Contents Preface Chapter 1 Preliminary 1 1.1 Galois-eld GF(p) 1 1.1.1 Galois-eld GF(p), characteristic number p 1 1.1.2 Algebraic extension-elds of Galois-eld GF(p) 3 1.2 Structures of local-elds 5 1.2.1 De-nitions of local-elds 5 1.2.2 Valued structure of a local-eld Kq 5 1.2.3 Haar measure and Harr integral on a local-eld Kq 7 1.2.4 Important subsets in a local-eld Kq 9 1.2.5 Base for neighborhood system of a local-eld Kq 10 1.2.6 Expressions of elements in Kq, operations 11 1.2.7 Important properties of balls in a local-eld Kp 14 1.2.8 Order structures in a local-eld Kp 15 1.2.9 Relationship between local-eld Kq and Euclidean space R 18 Exercises 19 Chapter 2 Character Group of Local Field Kp 21 2.1 Character groups of locally compact groups 21 2.1.1 Characters of groups 21 2.1.2 Characters and character groups of locally compact groups 21 2.1.3 Pontryagin dual theorem 22 2.1.4 Examples 23 2.2 Character group p of Kp 25 2.2.1 Properties of p 26 2.2.2 Character group of p-series-eld Sp 29 2.2.3 Character group of p-adic-eld Ap 32 2.3 Some formulas in local-elds 35 2.3.1 Haar measures of certain important sets in Kp 35 2.3.2 Integrals for characters in Kp 36 2.3.3 Integrals for some functions in Kp 38 Exercises 40 Chapter 3 Harmonic Analysis on Local Fields 41 3.1 Fourier analysis on a local-eld Kp 41 3.1.1 L1-theory 42 3.1.2 L2-theory 61 3.1.3 Lr-theory 1<r<2 68 3.1.4 Distribution theory on Kp 70 Exercises 83 3.2 Pseudo-dierential operators on local-elds 83 3.2.1 Symbol class S 84 3.2.2 Pseudo-dierential operator on local-elds 87 3.3 p-type derivatives and p-type integrals on local-elds 89 3.3.1 p-type calculus on local-elds 90 3.3.2 Properties of p-type derivatives and p-type integrals 91 3.3.3 p-type derivatives and p-type integrals 93 3.3.4 Background of establishing of p-type calculus 96 3.4 Operator and construction theory of function on local-elds 103 3.4.1 Operators on a local-eld Kp 104 3.4.2 Construction theory of function on a local-eld Kp 107 Exercises 131 Chapter 4 Function Spaces on Local Fields 132 4.1 B-type spaces and F-type spaces on local-elds 132 4.1.1 B-type spaces, F-type spaces 132 4.1.2 Special cases of B-type spaces and F-type spaces 138 4.1.3 H.older type spaces on local-elds 139 4.1.4 Lebesgue type spaces and Sobolev type spaces 144 Exercises 151 4.2 Lipschitz classes on local-elds 151 4.2.1 Lipschitz classes on local-elds 151 4.2.2 Chains of function spaces on Euclidean spaces 157 4.2.3 The cases on local-elds 161 4.2.4 Comparison of Euclidean space analysis with local-eld analysis 162 Exercises 165 4.3 Fractal spaces on local-elds 166 4.3.1 Fractal spaces on Kp 166 4.3.2 Completeness of space K((Kp);h) on Kp 168 4.3.3 Some useful transformations on Kp 174 Exercises 185 Chapter 5 Fractal Analysis on Local Fields 187 5.1 Fractal dimensions on local-elds 187 5.1.1 Hausdor measure and dimension 187 5.1.2 Box dimension 194 5.1.3 Packing measure and dimension 199 Exercises 203 5.2 Analytic expressions of dimensions of sets in local-elds 204 5.2.1 Borel measure and Borel measurable sets 204 5.2.2 distribution dimension 204 5.2.3 Fourier dimension 214 Exercises 216 5.3 p-type calculus and fractal dimensions on local-elds 217 5.3.1 Structures of Kp, 3-adic Cantor type set,3-adic Cantor type function 217 5.3.2 p-type derivative and integral ofon K3 222 5.3.3 p-type derivative and integral of Weierstrass type function on Kp 231 5.3.4 p-type derivative and integral of second Weierstrass type function on Kp 238 Exercises 247 Chapter 6 Fractal PDE on Local Fields 248 6.1 Special examples 248 6.1.1 Classical 2-dimension wave equation with fractal boundary 248 6.1.2 p-type 2-dimension wave equation with fractal boundary 260 6.2 Further study on fractal analysis over local-elds 272 6.2.1 Pseudo-di.erential operator 272 6.2.2 Further problems on fractal analysis over local-elds 288 Exercises 289 Chapter 7 Applications to Medicine Science 290 7.1 Determine the malignancy of liver cancers 290 7.1.1 Terrible havocs of liver cancer, solving idea 290 7.1.2 The main methods in studying of liver cancers 294 7.2 Examples in clinical medicine 297 7.2.1 Take data from the materials of liver cancers of patients 297 7.2.2 Mathematical treatment for data 301 7.2.3 Compute fractal dimensions 306 7.2.4 Induce to obtain mathematical models 310 7.2.5 Other problems in the research of liver cancers 310 References 312 Index 321
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