Formulas and Inequalities for Some Special Functions of a Complex Variable

2020 ◽  
pp. 299-318
Author(s):  
Arcadii Z. Grinshpan
Keyword(s):  
Complex Variable ◽  
Special Functions

scholarly journals Analytic Normalized Solutions of 2D Fractional Saint-Venant Equations of a Complex Variable

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Najla M. Alarifi ◽  
Rabha W. Ibrahim
Keyword(s):  
Complex Variable ◽  
Upper Bound ◽  
Special Functions ◽  
Univalent Functions ◽  
Open Unit ◽  
Pressure Surface ◽  
Operator Type ◽  
Saint Venant Equations ◽  
The Difference ◽  
Bed Slope

Saint-Venant equations describe the flow below a pressure surface in a fluid. We aim to generalize this class of equations using fractional calculus of a complex variable. We deal with a fractional integral operator type Prabhakar operator in the open unit disk. We formulate the extended operator in a linear convolution operator with a normalized function to study some important geometric behaviors. A class of integral inequalities is investigated involving special functions. The upper bound of the suggested operator is computed by using the Fox-Wright function, for a class of convex functions and univalent functions. Moreover, as an application, we determine the upper bound of the generalized fractional 2-dimensional Saint-Venant equations (2D-SVE) of diffusive wave including the difference of bed slope.

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.

Sign in / Sign up

Export Citation Format

Share Document