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Understanding phase transitions requires not only identifying order parameters but also characterizing how their correlations behave across scales. By quantifying how fluctuations at distinct spatial or temporal points are related,…
The brain can efficiently learn a wide range of tasks, motivating the search for biologically inspired learning rules for improving current artificial intelligence technology. Most biological models are composed of point neurons, and cannot…
We show that when the standard techniques for calculating fractal dimensions in empirical data (such as the box counting) are applied on uniformly random structures, apparent fractal behavior is observed in a range between physically…
We construct first a class of Moran fractals in R^d with countably many generators and non-stationary contraction rates; at each step n, the contractions depend on n-truncated sequences, and are related to asymptotic letter frequencies. In…
Many physical, biological, and engineered systems exhibit memory effects that challenge Markovian models. Fractional calculus provides nonlocal operators to capture hereditary dynamics. This survey connects modeling, analysis, and…
The frequency of occurrence of prime numbers at unit number spacing intervals exhibits selfsimilar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows,…
We introduce and develop a theory of fusion and statistical processes of gapped excitations in Abelian fracton phases. The key idea is to incorporate lattice translation symmetry via its action on superselection sectors, which results in a…
With the discovery of new superconductors there was a running to find the justifications for the new properties found in these materials. In order to describe these new effects some theories were adapted and some others have been tried. In…
We propose a novel extension to symmetrized neural network operators by incorporating fractional and mixed activation functions. This study addresses the limitations of existing models in approximating higher-order smooth functions,…
In this work, we rigorously investigate the robustness of graph neural fractional-order differential equation (FDE) models. This framework extends beyond traditional graph neural (integer-order) ordinary differential equation (ODE) models…
Information propagation characterizes how input correlations evolve across layers in deep neural networks. This framework has been well studied using mean-field theory, which assumes infinitely wide networks. However, these assumptions…
Reduced models of neuronal activity such as Integrate-and-Fire models allow a description of neuronal dynamics in simple, intuitive terms and are easy to simulate numerically. We present a method to fit an Integrate-and-Fire-type model of…
A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order $\alpha, 0 < \alpha \leq 1$, called $F^\alpha$-integral, is defined, which is suitable to integrate functions with fractal…
This study presents the formulation, the numerical solution, and the validation of a theoretical framework based on the concept of variable-order mechanics and capable of modeling dynamic fracture in brittle and quasi-brittle solids. More…
Nonlinear interactions in the dendritic tree play a key role in neural computation. Nevertheless, modeling frameworks aimed at the construction of large-scale, functional spiking neural networks, such as the Neural Engineering Framework,…
This paper proposes a fractional order gradient method for the backward propagation of convolutional neural networks. To overcome the problem that fractional order gradient method cannot converge to real extreme point, a simplified…
The study of balanced networks of excitatory and inhibitory neurons has led to several open questions. On the one hand it is yet unclear whether the asynchronous state observed in the brain is autonomously generated, or if it results from…
This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived…
We have performed a detailed analysis of the fast multipole method (FMM) in the adaptive case, in which the depth of the FMM tree is non-uniform. Previous works in this area have focused mostly on special types of adaptive distributions,…
As a general means of expression, audio analysis and recognition has attracted much attentions for its wide applications in real-life world. Audio emotion recognition (AER) attempts to understand emotional states of human with the given…