For the complex nonlinear elliptic partial differential equations,it is not easy to prove the Pogorelov type estimates.In this paper,we study a class of complex Hessian quotient type equations in convex cone Γ_(k),wh...For the complex nonlinear elliptic partial differential equations,it is not easy to prove the Pogorelov type estimates.In this paper,we study a class of complex Hessian quotient type equations in convex cone Γ_(k),which is of mixed Hessian type.For the Γ_(k)-admissible solutions,we establish the corresponding Pogorelov estimates with 0≤l<k<n and Pogorelov type estimates with l+1<k=n and l≥0.These extend the previous related results on the real Hessian quotient type equations.展开更多
Quick and accurate determination of the optimal synchrophase angle is crucial for synchrophasing control of multi-propeller aircraft with low noise.This paper proposes a novel noise prediction and optimization strateg...Quick and accurate determination of the optimal synchrophase angle is crucial for synchrophasing control of multi-propeller aircraft with low noise.This paper proposes a novel noise prediction and optimization strategy,developing a continuous and accurate noise prediction model and obtaining its minimum by solving the Hessian matrix and Fourier-Frobenius matrix.Firstly,a novel propeller noise prediction method uses acoustic simulation pressure signals and improved propeller signatures theory to accurately estimate noise for all synchrophase angles and receiving points.Secondly,a novel optimization approach is proposed to solve the analytical solution of the minimum propeller noise:(A)A noise objective function is established,and use its first derivatives’zeros and Hessian matrix to determine the function minimum.(B)A novel Euler formula transform method is proposed to convert trigonometric polynomials into algebraic polynomials,changing the zeros of the former into those of the latter.(C)Utilize the Fourier-Frobenius matrix method to solve the zeros of algebraic polynomials.To assess the computation time and accuracy,a turboprop aircraft with two six-bladed propellers was analyzed using the computational fluid dynamics and acoustic analogy method,providing acoustic pressure signals at 20 receivers for noise prediction and optimization.The Durand-Kerner and Fourier-Frobenius matrix methods were compared.Results demonstrate that improved propeller signatures theory is more accurate,and the Hessian matrix+Fourier-Frobenius matrix method is faster and more precise than the Hessian matrix+Durand-Kerner method.展开更多
本文主要对Hessian度量诱导的截面曲率展开分析。首先阐述Hessian度量和与之相关的Christoffor符号和曲率张量公式。接着介绍当定义域U为锥时,齐次函数的相关概念,以及Clebsch covariant S(f)和截面曲率的联系。利用R3中开子集U和超平面...本文主要对Hessian度量诱导的截面曲率展开分析。首先阐述Hessian度量和与之相关的Christoffor符号和曲率张量公式。接着介绍当定义域U为锥时,齐次函数的相关概念,以及Clebsch covariant S(f)和截面曲率的联系。利用R3中开子集U和超平面M = {f = 1}相切2-平面上一点处的截面曲率可由S(f)和Hessian行列式H(f)表示,其中f为R3上的齐次多项式,得到不变量S(f)为零与Witten-Dijkgraaf-Verlinde-Verlinde (WDVV)方程等价的条件,即f可以表示为两种特殊的形式。This paper primarily analyzes the sectional curvature induced by the Hessian metric. It begins by detailing the Hessian metric and its associated Christoffel symbols and curvature tensor formulas. It then introduces the concept of homogeneous functions when the domain U is a cone, as well as the relationship between the Clebsch covariant S(f) and the sectional curvature. Using an open subset U in R3 and a hypersurface M = {f = 1}, the sectional curvature at a point on a 2-plane tangent to M can be expressed in terms of S(f) and the Hessian matrix H(f), where f is a homogeneous polynomial on R3. The condition for the invariant S(f) to be zero is equivalent to the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations, which implies that f can be represented in two specific forms.展开更多
In this paper, we derive the a priori estimates for a class of more general (k, l)-Hessian quotient type equations involving u and Du on the right hand function. As an application we prove the Liouville theorem depend...In this paper, we derive the a priori estimates for a class of more general (k, l)-Hessian quotient type equations involving u and Du on the right hand function. As an application we prove the Liouville theorem depending on Pogorelov type estimates. On the other hand, we obtain the existence and uniqueness of the k-admissible solution for these general equations with the Neumann boundary condition, based on some growth conditions for the right hand function.展开更多
In this article,we are concerned with the C^(2)estimates for the k-convex solutions of a class of degenerate k-Hessian equations on closed Hermitian manifolds,whose function in the right-hand side is relevant to the u...In this article,we are concerned with the C^(2)estimates for the k-convex solutions of a class of degenerate k-Hessian equations on closed Hermitian manifolds,whose function in the right-hand side is relevant to the unknown function and its gradient.We will get C^(0)estimate by promoting others′results,and get the“HMW estimate”of this equation such that the conditions of using blow-up analysis are satisfied,and the gradient estimate and second-order estimate will be obtained.Such an estimate will be helpful to study the existence for the solution of the equation.展开更多
文摘For the complex nonlinear elliptic partial differential equations,it is not easy to prove the Pogorelov type estimates.In this paper,we study a class of complex Hessian quotient type equations in convex cone Γ_(k),which is of mixed Hessian type.For the Γ_(k)-admissible solutions,we establish the corresponding Pogorelov estimates with 0≤l<k<n and Pogorelov type estimates with l+1<k=n and l≥0.These extend the previous related results on the real Hessian quotient type equations.
基金supported by the National Natural Science Foundation of China(Nos.51576097,51976089)the Funding for Outstanding Doctoral Dissertation in Nanjing University of Aeronautics and Astronautics,China(No.BCXJ24-05)the Aeronautical Science Foundation of China(No.2023L060052001).
文摘Quick and accurate determination of the optimal synchrophase angle is crucial for synchrophasing control of multi-propeller aircraft with low noise.This paper proposes a novel noise prediction and optimization strategy,developing a continuous and accurate noise prediction model and obtaining its minimum by solving the Hessian matrix and Fourier-Frobenius matrix.Firstly,a novel propeller noise prediction method uses acoustic simulation pressure signals and improved propeller signatures theory to accurately estimate noise for all synchrophase angles and receiving points.Secondly,a novel optimization approach is proposed to solve the analytical solution of the minimum propeller noise:(A)A noise objective function is established,and use its first derivatives’zeros and Hessian matrix to determine the function minimum.(B)A novel Euler formula transform method is proposed to convert trigonometric polynomials into algebraic polynomials,changing the zeros of the former into those of the latter.(C)Utilize the Fourier-Frobenius matrix method to solve the zeros of algebraic polynomials.To assess the computation time and accuracy,a turboprop aircraft with two six-bladed propellers was analyzed using the computational fluid dynamics and acoustic analogy method,providing acoustic pressure signals at 20 receivers for noise prediction and optimization.The Durand-Kerner and Fourier-Frobenius matrix methods were compared.Results demonstrate that improved propeller signatures theory is more accurate,and the Hessian matrix+Fourier-Frobenius matrix method is faster and more precise than the Hessian matrix+Durand-Kerner method.
文摘本文主要对Hessian度量诱导的截面曲率展开分析。首先阐述Hessian度量和与之相关的Christoffor符号和曲率张量公式。接着介绍当定义域U为锥时,齐次函数的相关概念,以及Clebsch covariant S(f)和截面曲率的联系。利用R3中开子集U和超平面M = {f = 1}相切2-平面上一点处的截面曲率可由S(f)和Hessian行列式H(f)表示,其中f为R3上的齐次多项式,得到不变量S(f)为零与Witten-Dijkgraaf-Verlinde-Verlinde (WDVV)方程等价的条件,即f可以表示为两种特殊的形式。This paper primarily analyzes the sectional curvature induced by the Hessian metric. It begins by detailing the Hessian metric and its associated Christoffel symbols and curvature tensor formulas. It then introduces the concept of homogeneous functions when the domain U is a cone, as well as the relationship between the Clebsch covariant S(f) and the sectional curvature. Using an open subset U in R3 and a hypersurface M = {f = 1}, the sectional curvature at a point on a 2-plane tangent to M can be expressed in terms of S(f) and the Hessian matrix H(f), where f is a homogeneous polynomial on R3. The condition for the invariant S(f) to be zero is equivalent to the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations, which implies that f can be represented in two specific forms.
基金supported by the National Natural Science Foundation of China(No.11971157).
文摘In this paper, we derive the a priori estimates for a class of more general (k, l)-Hessian quotient type equations involving u and Du on the right hand function. As an application we prove the Liouville theorem depending on Pogorelov type estimates. On the other hand, we obtain the existence and uniqueness of the k-admissible solution for these general equations with the Neumann boundary condition, based on some growth conditions for the right hand function.
文摘In this article,we are concerned with the C^(2)estimates for the k-convex solutions of a class of degenerate k-Hessian equations on closed Hermitian manifolds,whose function in the right-hand side is relevant to the unknown function and its gradient.We will get C^(0)estimate by promoting others′results,and get the“HMW estimate”of this equation such that the conditions of using blow-up analysis are satisfied,and the gradient estimate and second-order estimate will be obtained.Such an estimate will be helpful to study the existence for the solution of the equation.
