The spectrum of arbitrary graph of finite order the exponential growth of the resolvent of graph G is one of the most investigated object during the last 50 years. In particular, the resolvent matrix is a matrix with property that all of its eigenvalues are outside the spectra of G. In this paper, we study the exponential growth of the resolvent of graph G. The exponential growth of resolvent energy of graph G was investigated.

According to most opinions of researchers, an increase in the population creates many restrictions in relation to water supply in the cities. Because an increase in the population causes climate change, water consumption in the sewage sector, lack of water infrastructure and rising demand for food production. Therefore, accurate planning is needed to predict population numbers in the future years. Therefore, in this research, a specific mathematical equation is introduced for predicting the Tabriz population in the future and two linear and quadratic equations are introduced for water consumption demand. Also, this study indicated that Tabriz's population growth rate followed on an exponential model from 1956 to 1975. Nevertheless, from 1976, later for various reasons, the population growth pattern turned out to be a logistics function model. In addition, by using this method, it is possible to better plan for the future of this city in terms of water scarcity, water demand and consumption, and water security. The used method can be applied to other cities and regions to predict the population numbers and to assess water scarcity.

Background: Childhood growth factors play an essential role in the process of child development. Providing an accurate measurement for determining the growth rate predictors longitudinally has an advantage over a growth chart that only measures cumulative growth. This study aimed to determine the trends in growth rate and its effective factors among infants using two different methods. Materials and Methods: This longitudinal study was conducted on 181 infants referred to healthcare centers of Zanjan, Iran, from April 2017 to April 2019. The growth rate using 9-time points was estimated for participants via Point Average Weighting (PAW) model and the exponential model. The generalized estimating equation (GEE) method was used to estimate predictors of infant's growth rate. The coincident and parallelism test was used to compare the growth rate between models. Results: The mean growth rate of children in the PAW model and the exponential model in the first month were 298. 2+109. 2 and 299. 4+115. 6 gr/kg per month. The results showed that the two models' growth rate prediction ability is almost identical over time. The trend of growth rate was decreased significantly (AAPC=22. 46% per month, P<0. 0001) with a change-point in month ninth. The estimated growth rates based on two models coincided (P=0. 885), and parallel (P=0. 898) across 24 months. The associations between growth rate with variables mother job (B=9. 4, P=0. 005), breastfeeding (B=-9. 3, P=0. 005), and multi-fetal pregnancy (B=-18. 9, P=0. 005) were significant. Conclusion: There was no difference between the two models when pediatricians apply them in office or other clinical settings. The multi-fetal pregnancy, job of mother, and exclusion breastfeeding were the most important predictor of growth rate, especially at the first nine months of age.

We introduce a new wrapped exponential distribution named transmuted wrapped exponential (TWE) distribution, for the modeling of circular datasets by using the Transmutation Rank-Map method. This method is employed for the first time for a wrapped distribution with this study. The introduced distribution is more flexible than traditional wrapped exponential distribution. The paper provides the explicit form of important distributional properties of the introduced distribution such as expectation, median, moments, characteristic function, quantile function, hazard rate function and stress-strength reliability. Ré nyi and Shannon entropies are also obtained. The statistical inference problem for the TWE distribution is investigated using maximum likelihood, least squares and weighted least squares and comparative numerical study results are presented. Furthermore, we present a real dataset analysis.

Welch & Peers (1963) used a root-information prior to obtain posterior probabilities for a scalar parameter exponential model and showed that these Bayes probabilities had the confidence property to second order asymptotically. An important undercurrent of this indicates that the constant information reparameterization provides location model structure, for which the confidence property was and is well known. This paper examines the role of the scalar-parameter exponential model for obtaining approximate probabilities and approximate confidence levels, and then addresses the extension for the vector-parameter exponential model.