# Denis L. Blackmore

### Contact Info

Title: | Professor |

Email: | denis.l.blackmore@njit.edu |

Office: | CULM 517 |

Hours: | Mon.: 11:30 - 12:30; Thurs.: 4:30 - 5:30 pm; Fri.: 1 - 2 pm |

Phone: | 973-596-3495 |

Dept: | Mathematical Sciences |

Webpage: | http://math.njit.edu/%7Edeblac/ |

### About Me

Denis Blackmore is a Professor of Mathematical Sciences at the New Jersey Institute of Technology(NJIT). He is a founding member of the Center for Applied Mathematics and Statistics, a member of the Center for Manufacturing 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 research in manufacturing science, fractal surface characterization, vortex breakdown, granular flow dynamics, metrology, biomathematics and his current work in 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 year.

In recognition for his work in differential equations and dynamical systems, Dr. Blackmore was invited in 1988 to give a series of lectures at 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 Prykarpatsky ) 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 organized a minisymposium on ´´Integrable Dynamical Systems and Their Applications´´ for the 1995 International Congress on Industrial and Applied Mathematics in Hamburg, Germany.

He is a member of the National Honor Societies Sigma Xi and Tau Beta Pi. In addition to his research, for which he received the Harlan Perlis Research Award from NJIT in 1993, Dr. Blackmore is devoted to instruction at both the undergraduate and graduate level and has won several awards for 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 currently the director.

### Education

- PhD, Mathematics, Polytechnic University
- MS, Mathematics, Polytechnic University
- BS, Aerospace Engineering, Polytechnic University

### Website

http://math.njit.edu/%7Edeblac/### Classes Taught

### Research Interests

Dynamical systems (nonlinear dynamics) theory is a rich amalgam of techniques from algebra, analysis, chaos theory, differential equations, differential geometry, differential topology, fractals, geometry, singularity theory, and topology, and has important applications in every branch of science and engineering. Denis Blackmore´s research is primarily in the theory and applications of dynamical systems and closely related fields. He has studied a plethora of applications in such areas as acoustics, automated assembly, biological populations, computer aided geometric design, fluid mechanics, granular flows, plant growth (phyllotaxis), relativistic and quantum physics, and rough surface analysis. His theoretical work includes fundamental results on solution properties and integrability of differential equations, and analysis of hypersurface singularities.

### Current Research

Among his current projects are acoustically generated particle flows, biocomplexity of marshes, competing species dynamics, dynamical models in economics, integrability of infinite-dimensional dynamical systems (PDEs), particle dynamics, phyllotaxis, virtual reality systems, vortex dynamics, and weak shock waves.

### Selected Publications

- "Morse index for autonomous linear Hamiltonian systems," (with C. Wang), Int. J. Diff. Eqs. and Appl. 7 (2003), 295-309.
- "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.
- "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.
- "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.
- " Higher order conditions for weak shocks: modified Prandtl relation," (with L. Ting), PAMM 1 (2002), 397-398.
- "A perspective on vibration-induced size segregation of granular materials," J. Chem. Eng. Sci. 57 (2002), 265-275.
- "On the exponentially self-regulating population model," (with J. Chen), Chaos, Solitons and Fractals 14 (2002), 1433-1450.
- "Hamiltonian structure for vortex filament flows," (with O. Knio), ZAMM 81S (2001), 45-48.
- ´´Dynamical properties of discrete Lotka-Volterra equations,´´ (with J. Chen, J. Perez and M. Savescu), Chaos, Solitons and Fractals 12 (2001), 2553-2568.
- ´´New mathematical models for particle flow dynamics,´´ (with R. Samulyak and A. Rosato), J. Nonlinear Math. Physics 6, (1999), 198-221. 4. ´´KAM theory analysis of the dynamics of three coaxial vortex rings´´(with O. Knio), Physica D 140, (2000), 321-348.

### CAD/CAM Related Publications

1. ´´Analysis of Swept Volumes via Lie Groups and Differential Equations´´ (with M.C. Leu), International Journal of Robotics Research, 11, 1992, pp. 516-537.

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, pp. 341-346.

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. 1191-1198.

7. ´´Application of Sweep Differential Equation Approach to Nonholonomic Motion Planning´´ (with Z. Deng and M.C. Leu), Proc. Japan-USA Symposium 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. International Conference on Intelligent Manufacturing, 1995, pp. 211-220. 9. ´´Analysis of Sweep Classes: An Application of Differential Topology to Manufacturing Science´´(with M.C. Leu), SIAM J. Applied Math. (to appear)

10. ´´A General Fractal Distribution Function for Rough Surface Profiles´´(with G. Zhou), SIAM J. Applied Math. 56, 1996, pp. 1694-1719.

11. ´´Hamiltonian Structure of Axial Benney-Type Hydrodynamic and Boltzmann-Vlasov Kinetic Equations with Applications to Manufacturing Science´´(with A. Prykarpatsky 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 Computations 5, 1997, pp. 81-102.

14. ´´A verification program for 5-axis NC machining with general APT tools´´ (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 deformed swept volume´´(with M.C. Leu and F. Lu), Annals of CIRP 47, 1998, pp. 441-446.

16. ´´Trimming swept volumes´´(with R. Samulyak and M.C. Leu), Computer-Aided 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. Abdel-Malek 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.

### Recent Publications

1. ´´The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra´´ (with Y. Prykarpatsky and R. Samulyak), J. Nonlin. Math. Phys. 5 (1998), 54-67.

2. ´´Fractal analysis of height distributions of anisotropic rough surfaces´´(with 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, (1999), 198-221. 4. ´´KAM theory analysis of the dynamics of three coaxial vortex rings´´(with 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 (2001), 2553-2568.

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. Wang), Int. J. Diff. Eqs. and Appl. 7 (2003), 295-309.

14. "A generalized Poincaré-Birkhoff theorem with applications to coaxial 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 (in press)

### Projects

** Manufacturing Automation**

- Analysis and Representation of Swept Volumes: We developed characterizations of swept volumes of general piecewise-smooth objects in terms of trajectories of differential equations that we call the sweep differential equation(SDE) and the sweep-envelope differential equation(SEDE). Both SDE and SEDE based methods have been applied to solve real world problems in NC machining, robot motion planning and virtual design and manufacturing.

- Swept Volume Algorithms and Software: Using SDE and SEDE theory as a framework, we developed fast, efficient and robust algorithms for computing and graphically representing swept volumes. These algorithms are being used to create computer software that can be interfaced with commercial software in order to obtain more useful software for a variety of manufacturing applications.