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1.
Let be the lattice of classes of reals. We show there are exactly two possible isomorphism types of end intervals, . Moreover, finiteness is first order definable in .

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2.
It is proved that if is a compact Hausdorff space of Lebesgue dimension , then the squaring mapping , defined by , is open if and only if . Hence the Lebesgue dimension of can be detected from openness of the squaring maps . In the case it is proved that the map , from the selfadjoint elements of a unital -algebra into its positive elements, is open if and only if is isomorphic to for some compact Hausdorff space with .

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3.
Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

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4.
A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).

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5.
For convex domains with diameter we prove


for any with zero mean value on . We also show that the constant in this inequality is optimal.

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6.
Recently Wu proved that for all primes ,


where runs over all normalized newforms of weight 2 and level . Here we show that can be replaced by .

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7.
Let a Banach space and a -algebra of subsets of a set . We say that a vector measure Banach space has the bounded Vitaly-Hahn-Sacks Property if it satisfies the following condition: Every vector measure , for which there exists a bounded sequence in verifying for all , must belong to . Among other results, we prove that, if is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of , then contains a copy of .

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8.
We prove: Let be a compact real-analytic variety in . Assume (i) is polynomially convex and (ii) every point of is a peak point for . Then . This generalizes a previous result of the authors on polynomial approximation on three-dimensional real-analytic submanifolds of .

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9.
Let be abelian groups where is a direct sum of countable -groups. A condition is given on the Ulm-Kaplansky -invariants of and such that .

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10.
Let be a compact connected orientable Riemannian manifold of dimension and let be the -th positive eigenvalue of the Laplacian acting on differential forms of degree on . We prove that the metric can be conformally deformed to a metric , having the same volume as , with arbitrarily large for all .

Note that for the other values of , that is and , one can deduce from the literature that, 0$">, the -th eigenvalue is uniformly bounded on any conformal class of metrics of fixed volume on .

For , we show that, for any positive integer , there exists a metric conformal to such that, , , that is, the first eigenforms of are all exact forms.

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