Axiomatic Way Research Articles

New axioms for top trading cycles

AbstractThe school choice problem is of great importance both in theory and practice. This paper studies the (student‐optimal) top trading cycles mechanism (TTCM) in an axiomatic way. We introduce two new axioms: MBG (mutual best group)‐quota‐rationality and MBG‐robust efficiency. While stability implies MBG‐quota rationality, MBG‐robust efficiency is weaker than robust efficiency, which is stronger than the combination of efficiency and group strategy proofness. The TTCM is characterized by MBG‐quota rationality and MBG‐robust efficiency. Our results construct a new basis to compare the TTCM with the other school choice mechanisms, in terms of both fairness and incentive‐related axioms.

Many valued topologies on L-sets

Many valued topologies on L-sets, providing a common framework for fuzzy topologies on fuzzy sets and the fixed-basis lattice-valued topologies on ordinary sets, are the main subject of the present study. In this paper, we have introduced many valued topological L-set-spaces, referring to L-sets equipped with many valued topologies, in an axiomatic way and focused on the vindication of their axiomatization in a categorical framework. In order to attain this objective, we apply the theory of Höhle of topological space objects formulated in terms of a partially ordered monad on an abstract category, and show that the category of many valued topological L-set-spaces is isomorphic to the category of topological space objects with respect to a partially ordered monad on a category of L-sets.

Functional biodynamics of human-body system: A mathematical axiomatics with functional learning and aging in life cycle.

"Systems neuroergonomics" (Mau, J. In: R. Wang and X. Pan, editors, Advances in Cognitive Neurodynamics (V), chapter 59, pages 431-437, Springer Science+Business Media, Singapore, 2016) showed a separation of human-body system's functional organization from its cellular material in order to open a holistic perspective that can comprise all body functions. This was achieved with a strictly hierarchical structure in drill-down from system's functional whole to base functional elements. Implied multi-scale functional dynamics with coherent up-scaling were mathematically described in an axiomatic way, but the daily wake/sleep dynamics cover mainly anticipated functional challenges. Here, the theory is expanded to permit "functional learning" in the sense of adjustment of functional capacities to meet higher demand. "Functional aging" as a whole-body frailty is derived from biodynamic entropy that eventually drives the living to meet its inevitable destiny, death. The ages at which frailty begins to dominate can currently be obtained only from official life tables as population averages and are different for men and women, accordingly. The objective is to develop a mathematical theory in medicine from an axiomatic approach to phenomenological biodynamics, in which cognition is viewed as an embodied activity.

Intersection Space Constructible Complexes

We present an obstruction theoretic inductive construction of intersection space pairs, which generalizes Banagl's construction of intersection spaces for arbitraty depth stratifications. We construct intersection space pairs for pseudomanifolds with compatible trivial structures at the link fibrations; this includes the case of toric varieties. We define intersection space complexes in an axiomatic way, similar to Goresky-McPherson axioms for intersection cohomology. We prove that if the intersection space pair exists, then the pseudomanifold has an intersection space complex whose hypercohomology recovers the cohomology of the intersection space pair. We characterize existence and uniqueness of intersection space complexes in terms of the derived category of constructible complexes. In the case of algebraic varieties we show that a parallel obstruction theory in the derived category of Mixed Hodge Modules endowes intersection space cohomology with a Mixed Hodge Structure if the obstruction vanishes. We find classes of examples admitting intersection space complex, and counterexamples not admitting them which illustrate the use of the previously developed obstruction theory (counterexamples were known previously by various specialists). We prove that the (shifted) Verdier dual of an intersection space complex is an intersection space complex. We prove a generic Poincaré duality theorem for intersection space complexes.

DESAIN PENGEMBANGAN KURIKULUM BAHASA ARAB: PENDEKATAN OTAK KANAN

This paper aims to discuss the development of the Arabic languagecurriculum based on the right brain approach. This paper relies on bibliographicsources in the form of books and articles (literature studies) which are in thescientific journal related to the subject matter. Reading the data of thought byacademics using a critical constructive approach and meaning of the substancewith content analysis. Based on the studies that have been conducted, the rightbrain approach in learning Arabic can be narrated as an axiomatic way ofthinking about how to learn Arabic by optimizing right brain functions whichinclude rhymes, rhythms, music, visual impressions, colors, and images. Thusthe purpose of learning Arabic is the students are able to obtain Arabic languageknowledge that is ready to be used in the field of scientific and practical social inthe community, in a way that is fun and full of enthusiasm. The method that canbe used is the psychological method. While in learning techniques in principle,it must be able to attract students’ interests and motivations, encourage studentsto think critically-creatively-holistically, and must encourage students to becooperative and interactive.

Data Discovery and Anomaly Detection Using Atypicality: Theory

A central question in the era of big data is what to do with the enormous amount of information. One possibility is to characterize it through statistics, e.g., averages, or classify it using machine learning, in order to understand the general structure of the overall data. The perspective in this paper is the opposite, namely that most of the value in the information in some applications is in the parts that deviate from the average, that are unusual, atypical. We define what we mean by atypical in an axiomatic way as data that can be encoded with fewer bits in itself rather than using the code for the typical data. We show that this definition has good theoretical properties. We then develop an implementation based on universal source coding, and apply this to a number of real world data sets.

Valuing streams of risky cashflows with risk-value models

Based on risk-value models, we introduce a multiperiod approach to the valuation of streams of risky cashflows. The valuation is based on the (expected) value of the output's or input's magnitude and the risk of the output cashflow, as captured by a risk measure. We derive three formulas for valuing single cashflows and utilize the principles of separate valuation and of cumulating the cashflows to derive a multiperiod valuation method. In an axiomatic way, the paper sets out the foundations for a new approach and suggests several directions for its further development.

Fredholm operators on C*-algebras

The aim of this note is to generalize the notion of Fredholm operator to an arbitrary $C^*$-algebra. Namely, we define "finite type" elements in an axiomatic way, and also we define Fredholm type element $a$ as such element of a given $C^*$-algebra for which there are finite type elements $p$ and $q$ such that $(1-q)a(1-p)$ is "invertible". We derive index theorem for such operators. In applications we show that classical Fredholm operators on a Hilbert space, Fredholm operators in the sense of Breuer, Atiyah and Singer on a properly infinite von Neumann algebra, and Fredholm operators on Hilbert $C^*$-modules over an unital $C^*$-algebra in the sense of Mishchenko and Fomenko are special cases of our theory.

A New Solution Concept for the Ultimatum Game leading to the Golden Ratio

The Ultimatum Game is a paradigmatic two-player game. A proposer can offer a certain fraction of some valuable good. A responder can accept the offer or reject it, implying that the two players receive nothing. The only subgame-perfect Nash equilibrium is to only offer an infinitesimal amount and to accept this. However, this equilibrium is not in agreement with experimental observations, which show varying accepted offers around 40%. While some authors suggest that the fairest split of 50% vs. 50% would be explainable on theoretical grounds or by computer simulation, a few authors (including myself) have recently suggested that the Golden Ratio, about 0.618 vs. about 0.382, would be the solution, in striking agreement with observations. Here we propose a solution concept, based on an optimality approach and epistemic arguments, leading to that suggested solution. The optimality principle is explained both in an axiomatic way and by bargaining arguments, and the relation to Fibonacci numbers is outlined. Our presentation complements the Economic Harmony theory proposed by R. Suleiman and is based on infinite continued fractions. The results are likely to be important for the theory of fair salaries, justice theory and the predictive value of game theory.

General Information Conditioned by a Variable Event

The aim of this paper is to present, by axiomatic way, an idea about the general conditional information of a single, fixed fuzzy set when the conditioning fuzzy event is variable. The properties of this conditional information are translated in a system of functional equations. Some classes of solutions of this functional system have been found.

Analytic monoids and factorization problems

The main goal of this paper is to initiate study of analytic monoids as a general framework for quantitative theory of factorization. So far the latter subject was developed either in concrete settings, for instance in orders of number fields, or abstractly, in an axiomatic way. Some of the abstract approaches are too general to address delicate problems concerning oscillatory nature of the related counting functions, or are too restrictive in the sense that they suffer from the lack of examples except classical ones i.e. the Hilbert monoids of algebraic integers and their products. The notion of an analytic monoid is enough flexible to allow constructions of many other examples, and also ensures sufficiently rich analytic structure. In particular, we construct examples of such monoids with the associated L-functions being products of classical Dirichlet L-functions and L-functions of twisted irreducible unitary cuspidal automorphic representations of GL_d({mathbb {A}}_{mathbb {Q}}) satisfying the Ramanujan conjecture and having real coefficients. Finally, to illustrate how a typical problem from the quantitative theory of factorization can be studied in the framework of analytic monoids, we formulate several results concerning oscillations of the remainder term in the asymptotic formula for the number of irreducible elements with norms less or equal x, as x tends to infinity.

Arithmetization and Rigor as Beliefs in the Development of Mathematics

With the arrival of the nineteenth century, a process of change guided the treatment of three basic elements in the development of mathematics: rigour, the arithmetization and the clarification of the concept of function, categorised as the most important tool in the development of the mathematical analysis. In this paper we will show how several prominent mathematicians contributed greatly to the development of these basic elements that allowed the solid underpinning of mathematics and the consideration of mathematics as an axiomatic way of thinking in which anyone can deduce valid conclusions from certain types of premises. This nineteenth century stage shares, possibly with the Heroic Age of Ancient Greece, the most revolutionary period in all history of mathematics.

Just Freedom?

As Philip Pettit notes in his acknowledgements, Just Freedom ‘draws together strands from [his] work in political philosophy over the past 20 years’ (241). Anybody who wishes to acquaint himor herself with Pettit’s work in a quick way can therefore easily do so using this book. It is written to be accessible to a general audience, and Pettit has indeed admirably succeeded in presenting the ideas he has developed over the past two decades in a very lucid and engaging way. But the book is also an accomplishment in its own right; it gives the reader Pettit’s distinctive development of the ‘republican’ conceptions of freedom and justice in a more systematic, simplified, and axiomatic way than any of his previous books have done. This is not to say that there is nothing that those who take similar views will want to object to, but it is to say that few recent books have captured this general sort of republican view of freedom and justice in as cogent a way as Pettit does here. Pettit aims to offer the reader a ‘moral compass for a complex world’, and he thinks that republican freedom as non-domination fits the bill. Freedom, he tells us, is not the only value, but it is a ‘gateway good’ (189). If we can secure freedom for all in their relations to their fellow citizens and to their governments, and we can also secure republican independence for free peoples in relation to other corporate entities (such as states or powerful non-state actors), this gives us, Pettit argues, a kind of basic justice from which all the rest of social, political, and international justice can flow or derive its content and justification. As Pettit summarizes his view, ‘justice is freedom, freedom justice’ (xxiii). Thus, Pettit writes that whereas John Rawls gives us two basic principles of justice, his own favored approach is

Indiscernibilidade e identidade em química: aspectos filosóficos e formais

Neste artigo tratamos, de um ponto de vista formal e filosófico, com alguns conceitos que fazem parte da química usual. As teorias da química, e seus conceitos, normalmente são apresentadas de um ponto de vista informal (não axiomatizada), e isso pode trazer dificuldades filosóficas (embora a química propriamente - bem como também as outras disciplinas da ciência aplicada - pareçam não sofrer qualquer restrição quanto a isso). Aqui estaremos ocupados em um aspecto parti-cu-lar, que diz respeito à indiscernibilidade de alguns objetos básicos da química, tais como átomos, moléculas, bem como de seus componentes. Começamos com uma visão geral da identidade destes compostos e seus componentes a partir de uma perspectiva filosófica e, em seguida, mostramos em que sentido o conceito de identidade dos compostos químicos é problemático em relação ao conceito correspondente de identidade na lógica e matemática clássicas (que, como em geral é suposto, estão 'alicerçando' as teorias químicas). Argumentamos que, por um lado, a química parece supor que esses objetos básicos precisam ser 'idênticos' (indistinguíveis) uns aos outros (como enfatizado há muito tempo por John Dalton, como veremos.) Por outro lado, do ponto de vista formal, se a lógica subjacente à teoria atômica for a lógica clássica (que também é pano de fundo da matemática padrão), coisas idênticas devem ser a mesma coisa e logo 'colapsar' em apenas uma (e este não é o caso em química, uma vez que temos uma enorme quantidade de átomos e moléculas semelhantes, mas não apenas um). Esta aparente contradição pode ser tratada de várias perspectivas e, aqui, propomos o uso de uma teoria de conjuntos não-clássica (a saber, a teoria de quase-conjuntos) para alicerçar uma formulação axiomática de certas teoria químicas, mostrando como essa diferente base matemática pode nos levar a uma visão que está mais perto da química em si. Como esta teoria não é conhecida em geral, uma breve revisão se faz necessária. Por último, mostramos como podemos construir modelos matemáticos para átomos e moléculas utilizando esta teoria de conjuntos alternativa evitando, assim, o problema lógico acima mencionado, e discutimos um pouco da abordagem mereológica dos compostos químicos a partir dessa perspectiva quase-conjuntista.

A Form of Information Entropy

In this paper, by axiomatic way, a form of information entropy will be presented on crisp and fuzzy setting. Information entropy is the unavailability of information about a crisp or fuzzy event. It will use measure of information defined without any probability or fuzzy measure: for this reason it is called general information.

The V-filtration for tame unit $$F$$ F -crystals

Let \(X\) be a smooth variety over an algebraically closed field of characteristic \(p > 0, Z\) a smooth divisor, and \(j: U=X {\setminus } Z \rightarrow X\) the natural inclusion. We introduce in an axiomatic way the notion of a \(V\)-filtration on unit \(F\)-crystals and prove such axioms determine a unique filtration. It is shown that if \(\mathcal M \) is a tame unit \(F\)-crystal on \(U\), then such a \(V\)-filtration along \(Z\) exists on \(j_*\mathcal M \). The degree zero component of the associated graded module is proven to be the (unipotent) nearby cycles functor of Grothendieck and Deligne under the Emerton–Kisin Riemann–Hilbert correspondence. A few applications to \(\mathbb A ^1\) and gluing are then discussed.

A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems

This paper analyzes risk capital allocation problems. For risk capital allocation problems, the aim is to allocate the risk capital of a firm to its divisions. Risk capital allocation is of central importance in risk-based performance measurement. We consider a case in which the aggregate risk capital is determined via a coherent risk measure. The academic literature advocates an allocation rule that, in game-theoretic terms, is equivalent to using the Aumann-Shapley value as solution concept. This value is however not well-defined in case a differentiability condition is not satisfied. As an alternative, we introduce an allocation rule inspired by the Shapley value in a fuzzy setting. We take a grid on a fuzzy participation set, define paths on this grid and construct an allocation rule based on a path. Then, we define a rule as the limit of the average over these allocations, when the grid size converges to zero. We introduce this rule for a broad class of coherent risk measures. We show that if the Aumann-Shapley value is well-defined, the allocation rule coincides with it. If the Aumann-Shapley value is not defined, which is due to non-differentiability problems, the allocation rule specifies an explicit allocation. It corresponds with the Mertens value, which is originally characterized in an axiomatic way (Mertens, 1988), whereas we provide an asymptotic argument.

The Logical Foundation of Fundamental Rights and their Universality

This paper offers a critical analysis of two central issues in Luigi Ferrajoli’s Principia iuris, and more generally of his theory of rights. One is the way in which ‘expectations’ play a crucial role in his deontic theory by establishing the logical basis for his guarantee-based conception of law and rights. The axiomatic way in which Ferrajoli arrives at his conception of fundamental rights is questioned, for it fails to give a full account of the nature of subjective rights. The other issues discussed here is Ferrajoli’s own defence of the universality of fundamental rights, and how this is made to depend on a ‘normative technique’ that one can associate with legal discourse and modern law in Western societies. This, however, poses the problem of how this formal universalism can be reconciled with the intercultural dialogue that Ferrajoli also advocates.

An Axiomatic Approach to the Roughness Measure of Rough Sets

In Pawlak's rough set theory, a set is approximated by a pair of lower and upper approximations. To measure numerically the roughness of an approximation, Pawlak introduced a quantitative measure of roughness by using the ratio of the cardinalities of the lower and upper approximations. Although the roughness measure is effective, it has the drawback of not being strictly monotonic with respect to the standard ordering on partitions. Recently, some improvements have been made by taking into account the granularity of partitions. In this paper, we approach the roughness measure in an axiomatic way. After axiomatically defining roughness measure and partition measure, we provide a unified construction of roughness measure, called strong Pawlak roughness measure, and then explore the properties of this measure. We show that the improved roughness measures in the literature are special instances of our strong Pawlak roughness measure and introduce three more strong Pawlak roughness measures as well. The advantage of our axiomatic approach is that some properties of a roughness measure follow immediately as soon as the measure satisfies the relevant axiomatic definition. (This work was supported by the National Natural Science Foundation of China under Grants 60821001, 60873191, and 60903152. The author would like to thank the anonymous reviewers for their helpful comments.)

Scale and cost efficiency analysis of networks of processes

In this paper a simple way of computing technical, scale, cost and allocative efficiency scores for homogeneous networks of processes is presented. The system Production Possibility Set (PPS) is formed through the composition of the PPS of the individual processes, which, in turn, are modelled in the conventional, axiomatic way using observed data. Firstly, the overall system scale and technical efficiency are computed using the relational network DEA approach. Local Returns To Scale (RTS) can also be estimated with these models. Secondly, assuming the prices of exogenous inputs are known, a minimum cost network DEA model is solved, from which cost and allocative efficiencies are derived. The proposed approach is illustrated with a two-stage problem from the literature, showing the usefulness of a more detailed problem assessment both in terms of technical and scale efficiency and RTS and in terms of cost and allocative efficiency.