An algebraic solution for the communalities

Factorial analysis begins with ann ×n correlation matrixR, whose principal diagonal entries are unknown. If the common test space of the battery is under investigation, the communality of each test is entered in the appropriate diagonal cell. This value is the portion of the test's variance shared with others in the battery. The communalities must be so estimated thatR will maintain the rank determined by its side entries, after the former have been inserted. Previous methods of estimating the communalities have involved a certain arbitrariness, since they depended on selecting test subgroups or parts of the data inR. A theory is presented showing that this difficulty can be avoided in principle. In its present form, the theory is not offered as a practical computing procedure. The basis of the new method lies in the Cayley-Hamilton theorem: Any square matrix satisfies its own characteristic equation.     

An algebraic treatment of the barwise compactness theory

A theorem on the extendability of certain subsets of a Boolean algebra to ultrafilters which preserve countably many infinite meets (generalizing Rasiowa-Sikorski) is used to pinpoint the mechanism of the Barwise proof in a way which bypasses the set theoretical elaborations.During the preparation of this work the second author was a postdoctoral visitor at the C. R. M. supported by a National Council of Canada operating grant.     

An algebraic solution for a class of subjective metrics models

It is shown that an obvious generalization of the subjective metrics model by Bloxom, Horan, Carroll and Chang has a very simple algebraic solution which was previously considered by Meredith in a different context. This solution is readily adapted to the special case treated by Bloxom, Horan, Carroll and Chang. In addition to being very simple, this algebraic solution also permits testing the constraints of these models explicitly. A numerical example is given.     

Hyperalgebraic primitive elements for relational algebraic and topological algebraic models

Using nonstandard methods, we generalize the notion of an algebraic primitive element to that of an hyperalgebraic primitive element, and show that under mild restrictions, such elements can be found infinitesimally close to any given element of a topological field.Presented by Robert Goldblatt     

An algebraic approach to intuitionistic modal logics in connection with intermediate predicate logics

Modal counterparts of intermediate predicate logics will be studied by means of algebraic devise. Our main tool will be a construction of algebraic semantics for modal logics from algebraic frames for predicate logics. Uncountably many examples of modal counterparts of intermediate predicate logics will be given. Dedicated to Prof. T. Umezawa on his 60th birthday     

Chang's modal operators in algebraic logic

Chang algebras as algebraic models for Chang's modal logics [1] are defined. The main result of the paper is a representation theorem for these algebras.     

An analogue study of intrusions

According to cognitive theorists, intrusive trauma memories have their origin in how information during the event is processed. Two studies investigated functional cognitive strategies during medical crises that might protect against intrusions. In Study 1, interviews with health-care professionals were used to identify cognitive strategies judged to be effective in controlling emotions and dealing with medical crises. Study 2 systematically manipulated the use of those strategies in a trauma analogue film paradigm. Experimental participants reported fewer intrusions, and less fear and avoidance of film-related stimuli during the subsequent week than controls. The manipulation did not affect anxiety during the film or memory disorganization. Implications for cognitive theories of intrusion development are discussed.     

Comprehension of algebraic expressions by experienced users of mathematics

Little is known about how people comprehend mathematical expressions. In the present study we investigate the internal representations used by experienced users of mathematics to encode algebraic expressions. Initially, a memory recognition task was conducted that examined the role of mathematical syntax in the encoding of algebraic expressions. The results indicate that participants could more readily identify those parts of a previously seen expression that were syntactically well formed than those that were not well formed, suggesting that syntax plays an important role. To determine the level of syntactic structure involved, a second recognition task was conducted. The results indicate that algebraic expressions are encoded into components that represent the phrasal constituents of the expression. However, the results of these experiments did not rule out the possibility that the visual processes of perceptual organization were used to encode the expressions, or that the data were a consequence of a mathematical “lexicon” of common mathematical “phrases”. Three further experiments were conducted to examine this, the results of which indicate that the encoding of algebraic expressions is based primarily on processes that occur beyond the level of visual or “lexical” processing. This is consistent with our finding that syntactic structure plays a key role.