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Studies of Aristotle’s syllogistic system, since Corcoran’s deductionist interpretation supplanted Łukasiewicz’ axiomaticist interpretation, misrepresent Aristotle’s logic in two important respects. Following Corcoran, they take indirect deduction to occur only once in a deduction discourse; they then obviate the system having a reductio rule. Second, they represent reduction as a deductive process for deriving ‘imperfect’ syllogisms from ‘perfect’ syllogisms to impose an axiomatic interpretation on the logic. Denying that Aristotle's logic admits of a reductio rule results from this misrepresentation of reduction. This discussion shows that the defects imputed to Aristotle's logic, and systems devised to resolve them, result from misunderstanding reduction, which itself results from misapprehending Prior Analytics expressly to identify in a metadiscourse deduction rules and not deductions per se. [ABSTRACT FROM AUTHOR]

Totalisation of operations and functions is a subject of great importance in the precise sciences at large, and in particular, in computer science, where computed functions must be totalised to account for edge cases and prevent system failures. In this paper, we investigate the totalisation of computable functions in the general and abstract setting of the axiomatic theory of computing systems, such as automata, computer networks, and algorithms, allowing us to define characteristics and discover relations which resonate true through many computational models – from individual automata through complex distributed systems with concurrent functioning. We explore the possible types of totalisations, the complexes of computing systems that do (or do not) have totalisations, and the behaviour of totalisation in various transformations and compositions. In addition, we point to a few interesting directions for future research. [ABSTRACT FROM AUTHOR]

In this paper, classes of automata that perform distributed computations with unconventional interaction are described and studied. These automata are called selective machines and they are more powerful than Turing machines while their high computing and recognising power can be achieved exclusively by interaction when a system of recursive algorithms (automata) becomes super-recursive due to their interaction. Computations of selective machines are described by selective algorithms, which are super-recursive allowing computations of functions that are incomputable by Turing machines. Examples of selective algorithms are grammars with prohibition, correction grammars and grammars with exclusion. The study of selective machines and selective algorithms is based on the axiomatic theory of algorithms, in which the results are obtained in the general situation of axiomatically defined classes of automata and algorithms. Then these results are specified for many concrete classes of automata and algorithms, such as finite automata or Turing machines, by checking the necessary axioms. [ABSTRACT FROM AUTHOR]

In this study, we consider different cities located along a tram line. Each city may have one or several stations and information is available about the flow of passengers between any pair of stations. A fixed cost (salaries of the executive staff, repair facilities, or fixed taxes) must be divided among the cities. This cost is independent of the number of passengers and the length of the line. We propose a mathematical model to identify suitable mechanisms for sharing the fixed cost. In the proposed model, we study, and characterize axiomatically, three rules, which include the uniform split, the proportional allocation and an intermediate situation. The analyzed axioms represent the basic requirements for fairness and elemental properties of stability. [ABSTRACT FROM AUTHOR]

If we start with continuous absolute geometry, remove Side-Angle-Side as an axiom and replace it with exactly one of Angle-Side-Angle, Side-Angle-Angle, or Side-Side-Side as a new axiom, then we again end up with continuous absolute geometry. Martin has asked the question: if we start with absolute geometry, remove Side-Angle-Side as an axiom and re- place it with Angle-Angle-Angle as a new axiom, then do we again end up with absolute geometry? In this paper, we give a negative answer to this question for non-continuous absolute geometry. In particular, we construct a non-continuous model A in which the Angle-Angle-Angle criterion for con- gruence of triangles holds, yet in which Side-Angle-Side does not hold. [ABSTRACT FROM AUTHOR]

Abstract: High‐density SNP microarrays (“SNP chips”) are a rapid, accurate and efficient method for genotyping several hundred thousand polymorphisms in large numbers of individuals. While SNP chips are routinely used in human genetics and in animal and plant breeding, they are less widely used in evolutionary and ecological research. In this article, we describe the development and application of a high‐density Affymetrix Axiom chip with around 500,000 SNPs, designed to perform genomics studies of great tit (Parus major) populations. We demonstrate that the per‐SNP genotype error rate is well below 1% and that the chip can also be used to identify structural or copy number variation. The chip is used to explore the genetic architecture of exploration behaviour (EB), a personality trait that has been widely studied in great tits and other species. No SNPs reached genomewide significance, including at DRD4, a candidate gene. However, EB is heritable and appears to have a polygenic architecture. Researchers developing similar SNP chips may note: (i) SNPs previously typed on alternative platforms are more likely to be converted to working assays; (ii) detecting SNPs by more than one pipeline, and in independent data sets, ensures a high proportion of working assays; (iii) allele frequency ascertainment bias is minimized by performing SNP discovery in individuals from multiple populations; and (iv) samples with the lowest call rates tend to also have the greatest genotyping error rates. [ABSTRACT FROM AUTHOR]

Based on the in-depth analysis of the essence and key features of vague phenomena, this paper focuses on establishing the axiomatical foundation of membership degree theory using for modeling vague phenomena, presents an axiomatic system to govern membership degrees and their interconnections. The concept of vague partition is introduced, on this basis, the concept of fuzzy set in Zadeh’s sense is redefined based on vague partition from the perspective of axiomatization. The thesis defended in this paper is that the mutual constraint relationship among vague attribute values in a vague partition should be the starting point to recognize and model vague phenomena by the quantitative analysis method. [ABSTRACT FROM AUTHOR]

Unified Temporal Logic (UTL) combines all characteristics of traditional Linear Temporal Logic (LTL) and Propositional Projection Temporal Logic (PPTL). It can be used to describe full regular and omega-regular properties, which are often encountered in the field of formal verification. To support formal verification with UTL, this paper proposes a proof system for UTL. First, the syntax and semantics of UTL are briefly introduced. Further, axioms and inference rules are formalized. Besides, a number of theorems are derived and proved to refine the system. Moreover, the soundness and completeness of the proof system are proved in detail. To facilitate the completeness proof, some auxiliary lemmas are introduced and proved in advance. Finally, an example is given to illustrate how to use this proof system for verifying properties of a system. [ABSTRACT FROM AUTHOR]

Revealed preference theory provides axiomatic tools for assessing whether individuals make observable choices "as if" they are maximizing an underlying utility function. The theory evokes a tradeoff between goods whereby individuals improve themselves by trading one good for another good to obtain the best combination. Preferences revealed in these choices are modeled as curves of equal choice (indifference curves) and reflect an underlying process of optimization. These notions have far-reaching applications in consumer choice theory and impact the welfare of human and animal populations. However, they lack the empirical implementation in animals that would be required to establish a common biological basis. In a design using basic features of revealed preference theory, we measured in rhesus monkeys the frequency of repeated choices between bundles of two liquids. For various liquids, the animals' choices were compatible with the notion of giving up a quantity of one good to gain one unit of another good while maintaining choice indifference, thereby implementing the concept of marginal rate of substitution. The indifference maps consisted of nonoverlapping, linear, convex, and occasionally concave curves with typically negative, but also sometimes positive, slopes depending on bundle composition. Out-of-sample predictions using homothetic polynomials validated the indifference curves. The animals' preferences were internally consistent in satisfying transitivity. Change of option set size demonstrated choice optimality and satisfied the Weak Axiom of Revealed Preference (WARP). These data are consistent with a version of revealed preference theory in which preferences are stochastic; the monkeys behaved "as if" they had well-structured preferences and maximized utility. [ABSTRACT FROM AUTHOR]

Islamic finance, as a complementary component among financial instruments, provides sustainable material and spiritual welfare with its fairer, more participative and more sharing structure, by encouraging private individuals and legal entities that wishes to make investments and facilitate access to financing in economy. Islamic finance is unique and alternative financial methods, adherence to religious rules, it is necessary in terms of legal and structural. At this point, latest improvements notice that Islamic finance is acting in a capitalist system apart from an Islamic economic axioms system. With the speedy evolution of Islamic finance, inquiry condensationing on the constitutive foundation has been highly disregarded. However, without setting the basic axioms, which is the morality system of this new current, it is not feasible to have a strong financial system, in particular of Islamic finance. This article, for this reason, investigates individual axioms in illuminating and alternate system address of Islamic financial in any case whether such a structure is probable or not. [ABSTRACT FROM AUTHOR]

The authors review theoretical rationale for creating a list of operational and tactical requirements, determine both necessary and sufficient conditions for developing alternative resources that enable the assigned functions. [ABSTRACT FROM AUTHOR]

This paper attempts to study the properties of Hsueh fuzzy matroids. Two equivalent descriptions of the fuzzy augmentation property are given and fuzzy submatroids are introduced. Moreover, the existence of fuzzy bases for Hsueh fuzzy matroids is proved. Based on the properties of fuzzy bases, fuzzy base axioms for Hsueh fuzzy matroids are obtained. Additionally, an application of fuzzy base axioms is presented. [ABSTRACT FROM AUTHOR]

LetSbe any semigroup anda,s ∈ S. Ifa = asa, thensis anassociateofa. A subgroupGofSis anassociate subgroupofSif everya ∈ Shas a unique associatea* inG. It turns out thatG = Hzfor some idempotentz, thezenithofS. The mappinga → a* is a unary operation onS. We say thatSismonogenicifSis generated, as a unary semigroup, by a single element. We embark upon the problem of the structure of monogenic semigroups in this sense by characterizing monogenic ones belonging to completely simple semigroups, normal cryptogroups, orthogroups, combinatorial semigroups, cryptic medial semigroups, cryptic orthodox semigroups, and orthodox monoids. In each of these cases, except one, we construct a free object. The general problem remains open. [ABSTRACT FROM AUTHOR]

We prove, in a formal way, that the Möbius configuration and one of its generalizations yield an elementary characterization of Pappian projective 3-space i.e. they close exactly in projective spaces coordinatized by fields (commutative division rings). [ABSTRACT FROM AUTHOR]

We consider unipotent, weakly cancellative, left cancellative, right cancellative, cancellative monoids and groups with the unary operation of mapping onto the identity element and bands with the identity mapping as unary operations. Then form Malcev products of each of the former with varieties of rectangular bands, semilattices, normal bands and all bands. Regarding the resulting semigroups with the induced unary operations, we characterize these classes, some of them we provide with a construction, and determine a basis for their implications, for they all turn out to be quasivarieties. We also determine their relationship which we clarify by an inclusion diagram. [ABSTRACT FROM AUTHOR]

In this paper, we will try to explain the importance of a mathematical proof. Mathematical logic, i.e., logical deduction are fundamental to understanding proofs. We will describe a logical language since it serves as the basis for deduction of proofs. We will also define basic types of statements as they also make part of the proving process. Finally, we will give several examples of mathematical proofs by using various techniques. [ABSTRACT FROM AUTHOR]

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a semigroup. Elements $a,b$ of $S$ are $\widetilde{\mathscr{R}}$-related if they have the same idempotent left identities. Then $S$ is weakly left ample if (1) idempotents of $S$ commute, (2) $\widetilde{\mathscr{R}}$ is a left congruence, (3) for any $a \in S$, $a$ is $\widetilde{\mathscr{R}}$-related to a (unique) idempotent, say $a^+$, and (4) for any element $a$ and idempotent $e$ of $S$, $ae=(ae)^+a$. Elements $a,b$ of $S$ are $\mathscr{R}^*$-related if, for any $x,y \in S^1$, $xa=ya$ if and only if $xb=yb$. Then $S$ is left ample if it satisfies (1), (3) and (4) relative to $\mathscr{R}^*$ instead of $\widetilde{\mathscr{R}}$. Further, $S$ is (weakly) ample if it is both (weakly) left and right ample. We establish several characterizations of these classes of semigroups. For weakly left ample ones we provide a construction of all such semigroups with zero all of whose nonzero idempotents are primitive. Among characterizations of weakly ample semigroups figure (strong) semilattices of unipotent monoids, and among those for ample semigroups, (strong) semilattices of cancellative monoids. This describes the structure of these two classes of semigroups in an optimal way, while, for the ‘one-sided’ case, the problem of structure remains open. [ABSTRACT FROM PUBLISHER]

The general axiomatics of information measures for the macro analysis of relations of an information-cybernetic system in the control is introduced. The general structure of a semantically marked graph of open and closed relations of an information-cybernetic system between the participants in the environment, as well as thenecessary axiomatic and technological information-cybernetic system conditions of controllability and observability of objects, for the case of a space vehicle in orbit, are justified. [ABSTRACT FROM AUTHOR]

Let S be a semigroup and s, t∈ S. We say that t is an associate of s if s= sts. If S has a maximal subgroup G such that every element s of S has a unique associate in G, say s, we say that G is an associate subgroup of S and consider the mapping s→ s as a unary operation on S. In this way, semigroups with an associate subgroup may be identified with unary semigroups satisfying three simple axioms. Among them, only those satisfying the identity ( st)= t s, called medial, have a structure theorem, due to Blyth and Martins. We introduce several relations germane to the presence of this unary operation. Next we characterize medial, orthodox, cryptic and medial, and cryptic and orthodox semigroups with an associate subgroup in terms of a construction and several choices of a basis of their identities. In addition, we characterize medial monoids and discuss F-inverse semigroups in terms of a construction and bases for their systems of identities. We establish an isomorphism between the lattice of varieties of cryptic medial monoids and the direct product of the lattice of band monoids and the lattice of group varieties. An embedding into idempotent generated semigroups is proved as well. [ABSTRACT FROM AUTHOR]

Abstract: This paper is a sequel to the 1995 paper On L-Tychonoff spaces. The embedding theorem for L-topological spaces is shown to hold true for L an arbitrary complete lattice without imposing any order reversing involution on L. Some results on completely L-regular spaces and on L-Tychonoff spaces, which have previously been known to hold true for a frame, are exhibited as ones holding for a meet-continuous lattice. For such a lattice an insertion theorem for completely L-regular spaces is given. Some weak forms of separating families of maps are discussed. We also clarify the dependence between the sub- separation axiom of Liu and the separation axiom of Rodabaugh. [Copyright &y& Elsevier]

Abstract: Consensus trees have been used routinely in systematic biology for over 30 years with the majority-rule consensus tree among the most popular. In this paper we give axioms that characterize a new consensus function, called majority-rule (+), that is based on a generalization of majority-rule to the supertree case. In addition, two other related consensus functions are characterized. [Copyright &y& Elsevier]

Let $\mathcal{U}$ be the class of all unipotent monoids and $\mathcal{B}$ the variety of all bands. We characterize the Malcev product $\mathcal{U} \circ \mathcal{V}$ where $\mathcal{V}$ is a subvariety of $\mathcal{B}$ low in its lattice of subvarieties, $\mathcal{B}$ itself and the subquasivariety $\mathcal{S} \circ \mathcal{RB}$, where $\mathcal{S}$ stands for semilattices and $\mathcal{RB}$ for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation $\widetilde{\mathcal{H}}= \widetilde{\mathcal{L}} \cap \widetilde{\mathcal{R}}$, where $a\;\,\widetilde{\mathcal{L}}\;\,b$ if and only if a and b have the same idempotent right identities, and $\widetilde{\mathcal{R}}$ is its dual. We also consider $(\mathcal{U} \circ \mathcal{RB}) \circ \mathcal{S}$ which provides the motivation for this study since $(\mathcal{G} \circ \mathcal{RB}) \circ \mathcal{S}$ coincides with completely regular semigroups, where $\mathcal{G}$ is the variety of all groups. All this amounts to a generalization of the latter: $\mathcal{U}$ instead of $\mathcal{G}$. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to study which quasi-metrizable spaces are monotonically normal. In particular, we provide a sufficient condition for a quasi-metrizable space to be monotonically normal. This enables us to prove the monotone normality of a certain amount of interesting examples of quasi-metric spaces; for instance, we show that the continuous poset of formal balls of a metric space, endowed with the Scott topology, is a monotonically normal quasi-metrizable space. [Copyright &y& Elsevier]

Human factors play a key role in many problems, especially those involve human. In this paper, we address the problem of quantifying private information with the consideration of human factors in data publishing for binary sensitive attributes. We first propose five axioms to capture the properties of private information in data publishing. In particular, the fifth axiom considers human factors in the view of social psychology. We believe that a function that quantifies private information has to satisfy all these five axioms. Therefore, we propose an expected gain model, which allows users to tune a weighting factor to reflect the importance of human factors in their applications. The proposed model has been analyzed and compared with some existing models, such as information gain. Moreover, experiments have been performed on real applications to study the practicality of the proposed model. [ABSTRACT FROM AUTHOR]

A set of axioms for classical absolute geometry is proposed that is accessible to students new to axioms. The metric axioms adopted are the ruler axiom, triangle inequality and the bisector axiom. Angle measure is derived from distance, and all properties needed to establish a consistent system are derived. In particular, the SAS congruence theorem is proved. The proofs are broken into many small steps suitable for the target audience. Some explorations with geometry drawing programmes are shown. [ABSTRACT FROM AUTHOR]

An orthogroup is a completely regular semigroup whose idempotents form a subsemigroup. For any semigroup S and a, x ∈ S, x is an associate of a if a = axa. A subgroup G of S is an associate subgroup of S if for every element a of S, G contains exactly one associate of a. An idempotent e of S is medial if for any element s of S which is a product of idempotents holds s = ses. We characterize orthogroups with an associate subgroup in several ways using the structural description of orthogroups as a semilattice of rectangular groups. Using the Blyth-Martins structure theorem for semigroups with an associate sub-group whose identity element is medial, we provide multiple characterizations of orthogroups, and some related semigroups, with an associate subgroup. [ABSTRACT FROM AUTHOR]

Abstract: An individual is asked to assess a real-valued variable y based on certain characteristics x =(x 1,…, x m ), and on a database consisting of n observations of (x 1,…, x m , y). A possible approach to combine past observations of x and y with the current values of x to generate an assessment of y is similarity-weighted averaging. It suggests that the predicted value of y, y n+1 s , be the weighted average of all previously observed values y i , where the weight of y i is the similarity between the vector x n+1 1,…, x n+1 m , associated with y n+1, and the previously observed vector, x i 1,…, x i m . This paper axiomatizes, in terms of the prediction y n+1, a similarity function that is a (decreasing) exponential in a norm of the difference between the two vectors compared. [Copyright &y& Elsevier]

Simplicity arguments are to be found in most geometrical works, from those of Proclus in his Commentaries on the First Book of Euclid’s Elements, up to those of contemporary manuals. Our goal is to read these arguments in their historical contexts to analyze agreements, disagreements and the multiplicity of points of view. For a better apprehension and a better understanding of the different conceptions, we will focus on the notion of angles and their measurements. We will study the notion of ≪ simplicity ≫ in various Elements of Geometry, in particular those of Euclid, Peletier du Mans (), Arnauld (), Lacroix () and Hoüel (). From there, we will examine French schoolbooks of geometry, beginning from the 1960s up to the 1990s, including those of the so-called period of ≪la réforme des mathématiques modernes≫ in France. [ABSTRACT FROM AUTHOR]

Abstract: We deal with allocation problems among sharing groups. There are n agents. The agents are divided into several sharing groups. A homogeneous good is allocated among sharing groups rather than among the agents. The good is a private good for sharing groups, and a public good for the members of each sharing group. That is, all of them in the same sharing group can consume it without rivalry. We introduce some allocation rules and axioms. The utilitarian allocation rule and the egalitarian allocation rule are characterized by some axioms. [Copyright &y& Elsevier]

Large-scale engineering systems provide important functions to the human society. The involvement of multiple, competing functionality requirements and lots of resources has imposed high expectations, and at the same time challenges, for achieving reliable, affordable design. Axiomatic design approach has demonstrated its strength in various types of large-scale system design, including vehicles, aircrafts, manufacturing facilities, and so on. However, several obstacles are yet to be overcome in proper application of the axioms to a real-world design problem, by both researchers and practitioners. In this work, a systematic methodology is presented for applying Suh’s axioms to evaluate and optimize large-scale engineering systems. When using the proposed methodology, design matrices are first obtained to map multiple, competing functionality requirements to their associated physical embodiments at different system hierarchical levels based on Axiom 1. An R / S analysis using surrogate modelling is then conducted on the design matrices to evaluate the existing system design. If the evaluation shows any areas for improvement, a combination of surrogate modelling-based R / S analysis and optimization is used to achieve a less functionally coupled design. An application in a nuclear reactor system design is used to demonstrate the use of the proposed methodology in dealing with real-world design problems. The results show that the proposed methodology provides a promising approach to efficiently evaluate and optimize a large-scale engineering system against multiple, competing design objectives. [ABSTRACT FROM AUTHOR]

The goal of this paper is to represent two approaches to the phenomenon of information, explicating its nature and essence. In this context, Mark Burgin demonstrates how the general theory of information (GTI) describes and elucidates the phenomenon of information by explaining the axiomatic foundations for information studies and presenting the comprising mathematical theory of information. The perspective promoted by Jaime F. Cárdenas-García is based on Gregory Bateson's description of information as "difference which makes a difference" and involves the process of info-autopoiesis as a sensory commensurable, self-referential feedback process. [ABSTRACT FROM AUTHOR]

Abstract: As an extension of separation axiom in general topology, axiom and s- axiom were introduced in I-fuzzy topological spaces by Yue and Fang [On separation axioms in I-fuzzy topological spaces, Fuzzy Sets and Systems 157 (2006) 780–793]. In this note, it is proved that the axiom and the s- axiom are equivalent to each other. [Copyright &y& Elsevier]

Everyone has some intuition as to what is morally right and what is morally wrong. However, such intuitions can prove inconsistent when questions of morality become increasingly numerical. One of the most consistent methods of determining which of a given set of decisions is the morally correct decision is to use a measure of utility. Such has led to the development of Utilitarianism and subsequently, Negative Utilitarianism. Unfortunately, despite Negative Utilitarianism solving some of the problems posed by Utilitarianism, it introduces additional problems and this article will propose a set of principles or axioms that resolve these new problems. The method of derivation involves considering possible distributions of utility over a sentient population. The proposed axioms will be used in an algorithmic fashion to gauge which decisions are to be considered optimal or moral. The axioms have been designed, in theory, to cover all scenarios of morality and ethics, specifically taking into account: selflessness, the disparity between individual and collective utility, equality over a population and over time, population size and even scenarios involving abortion. The proposed system of axioms has also been designed to accommodate differing intuitions, adding flexibility without compromising the mathematical structure of the system. [ABSTRACT FROM AUTHOR]