Research Contributions of Jeffrey S. Rosenthal

Hello! Bonjour! ¡Buenos días! Buon giorno! Guten Tag! Bom dia! Welcome to my research page. My research interests include probability theory, Markov chain analysis, stochastic processes, and statistical computation especially Markov chain Monte Carlo computer algorithms. See also a one-paragraph summary, an interdisciplinary research summary, my research Java applets, and my non-research writings.

Shortcuts to selected ... review papers: probsurv / computsimple / scalerev / bruns / gentheory / eigen

... research papers: amcmc / adaptex / adapt / determ / one-shot / period / hobertfinal / suffic / slice / hybrid / sens / lang / james / minor / varnew / rots

... books: Struck by Lightning / Science of Uncertainty / Rigorous Probability

... years: 2011 / 2010 / 2009 / 2008 / 2007 / 2006 / 2005 / 2004 / 2003 / 2002 / 2001 / 2000 / 1999 / 1998 / 1997 / 1996 / 1995 / 1994 / 1993 / 1992 / 1989 / Books / Other / Misc

My research papers are provided below, grouped by year of initial submission, with multiple authors listed alphabetically except where indicated by [N-A]. Most of the papers are provided in postscript format; to view them use e.g. ghostview. Most of them are also provided in pdf format; to view them use e.g. Adobe Reader or xpdf or ghostview.

2011

Y. Yunusova, J.S. Rosenthal, K. Rudy, M. Baljko, and J. Daskalogiannakis [N-A], Place of articulation for lingual consonants defined using electromagnetic articulography. [To be posted later.]
D.B. Woodard and J.S. Rosenthal [N-A], Convergence rate of Markov chain methods for genomic motif discovery. [pdf paper, pdf Appendix]
J.D. Nielsen, J.S. Rosenthal, Y. Sun, D.M. Day, I. Bevc, and T. Duchesne [N-A], Group-based Criminal Trajectory Analysis using Cross-Validation Criteria. [pdf, postscript] (See also the related software.)
K. Latuszynski, G.O. Roberts, and J.S. Rosenthal, Adaptive Gibbs samplers and related MCMC methods. [pdf, postscript] (Major revision of an earlier two-author paper.) Ann. Appl. Prob., to appear.
J.S. Rosenthal, Was the Conservative majority predictable? [pdf, postscript] Can. J. Stat. 39:721-733, 2011.
D.M. Day, J.D. Nielsen, A.K. Ward, Y. Sun, J.S. Rosenthal, I. Bevc, T. Duchesne, and L. Rossman [N-A], Trajectories of Criminal Activity in a Sample of 378 Adjudicated Ontario Youth. [pdf poster, pdf paper] Presented at the 72nd annual CPA conference / 2nd NACCJPC conference, Toronto, June 2, 2011.

2010

J.S. Rosenthal and A.H. Yoon, Judicial Ghostwriting: Authorship on the U.S. Supreme Court [in pdf only]. (Cornell Law Review, to appear. See also the NY Times mention.)
K. Latuszynski and J.S. Rosenthal, Adaptive Gibbs samplers [or in pdf]. (Later revised into this paper.)
A.K. Ward, D.M. Day, I. Bevc, Y. Sun, J.S. Rosenthal, and T. Duchesne [N-A], Criminal Trajectories and Risk Factors in a Canadian Sample of Offenders [in pdf only]. (See team photo.) Criminal Justice and Behavior, to appear.
S. Richardson, L. Bottolo, and J.S. Rosenthal [N-A], Bayesian models for sparse regression analysis of high dimensional data [in pdf only]. (Published in Valencia IX Bayesian Meeting conference proceedings, 2011.)
J. Chen and J.S. Rosenthal, Decrypting Classical Cipher Text Using Markov Chain Monte Carlo [or in pdf or published version]. (See also the related software.) Stat. and Comput. 22:397-413, 2011.
M.A. Turner, J.S. Rosenthal, J. Chen, and C. Hao [N-A], Adaption to Climate Change: Evidence from 18th and 19th Century Iceland [in pdf only]. (See also the conference proceedings version.)
D.M. Day, J.D. Nielsen, A.K. Ward, Y. Sun, J.S. Rosenthal, T. Duchesne, I. Bevc, and L. Rossman [N-A], Long-term Follow-up of Criminal Activity with Adjudicated Youth in Ontario [in pdf only].

2009

Y. Atchadé, G.O. Roberts, and J.S. Rosenthal, Towards Optimal Scaling of Metropolis-Coupled Markov Chain Monte Carlo [or in pdf]. See also my slightly-related Cambridge talk. (Stat. and Comput. 21:555-568, 2011.)
M.A. Proschan and J.S. Rosenthal, Beyond the Quintessential Quincunx [in pdf only]. The American Statistician 64:78-82, 2010. (See also the related Java applet.)
J.S. Rosenthal and A.H. Yoon, Detecting Multiple Authorship of United States Supreme Court Legal Decisions Using Function Words [or in pdf]. (See also the related software.) Ann. Appl. Stat. 5:283-308, 2011. (See the published version.)

2008

G.O. Roberts and J.S. Rosenthal, Quantitative Non-Geometric Convergence Bounds for Independence Samplers [or in pdf]. MCAP 13:391-403, 2011.
J.S. Rosenthal, Optimal Proposal Distributions and Adaptive MCMC [or in pdf]. (Chapter 4 of Handbook of MCMC, 2011 [amazon.ca, amazon.com]. See also the related software.)
J.S. Rosenthal, Optimising Monte Carlo Search Strategies for Automated Pattern Detection [or in pdf]. Far East J. Math. Sci. 32:311-328, 2009. (See also the related Java applet.)
Y. Bai, G.O. Roberts, and J.S. Rosenthal, On the Containment Condition for Adaptive Markov Chain Monte Carlo Algorithms [or in pdf]. (Significantly revised and expanded, July 2010; see the previous version [or in pdf]). Adv. Appl. Stat. 21:1-54, 2011.
R.V. Craiu, J.S. Rosenthal, and C. Yang, Learn From Thy Neighbor: Parallel-Chain Adaptive MCMC [or in pdf]. (JASA 488:1454-1466, 2009.)
J.S. Rosenthal, A mathematical analysis of the Sleeping Beauty problem [or in pdf]. Math. Intel. 31:32-37, 2009.
D.M. Day, I. Bevc, F. Theodor, J.S. Rosenthal, and T. Duchesne [N-A], Change and Continuity in Criminal Offending: Criminal Trajectories of the 'Toronto' Sample. (Unpublished report written for the Ontario Ministry of Children and Youth Services.)
J.S. Rosenthal, Markov Chain Monte Carlo Algorithms: Theory and Practice [or in pdf]. (For MCQMC'08 conference proceedings.)

2007

O. Häggström and J.S. Rosenthal, On Variance Conditions for Markov Chain CLTs [or in pdf]. (Elec. Comm. Prob. 12:454-464, 2007.)
M. Bedard and J.S. Rosenthal, Optimal Scaling of Metropolis Algorithms: Heading Towards General Target Distributions [or in pdf]. (Can. J. Stat. 36(4):483-503, 2008.)
J.P. Hobert and J.S. Rosenthal, Norm Comparisons for Data Augmentation [or in pdf]. (Adv. Appl. Stat. 7:291-302, 2007.)
J.S. Rosenthal, AMCMC: An R interface for adaptive MCMC [or in pdf, or published version]. (See also the related software.) (Comp. Stat. & Data Anal. 51:5467-5470, 2007.)
J.S. Rosenthal, Expressions for the Markov Chain CLT Variance [or in pdf]. (See also the earlier version.) [Unpublished notes, superceded by this paper.]
J.S. Rosenthal, Waiting time correlations on disorderly streetcar routes [or in pdf]. (See also the related software.) (Int. J. Pure Appl. Math. Sci., to appear.)
D.M. Day, I. Bevc, T. Duchesne, J.S. Rosenthal, Y. Sun, and F. Theodor (2007) [N-A], Criminal trajectories from adolescence to adulthood in an Ontario sample of offenders. (Proc. 2007 North American Correctional and Criminal Justice Psych. Conf., 143-148.)

2006

G.O. Roberts and J.S. Rosenthal, Variance bounding Markov chains [or in pdf or published version]. (Ann. Appl. Prob. 18:1201-1214, 2008.)
G.O. Roberts, J.S. Rosenthal, J. Segers, and B. Sousa, Extremal Indices, Geometric Ergodicity of Markov Chains, and MCMC [or in pdf]. (Extremes 9:213-229, 2006.)
G.O. Roberts and J.S. Rosenthal, Examples of Adaptive MCMC [or in pdf]. (See also the related software, a related talk and talk#2 and French-language talk, and this follow-up paper.) (J. Comp. Graph. Stat. 18:349-367, 2009.)

2005

G.O. Roberts and J.S. Rosenthal, Coupling and Ergodicity of Adaptive MCMC [or in pdf]. (See also the related Java applet and related French-language abstract.) (J. Appl. Prob. 44:458-475, 2007.)
D.M. Day, I. Bevc, T. Duchesne, J.S. Rosenthal, L. Rossman, and F. Theodor [N-A], Comparison of Adult Offense Prediction Methods based on Juvenile Offense Trajectories using Cross-Validation [or in pdf] (Adv. Appl. Stat. 7:1-46, 2007).

2004

G.O. Roberts and J.S. Rosenthal, Harris Recurrence of Metropolis-Within-Gibbs and Trans-Dimensional Markov Chains [or in pdf]. (Ann. Appl. Prob. 16:2123-2139, 2006.)
G.O. Roberts and J.S. Rosenthal, General state space Markov chains and MCMC algorithms [or in pdf / arXiv]. (Probability Surveys 1:20-71, 2004.) [See also related follow-up papers here and here and here and here and here.]
M. Bramson, J. Quastel, and J.S. Rosenthal, When Can Martingales Avoid Ruin? [or in pdf]. (Unpublished manuscript.)

2003

D.M. Day, I. Bevc, J.S. Rosenthal, T. Duchesne, L. Rossman, and F. Theodor [N-A], Predicting Adult Offenders' Criminal Trajectories From Their Juvenile Criminal Trajectories. (Poster, American Psychological Association annual convention, 2003.)
I. Bevc, T. Duchesne, J.S. Rosenthal, L. Rossman, F. Theodor, and E. Sowa [N-A], Young Offenders' Diagnoses As Predictors of Subsequent Adult Criminal Behaviour. (Poster, American Psychological Association annual convention, 2003.)
Y.F. Atchadé and J.S. Rosenthal, On Adaptive Markov Chain Monte Carlo Algorithms [or in pdf]. (See also the related Java applet.) (Bernoulli 11:815-828, 2005.)
G.O. Roberts and J.S. Rosenthal, Downweighting Tightly Knit Communities in World Wide Web Rankings [or in pdf] (Adv. Appl. Stat. 3:199-216, 2003.)
O.F. Christensen, G.O. Roberts, and J.S. Rosenthal, Scaling Limits for the Transient Phase of Local Metropolis-Hastings Algorithms [or in pdf]. (JRSSB 67:253-268, 2005.)

2002

A. Feuerverger and J.S. Rosenthal, Achieving Limiting Distributions for Markov Chains Using Back Buttons [or in pdf]. (Stat. and Comput. 14:131-141, 2004. See also the extended abstract version and the related C program.)
S.P. Brooks, Y. Fan, and J.S. Rosenthal, Perfect Forward Simulation via Simulated Tempering [or pdf]. (Comm. Stat.: Sim. and Comp. 35:683-713, 2006.) See also the related computer code.
J.S. Rosenthal, Geometric Convergence Rates for Time-Sampled Markov Chains [or in pdf]. (J. Theoretical Prob. 16:671-688, 2003.)
R. Douc, E. Moulines, and J.S. Rosenthal, Quantitative bounds on convergence of time-inhomogeneous Markov Chains [or in pdf]. (Ann. Appl. Prob. 14:1643-1665, 2004.)
J.S. Rosenthal, Quantitative convergence rates of Markov chains: A simple account [or in pdf]. (See also a related French-language abstract.) (Elec. Comm. Prob. 7 no. 13:123-128, 2002.)
T. Duchesne and J.S. Rosenthal, On the Collapsibility of Lifetime Regression Models [or in pdf]. (Adv. Appl. Prob. 35:755-772, 2003.) (See also the conference proceedings version.)

2001

G. Fort, E. Moulines, G.O. Roberts, and J.S. Rosenthal, On the geometric ergodicity of hybrid samplers [or pdf]. (J. Appl. Prob. 40:123-146, 2003.)
G.O. Roberts and J.S. Rosenthal, One-shot coupling for certain stochastic recursive sequences [or in pdf]. (Stoch. Proc. Appl. 99:195-208, 2002.)
G.O. Roberts and J.S. Rosenthal, Combinatorial identities associated with CFTP [or in pdf]. (See also the related C program.) Far East J. Math. Sci. 13:391-404, 2004.
G.O. Roberts and J.S. Rosenthal, Optimal scaling for various Metropolis-Hastings algorithms [or in pdf]. (Statistical Science 16:351-367, 2001.)
J.S. Rosenthal, Asymptotic Variance and Convergence Rates of Nearly-Periodic MCMC Algorithms [or in pdf]. (JASA 98:169-177, 2003.) (See also the original, longer version [or in pdf] and the follow-up paper [or in pdf].)
J.P. Hobert, G.L. Jones, B. Presnell, and J.S. Rosenthal, On the Applicability of Regenerative Simulation in Markov Chain Monte Carlo [or pdf]. (Biometrika 89:731-743, 2002.)

2000

G.O. Roberts and J.S. Rosenthal, Small and pseudo-small sets for Markov chains [or in pdf]. (Stochastic Models 17:121-145, 2001.)
J.J. Lu, J.S. Rosenthal, and A.E. Shaffer, A case study in the meta-reasoning procedure ND [or pdf]. (J. Exp. Th. Artificial Intelligence 15:47-71, 2003.) (See also the conference proceedings version in FTP'2000.)
L. Breyer, G.O. Roberts, and J.S. Rosenthal, A note on geometric ergodicity and floating-point roundoff error [or in pdf]. (Shortened version [or in pdf] published in Stat. Prob. Lett. 53:123-127, 2001.)
M.J. Gordon and J.S. Rosenthal, Capitalism's Growth Imperative. (Cambridge J. Economics 27:25-48, 2003.)
A. Borodin, G.O. Roberts, J.S. Rosenthal, and P. Tsaparas, Link Analysis Ranking: Algorithms, Theory, and Experiments [or in pdf]. (ACM TOIT 5:231-297, 2005.) (See also the conference proceedings version, published in WWW10, 2001, chosen as selected area highlight.)

1999

R.B. Israel, J.S. Rosenthal, and J.Z. Wei, Finding generators for Markov chains via empirical transition matrices, with applications to credit ratings [or in pdf]. (Mathematical Finance 11:245-265, 2000.)
G.O. Roberts and J.S. Rosenthal, Bayesian models with infinite hierarchies [or in pdf]. (Bernoulli 7:453-471, 2001.)
G.O. Roberts and J.S. Rosenthal, Recent progress on computable bounds and the simple slice sampler [or in pdf]. See also the Fields-format version. (In Monte Carlo Methods (N. Madras, ed.), 2000, 123-130.)
G.O. Roberts and J.S. Rosenthal, The polar slice sampler [or in pdf]. (Stochastic Models 18:257-280, 2002.) See also the related French-language slides (or in pdf) and Java applet.
J.S. Rosenthal, Parallel computing and Monte Carlo algorithms [or in pdf]. (Far East J. Theor. Stat. 4:207-236, 2000.) See also the related software.
J.S. Rosenthal, A review of asymptotic convergence for general state space Markov chains [or in pdf]. (Expository paper. Far East J. Theor. Stat. 5:37-50, 2001.)

1998

G.O. Roberts and J.S. Rosenthal, Markov chains and de-initialising processes [or in pdf] [formerly Sufficient Markov chains]. (Scandanavian J. Stat. 28:489-504, 2001.)
R. Pemantle and J.S. Rosenthal, Moment conditions for a sequence with negative drift to be uniformly bounded in L^r [or in pdf]. (Stoch. Proc. Appl. 82:143-155, 1999.)
J.A. Fill, M. Machida, D.J. Murdoch, and J.S. Rosenthal, Extension of Fill's perfect rejection sampling algorithm to general chains [or in pdf]. (Random Structures and Algorithms 17:290-316, 2000.) See also the extended abstract version [or in pdf] (in Monte Carlo Methods (N. Madras, ed.), 2000, 37-52) and the original two-author version [or in pdf].
D.J. Murdoch and J.S. Rosenthal, Efficient use of exact samples [or in pdf]. (Stat. and Comput. 10:237-243, 2000.) See also the associated poster, the related Java applet, and D.B. Wilson's annotated bibliography.
M.J. Osborne, J.S. Rosenthal, and M.A. Turner, Meetings with costly participation [or pdf]. (American Economic Review 90:927-943, 2000.) (See also Osborne's expository article.)
S. Petrone, G.O. Roberts, and J.S. Rosenthal, A note on convergence rates of Gibbs sampling for nonparametric mixtures [or in pdf]. (Far East J. Theor. Stat. 3:213-225, 1999.)

1997

M.K. Cowles, G.O. Roberts, and J.S. Rosenthal, Possible biases induced by MCMC convergence diagnostics [or in pdf]. (J. Stat. Comp. and Sim. 64:87-104, 1999.)
G.O. Roberts and J.S. Rosenthal, Convergence of slice sampler Markov chains [or in pdf]. (JRSSB 61:643-660 , 1999.) See also the related Java applet.
G.O. Roberts and J.S. Rosenthal, On convergence rates of Gibbs samplers for uniform distributions [or in pdf]. (Ann. Appl. Prob. 8:1291-1302, 1998.)
J.S. Rosenthal and P.O. Schwartz, Gambling systems and multiplication-invariant measures [or in pdf]. (Adv. Appl. Math. 22:303-311, 1999.)
G.O. Roberts and J.S. Rosenthal, Markov chain Monte Carlo: Some practical implications of theoretical results [or in pdf]. (Can. J. Stat. 26:5-31, 1998, with discussion and rejoinder [or in pdf].) See also the related Java applet and related talk [or in pdf].
G.O. Roberts and J.S. Rosenthal, Two convergence properties of hybrid samplers [or in pdf]. (Ann. Appl. Prob. 8:397-407, 1998.)

1996

G.O. Roberts and J.S. Rosenthal, Geometric ergodicity and hybrid Markov chains [or in pdf]. (Elec. Comm. Prob. 2, paper 2, 1997.)
M.K. Cowles and J.S. Rosenthal, A simulation approach to convergence rates for Markov chain Monte Carlo algorithms [or in pdf]. (Stat. and Comput. 8:115-124, 1998.)
G.O. Roberts and J.S. Rosenthal, Quantitative bounds for convergence rates of continuous time Markov processes [or in pdf]. (Elec. J. Prob. 1, paper 9, 1996.)

1995

G.O. Roberts, J.S. Rosenthal, and P.O. Schwartz, Convergence properties of perturbed Markov chains [or in pdf]. (J. Appl. Prob. 35:1-11, 1998.)
G.O. Roberts and J.S. Rosenthal, Optimal scaling of discrete approximations to Langevin diffusions [or in pdf]. (JRSSB 60:255-268, 1998.)
J.S. Rosenthal, Faithful couplings of Markov chains: now equals forever [or in pdf]. (Adv. Appl. Math. 18:372-381, 1997.)

1994

G.O. Roberts and J.S. Rosenthal, Shift-coupling and convergence rates of ergodic averages [or in pdf]. (Comm. in Stat. - Stoch. Models 13:147-165, 1997.) (See also the published version.)
J.S. Rosenthal, Markov chain convergence: from finite to infinite [or in pdf]. (Stoch. Proc. Appl. 62:55-72, 1996.)
J.S. Rosenthal, Analysis of the Gibbs Sampler for a model related to James-Stein Estimators [or in pdf]. (Stat. and Comput. 6:269-275, 1996.)
J.R. Baxter and J.S. Rosenthal, Rates of convergence for everywhere-positive Markov chains [or in pdf]. (Stat. Prob. Lett. 22:333-338, 1995.)

1993

J.S. Rosenthal, Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo [or in pdf]. (JASA 90:558-566, 1995.) See also the one-page supplement [or in pdf].
J.R. Hoffman and J.S. Rosenthal, Convergence of Independent Particle Systems [or in pdf]. (Stoch. Proc. Appl. 56:295-305, 1995.)
J.S. Rosenthal, Convergence rates of Markov chains [or in pdf]. (Expository paper. SIAM Review 37:387-405, 1995.)

1992

J.S. Rosenthal, On Generalizing the Cut-off Phenomenon for Random Walks on Groups [or in pdf]. (Adv. Appl. Math. 16:306-320, 1995.)
J.S. Rosenthal, On Duality of Probabilities for Card-dealing [or in pdf]. (Proc. AMS 123:559-561, 1995.)
J.S. Rosenthal, Random Rotations: Characters and Random Walks on SO(N) [or in pdf]. (Ann. Prob. 22:398-423, 1994.) [From my Ph.D. thesis.]
J.S. Rosenthal, Random Walks on Discrete and Continuous Circles [or in pdf]. (J. Appl. Prob. 30:780-789, 1993.) [From my Ph.D. thesis.]
J.S. Rosenthal, Rates of Convergence for Data Augmentation on Finite Sample Spaces [or in pdf]. (Ann. Appl. Prob. 3:819-839, 1993.) [From my Ph.D. thesis.]
J.S. Rosenthal, Rates of Convergence for Gibbs Sampling for Variance Components Models [or in pdf]. (Ann. Stat. 23:740-761, 1995.) [From my Ph.D. thesis.]
J.S. Rosenthal, Convergence of pseudo-finite Markov chains [or in pdf]. (Unpublished manuscript.)

1989

M.D. Choi, J.S. Rosenthal, and P. Rosenthal, Linear-Algebraic Results Associated with Antiferromagnetic Heisenberg Chains [or in pdf]. (SIAM J. Matrix Anal. Appl. 3:830-852, 1993.)

Books

J.S. Rosenthal, Struck by Lightning: The Curious World of Probabilities (book for the general public). HarperCollins Canada, 272 pages, 2005. (Later published in many other countries too.)
M.J. Evans and J.S. Rosenthal, Probability and Statistics: The Science of Uncertainty (undergraduate-level textbook). W.H. Freeman and Co., 685 pages, 2003.
J.S. Rosenthal, A First Look at Rigorous Probability Theory (graduate-level textbook). World Scientific Publishing Co., 177 pages, 2000. Second edition, 219 pages, 2006.

Other Research Writings

J.S. Rosenthal's Ph.D. thesis, Rates of Convergence for Gibbs Sampler and other Markov Chains, 1992. See acknowledgements, front matter, introduction, chapter 2, chapter 3, chapter 4, chapter 5, conclusion, citation, and math genealogy (source).
J.S. Rosenthal, Theoretical rates of convergence for Markov chain Monte Carlo [or in pdf]. (Interface '94 conference proceedings.)
J.S. Rosenthal, Active-Learning Strategies in Advanced Mathematics Classes [or in pdf]. (Education paper. Studies in Higher Ed. 20:223-228, 1995.)
J.S. Rosenthal, The de Branges proof of the Bieberbach conjecture [or in pdf]. (Master's thesis, 1990.)
U. Porod and J.S. Rosenthal, An open problem related to random walks on U(N) [or in pdf] (including prize offer!) and some related notes [or in pdf], 1994.
G.O. Roberts and J.S. Rosenthal, Markov chain Monte Carlo [or in pdf]. A short review article for Section 10 (Probability Theory) of the Encyclopedia of the Actuarial Sciences (S. Asmussen, ed.), 2003.
G.O. Roberts and J.S. Rosenthal, Understanding MCMC [or pdf] (slides for week-long U.K. post-graduate course), 2003. See also the exercises [or in pdf] and solutions [or in pdf] and computer lab [or in pdf] and R commands, and some photos.
J.S. Rosenthal, Statistics and MCMC [or pdf] (slides for survey talk at Markov Chains symposium in Durham, U.K.), 2003.
See also my non-research writings, and my C.V. in postscript / pdf.

Miscellaneous Items of Possible Interest

I have written some Java applets related to my research; see also my computing page and my interdisciplinary research page.
For the three-year period 2007-2009, I was an Associate Editor of the Canadian Journal of Statistics and of the Electronic Journal of Statistics.
See also my Mathematics Genealogy Project page (from which I constructed my personal math genealogy), and my MathSciNet listings, and my Google Scholar listings.
Thanks to P. Kim and J.Y. Koo for christening the "Rosenthal distribution" in this paper (J. Mult. Anal. 80:21-42, 2002)!
For research papers by others about similar topics, visit the MCMC Preprint Service, and the exact sampling annotated bibliography.
I also occasionally do consulting work. For example, in 2009-2010, I designed and programmed the Koodo Savings Calculator.
My Erdos number is 3, thanks to: Bruce A. Reznick (1) and Man-Duen Choi (2); or alternatively Svante Janson (1) and Robin Pemantle (2); or (if you count "comments" as papers) Fan Chung (or Ron Graham or Svante Janson) (1) and Persi Diaconis (2).
I am a very serious researcher. Here I am hard at work in Cambridge, England; in Tirano, Italy; in Monte Carlo, Monaco; in Ottawa, Canada; in Lancaster, England; in Bormio, Italy (with an audience); and in Hamilton Island, Australia (with more audience).

[contact me / My Home Page]