From the terrible loneliness to the wonderful agreement of human beings

What would be the ""terrible loneliness"" and what would be the ""wonderful agreement"" in the present paper? The ""terrible loneliness"" is the only reality that a person perceives and/or thinks during the now going on. For the person, an enormous quantity of occurrences is in the present moment absent. A very small quantity of occurrences is present. The person is the only being in having this. And, this is only during a little moment. The person never thinks about his loneliness in this moment. On the contrary, he thinks he is plenty of people and full of occurrences. But, if he were thinking about reality, he would live in a terrible loneliness. How does he escape himself from this loneliness? He thinks that the probable occurrences are real occurrences. He may be right in a plenty of times. Going through what I call opening hypotheses-basic hypotheses and non-basic but important hypotheses-and going through what I call simply hypotheses he is able to sanction a wonderful agreement of human beings about the known parts of the Universe. However, they are hypotheses, not absolute realities.

COMMUTATIVE AUTOMORPHIC LOOPS OF ORDER p(3)

A loop is said to be automorphic if its inner mappings are automorphisms. For a prime p, denote by A(p) the class of all 2-generated commutative automorphic loops Q possessing a central subloop Z congruent to Z(p) such that Q/Z congruent to Z(p) x Z(p). Upon describing the free 2-generated nilpotent class two commutative automorphic loop and the free 2-generated nilpotent class two commutative automorphic p-loop F-p in the variety of loops whose elements have order dividing p(2) and whose associators have order dividing p, we show that every loop of A(p) is a quotient of F-p by a central subloop of order p(3). The automorphism group of F-p induces an action of GL(2)(p) on the three-dimensional subspaces of Z(F-p) congruent to (Z(p))(4). The orbits of this action are in one-to-one correspondence with the isomorphism classes of loops from A(p). We describe the orbits, and hence we classify the loops of A(p) up to isomorphism. It is known that every commutative automorphic p-loop is nilpotent when p is odd, and that there is a unique commutative automorphic loop of order 8 with trivial center. Knowing A(p) up to isomorphism, we easily obtain a classification of commutative automorphic loops of order p(3). There are precisely seven commutative automorphic loops of order p(3) for every prime p, including the three abelian groups of order p(3).

Multiparameter twisted Weyl algebras

We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's q-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter twisted Weyl algebras. (C) 2011 Elsevier Inc. All rights reserved.