| This thesis investigates seveval problems about discrete fractal indices and their intrinsic relationships in Euclidean lattice and Euclidean space, and some problems in local bifurcation problems of one state variable with Z2 - symmetry. The thesis divided into two parts. The part one including the first two chapters investigates discrete fractals .and the second part including the last two chapters investigates the singularity theory with Z2 -symmetry and intrinsic ideals and intrinsic submodules in local bifurcation problems with one state variable.The purpose of chapter 1 is to give a systematic investigation of the definitions and their properties of discrete fractal indices of unbounded subsets both in Euclidean lattice Zd and Euclidean space Rd. Especially, for any A Rd ,(A) and A are defined in section 1. 1 , which are called the Z-set and fat set of A respectively. In section 1. 2, lower and upper mass dimensions of unbounded subset A of Rd ,dimLM (A) and dimUM(A),are defined. We conclude that by Theorem 1. 2. 1,dimLM( (A)) = dimLM((A)) = dimLM(A) = dimLM(A)and In sections 1. 3 and 1. 4, at first several classes of discrete fractal indices, such as discrete Hausdorff dimension and packing dimension etc. , are defined. Then we establish the equivalences or intrinsic relations among these discrete fractal indices from Theorem 1. 3. 1 to Theorem 1. 4. 11 as the following.and (the common values above are denoted as dimH(A) ,dimp(A) , (A) , (A) respectively);(Where dimP( . ) ,dimP( . ),A( . ) ,dimP( . ),A( . ) and ( . ) denote the common values ofrespectively for ) andfor . Especially we haveIn section 1. 5 , some results first developed by M. T. Barlow and S. J. Taylor (see [20]) are generalized here, we obtained the discrete analoques of density and Frostman Lemmas (see [5,18]) by using the concepts "m - cube" and "sm - cube" . Theorem 1. 5. 1 and Theorem 1. 5. 2,as the main results of this section, provided us with an efficient algorithm in the calculations of discrete fractal indices in Zd . As applications, two examples are given in section 1. 6.The purpose of section 1. 7 is to provide us a strict theoretical basis which illustrate the feasibility of the definitions of discrete packing dimension in Zd . Theorem 1. 7. 1 provide us, in fact, an equivalent definition of discrete packing dimension in Zd,i.e. ,Theorem 1. 7. 2 show that there exists some e06 (0,1), such that TP for a (0,d),and TP(m)a (Zd, e)< + for a (d, +00) and e (0, 1). By Theorem 1. 7. 1~1. 7. 2, Theorem 1. 7. 3 and 1. 7. 4 show us thatandfor any given nonempty subsetIn chapter 2, we introduced the idea, self - similarity, from classical fractal theory to Euclidean lattice Zd. We proved that, the discrete mass dimension, the discrete Hausdorff dimension and the discrete packing dimension etc. ,are of linear invariance properties ,i. e. ,for any x Zd and AeZ\{0,-l}.In chapter 3 we generalized some results of singularity theory with Z2 -symmetry by [23~27]. Theorem 3. 2.1 represented that if g(x,A) and h(x,) be Z2- equivalent germs in (Z2) ,thenwhereLet J be the projection from (Z2) to (ItrT(g,Z2)defined by J(f)(x,) then Theorem 3. 3. 1 gives us a formula to calculate the Z2-codimension of local bifurcation problem of one state variable with Z2 -symmetry,where Since the concept "intrinsic ideal" is one of the fundamental tools in both the classification problems for normal forms and the recognition problems for universal unfoldings [see 22~38,etc. ],then it's important to give a convenient and efficientalgorithm of Itr. , the largest intrinsic ideal in F. In section 4.1,a simple illustration show us that, each ideal finite codimension is generated by finite many monomials and finite many polynomials. Thus, the calculation of Itr F can be reduced 10 calculate the largest intrinsic ideal of an ideal generated by monomials. In section 4. 2. Theorem 4. 2. 4 gives a calculation formula as following,and k such thatAs application of Theorem 4. 2. 4, two neces... |
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