Blaschke products and proper holomorphic mappings

Suppose f: $\mathbb{D} \to V$ is a proper holomorphic map of the unit disk $\mathbb{D} \subset \mathbb{C}$ onto a subset $V \subset \mathbb{D}$ of degree d>0. We show that f is conjugate to either an affine map or a degree d Blaschke product. As an application we give a unified treatment of theorems of Böttcher and Schröder coordinates.     

Characterization of Finite Groups With Some <Emphasis Type="Italic">S</Emphasis>-quasinormal Subgroups

A subgroup of a finite group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we give a characterization of a finite group G under the assumption that every subgroup of the generalized Fitting subgroup of prime order is S-quasinormal in G.     

Gibbs state for some class of meromorphic functions

This paper is a continuation of our earlier works [1,2] on the fractal structure of expanding and subexpanding meromorphic functions of the form F = H o exp o Q, where H and Q are non-constant rational maps. Under some assumptions on the forward trajectories of asymptotic values ofF we define a class of summable potentials for the maps f of the punctured cylinder induced by F. We prove the existence and uniqueness of Gibbs states for these potentials.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Thermodynamic formalism of transcendental entire maps of finite singular type

We build a version of a thermodynamic formalism for maps of the form f(z) = ∑ _{j = 0} ^{p + q} a _j e ^(j−p)z where p, q > 0 and . We show in particular the existence and uniqueness of (t,α)-conformal measures and that the Hausdorff dimension HD(J _f ^r ) = h is the unique zero of the pressure function t ↦ P(t) for t > 1, where the set J _f ^r is the radial Julia set. Partially supported by NSF Grant DMS 0100078. Partially supported by Warsaw University of Technology Grant No. 504G11200023000, Polish KBN Grant No. 2PO3A03425 and Chilean FONDECYT Grant No. 11060280.     

On quasi-strongly regular graphs

We study the quasi-strongly regular graphs, which are a combinatorial generalization of the strongly regular and the distance regular graphs. Our main focus is on quasi-strongly regular graphs of grade 2. We prove a “spectral gap”-type result for them which generalizes Seidel's well-known formula for the eigenvalues of a strongly regular graph. We also obtain a number of necessary conditions for the feasibility of parameter sets and some structural results. We propose the heuristic principle that the quasi-strongly regular graphs can be viewed as a “lower-order approximation” to the distance regular graphs. This idea is illustrated by extending a known result from the distance-regular case to the quasi-strongly regular case. Along these lines, we propose a number of conjectures and open problems. Finally, we list the all the proper connected quasi-strongly graphs of grade 2 with up to 12 vertices.     

On an application of Lambert's W function to infinite exponentials

In this article we survey the basic results concerning the convergence of Infinite Exponentials; we use Lambert's W function to show convergence for the real and complex cases in a more elegant way and prove several incidental results about Infinite Exponentials. We also show how to extend analytically the Infinite Exponential function over the complex plane and how to derive exact expansions for finite and infinite power iterates of the hyperpower function. As a final application we derive several series identities involving Infinite Exponentials.     

On the pivot structure for the weighing matrix W(12,11)

In the present article we concentrate our study on the growth problem for the weighing matrix W(12,11) and show that the unique W(12,11) has three pivot structures. An improved algorithm for extending a k × k (0,+,-) matrix to a W(n,n-1), if possible, has been developed to simplify the proof. For the implementation of the algorithm special emphasis is given to the notions of data structures and parallel processing.     

On the regularizing properties of the GMRES method

Summary. The GMRES method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix. However, little is known about the behavior of this method when it is applied to the solution of nonsymmetric linear ill-posed problems with a right-hand side that is contaminated by errors. We show that when the associated error-free right-hand side lies in a finite-dimensional Krylov subspace, the GMRES method is a regularization method. The iterations are terminated by a stopping rule based on the discrepancy principle. Received November 10, 2000 / Revised version received April 11, 2001 / Published online October 17, 2001     

Normality and shared-value of derivatives

We show that a family F of functions holomorphic in a domain is normal, if for all ? ∈ F the derivative ?′ and ? share the value a(≠0) and if the set of values that ? _(k+1) takes the zeros of ??a, is bounded. As an application we obtain some new uniqueness theorems of entire functions which improve the former results [Amer. H.H. Al-khaladi (2000). On a question of Yi-Yang. J. Shandong Univ., 35(3), 162–167; G. Jank, E. Mues and L. Volkmann (1986 Jank, G., Mues, E. and Volkmann, L. 1986. Meromorphic functionen die mit ihrer ersten und zweiten Ableitung einen endlichen wert teilen. Complex Variables, 6: 51–71. [Taylor & Francis Online] , [Google Scholar]). Meromorphic functionen die mit ihrer ersten und zweiten Ableitung einen endlichen wert teilen. Complex Variables, 6, 51–71; P. Li and C.C. Yang (2001). Uniqueness theorem on entire functions and their derivatives. J. Math. Anal. Appl., 253, 50–57; W. Lin (2001). Uniqueness theorem and normal criteria with sharing array. Math. Theory Appl., 21, 9–16 (Chinese); H.X. Yi and C.C. Yang (1995). Uniqueness Theory of Meromorphic Functions, Pure and Applied Math. Monographs No. 32. Science Press, Beiling; H. Zhong (1995). Entire functions that share one values with their derivatives. Kodai Math. J., 18, 250–259, etc].     

Totally Permutable Products of Finite Groups Satisfying SC or PST

Finite groups G=AB factorized by two subgroups A and B such that every subgroup of A permutes with every subgroup of B are studied in this paper. The behaviour of such products with respect to the class of finite groups in which Sylow-permutability is transitive is analyzed.     

Geometric rigidity of transcendental meromorphic functions

The classes of dynamically and geometrically tame functions meromorphic outside a small set are introduced. The Julia sets of geometrically tame functions are proven to be either geometrical circle (in ) or to have Hausdorff dimension strictly larger than 1. Vast classes of dynamically and geometrically tame functions are identified. The research of both authors was supported in part by the Polish KBN Grant No 2 PO3A 034 25, the Warsaw University of Technology Grant no 504G 11200043000 and by the NSF/PAN grant INT-0306004. The research of the second author was supported in part by the NSF Grant DMS 0400481.     

On sums of roots of infinite order entire functions

A class of infinite order entire functions is considered. Estimates for sums of the roots are derived. These estimates supplement the Hadamard theorem. Moreover, we establish a new estimate for the counting function of the roots, which in appropriate situations can be more useful than the Jensen inequality.     

On Riesz distributions and integer powers of the Dirac operator on the hyperbolic unit ball

In this article the Dirac operator is defined on the m-dimensional hyperbolic unit ball and a fundamental solution for integer powers of this operator is determined, using Riesz's distributions. This fundamental solution is then expressed in terms of Gegenbauer functions of the second kind.     

Small functions and weighted sharing three values

This article studies the problem of the uniqueness of meromorphic functions that weighted sharing three values which improve some results given by Yi [Theorem 4, Yi, H.X., 1995, Unicity theorems for meromorphic functions that share three values. Kodai Mathematical Journal, 18, 300-314] and Ueda [Ueda, H., 1983, Unicity theorems for meromorphic or entire functions II. Kodai Mathematical Journal, 6, 26-36] and other authors. An application of these new results, if f and g are two distinct nonconstant meromorphic functions sharing 0, 1 and CM, and a is a nonconstant rational function, then N2)(r,1/(g-a))= [image omitted] An example shows that the latter result is not true for some transcendental small functions of f and g.     

On a nonstandard problem for heat conduction in a cylinder

Decay bounds are derived for the solution of a heat conduction problem in a semi-infinite cylinder when the lateral surface is held at zero temperature, a nonzero temperature is prescribed on the finite base, and the temperature at time T is prescribed to be a constant multiple of the temperature at initial time. Both energy and pointwise decay bounds are computed for a range of values of the constant multiple. Such problems were originally introduced as a means of stabilizing the backward-in-time problem for the heat equation.     

Aut(X, σ) and Aut(X/σ) are not isomorphic in the category of surfaces with nodes

A symmetric Riemann surface is a pair (X,?σ) where X is a Riemann surface and?σ?is an anticonformal involution. We denote by Aut(X,?σ) the subgroup of Aut(X) defined by the automorphisms commuting with σ. There is a natural isomorphism between Aut(X,?σ) and Aut(X/σ). In this article we shall show that this isomorphism does not stand if X is a Riemann surface with nodes.     

Trees and hairs for some hyperbolic entire maps of finite order

Let f be an entire transcendental map of finite order, such that all the singularities of f ⁻¹ are contained in a compact subset of the immediate basin B of an attracting fixed point. It is proved that there exist geometric coding trees of preimages of points from B with all branches convergent to points from . This implies that the Riemann map onto B has radial limits everywhere. Moreover, the Julia set of f consists of disjoint curves (hairs) tending to infinity, homeomorphic to a half-line, composed of points with a given symbolic itinerary and attached to the unique point accessible from B (endpoint of the hair). These facts generalize the corresponding results for exponential maps. Research supported by Polish KBN Grant No 2 P03A 034 25.