Since the function z 7→ 1+z 1−z is univalent with image the right half plane, we see that z 7→ ( 1+z 1−z )2 is univalent, so k ∈ S, and the image of k is the entire complex plane except for real numbers ≤ −14 . In 1916, L. Bieberbach [Bi] conjectured that the Koebe function was maximal with respect to the absolute value of the coefficients of its power series. More precisely, he conjectured the...

Let A be a finite-dimensional algebra over the complex numbers with nonzero identity element e. If x ∈ A, then the resolvent set associated to x is the set ρ(x) of complex numbers λ such that λ e − x is invertible, and the spectrum of x is the set σ(x) of complex numbers λ such that λ e − x is not invertible. For instance, if V is a finite-dimensional vector space over the complex numbers of po...

A colony of beavers, an unkindness of ravens, a murder of crows, a team of oxen, . . . each is an example of a set of things. Rather than define what a set is, we assume you have the “ordinary, human, intuitive (and frequently erroneous) understanding”1 of what a set is. 1 Paul Halmos, Naive Set Theory, Springer–Verlag, NY 1974. Sets have elements, often called members. The elements of a set ma...

We prove that the meandering set for $$f_a(z)=e^z+a$$ is homeomorphic to space of irrational numbers whenever a belongs Fatou $$f_a$$ . This extends recent results by Vasiliki Evdoridou and Lasse Rempe. It implies radial Julia has topological dimension zero all attracting parabolic parameters, including $$a\in (-\infty ,-1]$$ Similar are obtained Fatou’s function $$f(z)=z+1+e^{-z}$$

Definition W.1 An elliptic function f (z) is a non constant meromorphic function on C that is doubly periodic. That is, there are two nonzero complex numbers ω 1 , ω 2 whose ratio is not real, such that f (z + ω 1) = f (z) and f (z + ω 2) = f (z). Fix two real numbers β, γ > 0. The Weierstrass function with primitive periods γ and iβ is the function ℘ : C → C defined by ℘(z) = 1 z 2 + ω∈γZ Z⊕iβ...

Singular values and fixed points of one parameter family of generating function of Bernoulli’s numbers, gλ(z) = λ z ez−1 , λ ∈ R\{0}, are investigated. It is shown that the function gλ(z) has infinitely many singular values and its critical values lie outside the open disk centered at origin and having radius λ. Further, the real fixed points of gλ(z) and their nature are determined. The result...

Decisions are based on information. To be useful, information must be reliable. Basically, the concept of a Z-number relates to the issue of reliability of information. A Z-number, Z, has two components, Z=(A,B). The first component, A, is a restriction (generalized constraint) on the values which a real-valued uncertain variable, X, is allowed to take. The second component, B, is a measure of ...

Let p be a fixed odd prime. Throughout this paper, Zp,Qp,Cp will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp. The p-adic absolute value | |p on Cp is normalized so that |p|p 1/p. Let Z>0 be the set of natural numbers and Z≥0 Z>0 ∪ {0}. As is well known, the Bernoulli polynomials Bn x are defined by the ge...

where ((x)) = x − [x]G − 1 2 , if x / ∈ Z, ((x)) = 0, x ∈ Z, where [x]G is the largest integer ≤ x cf. ([1], [5], [9], [11], [12], [13]). In this paper, Zp, Qp, Cp, C and Z, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, the p-adic completion of the algebraic closure of Qp normalized by |p|p = p −1, and the complex field and integer numbers. Let q be an ...

This note contains some disconnected minor remarks on number theory . 1 . Let (1) Iz j I=1, 1<j<co be an infinite sequence of numbers on the unit circle . Put n s(k, n) _ z~, Ak = Jim sup I s(k, n) j=1 k=oo and denote by B k the upper bound of the numbers I s(k,n)j . If z j = e 2nij' a =A 0 then all the Ak 's are finite and if the continued fraction development of a has bounded denominators the...