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1.
Suppose that where are real numbers such that and The union is not assumed to be disjoint. It is shown that the translates , , tile the real line for some bounded measurable set if and only if the exponentials , , form an orthogonal basis for some bounded measurable set

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2.
We examine the question of when the -homomorphism of full amalgamated free product C-algebras, arising from compatible inclusions of C-algebras , and , is an embedding. Results giving sufficient conditions for to be injective, as well as classes of examples where fails to be injective, are obtained. As an application, we give necessary and sufficient conditions for the full amalgamated free product of finite-dimensional C-algebras to be residually finite dimensional.

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3.
We study the linear functional on the Cuntz algebra associated to a sequence of unit vectors in that is a generalization of the Cuntz state. We prove that is positive if and only if is a constant sequence.

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4.
Let be a nontrivial dilation. We show that every complete norm on that makes from into itself continuous is equivalent to . also determines the norm of both and with in a weaker sense. Furthermore, we show that even all the dilations do not determine the norm on .

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5.
Let be a finite group and the largest irreducible character degree of . In this note, we show the following results: if , then ; if and, in addition, is -solvable with abelian Sylow -subgroup, then .

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6.
Theorem. If are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets such that (i) is either a Kronecker set or (ii) for some , is a translate of a -set all of whose elements have order , and (iii) is isomorphic to the projective tensor product .

This extends what was previously known for groups such as or for the case to the general locally compact abelian group. Old results concerning the local existence of Kronecker and -sets are improved.

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7.
For each prime power , we realize the classical cyclotomic polynomial as one of a collection of different polynomials in . We show that the new polynomials are similar to in many ways, including that their discriminants all have the form . We show also that the new polynomials are more complicated than in other ways, including that their complex roots are generally fractal in appearance.

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8.
We show that there are polynomials with such that the ideal class group of the real quadratic extensions has an element of order .

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9.
Suppose that is a finite dimensional discrete quantum group and is a Hilbert space. This paper shows that if there exists an action of on so that is a modular algebra and the inner product on is -invariant, then there is a unique C*-representation of on supplemented by the The commutant of in is exactly the -invariant subalgebra of . As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.

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10.
Let be a quadratic extension of -adic fields. If is an admissible representation of that is parabolically induced from discrete series representations, then we prove that the space of -invariant linear functionals on has dimension one, where is the mirabolic subgroup. As a corollary, it is deduced that if is distinguished by , then the twisted tensor -function associated to has a pole at . It follows that if is a discrete series representation, then at most one of the representations and is distinguished, where is an extension of the local class field theory character associated to . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.

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