Let $T(S)$ be the Teichmüller space of a hyperbolic Riemann surface $S$. Suppose that $\mu $ is an extremal Beltrami differential at a given point $\tau $ of $T(S)$ and $\{ {\phi }_{n} \} $ is a Hamilton sequence for $\mu $. It is an open problem whether the sequence $\{ {\phi }_{n} \} $ is always a Hamilton sequence for all extremal differentials in $\tau $. S. Wu [‘Hamilton sequences for extremal quasiconformal mappings of the unit disk’, Sci. China Ser. A 42 (1999), 1033–1042] gave a positive answer to this problem in the case where $S$ is the unit disc. In this paper, we show that it is also true when $S$ is a doubly-connected domain.

We obtain lower bounds for the growth of solutions of higher order linear differential equations, with coefficients analytic in the unit disc of the complex plane, by localising the equations via conformal maps and applying known results for the unit disc. As an example, we study equations in which the coefficients have a certain explicit exponential growth at one point on the boundary of the unit disc and consider the iterated $M$-order of solutions.

In this paper a formula is obtained for the entries of the diagonal factor in the U D L factorisation of an invertible operator matrix in the case when its inverse has a chordal graph. As a consequence, in the finite dimensional case a determinant formula is obtained in terms of some key principal minors. After a cancellation process this formula leads to a determinant formula from an earlier paper by W.W. Barrett and C.R. Johnson, deriving in this way a different and shorter proof of their result. Finally, an algorithmic method of constructing minimal vertex separators of chordal graphs is presented.

Let $G$ be a locally compact amenable group and $A(G)$ and $B(G)$ be the Fourier and the Fourier–Stieltjes algebras of $G,$ respectively. For a power bounded element $u$ of $B(G)$, let ${\mathcal{E}}_{u}:=\{g\in G:|u(g)|=1\}$. We prove some convergence theorems for iterates of multipliers in Fourier algebras.

(a) If $\Vert u\Vert _{B(G)}\leq 1$, then $\lim _{n\rightarrow \infty }\Vert u^{n}v\Vert _{A(G)}=\text{dist}(v,I_{{\mathcal{E}}_{u}})\text{ for }v\in A(G)$, where $I_{{\mathcal{E}}_{u}}=\{v\in A(G):v({\mathcal{E}}_{u})=\{0\}\}$.

(b) The sequence $\{u^{n}v\}_{n\in \mathbb{N}}$ converges for every $v\in A(G)$ if and only if ${\mathcal{E}}_{u}$ is clopen and $u({\mathcal{E}}_{u})=\{1\}.$

(c) If the sequence $\{u^{n}v\}_{n\in \mathbb{N}}$ converges weakly in $A(G)$ for some $v\in A(G)$, then it converges strongly.

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is again finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.

We consider a special kind of structure resolvability and irresolvability for measurable spaces and discuss analogues of the criteria for topological resolvability and irresolvability.

Let I = [0, 1] and let f be an unimodal expanding map in C0(I, I). If f has an expanding constant for some integers m ≧ 0 and n ≧ 1, where λn is the unique positive zero of the polynomial x2n+1 − 2x2n−1 −1, then we show that f has a periodic point of period 2m(2n + 1). The converse of the above result is trivially false. The condition in the above result is the best possible in the sense that we cannot have the same conclusion if the number λn is replaced by any smaller positive number and the generalisation of the above result to arbitrary piecewise monotonic expanding maps in C0(I, I) is not possible.

A method is present for simplifying the calculation of steady Stokes flows. It is applicable for domains which are essentially half spaces and utilizes a special vector stream potential to decouple the boundary conditions. The results are applied to estimate the effect of no-shear zones on a shear flow.

In a paper dealing with trans-sonic jet flows Frankl (Bull. Acad. Sci. URSS Sér. Math. [Izv. Akad. Nauk SSSE] 9 (1945), 121–143) considered the following problem (T) by applying the condition

where k = k(y) is a monotone increasing function with a continuous second derivative, k(0) = 0, F(0) > 0, k′(y) ≠ 0 for y < 0. Consider an equation of the form

which is elliptic for y > 0, hyperbolic for y < 0, and parabolic for y = 0. Consider equation (2) in a bounded simply connected region D ⊂ R2 which is bounded by the following three curves: a piecewise smooth curve Γ0 lying in the half-plane y > 0 which intersects the line y = 0 at the points A(0, 0) and B(l, 0); for y < 0 by a smooth curve Γ2 through B which meets the characteristic of (2) issuing from A(0, 0) at the point P; and the curve Γ1 which consists of the portion PA of the characteristic through A. The problem (T) (or problem of Tricomi-Frankl) consists of finding a solution u = u(x, y) ∈ C2(D) assuming prescribed values on Γ0 ∪ Γ2. In the present paper we generalize Frankl's uniqueness theorem; our uniqueness theorem includes cases where F(y) may be negative.

A simple change of the operation in a commutative semigroup leads to a groupoid. It is shown that it has no finite basis for its laws.

In this paper we introduce regular topological distributively generated (d.g.) near-rings (distinct from d.g. regular near-rings) as the d.g. near-ring analogue of regular rings and develop a structure theory for this class of near-rings.

We show that any hyperbolic flow (X, π) on a metric space X is topologically stable by showing that it is expansive and has the chain-tracing property.

We give simple direct proofs of some hyperelliptic function identities conjectured by R.J. Baxter which arose in his study of the chiral Potts model.

Let k be a perfect field, X a smooth curve over k, and denote by Xc the subset of closed points of X. We show that for any non-constant element f of the function field k (X) there exists a natural homomorphism Where

We explain how this generalises the usual results on descents on Jacobians and Picard groups of curves.

We solve the question raised by Barbosa and do Carmo as to whether there exists a complete, noncompact stably immersed surface in R3 with nonzero constant mean curvature. We show that such a surface is necessarily minimal, that is, its mean curvature is zero.

This is proved: if B is a commutative Banach algebra with identity, then the non-invertible elements of B are weakly dense in B if and only if the maximal ideal space of B is infinite.

In this paper, we study two-point boundary value problems for the nonlinear second order difference equation We establish the relationship between the number of sign-variation of f on {0,…, T + 2} and the one of the solution u of the above problem.

Extending the results of An†onovskiĭ, Bol†janskiĭ, and Sarymsakov on semifield metric spaces, the authors define a regular semifield metric to be one in which the distance in the standard Tychonoff product representation of a point from a disjoint closed set is nonzero. It is shown that every completely regular topological space possesses a completely regular semifield metric and that there is an equivalent completely regular semifield metric for every semifield metric space. A normal semifield metric is defined to be one in which the distance between two disjoint closed sets is nonzero and it is shown that possessing a normal semifield metric is equivalent to being a normal topological space. Finally, Cauchy nets in semifield metric spaces are introduced leading to the notion of completeness. It is shown that a semifield metric space is complete iff every Cauchy net with the property that its directed set has cardinality less than or equal to the cardinality of the indexing set of the Tychonoff product representation of the semifield converges.

In this note about Genetic Algorithms (GA-), we study the 2-epistasis of a fitness function over a search space. This concept is a natural generalisation of that of epistasis, previously considered by Davidor in 1991 Suys and Verschoren in 1996 and Van Hove and Verschoren in 1994 for example. We completely characterise fitness functions whose 2-epistasis is minimal: these are exactly the second order functions. The validity of 2-epistasis as a measure of hardness with respect to genetic algorithms is checked over some classical laboratory functions. Finally, we obtain an upper bound of the maximal value of the 2-epistasis when we restrict attention to non-negative functions.

In this paper we prove the existence theorem for the implicit Darboux problem on the quarterplane x ≥ 0, y ≥ 0. Moreover, we study the topological structure of the solution set of this problem.