Axioms Of Theory Research Articles

Consolidation of an Efficient and a Non-Efficient Solution in a Cooperative Game: The Egalitarian Banzhaf Value

Objective: The principles of marginalism and egalitarianism are handy tools for sharing worth among the players of a group in a TU game. Here, our objective is to combine two such solutions to generate a new value that keeps in mind the need of each player required for the survival of a group in a TU game. Methods: Here, the Banzhaf value and the equal division value were merged to establish a new consolidated solution. Method of induction is used over unanimity games and symmetric games to characterize the proposed value using some well-known as well as some freshly defined axioms of cooperative game theory. Findings: To describe the new value, we first looked at a number of intuitive axioms linked to it and the uniqueness of the value is obtained by characterizing it using the defined axioms. The value is then extended to the class of simple games. Novelty: The proposed solution is a non-efficient solution that allocates a portion of the total worth to the players and keeps a portion undistributed to use for the purpose of further investment by the group. Keywords: Cooperative game; Banzhaf value; Egalitarian Value; Null Player; Solution Concept

Set-theoretic relations for metasets

ABSTRACTThe paper introduces partial set-theoretic relations for metasets and investigates their properties. Metaset is a concept of imprecise set designed to represent vague notions in Artificial Intelligence, whose idea stems from Boolean-valued techniques in classical set theory. The basic set relations are extended to functions valued in a Boolean algebra or unit interval. Important classical set theory axioms: extensionality and comprehension are formulated for metasets. The latter enables formal definitions of collections with blurred boundaries by using set-theory formulae. This facilitates representing and reasoning about imprecise data.

Context-effect bias in capuchin monkeys (Sapajus spp.): exploring decoy influences in a value-based food choice task

Decision making is known to be liable to several context effects. In particular, adding a seemingly irrelevant alternative (decoy) to a set of options can modify preferences: typically, by increasing choices towards whatever option clearly dominates the decoy (attraction effect), but occasionally also decreasing its appeal and generating a shift in the opposite direction (repulsion effect). Both types of decoy effects violate rational choice theory axioms and suggest dynamic processes of preference-formation, in which the value of each alternative is not determined a priori, but it is instead constructed by comparing options during the decision process. These effects are well documented, both in humans and in other species: e.g., amoebas, ants, honeybees, frogs, birds, cats, dogs. However, evidence of decoy effects in non-human primates remains surprisingly mixed. This study investigates decoy effects in capuchin monkeys (Sapajus spp.), manipulating time pressure across different conditions, to test whether such effects require time-consuming comparative processes among available alternatives. Whereas the time-dependent nature of decoy effects is a robust finding in the human literature, this is its first investigation in non-human animals. Our results show that capuchins exhibit an attraction effect with decoys targeting their preferred food, and that this effect disappears under time pressure; moreover, we observe preliminary evidence of a repulsion effect when decoys target instead the less-preferred food, possibly due to the larger distance between decoy and target in the attribute space. Taken together, these results provide valuable insight on the evolutionary roots of comparative decision making.

Extending Kolmogorov's Axioms for a Generalized Probability Theory on Collections of Contexts.

Kolmogorov’s axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason’s theorem.

Axiomatic (and non-axiomatic) mathematics

Axiomatizing mathematical structures and theories, or postulating them as Russell (1919) put it, is a goal of mathematical logic. Some axiomatic systems are mere definitions, such as the axioms of Group Theory; but some are much deeper, such as the axioms of complete ordered fields with which real analysis starts. Groups abound in the mathematical sciences, while by Dedekind’s theorem (1888) there exists only one complete ordered field, up to isomorphism. Cayley’s theorem (1854) in abstract algebra implies that the axioms of group theory completely axiomatize the class of permutation sets that are closed under composition and inversion. In this expository article, we survey some old and new results on the first-order axiomatizability of various mathematical structures. As we will see, axiomatizability of some structures are still unsolved questions in mathematics, and several results have been open problems in the past. We will also review identities over addition, multiplication, and exponentiation that hold in the set of positive real numbers; and will have a look at Tarski’s high school problem (1969) and its solution.

Солнечная энергетика: анализ зонной модели проводимости полупроводников с позиции квантовой теории

The article focuses on the challenges of improving the energy efficiency when using the solar energy for electrical power generation. The key point is that improvement of the energy efficiency in application of this method depends on enhancement of theoretical knowledge on physical phenomena and regularities in technical devices. The objective of this research is to clarify the cause-and-effect relations and to explain regularities of physical phenomena in devices that convert solar power into electric energy in terms of fundamental axioms of the quantum theory in molecular systems, which would open up the possibilities to enhance the energy efficiency of solar power engineering. The paper analyzes the notions that are common in the scientific and educational literature and are related to electric current, free and superfluous electrons in atoms, and the motion of charged particles under the action of electric current. The scientific novelty consists in conceptualization of scientific statements in the fields of physics, electrical engineering and chemistry. A new concept of the energy phenomena in multilayer structures of semiconductor-based devices for conversion of solar radiation into the electric current has been formed for the first time ever. A schematic diagram of electric power generation upon solar irradiation of these devices is presented. The electric current in a closed circuit is the total solar energy absorbed by the valence electrons of the device's semiconductors, which they emit in a concentrated manner within a closed electric circuit. The practical value of the new knowledge lies in improving the professional competences and enhancing the energy efficiency of technological devices used to convert solar energy into electric current.

The Topology of Quantum Theory and Social Choice

Based on the axioms of quantum theory, we identify a class of topological singularities that encode a fundamental difference between classic and quantum probability, and explain quantum theory’s puzzles and phenomena in simple mathematical terms so they are no longer ‘quantum paradoxes’. The singularities provide also new experimental insights and predictions that are presented in this article and establish a surprising new connection between the physical and social sciences. The key is the topology of spaces of quantum events and of the frameworks postulated by these axioms. These are quite different from their counterparts in classic probability and explain mathematically the interference between quantum experiments and the existence of several frameworks or ‘violation of unicity’ that characterizes quantum physics. They also explain entanglement, the Heisenberg uncertainty principle, order dependence of observations, the conjunction fallacy and geometric phenomena such as Pancharatnam–Berry phases. Somewhat surprisingly, we find that the same topological singularities explain the impossibility of selecting a social preference among different individual preferences: which is Arrow’s social choice paradox: the foundations of social choice and of quantum theory are therefore mathematically equivalent. We identify necessary and sufficient conditions on how to restrict experiments to avoid these singularities and recover unicity, avoiding possible interference between experiments and also quantum paradoxes; the same topological restriction is shown to provide a resolution to the social choice impossibility theorem of Chichilnisky.

L-Orthogonal Elements and L-Orthogonal Sequences

Abstract Given a Banach space $X$, we say that a sequence $\{x_n\}$ in the unit ball of $X$ is $L$-orthogonal if $\Vert x+x_n\Vert \rightarrow 1+\Vert x\Vert $ for every $x\in X$. On the other hand, an element $x^{**}$ in the bidual sphere is said to be $L$-orthogonal (to $X$) if $\|x+x^{**}\|= 1+\Vert x\Vert $ for every $x\in X$. The aim of this paper is to clarify the relation between $L$-orthogonal sequences and $L$-orthogonal elements. Namely, we study whether every $L$-orthogonal sequence contains $L$-orthogonal elements in its weak*-closure. We provide an affirmative answer whenever the ambient space has small density character. Nevertheless, we show that, surprisingly, the general answer is independent of the usual axioms of set theory.

Copulatory behaviour and seminal parameters of rapid ejaculator rats during competition and female pacing

Axioms of sperm competition theory (risk and intensity model) predict that males increase sperm expenditure when ongoing competitive copulating encounters with other males. Particularly, the intensity model that considers a real male confrontation has not been experimentally tested in laboratory rats. Furthermore, no study has explored the active role of the female (that occurs in natural conditions) in regulating the timing of copulation on the effects of competition on sexual behaviour and seminal parameters. Thus, the goal of this study was to analyse the copulatory behaviour and seminal parameters under male–male competition and female pacing. In order to control for possible variations due to the sexual behaviour phenotype, all experiments were done in male rats with short ejaculation latencies (rapid ejaculators). The results show that competition decreased by half the ejaculation latency and increased sperm quantity, but substantially decreased sperm motility. On the contrary, when competition is tested under conditions where the female regulates the timing of copulation (paced mating), the ejaculation latency increased to double, the sperm count did not change, and the sperm motility significantly decreased. The variations in the parameters of the ejaculate depend on the context, i.e., the confrontation with a competitor, and the active participation of the female during copulation. Probably, when there is more than one male during mating, the female – through cryptic choice – promotes mechanisms that interfere temporally with sperm motility.

ФОРМИРОВАНИЕ ИНДИКАТОРОВ, КРИТЕРИЕВ, МЕР И ШКАЛ ОЦЕНКИ РИСКОВ

The article discusses the architecture of the formation of indicators, criteria, measures and risk assessment scales necessary in risk management systems based on the fundamental axioms of management theory and the principles of risk theory to build a risk management structure. As the most effective modeling method, it is proposed to use a hybrid one using the advantages of both formal and informal methods and models in building an adaptive risk management system in enterprises on a bottomup basis. To remove the emerging contradictions of indicative risk indicators, it is proposed to use a multicriteria method of structuring characteristics in evaluation models according to different criteria of objective functions – minimizing time or costs, maximizing sales revenue or profit, which determines the risk of an unmissed opportunity. The structure of the hybrid adaptive model suggests a radical scheme for changing managerial behavior, which consists in the transition from conventional regulation from above to direct regulation of risk factors at the points of their manifestation.

Differential forms, Fukaya $A_\infty$ algebras, and Gromov–Witten axioms

Consider the differential forms $A^*(L)$ on a Lagrangian submanifold $L \subset X$. Following ideas of Fukaya-Oh-Ohta-Ono, we construct a family of cyclic unital curved $A_\infty$ structures on $A^*(L),$ parameterized by the cohomology of $X$ relative to $L.$ The family of $A_\infty$ structures satisfies properties analogous to the axioms of Gromov-Witten theory. Our construction is canonical up to $A_\infty$ pseudoisotopy. We work in the situation that moduli spaces are regular and boundary evaluation maps are submersions, and thus we do not use the theory of the virtual fundamental class.

МНОГОКРИТЕРИАЛЬНЫЕ МЕТОДЫ АНАЛИЗА ПОКАЗАТЕЛЕЙ ПРОДУКЦИИ В УПРАВЛЕНИИ РИСКАМИ

The article discusses the architecture of the formation of indicative risk indicators in management systems based on the fundamental axioms of management theory and the principles of risk theory in building an adaptive risk management structure. As the most effective modeling method, it is proposed to use the methods of structuring and quantifying product indicators in the form of a hybrid option using the advantages of both formal and informal methods and models in risk management systems at enterprises according to a mixed bottom-up and top-down principle. To remove the emerging contradictions of indicative risk indicators, it is proposed to use a multi-criteria method for quantifying characteristics to form an integral criterion as a risk assessment measure, which determines the risk of an unmissed opportunity. The hybrid adaptive model proposed in this article makes it possible to significantly transform the model of managerial behavior, guided by the mixed management algorithm, taking into account risk indicators at the places of their manifestation, and is implemented by all participants in the labor process.

Can redescriptions of outcomes salvage the axioms of decision theory?

The basic axioms or formal conditions of decision theory, especially the ordering condition put on preferences and the axioms underlying the expected utility (EU) formula, are subject to a number of counter-examples, some of which can be endowed with normative value and thus fall within the ambit of a philosophical reflection on practical rationality. Against such counter-examples, a defensive strategy has been developed which consists in redescribing the outcomes of the available options in such a way that the threatened axioms or conditions continue to hold. We examine how this strategy performs in three major cases: Sen's counterexamples to the binariness property of preferences, the Allais paradox of EU theory under risk, and the Ellsberg paradox of EU theory under uncertainty. We find that the strategy typically proves to be lacking in several major respects, suffering from logical triviality, incompleteness, and theoretical insularity (i.e., being cut off from the methods and results of decision theory). To give the strategy more structure, philosophers have developed “principles of individuation”; but we observe that these do not address the aforementioned defects. Instead, we propose the method of checking whether the strategy can overcome its typical defects once it is given a proper theoretical expansion (i.e., it is duly developed using the available tools of decision theory). We find that the strategy passes the test imperfectly in Sen's case and not at all in Allais's. In Ellsberg's case, however, it comes close to meeting our requirement. But even the analysis of this more promising application suggests that the strategy ought to address the decision problem as a whole, rather than just the outcomes, and that it should extend its revision process to the very statements it is meant to protect. Thus, by and large, the same cautionary tale against redescription practices runs through the analysis of all three cases. A more general lesson, simply put, is that there is no easy way out from the paradoxes of decision theory.

Pauling’s rules for oxide-based minerals: A re-examination based on quantum mechanical constraints and modern applications of bond-valence theory to Earth materials

AbstractSince their introduction in 1929, Pauling’s five rules have been used by scientists from many disciplines to rationalize and predict stable arrangements of atoms and coordination polyhedra in crystalline solids; amorphous materials such as silicate glasses and melts; nanomaterials, poorly crystalline solids; aqueous cation and anion complexes; and sorption complexes at mineral-aqueous solution interfaces. The predictive power of these simple yet powerful rules was challenged recently by George et al. (2020), who performed a statistical analysis of the performance of Pauling’s five rules for about 5000 oxide crystal structures. They concluded that only 13% of the oxides satisfy the last four rules simultaneously and that the second rule has the most exceptions. They also found that Pauling’s first rule is satisfied for only 66% of the coordination environments tested and concluded that no simple rule linking ionic radius to coordination environment will be predictive due to the variable quality of univalent radii.We address these concerns and discuss quantum mechanical calculations that complement Pauling’s rules, particularly his first (radius sum and radius ratio rule) and second (electrostatic valence rule) rules. We also present a more realistic view of the bonded radii of atoms, derived by determining the local minimum in the electron density distribution measured along trajectories between bonded atoms known as bond paths, i.e., the bond critical point (rc). Electron density at the bond critical point is a quantum mechanical observable that correlates well with Pauling bond strength. Moreover, a metal atom in a polyhedron has as many bonded radii as it has bonded interactions, resulting in metal and O atoms that may not be spherical. O atoms, for example, are not spherical in many oxide-based crystal structures. Instead, the electron density of a bonded oxygen is often highly distorted or polarized, with its bonded radius decreasing systematically from ~1.38 Å when bonded to highly electropositive atoms like sodium to 0.64 Å when bonded to highly electronegative atoms like nitrogen. Bonded radii determined for metal atoms match the Shannon (1976) radii for more electropositive atoms, but the match decreases systematically as the electronegativities of the M atoms increase. As a result, significant departures from the radius ratio rule in the analysis by George et al. (2020) is not surprising. We offer a modified, more fundamental version of Pauling’s first rule and demonstrate that the second rule has a one-to-one connection between the electron density accumulated between the bonded atoms at the bond critical point and the Pauling bond strength of the bonded interaction.Pauling’s second rule implicitly assumes that bond strength is invariant with bond length for a given pair of bonded atoms. Many studies have since shown that this is not the case, and Brown and Shannon (1973) developed an equation and a set of parameters to describe the relation between bond length and bond strength, now redefined as bond valence to avoid confusion with Pauling bond-strength. Brown (1980) used the valence-sum rule, together with the path rule and the valence-matching principle, as the three axioms of bond-valence theory (BVT), a powerful method for understanding many otherwise elusive aspects of crystals and also their participation in dynamic processes. We show how a priori bond-valence calculations can predict unstrained bond-lengths and how bond-valence mapping can locate low-Z atoms in a crystal structure (e.g., Li) or examine possible diffusion pathways for atoms through crystal structures.In addition, we briefly discuss Pauling’s third, fourth, and fifth rules, the first two of which concern the sharing of polyhedron elements (edges and faces) and the common instability associated with structures in which a polyhedron shares an edge or face with another polyhedron and contains high-valence cations. The olivine [α-(MgxFe1–x)2SiO4] crystal structure is used to illustrate the distortions from hexagonal close-packing of O atoms caused by metal-metal repulsion across shared polyhedron edges.We conclude by discussing several applications of BVT to Earth materials, including the use of BVT to: (1) locate H+ ions in crystal structures, including the location of protons in the crystal structures of nominally anhydrous minerals in Earth’s mantle; (2) determine how strongly bonded (usually anionic) structural units interact with weakly bonded (usually cationic) interstitial complexes in complex uranyl-oxide and uranyl-oxysalt minerals using the valence-matching principle; (3) calculate Lewis acid strengths of cations and Lewis base strengths of anions; (4) determine how (H2O) groups can function as bond-valence transformers by dividing one bond into two bonds of half the bond valence; (5) help characterize products of sorption reactions of aqueous cations (e.g., Co2+ and Pb2+) and oxyanions [e.g., selenate (Se6+O4)2− and selenite (Se4+O3)2−] at mineral-aqueous solution interfaces and the important role of protons in these reactions; and (6) help characterize the local coordination environments of highly charged cations (e.g., Zr4+, Ti4+, U4+, U5+, and U6+) in silicate glasses and melts.

The Proper Formulation of the Minimalist Theory of Truth

Abstract Minimalism about truth is one of the main contenders for our best theory of truth, but minimalists face the charge of being unable to properly state their theory. Donald Davidson incisively pointed out that minimalists must generalize over occurrences of the same expression placed in two different contexts, which is futile. In order to meet the challenge, Paul Horwich argues that one can nevertheless characterize the axioms of the minimalist theory. Sten Lindström and Tim Button have independently argued that Horwich’s attempt to formulate minimalism remains unsuccessful. We show how to properly state Horwich’s axioms by appealing to propositional functions that are given by definite descriptions. Both Lindström and Button discuss proposals similar to ours and conclude that they are unsuccessful. Our new suggestion avoids these objections.

Universal Bayes consistency in metric spaces

We extend a recently proposed 1-nearest-neighbor based multiclass learning algorithm and prove that our modification is universally strongly Bayes consistent in all metric spaces admitting any such learner, making it an “optimistically universal” Bayes-consistent learner. This is the first learning algorithm known to enjoy this property; by comparison, the k-NN classifier and its variants are not generally universally Bayes consistent, except under additional structural assumptions, such as an inner product, a norm, finite dimension or a Besicovitch-type property. The metric spaces in which universal Bayes consistency is possible are the “essentially separable” ones—a notion that we define, which is more general than standard separability. The existence of metric spaces that are not essentially separable is widely believed to be independent of the ZFC axioms of set theory. We prove that essential separability exactly characterizes the existence of a universal Bayes-consistent learner for the given metric space. In particular, this yields the first impossibility result for universal Bayes consistency. Taken together, our results completely characterize strong and weak universal Bayes consistency in metric spaces.

The Relativistic Nature of Entropy Changes

This paper presents a method to obtain the variations of the entropies of the phases of a chemical substance in its vapor state, which allows deriving, from thermodynamics, the axioms of a quantum theory that conforms to special relativity.

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

Bounded finite set theory

AbstractWe define an axiom schema for finite set theory with bounded induction on sets, analogous to the theory of bounded arithmetic, , and use some of its basic model theory to establish some independence results for various axioms of set theory over . Then we ask: given a model M of , is there a model of whose ordinal arithmetic is isomorphic to M? We show that the answer is yes if .

Martin's Maximum${}^{++}$ implies Woodin's axiom $(*)$

We show that Martin's Maximum${}^{++}$ implies Woodin's $\mathbb{P}_{\mathrm{max}}$ axiom $(*)$. This answers a question from the 1990s and amalgamates two prominent axioms of set theory which were both known to imply that there are $ℵ_2$ many real numbers.