Non-comparable Elements Research Articles

Using Mixed Methods to Interpret Differential Item Functioning

ABSTRACTAnalysis of differential item functioning (DIF) is often used to determine if cross-lingual assessments are equivalent across languages. However, evidence on the causes of cross-lingual DIF is still evasive. Expert appraisal is a qualitative method useful for obtaining detailed information about problematic elements in the different linguistic versions of items. In this article we propose and explore a mixed methods approach that integrates quantitative results from DIF analysis and qualitative findings from expert appraisal to discover reasons why items exhibit DIF across languages. First, polytomous DIF was analyzed in responses to the U.S. and Spanish version of scales from the PISA Student Questionnaire by Differential Step Functioning and Ordinal Logistic Regression. Items flagged by both methods were selected to be studied to interpret DIF causes. Secondly, experts were asked about non-comparable elements in items. Experts provided qualitative evidence on problematic issues (different interpretation patterns or response processes), that may have been the cause of the DIF. The integration of results from both methods was aimed at relating type of DIF to expert appraisal findings.

MONOTONIC INDEPENDENCE ASSOCIATED WITH PARTIALLY ORDERED SETS

A generalization of Muraki's notion of monotonic independence onto the case of partially ordered index set is given: algebras indexed by chains are monotonically independent, and algebras indexed by non-comparable elements are boolean independent. Examples of central limit theorem are shown in two cases. For the integral-points lattices ℕd the moments of the limit measure are related to the combinatorics of the finite heap-ordered labelled rooted trees (if d = 2). For the integral-points lattice ℕ × ℤd in Minkowski spacetime the limit measure is given by the recurrence of it's moments, which, for the case d = 1 is related to the inverse error function. Various formulas for computing mixed moments are shown to be related to the boolean-monotonic non-crossing pair partitions.

Density and unique decomposition theorems for the lattice of cellular classes

A class C of pointed spaces is called a cellular class if it is closed under weak equivalences, arbitrary wedges and pointed homotopy pushouts. The smallest cellular class containing X is denoted by C(X), and a partial order relation much less than is defined by: X much less than Y if Y is an element of C(X). In this text we investigate the sub partial order sets generated respectively by simply connected finite CW-complexes and by rational spaces. For rational spaces we prove a unique decomposition theorem, a density theorem and the existence of infinitely many non-comparable elements. We then prove the density theorem for a generic class of finite CW-complexes.

Evolution of the vertebrate jaw: homology and developmental constraints

In embryonic development of the vertebrate head, neural crest-derived ectomesenchyme contributes to a wide range of tissue types including oro-pharyngeal and ethmoidal cartilages. The evolution of the jaw, therefore, can be viewed as a change of developmental program for specification of the crest cells. Along the anteroposterior axis of the neural crest of amniote embryos, a series of homeobox genes are expressed in a nested pattern, and the jaw-forming mandibular arch receives crest cells expressing no Hox genes and midbrain-derived crest cells that express Otx2. Cognates of these regulatory genes are present in the lamprey, and are expressed in the comparable cell lineages of the embryo. Evolution of the jaw cannot be explained from such shared developmental mechanisms, but rather noncomparable elements have to be sought, if the jaw is truly an evolutionary novelty. By precise comparative morphology and gene expression analyses, a possibility was inferred that ammocoete lips may not be identical to gnathostome jaws.

About the correspondences between dual heyting arrow lattices of a distributive lattice and its sublattices for relational biologic systems

Heyting and dual Heyting arrow operations relating non-comparable elements of finite relatively pseudo-complemented lattices being pseudo-Boolean algebras ( L) gave place to new structures named Heyting arrow ( L F ) and dual Heyting arrow lattices ( L F) (though sometimes they are only posets). They were used for analyzing qualitative relations in biological systems by means of isomorphisms relating the lattice elements with energy states identified through abstract relational concepts describing the system being represented. This paper considers the problem of connecting the poset L F with the posets L F (κ) corresponding to the epimorphic images L κ of a pseudo-Boolean lattice L.

Correspondences between the Heyting arrow lattices of a distributive lattice and its sublattices in relational biologic systems

Qualitative relations in biological systems were studied by means of lattices. The first representations gave place to finite pseudo-Boolean algebras. The Heyting arrow operation relating noncomparable elements of the lattices ( L) allowed the definition of a new one: the Heyting arrow lattice ( L F ). Owing to the fact that the lattice elements are isomorphically related with states of the system being represented, then the Heyting operation is isomorphically related with enhancements of energy. This paper considers the problem of connecting the lattice L F with the lattices L F ( i) corresponding to the epimorphic images L i of L.

The Whitman condition in the Heyting arrow lattice for relational biologic processes

Qualitative relations in biological systems have been studied by means of lattices. The first representations gave place to finite pseudo-Boolean algebras. The Heyting arrow operation relating noncomparable elements of the lattices allowed the definition of a new one: the Heyting arrow lattice. Being the elements of the lattices expressed in energetic terms, enhancements of energy in the systems the represent were associated with the Heyting operation. The requirements of improving the knowledge about the meaning of this arrow in the representations, led to analyze the mathematical properties of the Heyting arrow lattice. The Whitman condition is considered in this paper.

Semidistributivity in the Heyting arrow lattice for relational biologic processes

Lattices have been used to study qualitative relations in biological systems. The resulting lattices are pseudo-Boolean algebras. The Heyting arrow operation between noncomparable elements of the original lattices gives place to a new one named the Heyting arrow lattice L F . The elements of these lattices are considered relational states of the systems being represented and they are expressed in terms of energy. In connection with this, the Heyting arrow has been related to enhancements of energy in the system. In order to know more about the meaning of this type of energetical variations, mathematical properties of L F are being studied. In this paper, the condition of semidistributivity is analyzed.

The dual Heyting arrow lattice for biologic relational processes

Qualitative relations in biological systems were studied by means of lattices. The first main representations gave place to lattices being double pseudo-Boolean algebras. So, close to the poset which defined the lattices, two operations appeared: the Heyting arrow and the dual Heyting arrow. Both are important to relate noncomparable elements in terms of processes concerned with increasing and decreasing energies in the system, as results from the order defining the poset which considers the lattice elements as relational energetic states. The present developments find out mathematical properties of lattice structures satisfying dual Heyting arrow operations between noncomparable elements.

The Heyting arrow lattice for qualitative relations in biological systems

Previous developments to study qualitative relations in biological systems were made by means of lattices, which resulted in pseudo-Boolean algebras. The Heyting arrow operation being obtained is considered particularly useful to study qualitative relations between non-comparable elements of the lattices. This operation involves enhancements of energy in the systems. Accordingly, the lattice built with the results of the Heyting arrow operation between non-comparable elements of pseudo-Boolean algebras is defined. The first properties of this lattice are found, and they are connected with the sets of closed and dense elements of the original algebra.

Conal orders on homogeneous spaces

In [6] and [7] Faraut raises the question whether the order intervals Ix, y] = {z: x ~ z ~ y} in an ordered symmetric space M = G/H of a Lie group G are compact or not. The compactness of the order intervals is necessary in order to define the algebra of Volterra kernels on such spaces. We give a first general answer to this question. Moreover we obtain very general results on the structure of ordered homogeneous spaces (Propos. 1.27) and on the foliations of such spaces by hypersurfaces consisting of non-comparable elements (Sect. 2). We begin by explaining how the differentiable structure of a homogeneous manifold can be linked with the presence of a partial order such as it exists on the space-times of general relativity. A homogeneous cone field on a homogeneous space M = G/H of a connected Lie group G assigns to each point x ~ M a pointed closed convex cone O(x) in the tangent space TAM), such that the action of G on M preserves the cone field. Such a field of cones gives rise to a conal order on M, defined by x x}. We say that M is globally orderable if =~ is a partial order. Let us denote the base point of the homogeneous space M with ~ := g(l) = H. Then the set

Ordered sets with small width and large jump number

The following result is proved in this note: For any positive integers w and t, if an ordered set P has jump number at least (t+1)w−1, then either the width of P is more than w, or P has a tower, i.e., a linear sum of pairs of noncomparable elements, of height more than t.

On Finitely Generated Lattices of Finite Width

The width of a lattice L is the maximum number of pairwise noncomparable elements in L.It has been known for some time ([5] ; see also [4]) that there is just one subdirectly irreducible lattice of width twro, namely the five-element nonmodular lattice N5. It follows that every lattice of width two is in the variety of N5, and that every finitely generated lattice of width two is finite.Beginning a study of lattices of width three, W. Poguntke [6] showed that there are infinitely many finite simple lattices of width three. Further studies on width three lattices were made in [3], where it was asked whether every finitely generated simple lattice of width three is finite. In this paper we will show that, in fact, more is true:THEOREM 1.1. Every finitely generated subdirectly irreducible lattice of width three is finite.

Partially ordered sets of 1-degrees, contained in recursively enumerable m-degrees

If A is a recursively enumerable (r.e.) nonrecursive set, then the family of all 1-degrees, contained in the m-degree of the set A, along with the natural relation ~ for the ldegrees, forms a partially ordered set L(A). In the present article, we investigate the structure of L(A) in terms of the properties of the set A. Let N={O,4... I" If A ~N, then f-N\A, and JA[ is the power of the set A. We denote by pf and ~f the domain of values and domain of definition of the one-place function f. We recall that, given any A, L(AI contains a maximal element. We will call aeL(A) a minimal element if it is not the least element in L(A), and, given any ~eL(A), ~ implies ~=a or that b is the least element of L(A). The concept of maximal element is defined in a similar way. We shall say that L(~) is dense if (~a)(~)(O,~EL(~)~a~==,~(3C)(a~c~). If L(A)con~ slats of a single element, then the m-degree of the set A is called irreducible. In this case, the m-degree of A contains only cylinders. It may be mentioned in this connection that simple and pseudosimple sets are not cylinders [i]. Some facts that are easily extracted from familiar results will be stated as propositions. Proposition I. If the m-degree of an r.e. set is not irreducible, then L(A) has no maximal elements and contains both an infinite chain and an infinite number of pairwise noncomparable elements. This follows from the results of [2, 3]. For, it was shown in [2] that, if A is not a cylinder, then A~A is likewise not a cylinder, and ~t~@A 9 When constructing the antichain, essential use was made of the recursive enumerability of the set A [3]. Proposition 2. If L(AJ contains minimal elements, then L(A) has a least element. This is obvious in the case when L(A} contains only one minimal element. If A ~ ~, ~m ~z, the 1-degrees of the sets Bz and B2 are minimal and noncomparable, and ~mBz by means of the general recursive function (g.r.f.) ~ , then let

A counterexample to the generalization of Sperner's theorem

It has been conjectured that the analog of Sperner's theorem on non-comparable subsets of a set holds for arbitrary geometric lattices, namely, that the maximal number of non-comparable elements in a finite geometric lattice is max w( k), where w( k) is the number of elements of rank k. It is shown in this note that the conjecture is not true in general. A class of geometric lattices, each of which is a bond lattice of a finite graph, is constructed in which the conjecture fails to hold.