Michel L. Lapidus, P. J. Pearse, M. L. Lapidus, M. Van Frankenhuijsen, Nd Revised, ...
and enlarged edition to appear shortly.) “Characterizing the measurability of fractal strings ” (a primer for [FGNT]), E.P.J.P., available on my web page (listed as my Oral Examination).
Michel L. Lapidus, M. L. Lapidus, L. Lapidus, ...
[SST] “Canonical self-similar tilings by IFS”, E. P. J.
Toward zeta functions and complex dimensions of multifractals (2008)
Lapidus, Michel L., Rock, John A.
Multifractals are inhomogeneous measures (or functions) which are typically described by a full spectrum of real dimensions, as opposed to a single real dimension. Results from the study of fractal...
A formula for the interior ε-neighbourhood of the classical von Koch snowflake curve is computed in detail. This function of ε is shown to match quite closely with earlier predictions from [La-vF1]...
E Rin, P. J. Pearse, Michel L. Lapidus
Upper and lower division honours.
Tube formulas for self-similar fractals (2007)
Lapidus, Michel L., Pearse, Erin P. J.
Tube formulas (by which we mean an explicit formula for the volume of an $\epsilon$-neighbourhood of a subset of a suitable metric space) have been used in many situations to study properties of the...
The Riemann Zeta-Function and the One-Dimensional Weyl-Berry Conjecture for Fractal Drums (2006)
Lapidus, Michel L., Pomerance, Carl
Based on his earlier work on the vibrations of ‘drums with fractal boundary’, the first author has refined M. V. Berry's conjecture that extended from the ‘smooth’ to the ‘fractal’ case...
Fractal Strings and Multifractal Zeta Functions (2006)
Lapidus, Michel L., Vehel, Jacques Levy, Rock, John A.
For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first...
Dirac operators and spectral triples for some fractal sets built on curves (2006)
Christensen, Erik, Ivan, Cristina, Lapidus, Michel L.
We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each...
Localization on Snowflake Domains (2006)
Daudert, Britta, Lapidus, Michel L.
The geometric features of the square and triadic Koch snowflake drums are compared using a position entropy defined on the grid points of the discretizations (pre-fractals) of the two domains....
Ihara's zeta function for periodic graphs and its approximation in the amenable case (2006)
Guido, Daniele, Isola, Tommaso, Lapidus, Michel L.
In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we...
A trace on fractal graphs and the Ihara zeta function (2006)
Guido, Daniele, Isola, Tommaso, Lapidus, Michel L.
Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a...
Ihara zeta functions for periodic simple graphs (2006)
Guido, Daniele, Isola, Tommaso, Lapidus, Michel L.
The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated...
Tube formulas and complex dimensions of self-similar tilings (2006)
Lapidus, Michel L., Pearse, Erin P. J.
We use the self-similar tilings constructed by the second author in ``Canonical self-affine tilings by iterated function systems'' to define a generating function for the geometry of a self-similar...
Michel L. Lapidus, Jacques Lévy-véhel, John A. Rock
Abstract. We define a one-parameter family of geometric zeta functions for a Borel measure on the unit interval and a sequence which tends to zero. The construction of this family is based on that of...
Michel L. Lapidus, M. L. Lapidus, P. J. Pearse, M. Van Frankenhuijsen, Fractal Strings [fgcd
Tube formulas and complex dimensions of self-similar tilings.
Michel L. Lapidus, Jacques Lévy-véhel, John A. Rock, Zeta Functions
Definition 1. A fractal string Ω is a bounded open subset of the real line, thus Ω = � ∞ j=1 (aj, bj). Let L = {ℓj} ∞ j=1 = {|(aj, bj)|} ∞ j=1 and let {ln} ∞ n=1 be the distinct...
A tube formula for the Koch snowflake curve, with applications to complex dimensions (2004)
Lapidus, Michel L., Pearse, Erin P. J.
A formula for the interior epsilon-neighborhood of the classical von Koch snowflake curve is computed in detail. This function of epsilon is shown to match quite closely with earlier predictions of...
Fractality, Self-Similarity and Complex Dimensions (2004)
Lapidus, Michel L., Van Frankenhuijsen, Machiel
We present an overview of a theory of complex dimensions of self-similar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical...
Complex Dimensions of Self-Similar Fractal Strings and Diophantine Approximation (2003)
Lapidus, Michel L., Van Frankenhuysen, Machiel
We study the solutions in s of a "Dirichlet polynomial equation'' $m_1r_1^s+\dots+m_Mr_M^s=1$. We distinguish two cases. In the lattice case, when $r_j=r^{k_j}$ are powers of a common base r, the...