'Ht moi, ... , si j'avait su comment en revenir, One lemce mathematics has rendered the je n'y serai. point aile.' human race. It has put common sense back Jule. Verne ..... "'" it belong., on the topmost shelf next to the dusty caniller labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'~re of this series.

Klappentext

1. Preliminary Information about Integration Theory.- Notation and Terminology.- Some Auxiliary Information about Sets and Functions in Rn.- General Information about Measures and Integrals.- Differentiation Theorems for Measures in Rn.- Generalized Functions.- 2. Functions with Generalized Derivatives.- Sobolev-Type Integral Representations.- Other Integral Representations.- Estimates for Potential-Type Integrals.- Classes of Functions with Generalized Derivatives.- Theorem on the Differentiability Almost Everywhere.- 3. Nonlinear Capacity.- Capacity Induced by a Linear Positive Operator.- The Classes W(T, p, V).- Sets Measurable with Respect to Capacity.- Variational Capacity.- Capacity in Sobolev Spaces.- Estimates of [l, p]-Capacity for Some Pairs of Sets.- Capacity in Besov-Nickolsky Spaces.- 4. Density of Extremal Functions in Sobolev Spaces with First Generalized Derivatives.- Extremal Functions for (l, p)-Capacity.- Theorem on the Approximation of Functions from Lpl by Extremal Functions.- Removable Singularities for the Spaces Lpl (G).- 5. Change of Variables.- Multiplicity of Mapping, Degree of Mapping, and Their Analogies.- The Change of Variable in the Integral for Mappings of Sobolev Spaces.- Sufficient Conditions of Monotonicity and Continuity for the Approximation Functions of the Class Lnl.- Invariance of the Spaces Lpl(G)(Lnl(G)) for Quasiisometric (Quasiconformal) Homeomorphisms.- 6. Extension of Differentiate Functions.- Arc Diameter Condition.- Necessary Extension Conditions for Seminormed Spaces.- Necessary Extension Conditions for Sobolev Spaces.- Necessary Extension Conditions for Besov and Nickolsky Spaces.- Sufficient Extension Conditions.- Comments.- References.

Inhalt

1. Preliminary Information about Integration Theory.- §1. Notation and Terminology.- §2. Some Auxiliary Information about Sets and Functions in Rn.- §3. General Information about Measures and Integrals.- §4. Differentiation Theorems for Measures in Rn.- §5. Generalized Functions.- 2. Functions with Generalized Derivatives.- §1. Sobolev-Type Integral Representations.- §2. Other Integral Representations.- §3. Estimates for Potential-Type Integrals.- §4. Classes of Functions with Generalized Derivatives.- §5. Theorem on the Differentiability Almost Everywhere.- 3. Nonlinear Capacity.- §1. Capacity Induced by a Linear Positive Operator.- §2. The Classes W(T, p, V).- §3. Sets Measurable with Respect to Capacity.- §4. Variational Capacity.- §5. Capacity in Sobolev Spaces.- §6. Estimates of [l, p]-Capacity for Some Pairs of Sets.- §7. Capacity in Besov-Nickolsky Spaces.- 4. Density of Extremal Functions in Sobolev Spaces with First Generalized Derivatives.- §1. Extremal Functions for (l, p)-Capacity.- §2. Theorem on the Approximation of Functions from Lpl by Extremal Functions.- §3. Removable Singularities for the Spaces Lpl (G).- 5. Change of Variables.- §1. Multiplicity of Mapping, Degree of Mapping, and Their Analogies.- §2. The Change of Variable in the Integral for Mappings of Sobolev Spaces.- §3. Sufficient Conditions of Monotonicity and Continuity for the Approximation Functions of the Class Lnl.- §4. Invariance of the Spaces Lpl(G)(Lnl(G)) for Quasiisometric (Quasiconformal) Homeomorphisms.- 6. Extension of Differentiate Functions.- §1. Arc Diameter Condition.- §2. Necessary Extension Conditions for Seminormed Spaces.- §3. Necessary Extension Conditions for Sobolev Spaces.- §4. Necessary Extension Conditions for Besov and Nickolsky Spaces.-§5. Sufficient Extension Conditions.- Comments.- References.