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2002 | Buch

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Mathematical Formulas for Economists

verfasst von: Professor Dr. Bernd Luderer, Professor Dr. Volker Nollau, Dr. Klaus Vetters

Verlag: Springer Berlin Heidelberg

Enthalten in: Professional Book Archive

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Über dieses Buch

This collection of formulas constitutes a compendium of mathematics for eco nomics and business. It contains the most important formulas, statements and algorithms in this significant subfield of modern mathematics and addresses primarily students of economics or business at universities, colleges and trade schools. But people dealing with practical or applied problems will also find this collection to be an efficient and easy-to-use work of reference. First the book treats mathematical symbols and constants, sets and state ments, number systems and their arithmetic as well as fundamentals of com binatorics. The chapter on sequences and series is followed by mathematics of finance, the representation of functions of one and several independent vari ables, their differential and integral calculus and by differential and difference equations. In each case special emphasis is placed on applications and models in economics. The chapter on linear algebra deals with matrices, vectors, determinants and systems of linear equations. This is followed by the representation of struc tures and algorithms of linear programming. Finally, the reader finds formu las on descriptive statistics (data analysis, ratios, inventory and time series analysis), on probability theory (events, probabilities, random variables and distributions) and on inductive statistics (point and interval estimates, tests). Some important tables complete the work.

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Inhaltsverzeichnis

Frontmatter

Sets and Propositions

Bernd Luderer, Volker Nollau, Klaus Vetters

Number Systems and their Arithmetic

Bernd Luderer, Volker Nollau, Klaus Vetters

Combinatorial Analysis

Abstract

For n given elements an arbitrary arrangement of all elements is called a permutation. If among the n elements there are p groups of the same elements, then one speaks about permutations with repetition. Let the number of elements in the i-th group be n _i, where it is assumed that n ₁ + n ₂+... +n _p = n.

Bernd Luderer, Volker Nollau, Klaus Vetters

Sequences and Series

Abstract

A mapping a: K → ℝ, K ⊂ ℕ,is called a sequence (of numbers) and denoted by {a_n}. For K = ℕ it consists of the elements (terms) a_n = a(n), n = 1, 2,... The sequence is said to be finite or infinite depending on whether the set K is finite or infinite.

Bernd Luderer, Volker Nollau, Klaus Vetters

Mathematics of Finance

Abstract

The most frequently period considered is the year, but a half-year, quarter, month etc. can be used either. The number of days per year or month differs from country to country. Usual usances are \(\frac{{30}}{{360}},\frac{{act}}{{360}},\frac{{act}}{{act}}\), where act means the actual number of days. In what follows, in all formulas containing the quantity T the underlying period is the year with 360 days, each month having 30 days (i. e., the first usance is used).

Bernd Luderer, Volker Nollau, Klaus Vetters

Functions of one Independent Variable

Abstract

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

Bernd Luderer, Volker Nollau, Klaus Vetters

Differential Calculus for Functions of one Variable

Abstract

If {x _n} is an arbitrary sequence of points converging to the point x ₀ such that x _n ∈ D _f, then the number a ∈ ℝ is called the limit of the function f at the point x ₀ if \(\mathop {\lim }\limits_{n \to \infty } f({x_n}) = a.\) Notation: \(\mathop {\lim }\limits_{x \to {x_0}} f(x) = a\;(or\,f(x) \to a\;for\;x \to {x_0}).\)

Bernd Luderer, Volker Nollau, Klaus Vetters

Integral Calculus for Functions of one Variable

Abstract

Every function F: (a, b) → ℝ satisfying the relation F′(x) = f(x) for all x ∈ (a, b) is called a primitive of the function f: (a, b) → ℝ The set of all primitives {F + C|C ∈ ℝ} is said to be the indefinite integral of f on (a, b); C is the integration constant. Notation: ∫ f(x) dx = F(x) + C.

Bernd Luderer, Volker Nollau, Klaus Vetters

Differential Equations

Abstract

Every n-times continuously differentiable function y(x) satisfying the differential equation for all x, a ≤ x ≤ b is called a (special) solution of the differential equation in the interval [a, b]. The set of all solutions of a differential equation or a system of differential equations is said to be the general solution.

Bernd Luderer, Volker Nollau, Klaus Vetters

Difference Equations

Bernd Luderer, Volker Nollau, Klaus Vetters

Differential Calculus for Functions of Several Variables

Abstract

A one-to-one mapping assigning to any vector x = (x₁ x ₂, ... , x _n)^Т ∈ D _f ⊂ ℝⁿ a real number f (x) = f (x _l, x ₂, ... , x _n is called a real function of several (real) variables; notation: f: D _f → ℝ, D _f ⊂ ℝⁿ.

Bernd Luderer, Volker Nollau, Klaus Vetters

Linear Algebra

Bernd Luderer, Volker Nollau, Klaus Vetters

Linear Programming and Transportation Problem

Abstract

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

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

and a given objective function z(x) = c ^⊤ x+c ₀ = c ₁ x ₁+c ₂ x ₂+...+c _n x _n+c ₀ attains its smallest value (minimum problem) or its greatest value (maximum problem) under all vectors x = (x ₁, x ₂,..., x _n)^⊤ fulfilling these conditions is called a linear programming (or optimization) problem. The conditions posed above are called the constraints or the restrictions of the problem. A vector x = (x ₁,..., x _n)^⊤ satisfying all constraints is said to be feasible. A variable x _i for which the relation x _i ≥ 0 (non-negativity requirement) fails to occur among the constraints is referred to as a free or unrestricted variable.

Bernd Luderer, Volker Nollau, Klaus Vetters

Descriptive Statistics

Abstract

Basis of a statistical analysis is a set (statistical mass) of objects (statistical unit), for which one (in the univariate case) or several (in the multivariate case) characters are tested. The results which can occur in observing a character are called the character values.

Bernd Luderer, Volker Nollau, Klaus Vetters

Calculus of Probability

Abstract

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.

Bernd Luderer, Volker Nollau, Klaus Vetters

Inductive Statistics

Abstract

By a mathematical sample of size n chosen from a parent population M _X one understands a n-dimensional random vector X = (X ₁,..., X _n) the components of which are independent and distributed like X. Every realization x = (x ₁,..., x _n) of X is called a concrete sample.

Bernd Luderer, Volker Nollau, Klaus Vetters

Backmatter

Titel: Mathematical Formulas for Economists
verfasst von: Professor Dr. Bernd Luderer
Professor Dr. Volker Nollau
Dr. Klaus Vetters
Copyright-Jahr: 2002
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-12431-4
Print ISBN: 978-3-540-42616-5
DOI: https://doi.org/10.1007/978-3-662-12431-4