Sharp Distortion Estimate for Locally Schlicht Bloch Functions

Let B be the space of locally schlicht Bloch functions f whichare analytic in the unit disc with f(0) = f'(0) – 1 =0 satisfying 0 < |f'(z)|(1 – |z|²) 1. For each fixedz₀ we shall determine the shape of the set {logf'(z₀): fB},that is, we shall give the sharp distortion estimate for locallyschilcht Bloch functions.     

Mobius Instability of Sampling for Small Weighted Spaces of Analytic Functions

In this paper, the space A (D)is considered, consisting of thoseholomorphic functions f on the unit disk D such that || f ||= sup_{z D} | f(z)|(|z|) < +, with (1) = 0. The sampling problemis studied for weights satisfying ln (r)/ln(1 – r) 0.Möbius stability of sampling is shown to fail in this space.2000 Mathematics Subject Classification 30H05 (primary), 30D60(secondary).     

Exceptional Functions and Normality

Yang proved in [10] that if f and f^(k) have no fix-points forevery fF, where F is a family of meromorphic functions in adomain G and k a fixed integer, then F is normal in G. In thispaper we prove normality for families F for which every fF omits₁ and f^(k) omits ₂, where ₁ and ₂ are analytic functions with. 1991 Mathematics SubjectClassification 30D35, 30D45.     

The structure of the normalisers of the congruence subgroups of the Hecke group G5

Let = 2cos (/5) and let []. Denote the normaliser ofG₀() of the Hecke group G₅ in PSL₂() by N(G₀()). Then N(G₀())= G₀(/h), where h is the largest divisor of 4 such that h² divides. Further, N(G₀())/G₀() is either 1 (if h = 1), ₂ x ₂ (if h= 2) or ₄ x ₄ (if h = 4).     

Volume of a small Extrinsic Ball in a Submanifold

For a submanifold M^p R, we determine a two-term asymptoticformula for vol (M^p B(x)) for x M^p as 0. The second termis a quadratic curvature invariant of the second fundamentalform of the imbedding. Imbedded spheres are characterized amongcompact hypersurfaces by this term.     

On a theorem of K. Schmidt

Let Y be a locally compact group, Aut(Y) be the group of topologicalautomorphisms of Y and (Y) be the set of continuous positivedefinite functions on Y which have unit value at the identity.A function (Y²) is said to be of product type if there aresuch functions _j (Y) that (u, v) = ₁(u)₂(v). Define the mappingT: Y² Y² by the formula T(u, v) = (A₁ uA₂ v, A₃ u A₄ v), whereA_j Aut(Y), and assume that T is a one-to-one transform. K.Schmidt proved: (i) if both (u, v) and (T(u, v)) are of producttype, then the functions _j are infinitely divisible; (ii) ifY is Abelian, both (u, v) and (T(u, v)) are of product type,and (u, v) 0, then the functions _j are Gaussian. We show thatstatement (i), generally, is not valid, but K. Schmidt's proofholds true if (u, v) 0. We also give another proof of statement(ii). Our proof uses neither the Levy–Khinchin formulafor a continuous infinitely divisible positive definite functionnor (i) on which K. Schmidt's proof is based.     

A Uniqueness Theorem for Bounded Analytic Functions

Suppose that K is a linear space of functions analytic in somedomain D in the complex plane. A sequence = (_k) of distinctpoints from D is said to be a set of uniqueness for K if fKand f(_k) = 0 for all k imply f0. Depending on the dispersionand the density of on the one hand, and the growth of the functionsin K on the other, one may often require only |f(_k)| a_k forsome sequence of positive numbers a_k, and still conclude thatf0 for fK. Of particular interest are sharp conditions on thedecay of a_k, which reflect the interplay between growth anddecay of analytic functions. 1991 Mathematics Subject Classification30A99, 31A05.     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, N 2, and let –0be the Dirichlet Laplacian defined in L²(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||₂ = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||₂=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.     

Classification Des Ideaux a Droite De A1(k)

Les études récentes sur les idéaux àdroite de A₁(k), la première algèbre de Weyl surun corps algébriquement clos et de caractéristiquenulle k, nous montrent que : pour tout idéal I 0 àdroite de A₁(k), il existe x Q = frac(A₁(k)), et V V telsque : I = xD(R, V) o V est l'ensemble des sous-espaces primairementdécomposables de k[t] = R, et D(R, V), l'idéalà droite {d A₁(k/d(R V}. Dans cet article nous montreronsprincipalement que: pour tout 0 I idéal à droitede A₁(k, !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),où X_n est la courbe d'algèbre des fonctions régulières: O(X_n = k+tⁿ⁺¹k[t]. La forme des idéaux décriteci-dessus permet de voir dans une hypothèse de Letzteret Makar-Limanov, pour deux courbes algébriques affinesX et X' on a : D(XD(X') co dim D(X = co dim D(X'). Recent studies on right ideals of the first Weyl algebra A₁(k)over an algebraic closed field k with characteristic zero showthat: for each right ideal I 0 of A₁(k), there exist x Q =fracA₁(k)) and a primary decomposable sub-space V of k[t] suchthat I=xD(R,V), where D(R,V) : = {d A₁(k)/d(R) V} is a rightideal of A₁(k). In this paper, we show that for all right idealsI 0 of A₁(k), !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),where X_n denotes the affine algebraic curve with ring of regularfunctions O(X_n=k+tⁿ⁺¹k[t]. With ideals as described above, onecan easily see, under a hypothesis given by Letzter and Makar-Limanov,that for two affine algebraic curves X and X', D(X)D(X') codim D(X) = co dim D(X'). 2000 Mathematics Subject Classification16S32.     

Normal families and omitted functions II

Let k 2 be an integer and let be a family of functions meromorphicon a domain D in , all of whose poles are multiple and whosezeros all have multiplicity at least k + 1. Let h be a functionmeromorphic on D, h 0, . Suppose that for each f , f^(k)(z) h(z) for z D. Then is a normal family on D.