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1.
    
We give the characterization of -homogeneous compacta in : Let be a locally compact (possibly nonclosed) subset of . Then is -homogeneous if and only if is a -submanifold of .

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2.
Let be a complex Lebesgue space with a unique duality map from to , the conjugate space of . Let be a compact operator on . This paper focuses on properties of and . We adapt an example due to Halmos to show that for , there is a compact operator on with the semi-open interval . So is not attained as a maximum. A corollary of the main result in this paper is that if , and , then is attained as a maximum.

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3.
A topological space is -resolvable if has disjoint dense subsets. In this paper, we prove that if is -resolvable for each positive integer , then is -resolvable.

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4.
A Banach space is not reflexive if and only if there exist a closed separable subspace of and a convex closed subset of with empty interior which contains translates of all compact sets in . If, moreover, is separable, then it is possible to put .

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5.
If is a finitely generated nilpotent group which is not abelian-by-finite, a field, and a finite dimensional separable division algebra over , then there exists a simple module for the group ring with endomorphism ring . An example is given to show that this cannot be extended to polycyclic groups.

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6.
Let , where is polar and compact and is a domain with Green function . We characterize those subsets of which have the following property: Every positive continuous function on can be written as , where and for each .

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7.
Let be a field of characteristic and a polynomial algebra in two variables. By a -generator of we mean an element of for which there exist and such that . We also define a -line of to mean any element of whose coordinate ring is that of a -generator. Then we prove that if is such that is a -line of (where is an indeterminate over ), then is a -generator of . This is analogous to the well-known fact that if is such that is a line of , then is a variable of . We also prove that if is a -line of for which there exist and such that , then is in fact a -generator of .

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8.
In this paper we prove some properties of the set of group-like elements of , , for a pointed minimal quasitriangular Hopf algebra over a field of characteristic 0, and for a pointed quasitriangular Hopf algebra which is indecomposable as a coalgebra. We first show that over a field of characteristic 0, for any pointed minimal quasitriangular Hopf algebra , is abelian. We show further that if is a quasitriangular Hopf algebra which is indecomposable as a coalgebra, then is contained in , the minimal quasitriangular Hopf algebra contained in . As a result, one gets that over a field of characteristic 0, a pointed indecomposable quasitriangular Hopf algebra has a finite abelian group of group-like elements.

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9.
We study the operators

where is the Hardy-Littlewood maximal function, the Hilbert transform or Carleson operator.

Under suitable conditions on the weight of exponential type, we prove boundedness of from spaces, defined on with respect to the measure to with the same density measure. These operators, that arise in questions of harmonic analysis on noncompact symmetric spaces, are bounded from to if and only if .

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10.
Let be a periodic distribution of period . Let be its Fourier series. We show that the distributional point value exists and equals if and only if the partial sums converge to in the Cesàro sense as for each .

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