The maximal Lp-regularity properties of nonlocal fractional elliptic equa- tions are studied. Particularly, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an ana- lytic semigroup. Moreover, maximal regularity properties of nonlocal fractional parabolic equation are established.

In this work we implement two numerical schemes namely continuous Galerkin-Petrov (cGP(2)) and Legendre Wavelet Collocation Method (LWCM) for the approximate solution of the mathematical model which describes the behavior of CD4+ T-cells, infected CD4+T-cells and free HIV virus particles after HIV infection. The present study discuss and analyse the effect of constant and different variable source terms (depending on the viral load) used for the supply of new CD4+ T-cells from thymus on the dynamics of CD4+ T-cells, infected CD4+ T-cells and free HIV virus. Furthermore, the model is also solve using fourth order Runge Kutta (RK4) method. Finally, the validity and reliability of the proposed schemes are verified by comparing the numerical and graphical results with the results of RK4-method. Comparison of the numerical and graphical results of cGP(2) and LWCM with RK4-method confirmed that cGP(2) and LWCM performs excellent accuracy. The present study highlights the accuracy and efficiency of the proposed schemes with the other traditional schemes such as the Laplace Adomian Decomposition Method (LADM), Variational Iteration Method (VIM), Homotopy Analysis Method (HAM), Homotopy Perturbation Method (HPM), Genetic Algorithm (GA), Interior Point Algorithm (IPA), Active Set Algorithm (ASA), Multistep Laplace Adomian Decomposition Method (MSLADM) etc.

In this paper, we have proved the indispensable inequalities of classical analysis for interval value functions: Hölder, Cauchy-Schwarz and Carlson inequalities. Additionally, we achieved generalization for Carlson’s inequality for interval-valued functions.

In this paper, the Boundary-Domain Integral Equations (BDIEs) for the mixed boundary value problem(BVP) for a compressible Stokes system of partial differential equation(PDE) with variable coefficient in 2D is considered . An appropriate parametrix is used to reduce this BVP to the BDIEs. Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The properties of corresponding potential operators are investigated. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems in appropriate Sobolev spaces are proved.

The Dirichlet, Neumann and mixed boundary value problems for the linear second-order scalar elliptic differential equation with variable coefficient in a bounded three-dimensional domain are considered. The PDE right-hand side belongs to $H^{-1}(\Omega)$ or $\widetilde{H}^{-1}(\Omega),$ when neither classical nor canonical co-normal derivatives are well defined. Using the two-operator approach and appropriate parametrix (Levi function) each problem is reduced to different systems of boundary domain integral equations (BDIEs). Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and as well as invertibility of the BDIE operators are analysed in appropriate Sobolev (Bessel potential) spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading to invertibility of the perturbed operators.

In order to measure the estimated error, the definition of E-MSE(expected mean squared error) is introduced based on the definition of E-Bayesian estimation. Moreover, under different loss functions(include: squared error loss, K-loss function, precautionary loss and entropy loss), the formulas of E-Bayesian estimation and E-MSE for reliability parameters of exponential distribution are given respectively. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and a real data set have been analysed for illustrative purposes(also using OpenBUGS), results are compared on the basis of E-MSE. When considering evaluating the E-Bayesian estimations under different loss functions, this paper proposed the E-MSE as evaluation standard.

In this paper a new so called Iterative laplace transform method is implemented to investigate the solution of certain important population models of non-integer order. The iterative procedure is combined effectively with Laplace transformation to develop the suggested methodology. The Caputo operator is applied to express non-integer derivative of fractional-order. the series form solution is obtained having components of convergent behavior toward the exact solution. For justification and verification of the present method some illustrative examples are discussed. The closed contact is observed between the obtained and exact solutions. Moreover, the suggested method has small volume of calculations and therefore it can be applied to handle the solutions of various problems with fractional-order derivatives.

In this paper, we deal with the equation (aⁿ-2)(bⁿ-2)=x²,2≤a3.1 We solve the equation (3.1) for (a,b)∈{(2,10),(4,100),(10,58),(3,45)}. Moreover, we show that (aⁿ-2)(bⁿ-2)=x² has no solution n,x if 2|n and gcd(a,b)=1. We also give a conjecture which says that the equation (2ⁿ-2)((2P_{k})ⁿ-2)=x² has only the solution (n,x)=(2,Q_{k}), where k>3 is odd and P_{k},Q_{k} are Pell and Pell Lucas numbers, respectively. We also conjecture that if the equation (aⁿ-2)(bⁿ-2)=x² has a solution n,x, then n≤6.

In this work, the fractional order thermoelasticity theory is used to investigate the thermoelastic problem of a thin slim strip considering the thermal conductivity is to be variable. The theory of thermal stresses based on the heat conduction equation with the Caputo time-fractional derivative of order α is used. The surface of the strip is subjected to a thermal shock and assumed to be traction free. By using the Laplace transform and numerical Laplace inversion, the governing equations are solved. The inverse of the Laplace transform is done numerically using a method based on Fourier expansion techniques. Numerical calculations for the considered variables are performed and the results obtained have been presented graphically. The effects of fractional order parameter and the variation of thermal conductivity on temperature, stress, and displacement are investigated.

We consider a linear RLC network, modeled by differential-algebraic equations, containing distributed semiconductor devices, modeled by multi-dimensional parabolic-elliptic equations. With appropriate coupling conditions, we obtain a system of multidimensional partial differential-algebraic equations. For the resulting system, we prove an existence and uniqueness result, and study the asymptotic behavior of the solutions.

We consider the Sturm-Liouville operator on quantum graphs with a loop with the standard matching conditions in the internal vertex and the jump conditions at the boundary vertex. Given the potential on the loop, we try to recover the potential on the boundary edge from the subspectrum. The uniqueness theorem and a constructive algorithm for the solution of this partial inverse problem are provided.

The main prospect of this proposed work is to confer an algebraic theory for analyzing fractional singular systems. A modern class of linear fractional singular delay systems with two orders are proposed. The key notion used in the enlargement is the decomposition form for matrix regular pencils. As crucial issue, a procedure for computation of reachable set, and control input of addressed system is acquainted. The considerations are illustrated by suitable examples.

A linear manifold ${\mathcal K}_2$ of evolutionary equations for a pseudovector field on ${\Bbb R}^3$ is described. An infinitisimal shift of each equation is determined by a second-order differential operator of divergent type. All operators are invariant with respect to space translations in ${\Bbb R}^3$, relative to time translations, and they are transformed by covariant way relative to rotations of ${\Bbb R}^3$. It is proved that the linear space ${\mathcal M}_2 \subset {\mathcal K}_2$ of differential operators preserving solenoidal property and unimodularity of the field is one-dimensional and an explicit form of such operators is found.

A novel machine, acronymed HEVSTOW (High Efficiency Vertically Stacked vermicomposting system for Treating Organic Waste), of which a patent has recently been granted, is described. The machine is based on the paradigm of high-rate vermicomposting earlier developed by the authors. It makes it possible to rapidly vermicompost not only animal manure but also substrates like phytomass and paper waste which the conventional vermireactors are not able to process without elaborate pre-treatment and/or augmentation of animal manure. The machine also makes it possible to simultaneously vermicompost different substrates as also use different species of earthworms without one batch influencing the other. The applicability of the machine is demonstrated by its application in simultaneous yet independent processing of paper waste and a weed, each with two species of earthworms.

Vulnerability assessment has become a critical issue and an important approach for regional sustainable development. The Qinghai Province, located in the northeastern part of the Qinghai-Tibet Plateau, suffers a severe grassland degradation caused by climate change and human activities. The degradation constrains the development of local animal husbandry and further affects the vulnerability of social system. In this study, based on the vulnerability assessment framework of exposure-sensitivity-adaptability, two index systems were established including ecological and social aspects. The entropy weight method was used to determine the index weight. The dynamic changes of the ecological vulnerability and social vulnerability of Qinghai province were assessed from 1995 to 2015. Results indicated that ecological vulnerability in Qinghai province increased from eastern part to the west, and decreased from northern to southern part, while social vulnerability showed an opposite trend. Key ecologically fragile areas were mainly located in the Qaidam Basin and western Three-River Headwaters Region (TRHR), while key social vulnerability areas were mainly distributed in the Qilian Mountains and eastern Qinghai province. The overall ecological vulnerability showed a decreasing trend through time, but increased in several local areas. Social vulnerability dropped significantly, especially in the eastern part of Qinghai province. The results will help to identify key vulnerable areas of Qinghai province and provide references for the ecological protection and restoration and the formulation of ecosystem management policies.

Circulating fluidized bed (CFB) boiler is applied for peak-load regulation nowadays, with the rapid increase of renewable energy generation in the power grid. However, due to the huge inertia in CFB system, it is important to investigate the operational characteristics of CFB boiler to achieve depth peak load cycling. In this study, a one-dimensional dynamic model for gas-solid hydrodynamics in a full-loop CFB boiler is established. The model was validated by the field test data of a 350 MW supercritical CFB boiler. The impact of the changing speed for extra operating conditions, the total bed inventory, and the size distribution of the feeding ash particles under the load reduction process were analyzed. The results show that there is an optimal load changing rate under load reduction process. The total bed inventory being higher and feeding relatively larger particles benefit to achieve a better performance in a load reduction process.

Particle-laden flows in a vertical channel were simulated using an Eulerian–Eulerian, Anisotropic-Gaussian (EE-AG) model. Two sets of cases varying the overall mass loading were done using particle sizes corresponding to either a large or small Stokes number. Primary and turbulent statistics were extracted from these results and compared with counterparts collected from Eulerian–Lagrangian (EL) simulations. The statistics collected from the small Stokes number particle cases correspond well between the two models, with the EE-AG model replicating the transition observed using the EL model from shear-induced turbulence to relaminarization to cluster-induced turbulence as the mass loading increased. The EE-AG model was able to capture the behavior of the EL simulations only at the largest particle concentrations using the large Stokes particles. This is due to the limitations involved with employing a particle-phase Eulerian model (as opposed to a Lagrangian representation) for a spatially intermittent system that has a low particle number concentration.

The mechanical properties play a vital role in the stability and behavior of clathrate hydrate. In this work, the structural and mechanical properties of some nitride gas (NH3, NO and N2O) hydrates were investigated using density functional theory calculations. The equilibrium lattice structures for these hydrates were obtained. The full second-order elastic constants were then determined by energy–strain analyses and the polycrystalline elastic properties were also predicted. It is found that these three gas hydrates have high elastic isotropy, but their shear properties are significantly different. This study may lay a theoretical foundation for future research on the structural evolution of clathrate hydrates under a mechanical field.

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This paper presents an automatic side stand retractor to ensure the safety of the motorcycle. Many motorcycle accidents take place due to the wrong position of side stand. The automatic side stands retractor prevents the starting of the motorcycle for ensuring safety. The proposed system uses an embedded controller, reed switch sensor, permanent magnet, buzzer and relay. The embedded controller detects the position of side stand using a reed sensor and a permanent magnet attached to it. Then embedded controller breaks the ignition circuit of the motorcycle using a relay in case of the wrong position of the side stand. A buzzer makes a sound to notify the rider to retract the side stand. This system is tested in motorcycle and the obtained result shows the effectiveness of the system for the safety of the motorcycle. The device not only ensure the safety of the driver but also ensure the mobility of the whole system of the motorcycle.

Curved ducts with non-circular cross-sectional geometry have significant applications in different industries. Hydrodynamic stability in these curved ducts is an interesting issue in field of fluid mechanics. In the present study, the linear hydrodynamics stability of fluid flow in the curved rectangular duct is investigated semi-analytically by Homotopy perturbation method (HPM). Then, for the first time, the hydrodynamic stability in these ducts is estimated via using artificial neural networks (ANNs). To this accomplishment, critical Dean number (Dnc) is estimated under various aspect ratios and curvature ratios. Based on the semi-analytical results, the Dnc is increased by curvature ratio enhancement. In addition, irregular variation on trend of Dnc is found by an enhancement in the aspect ratio. Moreover, maxima of mean square error and minima of correlation coefficient for intended ANN are obtained 0.00144 and 0.98621, respectively. Finally, predictive equation is suggested to estimate of Dnc using weights and bias of designed ANN.

The study describes the application and comparison of five artificial intelligence methods for PV system output power prediction. General Regression Neural Network (GRNN), Radial Basis Function network (RBF), Group Method of Data Handling (GMDH) network, Multilayer Perceptron neural network (MLP) and Linear Regression (LR). Measured values of temperature (T°C) and irradiance E (Kw/m^2) were used as inputs (independent variables) and PV output power P (Kw) was used as output (dependent variable). Predictive performances have been evaluated using statistical metrics. Comparison of the results provided by the five models has been conducted and commented. It was observed that predictive accuracy depend of the nature of data set used and the optimization parameters of each model. Response surfaces that represent the combined impact of simultaneous variation in temperature and irradiance on PV output power have been illustrated. Curves that showed how close were validation predicted values and actual values have been plotted. Relationship between output power and the two parameters have been illustrated and it was found to be nonlinear. Importance of each ambient parameter contribution to the PV output power has been demonstrated.

The Ngazi Method of Exponential Function CalculationMsc. Student, Research MethodsJomo Kenyatta University of Agriculture and Technology. Tel: +254-713 172 555 e-mail: tkuria19@gmail.com Abstract This paper presents an entirely new way of calculating exponential functions. This method uses the 2n sequence at its foundation. An example of the 2n sequence is 2,4,8,16,32,64 and so on. This method is vastly more time saving and energy saving because one performs very few multiplication operations once an exponent has been broken down into its 2n components. This method can be coded into a computer software and this will improve the speed with which computer libraries calculate exponential functions. This paper also explains how this new method of exponential function calculation can partly help to solve the discrete exponential and discrete logarithm problem through easier calculation of exponential functions.1.Introduction.“Neural network simulations often spend a large proportion of their time computing exponential functions. Since the exponentiation routines of typical math libraries are rather slow, their replacement with a fast approximation can greatly reduce the overall computation time.”[1]It is true that computer math libraries that deal with the exponential functions are rather slow but this is a function of using very archaic and long routine calculations which are time-consuming and energy wasting. If a more time-efficient method for calculating exponential functions can be found and then coded into a computer math library then this will save a lot of time for many mathematicians, engineers and computer scientists who use these exponential function libraries a lot. This author presents a new method of calculating exponential functions which is both accurate and time-saving. We no longer need to rely on time-saving approximations of exponential functions. We can now use time-saving accurate calculations of exponential functions and this could indeed change the field computational science.There is a relationship between exponential functions and logarithms. The power of logarithms as a computational device lies in the fact that by them multiplication and division are reduced to the simpler operations of addition and subtraction.[2]“There are many applications of exponential functions and logarithmic functions in science and technology. The voltage in a given circuit can be expressed using exponents. The value of money in an investment can be determined through the use of exponents. The intensity of earthquakes is measured by a logarithmic scale. The intensity of light related to the thickness of the material through which it passes can be expressed using exponents. The distinction between acids and bases in chemistry is measured in terms of logarithms.” [3]When it comes to exponential functions, the word exponent is often used instead of index, and functions in which the variable is in the index (such as 2x, 10sinx) are called exponential functions. [4] If b is a real number greater than zero, then for each real exponent x we assume bx is a unique real number. Since for each real x there is one and only one bx, the equation y=bx,(b>0)defines a function. We call such an equation an exponential function.[3]2. Body .To solve the exponential function f(x)=mx, the most common method of calculating exponential functions has been to directly multiply m, x number of times. This method of calculation is extremely slow especially if the exponent has a large value. For, example calculatingf(x) = 35123000 involves multiplying 35, 123,000 times. This is an extremely difficult task and is frankly almost impossible if calculated manually. Computers are best suited to calculate this function because computers do not get tired of performing repetitive tasks. However, there is a need to come up with a more efficient way of calculating the exponential function. This is important because exponential functions are very important in mathematics and engineering fields.This paper introduces a new, novel method of calculating the exponential function. This method is extremely efficient and it uses the 2n sequence at its foundation. One can solve the exponential function quicker if one uses the 2n table. A sample of the table will be displayed towards the end of the paper.I have decided to call this method the Ngazi method. Ngazi is a swahili word for ladder or stairs. Ngazi is a two syllable word which is pronounced as (nga-zi) where /n/ and /g/ are pronounced as one syllable.[5] I named this method ngazi or stairs because the results of the first multiplication are used in the second multiplication and the results of the second multiplication are used in the third multiplication and so on. Therefore the first multiplication is linked to the second multiplication and the second multiplication is linked to the third multiplication and this reminds the author of a flight of stairs where one stair leads to the next stair up to the final stair.

Dysphagia has physiological and psychosocial consequences for patients and their caregivers. Although the psychological impact of dysphagia can be devastating, research has privileged the physiological and clinical outcomes of the disorder. Provision of information may enhance patient’s knowledge and ability to cope with the disorder. In the present study, an original booklet and routine information procedures where compared regarding satisfaction with communication, subjective feeling of happiness, and knowledge about dysphagia in patients and caregivers. A convenience sample of 27 patients with neurological dysphagia was allocated to an experimental group (n = 14) who received the booklet, and a control group (n = 13) submitted to the routine information procedures. Evaluation of type and severity of dysphagia was performed in both groups, and participants were surveyed at two moments (T1 and T2) regarding subjective well-being – Subjective Happiness Scale (SHS) and satisfaction with the communication- Satisfaction Survey adapted from Patients Satisfaction with the Interview Assessment Questionnaire (PSIAC). Patient’s knowledge about dysphagia was evaluated at At T2. Statistically significant higher level of SHS and knowledge about dysphagia was found in the experimental group at T2. Conversely, Satisfaction was lower in the control group at T2 compared with T1. This study provides compelling arguments for combining verbal and written information in patients with neurological dysphagia. The use of the booklet made a significant contribution to patients’ knowledge.