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In this paper, the use of the MaxInf Principle in real optimization problems is investigated for engineering applications, where the current design solution is actually an engineering approximation. In industrial manufacturing, multibody system simulations can be used to develop new machines and mechanisms by using virtual prototyping, where an axiomatic design can be employed to analyze the independence of elements and the complexity of connections forming a general mechanical system. In the classic theories of Fisher and Wiener-Shannon, the idea of information is a measure of only probabilistic and repetitive events. However, this idea is broader than the probability alone field. Thus, the Wiener-Shannon's axioms can be extended to non-probabilistic events and it is possible to introduce a theory of information for non-repetitive events as a measure of the reliability of data for complex mechanical systems. To this end, one can devise engineering solutions consistent with the values of the design constraints analyzing the complexity of the relation matrix and using the idea of information in the metric space. The final solution gives the entropic measure of epistemic uncertainties which can be used in multibody system models, analyzed with an axiomatic design. [ABSTRACT FROM AUTHOR]

The aim of this paper is to initiate and discuss the properties and characterizations of soft semi-Ti and soft semi-Di(for i = 0; 1; 2) spaces at soft point by analyzing the relationship among them. We also introduce and explore the properties of soft S-continuous functions. These results will be useful to enhance the theoretical framework and to promote further study towards the daily life applications. [ABSTRACT FROM AUTHOR]

Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, but in constructive geometry, it must be done without a case distinction. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to different versions of the parallel postulate.We consider three versions of Euclid’s parallel postulate. The two most important are Euclid’s own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair’s axiom (dating from 1795), which says there cannot be two distinct parallels to line L through the same point p. These differ in that Euclid 5 makes an existence assertion, while Playfair’s axiom does not. The third variant, which we call the strong parallel postulate, isolates the existence assertion from the geometry: it amounts to Playfair’s axiom plus the principle that two distinct lines that are not parallel do intersect. The first main result of this paper is that Euclid 5 suffices to define coordinates, addition, multiplication, and square roots geometrically.We completely settle the questions about implications between the three versions of the parallel postulate. The strong parallel postulate easily implies Euclid 5, and Euclid 5 also implies the strong parallel postulate, as a corollary of coordinatization and definability of arithmetic. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The independence proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions. “Field elements” in these models are real-valued functions. [ABSTRACT FROM PUBLISHER]

Relativized common knowledge is a generalization of common knowledge proposed for public announcement logic by treating knowledge update as relativization. Among other things relativized common knowledge, unlike standard common knowledge, allows reduction axioms for the public announcement operators. Public announcement logic can be seen as one of the simplest special cases of action model logic (AML). However, so far no notion of relativized common knowledge has been proposed for AML in general. That is what we do in this paper. We propose a notion of action model relativized common knowledge for action model logic, and study expressive power and complete axiomatizations of resulting logics. Along the way we fill some gaps in existing expressivity results for standard relativized common knowledge. [ABSTRACT FROM AUTHOR]

The separation axioms are important and interesting concepts in topological spaces. In this paper we introduce and investigate some weak separation axioms by using the notion of ωδ-open set. [ABSTRACT FROM AUTHOR]

This paper addresses the issue of stipulation in three cases of implicit definitions (postulates of scientific terms, systems of axioms and abstraction principles). It argues that the alleged implicit definitions do not have a purely stipulative status. Stipulation of the vehicles of the implicit definitions in question should end up with true postulates. However, those postulates should not be taken to be true only in virtue of stipulation since they have extra commitments. Horwich's worry emerges in all three kinds of implicit definitions under consideration, since the existence of meanings so that the alleged postulates are true depends on extra requirements that should be fulfilled. Moreover, if Ramseyfication method is applied to the three kinds of implicit definition, they are split up into two components from which the first one is broadly factual while the second one is purely stipulative. The paper argues that their definitional task in each case should be assigned to their second component i.e. their Carnap-conditional. [ABSTRACT FROM AUTHOR]

In this paper, under the idea of L-T0 or sub-T0, we propose a set of new separation axioms in L-topological spaces, namely sub-separation axioms. And some of their properties are studied. In addition, the relation between the sub-separation axioms defined in the paper and other separation axioms is discussed. The results show that the subseparation axioms in this paper are weaker than other separation axioms that had appeared in literature. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to give an axiomatic characterization of the interval Baker-Thompson rule. [ABSTRACT FROM AUTHOR]

In this article, the author focuses on the development of mathematics towards exact precision that has led to formalization of large areas of it. It mentions two formal systems, Principia Mathematica (PM) and Zermelo-Fraenkelian axiom-system of set theory. It further mentions that these systems are formalized to provide few axioms and deduction rules to decide mathematical questions.

Abstract: A mean proximity rule is a voting rule having a mean proximity representation in Euclidean space. Legal ballots are represented as vectors that form the representing polytope. An output plot function determines a location for each possible election output in the same space, and these locations decompose the polytope into proximity regions according to which output is closest. The election outcome is then determined by which region(s) contain the mean position of all ballots cast. Mean neat rules are obtained by relaxing the requirement that the regions be determined by proximity, insisting only that they be neatly separable by a hyperplane. If each of these hyperplanes contains a dense set of rational points (vectors with all rational components), the mean neat voting rule is said to be rational . The aim of this article is to prove that consistency and connectedness are necessary and sufficient conditions for mean neat rationality of any voting rule that is anonymous . Connectedness can be viewed as a strong form of continuity, with an intuitive content related to the Intermediate Value Theorem (or to a discrete analogue of this theorem). The proof relies on a recent result in convexity theory [D. Cervone, W.S. Zwicker, Convex decompositions, J. Convex Anal. 2008 (in press)] and suggests a conjecture: if we relax connectedness to continuity, the class so characterized is that of the mean neat voting rules. This latter class properly contains all intuitive scoring rules. [Copyright &y& Elsevier]