Fourth Axiom Research Articles

Alternatives for the q-matroid axioms of independent spaces, bases, and spanning spaces

It is well known that in q-matroids, axioms for independent spaces, bases, and spanning spaces differ from the classical case of matroids, since the straightforward q-analogue of the classical axioms does not give a q-matroid. For this reason, a fourth axiom has been proposed. In this paper we show how we can describe these spaces with only three axioms, providing two alternative ways to do that. As an application, we show direct cryptomorphisms between independent spaces and circuits and between independent spaces and bases.

Khinchin’s Fourth Axiom of Entropy Revisited

The Boltzmann–Gibbs–Shannon (BGS) entropy is the only entropy form satisfying four conditions known as Khinchin’s axioms. The uniqueness theorem of the BGS entropy, plus the fact that Shannon’s mutual information completely characterizes independence between the two underlying random elements, puts the BGS entropy in a special place in many fields of study. In this article, the fourth axiom is replaced by a slightly weakened condition: an entropy whose associated mutual information is zero if and only if the two underlying random elements are independent. Under the weaker fourth axiom, other forms of entropy are sought by way of escort transformations. Two main results are reported in this article. First, there are many entropies other than the BGS entropy satisfying the weaker condition, yet retaining all the desirable utilities of the BGS entropy. Second, by way of escort transformations, the newly identified entropies are the only ones satisfying the weaker axioms.

Anonymity in sharing the revenues from broadcasting sports leagues

We study the problem of sharing the revenues from broadcasting sports leagues axiomatically. Our key axiom is anonymity, the classical impartiality axiom. Other impartiality axioms already studied in broadcasting problems are equal treatment of equals, weak equal treatment of equals and symmetry. We study the relationship between all impartiality axioms. Besides we combine anonymity with other axioms that have been considered in the literature. Some combinations give rise to new characterizations of well-known rules. The family of generalized split rules is characterized with anonymity, additivity and null team. The concede-and-divide rule is characterized with anonymity, additivity and essential team. Other and combinations characterize new rules that had not been considered before. We provide three characterizations in which three axioms are the same (anonymity, additivity, and order preservation) and the fourth one is different (maximum aspirations, weak upper bound, and non-negativity). Depending on the fourth axiom we obtain three different families of rules. In all of them concede-and-divide plays a central role.

The Myth of the Importance of Consistency in Analytical Hierarchical Processes

In this work, rather than relating different areas of knowledge, the authors make use of logic, which is intrinsic to mathematical models. They discuss one of the fallacies that hides behind the analytic hierarchy process (AHP). In particular, the analysis of the consistency index, sometimes referred as the Saaty consistency or simply the consistency, will be discussed. The AHP is based on three fundamental axioms: reciprocity, homogeneity, and synthesis. However, it gives a great relevance to consistency, which is managed like a fourth axiom. On each of these four axioms, some comments will be made. However, this work will focus mainly on discussing the importance of the consistency index, to which, undoubtedly, a large number of essays and many hours of work have been devoted. In trying to answer the question, the calculation of consistency is really important. The aim of the paper is to raise a discussion of one of the fallacies that hides behind the analytic hierarchy process and present a simple and direct way to handle Saaty consistency.

The Fourth Axiom of Similarity Measures

In this research, the fourth axiom to improve the well-defined examination of similarity measures is studied, where the measures have a symmetric structure. We first provide a theoretic enhancement of three correlation coefficient similarity measures that were proposed by a source paper. Second, we use the same numerical example proposed by the source paper for pattern recognition problems to illustrate that the weighted correlation coefficient similarity measure is dependent on the set of weights. Finally, we demonstrate that the correlation coefficient similarity measure in the intuitionistic fuzzy set environment can address the issue of practical fault diagnosis when solving the turbine engine problems using similarity measures with symmetric characteristics.

New similarity measure and distance measure for Pythagorean fuzzy set

Pythagorean fuzzy set (PFS), disposing the indeterminacy portrayed by membership and non-membership, is a more viable and effective means to seize indeterminacy. Due to the defects of existing Pythagorean fuzzy similarity measures or distance measures (cannot obey third or fourth axiom; have no power to differentiate positive and negative difference; have no power to deal the division by zero problem), the major key of this paper is to explore the novel Pythagorean fuzzy distance measure and similarity measure. Meanwhile, some interesting properties of distance measure and similarity measure are proved. Some counterintuitive examples are presented to state their availability of similarity measure among PFSs.

A Brief Review of Generalized Entropies.

Entropy appears in many contexts (thermodynamics, statistical mechanics, information theory, measure-preserving dynamical systems, topological dynamics, etc.) as a measure of different properties (energy that cannot produce work, disorder, uncertainty, randomness, complexity, etc.). In this review, we focus on the so-called generalized entropies, which from a mathematical point of view are nonnegative functions defined on probability distributions that satisfy the first three Shannon–Khinchin axioms: continuity, maximality and expansibility. While these three axioms are expected to be satisfied by all macroscopic physical systems, the fourth axiom (separability or strong additivity) is in general violated by non-ergodic systems with long range forces, this having been the main reason for exploring weaker axiomatic settings. Currently, non-additive generalized entropies are being used also to study new phenomena in complex dynamics (multifractality), quantum systems (entanglement), soft sciences, and more. Besides going through the axiomatic framework, we review the characterization of generalized entropies via two scaling exponents introduced by Hanel and Thurner. In turn, the first of these exponents is related to the diffusion scaling exponent of diffusion processes, as we also discuss. Applications are addressed as the description of the main generalized entropies advances.

Discounting axioms imply risk neutrality

Although most applications of discounting occur in risky settings, the best-known axiomatic justifications are deterministic. This paper provides an axiomatic rationale for discounting in a stochastic framework. Consider a representation of time and risk preferences with a binary relation on a real vector space of vector-valued discrete-time stochastic processes on a probability space. Four axioms imply that there are unique discount factors such that preferences among stochastic processes correspond to preferences among present value random vectors. The familiar axioms are weak ordering, continuity and nontriviality. The fourth axiom, decomposition, is non-standard and key. These axioms and the converse of decomposition are assumed in previous axiomatic justifications for discounting with nonlinear intraperiod utility functions in deterministic frameworks. Thus, the results here provide the weakest known sufficient conditions for discounting in deterministic or stochastic settings. In addition to the four axioms, if there exists a von Neumann-Morgenstern utility function corresponding to the binary relation, then that function is risk neutral (i.e., affine). In this sense, discounting axioms imply risk neutrality.

An axiomatization of the median procedure on the [formula omitted]-cube

The general problem in location theory deals with functions that find sites to minimize some cost, or maximize some benefit, to a given set of clients. In the discrete case sites and clients are represented by vertices of a graph, in the continuous case by points of a network. The axiomatic approach seeks to uniquely distinguish certain specific location functions among all the arbitrary functions that address this problem by using a list of intuitively pleasing axioms. The median function minimizes the sum of the distances to the client locations. This function satisfies three simple and natural axioms: anonymity, betweenness, and consistency. They suffice on tree networks (continuous case) as shown by Vohra (1996) [19], and on cube-free median graphs (discrete case) as shown by McMorris et al. (1998) [9]. In the latter paper, in the case of arbitrary median graphs, a fourth axiom was added to characterize the median function. In this note we show that the above three natural axioms still suffice for the hypercubes, a special instance of arbitrary median graphs.

Risk Neutrality and Ordered Vector Spaces

The following result clarifies when preferences over time and under risk correspond to discounting and are not risk neutral. If a binary relation on a real vector space V satisfies four axioms, then there is a utility function U=fu from V to R where u from V to R is linear as a map of vector spaces and f from R to R is continuous and weakly monotone. Three axioms are familiar: weak ordering, continuity, and non-triviality. The fourth axiom is X-Y is weakly preferred to 0 (where 0 is the zero element in V) implies X is weakly preferred to Y (X and Y in V). The function f from R to R is linear if and only if the binary relation also satisfies the converse of the fourth axiom. When V is a real vector space of stochastic processes and 0 is the zero process, it is known that the four axioms imply the existence of discount factors and the linearity of an intra-period utility function. So preferences correspond to discounting and are not risk neutral only if the converse of the fourth axiom is not satisfied.

Discounting and Risk Neutrality

Consider a binary relation on a real vector space of vector-valued discrete-time stochastic processes. If the binary relation satisfies four axioms, then there are unique discount factors such that preferences regarding stochastic processes induce prefences among present value random vectors. Three of the axioms are familiar: weak ordering, continuity, and non-triviality. The fourth axiom is decomposition. Also, if preferences satisfy the four axioms then the following properties are equivalent: the converse of decomposition, the existence of a felicity function on the set of random vectors, and risk neutrality. In this limited sense, discounting implies risk neutrality.

On structural descriptions of lower ideals of trees

AbstractA finite structural description for proper lower ideals of trees was found in [5] by Robertson, Seymour, and Thomas. They proved existence of a specific description satisfying three axioms. We present our result starting from this fundamental construction. In an attempt to make their description unique and efficient, they introduced a fourth axiom, which conflicts with one of the prior axioms. In this paper, we make use of their valid constructive result and prove the desired uniqueness and efficiency using a different set of axioms. © 2005 Wiley Periodicals, Inc. J Graph Theory

Stratified model categories

The fourth axiom of a model category states that given a commutative square of maps, say $i:A\to B,\ g:B\to Y,\ f:A\to X$, and $p:X\to Y$ such that $gi=pf$, if $i$ is a cofibration, $p$ a fibration and either $i$ or $p$ is a weak equivalence, then a lifti

A proof-irrelevant model of Martin-Löf's logical framework

We extend the proof-irrelevant model defined in Smith (1988) to the whole of Martin-Löf's logical framework. The main difference here is the existence of a type whose objects themselves represent types rather than proof-objects. This means that the model must now be able to distinguish between objects with different degree of relevance: those that denote proofs are irrelevant whereas those that denote types are not. In fact a whole hierarchy of relevance exists.Another difference is the higher level of detail in the formulation of the formal theory, such as the explicit manipulation of contexts and substitutions. This demands an equally detailed definition of the model, including interpreting contexts and substitutions.We are thus led to a whole reformulation of the proof-irrelevant model. We present a model that is built up from an arbitrary model of the untyped lambda calculus. We also show how to extend it when the logical framework itself is enlarged with inductive definitions. In doing so, a variant of Church numerals is introduced.As in Smith (1988), the model can only be defined in the absence of universes, and it is useful to obtain an elementary proof of consistency and to prove the independence of Peano's fourth axiom.

Evolution: an axiomatic perspective.

In this paper we present an axiomatic theory of evolution which is inspired by the reading of a paper written by G.V. Schiaparelli in 1898. Schiaparelli was a famous astronomer, but he also studied the Darwinian ideas. We propose five axioms which can characterize the theory of evolution. We have also written these axioms using the language of the logic of predicates of first order with some constant monadic and dyadic predicates and appropriate functionals. But we can use other types of logic. So we can examine the concepts of species and of speciation. Then we introduce the interesting notion of generation distance. Moreover we give a theorem which establishes a geometrical model of our theory. If we analyze further the fourth axiom (which concerns the notion of generation distance) we can propose an elementary dynamical model by which we can represent possible evolutionary dynamics. These dynamics partially depend on random quantities.

Something about random normed modules

The fourth axiom in the definition of real random normed module is superfluous. A necesmry and sufficient condition is given that continuous random linear functionals over complex random normed spaces are a. s. bounded, and furthermore, in the sense of inometrical isomorphism it is proved the fourth axiom in the complex random normed module is superfluous.

Five axioms for the functional design of peptide-based polymers as molecular machines and materials: Principle for macromolecular assemblies

Peptide-based polymers can be transformable hydrogels, elastomers, regular thermoplastics, or inverse thermoplastics, and can function as diverse molecular machines. When of the appropriate composition, these polymers exhibit hydrophobic folding and assembly transitions in the accessible aqueous range as the temperature is raised from below to above a critical value referred to as Tt, the temperature for the onset of an inverse temperature transition. In this article, the design of these Tt-type molecular machines and materials is systematized in terms of five axioms for peptide-based polymer engineering: The first axiom is represented by a hydrophobicity scale that provides the information with which to design peptide-based polymers with a particular value of Tt. The second axiom concerns thermomechanical transduction wherein raising the temperature from below to above Tt results in hydrophobic folding and assembly with the performance of mechanical work. The third axiom states that at constant temperature, diverse energy inputs lower Tt from above to below an operating temperature by acting on a functional moiety within the polymer to drive hydrophobic folding and assembly with the isothermal performance of mechanical work. The fourth axiom concerns polymer compositions having two different functional groups, with each responsive to a different energy input and with each part of a common hydrophobic folding domain. The two functional groups become coupled one to the other such that an energy input acting on one functional group becomes an energy output by having changed the property of the second functional group. The fifth axiom states that the efficiency of conversion from one form of energy to another by means of the fourth axiom increases nonlinearly with hydrophobicity of the common hydrophobic folding and assembly domain. These phenomenological axioms allow design of peptide-based polymers capable of diverse energy conversions involving the intensive variables of mechanical force, temperature, pressure, chemical potential, electrochemical potential, and light. Knowledge of the physical process underlying the five axioms enhances their use in the design of materials and devices for medical and nonmedical applications. The physical process is demonstrated by the use of cartoons. © 1998 John Wiley & Sons, Inc. Biopoly 47: 167–178, 1998

Parsimony neither maximizes congruence nor minimizes incongruence or homoplasy

Wilkinson is correct to point out inaccuracy between Scotland's (1992: 3, 8) fourth axiom of cladistics the principle of parsimony maximizes character and discussion of character congruence and consistency (Scotland, 1992: 8). Here I take opportunity to illustrate that parsimony does not maximize character congruence and therefore Patterson's (1982) congruence test of homology is potentially misleading in name. I illustrate that Wilkinson is wrong in his assertion that parsimony minimizes incongruence. I further re-assert that parsimony minimizes ad hoc assumptions of homoplasy (Farris, 1983) but that subtleties of Farris's (1983) argument are easily misunderstood.

The Metric Properties of DNA-DNA Hybridization Dissimilarity Measures

Under a simple model of DNA sequence dissociation, DNA-DNA hybridization dissimilarity measures are expected to behave like the complement of Gower's coefficient of similarity, calculated for each pair of taxonomic units (j, k) across i homologues (nucleotide positions) with four possible states, multiplied by a constant relating base-pair mismatch to the temperature of DNA dissociation. DNA dissimilarity measures that include information only from homologues, therefore, are expected to conform to the axioms of a metric. Empirically, we have found that measures of dissimilarity between single-copy nuclear genomes regularly con- form to 3 of the 4 metric axioms: identity, distinctness, and the triangle inequality. However, reciprocal values frequently fail to conform to the fourth axiom, symmetry. For some measures dependent upon the percent of reassociation (e.g., delta T5OH), failure of symmetry can occur when sequences present in one genome have no homologue in other genomes (e.g., as a result of sequence deletion or lateral transfer). For all measures, including delta mode and delta Tm, the presence of sequences in one genome that lack a homologue in other genomes differentially affects the effective concentration of shared sequences in reciprocal reactions, causing reciprocal measurements to represent comparisons of different sets of sequences. This effect also potentially obtains under certain conditions of sequence duplication and divergence. (Metrics; DNA evo- lution; DNA-DNA hybridization; phylogeny; symmetry; triangle inequality; additivity.)

The independence of Peano's fourth axiom from Martin-Löf's type theory without universes

In Hilbert and Ackermann [2] there is a simple proof of the consistency of first order predicate logic by reducing it to propositional logic. Intuitively, the proof is based on interpreting predicate logic in a domain with only one element. Tarski [7] and Gentzen [1] have extended this method to simple type theory by starting with an individual domain consisting of a single element and then interpreting a higher type by the set of truth valued functions on the previous type.I will use the method of Hilbert and Ackermann on Martin-Löf's type theory without universes to show that ¬Eq(A, a, b) cannot be derived without universes for any type A and any objects a and b of type A. In particular, this proves the conjecture in Martin-Löf [5] that Peano's fourth axiom (∀x ϵ N)¬ Eq(N, 0, succ(x)) cannot be proved in type theory without universes. If by consistency we mean that there is no closed term of the empty type, then the construction will also give a consistency proof by finitary methods of Martin-Löf's type theory without universes. So, without universes, the logic obtained by interpreting propositions as types is surprisingly weak. This is in sharp contrast with type theory as a computational system, since, for instance, the proof that every object of a type can be computed to normal form cannot be formalized in first order arithmetic.