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## Scholarly Works (719 results)

Let k be an integral domain, n a positive integer, X a generic n x n matrix over k (i.e., the matrix (x_{ij}) over a polynomial ring k[x_{ij}] in n^{2} indeterminates x_{ij}): and adj(X) its classical adjoint. For char k = 0, it is shown that if n is odd, adj(X) is not the product of two noninvertible n x n matrices over k[x_{ij}], while for n even, only one special sort of factorization occurs. Whether the corresponding results hold in positive characteristic is not known. The operation adj on matrices arises from the (n-1)st exterior power functor on modules; the analogous factorization question for matrix constructions arising from other functors is raised, as are several other questions.

In 1957, N. G. de Bruijn showed that the symmetric group Sym(Ω) on an infinite set Ω contains a free subgroup on 2^{card(Ω)} generators, and proved a more general statement, a sample consequence of which is that for any group *A* of cardinality ≤ card(Ω), the group Sym(Ω) contains a coproduct of 2^{card(Ω)} copies of *A*, not only in the variety of all groups, but in any variety of groups to which *A* belongs. His key lemma is here generalized to an arbitrary variety of algebras **V**, and formulated as a statement about functors **Set** → **V**. From this one easily obtains analogs of the results stated above with "group" and Sym(Ω) replaced by "monoid" and the monoid Self(Ω) of endomaps of Ω, by "associative *K*-algebra" and the *K*-algebra End_{K}(*V*) of endomorphisms of a *K*-vector-space *V* with basis Ω, and by "lattice" and the lattice Equiv(Ω) of equivalence relations on Ω. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(Ω), Self(Ω) and End_{K}(*V*) contains a coproduct of 2^{card(Ω)} copies of itself.

That paper also gave an example of a group of cardinality 2^{card(Ω)} that was *not* embeddable in Sym(Ω), and R. McKenzie subsequently established a large class of such examples. Those results are shown here to be instances of a general property of the lattice of solution sets in Sym(Ω) of sets of equations with constants in Sym(Ω). Again, similar results - this time of varying strengths - are obtained for Self(Ω), End_{K}(*V*) and Equiv(Ω), and also for the monoid Rel(Ω), of binary relations on Ω.

Many open questions and areas for further investigation are noted.

Envy is a pan-human phenomenon, universally feared, at least subconsciously, as a particularly dangerous emotion, since it implies hostility and aggression capable of destroying individuals and even societies. Especially in Western society, man has rather successfully repressed his true feelings about envy, which he is taught is the most shameful and reprehensible of all emotions. But even while denying it, man in all cultures has found devices, most but not all of which are symbolic, to cope with his fear of the consequences of envy. In this paper I distinguish between envy and jealousy (the terms are badly confused in English), note the objects that most frequently cause envy (food, children, and health), and analyze envy relationships between both conceptual equals and conceptual nonequals (concluding that one does not "envy down"). I then note the ways in which envy is expressed, including the symbolic "compliment," much feared in many societies because it is recognized as an expression of envy. Particularly striking is the fact that in those situations in which the envy of others is feared, culture dictates a strategy of evasive behavior based on a specific sequence of preferred choices. That is, when an individual fears envy he first attempts to conceal his good fortune; when this is not practical, he falls back on denial that there is reason to envy him; when this is not adequate, he symbolically shares, and only as a last resort does he truly share. This sequence is illustrated with a detailed comparative analysis of food-envy behavior, including tipping, which is explained as a symbolic device to buy off the envy of the waiter. Cultural forms used by an individual to cope with the fear that he is suspected of envy are then noted, as are those used by a person who fears to acknowledge his own envy. Finally, institutional devices to reduce envy are discussed briefly. The paper stresses the ways in which cultural forms have been developed to aid man in coping with psychological problems.

There have been inconsistent findings in previous second language research on the effect of vocabulary glossing on reading comprehension (Davis, 1989; Jacobs, Dufon, & Fong, 1994; Johnson, 1982; Pak, 1986). The present study was undertaken to extend this body of research in two ways: (a) by including another set of second language learners, another text, and another set of vocabulary glosses, accompanied by rigorous experimental procedures; and, (b) by considering the possible interaction of other variables with glossing. These other variables were: psychological type, tolerance of ambiguity, proficiency, frequency of gloss use, perceived value of gloss use, and time on task.

Glossing can be situated in the context of recent work on the reading process (Eskey, 1988; Lesgold & Perfetti, 1981; Rumelhart, 1980; Stanovich, 1980) and learning strategies (Cohen, 1990; O'Malley & Chamot, 1990; Ojrford, 1990; Wenden, 1991). Glossing strengthens the bottom-up component of the reading process. The use of glossing is one of several possible repair strategies that readers can use when they recognize comprehension breakdowns.

One hundred sixteen U.S. college students enrolled in a third-semester Spanish course participated in the study. They were randomly assigned to one of two conditions, with half reading an unglossed Spanish text and half reading the same text accompanied by English glosses. After reading the text, participants were asked to write as much of the text as they could recall. Results showed a significant effect for glossing but no significant interactions between the treatment and any of the other variables. Suggestions are made as to the optimal use of vocabulary glosses.

Let S = Sym(Omega) be the group of all permutations of an infinite set Omega. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, then there exists a positive integer n such that every element of S may be written as a group word of length at most n in the elements of U. Likewise, if U is a generating set for S as a monoid, then there exists a positive integer n such that every element of S may be written as a monoid word of length at most n in the elements of U. Some related questions and recent results are noted, and a brief proof is given of a result of Ore's on commutators, which is used in the proof of the above result.