Sextic anharmonic oscillators and orthogonal polynomials

Under certain constraints on the parameters a, b and c, it is known that Schroedinger's equation -y"(x)+(ax^6+bx^4+cx^2)y(x) = E y(x), a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function y is the generating function for a set of orthogonal polynomials P_n^{(t)}(x) in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can be reduced,by means of a simple scaling transformation, to a remarkable class of orthogonal polynomials, P_n(E)=P_n^{(0)}(E) recently discovered by Bender and Dunne.Comment: 11 page

Fibonacci and Lucas Differential Equations

The second-order linear hypergeometric differential equation and the hypergeometric function play a central role in many areas of mathematics and physics. The purpose of this paper is to obtain differential equations and the hypergeometric forms of the Fibonacci and the Lucas polynomials. We also write again these polynomials by means of Olver’s hypergeometric functions. In addition, we present some relations between these polynomials and the other well-known functions

An Introduction to Montmorillonite Purification

Purification of montmorillonite is a process to remove non clay minerals (gangue) such as calcite, feldspar, quartz, opal (C-T), and mica from montmorillonite ore. This is performed to make montmorillonite suitable for use in sensitive applications such as pharmaceutical, cosmetic, food, and advanced materials for nanotechnology. Gangue minerals in raw montmorillonite ores can cause serious health problems when used in pharmaceutics, cosmetic, and food industries and reduce material quality in advanced materials production. Montmorillonite purification can be divided into two main classes as physical and chemical purification. Physical purification processes are based on particle size difference between the gangue and montmorillonite minerals. Purification processes based on gravity separation are ineffective since the specific weights of gangue and montmorillonite minerals are very close to each other. Physical purification process includes sedimentation, centrifugal separation, aero separation, and sieving techniques. Chemical purification of montmorillonite is based on dissolution and so extraction of carbonates, metal hydroxides, organic materials, and silica, respectively, using different leaching techniques

Czy łagodny stan przedrzucawkowy powoduje sztywność tętnic i przebudowę komory serca poprzez zapalenie?

Background: A link between preeclampsia (PE) and excessive maternal morbidity and mortality is a commonly recognized fact. Moreover, it has been suggested that chronic inflammatory state connected with PE contributes to accelerated atherosclerosis. There is also an association between PE and maternal cardiac remodeling and biventricular diastolic dysfunction. The aim of the study was to investigate the presence of impaired myocardial performance and increased arterial stiffness in patients who experienced a mild case of PE five years previously. Methods: The study included forty PE patients (40 women; mean age 33.75±7.95) and 27 healthy volunteers (27 women; mean age 36.44±10.45)Transthoracic echocardiography, including Doppler echocardiography combined with tissue Doppler imaging (TDI), and aortic stiffness index (AoSI), aortic distensibility (AoD), and aortic elastic modulus (AoEM) values were measured in each study participant. Results: There was a statistically significant increase in hsCRP, aortic stiffness index, and aortic elastic modulus in PE patients as compared to controls (2.43±1.91 vs. 3.80±2.06, p=0.007; 3.09±2.41 vs. 7.32±6.89, p=0.001; 2.89±2.11 vs. 7.00± 6.83, p=0.001), while a significant decrease was observed in the aortic strain and distensibility (respectively, 22.35±15.99 vs. 12.24±9.22, p=0.005; 11.17±9.68 vs. 6.13±4.99, p=0.018). No differences between the two groups were observed with regard to the left ventricular myocardial performance index (MPI) (0.55±0.16 vs. 0.53± 0.19, p= 0.630). Conclusions: To the best of our knowledge, this has been the first study to demonstrate impaired aortic elasticity and unaffected myocardial performance index in patients with mild PE. Moreover, these effects turned out to be significantly correlated with inflammation.Wstęp: Istnieje powiązanie pomiędzy stanem przedrzucawkowym (PE) a nadmierną zachorowalnością i śmiertelnością. Ponadto, sugeruje się, że przewlekły stan zapalny udzielający się w PE przyczynia się do przyspieszenia miażdżycy. Istnieje również związek między PE przebudowy mięśnia sercowego ze strony matki i dwukomorową dysfunkcją rozkurczową. Zaplanowaliśmy ocenić w tym badaniu, czy nie została osłabiona wydolność mięśnia sercowego oraz zwiększenie sztywności tętnic u pacjentek, które pięć lat wcześniej miały łagodny przypadek PE. Metody: W badanie włączonych zostało czterdzieści pacjentek (40 kobiet; średnia wieku 33,75±7,95) oraz 27 zdrowych ochotniczek (27 kobiet: średnia wieku: 36,44±10,45). Każda z pacjentek została zbadana za pomocą echokardiografii przezklatkowej, w tym echokardiografii dopplerowskiej w połączeniu z tkankową echokardiografią dopplerowską (TDI). Zostały również zmierzone takie wartości, jak wskaźnik sztywności aorty (AoSI), rozciągliwość aorty, a także moduł sprężystości aorty (AoEM). Wyniki: Stwierdzono statystycznie istotny wzrost hsCRP, wskaźnika sztywności aorty i modułu sprężystości aorty u pacjentów z PE w porównaniu z grupą kontrolną (2,43±1,91vs. 3,80±2,06, p=0.007; 3,09±2,41 vs. 7,32±6,89, p=0,001; 2,89±2,11 vs. 7,00± 6,83, p=0,001), natomiast znaczne zmniejszenie zaobserwowano w odkształceniu aorty i jej rozciągliwości (odpowiednio 22,35±15,99 vs. 12,24±9,22, p= 0,005; 11,17±9,68 vs. 6,13±4,99, p=0.018). Nie wystąpiły różnice pomiędzy tymi dwoma grupami w odniesieniu do wskaźnika wydolności mięśnia sercowego lewej komory (0,55± 0,16 vs 0,53± 0,19, p=0,630). Wnioski: Stwierdziliśmy po raz pierwszy w tym badaniu, że wystąpiły osłabiona elastyczność aorty i niezmieniony wskaźnik wydolność mięśnia sercowego (MPI) u pacjentek z łagodnym PE, ponadto, efekty te były znacząco skorelowane ze stanem zapalnym

Solutions for certain classes of Riccati differential equation

We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are C^{\infty}-functions. We show that if \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1} and s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1}, n=1,2,..., then The Riccati equation has a solution given by y(x)=\mp s_{n-1}(x)/\lambda_{n-1}(x). Extension to the generalized Riccati equation y'(x)+P(x)y(x)+Q(x)y^2(x)=R(x) is also investigated.Comment: 10 page