This paper studied the invariance of the Cauchy mean with respect to the arithmetic mean when the denominator functions satisfy certain conditions. The partial derivatives of Cauchy’s mean on the diagonal are obtaine...This paper studied the invariance of the Cauchy mean with respect to the arithmetic mean when the denominator functions satisfy certain conditions. The partial derivatives of Cauchy’s mean on the diagonal are obtained by using the method of Wronskian determinant in the process of solving. Then the invariant equation is solved by using the obtained partial derivatives. Finally, the solutions of invariant equations when the denominator functions satisfy the same simple harmonic oscillator equation or the denominator functions are power functions that have been obtained.展开更多
It is impossible,mathematically, to use a time series which is regarded as a set of observational facts of a dynamicsystem to reconstruct the particular system.Physically, however, with a few assumptions put, a dynami...It is impossible,mathematically, to use a time series which is regarded as a set of observational facts of a dynamicsystem to reconstruct the particular system.Physically, however, with a few assumptions put, a dynamic system canbe rebuilt approximately by means of observational facts.This is the goal of the so called invariant quantity method(IQM),whose research and experiment are of potential significance to atmospheric sciences.展开更多
In this article, we study the (2+1)-extension of Burgers equation and the KPequation. At first, based on a known Baecklund transformation and corresponding Lax pair, aninvariance which depends on two arbitrary functio...In this article, we study the (2+1)-extension of Burgers equation and the KPequation. At first, based on a known Baecklund transformation and corresponding Lax pair, aninvariance which depends on two arbitrary functions for (2+1)-extension of Burgers equation isworked out. Given a known solution and using the invariance, we can find solutions of the(2+1)-extension of Burgers equation repeatedly. Secondly, we put forward an invariance of Burgersequation which cannot be directly obtained by constraining the invariance of the (2+1)-extension ofBurgers equation. Furthermore, we reveal that the invariance for finding the solutions of Burgersequation can help us find the solutions of KP equation. At last, based on the invariance of Burgersequation, the corresponding recursion formulae for finding solutions of KP equation are digged out.As the application of our theory, some examples have been put forward in this article and somesolutions of the (2+1)-extension of Burgers equation, Burgers equation and KP equation are obtained.展开更多
We construct radial and non-radial singular solutions u∈C^(4)(B∖{0})with a non-removable singular point x=0 for the conformal Q-curvature equation■where B={x∈ℝ^(4):∣x∣<1}.More precisely,we can construct two di...We construct radial and non-radial singular solutions u∈C^(4)(B∖{0})with a non-removable singular point x=0 for the conformal Q-curvature equation■where B={x∈ℝ^(4):∣x∣<1}.More precisely,we can construct two different types of singular solutions u∈C^(4)(B∖{0})of the equation,satisfying■for some D_(0)≥0 and any D>D_(0)≥0,and satisfying■Moreover,detailed asymptotic expansions near x=0 of these radial and non-radial singular solutions can be established.As an application,we can also obtain the existence of two types of solutions■to the problem■satisfying|x|^(−2)u(x)→−D<0 uniformly as|x|→∞for some D_(0)≥0 and any D>D_(0),and satisfying u(x)=o(|x|^(2))uniformly as|x|→∞.展开更多
Using every realization of the Virasoro-type symmetry algebra , we can obtain various high-dimensional integrable models under the meaning that they possess infinitely many symmetries. By means of a concrete realizati...Using every realization of the Virasoro-type symmetry algebra , we can obtain various high-dimensional integrable models under the meaning that they possess infinitely many symmetries. By means of a concrete realization, many -dimensional equations which possess Kac–Moody–Virasoro-type infinite dimensional symmetry algebras are obtained.展开更多
A function which is reflexive is called by pre-mean, a more generalized definition of a mean. In this paper, we define a new pre-mean and study its properties, and then using the given invariant curve we consider the ...A function which is reflexive is called by pre-mean, a more generalized definition of a mean. In this paper, we define a new pre-mean and study its properties, and then using the given invariant curve we consider the problem of convergence of Gauss iteration of a kind of pre-mean type mappings generated by the exponential and logarithmic functions.展开更多
Under some conditions on the functions and defined on I, the weighted Bajraktarević?mean is given by where . In this paper, we study the invariance of the weighted Bajraktarević?mean with respect to ...Under some conditions on the functions and defined on I, the weighted Bajraktarević?mean is given by where . In this paper, we study the invariance of the weighted Bajraktarević?mean with respect to Beckenbach-Gini means.展开更多
文摘This paper studied the invariance of the Cauchy mean with respect to the arithmetic mean when the denominator functions satisfy certain conditions. The partial derivatives of Cauchy’s mean on the diagonal are obtained by using the method of Wronskian determinant in the process of solving. Then the invariant equation is solved by using the obtained partial derivatives. Finally, the solutions of invariant equations when the denominator functions satisfy the same simple harmonic oscillator equation or the denominator functions are power functions that have been obtained.
文摘It is impossible,mathematically, to use a time series which is regarded as a set of observational facts of a dynamicsystem to reconstruct the particular system.Physically, however, with a few assumptions put, a dynamic system canbe rebuilt approximately by means of observational facts.This is the goal of the so called invariant quantity method(IQM),whose research and experiment are of potential significance to atmospheric sciences.
文摘In this article, we study the (2+1)-extension of Burgers equation and the KPequation. At first, based on a known Baecklund transformation and corresponding Lax pair, aninvariance which depends on two arbitrary functions for (2+1)-extension of Burgers equation isworked out. Given a known solution and using the invariance, we can find solutions of the(2+1)-extension of Burgers equation repeatedly. Secondly, we put forward an invariance of Burgersequation which cannot be directly obtained by constraining the invariance of the (2+1)-extension ofBurgers equation. Furthermore, we reveal that the invariance for finding the solutions of Burgersequation can help us find the solutions of KP equation. At last, based on the invariance of Burgersequation, the corresponding recursion formulae for finding solutions of KP equation are digged out.As the application of our theory, some examples have been put forward in this article and somesolutions of the (2+1)-extension of Burgers equation, Burgers equation and KP equation are obtained.
基金supported by the National Natural Science Foundation of China(Grant No.12471104)supported by the Natural Science Foundation of Henan(Grant No.252300421051)+2 种基金the National Natural Science Foundation of China(Grant Nos.12371111,12471104)the Open Research Fund of Key Laboratory of Nonlinear Analysis&Applications(Central China Normal University),Ministry of Education,P.R.China(Grant No.NAA2024ORG004)supported by the National Natural Science Foundation of China(Grant No.12401244)。
文摘We construct radial and non-radial singular solutions u∈C^(4)(B∖{0})with a non-removable singular point x=0 for the conformal Q-curvature equation■where B={x∈ℝ^(4):∣x∣<1}.More precisely,we can construct two different types of singular solutions u∈C^(4)(B∖{0})of the equation,satisfying■for some D_(0)≥0 and any D>D_(0)≥0,and satisfying■Moreover,detailed asymptotic expansions near x=0 of these radial and non-radial singular solutions can be established.As an application,we can also obtain the existence of two types of solutions■to the problem■satisfying|x|^(−2)u(x)→−D<0 uniformly as|x|→∞for some D_(0)≥0 and any D>D_(0),and satisfying u(x)=o(|x|^(2))uniformly as|x|→∞.
文摘Using every realization of the Virasoro-type symmetry algebra , we can obtain various high-dimensional integrable models under the meaning that they possess infinitely many symmetries. By means of a concrete realization, many -dimensional equations which possess Kac–Moody–Virasoro-type infinite dimensional symmetry algebras are obtained.
文摘A function which is reflexive is called by pre-mean, a more generalized definition of a mean. In this paper, we define a new pre-mean and study its properties, and then using the given invariant curve we consider the problem of convergence of Gauss iteration of a kind of pre-mean type mappings generated by the exponential and logarithmic functions.
基金Funded by Longshan academic talent research supporting program of SWUST(17LZXY12)Doctoral fund of SWUST(18zx7166,15zx7142).
文摘Under some conditions on the functions and defined on I, the weighted Bajraktarević?mean is given by where . In this paper, we study the invariance of the weighted Bajraktarević?mean with respect to Beckenbach-Gini means.
