Axiomatic Description Research Articles

The axiomatic characterization on fuzzy variable precision rough sets based on residuated lattice

Axiomatization is a lively research direction in fuzzy rough set theory. Fuzzy variable precision rough set (FVPRS) incorporates fault-tolerant factors to fuzzy rough set, so its axiomatic description becomes more complicated and difficult to realize. In this paper, we present an axiomatic approach to FVPRSs based on residuated lattice (L-fuzzy variable precision rough set (LFVPRS)). First, a pair of mappings with three axioms is utilized to characterize the upper (resp., lower) approximation operator of LFVPRS. This is distinct from the characterization on upper (resp., lower) approximation operator of fuzzy rough set, which consists of one mapping with two axioms. Second, utilizing the notion of correlation degree (resp., subset degree) of fuzzy sets, three characteristic axioms are grouped into a single axiom. At last, various special LFVPRS generated by reflexive, symmetric and transitive L-fuzzy relation and their composition are also characterized by axiomatic set and single axiom, respectively.

No second law of entanglement manipulation after all

Many fruitful analogies have emerged between the theories of quantum entanglement and thermodynamics, motivating the pursuit of an axiomatic description of entanglement akin to the laws of thermodynamics. A long-standing open problem has been to establish a true second law of entanglement, and in particular a unique function that governs all transformations between entangled systems, mirroring the role of entropy in thermodynamics. Contrary to previous promising evidence, here we show that this is impossible and that no direct counterpart to the second law of thermodynamics can be established. This is accomplished by demonstrating the irreversibility of entanglement theory from first principles. Assuming only the most general microscopic physical constraints of entanglement manipulation, we show that entanglement theory is irreversible under all non-entangling transformations. We furthermore rule out reversibility without significant entanglement expenditure, showing that reversible entanglement transformations require the generation of macroscopically large amounts of entanglement according to certain measures. Our results not only reveal fundamental differences between quantum entanglement transformations and thermodynamic processes, but also showcase a unique property of entanglement that distinguishes it from other known quantum resources.

Всегда в моей памяти…

The article publishes memories of a mentor, an outstanding mathematician, academician Tashmukhamad Alievich Sarimsakov, who for many years conducted scientific activities at the Tashkent State University and was engaged in scientific development of “Theory of semi fields and its applications in probability theory”. He always supported new directions in science and gave his students the freedom to choose the direction of research, and then he himself connected to these studies, feeling their promise. At one time, the academician put forward the idea of creating a non-commutative analogue of semi fields for an axiomatic description of the algebra of random variables in non-commutative quantum probability theory. Sarimsakov also wrote several scientific monographs, one of them «Introduction to Quantum Probability Theory», which was highly appreciated by specialists. Several of his monographs and textbooks in the field of operator algebra were republished with additions, and he prepared a number of high-class specialists.

Gift Game Metatheory: Social Interaction and Interpersonal Growth.

This paper analyzes Game Theory (GT) from the point of view of moral psychology and makes explicit some of its assumptions regarding the human person as a moral agent, as well as the ends of human action, and reciprocity. Using a largely philosophical methodology, we will argue that GT assumes an instrumental form of rationality underpinned by a logic of self-interest, hence placing individuals, communities, and their social practices in service of external goods and their maximization. Because of this, GT is not adequate to describe the entirety of human social existence and interaction. Nevertheless, by revealing these assumptions, GT can be amplified with another form of rationality based on realist ethics and a personalist anthropology reinforced by the logic of gift. This rationality values the singularity of each person as a holistic unity, as the center of the social realm and as an end in herself called to growth and flourishing with others, nurturing the human community through giving and receiving. We will thus provide a wider philosophical framework for GT with a series of non-mathematical axioms of what can be called a Game Metatheory (GMt). These axioms refer to society as a complex system, not to particular interactions. GMt axioms are not a model of social games, but rather an axiomatic description of social life as a game, revealing its systematic character, complexity, and possible deterioration.

Local Mathematics and No Information at a Distance; Some Effects on Physics and Geometry

Local mathematics consists of a collection of mathematical systems located at each space and time point. The collection is limited to the systems that include numbers in their axiomatic description. A scalar map between systems at different locations is based on the distinction of two conflated concepts, number and number value. The effect that this setup has on theory descriptions of physical and geometric systems is described. This includes a scalar spin 0 field in gauge theories, expectation values in quantum mechanics and path lengths in geometry. The possible relation of the scalar map to consciousness is noted.

Generalized triparametric correlation coefficient for Pythagorean fuzzy sets with application to MCDM problems

Pythagorean fuzzy set (PFS) is an advanced version of intuitionistic fuzzy set which generalizes fuzzy set. Consequently, PFS has a better applicative expression in real-life decision-making (RLDM) or multicriteria decision-making (MCDM) problems due to its capacity to curb uncertainties embedded in decision making. Correlation coefficient is a significant measuring tool applicable to solving RLDM/MCDM problems via Pythagorean fuzzy environment approach. The main aim of this paper is to reexamine Garg’s correlation coefficient for PFSs and generalize it for a better output in resolving MCDM problems. The axiomatic description of correlation coefficient for PFSs is proposed, and the generalized triparametric correlation coefficient for PFSs is characterized with some number of results. Numerical verification of the proposed correlation coefficient is given to validate the preeminence of the generalized correlation coefficient for PFSs over Garg’s approach. Lastly, some MCDM problems such as pattern recognition problem (e.g., classification of mineral fields) and diagnostic medicine in the framework of Pythagorean fuzzy pairs are discussed with the aid of the novel correlation coefficient. This proposed measuring tool could be exploited in other MCDM problems via object-oriented approach.

Ontology Model for Process Level Capabilities of Manufacturing Resources

The promise of distributed cloud manufacturing (CM) is to manufacture products using shared resources, both asset-full (machines, tools, vehicles) and asset-light (design, analysis, inspection, management, maintenance), which can be provisioned flexibly and rapidly with minimal management and service provider interaction. One of the key enabler of CM are virtual enterprises (VE), which offer manufacturing resources as virtual services (SaaS, HaaS, PaaS) in a cloud based marketplace. Currently, virtualization and provisioning of diverse array of manufacturing resources face challenges from the heterogeneity in representation and communication protocols, as well as lack of integration with legacy practices in the organizations. Aiming to increase interoperability, a number of formal ontologies were developed by researchers in the past, to leverage on semantic data integration and validation. In spite of their success in providing axiomatic description and common taxonomy to classify manufacturing resources from different domains, models of representing the capabilities of the resources (i.e. expected quality of services they offer) were often overlooked. In this research, we present an ontology model to represent capabilities of manufacturing machine-tools at the process level, often called process boundaries (measured by process capability index) in industries. The definition of the capability is derived based on the foundational ontology called ‘Basic Formal Ontology’ (BFO). The primary contribution of this extension is a set of OWL axioms which can be used to assert facts about modal future (possibilia) – ultimately enabling us to associate process specific performance metrics to the semantic models of virtual machine-tools.

Buckling and wrinkling of anisotropic sandwich plates

The global buckling and wrinkling behavior of sandwich plates with anisotropic facesheets is investigated by means of a linearized stability analysis. High-order plate models are formulated referring to a sublaminate variable-kinematic approach: an axiomatic through-thickness description of the displacement field is introduced, wherein the selected model is employed for an arbitrarily defined group of plies, i.e., the sublaminate. The two-dimensional governing equations of the plate model are solved in weak form by means of the Ritz method. The modeling approach is applied to sandwich plates with anisotropic laminated facesheets and orthotropic cores. A wide set of configurations is analyzed: critical loads and wrinkling patterns are determined for panels with foam and honeycomb cores, subjected to uniaxial as well as multiaxial loads. The proposed approach is shown to provide accurate, quasi-3D predictions for both long and short wavelength buckling with a reduced computational effort.

On the Relativistic Canonical Quantum Mechanics and Field Theory of Arbitrary Spin

The new relativistic equations of motion for the particles with arbitrary spin and nonzero mass, suggested by author in years 2014–2016, are under consideration. The complete version of brief paper in J. Phys: Conf. Ser., 670 (2016) 012047(1-16) is given. The axiomatic level description of the relativistic canonical quantum mechanics of an arbitrary mass and spin has been given. The 64-dimensional ClR(0,6) algebra in terms of Dirac gamma matrices has been suggested. The interpretation of the 28-dimensional gamma matrix representation of SO(8) algebra over the field of real numbers is given. The link between the relativistic canonical quantum mechanics of the arbitrary spin and the covariant local field theory in the form of extended Foldy–Wouthuysen transformation has been found. Different methods of the Dirac equation derivation have been reviewed. The manifestly covariant field equation for an arbitrary spin that follows from the corresponding equation of relativistic canonical quantum mechanics, has been considered. The found equations are without redundant components. The partial examples for spin s=3/2 and s=2 are presented. The covariant local field theory equations for spin s = (3/2,3/2) particle-antiparticle doublet and spin s = (2,2) particle-antiparticle doublet have been introduced. The Maxwell and slightly generalized Maxwell-like equations containing mass member have been considered as well.

Homology for one-dimensional solenoids

Smale spaces are a particular class of hyperbolic topological dynamical systems, defined by David Ruelle. The definition was introduced to give an axiomatic description of the dynamical properties of Smale's Axiom A systems when restricted to a basic set. They include Anosov diffeomeorphisms, shifts of finite type and various solenoids constructed by R. F. Williams. The second author constructed a homology theory for Smale spaces which is based on (and extends) Krieger's dimension group invariant for shifts of finite type. In this paper, we compute this homology for the one-dimensional generalized solenoids of R. F. Williams.

On the Axiomatic Definition of Generalized Maximin Principle

The paper develops the solution to the problem of axiomatic definition of the maxmin expected utility decision criteria under uncertainty. The axiomatic description of the corresponding preference relation is proposed. It is based on the specific form of the principle of guaranteed result.

Link between the relativistic canonical quantum mechanics of arbitrary spin and the corresponding field theory

The new relativistic equations of motion for the particles with arbitrary spin and nonzero mass have been introduced. The axiomatic level description of the relativistic canonical quantum mechanics of the arbitrary mass and spin has been given. The 64-dimensional ClR(0,6) algebra in terms of Dirac gamma matrices has been suggested. The link between the relativistic canonical quantum mechanics of the arbitrary spin and the covariant local field theory has been found. Different methods of the Dirac equation derivation have been reviewed. The manifestly covariant field equations for an arbitrary spin that follow from the quantum mechanical equations have been considered. The covariant local field theory equations for spin s = (1,1) particle-antiparticle doublet, spin s = (1,0,1,0) particle antiparticle multiplet, spin s = (3/2,3/2) particle-antiparticle doublet, spin s = (2,2) particle-antiparticle doublet, spin s = (2,0,2,0) particle-antiparticle multiplet and spin s = (2,1,2,1) particle-antiparticle multiplet have been introduced. The Maxwell-like equations for the boson with spin s = 1 and nonzero mass have been introduced as well.

The behavior of a finite-state automaton in a fuzzy environment: Theory and applications

The behavior of M.L. Tsetlin's automaton with linear tactics in a fuzzy environment is considered. The behavior of such an automaton was previously studied in a random environment, where the classical apparatus of probabilistic Markov chains could be used. This work is based on two results that were previously obtained by the author, namely, a theory that generalizes Markov chains and the axiomatic description of operations with fuzzy evidence. The explicit formulas for describing the behavior of the automaton that were obtained in this paper suggest a deep analogy with the results of M.L. Tsetlin. In particular, it is shown that this automaton in a fuzzy environment also has the property of asymptotic optimality. As an application, we discuss the possibility of using this property to measure membership functions for values that are similar to singletons or point functions. It is concluded that the results that are obtained in this paper make it possible to construct an analogue of the statistics collection procedure for the field of fuzzy systems.

Generalized Unified Formulation shell element for functionally graded Variable-Stiffness Composite Laminates and aeroelastic applications

Composite materials have been extensively used in engineering thanks to their lightweight, superior mechanical performances and possibility to tailor the structural behavior, increasing the available design space. Variable Angle Tow (VAT) structures exploits this advantage by adopting a curvilinear patterns for the fibers constituting the lamina.This work, for the first time, extends the Generalized Unified Formulation (GUF) to the case of fourth-order triangular shell elements and VAT composites. Functionally graded material properties in both the thickness and in-plane directions are also possible. The finite element has been formulated with layers of variable thickness with respect to the in-plane coordinates.GUF is a very versatile tool for the analysis of Variable Stiffness Composite Laminates (VSCLs): it is possible to select generic element coordinate systems and define different types of axiomatic descriptions (Equivalent Single Layer, Layer Wise, and Zig-Zag enhanced formulations) and orders of the thickness expansions. Each displacement is independently treated from the others. All the infinite number of theories that can be generated with GUF are obtained by expanding six theory-invariant kernels (formally identical for all the elements), allowing a very general implementation. Finally, the possibility of tailoring the theory/order to increase the accuracy in desired directions makes the GUF VAT capability a very powerful tool for the design of aerospace structures.

General Yang–Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical frameworkorFrom Bianchi identities to twisted Courant algebroids

Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into what is called a Q-structure or, equivalently, an L∞-algebroid. This has many technical and conceptual advantages: complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator Q squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines and is given by what is called the derived bracket construction. This paper aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are valid for arbitrary highest form degree p, we pay particular attention to p = 2, i.e. 1- and 2-form gauge fields coupled nonlinearly to scalar fields (0-form fields). The structural identities of the coupled system correspond to a Lie 2-algebroid in this case and we provide different axiomatic descriptions of those, inspired by the application, including e.g. one as a particular kind of a vector-bundle twisted Courant algebroid.

A Topological Approach in the Extended Fraenkel-Mostowski Model of Set Theory

Abstract Lattices of subgroups are presented as algebraic domains. Given an arbitrary group, we define the Scott topology over the subgroups lattice of that group. A basis for this topology is expressed in terms of finitely generated subgroups. Several properties of the continuous functions with respect the Scott topology are obtained; they provide new order properties of groups. Finally there are expressed several properties of the group of permutations of atoms in a permutative model of set theory. We provide new properties of the extended interchange function by presenting some topological properties of its domain. Several order and topological properties of the sets in the Fraenkel-Mostowski model remains also valid in the Extended Fraenkel-Mostowski model, even one axiom in the axiomatic description of the Extended Fraenkel-Mostowski model is weaker than its homologue in the axiomatic description of the Fraenkel-Mostowski model.

Utilitarianism on infinite utility streams: summable differences and finite averages

This paper provides axiomatic descriptions of social welfare relations, defined on infinite streams of utility, that are consistent with the utilitarian criterion on subsets where maximizing aggregate utility has a clear interpretation: the streams, or their differences, are summable. Besides standard assumptions on efficiency, equity and interpersonal comparability, two axioms are introduced and shown to be necessary and sufficient. A more general version of one axiom suffices to distinguish streams with different long-run averages.

Herding Cats

We propose an axiomatic generic framework for modelling weak memory. We show how to instantiate this framework for Sequential Consistency (SC), Total Store Order (TSO), C++ restricted to release-acquire atomics, and Power. For Power, we compare our model to a preceding operational model in which we found a flaw. To do so, we define an operational model that we show equivalent to our axiomatic model. We also propose a model for ARM. Our testing on this architecture revealed a behaviour later acknowledged as a bug by ARM, and more recently, 31 additional anomalies. We offer a new simulation tool, called herd, which allows the user to specify the model of his choice in a concise way. Given a specification of a model, the tool becomes a simulator for that model. The tool relies on an axiomatic description; this choice allows us to outperform all previous simulation tools. Additionally, we confirm that verification time is vastly improved, in the case of bounded model checking. Finally, we put our models in perspective, in the light of empirical data obtained by analysing the C and C++ code of a Debian Linux distribution. We present our new analysis tool, called mole, which explores a piece of code to find the weak memory idioms that it uses.

The complexity in sociological theory: an axiomatic description

This article puts forth an axiomatic description of the complexity of an object of sociological investigation. The proposed axioms allow us to determine complexity within the framework of mathematical sociology such as the variational principle, which is formed relative to the state of the object of sociological investigation. On the basis of this principle we can conclude the equation of state, which coincides with the stationary forward Kolmogorov equation.

Axiomatic of Fuzzy Complex Numbers

Fuzzy numbers are fuzzy subsets of the set of real numbers satisfying some additional conditions. Fuzzy numbers allow us to model very difficult uncertainties in a very easy way. Arithmetic operations on fuzzy numbers have also been developed, and are based mainly on the crucial Extension Principle. When operating with fuzzy numbers, the results of our calculations strongly depend on the shape of the membership functions of these numbers. Logically, less regular membership functions may lead to very complicated calculi. Moreover, fuzzy numbers with a simpler shape of membership functions often have more intuitive and more natural interpretations. But not only must we apply the concept and the use of fuzzy sets, and its particular case of fuzzy number, but also the new and interesting mathematical construct designed by Fuzzy Complex Numbers, which is much more than a correlate of Complex Numbers in Mathematical Analysis. The selected perspective attempts here that of advancing through axiomatic descriptions.