Research and Publications


Combinatorics: Coloring Maps

The Four Color Theorem for Small Maps
J. Comb. Theory B 19 (1975) 256-268.

Some Aspects of the Four Color Problem
Thesis, Harvard University, 1975. Advisor: Professor Andrew Gleason.
(The link is to a 7MB pdf file made by scanning the typed pages.)

The Four Color Theorem for Locally Planar Graphs (problem statement)
Graph Theory and Related Topics, Bondy and Murty (eds.), Academic Press (1979).

Michael O. Albertson and WRS
Locally Planar Toroidal Graphs are 5-Colorable,
Proceedings of the American Mathematical Society 84 (1982) 449-456 (JSTOR link).

Combinatorics: Combinatorial Games

WRS and D. Ullman
Sequential Compounds of Combinatorial Games
J. Theoretical Computer Science 119(2) (October 25, 1993) 311-321.

D. Loeb, A. J. S. Lazarus, J. Propp, WRS, and D. Ullman
Combinatorial Games under Auction Play
Games and Economic Behavior 27(2) (May, 1999) 229-264.

Winning Paths in N-by-Infinity Hex
Integers 7 (2007) G5

Combinatorics: Permuation Patterns

Packing layered posets into posets
July 18, 1993 memorandum (Word document with graphics, 10 pages)

M. H. Albert, M. D. Atkinson, C. C. Handley, D. A. Holton, WRS
On Packing Densities of Permutations
Electronic Journal of Combinatorics 9(1) (2002) R5 (link is to pdf at EJC).

Densities at Altitude (ppt)
Presentation at "Graph Theory With Altitude" conference, 2005.

Alexander Burstein, Matthieu Josuat-Vergès, and WRS
New Dumont permutations, Pure Mathematics and Applications 21 (2010) 177-206.
This paper incorporates an earlier version by Alexander Burstein and WRS,
Dumont permutations of the third kind, presented by the first author at FPSAC 2007.

Cathleen Battiste Presutti and WRS
Packing rates of measures and a conjecture for the packing density of 2413
In Permutation Patterns (2010), S. Linton, N. Ruškuc, and V. Vatter, Eds., vol. 376 of
London Math. Soc. Lecture Note Series, Cambridge University Press, pp. 3-40.
(First link is to PowerPoint presentation from Permutation Patterns 2007 conference,
St. Andrews, Scotland, June, 2007.)

Fair Division

How to Cut a Cake Fairly
American Mathematical Monthly Note, 87 (October, 1980) 640-644. Also addendum, October 1981 (JSTOR links).

WRS and D. R. Woodall
Sets on which Several Measures Agree
J. Math. Analysis and Applications 108 (1985) 241-248.

A pie that can't be cut fairly
Dagstuhl Seminar Proceedings 07261, Seminar on Fair Division, June, 2007.
That link is to Dagstuhl's archive. Local Link. Earlier version from seminar: Abstract, Paper, and Presentation.

Julius Barbanel, Steven J. Brams, and WRS
Cutting a pie is not a piece of cake
American Mathematical Monthly 116 (2009) 496-514.

Envy free cake divisions cannot be found by finite protocols
Electronic Journal of Combinatorics 15(1) (2008) R11.
Earlier versions: Dagstuhl Seminar Proceedings 07261 and conference version (June, 2007).

Geometry and Topology

The Maximum Distance Between Two-Dimensional Banach Spaces
Mathematica Scandinavica 48 (1981) 205-225.

Packing Unit Squares Inside Squares - I,  - II,   - III.
Unpublished memoranda 1984-85. (Version published in EJC, 2003. The original memoranda support results in EJC's DS7, a dynamic survey by E. Friedman.)

Inscribed Squares in Closed Curves
Mathematika 36 (1989) 187-197.

Jonathan Ashley, Branko Grünbaum, G. C. Shephard, and WRS
Self-duality groups and ranks of selfdualities
Applied geometry and discrete mathematics (the Victor Klee Festschrift)
AMS: DIMACS series on Discrete Mathematics and Theoretical Computer Science 4 (1991) 11-50.

Classification of Self-Dual Tilings
preprint, Special Session on Tilings, AMS Philadelphia meeting, 1991

Packing 10 or 11 Unit Squares in a Square
Electronic J. of Combinatorics 10(1) (2003) R8.

Applications: Tax Analysis

Federal Income Tax Treatment of Married and Single Taxpayers
Special Report, 8 Tax Notes 731 (June 11, 1979). Reprinted in 25 Years of Tax Notes, Tax Analysts, 1997 (local pdf link).

Roy A. Wyscarver with contibutors Ralph B. Bristol, Brenda G. James, WRS, Ben Okner
The Treasury Personal Individual Income Tax Model (local pdf link)
Published as OTA Paper 32 by the Office of Tax Analysis, Treasury Department, July, 1978.

Treasury Department (WRS among acknowledged contributors)
Blueprints for Basic Tax Reform
Treasury Department, January 17, 1977 (Treasury link).

Applications: Search and Tracking

WRS and L. D. Stone
Constrained Optimization of Functionals with Search Theory Applications
Math. of Operations Research 6 (1981) 518-529.

B. Belkin and WRS
The Need for Multiple-Hypothesis Data Correlation Methods in Multi-Target Tracking
Proceedings of the 8th MIT/ONR Workshop on C3 Systems, 1985.

Practitioner's Commentary: The Drug-Running Papers
UMAP Journal, December, 1988. Reprinted in "MCM: The First Ten Years," UMAP Journal Special Edition, COMAP, 1994.

C. A. Mauro and WRS
Probabilistic Treatment of Discrete Attributes in Correlation
Proc. Tri-Service Combat Identification Systems Conference (CISC-89), 1989.

G. Schweiter and WRS
The Effect of Sensor Quality on Tracker/Correlator Performance
Proc. Tri-Service Data Fusion Symposium, 1990.

Applications: Risk Management

Roots of Transition Matrices
Memorandum for Larry Forest, April 24, 1996.
This is the paper cited by Bluhm, Overbeck, and Wagner in An Introduction to Credit Risk Modeling (Chapman & Hall) and some web publications.

Uses and Limitations of Risk Analysis - Web (html) or Word (.doc)
Proceedings of the Risk Analysis Seminar, Royal Commission on the Ocean Ranger Marine Disaster, 1984.

Financial minicourse notes
From minicourse presented most recently at 2005 national meetings and 2005 Mathfest.

Other Topics

T. Andreescu, WRS, and Z. Sunik
Bandwidth Reduction Number of Square Grids
Algebra and Discrete Mathematics 2 (2007) 1-15. Also earlier version:, 26 May 2003.

Computing the Best Isotonic Approximation to a Function
Eugene Lukacs Symposium on Order-Restricted Inference, Bowling Green University, 1992.

R. C. Marshall, M. C. Meurer, J. F. Richard, and WRS
Numerical Analysis of Asymmetric First Price Auctions
Games and Economic Behavior 7 (1994) 193-220.

Modeling response to proposed OCS leasing schedules
Abstract of presentation, AAPG Bulletin, American Assoc. of Petroleum Geologists 71(5) (May, 1987) 619.