Point Z0 Research Articles

Two geometric characteristics of quasicircles

In this paper, the following two results are obtained: (1) If Γ is a Jordan curve of $$\bar R^2 $$ , ∞ e Γ, then Γ is a quasicircle if and only if there exists a constant k, 1 ≤ k<+ ∞, such that for any four points z 1,z 2,w 1,w 2 e Γ, there exists a k-quasiconformal mapping h of $$\bar R^2 $$ with h(∞)=∞, h(Γ)=Γ and h(z j)=w j(j=1,2). (2) If Γ is a Jordan curve of $$\bar R^2 $$ , then Γ is a quasicircle if and only if Γ is a bounded circular distortion curve.

Random iteration of isometries in unbounded metric spaces

We consider an iterated function system (with probabilities) of isometries on an unbounded metric space (X,d). Under suitable conditions it is proved that the random orbit {Zn}n⩾0 of the iterations corresponding to an initial point Z0∊X `escapes to infinity' in the sense that P(Zn∊K)→0, as n→∞ for every bounded set K⊂X. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point.

Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

LetX(-ϱB m ×C n be a compact set over the unit sphere ϱB m such that for eachz∈ϱB m the fiberX z ={ω∈C n ;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC n . The polynomial hull $$\hat X$$ ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z 0,w 0)∈Int $$\hat X$$ there exists a smooth up to the boundary analytic discF:Δ→B m ×C n with the boundary inX such thatF(0)=(z 0,w 0).

Connectedness of Julia Sets of Rational Functions

For a polynomial P it is well known that its Julia set \({\cal J_P}\) is connected if and only if the orbits of the finite critical points are bounded. But there is no such simple criteria for the connectedness of the Julia set of a rational function. Indeed, up to the very nice result of Shishikura that any rational function which has one repelling fixed point only has a connected Julia set almost nothing is known on the connectivity. In the first part of the paper we give constructive sufficient conditions for a basin of attraction to be completely invariant and the Julia set to be connected. Then it is shown that the connectedness of a basin of attraction depends heavily on the fact whether the critical points from the basin tend to the attracting fixed point z0 via a preimage of z0 or not. As a consequence we obtain for instance that rational functions with a finite postcritical set or with a Fatou set which contains no Herman rings and each component of which contains at most one critical point, counted without multiplicity, have a connected Julia set.

First-order coil-to-flower transition of a polymer chain pinned near a stepwise external potential: Numerical, analytical, and scaling analysis

A polymer chain near a penetrable interface is studied in the Gaussian model, in the lattice random walk model and by a scaling analysis. The interface is modeled as an external potential u of a Heaviside step-function form. One end of the chain is fixed at a distance z0 away from this interface. When the end point is fixed in the high potential region, a first-order coil-to-flower transition takes place upon variation of the distance z0. Here, the flower has a strongly stretched stem from the grafting point towards the interface and, on top of it, a crown composed of the remaining segments in a (perturbed) coil conformation. The coil-to-flower transition is analyzed in terms of the Landau free energy. The order parameter is taken to be related to the fraction of segments residing in the energetically favorable region. Exact analytical expressions for the Landau function are obtained in the Gaussian model for any distances z0 and potential strength u. A phase diagram in the z0 versus u coordinates is constructed. It contains a line of the first-order phase transitions (binodal line) ending at a critical point z0=u=0, and two spinodal lines. Numerical results are obtained for several chain lengths in the lattice random walk model demonstrating the effects of finite extensibility on the position of the transition point. Excluded volume effects are analyzed within the scaling approach.

Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables

In the first part, we generalize the classical result of Bohr by proving that an analogous phenomenon occurs whenever D is an open domain in Cm (or, more generally, a complex manifold) and (ϕn)∞n=0 is a basis in the space of holomorphic functions H(D) such that ϕ0=1 and ϕn(z0)=0, n≥1, for some z0∈D. Namely, then there exists a neighborhood U of the point z0 such that, whenever a holomorphic function on D has modulus less than 1, the sum of the suprema in U of the moduli of the terms of its expansion is less than 1 too. In the second part we consider some natural Hilbert spaces of analytic functions and derive necessary and sufficient conditions for the occurrence of Bohr's phenomenon in this setting.

Approximating the Conformal Maps of Elongated Quadrilaterals by Domain Decomposition

Let Q:={ Ω;z 1 ,z 2 ,z 3 ,z 4 } be a quadrilateral consisting of a Jordan domain Ω and four points z 1 , z 2 , z 3 , z 4 , in counterclockwise order on \partial Ω and let m(Q) be the conformal module of Q . Then Q is conformally equivalent to the rectangular quadrilateral {R m(Q) ;0,1,1+\mathop \it im (Q),\mathop \it im (Q)}, where R m(Q) := {(ξ,η): 0<ξ<1, 0 <η<m(Q)}, in the sense that there exists a unique conformal map f: Ω \rightarrow R m(Q) that takes the four points z 1 , z 2 , z 3 , z 4 , respectively, onto the four vertices 0 , 1 , 1+\mathop \it im (Q) , \mathop \it im (Q) of R m(Q) . In this paper we consider the use of a domain decomposition method (DDM) for computing approximations to the conformal map f , in cases where the quadrilateral Q is ``long.'' The method has been studied already but, mainly, in connection with the computation of m(Q) . Here we consider certain recent results of Laugesen \cite{La}, for the DDM approximation of the conformal map f: Ω \rightarrow R m(Q) associated with a special class of quadrilaterals (viz., quadrilaterals whose two opposite boundary segments (z 2 , z 3 ) and (z 4 , z 1 ) are parallel straight lines), and seek to extend these results to more general quadrilaterals. By making use of the available DDM theory for conformal modules, we show that the corresponding theory for f can, indeed, be extended to a much wider class of quadrilaterals than those considered by Laugesen.

Optimal Error of Analytic Continuation from a Finite Set with Inaccurate Data in Hilbert Spaces of Holomorphic Functions

We consider the problem of analytic continuation with inaccurate data from a finite subset U of a domain D of Cn to a point z0∈DU for the functions f belonging to a bounded correctness set V in a Hilbert space H(D) of analytic functions in D. In the case when H(D) is a Hilbert space with a reproducing kernel, we find constructive formulas for calculating the optimal error, the optimal function, and the optimal linear algorithm for extrapolation to a point z0 for functions in V whose approximate values are given on a set U. Moreover, we study the asymptotics of the optimal error in the case when the errors of initial data vanish.

On Functions Satisfying Modular Equations for Infinitely Many Primes

AbstractIn this paper we study properties of the functions which satisfy modular equations for infinitely many primes. The two main results are:1)every such function is analytic in the upper half plane;2)if such function takes the same value in two different points z1 and z2 then there exists an f -preserving analytic bijection between neighbourhoods of z1 and z2.

On the unique continuation property of elliptic divergence form equations in the plane

(2) a(z )ζα ζβ ≥ λ|ζ|2 for all z ∈ Ω, ζ ∈ IR2. Note that according to standard elliptic regularity theory, the solution φ is a Holder continuous function in Ω (see e.g. [GT], Theorem 8.22). We shall study the local behavior of φ near a point z0 ∈ Ω, which we assume to be the origin, that is, z0 = 0 ∈ Ω. Our main concern is to prove the following unique continuation property under the stated assumptions:

Randomness‐optimal oblivious sampling

We present the first efficient oblivious sampler that uses an optimal number of random bits, up to an arbitrary constant factor bigger than 1. Specifically, for any α>0, it uses (1+α)(m+log γ−1) random bits to output d=poly(ϵ−1, log γ−1, m) sample points z1,…,zd∈{0, 1}m such that for any function f: {0, 1}m→[0, 1], Pr[|(1/d)∑i=1df(zi)−Ef|≤ϵ]≥1−γ. Our proof is based on an improved extractor construction. An extractor is a procedure which takes as input the output of a defective random source and a small number of truly random bits, and outputs a nearly random string. We present the first optimal extractor, up to constant factors, for defective random sources with constant entropy rate. We give applications to constructive leader election and reducing randomness in interactive proofs. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11, 345–367 (1997)

Analytic Discs Attached to Manifolds with Boundary

Let X be a complex manifold of dimension n, M a closed half-space with boundary M, A an analytic disc of X to M, tangent to M at some point z0 of dA n M, and intersecting M + in any neighbourhood of z0. Then holomorphic functions extend from M + to a full neighborhood of z0. This theorem refines the results of [1] where the boundary dA (instead of the whole A) was supposed to intersect M . The argument of the proof consists in constructing a (closed) manifold with boundary W, contained in the envelop of holomorphy of M and such that A c W but A <£ d W. In this situation it is easy to find a new small disc A i c A with 8A l <£ dW. We are therefore in a situation similar to [1], and get the conclusion by exhibiting a disc transversal to d W at z0. Extension of holomorphic functions by the aid of tangent discs attached to M and of 0 is a particular case of a general theorem of wedge extendibility of CR-functions by A. Tumanov; the new part of our theorem is that no assumptions on defect are made. This paper is tightly inspired to the results and the techniques by A. Tumanov [7]. We also owe to A. Tumanov a great help during private communications.

An Operator Theoretic Approach to Degenerated Nevanlinna‐Pick Interpolation

In this paper we consider an interpolation problem of Nevanlinna-Pick type: If finitely many points z1, . . . , zn of the open upper half plane C + are given, we study the existence of functions f ∈ Nν assuming prescribed values w1, . . . , wn ∈ C in these points: f(zi) = wi for i = 1, . . . , n. (1) Furthermore a description of all solutions of (1) through selfadjoint extensions of a certain symmetric operator is given. We recall the definition of the classes Nν . If f is a complex function denote by ρ(f) the domain of holomorphy of f . Definition 1 Let π and ν be nonnegative integers or ∞. Denote by Nν the set of all functions f which are meromorphic in C, such that the kernel

Mappings of real algebraic hypersurfaces

A holomorphic function h(Z) defined in a neighborhood of a point Z0 E CN is algebraic if there are polynomials qj(Z), j = 0, ..., k, not all identically zero, such that qk(Z)h(Z)k + ... + qo(Z) _ 0. A holomorphic mapping is algebraic if all its components are algebraic. A smooth real hypersurface M in CN is algebraic if it is contained in the zero set of a nontrivial real-valued polynomial in Z and Z. We assume throughout this paper that N > 1. In [WI], Webster proved the following celebrated result: If M and M' are algebraic hypersurfaces in CN with nondegenerate Levi forms and if H is a biholomorphism defined in an open neighborhood of M and mapping M into M', then H is algebraic. Previously, it had long been known that if M and M are open subsets of spheres, then the components of H are in fact rational functions (Poincare [P], Tanaka [T]). In this paper we go a step further and give a complete characterization of algebraic hypersurfaces in CN such that any holomorphic mapping with nonvanishing Jacobian determinant between two such hypersurfaces must be algebraic. Our main result is stated in Theorem 1 below. ,N By a germ at p0 of a holomorphic vector field in C , we shall mean a

CHARACTERIZATION OF MODELS IN C2 BY THEIR AUTOMORPHISM GROUPS

Let Ω be a domain in C2. Assume that there exists a point z0 in Ω and a sequence (φp) of holomorphic automorphisms of Ω such that (φp (z0)) is accumulating at some point ζ0 where bΩ is pseudoconvex and finite type. We show that Ω is biholomorphically equivalent to its model at ζ0.

Polynomials with a Prescribed Zero and the Bernstein's Inequality

AbstractLet be the class of all polynomials/? of degree at most n such that In view of the example zn it follows from Bernstein's inequality for polynomials that at each point z0 of the unitcirle. It was shown by A. Giroux and Q. I. Rahman [2] that if denotes the subclass of polynomials in which vanish at 1, thenwhere c1 and c2 are constants not depending on n. Here we find the exact value of which has some special significance and also at certain other points of the unit circle.

Propagation of electromagnetic waves in a density-modulated plasma

The properties of electromagnetic wave propagation in a uniformly densitymodulated plasma are studied, starting from a unidimensional scalar wave (Hill) equation for the wave electric field. Introduction of the formalism of the spatial propagator Q(z2, z1), from the point z1 to the point z2 allows reduction of the problem to determination of the propagator relevant to a single plasma layer that constitutes the entire periodic structure. The transmission coefficient of a single layer can be computed for any kind of density profile by means of the Magnus approximation, satisfying energy flux conservation at each order in the relevant expansion. The appearance of ‘forbidden zones’ in parameter space leads to the possibility that the incident electromagnetic wave can be partially or completely reflected if a sufficient number of periods are present. The explicit computation of the transmission coefficient for a series of n successive layers confirms this effect as the result of a ‘resonant’ interaction of the incident wave and the ‘periodicity’ of the medium.

Experimental Study on Correction of Spherical Aberration in Electro-Magnetic Round Lens

As we know that the spherical aberration in a round electromagnetic lens can not be eliminated. Therefor,the correcting of such aberration is an important subject to improve the ultimate resolution of electron microscopes and the performance of other electron optical instruments, such as electron beam manufactureing machines, electron lithographic machine etc.A combination of quadrupoles and octupoles which was proposed by Scherzer is a reasonable way to correct this aberration, but it has been proved practically unsuccessful. Crewe suggested that the sextu-pole elements could be used as a device to correct the 3rd order aberration of a round lens. Later he has showed that a system of two sextupoles with a round lens placed in the middle of it can acturely act as an electromagnetical concave lens if the focal length of this middle lens be settled to satisfy certain condition. Instead of taking terms of z7 in series solution of motion equation of electron in sextu-pole which Crewe had done, we took it up to z20to compute the amplitude r and slope r' of electron trajectory at each specific point z1 , z2 , z3, and z4along the optical axis. The results show that the action of sex-tupole-lens-sextupole system does act as an optical concave lens. It produces a negative spherical aberration. Differing from optical concave lens, in this system the negative aberration can be varied in a wide rang by adjusting the strength of sextupole k. Fig.2 shows a series of computed performances of electron beam near Gaussian image point while the strength of sextupole k=0 incase(a) and k1&lt; k2in case (b) and (c). Evidently, this system does produce a negative spherical aberration while k is not zero.

A uniqueness characterization in terms of signed magnitude for functions in the polydisc algebra A(Un)

In an excellent paper written in 1983 Van Hove et al. [ IEEE Trans. Acoust. Speech Signal Process.ASSP-31, 1286 ( 1983)] proved that a real polynomial P(z), free of zeros in |z| = 1, is determined uniquely by its magnitude A(θ) and the sign of its real part u(θ) for z = ejθ. Van Hove et al. also indicated a possible extension to real polynomials in more than one variable and suggested an iterative procedure for the recovery of P(z). The restriction to polynomials (or rational functions), real or otherwise, is not crucial. In fact, on due reflection, it is seen that the problem continues to make sense for functions H(z) = H(z1, z2, …, zn) in n variables that are holomorphic in the open polydisc Un = {z:|z1| &lt; 1, |z2| &lt; 1, …, |zn| &lt; 1}, continuous in the closure Ūn = {z:|z1| ≤ 1, |z2| ≤ 1, …, |zn| ≤ 1}, and free of zeros on its distinguished boundary Tn = {z:|z1| = 1, |z2| = 1, …, |zn| = 1}. The collection A(Un) of functions holomorphic in Un and continuous in Ūn is Rudin’s polydisc algebra [ W. Rudin , Function Theory in Polydiscs ( Benjamin, NewYork, 1969)]. Our main result can be stated as follows. Let H(z) belong to A(Un) and be free of zeros in Tn. For real θ = (θ1, θ2, …, θn), write, in terms of real and imaginary parts, H(exp(jθ1), exp(jθ2), …, exp(jθn)) = H(ejθ) = u(θ) + jv(θ), and let A(θ) = |H(ejθ)|; then H(z) is determined uniquely by the functions A(θ) and sign u(θ), together with its value at one point z0 exp(jθ0) ∈ Tn for which u(θ0) ≠ 0. We also clarify the meaning and the scope of this theorem by the use of specific examples, but reconstruction techniques are not discussed.

Positive Superharmonic Functions and the Hölder Continuity of Conformal Mappings

We study the rate at which a positive superharmonic function u can tend to zero at a boundary point z0 of a plane domain G. In particular, if G is a quasidisk, and α > 0 is given, we show that the condition that lim inf u(z)/dist (z, ∂G)1/α > 0 as z0 in G for any such u is related to the condition that the conformal map f of the unit disk onto G witha f(1) = zo is Hölder continuous with exponent α at the point 1. This leads us to consider the problem of finding the best exponent α for which f is Hölder continuous. The answer depends on how we characterize quasidisks or quasicircles. In this connection we give a negative answer to a question of Näkki and Palka.