The Levinson-Type Formula for a class of Sturm-Liouville Equation

The boundary value problem $$-{\psi}''+q(x)\psi={\lambda}^2 \psi, \quad 0<x<\infty,$$ $${\psi}'(0)-(\alpha_{0}+\alpha_{1}\lambda){\psi}(0)=0$$ is considered, where $\lambda$ is a spectral parameter, $q(x)$ is real-valued function such that \begin{equation*} \int\limits_{0}^{\infty}(1+x)|q(x)|dx<\infty \end{equation*} with $\alpha_{0}, \alpha_{1}\geq0$ ( $\alpha_{0},\alpha_{1}\in \mathbb{R}$). In this paper, for the above-mentioned boundary value problem, the scattering data is considered and the characteristics properties (such as continuity of the scattering function $S(\lambda)$ and giving the Levinson-type formula) of this data are studied.{\small \bf Keywords. }{Scattering data, scattering function, Gelfand-Levitan-Marchenko equation, Levinson-type formula.

On The Uniform Convergence Of Spectral Expansions For A Spectral Problem With A Boundary Condition Rationally Depending On The Eigenparameter

The spectral problem −y ′′ + q(x)y = λy, 0 < x < 1, y(0) cos β = y ′ (0) sin β, 0 ≤ β < π; y ′ (1) y(1) = h(λ), is considered, where λ is a spectral parameter, q(x) is real-valued continuous function on [0, 1] and h(λ) = aλ + b − XN k=1 bk λ − ck , with the real coefficients and a ≥ 0, bk > 0, c1 < c2 < · · · < cN , N ≥ 0. The sharpened asymptotic formulae for eigenvalues and eigenfunctions of above-mentioned spectral problem are obtained and the uniform convergence of the spectral expansions of the continuous functions in terms of eigenfunctions are presented

Conformable fractional Dirac system on time scales

Abstract We study the conformable fractional (CF) Dirac system with separated boundary conditions on an arbitrary time scale T $\mathbb{T}$ . Then we extend some basic spectral properties of the classical Dirac system to the CF case. Eventually, some asymptotic estimates for the eigenfunction of the CF Dirac eigenvalue problem are obtained on  T $\mathbb{T}$ . So, we provide a constructive procedure for the solution of this problem. These results are important steps to consolidate the link between fractional calculus and time scale calculus in spectral theory

Uniform convergence of the spectral expansions in terms of root functions for a spectral problem

In this article, we consider the spectral problem \displaylines{ -y''+q(x)y=\lambda y,\quad 0 where $\lambda$ is a spectral parameter, a and b are real constants and a<0, q(x) is a real-valued continuous function on the interval [0,1]. The root function system of this problem can also consist of associated functions. We investigate the uniform convergence of the spectral expansions in terms of root functions

Obesity is not a descriptive factor for oxidative stress and viscosity in follicular fluid of in vitro fertilization patients.

Abstract Background Obesity’s impact on micro-environmental oxidative stress and follicular fluid (FF) viscosity and whether or not it has any effect on in vitro fertilization (IVF) success is a matter of debate. Aims In this study, our aim was to evaluate the levels of oxidative stress markers and the FF viscosity in obese and non-obese patients. Methods Eighty norm-responder patients undergoing IVF were prospectively grouped according to their body mass indexes (BMI). Group 1 (n = 40) and group 2 (n = 40) had BMI values of B24.9 and C25.0, respectively. Total sulfhydryl (RSH) levels (nmol/m) and the formation of thiobarbituric acid-reactive substances (malondialdehyde, or MDA) (lmol/ml) in FFs were quantified. For the first time in our study, FF viscosity with changing BMI values was also determined. Results The mean levels of MDA (lmol/ml) and RSH (nmol/ml) were not significantly different between groups (1.37 ± 0.51; 1.51 ± 0.51; p[0.05 for MDA and 0.42 ± 0.30; 0.41 ± 0.20; p[0.05 for RSH, respectively). Similarly, the FF viscosity (centipoise) was not different between groups (1.28 ± 0.28; 1.30 ± 0.19; p\0.05, respectively). Independent of BMI, no correlation was found between FF levels of oxidative markers and the number of oocytes retrieved or the fertilization rates. Conclusions In our study, we found no difference in the levels of follicular oxidative and anti-oxidative markers or the follicular fluid viscosity with changing BMI values. We also demonstrated that the levels of oxidative stress markers and the viscosity of follicular fluid did not affect clinical outcomes. Keywords Follicular fluid Oxidative stress Body mass index Malondialdehyde (MDA) Sulfhydryl group (RSH) Viscosit