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Vagueness in scientific studies poses a challenge. Intuitionistic Fuzzy Set (IFS) theory has emerged as a powerful and flexible tool to counter such challenge. So, there is a need to develop such measures which can not only measure the vagueness but also quantify the differences in underlying IFSs. The aim of this communication is to introduce one such divergence measure called Intuitionistic Fuzzy Jensen-Tsalli Divergence measure in the settings of IFS theory. The presence of parameters makes the proposed divergence measure flexible and competent for applications. Besides discussing some of its major properties, the findings are applied in pattern recognition problem and in medical diagnosis of some diseases with same set of symptoms. The performance of the proposed measure is genuinely compared with some other existing measures in literature through numerical examples based on medical diagnosis and pattern recognition. [ABSTRACT FROM AUTHOR]

In order to study the axiomatization of the construct over possibly non-halting programs and tests, this paper introduces the notion of -sets by considering the tests from an abstract -algebra. When the -algebra is an ada, the axiomatization is shown to be complete by obtaining a subdirect representation of -sets. Further, this paper considers the equality test with the construct and gives a complete axiomatization through the notion of agreeable -sets. [ABSTRACT FROM AUTHOR]

In the general context of presentations of monoids, we study normalization process that are determined by their restriction to length-two words. Garside's greedy normal forms and quadratic convergent rewriting systems, in particular those associated with the plactic monoids, are typical examples. Having introduced a parameter, called the class and measuring the complexity of the normalization of length-three words, we analyze the normalization of longer words and describe a number of possible behaviors. We fully axiomatize normalizations of class , show the convergence of the associated rewriting systems, and characterize those deriving from a Garside family. [ABSTRACT FROM AUTHOR]

Let ℬn be the variety of groups satisfying the identity xn = 1. It is proved that for every sufficiently large prime p, say p > 1010, the product ℬpℬp cannot be defined by a finite set of identities. This solves the problem formulated by C. K. Gupta and A. N. Krasilnikov in 2003. We also find the axiomatic and the basis ranks of the variety ℬpℬp. For this goal, we improve the estimate for the basis rank of the product of group varieties obtained by G. Baumslag, B. H. Neumann, H. Neumann and P. M. Neumann in 1964. [ABSTRACT FROM AUTHOR]

A median of a sequence π = (x1, x2, ..., xk) of elements of a finite metric space (X, d) is an element x for which is minimum. The function with domain the set of all finite sequences on X and defined by Med(π) = {x | x is a median of π} is called the Median function on X. In this note, an axiomatic characterization of the median function on finite trees is given. [ABSTRACT FROM AUTHOR]

Many researchers have put forward lots of entropy and distance measures for intuitionistic fuzzy sets (IFSs). This paper firstly reviews some basic concepts for IFSs and several widely used distance measures between IFSs. And then a series of distance measures are presented on the basis of above widely used distance measures. Based on such distance measures and intuitionistic index, a set of entropies for IFSs are proposed and proved to meet the axiomatic requirements given by Szmidt and Kacprzyk in 2001. Besides, two numerical examples are demonstrated to verify the efficiency of the proposed entropies for IFSs. Finally, the paper concludes with suggestions for future research. [ABSTRACT FROM AUTHOR]

In this paper, we study to what extent decidability is connected to universality. A natural context for such a study is provided by the axiomatic theory of computability, automata and algorithms. In Section 2, we introduce necessary concepts, constructions, axioms, postulates, conditions and problems. Section 3 contains the main results of the paper. In particular, it is demonstrated (Theorems 1 and 2) that undecidability of algorithmic problems does not depend on existence of universal algorithms and may be caused by weaker conditions. At the same time, results of Theorems 2 and 3 demonstrate that decidability is incompatible with universality. It is also proved (Theorem 5) that sufficiently big classes of total algorithms/automata, such as the class of all primitive recursive functions, cannot have universal algorithms/automata. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy (IF) rough sets are the generalization of traditional rough sets obtained by combining the IF set theory and the rough set theory. The existing research on IF rough sets mainly concentrates on the establishment of lower and upper approximation operators using constructive and axiomatic approaches. Less effort has been put on the attribute reduction of databases based on IF rough sets. This paper systematically studies attribute reduction based on IF rough sets. Firstly, attribute reduction with traditional rough sets and some concepts of IF rough sets are reviewed. Then, we introduce some concepts and theorems of attribute reduction with IF rough sets, and completely investigate the structure of attribute reduction. Employing the discernibility matrix approach, an algorithm to find all attribute reductions is also presented. Finally, an example is proposed to illustrate our idea and method. Altogether, these findings lay a solid theoretical foundation for attribute reduction based on IF rough sets. [ABSTRACT FROM AUTHOR]

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e. a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal 픪 in a commutative ring R, we show how solving (in)homogeneous linear systems over R픪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R픪. As a corollary, we obtain the computability of the category of finitely presented R픪-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings, we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm. [ABSTRACT FROM AUTHOR]

A ternary relation ${\hbox to 0pt{\hss\mid$\hss}\limits_\hbox to 0pt{\hss\smile$\hss}}$ between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be closely related to modular pairs in the lattice of algebraically closed sets. [ABSTRACT FROM AUTHOR]

We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thorn-forking, thorn-dividing, splitting or finite satisfiability share in all complete theories. We examine the relation between four additional axioms (extension, local character, full existence and symmetry) that one expects of a good notion of independence. We show that thorn-forking can be described in terms of local forking if we localize the number k in Kim's notion of "dividing with respect to k" (using Ben-Yaacov's "k-inconsistency witnesses") rather than the forking formulas. It follows that every theory with an M-symmetric lattice of algebraically closed sets (in Teq) is rosy, with a simple lattice theoretical interpretation of thorn-forking. [ABSTRACT FROM AUTHOR]

It is here shown how to use pieces of macroscopic thermodynamics to generate microscopic probability distributions for generalized ensembles, thereby directly connecting macro-state-axiomatics with microscopic results. [ABSTRACT FROM AUTHOR]

The implication of multivalued dependencies (MVDs) in relational databases has originally and independently been defined in the context of some fixed finite universe by Delobel, Fagin, and Zaniolo. Biskup observed that the original axiomatisation for MVD implication does not reflect the fact that the complementation rule is merely a means to achieve database normalisation. He proposed two alternative ways to overcome this deficiency: i) an axiomatisation that does represent the role of the complementation rule adequately, and ii) a notion of MVD implication in which the underlying universe of attributes is left undetermined together with an axiomatisation of this notion. In this paper we investigate multivalued dependencies with null values (NMVDs) as defined and axiomatised by Lien. We show that Lien's axiomatisation does not adequately reflect the role of the complementation rule, and extend Biskup's findings for MVDs in total database relations to NMVDs in partial database relations. Moreover, a correspondence between (minimal) axiomatisations in fixed universes that do reflect the property of complementation and (minimal) axiomatisations in undetermined universes is shown. [ABSTRACT FROM AUTHOR]

In 1908, Minkowski [13] used space-like binary velocity-field of a medium, relative to an observer. In 1974, Hestenes introduced, within a Clifford algebra, an axiomatic binary relative velocity as a Minkowski bivector [7, 8]. We propose to consider binary relative velocity as a traceless nilpotent endomorphism in an operator algebra. Any concept of a binary axiomatic relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of (ternary) relative velocities in isometric special relativity (loop structure). We consider an algebra of many time-plus-space splits, as an operator algebra generated by idempotents. The kinematics of relativity groupoid is ruled by associative Frobenius operator algebra, whereas the dynamics of categorical relativity needs the non-associative Frölicher–Richardson operator algebra. The Lorentz covariance is the cornerstone of physical theory. Observer-dependence within relativity groupoid, and the Lorentz-covariance within the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than Lorentz-invariant. [ABSTRACT FROM AUTHOR]

In a recent paper [International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems 8:611–618], Yager & Kreinovich characterize a certain proportional division rule in terms of the three axioms (1) symmetry, (2) participant mergability, and (3) continuity. This technical note tightens their characterization: The proportional division rule is already fully characterized by the first two axioms (1) symmetry and (2) participant mergability. [ABSTRACT FROM AUTHOR]

Many scientists have struggled to uncover the elusive origin of "complexity", and its many equivalent jargons, such as emergence, self-organization, synergetics, collective behaviors, nonequilibrium phenomena, etc. They have provided some qualitative, but not quantitative, characterizations of numerous fascinating examples from many disciplines. For example, Schrödinger had identified "the exchange of energy" from open systems as a necessary condition for complexity. Prigogine has argued for the need to introduce a new principle of nature which he dubbed "the instability of the homogeneous". Turing had proposed "symmetry breaking" as an origin of morphogenesis. Smale had asked what "axiomatic" properties must a reaction–diffusion system possess to make the Turing interacting system oscillate. The purpose of this paper is to show that all the jargons and issues cited above are mere manifestations of a new fundamental principle called local activity, which is mathematically precise and testable. The local activity theorem provides the quantitative characterization of Prigogine's "instability of the homogeneous" and Smale's quest for an axiomatic principle on Turing instability. Among other things, a mathematical proof is given which shows none of the complexity-related jargons cited above is possible without local activity. Explicit mathematical criteria are given to identify a relatively small subset of the locally-active parameter region, called the edge of chaos, where most complex phenomena emerge. [ABSTRACT FROM AUTHOR]

In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2ω = ω2 and that $L({\mathscr{P}}(\omega_1))$ satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that □(κ) fails for all regular κ > ω1. [ABSTRACT FROM AUTHOR]

Inspired by Borcherds' work on "G-vertex algebras," we formulateand study an axiomatic counterpart of Borcherds' notion of G-vertex algebra for the simplest nontrivial elementary vertex group, which we denote by G[SUB1]. Specifically, we formulate a notion of axiomatic G[SUB1]-vertex algebra, prove certain basic properties and give certain examples, where the notion of axiomatic G[SUB1]-vertex algebra is a nonlocal generalization of the notion of vertex algebra. We also show how to construct axiomatic G[SUB1]-vertex algebras from a set of compatible G[SUB1]-vertex operators. [ABSTRACT FROM AUTHOR]