Rigidity of Continuous Coboundaries

We consider the functional equation F_oT–F=f, where T isa measure-preserving transformation and f is a continuous function.We show that if there is an L function F which satisfies thisequation, then F is constrained to satisfy a number of regularityconditions, and, in particular, if T is a one-sided Bernoullishift, then we show that there is a continuous function F satisfyingthis equation. We show that this is not the case for the two-sidedshift. 1991 Mathematics Subject Classification 28D05, 58F11.     

Conformal Conjugacies in Baker Domains

Let F be a meromorphic function in the complex plane. We investigatethe behaviour of the iterates of F in a Baker domain B. In particular,we describe the dynamics of the orbits with the help of conformalconjugacies; that is, we determine a function which is univalentin a large simply connected subdomain of B such that (F(z))=T((z))holds throughout B. Here T is either a parabolic or hyperbolicMöbius transformation mapping either a half plane or Conto itself. This functional equation is always solvable ina Baker domain if F has only finitely many poles. Moreover,there is an example of a function with infinitely many poleswhere one cannot find an appropriate conformal conjugacy inan invariant Baker domain.     

Invariant Curves by Vector Fields on Algebraic Varieties

If C is a reduced curve which is invariant by a one-dimensionalfoliation F of degree d_F on the projective space then it isshown that d_F–1+a is a bound for the quotient of the twocoefficients of the Hilbert–Samuel polynomial for C, wherea is an integer obtained from a concrete problem of imposingsingularities to projective hypersurfaces, and so a bound isobtained for the degree of C when it is a complete intersection.Concrete values of a can be derived for several interestingapplications. The results are presented in the form of intersection-theoreticalinequalities for one-dimensional foliations on arbitrary smoothalgebraic varieties.     

Exact Rates of Convergence to a Stable Law

Let F be a non-lattice distribution function which lies in thedomain of attraction of a non-normal stable distribution. Exactuniform convergence rates are obtained for the convergence ofthe normalised partial sums of random variables with distributionF. Second order regular variation conditions are assumed relatedto the domain of attraction conditions.     

Multiquadratic Extensions, Rigid Fields and Pythagorean Fields

Let F be a field of characteristic other than 2. Let F⁽²⁾ denotethe compositum over F of all quadratic extensions of F, letF⁽³⁾ denote the compositum over F⁽²⁾ of all quadratic extensionsof F⁽²⁾ that are Galois over F, and let F^{3} denote the compositumover F⁽²⁾ of all quadratic extensions of F⁽²⁾. This paper showsthat F⁽³⁾ = F^{3} if and only if F is a rigid field, and thatF⁽³⁾ = K⁽³⁾ for some extension K of F if and only if F is Pythagoreanand . The proofs depend mainly on the behavior of quadratic forms over quadratic extensions,and the corresponding norm maps.     

Norms on Free Topological Groups

A norm on a group G is a function N mapping G into the set ofnon-negative real numbers such that for each x and y in G, N(xy^–1) N(x)+N(y) and N(e) = 0, where e is the identity element ofG. It is shown here that if F(X) is the free topological groupon any completely regular Hausdorff space X and H is a subgroupof F(X) generated by a finite subset of X, then any norm onH can be extended to a continuous norm on F(X).     

Crossed Product Orders Over Valuation Rings II: Tamely Ramified Crossed Product Algebras

Let V be a commutative valuation domain of arbitrary Krull-dimension,with quotient field F, let K be a finite Galois extension ofF with group G, and let S be the integral closure of V in K.Suppose that one has a 2-cocycle on G that takes values in thegroup of units of S. Then one can form the crossed product ofG over S, S*G, which is a V-order in the central simple F-algebraK*G. If S*G is assumed to be a Dubrovin valuation ring of K*G,then the main result of this paper is that, given a suitabledefinition of tameness for central simple algebras, K*G is tamelyramified and defectless over F if and only if K is tamely ramifiedand defectless over F. The residue structure of S*G is alsoconsidered in the paper, as well as its behaviour upon passageto Henselization. 2000 Mathematics Subject Classification 16H05,16S35.     

Tjurina and Milnor Numbers of Matrix Singularities

To gain understanding of the deformations of determinants andPfaffians resulting from deformations of matrices, the deformationtheory of composites f F with isolated singularities is studied,where f : YC is a function with (possibly non-isolated) singularityand F : XY is a map into the domain of f, and F only is deformed.The corresponding T¹(F) is identified as (something like) thecohomology of a derived functor, and a canonical long exactsequence is constructed from which it follows that = µ(f F) – ß₀ + ß₁, where is the length of T¹(F) and ß_i is the lengthof Tor_i^O_Y(O_Y/J_f, O_X). This explains numerical coincidences observedin lists of simple matrix singularities due to Bruce, Tari,Goryunov, Zakalyukin and Haslinger. When f has Cohen–Macaulaysingular locus (for example when f is the determinant function),relations between and the rank of the vanishing homology ofthe zero locus of f F are obtained.     

Primitive Representations by Spinor Genera of Ternary Quadratic Forms

Let a be primitively represented by the genus of a ternary quadraticlattice L defined over the ring of integers of an algebraicnumber field F. Criteria to determine whether a is primitivelyrepresented by every spinor genus in the genus of L involvecertain subgroups *(L_p, a) of the multiplicative groups of thelocalizations F_p of F with respect to the various nonarchimedeanprime spots p on F. In this paper these groups *(L_p, a) aredetermined explicitly for nondyadic and 2-adic prime spots.Examples are given which show how this information can, in someinstances, be used in combination with known results, to determineall integers primitively represented by a particular positivedefinite ternary quadratic form.     

The Hausdorff Dimension of Julia Sets of Entire Functions IV

It is known that, for a transcendental entire function f, theHausdorff dimension of J(f) satisfies 1 dimJ(f) 2. For eachd (1, 2), an example of a transcendental entire function fwith dimJ(f) = d is given. It is then indicated how this functioncan be modified to produce a transcendental meromorphic functionF with one pole with dimJ(F) = d. These appear to be the firstexamples of Julia sets with non-integer dimensions whose dimensionshave been calculated exactly.     

On the Norm Equation Over Function Fields

If K is an algebraic function field of one variable over analgebraically closed field k and F is a finite extension ofK, then any element a of K can be written as a norm of someb in F by Tsen's theorem. All zeros and poles of a lead to zerosand poles of b, but in general additional zeros and poles occur.The paper shows how this number of additional zeros and polesof b can be restricted in terms of the genus of K, respectivelyF. If k is the field of all complex numbers, then we use Abel'stheorem concerning the existence of meromorphic functions ona compact Riemann surface. From this, the general case of characteristic0 can be derived by means of principles from model theory, sincethe theory of algebraically closed fields is model-complete.Some of these results also carry over to the case of characteristicp>0 using standard arguments from valuation theory.     

Elliptic and Hyperelliptic Curves Over Supersimple Fields

It is proved that if F is an infinite field with characteristicdifferent from 2, whose theory is supersimple, and C is an ellipticor hyperelliptic curve over F with generic ‘modulus’,then C has a generic F-rational point. The notion of generityhere is in the sense of the supersimple field F.     

On multiply connected wandering domains of meromorphic functions

We describe conditions under which a multiply connected wanderingdomain of a transcendental meromorphic function with a finitenumber of poles must be a Baker wandering domain, and we discussthe possible eventual connectivity of Fatou components of transcendentalmeromorphic functions. We also show that if f is meromorphic,U is a bounded component of F(f) and V is the component of F(f)such that f(U)V, then f maps each component of U onto a componentof the boundary of V in . We give examples which show that our results are sharp; for example,we prove that a multiply connected wandering domain can mapto a simply connected wandering domain, and vice versa.     

Cerf Theory for Graphs

We develop a deformation theory for k-parameter families ofpointed marked graphs with fixed fundamental group F_n. Applicationsinclude a simple geometric proof of stability of the rationalhomology of Aut(F_n), computations of the rational homology insmall dimensions, proofs that various natural complexes of freefactorizations of F_n are highly connected, and an improvementon the stability range for the integral homology of Aut(F_n).     

Ergodicity and mixing of non-commuting epimorphisms

We study mixing properties of epimorphisms of a compact connectedfinite-dimensional abelian group X. In particular, we show thata set F, with |F| > dim X, of epimorphisms of X is mixingif and only if every subset of F of cardinality (dim X) + 1is mixing. We also construct examples of free non-abelian groupsof automorphisms of tori which are mixing, but not mixing oforder 3, and show that, under some irreducibility assumptions,ergodic groups of automorphisms contain mixing subgroups andfree non-abelian mixing subsemigroups.     

Best Approximation in an Asymmetrically Weighted L1 Measure

Let f(x) be a given, real-valued, continuous function definedon an interval [a,b]of the real line. Given a set of m real-valued,continuous functions _j(x) defined on [a,b], a linear approximatingfunction can be formed with any real setA = {a₁, a₂,..., a_m}. We present results for determining A sothat F(A, x) is a best approximation to(x) when the measureof goodness of approximation is a weighted sum of |F(A, x)–f(x)|,the weights being positive constants, w, when F(A, x) f(x)and w2 otherwise (when w, = w2 = 1, the measure is the L₁, norm).The results are derived from a linear programming formulationof the problem. In particular, we give a theorem which shows when such bestapproximations interpolate the function at fixed ordinates whichare independent of f(x). We show how the fixed points can becalculated and we present numerical results to indicate thatthe theorem is quite robust.     

Some Properties of Free Groups of Some Soluble Varieties of Groups

Let F be a free group, and let _n(F) be the nth term of the lowercentral series of F. It is proved that F/[_j(F), _i(F), _k(F)]and F/[_j(F), _i(F), _k(F), _l(F)] are torsion free and residuallynilpotent for certain values of i, j, k and i, j, k, l, respectively.In the process of proving this, it is proved that the analogousLie rings are torsion free.     

Values at s = 1 of L-functions for multi-quadratic extensions of number fields and annihilation of the tame kernel

Suppose that is a totally real number field which is the compositeof all of its subfields E that are relative quadratic extensionsof a base field F. For each such E with a ring of integers _E,assume the truth of the 2-primary part of the Birch–Tateconjecture relating the order of the tame kernel K₂(_E) to thevalue of the Dedekind zeta function of E at s=–1, andassume the same for F as well. Excluding a certain rare situation,we prove the annihilation of K₂() by a generalized Stickelbergerelement in the group ring of the Galois group of /F. Annihilationof the odd part of this group is proved unconditionally. Thisresult on the odd part establishes a special case of a conjecturestated by Snaith.     

Perimeter, Diameter and Area of Convex Sets in the Hyperbolic Plane

The paper studies the relation between the asymptotic valuesof the ratios area/length (F/L) and diameter/length (D/L) ofa sequence of convex sets expanding over the whole hyperbolicplane. It is known that F/L goes to a value between 0 and 1depending on the shape of the contour. In the paper, it is firstof all seen that D/L has limit value between 0 and 1/2 in strongcontrast with the euclidean situation in which the lower boundis 1/ (D/L = 1/ if and only if the convex set has constant width).Moreover, it is shown that, as the limit of D/L approaches 1/2,the possible limit values of F/L reduce. Examples of all possiblelimits F/L and D/L are given.