Lapidus, Michel L. (Michel Laurent) 1956
Overview
Works:  57 works in 238 publications in 2 languages and 4,690 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Author, Editor 
Classifications:  QA614.86, 514.742 
Publication Timeline
.
Most widely held works by
Michel L Lapidus
Fractal geometry, complex dimensions and zeta functions : geometry and spectra of fractal strings by
Michel L Lapidus(
Book
)
32 editions published between 2006 and 2014 in English and held by 350 WorldCat member libraries worldwide
Number theory, spectral geometry, and fractal geometry are interlinked in this indepth study of the vibrations of fractal strings, that is, onedimensional drums with fractal boundary. Key Features The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings Complex dimensions of a fractal string, defined as the poles of an associated zeta function, are studied in detail, then used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal Examples of such explicit formulas include a Prime Orbit Theorem with error term for selfsimilar flows, and a geometric tube formula The method of Diophantine approximation is used to study selfsimilar strings and flows Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic and other zeta functions Throughout new results are examined. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts, and discusses several open problems and extensions. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics. From Reviews of Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions, by Michel Lapidus and Machiel van Frankenhuysen, Birkhäuser Boston Inc., 2000. "This highly original selfcontained book will appeal to geometers, fractalists, mathematical physicists and number theorists, as well as to graduate students in these fields and others interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style." Mathematical Reviews "It is the reviewer's opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is selfcontained, intelligent and well paced." Bulletin of the London Mathematical Society
32 editions published between 2006 and 2014 in English and held by 350 WorldCat member libraries worldwide
Number theory, spectral geometry, and fractal geometry are interlinked in this indepth study of the vibrations of fractal strings, that is, onedimensional drums with fractal boundary. Key Features The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings Complex dimensions of a fractal string, defined as the poles of an associated zeta function, are studied in detail, then used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal Examples of such explicit formulas include a Prime Orbit Theorem with error term for selfsimilar flows, and a geometric tube formula The method of Diophantine approximation is used to study selfsimilar strings and flows Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic and other zeta functions Throughout new results are examined. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts, and discusses several open problems and extensions. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics. From Reviews of Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions, by Michel Lapidus and Machiel van Frankenhuysen, Birkhäuser Boston Inc., 2000. "This highly original selfcontained book will appeal to geometers, fractalists, mathematical physicists and number theorists, as well as to graduate students in these fields and others interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style." Mathematical Reviews "It is the reviewer's opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is selfcontained, intelligent and well paced." Bulletin of the London Mathematical Society
The Feynman integral and Feynman's operational calculus by
Gerald W Johnson(
Book
)
26 editions published between 2000 and 2010 in English and held by 347 WorldCat member libraries worldwide
This book provides the most comprehensive mathematical treatment to date of the Feynman path integral and Feynman's operational calculus. It is accessible to mathematicians, mathematical physicists and theoretical physicists. Including new results and much material previously only available in the research literature, this book discusses both the mathematics and physics background that motivate the study of the Feynman path integral and Feynman's operational calculus, and also provides more detailed proofs of the central results
26 editions published between 2000 and 2010 in English and held by 347 WorldCat member libraries worldwide
This book provides the most comprehensive mathematical treatment to date of the Feynman path integral and Feynman's operational calculus. It is accessible to mathematicians, mathematical physicists and theoretical physicists. Including new results and much material previously only available in the research literature, this book discusses both the mathematics and physics background that motivate the study of the Feynman path integral and Feynman's operational calculus, and also provides more detailed proofs of the central results
In search of the Riemann zeros : strings, fractal membranes and noncommutative spacetimes by
Michel L Lapidus(
Book
)
10 editions published in 2008 in English and held by 277 WorldCat member libraries worldwide
Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possibleor, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line
10 editions published in 2008 in English and held by 277 WorldCat member libraries worldwide
Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possibleor, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line
Fractal geometry and applications : a jubilee of Benoît Mandelbrot(
Book
)
12 editions published in 2004 in English and held by 257 WorldCat member libraries worldwide
12 editions published in 2004 in English and held by 257 WorldCat member libraries worldwide
Generalized Dyson series, generalized Feynman diagrams, the Feynman integral, and Feynman's operational calculus by
Gerald W Johnson(
Book
)
14 editions published between 1985 and 1986 in English and held by 232 WorldCat member libraries worldwide
14 editions published between 1985 and 1986 in English and held by 232 WorldCat member libraries worldwide
Generalized Minkowski content, spectrum of fractal drums, fractal strings, and the Riemannzetafunction by
Christina Q He(
Book
)
11 editions published in 1997 in English and held by 231 WorldCat member libraries worldwide
In this work, we will extend most of these results by using, in particular, the notion of generalized Minkowski content which is defined through some suitable 'gauge functions' other than the power functions. (This content is used to measure the irregularity (or 'fractality') of the boundary $\Gamma=\partial\Omega).$ In the situation when the power function is not the natural 'gauge function', this will enable us to obtain more precise estimates, with a broader potential range of applications than in the above papers
11 editions published in 1997 in English and held by 231 WorldCat member libraries worldwide
In this work, we will extend most of these results by using, in particular, the notion of generalized Minkowski content which is defined through some suitable 'gauge functions' other than the power functions. (This content is used to measure the irregularity (or 'fractality') of the boundary $\Gamma=\partial\Omega).$ In the situation when the power function is not the natural 'gauge function', this will enable us to obtain more precise estimates, with a broader potential range of applications than in the above papers
Harmonic analysis and nonlinear differential equations : a volume in honor of Victor L. Shapiro : November 35, 1995, University
of California, Riverside by Conference on Analysis and Partial Differential Equations(
Book
)
16 editions published in 1997 in English and held by 229 WorldCat member libraries worldwide
This volume is a collection of papers dealing with harmonic analysis and nonlinear differential equations and stems from a conference on these two areas and their interface held in November 1995 at the University of California, Riverside, in honor of V.L. Shapiro. There are four papers dealing directly with the use of harmonic analysis techniques to solve challenging problems in nonlinear partial differential equations. There are also several survey articles on recent developments in multiple trigonometric series, dyadic harmonic analysis, special functions, analysis on fractals, and shock waves, as well as papers with new results in nonlinear differential equations. These survey articles, along with several of the research articles, cover a wide variety of applications such as turbulence, general relativity and black holes, neural networks, and diffusion and wave propagation in porous media. A number of the papers contain open problems in their respective areas
16 editions published in 1997 in English and held by 229 WorldCat member libraries worldwide
This volume is a collection of papers dealing with harmonic analysis and nonlinear differential equations and stems from a conference on these two areas and their interface held in November 1995 at the University of California, Riverside, in honor of V.L. Shapiro. There are four papers dealing directly with the use of harmonic analysis techniques to solve challenging problems in nonlinear partial differential equations. There are also several survey articles on recent developments in multiple trigonometric series, dyadic harmonic analysis, special functions, analysis on fractals, and shock waves, as well as papers with new results in nonlinear differential equations. These survey articles, along with several of the research articles, cover a wide variety of applications such as turbulence, general relativity and black holes, neural networks, and diffusion and wave propagation in porous media. A number of the papers contain open problems in their respective areas
Fractal geometry and number theory : complex dimensions of fractal strings and zeros of zeta functions, with 26 illustrations by
Michel L Lapidus(
Book
)
3 editions published in 2000 in English and held by 226 WorldCat member libraries worldwide
3 editions published in 2000 in English and held by 226 WorldCat member libraries worldwide
Dynamical, spectral, and arithmetic zeta functions : AMS Special Session on Dynamical, Spectral, and Arithmetic Zeta Functions,
January 1516, 1999, San Antonio, Texas by Spectral and Arithmetic Zeta Functions AMS Special Session on Dynamical(
Book
)
12 editions published between 2001 and 2002 in English and held by 209 WorldCat member libraries worldwide
12 editions published between 2001 and 2002 in English and held by 209 WorldCat member libraries worldwide
Progress in inverse spectral geometry by
S. I Andersson(
Book
)
7 editions published in 1997 in English and German and held by 168 WorldCat member libraries worldwide
7 editions published in 1997 in English and German and held by 168 WorldCat member libraries worldwide
Fractal geometry and number theory : complex dimensions of fractal strings and zeros of zeta functions by
Michel L Lapidus(
Book
)
16 editions published between 1999 and 2012 in English and held by 103 WorldCat member libraries worldwide
16 editions published between 1999 and 2012 in English and held by 103 WorldCat member libraries worldwide
Fractal geometry and dynamical systems in pure and applied mathematics by fractal geometry, dynamical Systems and economics PISRS International conference on analysis(
Book
)
3 editions published in 2013 in English and held by 97 WorldCat member libraries worldwide
3 editions published in 2013 in English and held by 97 WorldCat member libraries worldwide
Fractal geometry and applications: a jubilee of Benoît Mandelbrot(
Book
)
5 editions published in 2004 in English and held by 13 WorldCat member libraries worldwide
5 editions published in 2004 in English and held by 13 WorldCat member libraries worldwide
Fractal geometry and applications: a jubilee of Benoît Mandelbrot(
Book
)
5 editions published in 2004 in English and held by 12 WorldCat member libraries worldwide
5 editions published in 2004 in English and held by 12 WorldCat member libraries worldwide
Fractal geometry and dynamical systems in pure and applied mathematics by fractal geometry, dynamical Systems and economics PISRS International conference on analysis(
Book
)
2 editions published in 2013 in English and held by 9 WorldCat member libraries worldwide
2 editions published in 2013 in English and held by 9 WorldCat member libraries worldwide
Multifractals, probability and statistical mechanics, applications(
Book
)
2 editions published in 2004 in English and held by 9 WorldCat member libraries worldwide
2 editions published in 2004 in English and held by 9 WorldCat member libraries worldwide
Fractal geometry and applications : a jubilee of Benoît Mandelbrot : analysis, number theory, and dynamical systems(
Book
)
4 editions published in 2004 in English and held by 4 WorldCat member libraries worldwide
4 editions published in 2004 in English and held by 4 WorldCat member libraries worldwide
Encyclopaedia of mathematics. Supplement III by
Shreeram Shankar Abhyankar(
Book
)
2 editions published in 2002 in English and held by 1 WorldCat member library worldwide
This is the third supplementary volume to Kluwer's highly acclaimed twelvevolume Encyclopaedia of Mathematics. This additional volume contains nearly 500 new entries written by experts and covers developments and topics not included in the previous volumes. These entries are arranged alphabetically throughout and a detailed index is included. This supplementary volume enhances the existing twelve volumes, and together, these thirteen volumes represent the most authoritative, comprehensive and uptodate Encyclopaedia of Mathematics available
2 editions published in 2002 in English and held by 1 WorldCat member library worldwide
This is the third supplementary volume to Kluwer's highly acclaimed twelvevolume Encyclopaedia of Mathematics. This additional volume contains nearly 500 new entries written by experts and covers developments and topics not included in the previous volumes. These entries are arranged alphabetically throughout and a detailed index is included. This supplementary volume enhances the existing twelve volumes, and together, these thirteen volumes represent the most authoritative, comprehensive and uptodate Encyclopaedia of Mathematics available
Fractal geometry and applications : a jubilee of Benoît Mandelbrot : multifractals, probability and statistical mechanics,
applications(
Book
)
3 editions published in 2004 in English and held by 1 WorldCat member library worldwide
3 editions published in 2004 in English and held by 1 WorldCat member library worldwide
Fractal geometry and dynamical systems in pure and applied mathematics : PISRS 2011 International Conference on Analysis,
Fractal Geometry, Dynamical Systems and Economics, November 2011: Messina, Sicily, Italy, AMS Special Session on Fractal Geometry
in Pure and Applied Mathematics: in memory of Benoit Mandelbrot, January 2012: Boston, Massachusetts, AMS Special Session
on Geometry and Analysis on Fractal Spaces, March 2012: Honolulu, Hawaii by PISRS (Conference)(
)
in English and held by 0 WorldCat member libraries worldwide
in English and held by 0 WorldCat member libraries worldwide
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Related Identities
 Van Frankenhuysen, Machiel 1967 Author Editor
 Johnson, Gerald W. 1939 Author
 Mandelbrot, Benoit B. Honoree Dedicatee
 He, Christina Q. 1970 Author
 Shapiro, Victor L. (Victor Lenard) 1924 Honoree Dedicatee
 Rumbos, Adolfo J. 1962 Editor
 Harper, Lawrence H. (Lawrence Hueston) 1938 Editor
 Andersson, S. I. (Stig Ingvar) 1945 Author Editor
 Carfi, David 1971 Editor
 Pearse, Erin P. J. 1975 Editor
Associated Subjects
Calculus, Operational Commutative algebra Differentiable dynamical systems Differential equations, Nonlinear Differential equations, Partial Differential equations, PartialNumerical solutions Ergodic theory Feynman diagrams Feynman integrals Fractals Functional analysis Functions, Zeta Geometry Geometry, Riemannian Harmonic analysis Integrals, Generalized Inverse problems (Differential equations) Mathematics Measure theory Number theory Operator theory Perturbation (Quantum dynamics) Riemannian manifolds Riemann surfaces Space and time Spectral geometry Spectral theory (Mathematics) String models