scholarly journals Développement en série de fonctions holomorphes des fonctions d'une classe de Gevrey sur l'intervalle [-1;1]

2015 ◽  
Vol 98 (112) ◽  
pp. 287-293 ◽  
Author(s):  
Elmostafa Bendib ◽  
Hicham Zoubeir
Keyword(s):  
Simple Proof ◽  
Holomorphic Functions ◽  
Unit Interval ◽  
Gevrey Classes

We characterize Gevrey functions on the unit interval [-1; 1] as sums of holomorphic functions in specific neighborhoods of [-1; 1]. As an application of our main theorem, we perform a simple proof for Dyn'kin's theorem of pseudoanalytic extension for Gevrey classes on [-1; 1].

scholarly journals On the weak limit law of the maximal uniform k-spacing

2016 ◽  
Vol 48 (A) ◽  
pp. 235-238 ◽  
Author(s):  
Aleksandar Mijatović ◽  
Vladislav Vysotsky
Keyword(s):  
Limit Theorem ◽  
Simple Proof ◽  
Weak Limit ◽  
Fixed Number ◽  
Unit Interval ◽  
Classical Theorem

AbstractIn this paper we give a simple proof of a limit theorem for the length of the largest interval straddling a fixed number of points that are independent and uniformly distributed on a unit interval. The key step in our argument is a classical theorem of Watson on the maxima of m-dependent stationary stochastic sequences.

THE LOG CANONICAL THRESHOLD OF HOLOMORPHIC FUNCTIONS

2012 ◽  
Vol 23 (11) ◽  
pp. 1250115 ◽  
Author(s):  
LE MAU HAI ◽  
PHAM HOANG HIEP ◽  
VU VIET HUNG
Keyword(s):  
Holomorphic Function ◽  
Simple Proof ◽  
Holomorphic Functions ◽  
Chain Condition ◽  
Zero Set ◽  
Log Canonical Threshold ◽  
Log Canonical ◽  
Ascending Chain Condition

In this paper we give the relation between the log canonical threshold c0(f) and the geometry of the zero set {f = 0} of a holomorphic function f. Applying the above relation we give a simple proof for the ascending chain condition in dimension two.

scholarly journals Holomorphic series expansion of functions of Carleman type on the interval [−1, 1]

2017 ◽  
Vol 102 (116) ◽  
pp. 247-262
Author(s):  
Hicham Zoubeir
Keyword(s):  
Series Expansion ◽  
Holomorphic Functions ◽  
Unit Interval ◽  
Alternative Construction ◽  
Carleman Type

We characterize the functions of some Carleman classes on the unit interval [?1,1] as sums of holomorphic functions in specific neighborhoods of [?1,1]. As an application of our main theorem, we perform an alternative construction of the Dyn?kin?s pseudoanalytic extension for these Carleman classes on [?1,1].

scholarly journals The Uniformisation of the Equation $$z^w=w^z$$

2021 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Positive Solutions ◽  
Complex Plane ◽  
Complex Variable ◽  
Holomorphic Functions ◽  
Parametric Form ◽  
Lambert Function ◽  
The Real ◽  
Complex Equation ◽  
Real Equation ◽  
Expository Paper

AbstractThe positive solutions of the equation $$x^y = y^x$$ x y = y x have been discussed for over two centuries. Goldbach found a parametric form for the solutions, and later a connection was made with the classical Lambert function, which was also studied by Euler. Despite the attention given to the real equation $$x^y=y^x$$ x y = y x , the complex equation $$z^w = w^z$$ z w = w z has virtually been ignored in the literature. In this expository paper, we suggest that the problem should not be simply to parametrise the solutions of the equation, but to uniformize it. Explicitly, we construct a pair z(t) and w(t) of functions of a complex variable t that are holomorphic functions of t lying in some region D of the complex plane that satisfy the equation $$z(t)^{w(t)} = w(t)^{z(t)}$$ z ( t ) w ( t ) = w ( t ) z ( t ) for t in D. Moreover, when t is positive these solutions agree with those of $$x^y=y^x$$ x y = y x .

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Coefficients problems for families of holomorphic functions related to hyperbola

2020 ◽  
Vol 70 (3) ◽  
pp. 605-616
Author(s):  
Stanisława Kanas ◽  
Vali Soltani Masih ◽  
Ali Ebadian
Keyword(s):  
Unit Disk ◽  
Holomorphic Functions ◽  
Hankel Determinant ◽  
Coefficient Problem

AbstractWe consider a family of analytic and normalized functions that are related to the domains ℍ(s), with a right branch of a hyperbolas H(s) as a boundary. The hyperbola H(s) is given by the relation $\begin{array}{} \frac{1}{\rho}=\left( 2\cos\frac{\varphi}{s}\right)^s\quad (0 \lt s\le 1,\, |\varphi| \lt (\pi s)/2). \end{array}$ We mainly study a coefficient problem of the families of functions for which zf′/f or 1 + zf″/f′ map the unit disk onto a subset of ℍ(s) . We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.

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