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    York University

       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

 

 

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 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)

2013:

·         The Pressure Exerted by Adsorbing Directed Lattice Paths and Staircase Polygons.  EJ Janse van Rensburg and T Prellberg. J Phys A: Math Theo 46 115202 (2013)

·         Adsorbed Self-Avoiding Walks Subject to a Force.  EJ Janse van Rensburg and SG Whittington. . J Phys A: Math Theo (2013)

·         The Entropic Pressure of a Lattice Polygon.  F Gassoumov and EJ Janse van Rensburg.  .J Stat Mech: Theo Expr (2013)

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)

·         On Trivial Words in Finitely Presented Groups.  M Elder, A Rechnitzer, EJ Janse van Rensburg and T Wong.  arXiv Preprint arXiv:1210.3425 (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 RechnitzerJ 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 WhittingtonJ 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 RensburgJ 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 TheorySeries 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 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)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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 COURSE WEBPAGES:

On Teaching Sabbatical


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Random Maze

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


 

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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.

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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.
 

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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.

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Die Oranje Vrystaat.

Die foto is in die Noord Vrystaat geneem.
 

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