Papers by Michel L Lapidus
The Geometry and the Spectrum of Fractal Strings
Birkhäuser Boston eBooks, 2000
Concluding Comments
Birkhäuser Boston eBooks, 2000
We study generalised prime systems P (1 < p 1 ≤ p 2 ≤ • • • , with p j ∈ R tending to infinity) a... more We study generalised prime systems P (1 < p 1 ≤ p 2 ≤ • • • , with p j ∈ R tending to infinity) and the associated Beurling zeta function . Under appropriate assumptions, we establish various analytic properties of ζ P (s), including its analytic continuation and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of ζ P (s). Further we study 'well-behaved' g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on N 2 . Some of the above results may be relevant to the second author's theory of 'fractal membranes', whose spectral partition functions are precisely given by Beurling zeta functions.
Spectral Theory for Noncommuting Operators
Oxford University Press eBooks, Aug 1, 2015
Towards a noncommutative fractal geometry? Laplacians and volume measures on fractals
Contemporary mathematics, 1997
ABSTRACT
Philosophical Transactions of the Royal Society A, Aug 6, 2015
This research expository article contains a survey of earlier work (in §2- §4) but also contains ... more This research expository article contains a survey of earlier work (in §2- §4) but also contains a main new result (in §5), which we first describe. Given c ≥ 0, the spectral operator a = ac can be thought of intuitively as the operator which sends the geometry onto the spectrum of a fractal string of dimension not exceeding c. Rigorously, it turns out to coincide with a suitable quantization of the Riemann zeta function ζ = ζ(s): a = ζ(∂), where ∂ = ∂c is the infinitesimal shift of the real line acting on the weighted Hilbert space L 2 (R, e -2ct dt). In this paper, we establish a new asymmetric criterion for the Riemann hypothesis, expressed in terms of the invertibility of the spectral operator for all values of the dimension parameter c ∈ (0, 1/2) (i.e., for all c in the left half of the critical interval (0, 1)). This corresponds (conditionally) to a mathematical (and perhaps also, physical) "phase transition" occurring in the midfractal case when c = 1/2. Both the universality and the non-universality of ζ = ζ(s) in the right (resp., left) critical strip {1/2 < Re(s) < 1} (resp., {0 < Re(s) < 1/2}) play a key role in this context. These new results are presented in §5. In §2, we briefly discuss earlier joint work on the complex dimensions of fractal strings, while in §3 and §4, we survey earlier related work of the author with H. Maier and with H. Herichi, respectively, in which were established symmetric criteria for the Riemann hypothesis, expressed respectively in terms of a family of natural inverse spectral problems for fractal strings of Minkowski dimension D ∈ (0, 1), with D = 1/2, and of the quasi-invertibility of the family of spectral operators ac (with c ∈ (0, 1), c = 1/2).
Multifractals, probability and statistical mechanics, applications
American Mathematical Society eBooks, 2004
Multifractals: Introduction to infinite products of independent random functions (Random multipli... more Multifractals: Introduction to infinite products of independent random functions (Random multiplicative multifractal measures, part I) by J. Barral and B. B. Mandelbrot Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures (Random multiplicative multifractal measures, part II) by J. Barral and B. B. Mandelbrot Techniques for the study of infinite products of independent random functions (Random multiplicative multifractal measures, part III) by J. Barral Wavelet techniques in multifractal analysis by S. P. Jaffard The 2-microlocal formalism by J. L. Vehel and S. Seuret A vectorial multifractal formalism by J. Peyriere Probability and statistical mechanics: Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries by B. M. Hambly and T. Kumagai Random fractals and Markov processes by Y. Xiao On the scaling limit of planar self-avoiding walk by G. F. Lawler, O. Schramm, and W. Werner Conformal fractal geometry & boundary quantum gravity by B. Duplantier Applications: Self-organized percolation power laws with and without fractal geometry in the etching of random solids by A. Desolneux, B. Sapoval, and A. Baldassarri Nature inspired chemical engineering-Learning from the fractal geometry of nature in sustainable chemical engineering by M.-O. Coppens Fractal forgeries of nature by F. K. Musgrave.
Analysis, number theory, and dynamical systems
American Mathematical Society eBooks, 2004
Fractal geometry and applications-An introduction to this volume by M. L. Lapidus Cherche Livre..... more Fractal geometry and applications-An introduction to this volume by M. L. Lapidus Cherche Livre... et plus si affinite/Looking for a book...and more, if affinity by J. Barral and S. Jaffard Benefiting from fractals by M. Berry Benoit Mandelbrot, wizard of science by M.-O. Coppens Mandelbrot's vision for mathematics by R. L. Devaney Benoit Mandelbrot and York by M. M. Dodson Nul n'entre ici s'il n'est geometre/Let no one ignorant of geometry enter here by B. Duplantier A decade of working with a maverick by M. L. Frame Breakfast with Mandelbrot by M. Frantz Old memories by J.-P. Kahane My encounters with Benoit Mandelbrot by D. B. Mumford Fractal geometry and the foundations of physics by L. Nottale Is randomness partially tamed by fractals? by B. Sapoval On knowing Benoit Mandelbrot by J. E. Taylor Analysis: Reflections, ripples and fractals by M. M. France Lacunarity, Minkowski content, and self-similar sets in $\mathbb{R}$ by M. Frantz Fractals and geometric measure theory: Friends and foes by F. Morgan Eigenmeasures, equidistribution, and the multiplicity of $\beta$-expansions by H. Furstenberg and Y. Katznelson Distances on topological self-similar sets by A. Kameyama Energy and laplacian on the Sierpinski gasket by A. Teplyaev Electrical networks, symplectic reductions, and application to the renormalization map of self-similar lattices by C. Sabot Notes on Bernoulli convolutions by B. Solomyak Number theory: Some connections between Bernoulli convolutions and analytic number theory by T. Hilberdink On Davenport expansions by S. Jaffard Hausdorff dimension and diophantine approximation by M. M. Dodson and S. Kristensen Fractality, self-similarity and complex dimensions by M. L. Lapidus and M. van Frankenhuijsen Dynamical systems: The invariant fractals of symplectic piecewise affine elliptic dynamics by B. Kahng Almost sure rotation number of circle endomorphisms by S. Crovisier Kneading determinants and transfer operators in higher dimensions by V. Baladi The spectrum of dimensions for Poincare recurrences for nonuniformly hyperbolic geometric constructions by V. Afraimovich, L. Ramirez, and E. Ugalde A survey of results in random iteration by M. Comerford On fibers and local connectivity of Mandelbrot and multibrot sets by D. Schleicher.
Stability Properties of Feynman’s Operational Calculi
Oxford University Press eBooks, Aug 1, 2015
The stability (or continuity) properties of Feynman’s operational calculus are considered in this... more The stability (or continuity) properties of Feynman’s operational calculus are considered in this chapter. In particular, we consider three types of stability: stability with respect to the time-ordering measures, stability with respect to the operator-valued functions, and joint stability (that is, stability with respect to the operator-valued functions and the time-ordering measures simultaneously).
Dynamical, Spectral, and Arithmetic Zeta Functions
American Mathematical Society eBooks, 2001
... geometry, and algebra: Interactions and new directions, 2001 278 Eric Todd Quinto, Leon Ehren... more ... geometry, and algebra: Interactions and new directions, 2001 278 Eric Todd Quinto, Leon Ehrenpreis, Adel Faridani, Fulton Gonzalez, and Eric Grinberg, Editors, Radon transforms and tomography, 2001 277 Luca Capogna and Loredana ... van Frank-enhuysen, Machiel, 1967 III ...
The Feynman integral, the Feynman-Kac formula with a Lebesgue-Stieltjes measure and Feynman's operational calculus
Disentangling via Continuous and Discrete Measures
Oxford University Press eBooks, Aug 1, 2015
Disentangling via Tensor Products and Ordered Supports
Oxford University Press eBooks, Aug 1, 2015
More involved types of disentanglings are considered in this chapter. We discuss the situation wh... more More involved types of disentanglings are considered in this chapter. We discuss the situation where the Banach space X can be written as a direct sum X=Y⊕Z, for some Banach spaces Y and Z, and each of the operators Ai can be decomposed as Ai=Bi⊕Ci, for i=1,…,n. A second type of disentangling discussed is that of a function which is symmetric in several of its variables. Next, consideration is given to the disentangling of certain analytic (i.e. holomorphic) functions which can be written as tensor products of analytic functions of one variable. Also discussed is the effect on the disentangling process of the use of time-ordering measures with ordered supports. The results obtained in this part of Chapter 3 play a role throughout the remainder of the book. Finally, the disentangling of so-called exponential factors is addressed; we track the occurrence of a particular operator in the time-ordered products that constitute the disentangling.
The Critical Zeros of Zeta Functions
Birkhäuser Boston eBooks, 2000
As we saw in the previous chapter, the complex dimensions of a generalized Cantor string form an ... more As we saw in the previous chapter, the complex dimensions of a generalized Cantor string form an arithmetic progression {D + inp}n∈ℤ (for D ∈ (0,1) and p > 0). In this chapter, we use this fact to study arithmetic progressions of critical zeros of zeta functions.
The Riemann Hypothesis, Inverse Spectral Problems and Oscillatory Phenomena
In this chapter, we provide a geometric reformulation of the Riemann Hypothesis in terms of a nat... more In this chapter, we provide a geometric reformulation of the Riemann Hypothesis in terms of a natural inverse spectral problem for fractal strings. After stating this inverse problem in Section 7.1, we show in Section 7.2 that its solution is equivalent to the nonexistence of critical zeros of the Riemann zeta function on a given vertical line. This was done earlier in [LapMal-2], but now we use the point of view of complex dimensions and the explicit formulas of Chapter 4. Then, in Section 7.3, we extend this characterization to a large class of zeta functions, including all the number-theoretic zeta functions for which the extended Riemann Hypothesis is expected to hold.
Fractal geometry and applications—An introduction to this volume
Proceedings of symposia in pure mathematics, 2004
Feynman's Operational Calculus and Beyond: Noncommutativity and Time-Ordering
1. Introduction 2. Disentangling: Definitions, Properties and Elementary Examples 3. Disentanglin... more 1. Introduction 2. Disentangling: Definitions, Properties and Elementary Examples 3. Disentangling via Tensor Products and Ordered Supports 4. Extraction of Multilinear Factors and Iterative Disentangling 5. Auxiliary Operations and Disentangling Algebras 6. Time-Dependent Feynman's Operational Calculus and Evolution Equations 7. Stability Properties of Feynman's Operational Calculi 8. Disentangling via Continuous and Discrete Measures 9. Derivational Derivatives and Feynman's Operational Calculi 10. Spectral Theory for Noncommuting Operators 11. Epilogue: Miscellaneous Topics and Possible Extensions
The Vibrations of Fractal Drums and Waves in Fractal Media
ABSTRACT
{0, . . . , d}, however, the tube formula for tilings is a power series which also includes one t... more {0, . . . , d}, however, the tube formula for tilings is a power series which also includes one term for each complex dimension. This further justifies the term 'complex dimensions' and takes a step toward defining curvature for a fractal. This dissertation should have applications to spectral asymptotics on domains which are fractal or have fractal boundaries. It may also lead to a robust notion of curvature (measures) for self-similar sets.
Memoirs of the American Mathematical Society, 1986
L'accès aux archives de la revue « Annales mathématiques Blaise Pascal » () implique l'accord ave... more L'accès aux archives de la revue « Annales mathématiques Blaise Pascal » () implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
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Papers by Michel L Lapidus