1,721,040 research outputs found
Invariant relations in a finite domain
Full text linkABSTRACT: Some results of an abstract inquiry
into the concept of invariance are presented. The objects to be
judged in their possible invariance are relations of arbitrary
degrees in one single domain, when they are represented as sets of
strings of elements in that domain. The domain itself is presumed
to be of finite size, and the transformations with respect to
which invariance is judged are injective functions between parts
of it. The study focuses on correspondences between algebraic
structures of possible relations and algebraic structures of
possible transformations, as they are dually interrelated through the invariance condition
Singolarità della visione. Spunti di formalizzazione nello studio fenomenologico del percepire.
No full textGerarchie intervallari binarie fra loro incompatibili
No full textRIASSUNTO: Una gerarchia intervallare è una famiglia nidificata di intervalli in un dato insieme finito linearmente ordinato. Le gerarchie intervallari binarie su un tale insieme sono le gerarchie di massima numerosità in esso. In questo articolo si dimostra che per una qualsiasi gerarchia binaria esiste almeno un’altra gerarchia binaria sul medesimo dominio la quale non possiede alcun membro in comune con essa, fatta eccezione per gli intervalli banali, ossia l’intero dominio ed i singoletti in esso. Questa proprietà viene dimostrata mediante la definizione e verifica di una procedura costruttiva la quale consiste in una catena di trasformazioni elementari (dette progressioni) su una adeguata rappresentazione tabellare (detta T-matrice) della gerarchia binaria di intervalli supposta data.
TITLE: On disjointness between binary hierarchies of intervals.
ABSTRACT: An interval hierarchy is a nested family of intervals in some finite linearly ordered set. The binary interval hierarchies on that set are the interval hierarchies of maximum size on it. Here it is shown that for any binary interval hierarchy at least one other binary interval hierarchy on the same domain can be found which has no member in common with it, except for the trivial intervals, i.e., the whole domain and the singletons in it. This result is proved by defining and testing a constructive procedure which consists in a chain of elementary transformations (called progressions) on a suitable tabular representation (called a T-matrix) of the presumed binary hierarchy of intervals
Induction of permutations and the invariance condition
No full textABSTRACT: Set-theoretic objects to be judged for invariance are of various kinds, some quite simple, others of high structural complexity. Formation and/or transformation processes may be conceived, which go through classes of objects of varying complexity; these processes are paralleled by induction of permutations which act on the classes and become critical in judging the invariance of objects. This paper studies the connection between formation and induction processes, by presuming four basic ways of passing from class to class (restriction, transformation, the domain and codomain of a set-theoretic power) and four corresponding ways of permutation induction. The theoretical system thus defined is then applied in examining specifications of the invariance condition for various kinds of set-theoretic constructs (subsets, families of subsets, relations, operations, patterns, statistical measures, decision rules, etc.) which are encountered when discussing, for example, meaningfulness in measurement theory, symmetry of perceptual patterns, and properties of rules for statistical inference, and are thus of concern to some parts of psychological science
Tree representations of betweenness relations defined by intersection and inclusion
Full text linkABSTRACT: On a family of sets, a ternary relation may be defined by stating that, for U,V,W members of the family, V is “between” U and W if and only if V includes the intersection of U and W. The relation is called “intersection-betweenness” and may be understood as the description of proximities between objects associated with sets in the family. The problem of using a tree graph for representing such a relation is discussed. Characterisations are proven both for full tree representation (there is a tree-betweenness identical to the given intersection-betweenness: Section 2) and for partial tree representation (there is a tree-betweenness included in the given intersection-betweenness: Section 3). Procedures for actually finding solutions to full and partial tree representation problems are illustrated in Section 4. In Section 5 some related paradigms of modern psychometrics are mentioned, to highlight the peculiar aspects of the proposed approach
Optimal linear bipartitions of two-colour sets of points in low-dimensional Euclidean spaces
No full textABSTRACT: A bipartition (a pair of complementary parts) of a set of elements is said to be linear if there is a point in the line so that the two parts are separately included in the two half-lines the point specifies (when the elements are in the line), if there is a line in the plane so that the two parts are separately included in the two half-planes the line specifies (when the elements are in the plane), and if there is a plane in the three-dimensional space so that the two parts are separately included in the two half-spaces the plane specifies (when the elements are in the three-dimensional space). This study develops a method for finding the whole set of linear bipartitions of any finite set of points in the line, plane, and three-dimensional space. The method acts in a progressive way, so that the problem regarding the two-dimensional case is solved in terms of its one-dimensional projections, and that regarding the three-dimensional case in terms of its two-dimensional projections (which in turn may be solved in terms of their one-dimensional projections). Rules for finding optimal linear bipartitions and separators when the points in the set are of two colours are defined and illustrated
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