rothstein and J. Vrbik,
  “A Green’s function used in diffusion Monte Carlo”
  J. of Compt. Phys. 87 (1987), 1902-03.
• J. Vrbik,
  “Monte Carlo simulation of general elliptic operator”
  J. of Phys. A. 20 (1987), 2693-98.
• J. Vrbik,
  “Calculating the pseudo-skin factor due to the partial well completion”
  J. of Canadian Petroleum Technology 25 #5 (1986), 57-61.
• J. Vrbik,
  “Monte Carlo technique for finding the lowest eigenvalue of a modified Shroedinger equation”
  J. of Phys. A. 18 (1985), 1327-35.
• J. Vrbik,
  “Asymptotic distribution of the coefficient of cross-association based on a simple Markov-chain model”
  J. of Appl. Prob. 22 (1985), 946-950.
• J. Vrbik,
  “Statistical properties of the number of runs of matches between two random stratigraphic sections”
  Math. Geol. 17 #1 (1985), 29-40.
• R.S. Dhaliwal, B.M. Singh, and J. Vrbik,
  “Diffraction of torsional waves by a flat annular crack at the interface of two bonded dissimilar elastic solids”
  Eng. Fract. Mech. 19 #4 (1984), 771-776.
• R.S. Dhaliwal, B.M. Singh, and J. Vrbik,
  “Torsional impact of a layer or a cylinder bonded to an elastic half-space”
  Int. J. of Eng. Sci. 21 #11 (1983), 1397-1408.
• J. Vrbik,
  “Test of independence by repeated matching of stratigraphic sections”
  Math. Geol. J. 15 #3 (1983), 427-444.
• M. Nosal and J. Vrbik,
  “Stratigraphic analysis and the asymptotic distribution of the coefficient of cross-association”
  Math. Geol. 14 #1 (1982), 11-36.
• R.S. Dhaliwal, B.M. Singh, and J. Vrbik,
  “Semi-infinite medium with a cylindrical inclusion twisted by an annular die”
  J. of Elasticity 10 #2 (1980), 121-134.

Past Teaching Assignments

• MATH 1P12 Introductory Linear Algebra
• MATH 1P40 Mathematics Integrated with Computers and Applications I
• MATH 1P97 Differential and Integral Methods
• MATH 1P98 Basic Statistical Methods
• MATH 2P12 Linear Algebra II
• MATH 2P95 Mathematics and Music
• MATH 4/5P96 Technology and Mathematics Education