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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.
A. Operations on sets and logical statements.
B. Fundamental inequalities.
C. Function spaces.
D. Measure theory.
E. Various types of convergence.
F. Signed measures. Absolutely continuous and singular measures. Radon-Nikodym theorem.
G. Lebesgue differentiation theorem.
H. Fubini theorem.
I. Arzela-Ascoli compactness theorem.
J. Partial ordering and maximal principle.
K. Taylor decomposition.
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.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Partial ordering and maximal principle.


efinition

The relation $\leq$ is said to be a partial ordering on the set A iff it has the following properties

1. MATH

2. MATH

The partial ordering is said to be a linear ordering iff either $x\leq y$ or $y\leq x$ for any two elements $x,y$ of the set A.

Any linearly ordered subset C of a partially ordered set A is called a chain. The chain C is said to be maximal if it is not a nontrivial subset of any other chain.

For a chain C we define the upper bound $\sup C$ as an element with the property $x\leq\sup C$ for any $x\in C$ .

The element $m$ of the partially ordered set A is called maximal iff MATH

Axiom

(Axiom of choice). Let $I$ be an arbitrary index set MATH . Suppose that for any $i\in I$ a set $A_{i}$ is given. Then there exists a map $\phi$ acting on $I$ such that MATH .

Lemma

(Hausdorff maximal principle). Every chain of a partially ordered set is contained in some maximal chain.

Lemma

(Zorn maximal principle). If every chain of a partially ordered set has a upper bound then there exists a maximal element.

Both versions of the maximal principle are consequences of the axiom of choice.





Notation. Index. Contents.


















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