For a physics system which exhibits memory,if memory is preserved only at points of random self-similar fractals,we define random memory functions and give the connection between the expectation of flux and the fracti...For a physics system which exhibits memory,if memory is preserved only at points of random self-similar fractals,we define random memory functions and give the connection between the expectation of flux and the fractional integral.In particular,when memory sets degenerate to Cantor type fractals or non-random self-similar fractals our results coincide with that of Nigmatullin and Ren et al.展开更多
In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R^n generated from an initial cube pattern with an(n-m)-dimensional hyperplane V in a fixed direction is discussed...In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R^n generated from an initial cube pattern with an(n-m)-dimensional hyperplane V in a fixed direction is discussed. The authors give a sufficient condition which ensures that the Hausdorff dimensions of the slices of the fractal sets generated by "multirules" take the value in Marstrand's theorem, i.e., the dimension of the self-similar sets minus one. For the self-similar fractals generated with initial cube pattern, this sufficient condition also ensures that the projection measure μVis absolutely continuous with respect to the Lebesgue measure L^m. When μV《 L^m, the connection of the local dimension ofμVand the box dimension of slices is given.展开更多
文摘For a physics system which exhibits memory,if memory is preserved only at points of random self-similar fractals,we define random memory functions and give the connection between the expectation of flux and the fractional integral.In particular,when memory sets degenerate to Cantor type fractals or non-random self-similar fractals our results coincide with that of Nigmatullin and Ren et al.
基金supported by the National Natural Science Foundation of China(Nos.11371329,11471124,11071090,11071224,11101159,11401188)K.C.Wong Magna Fund in Ningbo University,the Natural Science Foundation of Zhejiang Province(Nos.LR13A010001,LY12F02011)the Natural Science Foundation of Guangdong Province(No.S2011040005741)
文摘In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R^n generated from an initial cube pattern with an(n-m)-dimensional hyperplane V in a fixed direction is discussed. The authors give a sufficient condition which ensures that the Hausdorff dimensions of the slices of the fractal sets generated by "multirules" take the value in Marstrand's theorem, i.e., the dimension of the self-similar sets minus one. For the self-similar fractals generated with initial cube pattern, this sufficient condition also ensures that the projection measure μVis absolutely continuous with respect to the Lebesgue measure L^m. When μV《 L^m, the connection of the local dimension ofμVand the box dimension of slices is given.
