Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
9. Levy process.
10. Weak derivative. Fundamental solution. Calculus of distributions.
11. Functional Analysis.
12. Fourier analysis.
13. Sobolev spaces.
A. Convolution and smoothing.
B. Approximation by smooth functions.
C. Extensions of Sobolev spaces.
D. Traces of Sobolev spaces.
E. Sobolev inequalities.
F. Compact embedding of Sobolev spaces.
G. Dual Sobolev spaces.
H. Sobolev spaces involving time.
I. Poincare inequality and Friedrich lemma.
14. Elliptic PDE.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Traces of Sobolev spaces.


roposition

(Trace theorem). Assume that MATH is a bounded set and $\partial U$ admits a locally continuously differentiable parametrization. The $p$ is restricted $1\leq p<\infty$ . There exists a bounded linear operator MATH such that for any MATH MATH

Proof

The statement is true for a flat boundary MATH and MATH . If the boundary is not flat then there is a change of variables that makes it locally flat. Then procedure extends globally by partition of unity (see the proof of the proposition ( Global approximation by smooth functions ) for an example of the technique). The procedure extends to MATH by closure of MATH in MATH .

Proposition

(Trace-zero functions in $W^{1,p}$ ) Assume that MATH is a bounded set and $\partial U$ admits a locally continuously differentiable parametrization. The $p$ is restricted $1\leq p<\infty$ . Let MATH . Then MATH





Notation. Index. Contents.


















Copyright 2007