Related papers: Scaling Dimension
Analogical reasoning is a powerful qualitative reasoning tool that enables humans to connect two situations, and to generalize their knowledge from familiar to novel situations. Cognitive Science research provides valuable insights into the…
This paper is a tutorial on Formal Concept Analysis (FCA) and its applications. FCA is an applied branch of Lattice Theory, a mathematical discipline which enables formalisation of concepts as basic units of human thinking and analysing…
A Concept Tree is a structure for storing knowledge where the trees are stored in a database called a Concept Base. It sits between the highly distributed neural architectures and the distributed information systems, with the intention of…
Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that…
Concept Hierarchies and Formal Concept Analysis are theoretically well grounded and largely experimented methods. They rely on line diagrams called Galois lattices for visualizing and analysing object-attribute sets. Galois lattices are…
The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a high-dimensional space and concepts are represented by regions in this space. Our recent…
This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the…
This paper introduces "Semantic Scaling," a novel method for ideal point estimation from text. I leverage large language models to classify documents based on their expressed stances and extract survey-like data. I then use item response…
We relate two formerly independent areas: Formal concept analysis and logic of domains. We will establish a correspondene between contextual attribute logic on formal contexts resp. concept lattices and a clausal logic on coherent algebraic…
The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a high-dimensional space and concepts are represented by regions in this space. In this…
Scaling arguments provide valuable analysis tools across physics and complex systems yet are often employed as one generic method, without explicit reference to the various mathematical concepts underlying them. A careful understanding of…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the…
Mass scaling is widely used in finite element models of structural dynamics for increasing the critical time step of explicit time integration methods. While the field has been flourishing over the years, it still lacks a strong theoretical…
In this chapter tools and techniques from the mathematical theory of formal concept analysis are applied to hypertext systems in general, and the World Wide Web in particular. Various processes for the conceptual structuring of hypertext…
Although the notion of a concept as a collection of objects sharing certain properties, and the notion of a conceptual hierarchy are fundamental to both Formal Concept Analysis and Description Logics, the ways concepts are described and…
We describe a new cognitive ability, i.e., functional conceptual substratum, used implicitly in the generation of several mathematical proofs and definitions. Furthermore, we present an initial (first-order) formalization of this mechanism…
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…
Formal Concept Analysis starts from a very basic data structure comprising objects and their attributes. Sometimes, however, it is beneficial to also define attributes of attributes, viz., meta-attributes. In this paper, we use Triadic…
The main goal of this paper has a double purpose. On the one hand, we propose a new definition in order to compute the fractal dimension of a subset respect to any fractal structure, which completes the theory of classical box-counting…
Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue…