Related papers: Ultrametric and Generalized Ultrametric in Computa…
Convex polyhedral abstractions of logic programs have been found very useful in deriving numeric relationships between program arguments in order to prove program properties and in other areas such as termination and complexity analysis. We…
Data are often represented as graphs. Many common tasks in data science are based on distances between entities. While some data science methodologies natively take graphs as their input, there are many more that take their input in…
With the increasing interplay between experimental and computational approaches at multiple length scales, new research directions are emerging in materials science and computational mechanics. Such cooperative interactions find many…
The main purpose of data mining and analytics is to find novel, potentially useful patterns that can be utilized in real-world applications to derive beneficial knowledge. For identifying and evaluating the usefulness of different kinds of…
Graph analytics can lead to better quantitative understanding and control of complex networks, but traditional methods suffer from high computational cost and excessive memory requirements associated with the high-dimensionality and…
In applied mathematics and related disciplines, the modeling-simulation-optimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data,…
In this paper we consider aspects of geometric observability for hypergraphs, extending our earlier work from the uniform to the nonuniform case. Hypergraphs, a generalization of graphs, allow hyperedges to connect multiple nodes and…
Noncommutative geometry has seen remarkable applications for high energy physics, viz. the geometrical interpretation of the Standard Model. The question whether it also allows for supersymmetric theories has so far not been answered in a…
Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a…
Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops \textit{from the ground up} an abstract algebraic and qualitative notion of similarity based on the…
We present a conceptual framework that unifies a variety of evaluation metrics for different structured prediction tasks (e.g. event and relation extraction, syntactic and semantic parsing). Our framework requires representing the outputs…
While deep learning excels in natural image and language processing, its application to high-dimensional data faces computational challenges due to the dimensionality curse. Current large-scale data tools focus on business-oriented…
This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements. It also covers recent developments in the applications of digital geometry and topology including image…
In continuous logic, there are plenty of examples of interesting stable metric structures. However, on the other side of the SOP line, there are only a few metric structures where order is relevant, and orders often appear in different…
The advancement of large language models (LLMs) for real-world applications hinges critically on enhancing their reasoning capabilities. In this work, we explore the reasoning abilities of large language models (LLMs) through their…
In this paper, a new approximate syllogistic reasoning schema is described that expands some of the approaches expounded in the literature into two ways: (i) a number of different types of quantifiers (logical, absolute, proportional,…
In this survey, we explore recent literature on finding the cores of higher graphs using geometric and topological means. We study graphs, hypergraphs, and simplicial complexes, all of which are models of higher graphs. We study the notion…
Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ and $\Psi$ are combinatorially similar if there are bijections $f \colon \Phi(X^2) \to \Psi(Y^{2})$ and $g \colon Y…
An ultrametric defined on a subset S of a metric space X can be extended to X while roughly preserving distances between pairs in S x X.
We introduce derivation depth-a computable metric of the reasoning effort needed to answer a query based on a given set of premises. We model information as a two-layered structure linking abstract knowledge with physical carriers, and…