Preface |
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1 | (20) |
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1 | (4) |
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1.1.1 Galois field GF(p), characteristic number p |
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1 | (2) |
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1.1.2 Algebraic extension fields of Galois field GF(p) |
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3 | (2) |
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1.2 Structures of local fields |
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5 | (16) |
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1.2.1 Definitions of local fields |
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5 | (1) |
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1.2.2 Valued structure of a local field Kq |
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6 | (1) |
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1.2.3 Haar measure and Haar integral on a local field Kq |
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7 | (2) |
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1.2.4 Important subsets in a local field Kq |
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9 | (1) |
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1.2.5 Base for neighborhood system of a local field Kq |
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10 | (1) |
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1.2.6 Expressions of elements in Kq and operations |
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11 | (3) |
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1.2.7 Important properties of balls in a local field Kp |
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14 | (1) |
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1.2.8 Order structure in a local field Kv |
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15 | (2) |
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1.2.9 Relationship between local field Kp and Euclidean space R |
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17 | (2) |
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19 | (2) |
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2 Character Group Tp of Local Field Kp |
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21 | (20) |
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2.1 Character groups of locally compact groups |
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21 | (4) |
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2.1.1 Characters of groups |
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21 | (1) |
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2.1.2 Characters and character groups of locally compact groups |
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22 | (1) |
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2.1.3 Pontryagin dual theorem |
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23 | (1) |
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23 | (2) |
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2.2 Character group Γp of Kp |
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25 | (10) |
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2.2.1 Properties of χ ε Γp and Γp |
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26 | (3) |
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2.2.2 Character group of p-series field Sp |
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29 | (3) |
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2.2.3 Character groups of p-adic field Ap |
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32 | (3) |
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2.3 Some formulas in local fields |
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35 | (6) |
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2.3.1 Haar measures of certain important sets in Kp |
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35 | (1) |
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2.3.2 Integrals for characters in Kp |
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36 | (2) |
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2.3.3 Integrals for some functions in Kp |
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38 | (1) |
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39 | (2) |
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3 Harmonic Analysis on Local Fields |
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41 | (88) |
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3.1 Fourier analysis on a local field Kp |
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41 | (41) |
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42 | (18) |
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60 | (7) |
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3.1.3 Lr-Theory, 1 < r < 2 |
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67 | (2) |
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3.1.4 Distribution theory on Kp |
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69 | (12) |
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81 | (1) |
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3.2 Pseudo-differential operators on local fields |
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82 | (6) |
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3.2.1 Symbol class Sαps(Kp) = Sαps (Kp × Γp) |
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82 | (3) |
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3.2.2 Pseudo-differential operator Tσ on local fields |
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85 | (3) |
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3.3 p-type derivatives and p-type integrals on local fields |
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88 | (14) |
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3.3.1 p-type calculus on local fields |
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88 | (1) |
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3.3.2 Properties of p-type derivatives and p-type integrals of φ ε S(KP) |
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89 | (3) |
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3.3.3 p-type derivatives and p-type integrals of T ε S*(Kp) |
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92 | (2) |
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3.3.4 Background of establishing for p-type calculus |
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94 | (8) |
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3.4 Operator and construction theory of function on Local fields |
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102 | (27) |
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3.4.1 Operators on a local field Kp |
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102 | (3) |
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3.4.2 Construction theory of function on a local field Kp |
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105 | (23) |
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128 | (1) |
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4 Function Spaces on Local Fields |
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129 | (54) |
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4.1 B-type spaces and F-type spaces on local fields |
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129 | (19) |
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4.1.1 B-type spaces, F-type spaces |
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129 | (6) |
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4.1.2 Special cases of B-type spaces and F-type spaces |
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135 | (1) |
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4.1.3 Holder type spaces on local fields |
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136 | (5) |
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4.1.4 Lebesgue type spaces and Sobolev type spaces |
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141 | (6) |
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147 | (1) |
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4.2 Lipschitz class on local fields |
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148 | (14) |
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4.2.1 Lipschitz classes on local fields |
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148 | (5) |
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4.2.2 Chains of function spaces on Euclidean spaces |
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153 | (4) |
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4.2.3 The cases on a local field Kp |
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157 | (2) |
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4.2.4 Comparison of Euclidean space analysis and local field analysis |
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159 | (3) |
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162 | (1) |
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4.3 Fractal spaces on local feilds |
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162 | (21) |
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4.3.1 Fractal spaces on Kp |
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163 | (1) |
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4.3.2 Completeness of (K(Kp),h) on Kp |
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164 | (8) |
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4.3.3 Some useful transformations on Kp |
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172 | (10) |
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182 | (1) |
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5 Fractal Analysis on Local Fields |
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183 | (60) |
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5.1 Fractal dimensions on local fields |
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183 | (17) |
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5.1.1 Hausdorff measure and dimension |
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183 | (7) |
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190 | (6) |
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5.1.3 Packing measure and dimension |
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196 | (4) |
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200 | (1) |
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5.2 Analytic expressions of dimensions of sets in local fields |
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200 | (13) |
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5.2.1 Borel measure and Borel measurable sets |
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200 | (1) |
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5.2.2 Distribution dimension |
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201 | (9) |
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210 | (3) |
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213 | (1) |
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5.3 p-type calculus and fractal dimensions on local fields |
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213 | (30) |
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5.3.1 Structures of Kp, 3-adic Cantor type set, 3-adic Cantor type function |
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213 | (5) |
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5.3.2 p-type derivative and p-type integral of v(x) on K3 |
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218 | (8) |
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5.3.3 p-type derivative and integral of Weierstrass type function on Kp |
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226 | (7) |
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5.3.4 p-type derivative and integral of second Weierstrass type function on Kp |
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233 | (9) |
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242 | (1) |
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6 Fractal PDE on Local Fields |
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243 | (40) |
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244 | (22) |
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6.1.1 Classical 2-dimension wave equation with fractal boundary |
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244 | (11) |
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6.1.2 p-type 2-dimension wave equation with fractal boundary |
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255 | (11) |
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6.2 Further study on fractal analysis over local fields |
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266 | (17) |
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6.2.1 Pseudo-differential operator Tα |
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266 | (15) |
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6.2.2 Further problems on fractal analysis over local fields |
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281 | (1) |
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282 | (1) |
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7 Applications to Medicine Science |
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283 | (22) |
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7.1 Determine the malignancy of liver cancers |
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284 | (7) |
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7.1.1 Terrible havocs of liver cancer, solving idea |
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284 | (3) |
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7.1.2 The main methods in studying of liver cancers |
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287 | (4) |
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7.2 Examples in clinical medicine |
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291 | (14) |
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7.2.1 Take data from the materials of liver cancers of patients |
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291 | (1) |
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7.2.2 Mathematical treatment for data |
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291 | (9) |
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7.2.3 Compute fractal dimensions |
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300 | (3) |
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7.2.4 Induce to obtain mathematical models |
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303 | (1) |
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7.2.5 Other problems in the research of liver cancers |
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304 | (1) |
Bibliography |
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305 | (10) |
Index |
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315 | |