Department of Mathematical Sciences
Dr. Denis Blackmore
New Jersey Institute of Technology
Newark, New Jersey
Denis Blackmore is a Professor of Mathematical Sciences at the New
Jersey Institute of Technology(NJIT). He is a founding member of the
for Applied Mathematics and Statistics, a member of the Center for
Systems and a member of the Particle Technology Center, all at NJIT.
While conducting his own research in dynamical systems and differential
topology, he has also devoted considerable time to collaborative
research in various engineering and science disciplines, some of which
has been supported by grants
from the National Science Foundation and Office of Naval Research. His
in manufacturing science, fractal surface characterization, vortex
granular flow dynamics, metrology, biomathematics and his current work
swept volumes reflects his interests in applications of mathematics.
Professor Blackmore received his Ph.D. in Mathematics in 1971 from
the Polytechnic University of New York. He also received an M.S. in
Mathematics in 1966 and a B.S. in Aerospace Engineering from the same
institution. His research as a senior undergraduate and graduate
student was in the areas
of boundary layer theory in fluid mechanics and the qualitative theory
of ordinary differential equations. He was a Visiting Professor of
Mathematics at the Courant Institute of Mathematical Sciences during
the 1989-90 academic
In recognition for his work in differential equations and dynamical
systems, Dr. Blackmore was invited in 1988 to give a series of lectures
the Institute of Mathematics, Academia Sinica, Beijing, China and
several universities in X'ian and Guangzhou. Recently, he was invited
back to China to lecture on dynamical systems at the Nankei Institute,
and has been invited
to work with dynamical systems experts( including Mel'nikov and
) in Russia and Ukraine.
Dr. Blackmore organized the 834th Meeting of the American
Mathematical Society held at NJIT in April, 1987. He has also organized
numerous seminars and colloquia in the mathematical sciences, and
served on the organizing committee
of the 1992 Japan-USA Symposium on Flexible Automation. Recently, he
a minisymposium on ''Integrable Dynamical Systems and Their
for the 1995 International Congress on Industrial and Applied
in Hamburg, Germany.
He is a member of the National Honor Societies Sigma Xi and Tau
Pi. In addition to his research, for which he received the Harlan
Research Award from NJIT in 1993, Dr. Blackmore is devoted to
at both the undergraduate and graduate level and has won several awards
his teaching. He has developed perhaps a dozen graduate and
undergraduate courses and was a contributor to the development of the
Ph.D. program in mathematical
sciences( a joint NJIT - Rutgers/Newark program ) of which he is
CAD/CAM Related Publications
1. ''Analysis of Swept Volumes via Lie Groups and Differential
(with M.C. Leu), International Journal of Robotics Research, 11, 1992,
2. ''Applications of Flows and Envelopes to NC Machining'' (with M.C.
Leu and K.K. Wang), Annals of CIRP, 41, 1992, pp. 493-496.
3. ''Fractal Geometry Model for Wear Prediction'' (with G. Zhou and
M.C. Leu), Wear, 170/1, 1993, pp. 91-101.
4. ''The Flow Approach to CAD/CAM Modeling of Swept Volume''(with H.
Jiang and M.C. Leu), Advances in Manufacturing Systems, Elsevier, 1994,
5. ''Analysis and Modelling of Deformed Swept Volume''(with M.C. Leu
and F. Shih), Computer Aided Design, 26, 1994, pp. 315-326.
6. ''Improved Flow Approach for Swept Volumes'' (with M.C. Leu and D.
Qin), Proc. Japan-USA Symposium on Flexible Automation, 1994, pp.
7. ''Application of Sweep Differential Equation Approach to
Nonholonomic Motion Planning'' (with Z. Deng and M.C. Leu), Proc.
on Flexible Automation, 1994, pp. 1025-1034.
8. ''Implementation of the SDE Method to Represent Cutter Swept Volumes
in 5-Axis NC Milling''(with M.C. Leu, L. Wang and K. Pak), Proc.
Conference on Intelligent Manufacturing, 1995, pp. 211-220. 9.
of Sweep Classes: An Application of Differential Topology to
Science''(with M.C. Leu), SIAM J. Applied Math. (to appear)
10. ''A General Fractal Distribution Function for Rough Surface
G. Zhou), SIAM J. Applied Math. 56, 1996, pp. 1694-1719.
11. ''Hamiltonian Structure of Axial Benney-Type Hydrodynamic and
Kinetic Equations with Applications to Manufacturing Science''(with A.
and N. Bogoliubov), Nuovo Cimento. (to appear).
12. ''The Sweep-Envelope Differential Equation Algorithm and Its
Application to NC Machining Verification'' (with M.C. Leu and L. Wang),
Computer Aided Design 29, 1997, pp. 629-637.
13. ''Swept volumes: a retrospective and prospective view''(with M.C.
Leu, L.P. Wang, and H. Jiang), Neural, Parallel & Scientific
5, 1997, pp. 81-102.
14. ''A verification program for 5-axis NC machining with general APT
(with M.C. Leu and L.P. Wang), Annals of CIRP 46, 1997, pp. 419-424.
15. ''Simulation of NC machining with cutter deflection by modeling
swept volume''(with M.C. Leu and F. Lu), Annals of CIRP 47, 1998, pp.
16. ''Trimming swept volumes''(with R. Samulyak and M.C. Leu),
Design 31, 1999, pp. 215-223.
17. ''Swept volume computation for machining simulation in virtual
reality applications''(with B. Maiteh, M.C. Leu and L. Abdel-Malek), J.
Materials Processing & Manufacturing Science 7, 1999, pp. 380-390.
18. ''A singularity theory approach to swept volumes''(with R. Samulyak
and M.C. Leu), Int. J. Shape Modeling 6, 2000, pp. 105-129.
19. ''On swept volume formulations: implicit surfaces''(with K.
and J. Yang), Compuer-Aided Design 33, 2001, pp.113-121.
20. ''Creation of freeform solid models in virtual reality''(with M.C.
Leu and B. Maiteh), Annals of CIRP 50, 2001, pp. 73-76.
Some Recent Publications
1. ''The integrability of Lie-invariant geometric objects generated by
in the Grassmann algebra'' (with Y. Prykarpatsky and R. Samulyak), J.
Math. Phys. 5 (1998),
2. ''Fractal analysis of height distributions of anisotropic rough
G. Zhou), Fractals 6 (1998), 43-58.
3. ''New mathematical models for particle flow dynamics''(with R.
Samulyak and A. Rosato), J.
Nonlinear Math. Physics 6,
4. ''KAM theory analysis of the dynamics of three coaxial vortex
O. Knio), Physica D 140, (2000), 321-348.
5. ''Dynamical properties of discrete Lotka-Volterra equations''(with
J. Chen, J. Perez and M. Savescu), Chaos,
Solitons and Fractals 12
6. "Hamiltonian structure for vortex filament flows" (with O.
Knio), ZAMM 81S (2001), 45-48.
7. "On the exponentially self-regulating population model" (with J.
Chen), Chaos, Solitons and Fractals
14 (2002), 1433-1450.
8. "A perspective on vibration-induced size segregation of granular
materials, J. Chem. Eng. Sci.
57 (2002), 265-275.
9." Higher order conditions for weak shocks: modified Prandtl relation"
(with L. Ting), PAMM 1 (2002), 397-398.
10. "Fractionation and segregation of suspended particles using
acoustic and flow fields" (with N. Aboobaker and J. Meegoda), ASCE J. Environ. Eng. 129 (2003), 427-434.
11. "Vorticity jumps across shock surfaces" (with L. Ting), Proc. 2nd MIT Conf. on Computational
Fluids and Solid Mechanics, Vol. 1, K. J. Bathe, ed., Elsevier,
Amsterdam, 2003, pp. 847-849.
12. "The Lax solution to a Hamilton-Jacobi equation and its
generalizations: Part 2" (with Ya V. Mykytiuk and A. Prykarpatsky), Nonlin. Anal. 55 (2003), 629-640.
13. "Morse index for autonomous linear Hamiltonian systems" (with C.
Int. J. Diff. Eqs. and Appl. 7 (2003), 295-309.
14. "A generalized Poincaré-Birkhoff theorem with applications
vortex ring motion" (with J. Champanerkar and C. Wang), Disc. and Contin. Dyn. Systems-B (in press).
15. "A geometrical approach to quantum holonomic computing algorithms"
(with A. Samoilenko, Y. Prykarpatsky, U. Taneri and A. Prykarpatsky), Math. and Computers in Simulation
Projects in Manufacturing Automation
(1) Analysis and Representation of Swept Volumes: We developed
of swept volumes of general piecewise-smooth objects in terms of
of differential equations that we call the sweep differential
and the sweep-envelope differential equation(SEDE). Both SDE and SEDE
methods have been applied to solve real world problems in NC machining,
motion planning and virtual design and manufacturing.
(2) Swept Volume Algorithms and Software: Using SDE and SEDE theory as
a framework, we developed fast, efficient and robust algorithms for
and graphically representing swept volumes. These algorithms are being
to create computer software that can be interfaced with commercial
in order to obtain more useful software for a variety of manufacturing
Sample Exams Math222
Math 481/546 -001 (s04)
Projects in Computational