On fractional Langevin equation involving two fractional orders in different intervals

In this paper, we study a nonlinear Langevin equation involving two fractional orders  α ∈ (0; 1] and β ∈ (1; 2] with initial conditions. By means of an interesting fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. Some illustrative numerical examples are also discussed.&nbsp

Global age-sex-specific mortality, life expectancy, and population estimates in 204 countries and territories and 811 subnational locations, 1950–2021, and the impact of the COVID-19 pandemic: a comprehensive demographic analysis for the Global Burden of Disease Study 2021

Background: Estimates of demographic metrics are crucial to assess levels and trends of population health outcomes. The profound impact of the COVID-19 pandemic on populations worldwide has underscored the need for timely estimates to understand this unprecedented event within the context of long-term population health trends. The Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2021 provides new demographic estimates for 204 countries and territories and 811 additional subnational locations from 1950 to 2021, with a particular emphasis on changes in mortality and life expectancy that occurred during the 2020–21 COVID-19 pandemic period. Methods: 22 223 data sources from vital registration, sample registration, surveys, censuses, and other sources were used to estimate mortality, with a subset of these sources used exclusively to estimate excess mortality due to the COVID-19 pandemic. 2026 data sources were used for population estimation. Additional sources were used to estimate migration; the effects of the HIV epidemic; and demographic discontinuities due to conflicts, famines, natural disasters, and pandemics, which are used as inputs for estimating mortality and population. Spatiotemporal Gaussian process regression (ST-GPR) was used to generate under-5 mortality rates, which synthesised 30 763 location-years of vital registration and sample registration data, 1365 surveys and censuses, and 80 other sources. ST-GPR was also used to estimate adult mortality (between ages 15 and 59 years) based on information from 31 642 location-years of vital registration and sample registration data, 355 surveys and censuses, and 24 other sources. Estimates of child and adult mortality rates were then used to generate life tables with a relational model life table system. For countries with large HIV epidemics, life tables were adjusted using independent estimates of HIV-specific mortality generated via an epidemiological analysis of HIV prevalence surveys, antenatal clinic serosurveillance, and other data sources. Excess mortality due to the COVID-19 pandemic in 2020 and 2021 was determined by subtracting observed all-cause mortality (adjusted for late registration and mortality anomalies) from the mortality expected in the absence of the pandemic. Expected mortality was calculated based on historical trends using an ensemble of models. In location-years where all-cause mortality data were unavailable, we estimated excess mortality rates using a regression model with covariates pertaining to the pandemic. Population size was computed using a Bayesian hierarchical cohort component model. Life expectancy was calculated using age-specific mortality rates and standard demographic methods. Uncertainty intervals (UIs) were calculated for every metric using the 25th and 975th ordered values from a 1000-draw posterior distribution. Findings: Global all-cause mortality followed two distinct patterns over the study period: age-standardised mortality rates declined between 1950 and 2019 (a 62·8% [95% UI 60·5–65·1] decline), and increased during the COVID-19 pandemic period (2020–21; 5·1% [0·9–9·6] increase). In contrast with the overall reverse in mortality trends during the pandemic period, child mortality continued to decline, with 4·66 million (3·98–5·50) global deaths in children younger than 5 years in 2021 compared with 5·21 million (4·50–6·01) in 2019. An estimated 131 million (126–137) people died globally from all causes in 2020 and 2021 combined, of which 15·9 million (14·7–17·2) were due to the COVID-19 pandemic (measured by excess mortality, which includes deaths directly due to SARS-CoV-2 infection and those indirectly due to other social, economic, or behavioural changes associated with the pandemic). Excess mortality rates exceeded 150 deaths per 100 000 population during at least one year of the pandemic in 80 countries and territories, whereas 20 nations had a negative excess mortality rate in 2020 or 2021, indicating that all-cause mortality in these countries was lower during the pandemic than expected based on historical trends. Between 1950 and 2021, global life expectancy at birth increased by 22·7 years (20·8–24·8), from 49·0 years (46·7–51·3) to 71·7 years (70·9–72·5). Global life expectancy at birth declined by 1·6 years (1·0–2·2) between 2019 and 2021, reversing historical trends. An increase in life expectancy was only observed in 32 (15·7%) of 204 countries and territories between 2019 and 2021. The global population reached 7·89 billion (7·67–8·13) people in 2021, by which time 56 of 204 countries and territories had peaked and subsequently populations have declined. The largest proportion of population growth between 2020 and 2021 was in sub-Saharan Africa (39·5% [28·4–52·7]) and south Asia (26·3% [9·0–44·7]). From 2000 to 2021, the ratio of the population aged 65 years and older to the population aged younger than 15 years increased in 188 (92·2%) of 204 nations. Interpretation: Global adult mortality rates markedly increased during the COVID-19 pandemic in 2020 and 2021, reversing past decreasing trends, while child mortality rates continued to decline, albeit more slowly than in earlier years. Although COVID-19 had a substantial impact on many demographic indicators during the first 2 years of the pandemic, overall global health progress over the 72 years evaluated has been profound, with considerable improvements in mortality and life expectancy. Additionally, we observed a deceleration of global population growth since 2017, despite steady or increasing growth in lower-income countries, combined with a continued global shift of population age structures towards older ages. These demographic changes will likely present future challenges to health systems, economies, and societies. The comprehensive demographic estimates reported here will enable researchers, policy makers, health practitioners, and other key stakeholders to better understand and address the profound changes that have occurred in the global health landscape following the first 2 years of the COVID-19 pandemic, and longer-term trends beyond the pandemic

Global age-sex-specific mortality, life expectancy, and population estimates in 204 countries and territories and 811 subnational locations, 1950–2021, and the impact of the COVID-19 pandemic: a comprehensive demographic analysis for the Global Burden of Disease Study 2021

Background Estimates of demographic metrics are crucial to assess levels and trends of population health outcomes. The profound impact of the COVID-19 pandemic on populations worldwide has underscored the need for timely estimates to understand this unprecedented event within the context of long-term population health trends. The Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2021 provides new demographic estimates for 204 countries and territories and 811 additional subnational locations from 1950 to 2021, with a particular emphasis on changes in mortality and life expectancy that occurred during the 2020–21 COVID-19 pandemic period. Methods 22 223 data sources from vital registration, sample registration, surveys, censuses, and other sources were used to estimate mortality, with a subset of these sources used exclusively to estimate excess mortality due to the COVID-19 pandemic. 2026 data sources were used for population estimation. Additional sources were used to estimate migration; the effects of the HIV epidemic; and demographic discontinuities due to conflicts, famines, natural disasters, and pandemics, which are used as inputs for estimating mortality and population. Spatiotemporal Gaussian process regression (ST-GPR) was used to generate under-5 mortality rates, which synthesised 30 763 location-years of vital registration and sample registration data, 1365 surveys and censuses, and 80 other sources. ST-GPR was also used to estimate adult mortality (between ages 15 and 59 years) based on information from 31 642 location-years of vital registration and sample registration data, 355 surveys and censuses, and 24 other sources. Estimates of child and adult mortality rates were then used to generate life tables with a relational model life table system. For countries with large HIV epidemics, life tables were adjusted using independent estimates of HIV-specific mortality generated via an epidemiological analysis of HIV prevalence surveys, antenatal clinic serosurveillance, and other data sources. Excess mortality due to the COVID-19 pandemic in 2020 and 2021 was determined by subtracting observed all-cause mortality (adjusted for late registration and mortality anomalies) from the mortality expected in the absence of the pandemic. Expected mortality was calculated based on historical trends using an ensemble of models. In location-years where all-cause mortality data were unavailable, we estimated excess mortality rates using a regression model with covariates pertaining to the pandemic. Population size was computed using a Bayesian hierarchical cohort component model. Life expectancy was calculated using age-specific mortality rates and standard demographic methods. Uncertainty intervals (UIs) were calculated for every metric using the 25th and 975th ordered values from a 1000-draw posterior distribution. Findings Global all-cause mortality followed two distinct patterns over the study period: age-standardised mortality rates declined between 1950 and 2019 (a 62·8% [95% UI 60·5–65·1] decline), and increased during the COVID-19 pandemic period (2020–21; 5·1% [0·9–9·6] increase). In contrast with the overall reverse in mortality trends during the pandemic period, child mortality continued to decline, with 4·66 million (3·98–5·50) global deaths in children younger than 5 years in 2021 compared with 5·21 million (4·50–6·01) in 2019. An estimated 131 million (126–137) people died globally from all causes in 2020 and 2021 combined, of which 15·9 million (14·7–17·2) were due to the COVID-19 pandemic (measured by excess mortality, which includes deaths directly due to SARS-CoV-2 infection and those indirectly due to other social, economic, or behavioural changes associated with the pandemic). Excess mortality rates exceeded 150 deaths per 100 000 population during at least one year of the pandemic in 80 countries and territories, whereas 20 nations had a negative excess mortality rate in 2020 or 2021, indicating that all-cause mortality in these countries was lower during the pandemic than expected based on historical trends. Between 1950 and 2021, global life expectancy at birth increased by 22·7 years (20·8–24·8), from 49·0 years (46·7–51·3) to 71·7 years (70·9–72·5). Global life expectancy at birth declined by 1·6 years (1·0–2·2) between 2019 and 2021, reversing historical trends. An increase in life expectancy was only observed in 32 (15·7%) of 204 countries and territories between 2019 and 2021. The global population reached 7·89 billion (7·67–8·13) people in 2021, by which time 56 of 204 countries and territories had peaked and subsequently populations have declined. The largest proportion of population growth between 2020 and 2021 was in sub-Saharan Africa (39·5% [28·4–52·7]) and south Asia (26·3% [9·0–44·7]). From 2000 to 2021, the ratio of the population aged 65 years and older to the population aged younger than 15 years increased in 188 (92·2%) of 204 nations. Interpretation Global adult mortality rates markedly increased during the COVID-19 pandemic in 2020 and 2021, reversing past decreasing trends, while child mortality rates continued to decline, albeit more slowly than in earlier years. Although COVID-19 had a substantial impact on many demographic indicators during the first 2 years of the pandemic, overall global health progress over the 72 years evaluated has been profound, with considerable improvements in mortality and life expectancy. Additionally, we observed a deceleration of global population growth since 2017, despite steady or increasing growth in lower-income countries, combined with a continued global shift of population age structures towards older ages. These demographic changes will likely present future challenges to health systems, economies, and societies. The comprehensive demographic estimates reported here will enable researchers, policy makers, health practitioners, and other key stakeholders to better understand and address the profound changes that have occurred in the global health landscape following the first 2 years of the COVID-19 pandemic, and longer-term trends beyond the pandemic

On the new Hyers–Ulam–Rassias stability of the generalized cubic set-valued mapping in the incomplete normed spaces

We present a novel generalization of the Hyers–Ulam–Rassias stability definition to study a generalized cubic set-valued mapping in normed spaces. In order to achieve our goals, we have applied a brand new fixed point alternative. Meanwhile, we have obtained a practicable example demonstrating the stability of a cubic mapping that is not defined as stable according to the previously applied methods and procedures

Effects of Irrigation Levels, Plant Densities and Planting Patterns on Water Use Efficiency Corn (KSC700(

In order to evaluate the effects of different levels of irrigation, crop densities and cropping patterns on corn (KSC700 variety) water use efficiency using subsurface drip irrigation system, three field experiments were carried out in 2005 and 2006 in Karaj. Experimental design was split plot design based on randomized complete blocks with three replications. In the first experimental Field, main plots were Three irrigation levels: 50%, 75% and 100% ET and sub plots were three plant densities: 65000, 75000 and 85000 plant per hectare and sub-sub plots were two planting patterns: one and two row plants per bed. Results showed that increasing the levels of irrigation from 50% to 100% of the plant water requirement, has a significant effect on yield and yield components. Results obtained from two years experiment showed that irrigation levels of 50% and 100% ET had the minimum and maximum yield values of 3.65, 12.28 and 3.58, 12.89 ton per hectare in years of 2005 and 2006 respectfully. Calculation on water use efficiency showed that treatments located to 75% and 100% ET groups have maximum water use efficiency compared to 50% ET treatments. This means that corn is a plant which is highly sensitive to deficit irrigation. It is recommended in area where there is no limitation in water resources, application of 100% ET for maintaining crop water requirement is suggested. In area where water resources is limited, it is suggested to maintain only 75% of crop water requirement by using subsurface drip irrigation method for corn production

Evaluation of the Effect of Different Irrigation Levels of Drip Irrigation (Tape) on Yield and Yield Components of Corn

Introduction: One of the serious problems in the further development of maize cultivation is increasing irrigation efficiency. Using conventional irrigation causes a shortage of water resources to increase the acreage of the crop. With regard to the development of maize cultivation, agronomic and executable methods must be studied to reduce water consumption. Using drip irrigation system is most suitable for row crops. Hamedi et al. (2005) compared drip (tape) and surface irrigation systems on yield of maize in different levels of water requirement and indicated that drip irrigation increases the amount of yield to 2015 kg/ha and water use efficiency to 3 time. Kohi et al. (2005) investigated the effects of deficit irrigation use of drip (tape) irrigation on water use efficiency on maize in planting of one and two rows. The results showed that maximum water use efficiency related to crop density, water requirement and planting pattern 85000, 125% and two rows, respectively with 1.46 kg/m3. Jafari and Ashrafi (2011) studied the effects of irrigation levels, plant density and planting pattern in drip irrigation (tape) on corn. The results showed that the amount of irrigation water and crop density on the level of 1% and their interactions and method of planting were significant at the 5 and 10% on water use efficiency, respectively. The yield was measured under different levels of irrigation, crop density and method of planting and the difference was significant on the level of 1%. Lamm et al. (1995) studied water requirement of maize in field with silt loam texture under sub drip irrigation and reported that water use reduced to 75%; but yield of maize remained at maximum amount of 12.5 t/ha. The objective of this study was to evaluate the drip (tape) irrigation method for corn production practices in the Qazvin province in Iran. Materials and Methods: In this study, yield and yield components of corn (SC 704) were investigated under different levels of irrigation water in drip tape systems in one and two rows planting patterns with different plant densities. The experiment was conducted on randomized complete blocks as a split plot (Split block) design with 3 replicates in the Qazvin region. Four levels of irrigation including: 80, 100 and 120 percent of water requirement with drip irrigation (tape) and 100% water requirement with furrow irrigation (control treatment) as main plots and method of planting (one and two rows) with three levels of crop density including: 75000, 90000 and 105000 as subplots were considered. After harvesting, grain yield, number of rows per ear, number of kernels per ear row, number of grains per ear and 1000-kernel weigh were measured. Results and Discussion: The results of simple variance analysis of attributes showed that the method of planting has a significant difference on the level of 5% for grain yield, but on the other the measured attributes did not have any significant effect. The respective effect of planting method and crop density showed a significant difference on the level of 5% for grain yield, number of kernels per ear and the 1000-grain weight, whereas it did not have any significant effect on the other measured attributes. The respective effects of irrigation method, planting method and crop density showed a significant difference on the level of 1% for the attributes of the number of kernels per ear. The planting in one row resulted in significantly higher grain yields than the other planting patterns. In mean comparisons of the interactions between irrigation methods, crop density and planting method, grain yield in drip irrigation at a level of 120% water requirement in the two rows planting pattern and crop density equal to 75000 plants was shown in the lead on the level of 10%. The results showed that the yields of the treatments were only affected by the method of planting and planting of one row lead the planting of two rows. According to means comparison and water use efficiency in each of the treatments and limitation of water resources, one row planting pattern with crop density equal to 90000 under drip irrigation at 80% and 120% (If there is no water restrictions) of water requirement were suitable. Conclusion: According to the table of variance analysis, it can be seen that the effect of irrigation on corn grain yield was not significant. Research results of Sorensen and Butts (2005) and Azari et al. (2007) have also confirmed this subject. The grain yield in one row planting method was superior compared to two rows planting method. The superiority of one ton per hectare was statistically significant and substantial. Grain yields varied from 5360 to 12873 kg/ha among the treatments: in drip irrigation at a level of 120% water requirement in the two rows planting pattern and crop density equal to 75000 plants per hectare was 12873 kg/ha and the lowest yield was found in drip irrigation at a level of 80% water requirement in the two-row planting pattern and crop density equal to 75000 plants per hectare as 5360 kg/ha. With regard to mean comparisons of grain yield under the effects of interaction, and water use efficiency for each treatments, I1R1D2, I3R1D2, I1R1D1 and I2R2D3 treatments seem better than others