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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
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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.
14. Elliptic PDE.
A. Energy estimates for bilinear form B.
B. Existence of weak solutions for elliptic Dirichlet problem.
C. Elliptic regularity.
D. Maximum principles.
E. Eigenfunctions of symmetric elliptic operator.
F. Green formulas.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Eigenfunctions of symmetric elliptic operator.


roposition

(Eigenvalues of symmetric elliptic operator) Let $U$ be a bounded subset of $\QTR{cal}{R}^{n}$ with $C^{1}$ boundary. There exists an orthonormal basis MATH of MATH where each $w_{k}$ is a solution of the problem MATH where $L_{0}$ is the elliptic operator MATH with MATH and the numbers MATH are real and MATH MATH

Proof

We have established during the proof of the proposition ( Existence of weak solution for elliptic Dirichlet problem 2 ) that the operator MATH is compact in MATH . The $L_{0}$ is symmetric in MATH . The rest follows from the propositions ( Spectrum of compact operator ) and ( Eigenvalues of compact symmetric operator ).





Notation. Index. Contents.


















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