Generalization of Scott's formula for retractions from generalized Alexandroff's cube

In the paper [2] the following theorem is shown: Theorem (Th. 3,5, [2]), If α=0 or δ=∞ or α?δ, then a closure space X is an absolute extensor for the category of 〈α, δ〉 -closure spaces iff a contraction of X is the closure space of all 〈α, δ〉-filters in an 〈α, δ〉-semidistributive lattice. In the case when α=ω and δ=∞, this theorem becomes Scott's theorem: Theorem ([7]). A topological space X is an absolute extensor for the category of all topological spaces iff a contraction of X is a topological space of “Scott's open sets” in a continuous lattice. On the other hand, when α=0 and δ=ω, this theorem becomes Jankowski's theorem: Theorem ([4]). A closure space X is an absolute extensor for the category of all closure spaces satisfying the compactness theorem iff a contraction of X is a closure space of all filters in a complete Heyting lattice. But for separate cases of α and δ, the Theorem 3.5 from [2] is proved using essentialy different methods. In this paper it is shown that this theorem can be proved using, for retraction, one uniform formula. Namely it is proved that if α= 0 or δ= ∞ or α ? δ and \(F_{\alpha ,\delta } \left( L \right) \subseteq B_{\alpha ,\delta }^\mathfrak{n} \) and if L is an 〈α, δ〉-semidistributive lattice, then the function $$r:{\text{ }}B_{\alpha ,\delta }^\mathfrak{n} \to F_{\alpha ,\delta } \left( L \right)$$ such that for x ε ? ( \(\mathfrak{n}\) ): (*) $$r\left( x \right) = inf_L \left\{ {l \in L|\left( {\forall A \subseteq L} \right)x \in C\left( A \right) \Rightarrow l \in C\left( A \right)} \right\}$$ defines retraction, where C is a proper closure operator for \(B_{\alpha ,\delta }^\mathfrak{n} \) . It is also proved that the formula (*) defines retraction for all 〈α, δ〉, whenever L is an 〈α, δ〉 -pseudodistributive lattice. Moreover it is proved that when α=ω and δ=∞, the formula (*) defines identical retraction to the formula given in [7], and when α = 0 and δ=ω, the formula (*) defines identical retraction to the formula given in [4].     

Some results on extensions and modifications of the Theil — Sen regression estimator

Many robust regression estimators have been proposed that have a high, finite‐sample breakdown point, roughly meaning that a large porportion of points must be altered to drive the value of an estimator to infinity. But despite this, many of them can be inordinately influenced by two properly placed outliers. With one predictor, an estimator that appears to correct this problem to a fair degree, and simultaneously maintain good efficiency when standard assumptions are met, consists of checking for outliers using a projection‐type method, removing any that are found, and applying the Theil — Sen estimator to the data that remain. When dealing with multiple predictors, there are two generalizations of the Theil — Sen estimator that might be used, but nothing is known about how their small‐sample properties compare. Also, there are no results on testing the hypothesis of zero slopes, and there is no information about the effect on efficiency when outliers are removed. In terms of hypothesis testing, using the more obvious percentile bootstrap method in conjunction with a slight modification of Mahalanobis distance was found to avoid Type I error probabilities above the nominal level, but in some situations the actual Type I error probabilities can be substantially smaller than intended when the sample size is small. An alternative method is found to be more satisfactory.     

On the extension of intuitionistic propositional logic with Kreisel-Putnam's and Scott's schemes

LetSKP be the intermediate prepositional logic obtained by adding toI (intuitionistic p.l.) the axiom schemes:S = ((? ?α→α)→α∨ ?α)→ ?α∨ ??α (Scott), andKP = (?α→β∨γ)→(?α→β)∨(?α→γ) (Kreisel-Putnam). Using Kripke's semantics, we prove:

1. SKP has the finite model property;
2. SKP has the disjunction property.

In the last section of the paper we give some results about Scott's logic S = I+S.     

The Scott-Suppes theorem on semiorders

Luce introduced semiorders as a natural generalization of weak orders. The basic representation theorem for semiorders was established in 1958 by Scott and Suppes. In this note this theorem is given a simple contructive proof.     

Metrics on spaces of finite trees

With the increasing popularity of hierarchical clustering methods in behavioral science, there is a need for ways of quantitatively comparing different tree structures on the same set of items. We employ lattice-theoretic methods to construct a variety of metrics on spaces of trees and to analyze their properties. Certain of these metrics are applied to data from Fillenbaum and Rapoport (1971) on the semantic structure of common English kin terms. This application shows that tree metrics can be used to select a componential analysis which is maximally consistent with an empirically derived set of trees.     

Gaifman's theorem on categorial grammars revisited

The equivalence of (classical) categorial grammars and context-free grammars, proved by Gaifman [4], is a very basic result of the theory of formal grammars (an essentially equivalent result is known as the Greibach normal form theorem [1], [14]). We analyse the contents of Gaifman's theorem within the framework of structure and type transformations. We give a new proof of this theorem which relies on the algebra of phrase structures and exhibit a possibility to justify the key construction used in Gaifman's proof by means of the Lambek calculus of syntactic types [15].     

Finite forms of de Finetti's theorem on exchangeability

A geometrical interpretation of independence and exchangeability leads to understanding the failure of de Finetti's theorem for a finite exchangeable sequence. In particular an exchangeable sequence of length r which can be extended to an exchangeable sequence of length k is almost a mixture of independent experiments, the error going to zero like 1/k.     

A note on some modifications of latent roots and vectors

When an arbitrary positive scalar matrix is added to a correlation matrix the latent roots of the sum are equal to the corresponding roots of the correlation matrix plus an amount equal to the scalar number of the scalar matrix. The latent vectors of the sum are identical with those of the correlation matrix. An approximation to these relationships is suggested for the case in which the sum is of a correlation matrix and of a positive semidefinite diagonal matrix. The approximation is used to allow the solution of a characteristic problem for a correlation matrix with unities in the main diagonal to provide a family of solutions for the same correlation matrix.This research has been supported by a grant from the National Institute of Mental Health, MH 7864-01.     

Study on emotion spaces with centrality measure

In this paper, we study existing models of emotion space using centrality, which is borrowed from network theory, to identify key emotions as the central nodes in a network, for the purposes of understanding the existing emotion spaces better in a new way. With several different definitions of centrality, key emotions are identified for four existing emotion space models. We also propose a method for integrating existing spaces to build a refined space with more emotion terms. Each model identified different key emotions. When we reduced emotion spaces such that they each contained 21 common emotions, the key emotions identified remained different, implying fundamental structural differences among existing emotion space models. These findings call for further experimental verification and the refinement of emotion models for future research to make it more useful in emotion research.     

On a theorem of Feferman

Summary In this paper I have argued that Feferman's theorem does not signify the existence of skeptic-satisfying consistency proofs. However, my argument for this is much different than other arguments (most particularly Resnik's) for the same claim. The argument that I give arises form an analysis of the notion of expression, according to which the specific character of that notion is seen as varying from one context of application (of a result of arithmetic metamathematics) to another.I want to thank Dale Gottlieb, Mike Resnik, John Corcoran, Nicolas Goodman and Stewart Shapiro for careful and helpful commentary on an ancestor of this paper that was delivered at the Western Division Meetings of the APA in 1977.     

Probability spaces,hilbert spaces,and the axioms of test theory

A branch of probability theory that has been studied extensively in recent years, the theory of conditional expectation, provides just the concepts needed for mathematical derivation of the main results of the classical test theory with minimal assumptions and greatest economy in the proofs. The collection of all random variables with finite variance defined on a given probability space is a Hilbert space; the function that assigns to each random variable its conditional expectation is a linear operator; and the properties of the conditional expectation needed to derive the usual test-theory formulas are general properties of linear operators in Hilbert space. Accordingly, each of the test-theory formulas has a simple geometric interpretation that holds in all Hilbert spaces.