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Editorial Reviews
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"...by far the most accessible mathematical account available and therefore is an invaluable addition to the literature." Science
"...a lovely introduction to the mathematics of fractal sets for the pure mathematician." American Mathematical Monthly
Product Description
This book contains a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Questions of local density and the existence of tangents of such sets are studied, as well as the dimensional properties of their projections in various directions. In the case of sets of integral dimension the dramatic differences between regular 'curve-like' sets and irregular 'dust like' sets are exhibited. The theory is related by duality to Kayeka sets (sets of zero area containing lines in every direction). The final chapter includes diverse examples of sets to which the general theory is applicable: discussions of curves of fractional dimension, self-similar sets, strange attractors, and examples from number theory, convexity and so on. There is an emphasis on the basic tools of the subject such as the Vitali covering lemma, net measures and Fourier transform methods.
See all Editorial Reviews
"...by far the most accessible mathematical account available and therefore is an invaluable addition to the literature." Science
"...a lovely introduction to the mathematics of fractal sets for the pure mathematician." American Mathematical Monthly
Product Description
This book contains a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Questions of local density and the existence of tangents of such sets are studied, as well as the dimensional properties of their projections in various directions. In the case of sets of integral dimension the dramatic differences between regular 'curve-like' sets and irregular 'dust like' sets are exhibited. The theory is related by duality to Kayeka sets (sets of zero area containing lines in every direction). The final chapter includes diverse examples of sets to which the general theory is applicable: discussions of curves of fractional dimension, self-similar sets, strange attractors, and examples from number theory, convexity and so on. There is an emphasis on the basic tools of the subject such as the Vitali covering lemma, net measures and Fourier transform methods.
See all Editorial Reviews
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Inside This Book
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First Sentence:
This section contains a condensed account of the basic measure theory we require. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
measurable plane sets, weak tangent, measure zero containing, metric outer measure, condensation direction, convex density, open set condition, tangency properties, binary cubes, binary intervals, angular densities, net intervals, elementary triangles, arbitrarily small measure, rectifiable curve, irregular sets, congruent copy, covering theorem, net measures, curvilinear triangles, residual set, fractional dimension, density theorem, geometric measure theory, unit segment Key Phrases - Capitalized Phrases (CAPs): (learn more)
Calculation of Hausdorff New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
This section contains a condensed account of the basic measure theory we require. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
measurable plane sets, weak tangent, measure zero containing, metric outer measure, condensation direction, convex density, open set condition, tangency properties, binary cubes, binary intervals, angular densities, net intervals, elementary triangles, arbitrarily small measure, rectifiable curve, irregular sets, congruent copy, covering theorem, net measures, curvilinear triangles, residual set, fractional dimension, density theorem, geometric measure theory, unit segment Key Phrases - Capitalized Phrases (CAPs): (learn more)
Calculation of Hausdorff New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 24
books:
See all 24 books this book cites
96
books
cite this book:
See all 96 books citing this book
- Selected Problems on Exceptional Sets by L. Carleson in Back Matter
- Real and Complex Analysis (International Series in Pure and Applied Mathematics) by Walter Rudin in Back Matter
- Nonlinear Dynamics and Turbulence by G. I. Barenblatt in Back Matter
- Transcendental Number Theory (Cambridge Mathematical Library) by Alan Baker in Back Matter
- Geometric Measure Theory (Classics in Mathematics) by Herbert Federer in Back Matter
See all 24 books this book cites
- Fluid Mechanics of Viscoelasticity (Rheology Series) by R.R. Huilgol on page 102, page 104, and page 105
- Real Mathematical Analysis by Charles Chapman Pugh on page 412, page 414, and page 415
- Rough Sets by Lech Polkowski on page 358, and Back Matter
- Handbook of the Logistic Distribution (Statistics: a Series of Textbooks and Monogrphs) by Balakrishnan on page 520, and Back Matter
- Fractal Geometry and Stochastics III (Progress in Probability) (v. 3) by Christoph Bandt on page 76, and Back Matter
See all 96 books citing this book
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