Spectral operators on the Sierpinski gasket I

Allan, Adam; Bárány, Michael; Strichartz, Robert S.

doi:10.1080/17476930802272978

Cited by 16 publications

(26 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These log-periodic oscillations are represented on Fig.3 and we note that the higher order (with m) complex poles give much smaller contributions, a result related to the steep decrease of the Euler Γ function. A similar behavior has been found numerically for the Sierpinksi gasket [23]. For mathematical discussions of oscillations in heat kernel estimates see [52,53].…”

Section: Laplacian On Fractals -Heat Kernel and Spectral Zeta Functionsupporting

confidence: 76%

“…Since the early 1980's, mathematicians have opened new important directions by being able to define properly brownian motion on some classes of fractals ( [19]- [21]) and Laplacian operators on these structures [24,23]. Progress along these two directions have led to a vast literature and it would be a hopeless task to list it exhaustively.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Statistical Mechanics and Quantum Fields on Fractals

Akkermans¹

2013

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

View full text Add to dashboard Cite

This paper is dedicated to the memory of B. Mandelbrot.Abstract. Fractals define a new and interesting realm for a discussion of basic phenomena in quantum field theory and statistical mechanics. This interest results from specific properties of fractals, e.g., their dilatation symmetry and the corresponding absence of Fourier mode decomposition. Moreover, the existence of a set of distinct dimensions characterizing the physical properties (spatial or spectral) of fractals make them a useful testing ground for dimensionality dependent physical problems. This paper addresses specific problems including the behavior of the heat kernel and spectral zeta functions on fractals and their importance in the expression of spectral properties in quantum physics. Finally, we apply these results to specific problems such as thermodynamics of quantum radiation by a fractal blackbody.

show abstract

Section: Laplacian On Fractals -Heat Kernel and Spectral Zeta Functionsupporting

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Statistical Mechanics and Quantum Fields on Fractals

Akkermans¹

2013

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

View full text Add to dashboard Cite

show abstract

“…The third and last example stems from a relaxation of the approximation (2.3). On deterministic fractals, the heat kernel displays ripples with a logarithmic oscillatory pattern in diffusion time σ (e.g., [116][117][118][119][120][121][122]): P(σ) = (4πσ) −d S /2 F (σ), where F is periodic in ln σ [117,122]. As far as we known, these log-oscillations seem to be originated by the high degree of symmetry of these sets.…”

Section: Examples Of Multi-scale Measuresmentioning

confidence: 99%

Multi-scale gravity and cosmology

Calcagni

2013

J. Cosmol. Astropart. Phys.

209

View full text Add to dashboard Cite

The gravitational dynamics and cosmological implications of three classes of recently introduced multi-scale spacetimes (with, respectively, ordinary, weighted and qderivatives) are discussed. These spacetimes are non-Riemannian: the metric structure is accompanied by an independent measure-differential structure with the characteristics of a multi-fractal, namely, different dimensionality at different scales and, at ultra-short distances, a discrete symmetry known as discrete scale invariance. Under this minimal paradigm, five general features arise: (a) the big-bang singularity can be replaced by a finite bounce, (b) the cosmological constant problem is reinterpreted, since accelerating phases can be mimicked by the change of geometry with the time scale, without invoking a slowly rolling scalar field, (c) the discreteness of geometry at Planckian scales can leave an observable imprint of logarithmic oscillations in cosmological spectra and (d) give rise to an alternative mechanism to inflation or (e) to a fully analytic model of cyclic mild inflation, where near scale invariance of the perturbation spectrum can be produced without strong acceleration. Various properties of the models and exact dynamical solutions are discussed. In particular, the multi-scale geometry with weighted derivatives is shown to be a Weyl integrable spacetime.

show abstract

“…This paper will be concerned instead with the irregular domain being a fractal itself. Some notable works with this type of domain include [3,6,8,16,17,19] among others. Laakso's spaces were introduced in [11].…”

Section: Introductionmentioning

confidence: 99%

“…The papers [2,19] are devoted to finding and analyzing the heat kernel and the trace of the heat kernel. The notion of complex valued fractal dimensions and the accompanying oscillating behavior of the heat kernel were studied in [2,3].…”

Section: Introductionmentioning

confidence: 99%