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Understanding the behavior of a trained network and finding explanations for its outputs is important for improving the network's performance and generalization ability, and for ensuring trust in automated systems. Several approaches have…
This is an introduction to measure theory, integration and function spaces, with all the needed preliminaries included, and with some applications included as well. We first discuss some basic motivations, coming from discrete probability,…
In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is…
Many semantical aspects of programming languages, such as their operational semantics and their type assignment calculi, are specified by describing appropriate proof systems. Recent research has identified two proof-theoretic features that…
We obtain new results on the existence and multiplicity of fixed points of Hammerstein equations in very general cones. In order to achieve this, we combine a new formulation of cones in terms of continuous functionals with fixed point…
Whether it be in normal form games, or in fair allocations, or in voter preferences in voting systems, a certain pattern of reasoning is common. From a particular profile, an agent or a group of agents may have an incentive to shift to a…
The main aim of this paper is to study of fixed point theory in partial cone metric spaces. Infact, some common fixed point theorems for two mappings in partial cone metric spaces are obtained.
In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat…
An inductive inference system for proving validity of formulas in the initial algebra $T_{\mathcal{E}}$ of an order-sorted equational theory $\mathcal{E}$ is presented. It has 20 inference rules, but only 9 of them require user interaction;…
Existing class-agnostic counting models typically rely on a single type of prompt, e.g., box annotations. This paper aims to establish a comprehensive prompt-based counting framework capable of generating density maps for concerned objects…
Starting from pseudometrics and preorders on sets of integers, we extend the focus to sets of finite sequences of integers, in particular sequences of consecutive integers. We outline existing concepts for deriving centred pseudometrics and…
The process of cognition is analysed to adjust the set theory to physical description. Postulates and basic definitions are revised. The specific sets of predicates, called presets, corresponding to the physical objects identified by an…
In group representations several inductions given by tensoring with appropriate bimodules may be reconstructed via homology of $G$-posets with $G$-equivariant coefficients. For this purpose, we need various local categories of a finite…
Consensus clustering fuses diverse basic partitions (i.e., clustering results obtained from conventional clustering methods) into an integrated one, which has attracted increasing attention in both academic and industrial areas due to its…
We present a formalization of a version of Abadi and Plotkin's logic for parametricity for a polymorphic dual intuitionistic/linear type theory with fixed points, and show, following Plotkin's suggestions, that it can be used to define a…
Initial semantics aims to capture inductive structures and their properties as initial objects in suitable categories. We focus on the initial semantics aiming to model the syntax and substitution structure of programming languages with…
We propose a framework for reasoning about programs that manipulate coinductive data as well as inductive data. Our approach is based on using equational programs, which support a seamless combination of computation and reasoning, and using…
Fixed point theorems are ubiquitous in economic research. Many studies cite Smithson (1971) ``Fixed points of order preserving multifunctions,'' yet the original proof contains errors. This note presents a new, concise proof and explains…
The class of O-metric spaces generalize several existing metric-types in literature including metric spaces, b-metric spaces, and ultra metric spaces. In this paper, we discuss the properties of the topology induced by an O-metric and…
Concept embeddings offer a practical and efficient mechanism for injecting commonsense knowledge into downstream tasks. Their core purpose is often not to predict the commonsense properties of concepts themselves, but rather to identify…