The necessary optimality conditions with Lagrange multipliers  are studied and derived for a new class that includes the system of Caputo–Katugampola fractional derivatives to the optimal control problems with considering the end time free. The formula for the integral by parts has been proven for the left Caputo–Katugampola fractional derivative that contributes to the finding and deriving the necessary optimality conditions. Also, three special cases are obtained, including the study of the necessary optimality conditions when both the final time  and the final state  are fixed. According to convexity assumptions prove that necessary optimality conditions are sufficient optimality conditions.

   In this work, a novel technique to obtain an accurate solutions to nonlinear form by multi-step combination with Laplace-variational approach (MSLVIM) is introduced. Compared with the  traditional approach for variational it overcome all difficulties and enable to provide us more an accurate solutions with extended of the convergence region as well as covering to larger intervals which providing us a continuous representation of approximate analytic solution and it give more better information of the solution over the whole time interval. This technique is more easier for obtaining the general Lagrange multiplier with reduces the time and calculations. It converges rapidly to exact formula with simply computable terms wit

This study aims to investigate the possible role of circulating microRNA-142-3p (miR-142-3p) in the
development of graves disease (GD) and its association with the antibody directed against thyroid
stimulating hormone receptor (TSHR-Ab) production in patients with GD. Forty patients with positive
TSHR-Ab enrolled in this study were divided ,based on treatment, into (22 untreated (newly diagnosed) and
18 treated patients) and based on family history (30 with positive family history and 10 with negative family
history). In addition to forty healthy subjects with sex and age matching as a control group. The expression
level of circulating miR-142-3p was determined by two steps reverse transcription polymerase c

The alfalfa plant, after harvesting, was washed, dried, and grinded to get fine powder used in water treatment. We used the alfalfa plant with ethanol to make the alcoholic extract characterized by using (GC-Mass, FTIR, and UV) spectroscopy to determine active compounds. Alcoholic extract was used to prepare zinc nanoparticles. We characterized Zinc nanoparticles using (FTIR, UV, SEM, EDX Zeta potential, XRD, AFM). Zinc nanoparticle with Alfalfa extract and alfalfa powder were used in the treatment of water polluted with inorganic elements such as Cr, Mn, Fe, Cu, Cd, Ag by (Batch processing). The batch process with using alfalfa powder gets treated with Pb (51.45%), which is the highest percentage of treatment. Mn (13.18%), which is the

Gross domestic product (GDP) is an important measure of the size of the economy's production. Economists use this term to determine the extent of decline and growth in the economies of countries. It is also used to determine the order of countries and compare them to each other. The research aims at describing and analyzing the GDP during the period from 1980 to 2015 and for the public and private sectors and then forecasting GDP in subsequent years until 2025. To achieve this goal, two methods were used: linear and nonlinear regression. The second method in the time series analysis of the Box-Jenkins models and the using of statistical package (Minitab17), (GRETLW32)) to extract the results, and then comparing the two methods, T

The current study aims at identifying of the differences of life quality level for the academic instructors at Omar Al-Mukhtar University / Libya according to the city variable that the instructor belong to ( Libya, Iraq, Egypt ), and also according to gender Variable (Male - Female),To achieve the aims of study ,the researcher chose a sample consists of (210) instructors ( 170 males and 40 females) from four colleges at Omar Al- Mukhtar University/ Darna . The researcher built the scale of life quality which consists of (42) items in last version . The researcher has conducted the validity and reliability of the scale. and by using Analysis of Variance via (SPSS program). The research reaches the following results : 1-The life quality o

Generally, statistical methods are used in various fields of science, especially in the research field, in which Statistical analysis is carried out by adopting several techniques, according to the nature of the study and its objectives. One of these techniques is building statistical models, which is done through regression models. This technique is considered one of the most important statistical methods for studying the relationship between a dependent variable, also called (the response variable) and the other variables, called covariate variables. This research describes the estimation of the partial linear regression model, as well as the estimation of the “missing at random” values (MAR). Regarding the

The 3-parameter Weibull distribution is used as a model for failure since this distribution is proper when the failure rate somewhat high in starting operation and these rates will be decreased with increasing time .

In practical side a comparison was made between (Shrinkage and Maximum likelihood) Estimators for parameter and reliability function using simulation , we conclude that the Shrinkage estimators for parameters are better than maximum likelihood estimators but the maximum likelihood estimator for reliability function is the better using statistical measures (MAPE)and (MSE) and for different sample sizes.

Note:- n_s : small sample ; n_m=median sample

Mixture experiments are response variables based on the proportions of component for this mixture. In our research we will compare the scheffʼe model with the kronecker model for the mixture experiments, especially when the experimental area is restricted.

     Because of the experience of the mixture of high correlation problem and the problem of multicollinearity between the explanatory variables, which has an effect on the calculation of the Fisher information matrix of the regression model.

     to estimate the parameters of the mixture model, we used the (generalized inverse ) And the Stepwise Regression procedure

Multilevel models are among the most important models widely used in the application and analysis of data that are characterized by the fact that observations take a hierarchical form, In our research we examined the multilevel logistic regression model (intercept random and slope random model) , here the importance of the research highlights that the usual regression models calculate the total variance of the model and its inability to read variance and variations between levels ,however in the case of multi-level regression models, the calculation of  the total variance is inaccurate and therefore these models calculate the variations for each level of the model, Where the research aims to estimate the parameters of this m