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1.
Let be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by for each module and for each integer . We propose a conjecture asking if for any positive integers and . We prove that this is true provided the associated graded ring of has depth not less than . Furthermore we show that there are only finitely many possibilities for a pair of positive integers for which .

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2.
Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of consisting of ordinals of cofinality for each , then there exist abelian groups such that and for each . (2) There exist abelian groups such that for each and for each . The groups are the groups of -valued continuous functions on a topological space and their dual groups.

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3.
Let be a prime algebra over a commutative ring with unity and let be a multilinear polynomial over . Suppose that is a nonzero derivation on such that for all in some nonzero ideal of , with fixed. Then is central--valued on except when char and satisfies the standard identity in 4 variables.

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4.
We generalize the classical Wolff-Denjoy theorem to certain infinitely connected Riemann surfaces. Let be a non-parabolic Riemann surface with Martin boundary . Suppose each Martin function , , extends continuously to and vanishes there. We show that if is an endomorphism of and the iterates of converge to the point at infinity, then the iterates converge locally uniformly to a point in . As an application, we extend the Wolff-Denjoy theorem to non-elementary Gromov hyperbolic covering spaces of compact Riemann surfaces. Such covering surfaces are of independent interest. Finally, we use the theory of non-tangential boundary limits to give a version of the Wolff-Denjoy theorem that imposes certain mild restrictions on but none on itself.

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5.
For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear -width of a subset in a normed linear space . We proved by a topological method that for and the well-known Aleksandrov -width in a Banach space the following inequalities hold: . Let be the unit ball of Besov space , of multivariate periodic functions. Then for approximation in , with some restriction on and , we established the asymptotic degree of these -widths: .

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6.
For a Banach algebra , amenability of necessitates amenability of , and similarly for weak amenability provided is a left ideal in . For a locally compact group, indeed more generally, is amenable if and only if is finite. If is weakly amenable, then is weakly amenable.

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7.
This paper investigates the local integrals

where represents the integers of a composition algebra over a non-archimedean local field and is a non-trivial character on the units in the ring of integers of extended to by setting . The local zeta function for the trivial character is known for all composition algebras . In this paper, we show in the quaternion case that for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for equal to the quadratic character. In this latter case, for any character of order greater than .

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8.
In this paper we study the Foias-Williams operator

where , and is a Hankel operator with symbol . We exhibit a relationship between the similarity of to a contraction and the rate of decay of , the absolute values of the Fourier coefficients of the symbol .

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9.
For the Lie group , let be the open orbit of Lagrangian planes of signature in the generalized flag variety of Lagrangian planes in . For a suitably chosen maximal compact subgroup of and a base point we have that the orbit of is a maximal compact subvariety of . We show that for the connected component containing in the space of translates of which lie in is biholomorphic to , where denotes with the opposite complex structure.

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10.
We show that if and are Matlis reflexive modules over a complete Gorenstein local domain and is an ideal of such that the dimension of is one, then the modules are Matlis reflexive for all and if . It follows that the Bass numbers of are finite. If is not a domain, then the same results hold for .

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