Effective photon mass and exact translating quantum relativistic structures

Using a variation of the celebrated Volkov solution, the Klein-Gordon equation for a charged particle is reduced to a set of ordinary differential equations, exactly solvable in specific cases. The new quantum relativistic structures can reveal a localization in the radial direction perpendicular to the wave packet propagation, thanks to a non-vanishing scalar potential. The external electromagnetic field, the particle current density and the charge density are determined. The stability analysis of the solutions is performed by means of numerical simulations. The results are useful for the description of a charged quantum test particle in the relativistic regime, provided spin effects are not decisive

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

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

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

Quantum Circulant Preconditioner for Linear System of Equations

We consider the quantum linear solver for $Ax=b$ with the circulant preconditioner $C$. The main technique is the singular value estimation (SVE) introduced in [I. Kerenidis and A. Prakash, Quantum recommendation system, in ITCS 2017]. However, some modifications of SVE should be made to solve the preconditioned linear system $C^{-1} Ax = C^{-1} b$. Moreover, different from the preconditioned linear system considered in [B. D. Clader, B. C. Jacobs, C. R. Sprouse, Preconditioned quantum linear system algorithm, Phys. Rev. Lett., 2013], the circulant preconditioner is easy to construct and can be directly applied to general dense non-Hermitian cases. The time complexity depends on the condition numbers of $C$ and $C^{-1} A$, as well as the Frobenius norm $\|A\|_F$

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

Effect of Addition of Rosemary Leaves Powder on the Rheological Characteristics of Dough in Addition to the Quality Attributes of Bread Manufactured from to Local Wheat

        أضيفت تراكيز مختلفة من مسحوق نبات اكليل بنسب (2,5% و 5% و 7,5%) إلى دقيق الحنطة المحلية (استخلاص 80%) لمعرفة تاثير اضافة هذه النسب على الخواص الريولوجية باستخدام جهاز الفارينوجراف وقورنت النتائج مع دقيق القمح المحلي بدون اضافه. أوضحت النتائج المتحصلة أن دقيق القمح المحلي كان مقارب فى نسبة ثباتية العجبنة ونسبة الامتصاصية وزمن الوصول. بعد اضافة تركيزات (2,5% و5%) من مسحوق نبات اكليل الجبل وتحسنت صفات العجينة الناتجة من حيث الثباتية والامتصاصية ومدى تحمل العجينة للخلط. ويمكن ان نستنتج بان مسحوق أوراق إكليل الجبل له القدرة على تحسين الخصائص الريولوجية لطحين القمح المحلي عند (2.5٪و5٪) والى رغيف ذو مواصفات جيده.Different percentages (2.5%, 5% and 7.5%) of  rosemary leaves powder were added to local wheat flour(80% extraction), to study the result of adding this herb on the rheological properties of dough. To reach this target, Farinograph  was used to study the water absorption, dough development time, dough stability and degree of dough softening, the out come was compared with local wheat flour with out addition. The obtained results showed that local wheat had values of water absorption, dough stability approximate to those of the local wheat flour after adding the percent of 2.5 and 5% of degree dough softening from rosemary leaves powder. The water absorption, dough development time of local wheat flour was improved as a function of adding rosemary to the flour. The increase rosemary is forming, which helped include The water absorption, dough development time of local wheat flour. It can be concluded that rosemary leaves powder was able to improve the rheological properties of local wheat flour at (2.5% and 5%)and a good quality loaf

Impact of nitrogen regime on fatty acid profiles of Desmodesmus quadricaudatus and Chlorella sp. and ability to produce biofuel

Abstract Microalgae have emerged as one of the most promising sources for fatty acid production. Since the various fatty acid profiles (chain length, degree of unsaturation, and branching of the chain) of the different sources influence biodiesel fuel properties, it is important to possess data on how the presence of NaNO3 as nitrogen source can influence the profile of produced fatty acids from algae. The fatty acid profiles of Desmodesmus quadricaudatus and Chlorella sp. were detected in pure batch cultures experiments. BG-11 nitrogen free medium and the medium contained 1.5 g NaNO3 l−1 were used in this investigation. At late stationary growth phase in nitrogen free medium, Chlorella sp. produced 58.39% saturated fatty acids and 41.60% unsaturated fatty acids. While in medium contained 1.5 g NaNO3 l−1Chlorella sp. produced 62.08% saturated fatty acids and 37.92% unsaturated fatty acids. In nitrogen free medium D. quadricaudatus produced 66.92% saturated fatty acids and 33.07% unsaturated fatty acids. While in cultures contained 1.5 g NaNO3 l−1D. quadricaudatus produced 51.62% saturated fatty acids and 48.37% unsaturated fatty acids. The fatty acid profile of Chlorella sp. and D. quadricaudatus that isolated from Egyptian water body and grown in nitrogen free medium may be suitable for biodiesel production. The results discussed and compared to fatty acid profiles produced by other algal species