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Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable…
We introduce a geometric dynamical system where iteration is defined as a cycling composition of different maps acting on a space composed of three or more lines in $\mathbb{R}^2$. This system is motivated by the dynamics of iterated…
A complex system comprises multiple interacting entities whose interdependencies form a unified whole, exhibiting emergent behaviours not present in individual components. Examples include the human brain, living cells, soft matter, Earth's…
Reliably simulating two-dimensional many-body quantum dynamics with projected entangled pair states (PEPS) has long been a difficult challenge. In this work, we overcome this barrier for low-energy quantum dynamics by developing a stable…
Vision Transformers (ViTs) lack the hierarchical inductive biases inherent to Convolutional Neural Networks (CNNs), theoretically allowing them to maintain high-dimensional representations throughout all layers. However, recent observations…
The discrete wavelet packet transform (DWPT) and discrete wavelet transform (DWT) are used to extract and study the dynamics of coherent structures in a turbulent rotating fluid. Three-dimensional (3D) turbulence is generated by strong…
Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from detailed microscale dynamics. We explore such coordinate…
In recent years, machine learning methods have been widely used to study physical systems that are challenging to solve with governing equations. Physicists and engineers are framing the data-driven paradigm as an alternative approach to…
In this paper the theory of 2-Variable Boolean Operation (2-VBO) has been discussed on a pair of n-bit strings. 2-VBO serves to bring out the relation between numbers which when plot on a 2-D surface form interesting patterns; patterns that…
In this paper, we study the fractal dimension of the graph of a fractal transformation and also determine the quantization dimension of a probability measure supported on the graph of the fractal transformation. Moreover, we estimate the…
We present two interactive visualisations of 2x2 real matrices, which we call v1 and v2. v1 is only valid for PSD matrices, and uses the spectral theorem in a trivial way -- we use it as a warm-up. By contrast, v2 is valid for *all* 2x2…
The dynamics of many-body systems can often be captured in terms of only a few relevant variables. Mathematical and numerical approaches exist to identify these variables by exploiting a separation of time scales between slow relevant and…
The main objective of this article are two-fold. First, we introduce some general principles on phase transition dynamics, including a new dynamic transition classification scheme, and a Ginzburg-Landau theory for modeling equilibrium phase…
We analyze a variety of Weyl invariant dynamical problems in three dimensions.
Learning a model of dynamics from high-dimensional images can be a core ingredient for success in many applications across different domains, especially in sequential decision making. However, currently prevailing methods based on…
A wide class of Hamiltonian systems with N degrees of freedom and endowed with, at least, (N-2) functionally independent integrals of motion in involution is constructed by making use of the two-photon Lie-Poisson coalgebra. The set of…
Data valuation has become a cornerstone of the modern data economy, where datasets function as tradable intellectual assets that drive decision-making, model training, and market transactions. Despite substantial progress, existing…
A remarkable feature of fluid dynamics is its relationship with classical dynamics and statistical mechanics. This has motivated in the past mathematical investigations concerning, in a special way, the "derivation" based on kinetic theory,…
There is a classical geometric construction which uses a binary quadratic form to define an involution on the space of binary d-ics. We give a complete characterization of a general class of such involutions which are definable using…
Dynamical phase transitions (DPTs) in the space of trajectories are one of the most intriguing phenomena of nonequilibrium physics, but their nature in realistic high-dimensional systems remains puzzling. Here we observe for the first time…