共查询到20条相似文献,搜索用时 31 毫秒
1.
Identity Theorems for Functions of Bounded Characteristic 总被引:1,自引:0,他引:1
Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that
where h1(z), h2(z) are holomorphic in and have positive realpart there, while 1(z), 2(z) are Blaschke products, that is,
where p is a positive integer or zero, 0<|aj|<1, c isa constant and (1|aj|)<. We note in particular that, if c0, so that f(z)0,
(1.1) so that f(z)=0 only at the points aj. Suppose now that zj isa sequence of distinct points in such that |zj|1 as j and (1|zj|)=. (1.2) If f(zj)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(zj)0 sufficiently rapidly as j. Let j, j be sequences of positivenumbers such that j< and j as j. Danikas then defines
and proves Theorem A. 相似文献
2.
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 ßi is the lengthof ToriOY(OY/Jf, OX). This explains numerical coincidences observedin lists of simple matrix singularities due to Bruce, Tari,Goryunov, Zakalyukin and Haslinger. When f has CohenMacaulaysingular 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. 相似文献
3.
In this paper we define higher pre-Bloch groups n(F) of a fieldF. When the base field is algebraically closed, we study itsconnection to the homology of the general linear groups withcoefficients in /l , where l is a positive integer. As a resultof our investigation we give a necessary and sufficient conditionfor the natural map Hn(GLn–1(F), /l ) Hn(GLn(F), /l )to be bijective. We prove that this map is bijective for n4.We also demonstrate that a certain property of n() is equivalentto the validity of the Friedlander–Milnor isomorphismconjecture for (n+1)th homology of GLn(). 相似文献
4.
Buff Xavier; Fagella NuRia; Geyer Lukas; Henriksen Christian 《Journal London Mathematical Society》2005,72(3):689-716
For (,a) C* x C, let f,a be the rational map defined by f,a(z)= z2 (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f,a has a fixed Hermanring with rotation number (we consider that (e2i,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e2i ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e2i z(1+z). As a consequence, we show that for a (0,1/3), if fl,a has afixed Herman ring with rotation number and if ma is the modulusof the Herman ring, then, as a0, we have e ma=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e(a/2)(z-1/z). 相似文献
5.
A family of transcendental meromorphic functions, fp(z), p N is considered. It is shown that, if p 6, then the Hausdorffdimension of the Julia set of fp satisfies dim J(fp) 1/p, for0 < < 1/6p, and dim J(fp) 1(30 ln ln p/ln p),for p4p1/105 ln p < < p4p1/104 ln p. Theseresults are used elsewhere to show that, for each d (0, 1),there exists a transcendental meromorphic function for whichdim J(f) = d. 相似文献
6.
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. 相似文献
7.
Let =(n)n1 be a log concave sequence such that lim infn+n/nc>0for some c>0 and ((log n)/n)n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||Tn||=O(n)as n tends to + with n11/(n log n)=+, then T is unitary. Onthe other hand, if n11/(n log n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||Tn||=O(n) as n tends to +, andlim supn+||Tn||=+. 相似文献
8.
Let T = {T(t)}t0 be a C0-semigroup on a Banach space X. Thefollowing results are proved. (i) If X is separable, there exist separable Hilbert spacesX0 and X1, continuous dense embeddings j0:X0 X and j1:X X1,and C0-semigroups T0 and T1 on X0 and X1 respectively, suchthat j0 T0(t) = T(t) j0 and T1(t) j1 = j1 T(t) for all t 0. (ii) If T is -reflexive, there exist reflexive Banach spacesX0 and X1 , continuous dense embeddings j:D(A2) X0, j0:X0 X, j1:X X1, and C0-semigroups T0 and T1 on X0 and X1 respectively,such that T0(t) j = j T(t), j0 T0(t) = T(t) j0 and T(t) j1 = j1 T(t) for all t 0, and such that (A0) = (A) = (A1),where Ak is the generator of Tk, k = 0, Ø, 1. 相似文献
9.
Interpolating Blaschke Products and Factorization Theorems 总被引:1,自引:0,他引:1
Let M(H) be the maximal ideal space of H the Banach algebraof bounded analytic functions on the open unit disk. Let G bethe set of nontrivial points in M(H). By Hoffman's work, G hasdeep connections with the zero sets of interpolating Blaschkeproducts. It is proved that for a closed -separated subset Eof M(H) with E G, there exists an interpolating Blaschke productwhose zero set contains E. This is a generalization of Lingenberg'stheorem. Let f be a continuous function on M(H). Suppose thatf is analytic on a nontrivial Gleason part P(x), f(x) = 0, andf 0 on P(x). It is proved that there is an interpolating Blaschkeproduct b with zeros {zn}n such that b(x) = 0 and f(zn) = 0for every n. This fact can be used for factorization theoremsin Douglas algebras and in algebras of functions analytic onGleason parts. 相似文献
10.
We discuss the dynamics as well as the structure of the parameterplane of certain families of rational maps with few criticalorbits. Our paradigm is the family Rt(z) = (1 + (4/27)z3/(1– z)), with dynamics governed by the behaviour of thepostcritical orbit (Rn())n. In particular, it is shown thatif escapes (that is, Rn() tends to infinity), then the Juliaset of R is a Cantor set, or a Sierpiski curve, or a curve withone or else infinitely many cut-points; each of these casesactually occurs. 相似文献
11.
Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C*-algebraC*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w*-strongly exposed point of P(G)0, where P(G)0={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w*-strongly exposed point of P(G)0, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w*-strongly exposed point of P(G)0. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w*-stronglyexposed point of P(G)0, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the RadonNikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F2 on two generators), there isno closed point in its reduced dual space . 相似文献
12.
13.
If = {1, 2, ..., s}, where 1 2 ... s > 0, is a partitionof n then denotes the associated irreducible character of Sn,the symmetric group on {1, 2, ..., n}, and, if cCSn, the groupalgebra generated by C and Sn, then dc(·) denotes thegeneralized matrix function associated with c. If c1, c2 CSnthen we write c1 c2 in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCSn and c(e) 0, wheree denotes the identity in Sn, then or denotes (c(e))–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {1, 2, ..., s, 1t} isa partition of n such that s > 1 and s = 2, and ' denotes{1, 2, ..., s-1, 1t} then (, {2}) where denotes characterinduction from Sn–2 x S2 to Sn. This in turn implies thatif = {1, 2, ..., s, 1t} with s > 1, s = 2, and ßdenotes {1 + 2, 2, ..., s-1, 1t} then ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that ß and ß are both false. 相似文献
14.
15.
In 1940 Nisnevi published the following theorem [3]. Let (G) be a family of groups indexed by some set and (F) a family of fields of the same characteristic p0. Iffor each the group G has a faithful representation of degreen over F then the free product* G has a faithful representationof degree n+1 over some field of characteristic p. In [6] Wehrfritzextended this idea. If (G) GL(n, F) is a family of subgroupsfor which there exists ZGL(n, F) such that for all the intersectionGF.1n=Z, then the free product of the groups *ZG with Z amalgamatedvia the identity map is isomorphic to a linear group of degreen over some purely transcendental extension of F. Initially, the purpose of this paper was to generalize theseresults from the linear to the skew-linear case, that is, togroups isomorphic to subgroups of GL(n, D) where the D are divisionrings. In fact, many of the results can be generalized to ringswhich, although not necessarily commutative, contain no zero-divisors.We have the following. 相似文献
16.
A Torsion-Free Milnor-Moore Theorem 总被引:1,自引:0,他引:1
Let X be the space of Moore loops on a finite, q-connected,n-dimensional CW complex X, and let R Q be a subring containing1/2. Let (R) be the least non-invertible prime in R. For a gradedR-module M of finite type, let FM = M/Torsion M. We show thatthe inclusion P FH*(X;R) of the sub-Lie algebra of primitiveelements induces an isomorphism of Hopf algebras
provided that (R) n/q. Furthermore, the Hurewiczhomomorphism induces an embedding of F(*(X) R) in P, with P/F(*(X)R)torsion. As a corollary, if X is elliptic, then FH*(X;R) isa finitely generated R-algebra. 相似文献
17.
On Restrictions and Extensions of the Besov and Triebel-Lizorkin Spaces with Respect to Lipschitz Domains 总被引:8,自引:0,他引:8
The restrictions Bspq() and Fspq() of the Besov and TriebelLizorkinspaces of tempered distributions Bspq(Rn) and Fspq(Rn) to Lipschitzdomains Rn are studied. For general values of parameters (sR,p>0, q>0) a universal linear bounded extensionoperator from Bspq() and Fspq() into the corresponding spaceson Rn is constructed. The construction is based on a new variantof the Calderón reproducing formula with kernels supportedin a fixed cone. Explicit characterizations of the elementsof Bspq() and Fspq() in terms of their values in are also obtained. 相似文献
18.
Inverse SturmLiouville problems with eigenparameter-dependentboundary conditions are considered. Theorems analogous to thoseof both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)((py')'+qy), ,
where det = > 0, c 0, det > 0, t 0 and (cs + dr au tb)2 < 4(cr ta)(ds ub). Denoteby (l; ; ) the eigenvalue problem ly = y with boundary conditionsy(0)cos+y'(0)sin = 0 and (a+b)y(1) = (c+d)(py')(1). Define (; ; ) as above but with l replacedby . Let wn denote the eigenfunctionof (l; ; ) having eigenvalue n and initial conditions wn(0)= sin and pw'n(0) = cos and let n = awn(1)+cpw'n(1).Define n and n similarly. As sample results, it is proved that if (l; ; ) and (; ; ) have the same spectrum, and (l;; ) and (; ; ) have the samespectrum or for all n, thenq/r = /. 相似文献
19.
Interpolation of Vector-Valued Real Analytic Functions 总被引:2,自引:0,他引:2
Let Rd be an open domain. The sequentially complete DF-spacesE are characterized such that for each (some) discrete sequence(zn) , a sequence of natural numbers (kn) and any family the infinite system of equations
has an E-valued real analytic solution f. 相似文献
20.
The existence of 2-periodic solutions of the second-order differentialequation
where a, b satisfy and p(t)=p(t+2),t R, is examined. Assume that limits limx±F(x)=F(±)(F(x)=) and limx±g(x)=g(±)exist and are finite. It is proved that the equation has atleast one 2-periodic solution provided that the zeros of thefunction 1 are simple and the zeros of the functions 1, 2 aredifferent and the signs of 2 at the zeros of 1 in [0,2/n) donot change or change more than two times, where 1 and 2 aredefined as follows:
Moreover, it is also proved that the given equation has at leastone 2-periodic solution provided that the following conditionshold:
with 1 p < q 2. 相似文献