An Analog of Bolzano's Theorem for Functions of a Complex Variable

1982 ◽  
Vol 89 (3) ◽  
pp. 210 ◽  
Author(s):  
Mau-Hsiang Shih
Keyword(s):  
Complex Variable ◽  
Bolzano’S Theorem

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

scholarly journals A novel stepwise integrative analysis pipeline reveals distinct microbiota-host interactions and link to symptoms in irritable bowel syndrome

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Annikka Polster ◽  
Lena Öhman ◽  
Julien Tap ◽  
Muriel Derrien ◽  
Boris Le Nevé ◽  
...  
Keyword(s):  
Irritable Bowel Syndrome ◽  
Complex Variable ◽  
16S Rrna Gene Sequencing ◽  
Integrative Analysis ◽  
Rrna Gene ◽  
Analysis Pipeline ◽  
Host Interactions ◽  
Functional Relationships ◽  
Irritable Bowel ◽  
Host Genes

AbstractAlthough incompletely understood, microbiota-host interactions are assumed to be altered in irritable bowel syndrome (IBS). We, therefore, aimed to develop a novel analysis pipeline tailored for the integrative analysis of microbiota-host interactions and association to symptoms and prove its utility in a pilot cohort. A multilayer stepwise integrative analysis pipeline was developed to visualize complex variable associations. Application of the pipeline was demonstrated on a dataset of IBS patients and healthy controls (HC), using the R software package to analyze colonic host mRNA and mucosal microbiota (16S rRNA gene sequencing), as well as gastrointestinal (GI) and psychological symptoms. In total, 42 IBS patients (57% female, mean age 33.6 (range 18–58)) and 20 HC (60% female, mean age 26.8 (range 23–41)) were included. Only in IBS patients, mRNA expression of Toll-like receptor 4 and genes associated with barrier function (PAR2, OCLN, TJP1) intercorrelated closely, suggesting potential functional relationships. This host genes-based “permeability cluster” was associated to mucosa-adjacent Chlamydiae and Lentisphaerae, and furthermore associated to satiety as well as to anxiety, depression and fatigue. In both IBS patients and HC, chromogranins, secretogranins and TLRs clustered together. In IBS patients, this host genes-based “immune-enteroendocrine cluster” was associated to specific members of Firmicutes, and to depression and fatigue, whereas in HC no significant association to microbiota was identified. We have developed a stepwise integrative analysis pipeline that allowed identification of unique host-microbiota intercorrelation patterns and association to symptoms in IBS patients. This analysis pipeline may aid in advancing the understanding of complex variable associations in health and disease.

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 .

A Methodology to Teach Complex Variable Concepts based on Visualization

2020 ◽  
Author(s):  
Rubén Darío A. Santiago Acosta ◽  
Ernesto Manuel B. Hernández Cooper ◽  
Juan Carlos C. Del Valle
Keyword(s):  
Complex Variable
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