Tarnavas established mixed weighted power mean inequality in 1999. A separation of weighted power mean inequslity was derived in this paper. As its applications, some separations of other inequalities were given.
In this paper, we find the greatest value p = log 2/（log Tr - log 2） = 1.53.- and the least value q -- 5/3 - 1.66.. such that the double inequality Mp（a,b） 〈 T（a,b） 〈 Mq（a,b） holds for all a, b 〉 0 with a #...In this paper, we find the greatest value p = log 2/（log Tr - log 2） = 1.53.- and the least value q -- 5/3 - 1.66.. such that the double inequality Mp（a,b） 〈 T（a,b） 〈 Mq（a,b） holds for all a, b 〉 0 with a # b. Here, Mp（a, b） and T（a, b） are the p-th power and Seiffertmeans of two positive numbers a and b, respectively.展开更多
The main purpose of this paper is using estimates for trigonometric sums and properties of congruence to study the computation of one kind of fourth power mean of a generalized three-term exponential sum, and give an ...The main purpose of this paper is using estimates for trigonometric sums and properties of congruence to study the computation of one kind of fourth power mean of a generalized three-term exponential sum, and give an interesting identity for it.展开更多
The computational problem of fourth power mean of generalized three-term exponential sums is studied by using the trigonometric identity and the properties of the reduced residue system. Some explicit formulas for the...The computational problem of fourth power mean of generalized three-term exponential sums is studied by using the trigonometric identity and the properties of the reduced residue system. Some explicit formulas for the fourth power mean of generalized three-term exponential sums under different conditions are given.展开更多
The main purpose of this paper is using the properties of the classical Gauss sum and the analytic methods to study the computational problem of one kind of hybrid power mean involving the character sum of polynomials...The main purpose of this paper is using the properties of the classical Gauss sum and the analytic methods to study the computational problem of one kind of hybrid power mean involving the character sum of polynomials and a sum analogous to Kloosterman sum mod p,an odd prime,and give two sharp asymptotic formulae for them.展开更多
In this research work, we consider the below inequalities: (1.1). The researchers attempt to find an answer as to what are the best possible parameters <i><i>α</i></i>, <i><i&...In this research work, we consider the below inequalities: (1.1). The researchers attempt to find an answer as to what are the best possible parameters <i><i>α</i></i>, <i><i>β</i></i> that (1.1) can be held? The main tool is the optimization of some suitable functions that we seek to find out. Without loss of generality, we have assumed that <i>a</i> > <i>b</i> and let <img src="Edit_26c0f99b-93dd-48ff-acdb-f1c8047744f1.bmp" alt="" /> for 1) and <i>a</i> < <i>b</i>, <img src="Edit_15c32a7a-e9ae-41d3-8f49-c6b9c01c7ece.bmp" alt="" />(<i>t</i> small) for 2) to determine the condition for <i><i>α</i></i> and <i><i>β</i></i> to become <i>f</i>(<i>t</i>) ≤ 0 and <i>g</i>(<i>t</i>) ≥ 0.展开更多
The main purpose of this paper is to study the mean value properties of the character sums over the interval [1,p/8） by using the mean value theorems of the Dirichlet L-functions, and give an interesting mean value f...The main purpose of this paper is to study the mean value properties of the character sums over the interval [1,p/8） by using the mean value theorems of the Dirichlet L-functions, and give an interesting mean value formula for this study.展开更多
The main purpose of this article is to study the calculating problem of the sixth power mean of the two-term exponential sums,and give an interesting calculating formula for it.At the same time,the paper also provides...The main purpose of this article is to study the calculating problem of the sixth power mean of the two-term exponential sums,and give an interesting calculating formula for it.At the same time,the paper also provides a new and effective method for the study of the high order power mean of the exponential sums.展开更多
The main purpose of this paper is to study the asymptotic property of the fourth power mean of the general k-th Kloosterman sums, and give an interesting asymptotic formula.
The main purpose of this paper is to use elementary methods and properties of the classical Gauss sums to study the computational problem of one kind of fourth power mean of the generalized quadratic Gauss sums mod q ...The main purpose of this paper is to use elementary methods and properties of the classical Gauss sums to study the computational problem of one kind of fourth power mean of the generalized quadratic Gauss sums mod q （a positive odd number）, and give an exact computational formula for it.展开更多
The main purpose of this paper is using the analytic method and the properties of trigonometric sums and character sums to study the computational problem of one kind hybrid power mean involving two-term exponential s...The main purpose of this paper is using the analytic method and the properties of trigonometric sums and character sums to study the computational problem of one kind hybrid power mean involving two-term exponential sums and polynomial character sums.Then the authors give some interesting calculating formulae for them.展开更多
Japan, which is increasingly frustrated that it has not been granted a permanent seat on the UN Security Council, has found a new outlet of displeasure against the world body. On March 10, Japan proposed to the UN Gen...Japan, which is increasingly frustrated that it has not been granted a permanent seat on the UN Security Council, has found a new outlet of displeasure against the world body. On March 10, Japan proposed to the UN General Assembly budget committee,展开更多
The main purpose of this paper is using the analytic methods, the solutions of the congruence equation mod p and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the gen...The main purpose of this paper is using the analytic methods, the solutions of the congruence equation mod p and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 3-dimensional Kloostermann sums mod p, and give a sharp asymptotic formula for it.展开更多
With the continuous increase of solar penetration rate, it has brought challenges to the smooth operation of the power grid. Therefore, to make photovoltaic power generation not affect the smooth operation of the grid...With the continuous increase of solar penetration rate, it has brought challenges to the smooth operation of the power grid. Therefore, to make photovoltaic power generation not affect the smooth operation of the grid, accurate photovoltaic power prediction is required. And short-term forecasting is essential for the deployment of daily power generation plans. In this paper, A short-term photovoltaic power generation forecast method based on K-means++, grey relational analysis (GRA) and support vector regression (SVR) (Hybrid Kmeans-GRA-SVR, HKGSVR) was proposed. The historical power data was clustered through the multi-index K-means++ algorithm. And the similar days and the nearest neighbor similar day of the prediction day were selected by the GRA algorithm. Then, similar days and nearest neighbor similar days were used to train SVR to obtain an accurate photovoltaic power prediction model. Under ideal weather, the average values of MAE, RMSE, and R<sup>2</sup> were 0.8101 kW, 0.9608 kW, and 99.66%, respectively. The average computation time was 1.7487 s, which was significantly better than the SVR model. Thus, the demonstrated numerical results verify the effectiveness of the proposed model for short-term PV power prediction.展开更多
In order to ensure that the large-scale application of photovoltaic power generation does not affect the stability of the grid, accurate photovoltaic (PV) power generation forecast is essential. A short-term PV power ...In order to ensure that the large-scale application of photovoltaic power generation does not affect the stability of the grid, accurate photovoltaic (PV) power generation forecast is essential. A short-term PV power generation forecast method using the combination of K-means++, grey relational analysis (GRA) and support vector regression (SVR) based on feature selection (Hybrid Kmeans-GRA-SVR, HKGSVR) was proposed. The historical power data were clustered through the multi-index K-means++ algorithm and divided into ideal and non-ideal weather. The GRA algorithm was used to match the similar day and the nearest neighbor similar day of the prediction day. And selected appropriate input features for different weather types to train the SVR model. Under ideal weather, the average values of MAE, RMSE and R2 were 0.8101, 0.9608 kW and 99.66%, respectively. And this method reduced the average training time by 77.27% compared with the standard SVR model. Under non-ideal weather conditions, the average values of MAE, RMSE and R2 were 1.8337, 2.1379 kW and 98.47%, respectively. And this method reduced the average training time of the standard SVR model by 98.07%. The experimental results show that the prediction accuracy of the proposed model is significantly improved compared to the other five models, which verify the effectiveness of the method.展开更多
For p ∈ R, the generalized logarithmic mean Lp（a, b） and Seiffert＇s mean T（a, b） of two positive real numbers a and b are defined in （1.1） and （1.2） below respectively. In this paper, we find the greatest p ...For p ∈ R, the generalized logarithmic mean Lp（a, b） and Seiffert＇s mean T（a, b） of two positive real numbers a and b are defined in （1.1） and （1.2） below respectively. In this paper, we find the greatest p and least q such that the double-inequality Lp（a, b） 〈 T（a,b） 〈 Lq（a,b） holds for all a,b 〉 0 and a ≠ b.展开更多
基金Project supported by National Natural Science Foundation of China （Grant No. 10271071）
文摘Tarnavas established mixed weighted power mean inequality in 1999. A separation of weighted power mean inequslity was derived in this paper. As its applications, some separations of other inequalities were given.
基金Supported by the National Natural Science Foundation of China(61174076,61374086,11171307)the Natural Science Foundation of Zhejiang Province(LY13A010004)
文摘In this paper, we find the greatest value p = log 2/（log Tr - log 2） = 1.53.- and the least value q -- 5/3 - 1.66.. such that the double inequality Mp（a,b） 〈 T（a,b） 〈 Mq（a,b） holds for all a, b 〉 0 with a # b. Here, Mp（a, b） and T（a, b） are the p-th power and Seiffertmeans of two positive numbers a and b, respectively.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1137129161202437)
文摘The main purpose of this paper is using estimates for trigonometric sums and properties of congruence to study the computation of one kind of fourth power mean of a generalized three-term exponential sum, and give an interesting identity for it.
基金Supported by the National Natural Science Foundation of China(Grant No.11571277)the Science and Technology Program of Shaanxi Province(Grant Nos.2014JM1007+2 种基金2014KJXX-612016GY-0802016GY-077)
文摘The computational problem of fourth power mean of generalized three-term exponential sums is studied by using the trigonometric identity and the properties of the reduced residue system. Some explicit formulas for the fourth power mean of generalized three-term exponential sums under different conditions are given.
基金Supported by NSFC(No.12126357)Natural Science Basic Research Plan in Shaanxi Province of China(No.2023-JC-QN-0058)。
文摘The main purpose of this paper is using the properties of the classical Gauss sum and the analytic methods to study the computational problem of one kind of hybrid power mean involving the character sum of polynomials and a sum analogous to Kloosterman sum mod p,an odd prime,and give two sharp asymptotic formulae for them.
文摘In this research work, we consider the below inequalities: (1.1). The researchers attempt to find an answer as to what are the best possible parameters <i><i>α</i></i>, <i><i>β</i></i> that (1.1) can be held? The main tool is the optimization of some suitable functions that we seek to find out. Without loss of generality, we have assumed that <i>a</i> > <i>b</i> and let <img src="Edit_26c0f99b-93dd-48ff-acdb-f1c8047744f1.bmp" alt="" /> for 1) and <i>a</i> < <i>b</i>, <img src="Edit_15c32a7a-e9ae-41d3-8f49-c6b9c01c7ece.bmp" alt="" />(<i>t</i> small) for 2) to determine the condition for <i><i>α</i></i> and <i><i>β</i></i> to become <i>f</i>(<i>t</i>) ≤ 0 and <i>g</i>(<i>t</i>) ≥ 0.
文摘The main purpose of this paper is to study the mean value properties of the character sums over the interval [1,p/8） by using the mean value theorems of the Dirichlet L-functions, and give an interesting mean value formula for this study.
基金Supported by NSFC(Grant No.11771351)NSBRP(Grant No.2019JM-207)。
文摘The main purpose of this article is to study the calculating problem of the sixth power mean of the two-term exponential sums,and give an interesting calculating formula for it.At the same time,the paper also provides a new and effective method for the study of the high order power mean of the exponential sums.
基金the National Natural Science Foundation of China (No.10271093) the Shanxi Provincial Natural Science Foundation of China.
文摘The main purpose of this paper is to study the asymptotic property of the fourth power mean of the general k-th Kloosterman sums, and give an interesting asymptotic formula.
基金Supported by the NSF of China(Grant No.11771351)
文摘The main purpose of this paper is to use elementary methods and properties of the classical Gauss sums to study the computational problem of one kind of fourth power mean of the generalized quadratic Gauss sums mod q （a positive odd number）, and give an exact computational formula for it.
基金the National Natural Science Foundation of China(No.11771351)the Natural Science Basic Research Plan in Shaanxi Province(No.2018JQ1093)。
文摘The main purpose of this paper is using the analytic method and the properties of trigonometric sums and character sums to study the computational problem of one kind hybrid power mean involving two-term exponential sums and polynomial character sums.Then the authors give some interesting calculating formulae for them.
文摘Japan, which is increasingly frustrated that it has not been granted a permanent seat on the UN Security Council, has found a new outlet of displeasure against the world body. On March 10, Japan proposed to the UN General Assembly budget committee,
文摘The main purpose of this paper is using the analytic methods, the solutions of the congruence equation mod p and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 3-dimensional Kloostermann sums mod p, and give a sharp asymptotic formula for it.
文摘With the continuous increase of solar penetration rate, it has brought challenges to the smooth operation of the power grid. Therefore, to make photovoltaic power generation not affect the smooth operation of the grid, accurate photovoltaic power prediction is required. And short-term forecasting is essential for the deployment of daily power generation plans. In this paper, A short-term photovoltaic power generation forecast method based on K-means++, grey relational analysis (GRA) and support vector regression (SVR) (Hybrid Kmeans-GRA-SVR, HKGSVR) was proposed. The historical power data was clustered through the multi-index K-means++ algorithm. And the similar days and the nearest neighbor similar day of the prediction day were selected by the GRA algorithm. Then, similar days and nearest neighbor similar days were used to train SVR to obtain an accurate photovoltaic power prediction model. Under ideal weather, the average values of MAE, RMSE, and R<sup>2</sup> were 0.8101 kW, 0.9608 kW, and 99.66%, respectively. The average computation time was 1.7487 s, which was significantly better than the SVR model. Thus, the demonstrated numerical results verify the effectiveness of the proposed model for short-term PV power prediction.
文摘In order to ensure that the large-scale application of photovoltaic power generation does not affect the stability of the grid, accurate photovoltaic (PV) power generation forecast is essential. A short-term PV power generation forecast method using the combination of K-means++, grey relational analysis (GRA) and support vector regression (SVR) based on feature selection (Hybrid Kmeans-GRA-SVR, HKGSVR) was proposed. The historical power data were clustered through the multi-index K-means++ algorithm and divided into ideal and non-ideal weather. The GRA algorithm was used to match the similar day and the nearest neighbor similar day of the prediction day. And selected appropriate input features for different weather types to train the SVR model. Under ideal weather, the average values of MAE, RMSE and R2 were 0.8101, 0.9608 kW and 99.66%, respectively. And this method reduced the average training time by 77.27% compared with the standard SVR model. Under non-ideal weather conditions, the average values of MAE, RMSE and R2 were 1.8337, 2.1379 kW and 98.47%, respectively. And this method reduced the average training time of the standard SVR model by 98.07%. The experimental results show that the prediction accuracy of the proposed model is significantly improved compared to the other five models, which verify the effectiveness of the method.
基金supported by the National Natural Science Foundation of China (11071069 and 11171307)Natural Science Foundation of Hunan Province(09JJ6003)Innovation Team Foundation of the Department of Education of Zhejiang Province (T200924)
文摘For p ∈ R, the generalized logarithmic mean Lp（a, b） and Seiffert＇s mean T（a, b） of two positive real numbers a and b are defined in （1.1） and （1.2） below respectively. In this paper, we find the greatest p and least q such that the double-inequality Lp（a, b） 〈 T（a,b） 〈 Lq（a,b） holds for all a,b 〉 0 and a ≠ b.
