In this paper, we study two Diophantine equations of the type p<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> , where p is a prime number. We find that the equation 2<sup>x</...In this paper, we study two Diophantine equations of the type p<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> , where p is a prime number. We find that the equation 2<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> has exactly two solutions (x, y, z) in non-negative integer i.e., {(3, 0, 3),(4, 1, 5)} but 5<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> has no non-negative integer solution.展开更多
In this paper,we will discuss smoothness of weak solutions for the system of second order differential equations eith non-negative characteristies.First of all,we establish boundary,and interior estimates and then we ...In this paper,we will discuss smoothness of weak solutions for the system of second order differential equations eith non-negative characteristies.First of all,we establish boundary,and interior estimates and then we prove that solutions of regularization problem satisfy Lipschitz condition.展开更多
To properly describe and solve complex decision problems, research on theoretical properties and solution of mixed-integer quadratic programs is becoming very important. We establish in this paper different Lipschitz-...To properly describe and solve complex decision problems, research on theoretical properties and solution of mixed-integer quadratic programs is becoming very important. We establish in this paper different Lipschitz-type continuity results about the optimal value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of the linear constraints. The obtained results extend some existing results for continuous quadratic programs, and, more importantly, lay the foundation for further theoretical study and corresponding algorithm analysis on mixed-integer quadratic programs.展开更多
Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|x...Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|xyz, x<y<z, then y>p^(6p-2)/2.展开更多
In this article, we consider a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and one sign- changing function. The existence and multiplic...In this article, we consider a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and one sign- changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.展开更多
In this article,we study the following fractional(p,q)-Laplacian equations involving the critical Sobolev exponent:(Pμ,λ){(−Δ)s 1 p u+(−Δ)s 2 q u=μ|u|q−2 u+λ|u|p−2 u+|u|p∗s 1−2 u,u=0,inΩ,in R N∖Ω,whereΩ⊂R N i...In this article,we study the following fractional(p,q)-Laplacian equations involving the critical Sobolev exponent:(Pμ,λ){(−Δ)s 1 p u+(−Δ)s 2 q u=μ|u|q−2 u+λ|u|p−2 u+|u|p∗s 1−2 u,u=0,inΩ,in R N∖Ω,whereΩ⊂R N is a smooth and bounded domain,λ,μ>0,0<s 2<s 1<1,1<q<p<Ns 1.We establish the existence of a non-negative nontrivial weak solution to(Pμ,λ)by using the Mountain Pass Theorem.The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.展开更多
For any positive integer n, the famous Smarandache power function SP（n） is defined as the smallest positive integer m such that n｜m^m, where m and n have the same prime divisors. The main purpose of this paper is u...For any positive integer n, the famous Smarandache power function SP（n） is defined as the smallest positive integer m such that n｜m^m, where m and n have the same prime divisors. The main purpose of this paper is using the elementary methods to study the positive integer solutions of an equation involving the Smarandache power function SP（n） and obtain some interesting results. At the same time, we give an open problem about the related equation.展开更多
In the recent past maily results have been established on non-negative solu tions to boundry value problems of the form u(0) = 0= u(1) where λ > 0, f(0) > 0 (positone problem). In this paper we consider the imp...In the recent past maily results have been established on non-negative solu tions to boundry value problems of the form u(0) = 0= u(1) where λ > 0, f(0) > 0 (positone problem). In this paper we consider the impact on the non-negative solutions when f(0) <0. We find that we need f(u) to be convex to guarantee uniquenness of positive solutions and f(u) to be appropriately concave for multiple positive solutions.展开更多
车载自组织网络(vehicular ad hoc network,VANET)是构建智能交通系统的基础,其能够有效提高交通效率以及增加道路安全性.但由于其自身无线通信环境的开放性,车载自组织网络中的边缘节点易遭受到各种外部攻击.为了兼顾隐私保护和可追溯...车载自组织网络(vehicular ad hoc network,VANET)是构建智能交通系统的基础,其能够有效提高交通效率以及增加道路安全性.但由于其自身无线通信环境的开放性,车载自组织网络中的边缘节点易遭受到各种外部攻击.为了兼顾隐私保护和可追溯性并确保安全认证,一种有效的方法是采用车联网条件隐私保护认证方案.虽然目前已有各种车联网条件隐私保护认证方案被相继提出,但这些方案的安全性几乎都依赖于传统数论困难假设,难以具备后量子安全特性.因此,针对现有基于格的车联网条件隐私保护认证方案的安全和效率问题,同时为解决密钥托管问题,利用结构化格上的环上带误差学习决策困难假设以及环上小整数解困难假设,并结合拒绝抽样技术,设计了一种车联网中基于格的无抽样条件隐私保护聚合认证方案,并在随机预言机模型下证明了所提方案能够达到适应性安全.此外,通过性能评估展示了该方案在计算开销方面优于基于一般格的聚合认证方案.展开更多
文摘In this paper, we study two Diophantine equations of the type p<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> , where p is a prime number. We find that the equation 2<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> has exactly two solutions (x, y, z) in non-negative integer i.e., {(3, 0, 3),(4, 1, 5)} but 5<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> has no non-negative integer solution.
文摘In this paper,we will discuss smoothness of weak solutions for the system of second order differential equations eith non-negative characteristies.First of all,we establish boundary,and interior estimates and then we prove that solutions of regularization problem satisfy Lipschitz condition.
基金Supported by the National Natural Science Foundation of China(10571141,70971109)the Key Projectof the National Natural Science Foundation of China(70531030)
文摘To properly describe and solve complex decision problems, research on theoretical properties and solution of mixed-integer quadratic programs is becoming very important. We establish in this paper different Lipschitz-type continuity results about the optimal value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of the linear constraints. The obtained results extend some existing results for continuous quadratic programs, and, more importantly, lay the foundation for further theoretical study and corresponding algorithm analysis on mixed-integer quadratic programs.
文摘Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|xyz, x<y<z, then y>p^(6p-2)/2.
文摘In this article, we consider a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and one sign- changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.
基金National Natural Science Foundation of China(11501252 and 11571176)。
文摘In this article,we study the following fractional(p,q)-Laplacian equations involving the critical Sobolev exponent:(Pμ,λ){(−Δ)s 1 p u+(−Δ)s 2 q u=μ|u|q−2 u+λ|u|p−2 u+|u|p∗s 1−2 u,u=0,inΩ,in R N∖Ω,whereΩ⊂R N is a smooth and bounded domain,λ,μ>0,0<s 2<s 1<1,1<q<p<Ns 1.We establish the existence of a non-negative nontrivial weak solution to(Pμ,λ)by using the Mountain Pass Theorem.The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.
基金Supported by the Natural Science Foundation of China（10671155）
文摘For any positive integer n, the famous Smarandache power function SP（n） is defined as the smallest positive integer m such that n｜m^m, where m and n have the same prime divisors. The main purpose of this paper is using the elementary methods to study the positive integer solutions of an equation involving the Smarandache power function SP（n） and obtain some interesting results. At the same time, we give an open problem about the related equation.
文摘In the recent past maily results have been established on non-negative solu tions to boundry value problems of the form u(0) = 0= u(1) where λ > 0, f(0) > 0 (positone problem). In this paper we consider the impact on the non-negative solutions when f(0) <0. We find that we need f(u) to be convex to guarantee uniquenness of positive solutions and f(u) to be appropriately concave for multiple positive solutions.
文摘车载自组织网络(vehicular ad hoc network,VANET)是构建智能交通系统的基础,其能够有效提高交通效率以及增加道路安全性.但由于其自身无线通信环境的开放性,车载自组织网络中的边缘节点易遭受到各种外部攻击.为了兼顾隐私保护和可追溯性并确保安全认证,一种有效的方法是采用车联网条件隐私保护认证方案.虽然目前已有各种车联网条件隐私保护认证方案被相继提出,但这些方案的安全性几乎都依赖于传统数论困难假设,难以具备后量子安全特性.因此,针对现有基于格的车联网条件隐私保护认证方案的安全和效率问题,同时为解决密钥托管问题,利用结构化格上的环上带误差学习决策困难假设以及环上小整数解困难假设,并结合拒绝抽样技术,设计了一种车联网中基于格的无抽样条件隐私保护聚合认证方案,并在随机预言机模型下证明了所提方案能够达到适应性安全.此外,通过性能评估展示了该方案在计算开销方面优于基于一般格的聚合认证方案.
