Vita & publications

2009

190 M & Michael FRAME 2009a. A primer of negative test dimensions and degrees of emptiness of latent sets. Fractals: 17 (1) 1-14.

191 M 2009. New methods of statistical economics, revisited: Short vs. long tails,
Gaussian vs. power-Law distribution. Complexity : 14 (3) 55-65.
192 Julien Barral & M. 2009. Fractional multiplicative processes. Annales de l’Institut Henri Poincaré: Probabilités et Statistiques, Nov, 2009.
2010

193 Julien Barral, Xiong Jin, & M. Construction of signed b-adic multiplicative cascades. Annals of Applied Probablitity

194 Julien Barral, Xiong Jin, & M. Convergence of signed b-adic multiplicative cascades. Annals of Applied Probablitity
195 M. Fractal sums of pulses: self-affine global mesodiffusion and lateral limit theorems.


LIMITED DISTRIBUTION ITEMS

1949 California Institute of Technology, Pasadena CA

Some Problems of Potential Theory (Aeronautical Engineering Thesis).
1953 Institute for the Unity of Science, Boston MA

The Brain as a Speaking Machine (Handout at the meeting).


NON-SUPERSEDED REPORTS AND NOTES, IBM RESEARCH CENTER, YORKTOWN HEIGHTS NY.
RC-421 Five place tables of certain stable distributions (with F. Zarnfaller). Dec 31, 1959.

NC-107 A Critical Note on Information Theory and Statistical Mechanics. Apr 16, 1962.

NC-265 On the Description of Word Frequencies by the Log-Normal Function. May 15, 1963.

RC-1344 Self-Similar Turbulence and Non-Wienerian Conditioned Spectra. Dec 1, 1964.

EXPOSITORY PUBLICATIONS, AND THE LIKE
REPRINTS FROM THE BOOKS AND UNCOMPRESSED OR ELABORATED VARIANTS
RELATED TO LOGIQUE, LANGAGE ET THÉORIE DE L’INFORMATION (1957)

• 
  Macroscopic statistical linguistics. English Translation of Part I, by J. Cheek, Jr.
  

Cambridge MA: Harvard Computation Laboratory, 1957.

RELATED TO LES OBJETS FRACTALS, TROISIÈME ÉDITION (1989)

• 
  Original: Survol du langage fractal: Book B3, 185-240.
  
• 
  Excerpt from “Survol” translated into English: Fractal geometry: what is it and what does it do? Proc. of the Royal Society of London: A423, May 8 1989, 3-16.
  
• 
  Exact reproduction of the excerpt. Fractals in the Natural Sciences, Ed M. Fleischmann, D. J. Tildesley, & Robin C. Ball, Princeton NJ: The University Press, 1990, 3-9.
  
• 
  Exact reproduction of the excerpt. Fractals: Selected Reprints, Ed Alan J. Hurd, College Park MD: American Association of Physics Teachers, 1989, 8-14.
  
• 
  Shorter excerpt: An overview of the language of fractals. The Fractal Approach to Heterogeneous Chemistry. Ed David Avnir, New York NY: Wiley, 1989, 3-9.
  
• 
  Updated versions, in the form of several essays.
  

RELATED TO THE FRACTAL GEOMETRY OF NATURE (1982)

• 
  Reprint Chapter 13, with additions. Percolation Structures and Processes. Ed Guy Deutscher, Richard Zaller, & Joan Adler. Annals of the Israel Physical Society: 5, 1983, 59–80. Distributed by Adam Hilger, Bristol, UK.
  
• 
  Large excerpts in paraphrase, Les Fractals (les chroniques de Rose Polymath) a “comic book” by Ian C. Stewart. Paris, FR: Belin. 1983.
  
• 
  Edited excerpt: Bachelier, The New Palgrave: A Dictionary of Economic Theory and Doctrine, in Three Volumes. London, UK: The Macmillan Press Ltd., 1987.
  
• 
  The World Treasury of Physics, Astronomy and Mathematics. Ed Timothy Ferris. Boston MA: Little, Brown, 1991, 447-455.
  
• 
  Russian translation of the preceding: Chemistry and Life (A journal of the Russian Academy of Sciences).
  
• 
  Revised Chapters 1, 2, 3, 4, and 11. Wolf Prizes for Physics. Singapore, SG: World Scientific, in progress.
  
• 
  Uncompressed draft of a section. Two heirs to the great chain of being. circa 1982.
  
• 
  Uncompressed draft of a section. A crisis of intuition as viewed by Felix Klein and Hans Hahn and its resolution by fractal geometry. circa 1982.
  
• 
  Excerpts read by BBM. Audio-cassette on Complexity and Chaos. Nashville TN: Knowledge Products, 1993.
  
• 
  Edited excerpt: Harold Edwin Hurst. Statisticians of the Centuries. Ed Christopher C. Heyde & Ernest Seneta. New York NY: Springer, 2001, 335-338.
  
• 
  Large excerpts in paraphrase. In Praise of Fractals, a poem by Emily Grosholz. Hudson (a magazine of literature and the arts): LVI (1), 2003, 117–118.
  
• 
  Minimally edited reproductions of Chapters 18 and 19. Fractals and Chaos: The Mandelbrot Set and Beyond. New York NY: Springer, 2004, 52-72 and 178-192.
  

RELATED TO FRACTALS AND SCALING IN FINANCE (1997)

P. Three Fractal Models in Finance: Discontinuity, Concentration, Risk.

Economic Notes. Siena, IT: 26, 1997, 139-178.

RELATED TO THE (MIS)BEHAVIOR OF MARKETS (2004)

• 
  Excerpt: A look at Market-Moving Numbers – Literally. The Wall Street Journal, Jul 27, 2004.
  
• 
  Excerpt: Do you see the pattern here? An open letter to the wizards of Wall Street.
  Wired, Aug 2004.
  
• 
  Great confusion since the Great Crash. by Richard Hudson & Benoit Mandelbrot,
  Op-ed comment. The Financial Times, Oct 22, 2004.
  
• 
  English Abstract of the whole book, with comments by R. Dobelli and his staff, getAbstract compressed knowledge.
  
• 
  For lack of a nail… Money Science Financial Intelligence Network (on the web), Oct 2004.
  
• 
  Long live the corporate lab. Science/Business, Dec 2004, 24-25.
  
• 
  Lonely pioneer of an alternative and opposite approach. Letter to the Editor. Financial Times, Aug 25, 2007, 8.
  

THEORY OF ART IN THE CONTEXT OF FRACTALS

Les fractales: objets mathématiques, modèles physiques et créations artistiques.

Actes du forum int’l des formes nouvelles (Monte Carlo, Février 1983). Textes réunis par André Martin.

• 
  Actes du premier colloque image (Biarritz, Mai 1984). Textes réunis par B. Picinbono, Cagnes-sur-Mer, FR: Gretsi, 1-10.
  

Flare: A by-product of the study of a two-dimensional dynamical system,

IEEE Transactions on Circuits and Systems: 36, 1988, 768-769.
Fractals: an art for the sake of science. Leonardo: 1989, Supplemental Issue, 21-24.

• 
  F. Version abrégée en français: Les images fractales: un art pour l'amour de la science
  et ses applications. Sciences et Techniques: Mai 1984, 16-19, 34-35 & 65.
  
• 
  German translation of the French version: Eine Kunst zur Liebe der Wissenschaft.
  Artware: Kunst und Elektronik. Ed David Galloway. Düsseldorf, DE: Econ, 1987, 69-78.
  
• 
  Reprint: The Visual Mind: Art and Mathematics. Ed Michele Emmer. Leonardo Books.
  Cambridge MA: MIT Press, 1993, 11-14 and color Plate I.
  
• 
  Portuguese translation: Fractais: uma forma de arte a bem da ciência
  

(Tradução de Claudio da Costa). Imagem-máquina; a era das tecnologias do virtual. Textes réunis par André Parente, Rio de Janeiro, BR: Editora 34, 1993, 195-200.
Natura ex Machina (with F.K. Musgrave), IEEE Computer Graphics and its Applications: Jan 1989, Front cover & 4-7.
The art of fractal landscapes (with F. K. Musgrave). IBM Journal of Research and Development:
35, 1991, front and back covers & 535-540.
Was ist Schönheit? NZZ Folio. Neuer Zürcher Zeitung: Mai 1993, 15.
F. Les fractales, l'art algorithmique et le test de Turing. La science et la métamorphose des arts. Textes réunis par Raymond Daudel. Paris, FR: Presses Universitaires de France, 1994, 39-52.

• 
  Italian translation by Michele Emmer. L’Unità: 22 Apr 1994, 2-5.
  

INFORMAL EXPOSITIONS OF OWN WORK

Phenomenological principles and statistical thermodynamics. Bibliographical and historical footnote.

Thermodynamics: A Unifying Science. Ed George N. Hatsopoulos and Joseph H. Keenan.
Cambridge MA: MIT Department of Mechanical Engineering, 1964.

• 
  Partial reprint: Principles of General Thermodynamics, Ed George N. Hatsopoulos
  and Joseph H. Keenan, New York NY: J. Wiley, 1965, XXXIX-XL.
  

P. C. Some general comments upon the so-called macro-statistical models. Mathematics and

Social Science I: Proc. of the UNESCO Seminars. Menthon-Saint-Bernard, 1960, and Gösing, 1962.
Ed Saul Sternberg & al. The Hague, NL: Mouton & Co., 1965, 213-215.
P. Is there persistence in stock price movements? Seminar on the Analysis of Security Prices,
held November 12-13, 1966 at the Graduate School of Business of the University of Chicago.

• 
  Reprint: Staff/Board Conf. of the National Bureau of Economic Research, Inc. held Sep 17-20, 1969 at Montauk, Long Island, NY.
  

Operational hydrology using self-similar processes (with James R. Wallis).

Proc. of the Fifth Int’l Operations Research Conf., Venice 1969. Ed John Lawrence.

London, UK: Tavistock, and New York NY: Barnes & Noble, 1970, 265-282.

FE3.1 Le syndrome de la variance infinie et ses rapports avec la discontinuité des prix,
Economie Appliquée: 26, 1973, 321-348.
FE3.2 Le problème de la réalité des cycles lents et le syndrome de Joseph.
Economie Appliquée: 26, 1973, 349-365.
Physical objects with fractional dimension: seacoasts, galaxy clusters, turbulence and soap.

The Institute of Mathematics and its Applications. Great Britain, Bulletin: 13, Nos. 7/8, Jul-Aug 1977, 189-196.

• 
  Reprint: Fluid Dynamics. Les Houches, 1973. Ed Roger Balian & Jean-Laurent Peube.
  London, UK: Gordon & Breach, 1977, 557-578.
  

The fractal geometry of percolation, polymers and almost everything else.

Statistical Mechanicsand Statistical Methods in Theory and Application, a Tribute to Elliott W. Montroll.

Ed Uzi Landman. New York NY: Plenum, 1977, 331-342.

The fractal geometry of 1/f fluctuations. Proc. of the Symposium on 1/f Fluctuations.
Tokyo, 1977. Ed T. Musha. Tokyo, JP: Tokyo Institute of Technology, 1977, 206-212.
Getting snowflakes into shape. New Scientist: 78, 1978, 808-810.

• 
  Letter to the Editor: The Menger Sponge. Mathematical Intelligencer: 1, Summer 1978, 126.
  

F. Des monstres de Cantor et Peano à la géométrie fractale de la nature.

Penser les mathématiques. (Séminaire Jean Dieudonné, Maurice Loi & René Thom).

Textes préparés par F. Guénara & G. Leliève.

Collection Points-Sciences, Paris, FR: Editions du Seuil, 1982, 226-251.

• 
  F. Variant available on this web site: Des monstres mathématiques (de Bolzano, Cantor et Peano) au tohu-bohu de la nature.
  
• 
  Spanish translation: De los monstruos de Cantor y Peano a la geometría fractal de la naturaleza.
  Pensar la matemática, Barcelona, ES: Tusquets, 1984, 111-138.
  
• 
  Japanese translation: Shizen no fraktal kikagaku ni okeru Cantor to Peano no monster.
  Sugaku Gengo Genjitsu 2. Sugaku Seminar Books, 11, Tokyo, JP: Nihon Hyoron Sha, 1984, 111-141.
  
• 
  Italian translation: La geometria della natura, Book p.c., 1987, 49-91; 1949, 31-78.
  

The many faces of scaling: fractals, geometry of nature, and economics.
Self-Organization and Dissipative Structures. Ed William C. Schieve & Peter M. Allen.
Austin TX: University of Texas Press, 1982, 91-109.
F. Les fractales, les monstres et la beauté. Le débat: histoire, politique et société: Mar 1983, 54-72.

• 
  Brief version: L'ordinateur aidant, les fractales domptent les monstres mathématiques et transforment l'intuition géométrique. La Jaune et la Rouge, Déc 1982, 67-78.
  
• 
  Also: Diplômés 83 de l'Ecole Polytechnique Fédérale de Lausanne. Cover and 18-28.
  
• 
  Also: Mosaic. Zürich, 5/83, 18-20 and 6/83, 18-20.
  
• 
  Reprint: L’irruption des géométries fractales dans les sciences, textes réunis par Irène Herpe-Litwin. Paris, FR: Éditions de l’AEIS.
  

Intergalactic voids and filaments, and why they necessarily occur in fractal models.
Handout at the Annual Meeting of The American Association for the Advancement of Science,
Detroit MI, May 1983.
Le géomètre qui refait le monde. ça m'intéresse: 29 Juillet 1983, 72-75.
Why is nature fractal and when should noises be scaling (with Richard F. Voss).

Proc. of the 7th Int’l Conf. on Noise in Physical Systems and the 3rd Int’l Conf. on 1/f Noise,

Montpellier, 17-20 May, 1983. Ed M. Savelli, G. Lecoy & J. P. Nougier, Amsterdam, NL:

North-Holland, 1984, 31-39.
The fractal vision (Letter to the Editor). The Sciences: 24, 3, May/June 1984.
Order in the geometric chaos of roughness and fragmentation. Introduction to b, 1-3 and to c, 3-5.

• 
  Shorter reprint: Fractals in Engineering (Abstracts) / Les fractales et l’ingénieur. Montréal, 1992.
  Ed Claude Tricot & Francois Normant, 1992.
  

Der Computer macht eine neue Geometrie möglich. 75 Jahre IBM DE, 1985, 122-127.

Fractals and the rebirth of iteration theory. The Beauty of Fractals, by Heinz-Otto Peitgen &
Peter H. Richter, New York NY: Springer, 1986, 151-160.

• 
  Preliminary version, together with a German translation:
  Catalogue of the “Frontiers of Chaos” Exhibit, Bremen, DE: Mapart, 1985, 11-19.
  

Fractals: how to imitate the mountains, the clouds and the trees, and to create weird and wonderful new shapes. Teaching Non-Linear Phenomena II: Chaos in Education. Balatonfüred, HU, 1987.

Ed George Marx, Veszprem, 1988, 5-14.

• 
  Landscapes on an imaginary planet. Bioastronomy — The Next Steps.
  Ed George Marx. Dordrecht-Boston: Kluwer, 1988, 67-76.
  
• 
  Fractal mountains and dragons, with an excursion through fractal music.
  The State of the Art and the Future Trends of Acoustic Research. Roma, Apr 1987.
  Roma, IT: Institute of Acoustics O.M. Corbino, 1989, 57-82.
  
• 
  Montañas y dragones fractales: la intuición en la matemática y en las ciencias. Sobre la imaginación científica. Barcelona, Mayo 1987. Ed Jorge Wagensberg, Barcelona, ES: Tusquets 1990, 177-204.
  
• 
  Eine Computerformel fürs Matterhorn. Chaos und Kreativität. Ed Gottlieb Guntern.
  Zürich, CH - Berlin, DE - New York NY: Scalo, 1995, 167-214.
  

Fractals: a geometry of nature. The New Scientist: Sep 15, 1990, cover & 38-43.

• 
  Reprint: The New Scientist Guide to Chaos. Ed Nina Hall. London, UK: Penguin, 122-135.
  
• 
  Reprint: Exploring Chaos: a Guide to the New Science of Disorder. Ed Nina Hall.
  New York NY: Norton, 1993, 122-135.
  

Fractal craft (Letter to the Editor). The New Scientist: Sep 14, 1991.

Fractals. Chaos: The New Science (Gustavus Adolphus Nobel Conf. XXVI).
Ed John M. Holte. Lanham MD: University Press of America, 1993, 1-27.

• 
  Shortened Japanese translation by S. Miyajima & M. Suzuki.
  Suuri Kagaku (Mathematical Sciences): 386, Aug 1995, 74-83.
  
• 
  Shortened English text, together with the above Japanese translation.
  

The Honda Prize for 1994. Tokyo, JP: Honda Foundation Report: 79, 1995, 31 pp.

• 
  Another shortened variant: Fractals as a morphology of the amorphous. Introduction to Fractal Landscapes from the Real World, by William Hirst. Manchester, UK: Cornerhouse Publications, 1994.
  
• 
  Italian translation. Frattali. Dalla tribù alla conquista dell’universo: Scienza, tecnologia e società.
  

A cura di Umberto Colombo e Giuseppe Lanzavecchia. Milano, IT: Libri Schweiwiller, 2000, 195-204.

• 
  Reprint: The Colors of Infinity, Ed Nigel Gordon. Bath, UK: Clearpress, 2004, 46-65.
  
• 
  Portuguese translation. Fractais. Frontieras da Ciência. Coordenadores Rui Fausto, Carlos Fiolhais & João Filipe Queiró. Lisboa, PT: Gradiva & Imprensa da Universidade de Coimbra, 2003, 63-89.
  

F. Les inattendus des fractales. Pour la Science: 234, Avril 1997, 10-12.
A Multifractal Walk Down Wall Street. Scientific American, Feb 1999, 50-53.

Responses to “Letters to the Editor.” Scientific American, Jun 1999, 6.

Short advance version: Of dollars, deutschmarks and multifractality.
Convergence, Supplement to the Wall Street Journal Europe of 17 Nov 1997, 10-11.

Version abrégée en français: Les fractales et la bourse, Pour la Science: 242, Déc 1997, 16-17.

Not quite so short advanced version: ASX Perspective (Australian Stock Exchange), 1998, 4-7.

Version élargie en français: Randonnées multifractales à Wall Street. Les mathématiques sociales: Dossier Pour la Science. Paris: Belin, 1999, 126-130.

Edited reprint: Multifractal structure of financial prices and its implications. Scientific bridges for 2000 and Beyond: A virtual colloquium by the Elf Professeurs de l’Académie des Sciences.

Paris, FR: Tec et Doc, Librairie Lavoisier, 1999, 131-148.

Edited reprint: Fractal financial fluctuations: do they threaten sustainability? Science for Survival and Sustainable Development (Scripta Varia 98).

Città di Vaticano: Pontificia Academia Scientiarvm, MM, 299-319.

Edited reprint: Risks in Investment Accumulation Products of Financial Institutions.
Schaumburg IL: The Society of Actuaries, 2001, 19-24.
Fractal geometry and the representation of nature. New Ideas in Science and Arts.
Strasbourg, FR: Council of Europe. 1997, 28-29.
Is nature fractal? (Letter to the Editor) Science: 279, Feb 6, 1998, 783.
F. L’anneau fractal: de l’art à l’art, à travers la géométrie, la finance et les sciences.

Le Monde (quotidien): 29 août 2000, 12.

Short English version: Book y.

Version française illustrée. Université de tous les savoirs. Sous la direction d’Yves Michaud.
Qu’est-ce que l’Univers? Paris, FR: Odile Jacob, 4, 2001, 164-174.
The fractal universe. The Origins of Creativity. Ed Karl H. Pfenninger & Valerie R. Shubik,
Oxford, UK: University Press, 2001, 185-206.

Arabic translation, in progress.

Mouvement brownien, 4/3 et le rôle de l’oeil en mathématiques. Courrier des lecteurs.
Pour la Science: 2002.
C. Symmetry by dilation/reduction, fractals, and roughness, Symmetry 2000.
Ed I. Hargittai and T. Laurent, I, 2002, 133-141.
The garden of fractal orders. Il Corno du Heimdal. Pio Manzu. 2003, pp 142-149.

Italian translation: Il giardino degli ordini frattali, ibid.

Fractals and their role in education. Japanese translation by Mitsugu Matsushita.
Sugaku Seminar, Special Issue on Complex Dynamical Systems, 42, 2003, 8-16.
Fractals. Japan Prize 2003. Tokyo: The Science and Technology Foundation of Japan. pp.
Mathematics, art, and the story of the invincible Antaeus. Fractal Art: Beauty and Mathematics
Special Catalog of an Exhibit at the Int’l Congress of Mathematicians, 2006, Madrid, ES, p 3,

Spanish translation: Las matemáticas, el arte y la historia del invencible Anteo, ibid. p 1.

What’s the Greatest Innovation? Spiked.


FOREWORDS AND PREFACES TO BOOKS, AND THE LIKE (INCLUDING BLURBS MARKED *)

Hasard et prévision, par Daniel Zajdenweber. Paris, FR: Economica, 1976.
Fractals, by Jens Feder. New York NY: Plenum, 1988.
People and events behind the science of fractal images. The Science of Fractal Images.

Ed Heinz-Otto Peitgen & Dietmar Saupe, New York NY: Springer, 1988, 1-19.

Fractal Growth Processes, by Tamas Vicsek. Singapore, SG: World Publishing, 1989.
Fractals in geophysics, by Christopher H. Scholz and BBM. Basel & Boston: Birkhäuser, 1989, 1-4.
*Norbert Wiener (1894-1964), by Pesi R. Masani. Basel & Boston: Birkhäuser, 1990.
An Eye for Fractals: a Graphic/Photographic Essay, by Michael McGuire. Reading MA: Addison Wesley, 1991.
The grey and the green. Fractal Forms. Ed Étienne Guyon & H. Eugene Stanley. Amsterdam, NL: North-Holland, 1991.

Le gris et le vert. Les formes fractales. Rédaction Étienne Guyon & H. Eugene Stanley.

Paris, FR: Palais de la Découverte, 1991.
Fractals for the Classroom: Strategic Activities, by Heinz-Otto Peitgen, Hartmut Jürgens,

& Dietmar Saupe, Evan M. Matelski, Terry Perciante, & Lee E. Yunker. New York NY: Springer, 1991.

Fractals and the rebirth of experimental mathematics. Fractals for the Classroom, by Heinz-Otto Peitgen, Hartmut Jürgens, & Dietmar Saupe, Evan M. Matelski, Terry Perciante, & Lee E. Yunker.

New York NY: Springer, 1991.

Dieu joue-t-il aux dés?, par Ian Stewart. Pari, FR: Flammarion, 1992, 7-13.
Physique et structures fractales, par Jean-François Gouyet. Paris: Masson, 1992.

Physics and Fractal Structures, by Jean-François Gouyet. New York NY: Springer, 1994.
Non-Equilibrium States in Molecular Aggregation and Fractals in Chemistry. Croatica Chemica Acta. Special Issue. 65, 1992, v.
Chaos under Control: The Art and Science of Complexity, by David Peak & Michael L. Frame. New York NY: W. H. Freeman, 1993.
Foreword to the First Issue. Fractals: an Interdisciplinary Journal on the Complex Geometry of Nature: 1, 1993, v-vi.
Abraham Robinson: The Creation of Nonstandard Analysis; a Personal and Mathematical Odyssey, by Joseph W. Dauben. Princeton University Press, 1994, ix-xi.

Chinese translation, 2005.

The Paul Lévy I knew. Lévy Flights and Related Phenomena in Physics. Nice, 1994.

Ed Michael F. Shlesinger, George Zaslawsky, & Uriel Frisch, Lecture Notes in Physics.

New York NY: Springer, 1995, ix-xii.


Fractal Geometry and its Uses in Petroleum Geology and Earth Sciences. Ed Christopher C. Barton & P. R. La Pointe. New York NY: Plenum, 1995, vii-xi.
Universalités et fractales; jeu d'enfants ou délit d'initié?, par Bernard Sapoval. Paris, FR: Flammarion, 1997.
Discovery of Cosmic Fractals, by Yurij Baryshev & Pekka Teerikorpi. Singapore, SG: World Scientific, 2002.
*An Introduction to High-Frequency Finance, by Michel M. Dacorogna, Ramazan Gençay, Ulrich A. Müller, Richard B. Olsen, Olivier V. Pictet. Academic Press, 2001.
Marchés fractals, par Jacques Lévy-Véhel & Christian Walter. Paris, FR: Presses Universitaires de France, 2002.
*Heaven’s Fractal Net, by William J. Jackson. Bloomington IN: Indiana University Press, 2004.
*The Fractal Murders, by Mark Cohen. New York, NY: Mysterious Books, 2004; Warner Books, 2005.
*Dark Hero of the Information Age: In Search of Norbert Wiener, the Father of Cybernetics, by Flo Conway & Jim Siegelman. New York NY: Basic Books, 2004.