Related papers: Discrete scale invariance and stochastic Loewner e…
The purpose of the present paper is to present the main applications of a new method for the determination of the fractal structure of plane curves. It is focused on the inverse problem, that is, given a curve in the plane, find its fractal…
In part 1 (Chapter 2) we present the basic notions of Loewner theory. Here we use a modern form which was developed by F. Bracci, M. Contreras, S. D\'iaz-Madrigal et al. and which can be applied to certain higher dimensional complex…
We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration. The theory provides a differential structure which describes the infinitesimal evolution of Wiener functionals at very small…
Abstract Self-similar, fractal nature of turbulence is discussed in the context of two dimensional turbulence, by considering the fractal structure of the wave-number domain using spirals. In loose analogy with phyllotaxis in plants, each…
We investigate fractal aspects of elliptical polynomial spirals; that is, planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis of these spirals, computing explicitly their…
In many low-dimensional dynamical systems transport coefficients are very irregular, perhaps even fractal functions of control parameters. To analyse this phenomenon we study a dynamical system defined by a piece-wise linear map and…
The change of the effective dimension of spacetime with the probed scale is a universal phenomenon shared by independent models of quantum gravity. Using tools of probability theory and multifractal geometry, we show how dimensional flow is…
This paper conceptualizes the Deep Weight Spaces (DWS) of neural architectures as hierarchical, fractal-like, coarse geometric structures observable at discrete integer scales through recursive dilation. We introduce a coarse group action…
Diffusion-weighted magnetic resonance imaging (DW-MRI) is a critical imaging method for capturing and modeling tissue microarchitecture at a millimeter scale. A common practice to model the measured DW-MRI signal is via fiber orientation…
Recently, we pointed out that on a class on non exactly decimable fractals two different parameters are required to describe diffusive and vibrational dynamics. This phenomenon we call dynamical dimension splitting is related to the lack of…
Identification of nonlinear dynamical systems is crucial across various fields, facilitating tasks such as control, prediction, optimization, and fault detection. Many applications require methods capable of handling complex systems while…
The gravitational dynamics and cosmological implications of three classes of recently introduced multi-scale spacetimes (with, respectively, ordinary, weighted and q-derivatives) are discussed. These spacetimes are non-Riemannian: the…
This paper describes joint work with Oded Schramm and Wendelin Werner establishing the values of the planar Brownian intersection exponents from which one derives the Hausdorff dimension of certain exceptional sets of planar Brownian…
This paper reviews a class of univariate piecewise polynomial functions known as discrete splines, which share properties analogous to the better-known class of spline functions, but where continuity in derivatives is replaced by (a…
Growth fronts of slime molds are characterized through a direct geometric analysis based on Loewner evolutions, using experimentally acquired time-resolved images. The associated Loewner driving functions reconstructed from expanding…
After a brief review of the historical role of analyticity in the study of critical phenomena, an account is given of recent discoveries of discretely holomorphic observables in critical two-dimensional lattice models. These are objects…
A dynamic scaling of a diffusion process involving the Langmuir type adsorption is studied. We find dynamic scaling functions in one and two dimensions and compare them with direct numerical simulations, and we further study the dynamic…
This work focuses on multivalued stochastic differential equations with jumps. First, by employing the weak convergence approach, we establish the Freidlin-Wentzell uniform large deviation principle and the Dembo-Zeitouni uniform large…
This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Loewner Evolution (SLE) by Oded…
Intermittency is one of central obstacles for understanding small-scale dynamics in the fully developed hydrodynamic turbulence. The modern approach is largely based on the multifractal theory of Parisi and Frisch which is, however,…