Prof. Buks van Rensburg

Mathematics and Statistics

Professor E.J. Janse van Rensburg

Applied Mathematics Section
Faculty of Science and Engineering

OFFICIAL WEB SITE


Contact: Prof. E.J. Janse van Rensburg Petrie 215 Department of Mathematics and Statistics York University Toronto, Ontario M3J 1P3 Canada

Phone: (416)-736-2100 X33837

Email: rensburg@yorku.ca

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· Statistical Mechanics

· Combinatorics

· Monte Carlo Simulations

· Statistical Knot Theory

Selected Papers (With links to journals)

2012:

· Directed Path Models of Adsorbing and Pulled Copolymers. GK Iliev and EJ Janse van Rensburg. J Stat Mech: Theo Expr P01019 (2012)

· The Compressibility of Minimal Lattice Knots. EJ Janse van Rensburg and A Rechnitzer. J Stat Mech: Theo Expr P05003 (2012)

· Lattice Knots in a Slab. D Gasumova, EJ Janse van Rensburg and A Rechnitzer. J Stat Mech: Theo Expr P09004 (2012)

2011:

· BFACF-Style Algorithms for Polygons in the BCC and FCC Lattices. EJ Janse van Rensburg and A Rechnitzer. J Phys A: Math Theo 44 165001 (2011)

· On the Universality of Knot Probability Ratios. EJ Janse van Rensburg and A Rechnitzer. J Phys A: Math Theo 44 162002 (2011)

· Generalised Atmospheric Sampling of Knotted Polygons. EJ Janse van Rensburg and A Rechnitzer. J Knot Theo Ram 20 1145 (2011)

· Adsorption of Pulled Directed Paths in a Slanted Boundary. EJ Janse van Rensburg. J Phys A: Math Theo 44 325001 (2011)

· Short and Long Ranged Adsorption of Directed Paths. G Iliev and EJ Janse van Rensburg. J Stat Mech: Theo Expr P11013 (2011)

· Minimal Knotted Polygons in Cubic Lattices. EJ Janse van Rensburg and A Rechnitzer. J Stat Mech: Theo Expr P09008 (2011)

2010:

· Adsorbing Motzkin Paths. EJ Janse van Rensburg. J Phys A: Math Theo 43 485006 (2010)

· The Adsorption Transition in Directed Paths. EJ Janse van Rensburg. J Stat Mech: Theo Expr P08030 (2010)

· Pulled Motzkin Paths. EJ Janse van Rensburg. J Phys A: Math Theo 43 335001 (2010)

· Pulled Directed Lattice Paths. EJ Janse van Rensburg. J Phys A: Math Theo 43 215001 (2010)

· Approximate Enumeration of Self-Avoiding Walks. EJ Janse van Rensburg. Cont Math 520 127 (2010)

2009:

· Atmospheric Collapse in Self-Avoiding Walks: A Numerical Study using GARM. J Alvarez, M Gara, EJ Janse van Rensburg and A Rechnitzer. J Stat. Mech: Theo Expr P12005 (2009)

· Thermodynamics and Entanglements of Walks under Stress. EJ Janse van Rensburg, E Orlandini, MC Tesi and SG Whittington. J Stat Mech: Theo Expr P07014 (2009)

· Monte Carlo Methods for the Self-Avoiding Walk. EJ Janse van Rensburg. J Phys A: Math Theo 42 323001 (2009)

· Generalised Atmospheric Sampling of Self-Avoiding Walks. EJ Janse van Rensburg and A Rechnitzer. J Phys A: Math Theo 42 335001 (2009)

· Monte Carlo Methods for Lattice Polygons. E.J. Janse van Rensburg. In Polygons, Polyominoes and Polyhedra (Lect Notes Phys 775 (2009)) Ed AJ Guttmann. Canopus Publishing Ltd.

· Thoughts on Lattice Knot Statistics. EJ Janse van Rensburg. J Math Chem 45(1) 7 (Commemorative issue in honour of Stu Whittington and Ray Kapral) (2009)

2008:

· Generalised Atmospheric Rosenbluth Methods (GARM). A Rechnitzer and EJ Janse van Rensburg. JPhys A: Math Theo 41 442002 (2008)

· Directed Paths in a Layered Environment. J Alvarez and EJ Janse van Rensburg. J Phys A: Math Theo 41 465003 (2008)

· Self-Avoiding Walks and Polygons in Slits. EJ Janse van Rensburg, CE Soteros and SG Whittington. J Phys A: Math Theo 41 185004 (2008)

· Atmospheres of Polygons and Knotted Polygons. EJ Janse van Rensburg and A Rechnitzer. J Phys A: Math Theo 41 105002 (2008)

· Knotting in Stretched Polygons. EJ Janse van Rensburg, E Orlandini, MC Tesi and SG Whittington. J Phys A: Math Theo 41 015003 (2008)

· Knot Probability of Polygons Subjected to a Force: A Monte Carlo Study. EJ Janse van Rensburg, E Orlandini, MC Tesi and SG Whittington. J Phys A: Math Theo 41 025003 (2008)

· Partially Directed Paths in a Wedge. EJ Janse van Rensburg, T Prellberg and AR Rechnitzer. J Combl Theo Ser A 115(4) 623 (2008)

2007:

· Directed Paths in a Wedge. EJ Janse van Rensburg, T Prellberg and A Rechnitzer. J Phys A: Math Theo 40 14069 (2007)

· Squeezing Knots. EJ Janse van Rensburg. J Stat Mech: Theo Expr P03001 (2007)

· Partially Directed Paths in a Wedge. EJ Janse van Rensburg, T Prellberg and A Rechnitzer. FPSAC’07, Nankai University, China (2007)

2006:

· Self-Avoiding Walks in a Slab: Rigorous Results. EJ Janse van Rensburg, E Orlandini and SG Whittington. J Phys A: Math Gen 39 13869 (2006)

· Plane Partition Vesicles. EJ Janse van Rensburg and J Ma J Phys A: Math Gen 39 11171 (2006)

· Moments of Directed Paths in a Wedge. EJ Janse van Rensburg. J Phys A: Math Gen Conf Ser 42 147 (2006)

· Forces in Motzkin Paths in a Wedge. EJ Janse van Rensburg. J Phys A: Math Gen 39 1581 (2006)

2005:

· Self-avoiding Walks in a Slab with Attractive Walls. EJ Janse van Rensburg, E Orlandini, AL Owczarek, A Rechnitzer, and SG Whittington. J Phys A: Math Gen 38 L823 (2005)

· Adsorbing Bargraph Paths in a q-Wedge. EJ Janse van Rensburg. J Phys A: Math Gen 38 8505 (2005). Corrigendum: J Phys A: Math Gen 39 3847 (2006)

· Forces in Square Lattice Directed Paths in a Wedge. EJ Janse van Rensburg and Y Le. J Phys A: Math Gen 38 8493 (2005)

· Square Lattice Directed Paths Adsorbing on the Line Y=qX. EJ Janse van Rensburg. J Stat Mech: Theo Expr P09010 (2005)

· Rectangular Vesicles in Three Dimensions. J Ma and EJ Janse van Rensburg. J Phys A: Math Gen 38 4115 (2005)

· A Tutorial on Knot Energies. EJ Janse van Rensburg. Contributed to Physical and Numerical Models in Knot Theory. Series on Knots and Everything 36 (2004) Eds. J Calvo, K Millett, E Rawdon and A Stasiak. World Scientific, Singapore.

2004:

· Inflating Square and Rectangular Lattice Vesicles. EJ Janse van Rensburg. J Phys A: Math Gen 37 3903 (2004)

· Multiple Markov Chain Monte Carlo Study of Adsorbing Self Avoiding Walks in Two and Three Dimensions. EJ Janse van Rensburg and A Rechnitzer. J Phys A: Math Gen 37 6875 (2004)

· Exchange Relations, Dyck Paths and Copolymer Adsorption. A Rechnitzer and EJ. Janse van Rensburg. Disc Appl Math140 49 (2004)

2003:

· Statistical Mechanics of Directed Models of Polymers in the Square Lattice. EJ Janse van Rensburg. J Phys A: Math Gen 36 R11 (2003)

· High Precision Canonical Monte Carlo Determination of the Growth Constant of Square Lattice Trees. EJ Janse van Rensburg and A Rechnitzer. Phys Rev E 67 036116-1 (2003)

2002:

· Knotting in Adsorbing Lattice Polygons. EJ Janse van Rensburg. Cont Math 304 137 (2002)

· The Probability of Knotting in Lattice Polygons. EJ Janse van Rensburg. Cont Math 304 125 (2002)

· Canonical Monte Carlo Determination of the Connective Constant of Self Avoiding Walks. A Rechnitzer and EJ Janse van Rensburg. J Phys A: Math Gen 35 L605 (2002)

· Exchange Symmetries in Motzkin Path and Bargraph Models of Copolymer Adsorption. A Rechnitzer and EJ Janse van Rensburg. Elect J Comb 9 R20 (2002)

· Upper Bounds on Linking Numbers of Thick Links. Y Diao, C Ernst and EJ Janse van Rensburg. J Knot Theo Ram 11 199 (2002)

2001:

· Self-Averaging Sequences in the Statistical Mechanics of Random Copolymers. EJ Janse van Rensburg, A Rechnitzer, M Causo and SG Whittington. J Phys A: Math Gen 34 6381 (2001)

· Adsorbing and Collapsing Directed Animals. EJ Janse van Rensburg and A Rechnitzer. J Stat Phys 105 49 (2001)

· Adsorbing Trees in Two Dimensions – A Monte Carlo Study. S You and EJ Janse van Rensburg. Phys Rev E 64(4) 046101-1 (2001)

· Self-Averaging in Random Self-Interacting Polygons. EJ Janse van Rensburg, E Orlandini, MC Tesi and SG Whittington. J Phys A: Math Gen 34 L37 (2001)

· Trees at an Interface. EJ Janse van Rensburg. J Stat Phys 102 1177 (2001)

2000:

· Interacting Columns: Generating Functions and Scaling Exponents. EJ Janse van Rensburg. J Phys A: Math Gen 33 7541 (2000)

· The Cluster Structure in Collapsing Animals. EJ Janse van Rensburg. J Phys A: Math Gen 33 3653 (2000)

· A Lattice Tree Model of Branched Copolymer Adsorption. J Phys A: Math Gen 33 1171 (2000)

1999:

· Adsorbing Staircase Walks and Polygons. EJ Janse van Rensburg. Ann Comb 3 451 (1999)

· Composite Models of Polygons. EJ Janse van Rensburg. J Phys A: Math Gen 32 4351 (1999)

· The Curvature of Lattice Knots. EJ Janse van Rensburg and SD Promislow. J Knot Theo Ram 8 463 (1999)

· Collapsing Animals. EJ Janse van Rensburg, E Orlandini and MC Tesi. J Phys A: Math Gen 32 1567 (1999)

· Thicknesses of Knots. Y Diao, C Ernst and EJ Janse van Rensburg. Math Proc Camb Phil Soc 126 293 (1999)

· The Writhe of Knots and Links. EJ Janse van Rensburg, DW Sumners and SG Whittington. Contributed to Ideal Knots. Series on Knots and Everything 19 (1999) Eds. A Stasiak, V Katrich and LH Kauffman. World Scientific, Singapore.

· Knots with Minimal Energies. Y Diao, C Ernst and EJ Janse van Rensburg. Contributed to Ideal Knots. Series on Knots and Everything 19 (1999) Eds. A Stasiak, V Katrich and LH Kauffman. World Scientific, Singapore.

· Minimal Lattice Knots. EJ Janse van Rensburg. Contributed to Ideal Knots. Series on Knots and Everything 19 (1999) Eds. A Stasiak, V Katrich and LH Kauffman. World Scientific, Singapore.

1998:

· Collapsing and Adsorbing Polygons. EJ Janse van Rensburg. J Phys A: Math Gen 31 8295 (1998)

· Adsorbing and Collapsing Trees. EJ Janse van Rensburg and S You. J Phys A: Math Gen 31 8635 (1998)

· Critical Exponents and Universal Amplitude Ratios in Lattice Trees. S You and EJ Janse van Rensburg. Phys Rev E 58 3971 (1998)

· Asymptotics of Knotted Lattice Polygons. E Orlandini, MC Tesi, EJ Janse van Rensburg and SG Whittington. J Phys A: Math Gen 31 5953 (1998)

· Minimal Links in the Cubic Lattice. R Uberti, EJ Janse van Rensburg, E Orlandini, MC Tesi and SG Whittington. Contributed to Topology and Geometry in Polymer Science. IMA Volume 103. Eds. SG Whittington, DW Sumners and T Lodge. (Proc 1995-96 IMA Prog Math Meth in Mat Sci, June 1996).

· A Model of Lattice Vesicles. EJ Janse van Rensburg. Contributed to Topology and Geometry in Polymer Science. IMA Volume 103. Eds. SG Whittington, DW Sumners and T Lodge. (Proc 1995-96 IMA Prog Math Meth in Mat Sci, June 1996).

· Energies of Knots. Y Diao, C Ernst and EJ Janse van Rensburg. Contributed to Topology and Geometry in Polymer Science. IMA Volume 103. Eds. SG Whittington, DW Sumners and T Lodge. (Proc 1995-96 IMA Prog Math Meth in Mat Sci, June 1996).

· Percolation of Linked Circles. Y Diao and EJ Janse van Rensburg. Contributed to Topology and Geometry in Polymer Science. IMA Volume 103. Eds. SG Whittington, DW Sumners and T Lodge. (Proc 1995-96 IMA Prog Math Meth in Mat Sci, June 1996).

· Topological Entanglement Complexity of Polymer Chains in Confined Geometries. MC Tesi, EJ Janse van Rensburg, E Orlandini and SG Whittington. Contributed to Topology and Geometry in Polymer Science. IMA Volume 103. Eds. SG Whittington, DW Sumners and T Lodge. (Proc 1995-96 IMA Prog Math Meth in Mat Sci, June 1996).

· Entropic Exponents of Knotted Lattice Polygons. E Orlandini, EJ Janse van Rensburg, MC Tesi and SG Whittington. Contributed to Topology and Geometry in Polymer Science. IMA Volume 103. Eds. SG Whittington, DW Sumners and T Lodge. (Proc 1995-96 IMA Prog Math Meth in Mat Sci, June 1996).

· Monte Carlo Simulation of the Theta-Point in Lattice Trees. EJ Janse van Rensburg and N Madras. Contributed to Numerical Methods for Polymeric Systems. IMA Volume 102. Eds. SG Whittington. (Proc 1995-96 IMA Prog Math Meth in Mat Sci, May 1996).

1997:

· Metropolis Monte Carlo Simulation of Lattice Animals. EJ Janse van Rensburg and N Madras. J Phys A: Math Gen 30 8035 (1997)

· The Shapes of Self-Avoiding Polygons with Torsion. E Orlandini, MC Tesi, EJ Janse van Rensburg and SG Whittington. J Phys A: Math Gen 30 L693 (1997)

· Torsion of Polygons in Z^3. MC Tesi, EJ Janse van Rensburg, E Orlandini and SG Whittington. J Phys A: Math Gen 30 5179 (1997)

· In Search of a Good Polygonal Knot Energy. Y Diao, C Ernst and EJ Janse van Rensburg. J Knot Theo Ram 6(5) 633 (1997)

· Knot Energies by Ropes. Y Diao, C Ernst and EJ Janse van Rensburg. J Knot Theo Ram 6(6) 799 (1997)

· Crumpling Self-Avoiding Surfaces. EJ Janse van Rensburg. J Stat Phys 88 177 (1997)

· Monte Carlo Study of the Theta-Point for Collapsing Trees. N Madras and EJ Janse van Rensburg. J Stat Phys 86 1 (1997)

· The Writhe of Knots in the Cubic Lattice. EJ Janse van Rensburg, E Orlandini, DW Sumners, MC Tesi and SG Whittington. J Knot Theo Ram 6(1) 31 (1997)

1996:

· Entropic Exponents of Lattice Polygons with Specified Knot Type. E Orlandini, MC Tesi, EJ Janse van Rensburg and SG Whittington. J Phys A: Math Gen 29 L299 (1996)

· Critical Evaluation of the VSC Model for Tip Growth. IB Heath and EJ Janse van Rensburg. Mycoscience 37 71 (1996)

· Entanglement Complexity of Lattice Ribbons. EJ Janse van Rensburg, E Orlandini, DW Sumners, MC Tesi and EJ Janse van Rensburg. J Stat Phys 85 103 (1996)

· A Monte Carlo Algorithm for Lattice Ribbons. E Orlandini, EJ Janse van Rensburg and SG Whittington. J Stat Phys 82 1159 (1996)

· Monte Carlo Study of the Interacting Self-Avoiding Walk Model in Three Dimensions. MC Tesi, EJ Janse van Rensburg, E Orlandini and SG Whittington. J Stat Phys 82 155 (1996)

· Lattice Invariants for Knots. EJ Janse van Rensburg. Contributed to Mathematical Approaches to Biomolecular Structure and Dynamics. IMA Volume 82. Eds. JP Mesirov, K SChulten and DW Sumners. (Proc 1994 IMA Summer Prog in Mol Biol, July 1994).

· Topology and Geometry of Biopolymers. EJ Janse van Rensburg, E Orlandini, DW Sumners, MC Tesi and SG Whittington. Contributed to Mathematical Approaches to Biomolecular Structure and Dynamics. IMA Volume 82. Eds. JP Mesirov, K SChulten and DW Sumners. (Proc 1994 IMA Summer Prog in Mol Biol, July 1994).

1995:

· Interacting Self-Avoiding Walks and Polygons in Three Dimensions. MC Tesi, EJ Janse van Rensburg, E Orlandini and SG Whittington. J Phys A: Math Gen 29 2451 (1995)

· Twist in an Exactly Solvable Directed Lattice Ribbon. E Orlandini and EJ Janse van Rensburg. J Stat Phys 80 781 (1995)

· Minimal Knots in the Cubic Lattice. EJ Janse van Rensburg and SD Promislow. J Knot Theo Ram 4(1) 115 (1995)

1994:

· Lattice Ribbons: A Model of Double-Stranded Polymers. EJ Janse van Rensburg, E Orlandini, DW Sumners, MC Tesi and SG Whittington. Phys Rev E 50 4279 (1994)

· The Writhe of a Self-Avoiding Walk. E Orlandini, MC Tesi, SG Whittington, DW Sumners and EJ Janse van Rensburg. J Phys A: Math Gen 27 L333 (1994)

· Statistical Mechanics and Topology of Surfaces in Z^d. EJ Janse van Rensburg. J Knot Theo Ram 3(3) 365 (1994)

· Knot Probability for Lattice Polygons in Confined Geometries. MC Tesi, EJ Janse van Rensburg, E Orlandini and SG Whittington. J Phys A: Math Gen 27 347 (1994)

· Random Linking of Lattice Polygons. E Orlandini, EJ Janse van Rensburg, MC Tesi and SG Whittington. J Phys A: Math Gen 27 335 (1994)

· Knotting and Supercoiling in Circular DNA: A Model Incorporating the Effect of Added Salt. MC Tesi, EJ Janse van Rensburg, E Orlandini, DW Sumners and SG Whittington. Phys Rev E 49 868 (1994)

1993:

· The Writhe of a Self-Avoiding Polygon. EJ Janse van Rensburg, E Orlandini, DW Sumners, MC Tesi and SG Whittington. J Phys A: Math Gen 26 L981 (1993)

· Virial Coefficients for Hard Discs and Hard Spheres. EJ Janse van Rensburg. J Phys A: Math Gen 26 4805 (1993)

· Estimation of Multidimensional Integrals: Is Monte Carlo the Best Method? EJ Janse van Rensburg and GM Torrie. J Phys A: Math Gen 26 943 (1993)

· A Numerical Study of the Gel Electrophoresis of Knotted DNA. HA Lim and EJ Janse van Rensburg. Math Mod & Sci Comp 1 153 (1993)

· Electrophoresis of Circular and Knotted Polymers/DNA. HA Lim and EJ Janse van Rensburg. Contributed to Proc 8 Int Conf Math Comp Mod. Math Mod and Sci Comp 2 622 (1993)

· Random Knots in Ring Polymers. SG Whittington and EJ Janse van Rensburg. Contributed to Proc 8 Int Conf Math Comp Mod. Math Mod and Sci Comp 2 741 (1993)

1992:

· On the Number of Lattice Trees. EJ Janse van Rensburg. J Phys A: Math Gen 25 3523 (1992)

· Surfaces in Hypercubic Lattices. EJ Janse van Rensburg. J Phys A: Math Gen 25 3529 (1992)

· Entanglement Complexity of Self-Avoiding Walks. EJ Janse van Rensburg, DW Sumners, E Wasserman and SG Whittington. J Phys A: Math Gen 25 6557 (1992)

· Ergodicity of the BFACF Algorithm in Three Dimensions. EJ Janse van Rensburg. J Phys A: Math Gen 25 1031 (1992)

· A Non-local Algorithm for Lattice Trees. EJ Janse van Rensburg and N Madras. J Phys A: Math Gen 25 303 (1992)

· Electrophoresis of Knotted DNA in a Regular and Random Electrophoretic Medium. HA Lim, MT Carroll and EJ Janse van Rensburg. Contributed to Biomedical Modeling and Simulation. Eds J Eisenfeld, DS Levine and M Witten. (1992)

1991:

· The Dimensions of Knotted Polygons. EJ Janse van Rensburg and SG Whittington. J Phys A: Math Gen 24 3935 (1991)

· The BFACF Algorithm and Knotted Polygons. EJ Janse van Rensburg and SG Whittington. J Phys A: Math Gen 24 5553 (1991)

· Energy Transfer on Knotted Polygons. EJ Janse van Rensburg and SG Whittington. Macromol 24 1969 (1991)

1990:

· The Topology of Interfaces. EJ Janse van Rensburg. J Phys A: Math Gen 23 5879 (1990)

· Self-Avoiding Surfaces with Knotted Boundaries. EJ Janse van Rensburg and SG Whittington. J Phys A: Math Gen 23 2495 (1990)

· The Knot Probability of Lattice Polygons. EJ Janse van Rensburg and SG Whittington. J Phys A: Math Gen 23 3573 (1990)

· The Pivot Algorithm and Polygons: Results on the FCC Lattice. EJ Janse van Rensburg, SG Whittington and N Madras. J Phys A: Math Gen 23 1589 (1990)

· Punctured Discs on the Square Lattice. EJ Janse van Rensburg and SG Whittongton. J Phys A: Math Gen 23 1287 (1990)

· Graphs and Surfaces in the Regular Lattice. SG Whittington, CE Soteros and EJ Janse van Rensburg. Comp and Chem 14 281 (1990)

1989:

· Self-Avoiding Surfaces. EJ Janse van Rensburg and SG Whittington. J Phys A: Math Gen 22 4939 (1989)

· Exciton Migration on Polymers. EJ Janse van Rensburg, JE Guillet and SG Whittington. Macromol 22 4212 (1989)

1988:

· Resistance of the Edwards Walk. EJ Janse van Rensburg. J Phys A: Math Gen 21 147 (1988)

· Simulation of Random Walks in Field Theory. EJ Janse van Rensburg. J Phys G: Nucl Phys 14 397 (1988)

1987:

· A Non-Self Adjoint General Matrix Eigenvalue Problem. G Delic, EJ Janse van Rensburg and G Welke. J Comp Phys 69 325 (1987)

· Monte Carlo Simulations of Random Walks and Surfaces in Parallel. EJ Janse van Rensburg. J Phys A: Math Gen 20 L637 (1987)

COURSE WEBPAGES:

On Teaching Sabbatical

Random Maze

Here is a random maze I have generated by Monte Carlo. Can you find the way out?

In reality, a random maze would be much more convoluted than the above example. Such mazes are less pleasing to the eye. In the case above a maze consisting of walls that spiral from the origin was generated, and then subjected to a Monte Carlo simulation for a short time (before the maze would randomize completely).

Random Tree:

A maze is a special kind of random tree: in particular, it is a spanning tree of a square in the square lattice (such a maze would have only one way out from the center). Below is a random tree in the square lattice.

Towards the limiting random lattice tree

If the number of edges in the tree above should be multiplied, while the length scale is shrunk appropriately, a limiting random tree will be seen. In the picture below there are 10000 edges, each too small to be seen here, in a lattice tree generated by a Monte Carlo program. The tree begins to appear like a fractal object. The existence of a scaling limit is known in high dimensions (above 8). There is general consensus that it also exists in dimensions below 9.

Random Disks

The interior of a closed loop in the square lattice (or a polygon) is a Disk. In this example, a square lattice polygon was randomized by subjecting it to a Metropolis Monte Carlo algorithm. The interior of the polygon is a disk.

Die Oranje Vrystaat.

Die foto is in die Noord Vrystaat geneem.