Solvable Systems of Linear Differential Equations

The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear differential equations. The termination criteria of AIM will be re-examined and the whole theory is re-worked in order to fit this new application. As a result of our investigation, an interesting connection between the solution of linear systems and the solution of Riccati equations is established. Further, new classes of exactly solvable systems of linear differential equations with variable coefficients are obtained. The method discussed allow to construct many solvable classes through a simple procedure.Comment: 13 page

Enhancing the bioaccessibility of lycopene from tomato processing byproducts via supercritical carbon dioxide extraction

Tomato peel and seed from tomato processing industry are treated as waste; however, they contain lycopene, a high-value bioactive compound. In this study, lycopene was extracted from tomato peel and seed using supercritical carbon dioxide (SC–CO2) and hexane, and the bioaccessibilities of lycopene in the SC-CO2- and hexaneextracted oleoresins were investigated for the first time. The (Z)-lycopene content of the SC-CO2-extracted oleoresin (69%) was higher than that of hexane-extracted oleoresin (45%). Separation of the insoluble fraction from the oleoresins increased the (Z)-lycopene contents of the SC-CO2- and hexane-extracted oil fractions to 80% and 49%, respectively. The bioaccessibility of total-lycopene in the oleoresins was increased by 3.3-fold via SCCO2 extraction, which was attributed to higher (Z)-lycopene content, and small-sized uniform distribution of lycopene in the oleoresin. SC-CO2 extraction is not only a green method for extraction of bioactive compounds, but also has the potential to improve health benefits of bioactive compounds

Criterion for polynomial solutions to a class of linear differential equation of second order

We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where \lambda_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then \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}\hbox{and}\quad s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1},\quad n=1,2,.... Conversely (ii) if \lambda_n\lambda_{n-1}\ne 0 and \delta_n=0, then the differential equation has a polynomial solution of degree at most n. We show that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first and second kind), Gegenbauer, and the Hypergeometric type, etc, obey this criterion. Further, we find the polynomial solutions for the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations.Comment: 12 page

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

Hydrothermal Deposition and Characterization of Heteroepitaxial BaTiO₃ Films on SrTiO₃ and LaAlO₃ Single Crystals

Heteroepitaxial BaTiO3 thin films were deposited in an aqueous solution under hydrothermal conditions on single crystal substrates of (100) SrTiO3 and (012) LaAlO3. The reactants consisted of fine TiO2 particles in a strongly alkaline solution of Ba(OH)2 at a temperature of 150°C. The growth of the films was studied by atomic force microscopy, high resolution scanning electron microscopy, and X-ray diffraction. The formation of the films occurred by nucleation of {001} faceted islands followed by three-dimensional growth of the islands to cover the substrate. Repeated hydrothermal treatment improved the film thickness and the surface coverage of the substrate at the expense of increased surface roughness. X-ray diffraction coupled with pole figure analysis showed that the films had the same in-plane and out-of-plane orientation as the substrate

Physical applications of second-order linear differential equations that admit polynomial solutions

Conditions are given for the second-order linear differential equation P3 y" + P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of degree n. Several application of these results to Schroedinger's equation are discussed. Conditions under which the confluent, biconfluent, and the general Heun equation yield polynomial solutions are explicitly given. Some new classes of exactly solvable differential equation are also discussed. The results of this work are expressed in such way as to allow direct use, without preliminary analysis.Comment: 13 pages, no figure

Development of omega-3-rich \u3ci\u3eCamelina sativa\u3c/i\u3e seed oil emulsions

Camelina sativa seed is an underutilized oil source rich in omega-3 fatty acids; however, camelina oil is not fully explored for food applications. Its high omega-3 content makes it susceptible to oxidation, which may limit food applications. Therefore, the main objective of this study was to investigate the potential of camelina seed oil to form physically and oxidatively stable emulsions as a potential delivery system for omega-3 fatty acids. Effects of homogenization conditions, namely, pressure (15 MPa-30 MPa), number of passes (1,3,5, and 7), and type of homogenizers (high pressure and high shear) on the structural properties and stability of camelina seed oil emulsions stabilized with whey protein isolate were studied. High homogenization pressure (30 MPa) and number of passes (\u3e3) reduced the particle size (278 nm) and formed more physically and oxidatively stable emulsions compared to high shear homogenization; high shear homogenization generated bigger oil particles (~2,517 nm). Apparent viscosity and consistency index (k) decreased with increasing pressure, number of passes, and shear rate. Emulsions prepared with high pressure homogenization at both 15 and 30 MPa with 3 and more passes did not exhibit any peroxide formation over 28 days. Results indicated that camelina seed oil is a promising alternative oil source to form stable omega-3- rich emulsions for food applications

Review of Linac-Ring Type Collider Proposals

There are three possibly types of particle colliders schemes: familiar (well known) ring-ring colliders, less familiar however sufficiently advanced linear colliders and less familiar and less advanced linac-ring type colliders. The aim of this paper is two-fold: to present possibly complete list of papers on linac-ring type collider proposals and to emphasize the role of linac-ring type machines for future HEP research.Comment: quality of figures is improved, some misprints are correcte

Construction of exact solutions to eigenvalue problems by the asymptotic iteration method

We apply the asymptotic iteration method (AIM) [J. Phys. A: Math. Gen. 36, 11807 (2003)] to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue problems which includes Schroedinger problems with Coulomb, harmonic oscillator, or Poeschl-Teller potentials, as well as the special eigenproblems studied recently by Bender et al [J. Phys. A: Math. Gen. 34 9835 (2001)] and generalized in the present paper to higher dimensions.Comment: 10 page