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In this paper we apply a scaling invariance analysis to reduce a class of parabolic moving boundary problems to free boundary problems governed by ordinary differential equations. As well known free boundary problems are always non-linear…
A stochastic search method, the so-called Adaptive Subspace (AdaSub) method, is proposed for variable selection in high-dimensional linear regression models. The method aims at finding the best model with respect to a certain model…
Self-organizing systems demonstrate how simple local rules can generate complex stochastic patterns. Many natural systems rely on such dynamics, making self-organization central to understanding natural complexity. A fundamental challenge…
A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb{R}^d$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the…
In this paper, we investigate parameter families of iterated function systems and continuity. Specifically, if we have a set of iterated function systems that depend continuously on a parameter, which properties of the invariant sets will…
Metric-based meta-learning has attracted a lot of attention due to its effectiveness and efficiency in few-shot learning. Recent studies show that metric scaling plays a crucial role in the performance of metric-based meta-learning…
We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension differs…
The varying-coefficient model is an important nonparametric statistical model that allows us to examine how the effects of covariates vary with exposure variables. When the number of covariates is big, the issue of variable selection…
The problem of the effects of compressibility and large-scale anisotropy on anomalous scaling behavior is considered for two models describing passive advection of scalar density and tracer fields. The advecting velocity field is Gaussian,…
Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the…
We present a method for dimensionality reduction of an affine variational inequality (AVI) defined over a compact feasible region. Centered around the Johnson Lindenstrauss lemma, our method is a randomized algorithm that produces with high…
Multidimensional scaling (MDS) is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances…
Symmetric Positive Definite (SPD) matrices have been widely used in medical data analysis and a number of different Riemannian met-rics were proposed to compute with them. However, there are very few methodological principles guiding the…
Differentiable physics is a powerful approach to learning and control problems that involve physical objects and environments. While notable progress has been made, the capabilities of differentiable physics solvers remain limited. We…
This paper introduces the concept of dimensional stability for spline spaces over T-meshes, providing the first mathematical definition and a preliminary classification framework. We define dimensional stability as an invariant within the…
Estimating rare events in complex systems is a key challenge in reliability analysis. The challenge grows in multimodal problems, where traditional methods often rely on a small set of design points and risk overlooking critical failure…
Physical understanding of how the interplay between symmetries and nonlinear effects can control the scaling and multiscaling properties in a coupled driven system, such as magnetohydrodynamic turbulence or turbulent binary fluid mixtures,…
Solutions of symbolic regression problems are expressions that are composed of input variables and operators from a finite set of function symbols. One measure for evaluating symbolic regression algorithms is their ability to recover…
In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs…
Units equivariance (or units covariance) is the exact symmetry that follows from the requirement that relationships among measured quantities of physics relevance must obey self-consistent dimensional scalings. Here, we express this…