Axiomatic Approach Research Articles

An axiomatic approach to measure the ‘Leave no one behind’ principle

‘Leave no one behind’ has become an important principle of the entire 2030 Sustainable Development Agenda and the Sustainable Development Goals that brings inequality to the centre stage. Although a precise definition of what it is to lag behind is essential to assess sustainable economic growth and social progress a discussion on the desirable properties of such a measure is absent in the literature. This paper fills that gap by proposing and discussing a number of normative and analytical properties a measure of the ‘Leave no one behind’ principle should satisfy. The axioms proposed are necessary and sufficient to characterize the fuzzy measure by García-Pardo et al. [8] and together with other additional axioms make the structure of this measure more transparent.

Shrinkage of regional differences in the cost‐of‐living index in Japan

AbstractThis study aims to clarify the existence of regional differences and the impact of the 2008 Lehman (Brothers) shock, the collapse of a major U.S. securities firm that triggered the global financial crisis, on these differences based on the regional cost‐of‐living index (RCLI) in Japan. We apply a methodology developed by Kakwani and Hill, which is an axiomatic approach for constructing the multilateral spatial cost‐of‐living indices, to compare the price levels of households between regions of Japan. The results show, overall, a shrinkage of regional differences after the Lehman shock. However, the impact of the Lehman shock on the cost of living varies from region to region. In Japan, the Lehman shock lowered the RCLI in urban areas rather than rural areas. This finding implies that urban areas in Japan are more susceptible to macro shocks, which reduce regional differences with rural areas. Furthermore, based on the results of the cost‐of‐living index by group commodity, we offer a policy implication for the fundamental elimination of regional differences between Japan's urban and rural areas. This would encourage, for example, an increase in the cost of living in rural areas through increased public transport investment and costs, an accompanying increase in rent of housing in surrounding areas and an increase in communication costs owing to the spread of high‐speed internet services, such as 5G.

Independence of the axioms of hypergroup over the group

The independence of the axioms of hypergroup over the group is proved. The proof is composed of two parts. In the first part, the independence of the axioms $(P3)$, $(A1)$, $(A3)$, $(A5)$ in the system of axioms of hypergroup over the group is shown by fixing the structural mappings $\Phi$ and $\Xi$. In the same way, in the second part of the proof, the independence of the axioms $(P1)$, $(P2)$, $(A2)$, $(A4)$ is shown by fixing $\Psi$ and $\Lambda$.

Денотация и коннотация как логический приём в юридической технике

The article discusses the specifics of legal techniques in the formulation of legal definition. A comparative analysis of approaches to understanding the definition procedure in jurisprudence, philosophy, logic, mathematics and philology is carried out.
 The relevance of the topic under study lies in the absence in modern legal tech-nology of generally accepted approaches to understanding the methodological speci-ficity of the implementation of the definition procedure. The purpose of this article is to determine the methodological foundations of differentiation of axiomatic and con-textual definitions, the implementation of the critical analysis of the classifications of types of definitions in the modern scientific literature.
 The basis of the concept of definition proposed by the authors is the analytical study of the logicallinguistic operation of determination on the example of the defi-nition of «corruption» from the Federal Law of December 25, 2008 No. 273-FZ.
 There is a lack of legal recognition of corruption as a bribe, and criticism of the narrow definition. Corruption is defined as a criminal act of a social and legal nature, which is constrained by the nature of official crimes in conflict with the interests of society and the State, the essence of which is an obvious mercenary motive in personal en-richment (material and non-material) through the use of his official position (authori-ty) for mercenary purposes.
 The study is based on the methodological tools of systemic, structural-functional and comparative cognition methods.

Constructions of new q-cryptomorphisms

In the theory of classical matroids, there are several known equivalent axiomatic systems that define a matroid, which are described as matroid cryptomorphisms. A q-matroid is a q-analogue of a matroid where subspaces play the role of the subsets in the classical theory. In this article we establish cryptomorphisms of q-matroids. In doing so we highlight the difference between classical theory and its q-analogue. We introduce a comprehensive set of q-matroid axiom systems and show cryptomorphisms between them and existing axiom systems of a q-matroid. These axioms are described as the rank, closure, basis, independence, dependence, circuit, hyperplane, flat, open space, spanning space, non-spanning space, and bi-colouring axioms.

Distance measure on intuitionistic fuzzy sets and its application in decision‐making, pattern recognition, and clustering problems

Decision-making under uncertainty is consistently an essential fear and the most challenging circle of exploration. To manage the uncertainty, the intuitionistic fuzzy set (IFS) assumes a critical part in taking care of the conditions wherein decision-makers furnish an alternative with a grade of membership and a nonmembership. Distance measures of IFSs are apparatuses used in different decision-making problems, such as medical investigation, pattern recognition, multicriteria decision-making, clustering problems, and other real-world problems. As such, various distance measures were developed by different researchers and applied to decision-making problems with situation-based deficiencies. Motivated by this, in this paper, a symmetric distance formula is being proposed for effectively determining the distance between the information held by IFSs. The distance formula involves membership degree, nonmembership degree, the difference of the minimum of the cross-evaluation factor, and the difference of the maximum of the cross-evaluation factor. Furthermore, it is being proved that the proposed distance formula follows all the axiomatic definitions of a distance measure. Numerical examples depict the efficiency of the proposed distance measure. Hence, this measure is being applied to practical problems of decision-making, pattern recognition, and clustering problems. This measure is not restricted to a particular domain of study; it can be effectively applied in diverse decision-making problems.

Monadic NM-algebras: an algebraic approach to monadic predicate nilpotent minimum logic

AbstractIn this paper, we further study the variety of monadic nilpotent minimum (NM)-algebras and their corresponding logic. In order to solve the drawback of monadic NM-algebras, we review some well-known classes of monadic t-norm-based fuzzy logical algebras and then revise the axiomatic system of monadic NM-algebras. Then we show that the variety of monadic NM-algebras is the equivalent algebraic semantics of monadic predicate fuzzy logic $\textbf {mNM}_{\forall }$, which is equivalent to the modal fuzzy logic $\textbf {S5(NM)}$. Moreover, we show that the propositional case of the modal fuzzy logic $\textbf {S5(NM)}$, which is $\textbf{S5}^{\prime}\textbf{(NM),}$ is also complete with respect to the variety of monadic NM-algebras in the sense of Blok and Pigozzi and obtain a necessary and sufficient condition for this logic to be semilinear. Finally, we give some representations of monadic NM-algebras. In particular, we give some characterizations of representable and directly indecomposable monadic NM-algebras.

Geometric structures, the Gromov order, Kodaira dimensions and simplicial volume

We introduce an axiomatic definition for the Kodaira dimension and classify Thurston geometries in dimensions $\leq 5$ in terms of this Kodaira dimension. We show that the Kodaira dimension is monotone with respect to the partial order defined by maps of non-zero degree between 5-manifolds. We study the compatibility of our definition with traditional notions of Kodaira dimension, especially the highest possible Kodaira dimension. To this end, we establish a connection between the simplicial volume and the holomorphic Kodaira dimension, which in particular implies that any smooth K\"ahler 3-fold with non-vanishing simplicial volume has top holomorphic Kodaira dimension.

Designing the mathematics educational process in the context of education metadisciplinarity

The article is devoted to one of the current issues – the problem of educational process design. Among the Russian educational space innovations, a special place belongs to the metadisciplinary direction, new for domestic pedagogy. Comprehension of the metadisciplinarity category both in the theoretical aspect and applied from the position of the modern view becomes especially important. To some extent, the complexity of the problem is determined by the ambiguous translation of the prefix “meta.” Recently, there have been many studies devoted to metadisciplinarity. The analysis of scientific works on this problem showed no single point of view on this issue. The axiomatic approach is proposed as one of the options for building an educational strategy in the metadisciplinary field. This study aims to show the possibility of using the latter in designing the mathematics educational process as part of implementing the metadisciplinary direction of the educational standards.

Object-based unawareness: Axioms

This paper provides foundations for a model of unawareness, called object-based unawareness (OBU) structures, that can be used to distinguish between what an agent is unaware of and what she simply does not know. At an informal level, this distinction plays a key role in a number of papers such as Tirole (2009) and Chung & Fortnow (2016). In this paper, we give the model-theoretic description of OBU structures by showing how they assign truth conditions to every sentence of the formal language used. We then prove a model-theoretic sound and completeness theorem, which characterizes OBU structures in terms of a system of axioms. We then verify that agents in OBU structures do not violate any of the introspection axioms that are generally considered to be necessary conditions for a plausible notion of unawareness. Applications are provided in our companion paper.

The Axiomatics of Free Group Rings

In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring $R[G]$ and the corresponding universal and elementary theory of the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is a commutative ring with identity $1 \ne 0$. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it was proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to $L_{2}$, then simultaneously the group $G$ is elementarily equivalent to the group $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalent to the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ free group and $\mathbb{Z}$ be the ring of integers. Examining the universal theory of the free group ring ${\mathbb Z}[F]$ the hazy conjecture was made that the universal sentences true in ${\mathbb Z}[F]$ are precisely the universal sentences true in $F$ modified appropriately for group ring theory and the converse that the universal sentences true in $F$ are the universal sentences true in ${\mathbb Z}[F]$ modified appropriately for group theory. In this paper we show this conjecture to be true in terms of axiom systems for ${\mathbb Z}[F]$.

Quantum thermo-dynamical construction for driven open quantum systems

Quantum dynamics of driven open systems should be compatible with both quantum mechanic and thermodynamic principles. By formulating the thermodynamic principles in terms of a set of postulates we obtain a thermodynamically consistent master equation. Following an axiomatic approach, we base the analysis on an autonomous description, incorporating the drive as a large transient control quantum system. In the appropriate physical limit, we derive the semi-classical description, where the control is incorporated as a time-dependent term in the system Hamiltonian. The transition to the semi-classical description reflects the conservation of global coherence and highlights the crucial role of coherence in the initial control state. We demonstrate the theory by analyzing a qubit controlled by a single bosonic mode in a coherent state.

Two Myths of Representational Measurement

Abstract Axiomatic measurement theories are commonly interpreted as claiming that, in order to quantify an empirical domain, the qualitative structure of data about that domain must be mapped to a numerical structure. Such mapping is supposed to be established independently, i.e., without presupposing that the domain can be quantified. This interpretation is based on two myths: that it is possible to independently infer the qualitative structure of objects from empirical data, and that the adequacy of numerical representations can only be justified by mapping such qualitative structures to numerical ones. I dispel the myths and show that axiomatic measurement theories provide an inadequate characterization of the kind of evidence required to detect quantities.

Axiomatization of Some Basic and Modal Boolean Connexive Logics

Boolean connexive logic is an extension of Boolean logic that is closed under Modus Ponens and contains Aristotle’s and Boethius’ theses. According to these theses (i) a sentence cannot imply its negation and the negation of a sentence cannot imply the sentence; and (ii) if the antecedent implies the consequent, then the antecedent cannot imply the negation of the consequent and if the antecedent implies the negation of the consequent, then the antecedent cannot imply the consequent. Such a logic was first introduced by Jarmużek and Malinowski, by means of so-called relating semantics and tableau systems. Subsequently its modal extension was determined by means of the combination of possible-worlds semantics and relating semantics. In the following article we present axiomatic systems of some basic and modal Boolean connexive logics. Proofs of completeness will be carried out using canonical models defined with respect to maximal consistent sets.

Formalizing rough sets using a new noncontingency axiomatic system

Formalization of rough sets is a key issue in rough set theory. When rough sets are formalized by propositional logic, predicate logic, or modal propositional logic, it easily suffers from some problems. For instance, an incomplete system is obtained. The concepts of “ p r e c i s e” or “ r o u g h” of rough sets cannot be described. To tackle these issues, a new noncontingency axiomatic system is proposed for formalizing rough sets in this paper. First, a new concise accessibility relation is defined for the axiomatic system; then, two simpler axiom schemas of the axiomatic system are designed to replace the axiom schema K. This is helpful to prove the soundness and completeness theorems for the axiomatic system. Finally, rough sets can be perfectly formalized by our proposed axiomatic system. Theoretical analysis proves that a complete formal system is achieved. In addition, the concepts of “ p r e c i s e” or “ r o u g h” of rough sets can be described without the help of semantics functions of metalanguage.

Issues in clustering algorithm consistency in fixed dimensional spaces. Some solutions for k-means

Kleinberg introduced an axiomatic system for clustering functions. Out of three axioms, he proposed, two (scale invariance and consistency) are concerned with data transformations that should produce the same clustering under the same clustering function. The so-called consistency axiom provides the broadest range of transformations of the data set. Kleinberg claims that one of the most popular clustering algorithms, k-means does not have the property of consistency. We challenge this claim by pointing at invalid assumptions of his proof (infinite dimensionality) and show that in one dimension in Euclidean space the k-means algorithm has the consistency property. We also prove that in higher dimensional space, k-means is, in fact, inconsistent. This result is of practical importance when choosing testbeds for implementation of clustering algorithms while it tells under which circumstances clustering after consistency transformation shall return the same clusters. Two types of remedy are proposed: gravitational consistency property and dataset consistency property which both hold for k-means and hence are suitable when developing the mentioned testbeds.

Centralized Control System of the Flotation Department: Ross-Eshby Principle

The aim of the study is to scientifically substantiate the need to use the axiomatic approach when choosing and studying the processes of centralized dispatching control of technological parameters for a complex of flotation machines. During the research, the au-thors performed: comparison of a real object and its properties with the classical control axioms of system analysis; substantiated the use of a hierarchical structure of interrelated technological parameters and their application for situational management. Methods of classical control axioms of system analysis were used for the research. The authors formu-lated the criteria for situational management, the concept of visualization in the dispatching system. The results obtained allow us to assert that the implementation of the Ross-Ashby principle significantly increases the controllability of technological processes with a large number of units and measured parameters.

A Needle-Free Jet Injection System for Controlled Release and Repeated Biopharmaceutical Delivery.

Swift vaccination is necessary as a response to disease outbreaks and pandemics; otherwise, the species under attack is at risk of a high fatality rate or even mass extinction. Statistics suggest that at least 16 billion injections are administered worldwide every year. Such a high rate of needle/syringe injection administration worldwide is alarming due to the risk of needle-stick injuries, disease spread due to cross-contamination and the reuse of needles, and the misuse of needles. In addition, there are production, handling, and disposal costs. Needle phobia is an additional issue faced by many recipients of injections with needles. In addition to a detailed literature review highlighting the need for needle-free injection systems, a compressed air-driven needle-free jet injection system with a hydro-pneumatic mechanism was designed and developed by employing an axiomatic design approach. The proposed injection system has higher flexibility, uninterrupted force generation, and provides the possibility of delivering repeated injections at different tissue depths from the dermis to the muscle (depending on the drug delivery requirements) by controlling the inlet compressed air pressure. The designed needle-free jet injector consists of two primary circuits: the pneumatic and the hydraulic circuit. The pneumatic circuit is responsible for driving, pressurizing, and repeatability. The hydraulic circuit precisely injects and contains the liquid jet, allowing us to control the volume of the liquid jet at elevated pressure by offering flexibility in the dose volume per injection. Finally, in this paper we report on the successful design and working model of an air-driven needle-free jet injector for 0.2–0.5 mL drug delivery by ex vivo experimental validation.

Finite Arithmetic Axiomatization for the Basis of Hyperrational Non-Standard Analysis

The standard elementary number theory is not a finite axiomatic system due to the presence of the induction axiom scheme. Absence of a finite axiomatic system is not an obstacle for most tasks, but may be considered as imperfect since the induction is strongly associated with the presence of set theory external to the axiomatic system. Also in the case of logic approach to the artificial intelligence problems presence of a finite number of basic axioms and states is important. Axiomatic hyperrational analysis is the axiomatic system of hyperrational number field. The properties of hyperrational numbers and functions allow them to be used to model real numbers and functions of classical elementary mathematical analysis. However hyperrational analysis is based on well-known non-finite hyperarithmetic axiomatics. In the article we present a new finite first-order arithmetic theory designed to be the basis of the axiomatic hyperrational analysis and, as a consequence, mathematical analysis in general as a basis for all mathematical application including AI problems. It is shown that this axiomatics meet the requirements, i.e., it could be used as the basis of an axiomatic hyperrational analysis. The article in effect completes the foundation of axiomatic hyperrational analysis without calling in an arithmetic extension, since in the framework of the presented theory infinite numbers arise without invoking any new constants. The proposed system describes a class of numbers in which infinite numbers exist as natural objects of the theory itself. We also do not appeal to any “enveloping” set theory.

Cooperative and axiomatic approaches to the knapsack allocation problem

In the knapsack problem a group of agents want to fill a knapsack with several goods. Two issues must be considered. The first is to decide optimally what goods to select for the knapsack. This issue has been studied in many papers in the literature on Operations Research and Management Science. The second issue is to divide the total revenue among the agents. This issue has been studied in only a few papers, and this is one of them. For each knapsack problem we consider three associated cooperative games. One of them (the pessimistic game) has already been considered in the literature. The other two (realistic and optimistic games) are defined in this paper. The pessimistic and realistic games have non-empty cores but the core of the optimistic game could be empty. We then follow the axiomatic approach. We propose two rules: The first is based on the optimal solution of the knapsack problem. The second is the Shapley value of the so called optimistic game. We offer axiomatic characterizations of both rules.