It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff di...It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.展开更多
The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In ...The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.展开更多
The Nagumo equation ut = △u+ bu(u-a)(1-u), t＞0is investigated with initial data and zero Neumann boundary conditions on post-critically finite (p.c.f.) self-similar fractals that have regular harmonic structures and...The Nagumo equation ut = △u+ bu(u-a)(1-u), t＞0is investigated with initial data and zero Neumann boundary conditions on post-critically finite (p.c.f.) self-similar fractals that have regular harmonic structures and satisfy the separation condition. Such a nonlinear diffusion equation has no travelling wave solutions because of the 'pathological' property of the fractal. However, it is shown that a global Holder continuous solution in spatial variables exists on the fractal considered. The Sobolev-type inequality plays a crucial role, which holds on such a class of p.c.f self-similar fractals. The heat kernel has an eigenfunction expansion and is well-defined due to a Weyl's formula. The large time asymptotic behavior of the solution is discussed, and the solution tends exponentially to the equilibrium state of the Nagumo equation as time tends to infinity if b is small.展开更多
This paper defines the upper capacity densities of the subsets of R￣n, gets uniform lower bound of the upper capacity densities for -almost all points of the Hausdorff s-sets or the analytic sets with Hausdorff dimen...This paper defines the upper capacity densities of the subsets of R￣n, gets uniform lower bound of the upper capacity densities for -almost all points of the Hausdorff s-sets or the analytic sets with Hausdorff dimension s in R￣n which improves the results of Wen Zhiying and Zhang Yiping's paper in [1].展开更多
[1] has proved that the dissipative Zakharov system has an ε2-weak compact attractor. In this paper, we further show that the dissipative Langmuir waves in plasmas admit an inertial fractal set of (ε2,ε1)-type. We ...[1] has proved that the dissipative Zakharov system has an ε2-weak compact attractor. In this paper, we further show that the dissipative Langmuir waves in plasmas admit an inertial fractal set of (ε2,ε1)-type. We also make the estimates on its fractal dimension and exponential attraction.展开更多
Based on the concepts of fractal super fibers, the （3, 9＋2）-circle and （9＋2, 3）-circle binary fractal sets are abstracted form such prototypes as wool fibers and human hairs, with the （3）-circle and the （9＋2...Based on the concepts of fractal super fibers, the （3, 9＋2）-circle and （9＋2, 3）-circle binary fractal sets are abstracted form such prototypes as wool fibers and human hairs, with the （3）-circle and the （9＋2）-circle fractal sets as subsets. As far as the （9＋2） topological patterns are concerned, the following propositions are proved： The （9＋2） topological patterns accurately exist, but are not unique. Their total number is 9. Among them, only two are allotropes. In other words, among the nine topological patterns, only two are independent （or fundamental）. Besides, we demonstrate that the （3, 9＋2）-circle and （9＋2, 3）-circle fractal sets are golden ones with symmetry breaking.展开更多
In this paper we consider a class of equations for the flow and magnetic field within the earth, for initial-boundary value problem, we prove existence of inertial sets and it's fractal dimension has been given, t...In this paper we consider a class of equations for the flow and magnetic field within the earth, for initial-boundary value problem, we prove existence of inertial sets and it's fractal dimension has been given, the squeezing rate to inertial sets from trajectory of absorbing set has been estimated.展开更多
In this paper, we study the long-time behavior of a class of generalized nonlinear Kichhoff equation under the condition of n dimension. Firstly, the Lipschitz property and squeezing property of the nonlinear semigrou...In this paper, we study the long-time behavior of a class of generalized nonlinear Kichhoff equation under the condition of n dimension. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup related to the initial-boundary value problem are proved, and then the existence of its exponential attractor is obtained. By extending the space <em>E</em><sub>0</sub> to <em>E<sub>k</sub></em>, a family of the exponential attractors of the initial-boundary value problem is obtained. In the second part, we consider the long-time behavior for a system of generalized Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectrum interval condition.展开更多
In the paper ,we study longtime dynamic behavior of dissipative soliton equation existence of attractor ,geometrical structure of attractor dynamic behavior under the parametric perturbation of dissipative soliton equ...In the paper ,we study longtime dynamic behavior of dissipative soliton equation existence of attractor ,geometrical structure of attractor dynamic behavior under the parametric perturbation of dissipative soliton equation.estimate of fractal dimension of attractor.展开更多
Cutsets of series form an important class of fractal sets. In this paper, the author obtains the Hausdoff dimension of outset of complex valued Rademacher serise.
文摘It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.
文摘The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.
文摘The Nagumo equation ut = △u+ bu(u-a)(1-u), t＞0is investigated with initial data and zero Neumann boundary conditions on post-critically finite (p.c.f.) self-similar fractals that have regular harmonic structures and satisfy the separation condition. Such a nonlinear diffusion equation has no travelling wave solutions because of the 'pathological' property of the fractal. However, it is shown that a global Holder continuous solution in spatial variables exists on the fractal considered. The Sobolev-type inequality plays a crucial role, which holds on such a class of p.c.f self-similar fractals. The heat kernel has an eigenfunction expansion and is well-defined due to a Weyl's formula. The large time asymptotic behavior of the solution is discussed, and the solution tends exponentially to the equilibrium state of the Nagumo equation as time tends to infinity if b is small.
文摘This paper defines the upper capacity densities of the subsets of R￣n, gets uniform lower bound of the upper capacity densities for -almost all points of the Hausdorff s-sets or the analytic sets with Hausdorff dimension s in R￣n which improves the results of Wen Zhiying and Zhang Yiping's paper in [1].
文摘[1] has proved that the dissipative Zakharov system has an ε2-weak compact attractor. In this paper, we further show that the dissipative Langmuir waves in plasmas admit an inertial fractal set of (ε2,ε1)-type. We also make the estimates on its fractal dimension and exponential attraction.
基金supported by the National Natural Science Foundation of China (Nos. 10572076 and10872114)the Natural Science Foundation of Jiangsu Province (No. BK2008370)
文摘Based on the concepts of fractal super fibers, the （3, 9＋2）-circle and （9＋2, 3）-circle binary fractal sets are abstracted form such prototypes as wool fibers and human hairs, with the （3）-circle and the （9＋2）-circle fractal sets as subsets. As far as the （9＋2） topological patterns are concerned, the following propositions are proved： The （9＋2） topological patterns accurately exist, but are not unique. Their total number is 9. Among them, only two are allotropes. In other words, among the nine topological patterns, only two are independent （or fundamental）. Besides, we demonstrate that the （3, 9＋2）-circle and （9＋2, 3）-circle fractal sets are golden ones with symmetry breaking.
文摘In this paper we consider a class of equations for the flow and magnetic field within the earth, for initial-boundary value problem, we prove existence of inertial sets and it's fractal dimension has been given, the squeezing rate to inertial sets from trajectory of absorbing set has been estimated.
文摘In this paper, we study the long-time behavior of a class of generalized nonlinear Kichhoff equation under the condition of n dimension. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup related to the initial-boundary value problem are proved, and then the existence of its exponential attractor is obtained. By extending the space <em>E</em><sub>0</sub> to <em>E<sub>k</sub></em>, a family of the exponential attractors of the initial-boundary value problem is obtained. In the second part, we consider the long-time behavior for a system of generalized Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectrum interval condition.
文摘In the paper ,we study longtime dynamic behavior of dissipative soliton equation existence of attractor ,geometrical structure of attractor dynamic behavior under the parametric perturbation of dissipative soliton equation.estimate of fractal dimension of attractor.
基金the National Natural Science Foundation of China !19771031
文摘Cutsets of series form an important class of fractal sets. In this paper, the author obtains the Hausdoff dimension of outset of complex valued Rademacher serise.
