Related papers: Complex Scaling and Geometric Analysis of Several …
This chapter describes the progress made during the past three decades in the finite size scaling analysis of the critical phenomena of the Anderson transition. The scaling theory of localisation and the Anderson model of localisation are…
Multidimensional scaling is an important dimension reduction tool in statistics and machine learning. Yet few theoretical results characterizing its statistical performance exist, not to mention any in high dimensions. By considering a…
In this paper we sum up our works on multiscale programs, mainly simulations. We first start with describing what multiscaling is about, how it helps perceiving signal from a background noise in a ?ow of data for example, for a direct…
We use deposition models of kinetic roughening of a growing surface to introduce the concepts of universality and scaling and to analyze the qualitative and quantitative role of different parameters. In particular, we focus on two classes…
Conceptual Scaling is a useful standard tool in Formal Concept Analysis and beyond. Its mathematical theory, as elaborated in the last chapter of the FCA monograph, still has room for improvement. As it stands, even some of the basic…
We introduce a generalized framework for studying higher-order versions of the multiscale method known as Localized Orthogonal Decomposition. Through a suitable reformulation, we are able to accommodate both conforming and nonconforming…
This article presents an exhaustive classification of metric-affine theories according to their scale symmetries. First it is clarified that there are three relevant definitions of a scale transformation. These correspond to a projective…
We present two geometric interpretations for complex multivectors and determinants: a little known one in terms of square roots of volumes, and a new one which uses fractions of volumes and allows graphical representations. The fraction…
Clustering algorithms remain valuable tools for grouping and summarizing the most important aspects of data. Example areas where this is the case include image segmentation, dimension reduction, signals analysis, model order reduction,…
The exact determination of ground states of small systems is used in a scaling study of the random-field Ising model. While three variants of the model are found to be in the same universality class in 3 dimensions, the Gaussian and bimodal…
This work studies two dimensional local skew fields and their automorphisms.
Understanding large amounts of spatiotemporal data from particle-based simulations, such as molecular dynamics, often relies on the computation and analysis of aggregate measures. These, however, by virtue of aggregation, hide structural…
The purpose of this survey is to describe how locally compact groups can be studied as geometric objects. We will emphasize the main ideas and skip or just sketch most proofs, often referring the reader to our much more detailed book…
For each type of number, structures that differ by arbitrary scaling factors and are isomorphic to one another are described. The scaling of number values in one structure, relative to the values in another structure, must be compensated…
The article reviews some of the (fairly scattered) information available in the mathematical literature on the subject of angles in complex vector spaces. The following angles and their relations are considered: Euclidean, complex, and…
We review some recent learning approaches in variational imaging, based on bilevel optimisation, and emphasize the importance of their treatment in function space. The paper covers both analytical and numerical techniques. Analytically, we…
This paper introduces a new fundamental characteristic, \ie, the dynamic range, from real-world metric tools to deep visual recognition. In metrology, the dynamic range is a basic quality of a metric tool, indicating its flexibility to…
We develop a framework to study the phase space of a system consisting of a scalar field rolling down an arbitrary potential with varying slope and a background fluid, in a cosmological setting. We give analytical approximate solutions of…
This paper studies and critically discusses the construction of nonparametric confidence regions for density level sets. Methodologies based on both vertical variation and horizontal variation are considered. The investigations provide…
The monography examines the problem of constructing a group of automorphisms of a graph. A graph automorphism is a mapping of a set of vertices onto itself that preserves adjacency. The set of such automorphisms forms a vertex group of a…