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1.
Let be an mp arrangement in a complex algebraic variety with corresponding complement and intersection poset . Examples of such arrangements are hyperplane arrangements and toral arrangements, i.e., collections of codimension 1 subtori, in an algebraic torus. Suppose a finite group acts on as a group of automorphisms and stabilizes the arrangement setwise. We give a formula for the graded character of on the cohomology of in terms of the graded character of on the cohomology of certain subvarieties in .

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2.
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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3.
Maps of the nonnegative cone of into itself are considered where are nonnegative, primitive matrices with nondecreasing entries and at least one increasing entry. Let denote the dominant eigenvalue of and . These maps are shown to exhibit a dynamical trichotomy. First, if , then for all nonzero . Second, if , then for all . Finally, if and 1$">, then there exists a compact invariant hypersurface separating . For below , , while for above, . An application to nonlinear Leslie matrices is given.

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4.
W. T. Gowers' theorem asserts that for every Lipschitz function and 0$">, there exists an infinite-dimensional subspace of such that the oscillation of on is at most . The proof of this theorem has been reduced by W. T. Gowers to the proof of a new Ramsey type theorem. Our aim is to present a proof of the last result.

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5.
In this paper, the following problem is studied. Let and be two domains in the complex plane with . Suppose that are two quasiconformal mappings satisfying . Let be the mapping in defined by (). If both and are uniquely extremal, is always uniquely extremal? It is shown in this paper that the answer to this problem is no.

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6.
Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series in the standard fundamental domain for lie on . In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc . Using this result we prove a speculation of Ono, namely that the zeros of the unique ``gap function" in , the modular form with the maximal number of consecutive zero coefficients in its -expansion following the constant , has zeros only on . In addition, we show that the -invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight .

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7.
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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8.
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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9.
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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10.
A Seifert matrix is a square integral matrix satisfying


To such a matrix and unit complex number there corresponds a signature,


Let denote the set of unit complex numbers with positive imaginary part. We show that is linearly independent, viewed as a set of functions on the set of all Seifert matrices.

If is metabolic, then unless is a root of the Alexander polynomial, . Let denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices.

To each knot one can associate a Seifert matrix , and induces a knot invariant. Topological applications of our results include a proof that the set of functions is linearly independent on the set of all knots and that the set of two-sided averaged signature functions, , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if is the root of some Alexander polynomial, then there is a slice knot whose signature function is nontrivial only at and . We demonstrate that the results extend to the higher-dimensional setting.

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