Reciprocal relations are binary relations Q with entries Q(i,j)∈0,1, and such that Q(i,j)+Q(j,i)=1. Relations of this kind occur quite naturally in various domains, such as preference modeling and ...preference learning. For example, Q(i,j) could be the fraction of voters in a population who prefer candidate i to candidate j. In the literature, various attempts have been made at generalizing the notion of transitivity to reciprocal relations. In this paper, we compare three important frameworks of generalized transitivity: g-stochastic transitivity, T-transitivity, and cycle-transitivity. To this end, we introduce E-transitivity as an even more general notion. We also use this framework to extend an existing hierarchy of different types of transitivity. As an illustration, we study transitivity properties of probabilities of pairwise preferences, which are induced as marginals of an underlying probability distribution on rankings (strict total orders) of a set of alternatives. In particular, we analyze the interesting case of the so-called Babington Smith model, a parametric family of distributions of that kind.
For a reciprocal relation
Q on a set of alternatives
A, two transitivity frameworks which generalize both
T-transitivity and stochastic transitivity are compared: the framework of cycle-transitivity, ...introduced by the present authors (Soc. Choice Welf., to appear) and which is based upon the ordering of the numbers
Q
(
a
,
b
)
,
Q
(
b
,
c
)
and
Q
(
c
,
a
)
for all
(
a
,
b
,
c
)
∈
A
3
, and the framework of
FG
-transitivity, introduced by Switalski (Fuzzy Sets and Systems 137 (2003) 85) as an immediate generalization of stochastic transitivity. The rules that enable to express
FG
-transitivity in the form of cycle-transitivity and cycle-transitivity in the form of
FG
-transitivity, illustrate that for reciprocal relations the concept of cycle-transitivity provides a framework that can cover more types of transitivity than does the concept of
FG
-transitivity.
The mutual rank probability relation associated with a finite poset is a reciprocal relation expressing the probability that a given element succeeds another one in a random linear extension of that ...poset. We contribute to the characterization of the transitivity of this mutual rank probability relation, also known as proportional probabilistic transitivity, by situating it between strong stochastic transitivity and moderate product transitivity. The methodology used draws upon the cycle-transitivity framework, which is tailor-made for describing the transitivity of reciprocal relations.
It is well known that Lie algebra methods are the leading methods for studying controllability of continuous-time nonlinear systems including bilinear systems, where the controllability problems are ...usually transformed into the transitivity problems of the corresponding Lie algebras. Unfortunately, it is in general a difficult task to check transitivity of Lie algebras, especially in the high-dimensional cases. In this paper, we propose a new notion, called near-transitivity. We focus on unconstrained bilinear systems and show that the systems are nearly-controllable if and only if their corresponding Lie algebras are nearly-transitive. That is, even if the Lie algebras are not transitive, they can be nearly-transitive and the systems can still own a very large controllable region nearly covering the whole state space. More importantly, we demonstrate that near-transitivity is easier to check than transitivity. This will be useful in both the theory and applications of controllability of nonlinear systems since verifying near-controllability may suffice for most nonlinear systems. Sufficient algebraic conditions as well as algorithms for checking near-transitivity of unconstrained bilinear systems are presented, which are also generalized to inhomogeneous bilinear systems to derive near-controllability. Furthermore, we apply the presented near-transitivity results to structural bilinear systems to derive necessary and sufficient conditions on structural near-controllability. Examples are given to demonstrate the proposed near-transitivity of this paper.
In a discussion of Parfit's Drops of Water case, Zach Barnett has recently proposed a novel argument against “No Small Improvement”; that is, the claim that a single drop of water cannot affect the ...magnitude of a thirsty person's suffering. We first show that Barnett's argument can be significantly strengthened, and also that the fundamental idea behind it yields a straightforward argument for the transitivity of equal suffering (a much stronger and more important conclusion than Barnett's). We then suggest that defenders of No Small Improvement could reject a Pareto principle that is presupposed in Barnett's argument and our developments of it. However, this does not save No Small Improvement, since there is a convincing argument against this claim that does not presuppose the Pareto principle.
Navigated Transcranial Magnetic Stimulation (nTMS) is commonly used to causally identify cortical regions involved in language processing. Combining tractography with nTMS has been shown to increase ...induced error rates by targeting stimulation of cortical terminations of white matter fibers. According to functional Magnetic Resonance Imaging (fMRI) data, bilateral cortical areas connected by the arcuate fasciculus (AF) have been implicated in the processing of transitive compared to unergative verbs. To test this connection between transitivity and bilateral perisylvian regions, we administered a tractography-based inhibitory nTMS protocol during action naming of finite transitive (The man reads) and unergative (The man sails) verbs. After tracking the left and right AF, we stimulated the cortical terminations of the tract in frontal, parietal and temporal regions in 20 neurologically healthy native speakers of German. Results revealed that nTMS induced more errors during transitive compared to unergative verb naming when stimulating the left (vs right) AF terminations. This effect was specific to the left temporal terminations of the AF, whereas no differences between the two verb types were identified when stimulating inferior parietal and frontal AF terminations. Induced errors for transitive verbs over left temporal terminations mostly manifested as access errors (i.e., hesitations). Given the inhibitory nature of our nTMS protocol, these results suggest that temporal regions of the left hemisphere play a crucial role in argument structure processing. Our findings align with previous data on the role of left posterior temporal regions in language processing and by providing further evidence from a language production experiment using tractography-based inhibitory nTMS.
In this paper, we characterize the F-transitive and the dF-transitive families of composition operators on Lp(X,B,μ), where (X,B,μ) is a σ-finite measure space. In particular, both necessary and ...sufficient conditions on the inducing mappings for F-transitive and dF-transitive composition operators are presented, which includes both Theorem 1.1 and Theorem 1.2 given in 1 in a more general frame. Moreover, we provide the characterizations of F-transitive and dF-transitive semigroups induced by semiflows on Lρp(X,K). And as a result, the sufficient condition of d-transitive semigroups induced by semiflows on Lρp(Ω,K) appears to be necessary, which answers a recent question posed by Kostić in 16. Further, we provide the characterizations on the weight function ρ for F-transitive translation semigroups on Lρp(Δ,K), which extends the recent results in 10 and 13. Some related examples of dF-transitive composition operators and C0-semigroups that are the solution semigroups of certain Cauchy problems are provided. In particular, we present the properties of dF-transitive composition operators on Lp(T,B,λ) induced by automorphisms on the complex unit disk D and show the equivalence of d-transitivity and d-mixing property for these operators.
A graph or hypergraph is said to be vertex‐transitive if its automorphism group acts transitively upon its vertices. A classic theorem of Mader asserts that every connected vertex‐transitive graph is ...maximally edge‐connected. We generalise this result to hypergraphs and show that every connected linear uniform vertex‐transitive hypergraph is maximally edge‐connected. We also show that if we relax either the linear or uniform conditions in this generalisation, then we can construct examples of vertex‐transitive hypergraphs which are not maximally edge‐connected.
The classical notions of transitivity and full transitivity in Abelian p-groups have natural extensions to concepts called Krylov and weak transitivity. The interconnections between these four types ...of transitivity are determined for Abelian p-groups; there is a marked difference in the relationships when the prime p is equal to 2. In the final section the relationship between full and Krylov transitivity is examined in the case of mixed Abelian groups which are p-local in the sense that multiplication by an integer relatively prime to p is an automorphism.
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