**JACOB T. SCHWARTZ**

**Education**: Born:
January 9,1930, New York City

B.S.:
C.C.N.Y., 1948

Ph.D.:
Yale University, 1951

**Academic
Employment**:

Instructor: Yale
University, 1951-1953

Assistant
Professor: Yale
University, 1953 -1956

Associate
Professor: New
York University, 1957 -1958

Professor: New
York University, 1958 - present

**Currently**: Professor, Mathematics and Computer Science,
Courant Institute,

New
York University

**Administrative
Service**:

Former
Director, NYU Multimedia Center, 1995-96

Chairman,
Computer Science Department, Courant Institute of Mathematical Sciences, New
York University, 1969-1977

Former
Director, DARPA/ISTO, Arlington VA, 1987-89.

Former
Chairman, Computer Science Board, National Research Council

Technical
Program Chairman, AFIPS Spring Joint Computer Conference, 1972

**Honors**: Sloane
Fellow, 1961-1962

Wilbur
Cross Medal, Yale University

Townsend
Harris Medal, City University of New York

Steele
Prize, American Mathematical Society, 1981

(shared
with N. Dunford)

Mayor's
Medal for Contributions to Science and Technology, New
York City,1986

Member,
National Academy of Sciences

Member,
American Academy of Arts and Sciences

Corporate
Fellow, Thinking Machines Corp.

** **

** **

**Recent Grants: **Graduate Traineeeships
Grant, National Science Foundation, 1994-present. $750,000 over 5 years.

** **

Center for Advanced Technology Grant, New York State
Science and Technology Foundation, , 1994-present. $5,000,000 over 5 years. I was the lead developer of this major
grant, and, as initial Director of the Center for Advanced Technology which it created,
responsible for raising $1,000,000 in matching funds annually for the first two
years of the Center's existence. This involved a wide variety of activities,
including creation of a program of industrial training Seminars, which
metamorphosed into the substantial Multimedia program now offered by NYU's School
of Continuing Education.

** **

**Current Interests and Activities. **I hope at this point to concentrate my attention** on **a limited number of larger projects: (i) Development
of multimedia educational materials. Following on a preliminary design study of
a complete set of interactive multimedia materials for a much expanded version
of the current high-school mathematics curriculum (to include matrices,
splines, algebra thru Sturm's theorem and some calculus), I have developed a
plan for realizing the whole in about 5,000 interactive screens. Since current
authoring tools are not adequate for a project of this size, this led to a
second project, (ii) development of a new authoring tool suite, incorporating systematic
notions of keyboardless programming' and some new database ideas. This is
joint work with Salvatore Paxia and Giuseppe DiMauro, actively in progress for
the last 2 years, and now well advanced. An expanded edition of the book *Higher Level Programming:
An Introduction to the Set-Theoretic Language SETL, *describing this work (probably in 2 volumes,) is under
development. (iii) Studies of visual perception. My increasingly active involvement
in this area reflects the fact that such studies can be conducted very
effectively using current multimedia tools. (iv) Legion of Science': I have
developed some innovative ideas concerning the potential use of desktop computers
for mass (national scale) scientific research activities, and have begun
working with a few New York area high schools to get this started. (v) Computational
logic. Using the new set-based multimedia tools resulting from project (ii), I
hope to complete previously initiated work on a major new proof-verifier system
growing out of the series of papers with Ferro, Cantone, Omodeo, et al listed
below.

**Editorial Board Memberships**:

Advances
in Applied Mathematics;

Journal
of Programming Languages;

Discrete
and Computational Geometry;

Computers
and Mathematics with Applications;

Psychoanalysis
and Contemporary Thought

** **

** **

**Publications**

(Chronological order)

**Books**
(Mathematics and Computer Science)

*Linear Operators *(three volumes), Interscience Publishers.

vol.
I. *General Theory, *1958.

vol.
II. *Spectral Theory in Hilbert Space, *1963.

vol.
III. *Spectral Theory in Banach Spaces, *1970.

Republished by John Wiley & Sons in three volumes
as part of the Wiley Classics

Library, New York, 1988.

*Matrices and Vectors for High-Schools and
Colleges, *McGraw-Hill Publishing
Co., New York, 1961.

*Relativity in Illustrations, *NYU Press, 1962.

*Lectures on the Mathematical Method in Analytical
Economics, *Gordon and Breach
Publishers, New York, 1962.

*Theory of Money, *Gordon and Breach Publishers, New York, 1965.

*W-* Algebras*, Gordon and Breach Publishers, New York, 1967.

*Nonlinear Functional Analysis, *Gordon and Breach Publishers, New York, 1968.

*Differential Geometry and Topology, *Gordon and Breach Publishers, New York, 1969.

*Programming Languages and Their Compilers *(with John Cocke),

Courant Institute Lecture Notes, 1968-69.

*On Programming: An Interim Report on the SETL
Project.*

* Installment
I: Generalities. *New York
University, 1973.* *

* Installment
II: The SETL Language and Examples of its Use. *

* *Courant Institute Lecture Notes, November 1973.

* *

* *

*Higher Level Programming: An Introduction to the
Set-Theoretic Language SETL *(with E.
Schonberg, E. Dubinsky, and R. Dewar), Springer-Verlag Publishers, 1986.

*Discrete Thoughts: Essays on Mathematics, Science,
and Philosophy *(anthology of
previous papers; with M. Kac and G. C. Rota), Birkhauser Publishers, Boston MA,
1986.

*Planning, Geometry, and Complexity of Robot
Motion,* (anthology of previous
papers; with

M.
Sharir and J. Hopcroft), Ablex Publishing Corp., Norwood NJ, 1987.

*Advances in Robotics, vol. 1, *J. T. Schwartz, C. K. Yap, editors. Lawrence Erlbaum
Associates, Hillsdale NJ, 1987.

**Multimedia Electronic Publications**

*Animated Relativity*, (with Diana Schwartz; Multimedia version of *Relativity
in Illustrations*). The Multimedia
Library, New York 1992.

*The Electronic Dictionary of Musical Themes*, (with Diana Schwartz; Multimedia version of *The
Dictionary of Musical Themes* by H.
Barlow and S. Morgenstern, Crown Publishers, New York). The Multimedia Library,
New York 1992.

**Papers**, **Computer Science**

*Large Parallel Computers, *J.A.C.M., vol. 13, pp. 25-32, 1966.

*Sharer: A Time Sharing for the CDC 6600 *(with M. Harrison), C.A.C.M., vol. 10, pp. 659-665,
1967.

*Programming Considerations for Parallel Computers *(with E. Draughon, R. Grishman, and A. Stein), NYU
Technical Report, 1967 (Became basis for NYU Ultracomputer' project.)

*An Overview of Bugs, *Courant Computer Science Symposium 1, pp. 1-16,
Prentice-Hall Publishers, 1970.

*Semantic Definition Methods and The Evolution of
Programming Languages, *Courant
Computer Symposium 2, pp. 1- 23, Prentice-Hall Publishers, 1971.

*Principles of Specification Language Design, with
some Observations Concerning the Utility of Specification Languages, *Courant Computer Symposium 4, pp. 1-37, Prentice-Hall
Publishers, 1971.

*Abstract and Concrete Problems in the Theory of
Files, *Courant Computer Symposium 6,
pp. 1- 22, Prentice-Hall
Publishers, 1971.

*Semantic and Syntactic Issues in Programming, *American Mathematical Society Bulletin, vol. 88, pp.
195-206, 1974.

*Automatic and Semiautomatic Optimization of SETL, *Proc. of 1974 Symposium on Very High Level Languages,
SIGPLAN Notices, vol. 9, no. 4, pp. 43-49, 1974.

*An Introduction to the SETL Programming Language *(with K. Kennedy), Computing and Mathematics with
Applications, vol. 1, pp. 97-119,
1975.

* *

*Optimization of Very High Languages: Part I. Value
Transmission and its Corollaries, *Journal
of Computer Languages, vol. 1, pp. 161-194, 1975.

*Optimization of Very High Level Languages: Part
II. Deducing Relationships of Inclusion and Membership, *Journal of Computer Languages, vol. 1, pp. 197-218,
1976.

*Automatic Data Structure Choice in a Language of
Very High Level, *C.A.C.M.,
pp.722-728, 1975.

*Program Genesis and the Design of Programming
Languages. *In: *Current Trends in Programming
Methodology*, Raymond T. Yeh, ed.,
vol. IV, pp. 185-215, 1976.

*What Programmers Should Know, *Journal of Programming Languages, vol. 2, pp. 21-25,
1976.

*A Fast Switch *(with S. Coppage), American Math. Monthly, vol. 83, pp. 711-718, 1976.

* *

*A History of Computers *(review article), Enciclopedia Italiana, 1976.

*Expression Continuity and the Formal
Differentiation of Algorithms *(with
Robert Paige), Proceedings of the SIGPLAN/SIGACT Symposium on Principles of
Programming Languages, pp. 58-71, 1977.

*Correct-Program Technology Extensibility of
Verifiers, *(with M. Davis: Appendix
by E. Deak), NYU Courant Computer Science Report No. 12, 1977.

*Metamathematical Extensibility for Theorem
Verifiers *(with M. Davis), Courant
Computer Science Report 12B, 1977. (Republished in Journal of Computer
Mathematics, vol. 5, pp. 217-230, 1979.)

*SETL as a Tool for Generation of Quality Software *(with R. Dewar, A. Grand, S.C. Liu, and E.
Schonberg), Proc. of IFIP Working Conf. on Constructing Quality Software,
Novosibirsk, U.S.S.R., May 1977. (Proceedings republished by North-Holland
Publishing Co., December 1978).

*A Note on Monte Carlo Primality Tests and
Algorithmic Information Theory *(with
G. Chaitin), Comm. Pure and Applied Mathematics, vol. 31, pp. 521-527, 1978.

*Realizing the Potential of Microstructures:
Connection Patterns, Algorithms, Languages, and Limitations, *Proceedings of NSF Workshop on Opportunities in
Microelectronics, November, 1978 (Revived the NYU Ultracomputer' project).

*A Design for Optimizations of the Bitvectoring
Class *(with M. Sharir), NYU Courant
Computer Science Report No. 17, 1979. (Subsequently incorporated in the SETL
optimizer implementation).

*Programming by Refinement, as Exemplified by the
SETL Representation Sublanguage *(with
R. Dewar, A. Grand, S.D. Liu, and E. Schonberg), ACM TOPLAS, vol. 1, pp. 27-49,
1979.

*Automatic Data Structure Selection in SETL *(with M. Sharir), Proc. ACM Conf. on Principles of
Programming Languages, 1979.

*Fast Probabilistic Algorithms for Verification of
Polynomial Identities, *J.A.C.M.,
vol. 27, pp. 701-717, 1980.

*Decision Procedures for Elementary Sublanguages of
Set Theory*

Part I: *Multi-Level Syllogistic and its Extensions
*(with A. Ferro and E. Omodeo), Comm.
Pure and Appl. Math., vol. 33, pp. 599-608, 1980.

Part II: *A Decidable Class of Quantified Formulae *(with A. Ferro, M. Breban and E. Omodeo), Comm. Pure
and Appl. Math., vol. 34, pp. 177-195,1981.

Part V: *Multilevel Syllogistic Extended by the
General Union Operator, *(with D.
Cantone and A. Ferro), J. Comp. System Sciences, vol. 34, pp. 1-18, 1987.

Part VI: *Multilevel Syllogistic Extended by the
Powerset Operator, *(with D. Cantone
and A. Ferro), Comm. Pure Appl. Math., vol. 38, pp. 549-571, 1985.

*Ultracomputers. *ACM TOPLAS, vol. 2, pp.484-521, 1980.

*An
Automatic Technique for Selection of Data Representations in SETL
Programs *(with E. Schonberg and M.
Sharir), ACM TOPLAS, vol. 3, pp. 126-143,1981.

*Finding the Minimum Distance Between Two Convex
Polygons, *Info. Processing Letters,
vol. 15, pp. 168-170, 1981.

*On the Piano Movers' Problem: I. The special Case
of a Rigid Polygonal Body Moving Amidst Polygonal Barriers *(with M. Sharir), NYU Technical Report No. 39,
September, 1981. Also in Comm.
Pure Appl. Math., vol. 36, pp. 345-398, 1983. Also in *Office of Naval Research: Forty Years of Excellence in Support of
Naval Science; *40th anniversary,
1946-1986, ONR, Dept. of the Navy,
pp. 404-457.

*Networks and Algorithms for Large-Scale Parallel
Computation *(with A. Gottlieb), IEEE Computer Magazine, January,
1982, pp. 27-36.

*Artificial Intelligence, *appears as Chapter 17, pp. 747-753, ³Prospects for New Technologies², in *Outlook
for Science and Technology: The Next Five Years, *a report of

the National Research Council, National Academy of
Sciences, W.H. Freeman and Co., Publishers, 1982.

*On the Piano Movers' Problem: II. General Techniques
for Computing Topological Properties of Real Algebraic Manifolds *(with M. Sharir), Adv. Appl. Math., vol. 4,

pp. 298-351, 1983.

*Experience with the SETL Optimizer *(with S. Freudenberger and M. Sharir), ACM TOPLAS,
vol. 5, pp. 26-45, 1983.

*Languages and System for Computer-Aided
Instruction, *Machine-Mediated
Learning, vol. 1, No. 1, 1983.

*On the Piano Movers' Problem: III. Coordinating
the Motion of Several Independent Bodies. The Special Case of Circular Bodies
Moving Amidst Polygonal Barriers *(with
M. Sharir), Robotics Research vol. 2, pp. 46-75, 1983.

*The Next Generation of Robots *(with D. Grossman), in *Frontiers in Science and
Technology: A Report by the Committee on Science, Engineering, and Public
Policy of the National Academy of Sciences, *pp. 185-209*, *W.H. Freeman
and Co., 1983.

* *

*Mathematical Problems and Training in Robotics, *(with M. Sharir), Notices Amer. Math. Soc., vol. 30,
pp. 478-481, 1983.

*Software Prototyping Using the SETL Programming
Language *(with P. Kruchten and
E.Schonberg), IEEE Software, vol. 1, no. 4, pp. 66-75, 1984.

*On the Piano Movers' Problem: IV. The Case of a
Rod Moving in Three-Dimensional Space Amidst Polyhedral Obstacles *(with M. Sharir), Comm. Pure Appl. Math., vol. 37,

pp. 815-848, 1984.

*On the Complexity of Motion Planning for Multiple
Independent Objects; PSPACE hardness of the ³Warehouseman's Problem² *(with J.E. Hopcroft, and M. Sharir), Intl. J.
Robotics Research, vol. 3, pp. 76-88, 1984.

* *

*Large-scale Parallel Computers, *in: *Science, Computers, and the Information
Onslaught*, pp. 129-139, Academic
press, New York City, 1984.

*Efficient Detection of Intersections among Spheres
*(with J. Hopcroft and M. Sharir),
NYU Technical Report S9, Robotics Report 7. Also in Intl. J. of Robotics Research, vol. 2, No. 4, pp.
77-80, 1985.

*Preliminary Implementation of a Ratio Image Depth
Sensor *(with C. M. Bastuscheck), NYU
Technical Report No. 124, June, 1984.
Also in *Techniques for 3-D Machine Perception, *ed. by A. Rosenfeld, Elsevier Science Publishers,
1986.

*Decision Procedures for some Fragments of Set
Theory* (with A. Ferro and E.
Omodeo), Proceedings of the 5th Conference on Automated Deduction, Lecture
Notes on Computer Science, vol. 87, pp. 88-96, 1986, Springer Verlag, New York.

*Two Dimensional Model-Based Boundary Matching
Using Footprints *(with A. Kalvin, E.
Schonberg, and M. Sharir), NYU Technical Report No. 162, May, 1985. Also in
Intl. J. Robotics Research, vol. 5, pp. 38-55, 1986.

*Identification of Partially Obscured Objects in
Two Dimensions by Matching of Noisy Characteristic Curves', *(with M. Sharir), NYU Technical Report No. 165, June,
1985.

Also in Intl. J. Robotics Research, vol. 6, pp.
29-44, 1987.

*Object Recognition by 3-Dimensional Curve
Matching, *(with M. Sharir, E.
Schonberg, and C. M. Bastuscheck), NYU Technical Report No. 203, January,
1986), also in Intl. J. of Intell. Systems, vol. 1, pp. 105-132, 1986.

*Mathematical Problems in Robotics, *in *Frontiers of Supercomputing,* Edited by N. Metropolis, D.H. Sharp, W.J. Worlton,
and K.R. Ames, pp. 263-282, University of California Press, 1986.

*On the Existence and Synthesis of Multifinger
Positive Grips *(with B. Mishra and
M. Sharir). NYU Technical Report No. 259, 1986. Also in *Algorithmica,* vol. 2, pp. 541-558, 1987.

*Problems and Perspectives in Robotics, *in Mathematics and Computer Science, Proceedings of
the CWI Symposium of November 1983, (ed. J. W. De Bakker, M. Hazewinkel, and J.
K. Lenstra), pp. 66-87, Amsterdam, Elsevier Science Publishing Company, 1986.

*Improved Shape-Signature and Matching Methods for
Model-Based Robotic Vision *(with H.
Wolfson), in Proc. of the Workshop on Space Telerobotics, (ed. by G.
Rodriguez), vol. II, pp. 103-109, NASA, Jet Propulsion Lab, Pasadena CA, 1987.

*T-Theorem Proving* (with Alberto Policriti), Journal of Symbolic
Computation, 1996.

*Decision Algorithms for Some Fragments of Analysis
and Related Areas *(with D. Cantone,
A. Ferro, and E. Omodeo). Comm. Pure Appl. Math., vol. 40, pp. 281-300, 1987.

*On the Lower Envelope of Bivariate Functions and
Its Applications *(with M. Sharir),*
*Proc. of 28th IEEE Symp. of
Foundations of Computer Science, 1987, pp. 27-37.

*Precise Implementation of CAD Primitives Using
Rational Parametrizations of Standard Surfaces *(with S. Ocken and M. Sharir), in *Planning,
Geometry, and Complexity of Robot Motion, *(J. Hopcroft, J. Schwartz and M. Sharir, eds.), 1987.

*Identification of Partially Obscured Objects in
Two Dimensions by Matching of Noisy Characteristic Curves'*, Intl. J. Robotics Research, vol. 6, pp. 29-44,
1987.

*Dreamworld. *Daedalus Magazine, vol. 116, No. 3, pp. 165-185, 1987.

*The Limits of Artificial Intelligence, *in the *Encyclopedia of Artificial Intelligence, *pp. 488-503,* *John Wiley & Sons, 1987.

*On the Bivariate Function Minimization Problem and
Its Applications to Motion Planning, *(with
M. Sharir) in Lecture Notes in Computer Science, 267. Automata, languages, and programming. 14th Intl. Colloquium,
Karlsruhe, 1987, (ed. by T. Ottmann), pp. 357-363, Springer Verlag Publishers,
1987.

*Issues for the New Connectionism': Developing
Relationships Between Neuroscience and Artificial Intelligence. *Daedalus Magazine, vol. 117, pp. 123-142, 1988.

*Decision Procedures for Elementary Languages of
Set Theory, XI. Multilevel Syllogistic Extended by Some Elementary Map
Constructs *(with D. Cantone),
Technical report 374, New York University, 1988.

*On Recognition of 3-D Objects from 2-D Images, *(with H. Wolfson and Y. Lamdan), NYU Robotics Report
122, Technical Report 322. Also in Proc. of 1988 IEEE Intl. Conf. on Robotics
and Automation, pp. 1407-1413, IEEE Computer Society Press, Washington, D.C.

*Control of Multifinger Manipulation: Theoretical
and Experimental Studies Using a Planar Manipulator *(with J. Demmel, G. Lafferriere, and M. Sharir),
Robotics Report 126, Technical Report 329. Also in Proc. of 1988 IEEE Intl.
Conf. on Robotics and Automation, IEEE Computer Society Press, Washington DC,
pp. 390-395.

*Finding Effective Force-Targets' for
Two-Dimensional Multifinger Frictional Grips, (*with M. Sharir). NYU Computer Science Technical
Report 379, Robotics Report 156, June, 1988.

Also in
Proc. 25th Allerton Conf. on Communication, Control, and Computing, Sept. 1987.

*A Survey of Motion Planning* *and Related Geometric Algorithms *(with M. Sharir),* *Artificial Intelligence, vol. 37, pp. 157-169,
December, 1988; see also *Motion Planning* *and Related Geometric Algorithms in Robotics *(with M. Sharir),* *Proceedings of the International Congress of
Mathematicians, Berkeley 1986, pp. 1594-1161.

*The Practical and Not-Yet-Practical in Software
Engineering,* in: *Computing Tools
for Scientific problem Solving,* pp.
23-38, Academic Press, New York, 1990.

*Affine Invariant Model-Based Object Recognition,* (with Y. Lamdan and H. J. Wolfson), IEEE
Transactions on Robotics and Automation, vol. 6, pp. 578-589, 1990.

*Decidable Languages of Set Theory: MLS with
Monotone Functions*. (with Domenico
Cantone and Calogero Zarba) PPDP'99 Workshop on Declarative Programming with
Sets, Paris, France. September 28, 1999

** **

**Papers, Mathematics**

*A Note on the Space Lp*, *Proc. Amer. Math. Soc., vol. 2, No. 2,1951.

*Perturbations of Spectral Operations and
Applications, I. Bounded Perturbations. *

Pacific J. Math., vol. 4, pp. 270-275, 1954.

*The Formula for Change of Variables in a Multiple
lntegral, *Amer. Math. Monthly, vol.
61, pp. 415-488, 1954.

*De Rham's Theorem for Arbitrary Spaces, *Amer. J. of Math., vol. 77, pp. 29-44, 1955.

*Two Perturbation Formulas, *Comm. on Pure and Applied Math., vol. 8, pp. 371-376,
1955.

*Weak Compactness and Vector Measures, *(with N. Dunford and R. Bartle), Canadian J. Math.,
vol. 7, pp. 289-305, 1955.

*Convergence Almost Everywhere of Operator
Averages, *(with N. Dunford), J. of Rat. Mech., vol. 5, pp. 129-178,
1956.

* *

*On Mautner's Eigenfunction Expansion, *(with W.G. Bade), Proc. Natl. Acad. Sci., vol. 42,
No. 8, pp.519-525, 1956.

*Riemann's Method in the Theory of Special
Functions, *Bull. Amer. Math. Soc.,
vol. 62,

pp. 531-540, 1956.

*A Vector-Valued Random Ergodic Theorem *(with Anatole Beck), Proc. Amer. Math. Soc., vol 8,
pp. 1049-1059, 1958.

*Another Proof of E. Hopf's Ergodic Lemma, *Comm. Pure and Applied Math., Vol 12,

pp.
399-401, 1959.

*On Nash's Implicit Function Theorem, *Comm. Pure and Applied Math., Vol. 13, pp. 509-530,
1960.

*Some Non-Selfadjoint Operators, I. *Comm. Pure and Applied Math., Vol. 13, pp. 609-641, 1960.

*Free Quantized Lorentzian Fields, *Journal of Mathematical Physics, Vol. 2, pp. 271-290,
1961.

*Some Non-Selfadjoint Operators, II. A Class of
Operators Amenable to Friedrichs Method, *Comm. Pure and Applied Math., Vol. 14, pp. 619-626, 1961.

*Compact Positive Mappings in Lebesgue Spaces, *Comm. Pure and Applied Math., Vol. 14, pp. 693-705,
1961.

*A Remark on Inequalities of Calderon-Zygmund Type for Vector-Valued
Functions, *Ibid., Vol. 14, pp.
785-799, 1961.

*The Pernicious Influence of Mathematics on
Science,* in: Proc. of 1960
International Congress on Logic, Philosophy, and the Methodology of Science,
Stanford University Press, pp. 356-360, 1961.

*On Spectral Operators in Hilbert Space With
Compact Imaginary Part, *Comm. Pure
and Applied Math., Vol. 15, pp. 95-97, 1962.

*Some Results on the Spectra and Spectral
Resolution of Singular Integral Operators, *Comm. Pure and Applied Math., Vol. 15, pp. 159-172, 1962.

*Subdiagonalisation of Operators in Hilbert Space
with Compact Imaginary Part, *Comm.
Pure and Applied Math., Vol. 15, pp. 75-90, 1962.

*Two Finite, non-Hyperfinite, Non-isomorphic Factors, *Comm. Pure and Applied Math., Vol. 16, pp. 19-26, 1963.

*Non-isomorphic Factors of Type III, *Comm. Pure and Applied Math., Vol. 16, pp. 111-120,
1963.

* *

*Type II Factors in a Direct Integral Composition, *Comm. Pure and Applied Math., Vol. 16, pp. 247-252,
1963.

*Compact Nonlinear Mappings of Complex B-spaces and
a Lemma of Jane Cronin, *Comm. Pure
and Applied Math., Vol. 16, pp.
253-260, 1963.

* *

*Generators of the Convolution Algebra L1, *(with H. S. Shapiro and D. J. Newman), Trans. Amer.
Math. Soc., vol. 107, pp. 466-484, 1963.

* *

*Generalizing the Lusternik-Schnirelman Theory of
Critical Points, *Comm. Pure and
Applied Math., Vol. 17, pp. 307-315, 1964.

*On the Norm of a Finite Boolean Algebra of
Projections, with Applications to a Theorem of Kreiss and Morton *(with C. McCarthy), Comm. Pure and Applied
Math., Vol. 18,

pp. 191-201, 1965.

*A Remark on Morse Theory, *Comm. Pure and Applied Math., Vol. 19, pp. 139-143,
1966.

*Intersection-Theoretic Principles for the
Existence of Fixed Points and Critical Points, *Proceedings of the Georgetown Symposium on
Differential Equations, 1966.

*Representations of the Algebra of All Operators in
Hilbert Space, and Related Analytic Function Algebras, *Comm. Pure and Applied Math., Vol. 20, pp. 457-492,
1967.

*Recent Progress in the Structure Theory of Factors*, in: *Functional Analysis, Proceedings of a
Symposium,* pp. 37-52, Academic
Press, New York, 1970.

**Papers, General Interest**

*America's Economic-Technical Agenda for the 1990's*, Daedalus Magazine, vol. 131, No. 1, pp. 139-165,
1992.

*Mathematics as a Tool for Economic Understanding*, *in Mathematics Today*, Lynn Steen (ed), Springer Verlag 1978.