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In complex systems with fractal properties the scale invariance has an important rule to classify different statistical properties. In two dimensions the Loewner equation can classify all the fractal curves. Using the Weierstrass-Mandelbrot…
Wavelets are closely related to the Schr\"odinger's wave functions and the interpretation of Born. Similarly to the appearance of atomic orbital, it is proposed to combine anti-symmetric wavelets into orbital wavelets. The proposed approach…
For many complex systems the interaction of different scales is among the most interesting and challenging features. It seems not very successful to extract the physical properties in different scale regimes by the existing approaches, such…
Natural images are characterized by the multiscaling properties of their contrast gradient, in addition to their power spectrum. In this work we show that those properties uniquely define an {\em intrinsic wavelet} and present a suitable…
The fractal properties of models of randomly placed $n$-dimensional spheres ($n$=1,2,3) are studied using standard techniques for calculating fractal dimensions in empirical data (the box counting and Minkowski-sausage techniques). Using…
To unify general relativity and quantum theory is hard in part because they are formulated in two very different mathematical languages, differential geometry and functional analysis. A natural candidate for bridging this language gap, at…
This paper is devoted to the study of dimension theory, in particular multifractal analysis, for multimodal maps. We describe the Lyapunov spectrum, generalising previous results by Todd. We also study the multifractal spectrum of pointwise…
Concise introduction to a relatively new subject of non-linear algebra: literal extension of text-book linear algebra to the case of non-linear equations and maps. This powerful science is based on the notions of discriminant…
Mesoscale structures can often be described as fractional dimensional across a wide range of scales. We consider a $\gamma$ dimensional measure embedded in an $N$ dimensional space and discuss how to determine its dimension, both in $N$…
In the chapter "Multiresolution Analysis on Compact Riemannian Manifolds" Isaac Pesenson describes multiscale analysis, sampling, interpolation and approximation of functions defined on manifolds. His main achievements are: construction on…
Wavelet Transforms are a widely used technique for decomposing a signal into coefficient vectors that correspond to distinct frequency/scale bands while retaining time localization. This property enables an adaptive analysis of signals at…
Multifractal analysis has revealed regularities in many self-seeding phenomena, yet its use in modern deep learning remains limited. Existing end-to-end multifractal methods rely on heavy pooling or strong feature-space decimation, which…
Data series generated by complex systems exhibit fluctuations on many time scales and/or broad distributions of the values. In both equilibrium and non-equilibrium situations, the natural fluctuations are often found to follow a scaling…
If our aesthetic preferences are affected by fractal geometry of nature, scaling regularities would be expected to appear in all art forms, including music. While a variety of statistical tools have been proposed to analyze time series in…
Extensive confusion exists and persists in the literature on dynamical systems theory, cosmology, and other fields over spectra of fractal dimensions. Entirely different generating functions have been treated as if they should yield…
The creativity and emergence of biological and psychological behavior are nonlinear. However, that does not necessarily mean only that the measurements of the behaviors are curvilinear. Furthermore, the linear model might fail to reduce…
This paper introduces a novel framework for measuring multidimensional inequality based on a statistical physics reinterpretation of centrifugal and centripetal forces in rotating systems. Inspired by the mechanics of the Gravitron and…
We provide a comprehensive theory of multiple variants of ordinal multidimensional scaling,including internal unfolding and external unfolding. We first follow Shepard (1966) and work in a continuum model to gain insight. We then follow…
In cluster analysis, a common first step is to scale the data aiming to better partition them into clusters. Even though many different techniques have throughout many years been introduced to this end, it is probably fair to say that the…
Scale invariance is a central organizing principle in physics, underlying phenomena that range from critical behaviour in statistical mechanics to transport and chaos in nonlinear dynamical systems. Here we present a unified and physically…