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1.
Given a complex Borel measure with compact support in the complex plane the sesquilinear form defined on analytic polynomials and by , determines an operator from the space of such polynomials to the space of linear functionals on . This operator is called the Toeplitz operator with symbol . We show that has finite rank if and only if is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.

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2.
We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if and are operator algebras, then any bounded epimorphism of onto is completely bounded provided that contains a norming -subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a -algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a -algebra is similar to a -representation precisely when the image operator algebra -norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras of -diagonals () satisfying extends uniquely to a -isomorphism of the -algebras generated by and ; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

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3.
We prove that if and , then

for all . This polarized partition relation holds if for every partition either there are and with or there are and with .

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4.
Let be an imaginary quadratic field with ring of integers , where is a square free integer such that , and let is a linear code defined over . The level theta function of is defined on the lattice , where is the natural projection. In this paper, we prove that:

i) for any such that , and have the same coefficients up to ,

ii) for , determines the code uniquely,

iii) for , there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to .

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5.
Consider arbitrary elements. We characterize those functions that decompose into the sum of -periodic functions, i.e., with . We show that has such a decomposition if and only if for all partitions with consisting of commensurable elements with least common multiples one has .

Actually, we prove a more general result for periodic decompositions of functions defined on an Abelian group ; in fact, we even consider invariant decompositions of functions with respect to commuting, invertible self-mappings of some abstract set .

We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.

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6.
Let be the group of isotopy classes of orientation-preserving homeomorphisms of that preserve a Heegaard splitting of genus two. In this paper, we construct a tree in the barycentric subdivision of the disk complex of a handlebody of the splitting to obtain a finite presentation of .

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7.
In this note we prove a monotonicity result related to the principal eigenvalue of the -Laplacian in an annulus in .

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8.
Let and be bounded solid domains such that their associated volume potentials agree outside . Under the assumption that one of the domains is convex, it is deduced that .

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9.
For integers , , , with , and Dirichlet character , we define a mixed exponential sum

where , and denotes the summation over all with . The main purpose of this paper is to study the mean value of

and to give a related identity on the mean value of the general Kloosterman sum

where .

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10.
We show that the boundary of an -dimensional closed convex set , possibly unbounded, is a convex quadric surface if and only if the middle points of every family of parallel chords of lie in a hyperplane. To prove this statement, we show that the boundary of is a convex quadric surface if and only if there is a point such that all sections of by 2-dimensional planes through are convex quadric curves. Generalizations of these statements that involve boundedly polyhedral sets are given.

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11.
Let be an immersion of an -dimensional connected manifold in an -dimensional connected complete Riemannian manifold without conjugate points. Assume that the union of geodesics tangent to does not cover . Under these hypotheses we have two results. The first one states that is simply connected provided that the universal covering of is compact. The second result says that if is a proper embedding and is simply connected, then is a normal graph over an open subset of a geodesic sphere. Furthermore, there exists an open star-shaped set such that is a manifold with the boundary .

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12.
In this paper, it is shown that the number of partitions of a nonnegative integer with parts can be described by a set of polynomials of degree in , where denotes the least common multiple of the integers and denotes the quotient of when divided by . In addition, the sets of the polynomials are obtained and shown explicitly for and .

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13.
In most previous works on the existence of solutions to the -Yamabe problem, one assumes that the initial metric is -admissible. This is a pointwise condition. In this paper we prove that this condition can be replaced by a weaker integral condition.

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14.
A rainbow coloring of a graph is a coloring of the edges with distinct colors. We prove the following extension of Wilson's Theorem. For every integer there exists an so that for all , if

then every properly edge-colored contains pairwise edge-disjoint rainbow copies of .

Our proof uses, as a main ingredient, a double application of the probabilistic method.

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15.
In this article we study normal generation of irrational ruled surfaces. When is a smooth curve of genus , Green and Lazarsfeld proved that a very ample line bundle    Pic with deg   Cliff is normally generated where Cliff denotes the Clifford index of the curve (Green and Lazarsfeld, 1986). We generalize this to line bundles on a ruled surface over .

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16.
Answering a question of Sakai, we show that the minimal cardinality of a set of reals such that does not have the Pytkeev property is equal to the pseudo-intersection number . Our approach leads to a natural characterization of the Pytkeev property of by means of a covering property of , and to a similar result for the Reznichenko property of .

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17.
We prove that the Birkhoff map for KdV constructed on can be interpolated between and . In particular, the symplectic phase space can be described in terms of Birkhoff coordinates.

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18.
Let be a Riemannian compact -manifold. We know that for any , there exists such that for any , , being the smallest constant possible such that the inequality remains true for any . We call the ``first best constant'. We prove in this paper that it is possible to choose and keep a finite constant. In other words we prove the existence of a ``second best constant' in the exceptional case of Sobolev inequalities on compact Riemannian manifolds.

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19.
We prove the uniqueness for the solutions of the singular nonlinear PDE system:

(1)

In the special case when and , we classify all the solutions and thus obtain the best constant in the corresponding weighted Hardy-Littlewood-Sobolev inequality.

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20.
For an irrational number , let denote its -th continued fraction inverse complete quotient, obtained by deleting the first partial quotients. For any positive real number , we establish the optimal linear bound on the sum of the -th powers of the first complete quotients. That is, we find the smallest constants such that for all and all irrationals .

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