The main purpose is to characterise continuous maps that are n-branched coverings in terms of induced maps on the rings of functions. The special properties of Frobenius n-homomorphisms between two function spaces that correspond to n-branched coverings are determined completely. Several equivalent definitions of a Frobenius n-homomorphism are compared and some of their properties are proved. An axiomatic treatment of n-transfers is given in general and properties of n-branched coverings are studied and compared with those of regular coverings.

We study the analogues of the problems of averages and maximal averages over a surface in when the euclidean structure is replaced by that of a vector space over a finite field, and obtain optimal results in a number of model cases.

We show that the number of distinct non-parallel lines passing through two conjugates of an algebraic number α of degree d ≥ 3 is at most [d2/2]-d+2, its conjugates being in general position if this number is attained. If, for instance, d ≥ 4 is even, then the conjugates of α ∈ of degree d are in general position if and only if α has 2 real conjugates, d-2 complex conjugates, no three distinct conjugates of α lie on a line and any two lines that pass through two distinct conjugates of α are non-parallel, except for d/2-1 lines parallel to the imaginary axis. Our main result asserts that the conjugates of any Salem number are in general position. We also ask two natural questions about conjugates of Pisot numbers which lead to the equation α1+α2=α3+α4 in distinct conjugates of a Pisot number. The Pisot number shows that this equation has such a solution.

We show that the 2-category of partial combinatory algebras, as well as various related categories, admit a certain type of lax comma objects. This not only reveals some of the properties of such categories, but it also gives an interpretation of iterated realizability, in the following sense. Let φ: A → B be a morphism of PCAs, giving a comma object A ⋉φB. In the realizability topos RT(B) over B, the object (A, φ) is an internal PCA, so we can construct the realizability topos over (A, φ). This topos is equivalent to the realizability topos over the comma-PCA A ⋉φB. This result is both an analysis and a generalization of a special case studied by Pitts in the context of the effective monad.

We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellable idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of left-right constraints, and is motivated by higher category theory. To start, we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and observe that it is contractible (if non-empty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair monoidal category, where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered. The paper is self-contained. All arguments are elementary, some of them of a certain beauty.

We show that in a special Moufang set, either the root groups are elementary abelian 2-groups, or the Hua subgroup H (= the Cartan subgroup) acts “irreducibly” on U, i.e. the only non-trivial H-invariant subgroup of a root group normalized by H is the whole root group.

Generalizing the notion of left amenability for so-called F-algebras [12], we study the concept of ϕ-amenability of a Banach algebra A, where ϕ is a homomorphism from A to ℂ. We establish several characterizations of ϕ-amenability as well as some hereditary properties. In addition, some illuminating examples are given.

We prove that a complex Banach space X is a Hilbert space if (and only if) the Banach algebra (of all bounded linear operator on X) is unitary and there exists a conjugate-linear algebra involution • on satisfying T• = T−1 for every surjective linear isometry T on X. Appropriate variants for real spaces of the result just quoted are also proven. Moreover, we show that a real Banach space X is a Hilbert space if and only if it is a real JB*-triple and is -unitary, where stands for the dual weak-operator topology.

Let ϵ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of ϵ whose zero locus Z is a smooth subvariety of dimension n-r ≥ 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, ϵ, H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).

A central problem motivated by Diophantine approximation is to determine the size properties of subsets of of the formwhere ‖⋅‖ denotes an arbitrary norm, I a denumerable set, (xi,ri)i∈ I a family of elements of × (0, ∞) and ϕ a nonnegative nondecreasing function defined on [0, ∞). We show that if FId, where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subset V of , the set Fϕ belongs to a class Gh(V) of sets with large intersection in V with respect to a given gauge function h. We establish that this class is closed under countable intersections and that each of its members has infinite Hausdorff g-measure for every gauge function g which increases faster than h near zero. In particular, this yields a sufficient condition on a gauge function g such that a given countable intersection of sets of the form Fϕ has infinite Hausdorff g-measure. In addition, we supply several applications of our results to Diophantine approximation. For any nonincreasing sequence ψ of positive real numbers converging to zero, we investigate the size and large intersection properties of the sets of all points that are ψ-approximable by rationals, by rationals with restricted numerator and denominator and by real algebraic numbers. This enables us to refine the analogs of Jarník's theorem for these sets. We also study the approximation of zero by values of integer polynomials and deduce several new results concerning Mahler's and Koksma's classifications of real transcendental numbers.

In this paper we describe methods for dealing with the trace spectrum of a subgroup of PSL(2, ) generated by four elliptic elements α, β, γ, δ of respective orders 2, 2, 3, 3, satisfying αβγδ = 1. We give a parametrization and a fundamental domain in the parameter space of such groups. Furthermore we construct an algorithm that decides whether or not a given group is discrete and which moves the discrete groups into the fundamental domain. Our main result is that any two discrete such groups are isospectral if and only if they are conjugate in (2, ).

In the Appendix we consider pairs of subgroups of (2, ) that arise from non-conjugate maximal orders in a quaternion algebra over a number field. We show that for the isospectrality of such pairs there is a peculiar exception in the case where the groups contain elements of both orders 2 and 3.

All -simple corank-1 germs from to , where n ≠ 4, have an M-deformation, that is a deformation in which the maximal numbers of isolated stable singular points are simultaneously present in the image.

We study homeomorphisms of finite exponentially integrable distortion of the unit ball Bn onto a domain Ω of finite volume. We show that under such a mapping the images of almost all radii (in terms of a gauge dimension) have finite discrete length. We also show that our dimension estimate is essentially sharp.

By considering a tree-like decomposition of an arbitrary set we prove the existence of an associated series with the property that its partial sums approximate uniformly all elements in a relevant space of bounded functions. In a topological setting we show that these sums are dense in the space of continuous functions, hence in turn any Borel measurable function is the almost everywhere limit of an appropriate sequence of partial sums of the same series. The coefficients of the series may be chosen in c0, or in a weighted ℓp with 1 < p < ∞, but not in the corresponding weighted ℓ1.

We give a graphical calculus for the invariant tensors of the eight dimensional spin representation of the quantum group Uq(B3). This leads to a finite confluent presentation of the centraliser algebras of the tensor powers of this representation and a construction of a cellular basis.

In 1858 Chebyshev, and some years later his students Korkin and Zolotarev, determined the polynomial which deviates least from zero among all polynomials of degree n with leading coefficient one with respect to the maximum- and the L1-norm, respectively; these are now called the Chebyshev polynomial of first and second kind.

The next natural step which is to find, at least asymptotically, the minimal polynomial with respect to a given weight function has not been settled until today. Indeed, Bernstein gave asymptotics for the minimum deviation of weighted minimal polynomials, Fekete and Walsh found nth root asymptotics and, recently, Lubinsky and Saff provided asymptotics outside [−1, 1]. But the main point of interest: the asymptotic representation of the weighted minimal polynomials on the interval of approximation [−1, 1] remained open. Here we settle this problem with respect to the maximum norm for weight functions whose second derivative is Lipα, α ∈ (0, 1), and with respect to the L1-norm under somewhat stronger differentiability conditions.