Volker Nollau

Calculus of Probability

A trial is an attempt (observation, experiment) the result of which is uncertain within the scope of some possibilities and which is, at least in ideas, arbitrarily often reproducible when remaining unchanged the external conditions characterizing the attempt.

Differential Calculus for Functions of Several Variables

Functions y = f(x 1, x 2) of two independent variables x 1, x 2 can be visualized in a three-dimensional representation by a (x 1, x 2, y)-system of co-ordinates. The set of points (x 1, x 2, y) forms a surface provided that the function f is continuous. The set of points (x 1, x 2) such that f(x 1, x 2) = C = const is called a height line or level line of the function f to the height (level) C. These lines are located in the x 1, x 2-plane.

Sequences and Series

The sequence is said to be finite or infinite depending on whether the set K is finite or infinite.

Linear Programming. Transportation Problem

The problem to find a vector x *= (x * 1, x * 2,…, x * n)Tsuch that its components satisfy the conditions.

Functions of one Variable: Integral Calculus

Polynom division and partial fraction decomposition lead to integrals over polynomials and special partial fractions. The partial fractions can be integrated by the use of formulas from the ▸ table of indefinite integrals.

Functions of one Variable: Differential Calculus

If in addition to the above conditions the restricting requirement x n > x 0 (x n < x 0) is true, then one speaks about the limit from the right (from the left).

Functions of one Independent Variable

A real function f of one independent variable x ∈ℝis a mapping (rule of assignment) y = f(x) which relates to every number x of the domain D f ⊂ℝone and only one number y ∈ℝ Notation:f: D f →ℝ

Linear Programming and Transportation Problem

The problem to find a vector x* = (x 1 * , x 2 * ,..., x n * )⊤ such that its components satisfy the conditions

Number Systems and Their Arithmetic

\begin{gathered} {\alpha _{11}}{x_1} + {\alpha _{12}}{x_2} + \ldots + {\alpha _{1n}}{x_n} \leqslant {\alpha _1} \hfill \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \hfill \\ {\alpha _{r1}}{x_1} + {\alpha _{r2}}{x_2} + \ldots + {\alpha _{rn}}{x_n} \leqslant {\alpha _r} \hfill \\ {\beta _{11}}{x_1} + {\beta _{12}}{x_2} + \ldots + {\beta _{1n}}{x_n} \geqslant {\beta _1} \hfill \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \hfill \\ {\beta _{s1}}{x_1} + {\beta _{s2}}{x_2} + \ldots + {\beta _{sn}}{x_n} \geqslant {\beta _s} \hfill \\ {\gamma _{11}}{x_1} + {\gamma _{12}}{x_2} + \ldots + {\gamma _{1n}}{x_n} = {\gamma _1} \hfill \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \hfill \\ {\gamma _{t1}}{x_1} + {\gamma _{t2}}{x_2} + \ldots + {\gamma _{tn}}{x_n} = {\gamma _t} \hfill \\ \end{gathered}