Dynamics of Learning with Restricted Training Sets I: General Theory

We study the dynamics of supervised learning in layered neural networks, in the regime where the size $p$ of the training set is proportional to the number $N$ of inputs. Here the local fields are no longer described by Gaussian probability distributions and the learning dynamics is of a spin-glass nature, with the composition of the training set playing the role of quenched disorder. We show how dynamical replica theory can be used to predict the evolution of macroscopic observables, including the two relevant performance measures (training error and generalization error), incorporating the old formalism developed for complete training sets in the limit $\alpha=p/N\to\infty$ as a special case. For simplicity we restrict ourselves in this paper to single-layer networks and realizable tasks.Comment: 39 pages, LaTe

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

Wet-Process Phosphoric Acid Interaction with High Grade Phosphate Sediments: Statistical Modeling.

The most important factors affecting the preparation of triple superphosphate fertilizer (TSP) in terms of P2O5., %conversion efficiency were studied using both changing one factor at a time (OFAT) and the multivariate 24 full factorial methodologies. The obtained results were statistically analyzed using analysis of variances (ANOVA) to measure the adequacy of the fitted model. The first order regression model was built to approximate the preparation of triple superphosphate fertilizer based on the design of experiments (DOE).   It was noticed that a low H3PO4 acid concentration with a relatively long reaction time was more favorable for improving both water soluble phosphate (W/S) and P2O5 conversion efficiency during the preparation of triple superphosphate (TSP). The (DOE) methodology has also been shown to be more effective due to its economic feasibility and reduced time; additionally, the model built for P2O5 % conversion efficiency to produce (TSP) when low concentrated phosphoric acid was utilized was judged accurate and reliable. 95 % (-100) mesh, 20% H3PO4 solution, 1:4 S/L ratio, and a 20-minute reaction period were the optimal conditions for W/S and P2O5 conversion efficiency. Under these ideal conditions, a P2O5 conversion of 86.61% was effectively accomplished during the preparation of triple superphosphate fertilizer. These results backed up the model's experimental validity and the presence of ideal conditions. This verified that the developed model for P2O5., % conversion was accurate and trustworthy

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 $\|A\|_F$

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