Point Axioms Research Articles

ABOUT SOME FIXED POINT AXIOMS AND RELATED PRINCIPLES IN KRIPKE–PLATEK ENVIRONMENTS

AbstractStarting points of this article are fixed point axioms for set-bounded monotone Σ1 definable operators in the context of Kripke–Platek set theory $KP$. We analyze their relationship to other principles such as maximal iterations, bounded proper injections, and Σ1 subset-bounded separation. One of our main results states that in $KP + (V\, = \,L)$ all these principles are equivalent to Σ1 separation.

Non-equidistant GM(1,1) model based on GCHM_WBO and its application in corrosion rate prediction

Purpose The purpose of this paper is to present a new model which combines the non-equidistant GM(1,1) model with GCHM_WBO (generalized contra-harmonic mean (GCHM); weakening buffer operator (WBO)). The authors use the model to solve the deadlock that for a large number of non-equidistant corrosion rate, it is difficult to establish a reasonable prediction model and improve the prediction accuracy. Design/methodology/approach This research consists of three parts: non-equidistant GM(1,1) model, GCHM_WBO operator, and the optimization of morphing parameter (contained in GCHM, control the intensity of the weakening operator). The methodology is explained as follows. First, the authors built a non-equidistant GM(1,1) model with GCHM_WBO weakened data, of which morphing parameter was randomly selected. Next, the authors calculated the error between prediction data of model and the real data, and adjusted the morphing parameter according to the error and property of GCHM. Then, the authors generated a new non-equidistant GM(1,1) based on new morphing parameter, and repeated the previous step until the termination condition was satisfied. Finally, the model with appropriate morphing parameter was used to implement the prediction of new data. Findings This paper finds a property of GCHM, which is a monotonic increasing function of morphing parameter in some specific conditions. Based on the property and the fixed point axiom of WBO, an algorithm was designed to search an appropriate morphing parameter. The appropriate morphing parameter was implemented for the purpose of improving the accuracy of the model. The model was applied to predict the corrosion rate of six steels at Guangzhou experimental station. The results showed that the proposed method can get more accuracy in prediction capability compared to the models with the original data and AWBO weakened data. The method is applicable to long-term forecasts in case of data scarcity. Practical implications Corrosion will cause huge economic loss to a country; therefore, it is important to judge the remaining useful life of a material or equipment; the foundation for judgement of which is the prediction of material corrosion rate. However, the prediction of corrosion rate is very difficult because of corrosion data’s features, such as small sample size, non-equidistant, etc. The proposed method can be used to implement long-term forecast of corrosion data with only one sample and non-equidistant samples. Originality/value This paper presented a model which combines the non-equidistant GM(1,1) model with GCHM_WBO to handle the problem of long-term forecasting of corrosion data. In the modelling process, the proposed morphing parameter searched through algorithm can improve the prediction accuracy of the model. Therefore, the model can provide effective and reliable result when data are of a small sample size and non-equidistant.

Point axioms and related conditions in Dedekind categories

A Dedekind category is a convenient algebraic framework to manipulate (binary) relations. Concepts of points, point axioms and related conditions such as the axiom of totality, the axiom of subobject, the axiom of complement, and the relational axiom of choice are introduced in Dedekind categories in order to connect abstract notions to set-theoretical intuition. This paper summarises logical interrelations of these axioms and provides some ideas for using them.

The common point problem in constructive projective geometry

Using intuitionistic methods, an extension of an incidence plane was constructed by Heyting in 1959; however, a central question, the validity of the projective axiom that any two lines have a common point, was left open. A Brouwerian counterexample demonstrates that in the Heyting extension the common point axiom is constructively invalid.

Dedekind categories with cutoff operators

Dedekind categories provide a suitable categorical framework for lattice-valued binary relations. It is known that the notion of crispness cannot be described by the basic tools of this theory only. In this paper we will study Dedekind categories with a cutoff operator in order to circumvent this shortage. We will introduce and investigate the properties of the operator and its relationship with other tools previously used for the same purpose. The main result of this paper is a representation theorem for Dedekind categories with a cutoff operator satisfying the point axiom.

Disagreement point axioms and the egalitarian bargaining solution

We provide new characterizations of the egalitarian bargaining solution on the class of strictly comprehensive n-person bargaining problems. The main axioms used in all of our results are Nash’s IIA and disagreement point monotonicity—an axiom which requires a player’s payoff to strictly increase in his disagreement payoff. For n = 2 these axioms, together with other standard requirements, uniquely characterize the egalitarian solution. For n > 2 we provide two extensions of our 2-person result, each of which is obtained by imposing an additional axiom on the solution. Dropping the axiom of anonymity, strengthening disagreement point monotonicity by requiring player i’s payoff to be a strictly decreasing function of the disagreement payoff of every other player j ≠ i, and adding a “weak convexity” axiom regarding changes of the disagreement point, we obtain a characterization of the class of weighted egalitarian solutions. This “weak convexity” axiom requires that a movement of the disagreement point in the direction of the solution point should not change the solution point. We also discuss the so-called “transfer paradox” and relate it to this axiom.

Disagreement Point Axioms and the Egalitarian Bargaining Solution

Several characterizations of the egalitarian bargaining solution are derived; the main axioms involved refer to disagreement point changes. In addition, transfer paradoxes'' are discussed.

Extreme point axioms for closure spaces

A pair ( X , τ ) of a finite set X and a closure operator τ : 2 X → 2 X is called a closure space. The class of closure spaces includes matroids as well as antimatroids. Associated with a closure space ( X , τ ) , the extreme point operator ex : 2 X → 2 X is defined as ex ( A ) = { p | p ∈ A , p ∉ τ ( A - { p } ) } . We give characterizations of extreme point operators of closure spaces, matroids and antimatroids, respectively.

Inside Every Model of Abstract Stone Duality Lies an Arithmetic Universe

The first paper published on Abstract Stone Duality showed that the overt discrete objects (those admitting ∃ and = internally) form a pretopos, i.e. a category with finite limits, stable disjoint coproducts and stable effective quotients of equivalence relations. Using an N-indexed least fixed point axiom, here we show that this full subcategory is an arithmetic universe, having a free semilattice (“collection of Kuratowski-finite subsets”) and a free monoid (“collection of lists”) on any overt discrete object. Each finite subset is represented by its pair (□, ◊) of modal operators, although a tight correspondence with these depends on a stronger Scott-continuity axiom. Topologically, such subsets are both compact and open and also involve proper open maps. In applications of ASD this can eliminate lists in favour of a continuation-passing interpretation.

An algebraic formalization of fuzzy relations

This paper provides an algebraic formalization of mathematical structures formed by fuzzy relations with sup-min composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom has been given by Schmidt and Ströhlein. Unlike Boolean relation algebras, fuzzy relation algebras are not Boolean but equipped with semi-scalar multiplication. First, we present a set of axioms for fuzzy relation algebras and improve the definition of point relations. Then by using relational calculus, a representation theorem for such relation algebras is deduced without Tarski rule.

Characterizing the Nash and Raiffa Bargaining Solutions by Disagreement Point Axioms

We provide a new characterization of the n-person Nash bargaining solutions which does not involve Nash's Independence of Irrelevant Alternatives axiom, but mainly uses axioms which concern changes in the disagreement point and leave the feasible set fixed. The main axiom requires a convex combination of a disagreement point and the corresponding solution point to give rise to that same solution point. Further, we describe how the disagreement point approach can be applied to other bargaining solutions. The main result of the latter part is a first characterization of the so-called Continuous Raiffa solution.

Relation algebras: Concept of points and representability

In the axiomatization of relation algebras by Chin and Tarski certain elements are called right ideals. Aiming at applications in the relational theory of graphs and programs, we call such ideals ‘points’ and investigate an additional point axiom. First we prove a point insertion theorem. Then a representation theorem for such relation algebras is deduced by inherently relational methods, simplifying the proof of a similar result from Jónsson, Maddux and Tarski. Some historical remarks are inserted and an extended bibliography is added.

Non-Desarguesian Projective Plane Geometries Which Satisfy The Harmonic Point Axiom

1. Introduction and summary. In her papers (12) and (13) R. Moufang discusses projective plane geometries which satisfy the axiom of the uniqueness of the fourth harmonic point. Her main result is that in such geometries non-homogeneous co-ordinates may be assigned to the points of the plane (except for the “line at infinity”) in such a way that straight lines have equations of the forms aαx + y + β = 0, or x + γ − 0.