Cooperative and axiomatic approaches to the knapsack allocation problem

In the knapsack problem a group of agents want to fill a knapsack with several goods. Two issues must be considered. The first is to decide optimally what goods to select for the knapsack. This issue has been studied in many papers in the literature on Operations Research and Management Science. The second issue is to divide the total revenue among the agents. This issue has been studied in only a few papers, and this is one of them. For each knapsack problem we consider three associated cooperative games. One of them (the pessimistic game) has already been considered in the literature. The other two (realistic and optimistic games) are defined in this paper. The pessimistic and realistic games have non-empty cores but the core of the optimistic game could be empty. We then follow the axiomatic approach. We propose two rules: The first is based on the optimal solution of the knapsack problem. The second is the Shapley value of the so called optimistic game. We offer axiomatic characterizations of both rules.

ЕКОНОМІЧНА РЕТРОСПЕКТИВА РОЗВИТКУ ТЕОРІЇ ВЕРТИКАЛЬНОЇ СПЕЦІАЛІЗАЦІЇ

We explore the development of vertical specialization theory which is trade in goods across multiple stages of production on the relationship between trade and business cycle synchronization across countries. We study various papers about international trade model in which explores the degree of vertical specialization varying with trade barriers. We examine how and why these vertical specialization models created by economists and compare differences of those models. Since VS one of the most important changes involves the increasing interconnectedness of production processes in a vertical trading chain that stretches across many countries, with each country specializing in particular stages of a good’s production sequence. The purpose of the article is to form a theoretical and methodological basis for an economic retrospective of the development of the theory of vertical specialization. The subject of the research is theoretical and methodological aspects of the economic retrospective of the development of the theory of vertical specialization. Methods used in the study: historical method, methods of verification of theoretical positions (morphological analysis of the content and relationship of categories, principles and laws, assessment of the correspondence of historical facts to theoretical hypotheses), methods of analysis and synthesis, induction and deduction, logical method (hypothetical and axiomatic approaches), comparison method. Research hypothesis. In the context of global challenges, it is necessary to change the nature of interaction, for which it is necessary to form a theoretical and methodological basis for an economic retrospective of the development of the theory of vertical specialization. Presentation of the main material. Vertical specialization occurs when a country uses imported intermediate parts to produce goods that it later exports. This definition reflects the idea that countries are consistently linked to each other to produce the final good. The fulfillment of the conditions is justified: the product must be produced in several successive stages, two or more countries must specialize in the production of some, but not all, stages, and at least one stage must cross the international border more than once. It is emphasized that vertical specialization occurs when a country uses imported intermediate parts to produce goods that it later exports. Originality and practical significance of the research. The study of the economic retrospective of the development of the theory of vertical specialization includes four interrelated blocks – the theoretical and exploratory basis of interaction, the substantive (paradigmatic and methodological), applied and managerial basis of the interaction of business entities, which will allow comprehensively cover possible aspects of interaction. Conclusions of the study. The article examines the theoretical and methodological basis of vertical trade also connects the growth of international trade with an increase in international production. The forces that have driven increased vertical trade – lowering trade barriers and improving transport and communication technologies – are likely to persist. Thus, we can conclude that the value of VS and vertical trade in world trade will grow from year to year.

Currencies in Resource Theories

How may we quantify the value of physical resources, such as entangled quantum states, heat baths or lasers? Existing resource theories give us partial answers; however, these rely on idealizations, like perfectly independent copies of states or exact knowledge of a quantum state. Here we introduce the general tool of “currencies” to quantify realistic descriptions of resources, applicable in experimental settings when we do not have perfect control over a physical system, when only the neighbourhood of a state or some of its properties are known, or when slight correlations cannot be ruled out. Currencies are a subset of resources chosen to quantify all the other resources—like Bell pairs in LOCC or a lifted weight in thermodynamics. We show that from very weak assumptions in the theory we can already find useful currencies that give us necessary and sufficient conditions for resource conversion, and we build up more results as we impose further structure. This work generalizes axiomatic approaches to thermodynamic entropy, work and currencies made of local copies. In particular, by applying our approach to the resource theory of unital maps, we derive operational single-shot entropies for arbitrary, non-probabilistic descriptions of resources.

Arabic Science of Light: The birth of modern optics and of the experimental method

A millenium ago, the world witnessed a major revolution via the field of Optics. The study of Light, freed from postulates and axiomatic approaches, by the Arab Scholar Ibn-Al-Haytham marked the birth of the experimental method with its observational and analytical protocols, and the use of mathematics in formalising them. This revolution lead to the awakening of Sciences in Europe throughout the middle ages till the Renaissance.

Source Codes Oriented Software Trustworthiness Measure Based on Validation

Validation is critical to the success of software trustworthiness measurement. A large number of software trustworthiness measures are proposed; however, most of them are not validated from a theory perspective. Therefore, they lack theoretical foundation and will induce unnecessary cost and useless calculation. In this paper, we bring measurement theory into software trustworthiness measurement, construct a source codes oriented software trustworthiness measure based on extensive structure in the measurement theory, and validate the developed measure by use of axiomatic approaches. Compared with some software trustworthiness measures that are already presented, this measure can evaluate software trustworthiness better from a theory perspective.

The Reconciliation of Multiple Conflicting Estimates: Entropy-Based and Axiomatic Approaches

When working with economic accounts it may occur that multiple estimates of a single datum exist, with different degrees of uncertainty or data quality. This paper addresses the problem of defining a method that can reconcile conflicting estimates, given best guess and uncertainty values. We proceeded from first principles, using two different routes. First, under an entropy-based approach, the data reconciliation problem is addressed as a particular case of a wider data balancing problem, and an alternative setting is found in which the multiple estimates are replaced by a single one. Afterwards, under an axiomatic approach, a set of properties is defined, which characterizes the ideal data reconciliation method. Under both approaches, the conclusion is that the formula for the reconciliation of best guesses is a weighted arithmetic average, with the inverse of uncertainties as weights, and that the formula for the reconciliation of uncertainties is a harmonic average.

A new characterization of Kac–Moody–Malcev superalgebras

In the past two decades there has been great attention to Lie (super)algebras, which are extensions of affine Kac–Moody Lie (super)algebras, in certain typical or axiomatic approaches. These Lie (super)algebras have been mostly studied under variations of the name “extended affine Lie (super)algebras”. We show that certain classes of Malcev (super)algebras also can be put in this framework. This in particular allows us to provide new examples of Malcev (super)algebras which extend the known Kac–Moody Malcev (super)algebras.