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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.

Convolution and smoothing.


his section contains a recipe for building a smooth approximation to a measurable function.

The $U$ is a subset of $\QTR{cal}{R}^{n}$ . The $\partial U$ is the boundary of $U$ . For an $\varepsilon>0$ we denote MATH , where the MATH is the distance from $x$ to the boundary $\partial U$ .

Definition

(Standard mollifier definition).

1. MATH

2. For $\varepsilon>0$ MATH

$\eta$ is the "standard mollifier".

3. If MATH is locally integrable then we define the "mollification" MATH MATH for MATH .

Proposition

(Properties of mollifiers).

1. MATH .

2. MATH almost surely as MATH .

3. MATH uniformly on every compact subset of $U$ .

4. MATH $\Rightarrow$ MATH in MATH .

Definition

(Partition of unity definition). Let $U$ be a subset of $\QTR{cal}{R}^{n}$ and MATH be a finite or countable collection of subsets of $\QTR{cal}{R}^{n}$ such that MATH The collection of functions MATH MATH is called a smooth partition of unity subordinated to MATH .





Notation. Index. Contents.


















Copyright 2007