Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

2015 | Buch

Overview Kapitel lesen Erstes Kapitel lesen

On Meaningful Scientific Laws

verfasst von: Jean-Claude Falmagne, Christopher Doble

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

The authors describe systematic methods for uncovering scientific laws a priori, on the basis of intuition, or “Gedanken Experiments”. Mathematical expressions of scientific laws are, by convention, constrained by the rule that their form must be invariant with changes of the units of their variables. This constraint makes it possible to narrow down the possible forms of the laws. It is closely related to, but different from, dimensional analysis. It is a mathematical book, largely based on solving functional equations. In fact, one chapter is an introduction to the theory of functional equations.

Inhaltsverzeichnis

Frontmatter
1. Overview
Abstract
The mathematical expression of a scientific or geometric law typically does not depend on the units of measurement. For example, the statement of the Pythagorean Theorem, the equation of a parabola or the law of gravity do not depend on the units of measurement. The most important rationale for this convention is that measurement units do not appear in nature. Thus, any mathematical model or law whose form would be fundamentally altered by a change of units would be a poor representation of the empirical world.
Jean-Claude Falmagne, Christopher Doble
2. Extensive Measurement
Abstract
The title of this chapter refers to the measurement of the fundamental variables, they are called ‘scales’, entering in the equations of physics and geometry. We limit our discussion to those variables that are specified by their unit, such as mass, time, or length. The unit of mass may be one gram, or one kilogram, or one pound. It does not matter: the equations remain the same. In all such cases, the ratio of two values does not depend upon the chosen unit.
Jean-Claude Falmagne, Christopher Doble
3. Functional Equations
Abstract
There are situations, in the sciences or in mathematics, when the researcher’s intuition about a phenomenon leads to an equation involving one or more unknown functions. This may happen at the early stage of the investigation, when the scientist is reluctant to make specific assumptions about the form of the functions. However, the equations themselves, or some side conditions, may sometimes reduce the possibilities. The field of mathematics dealing with such derivations is called functional equations.
Jean-Claude Falmagne, Christopher Doble
4. Abstract Axioms and their Representations
Abstract
Except for the content of Section 4.7, which is relatively recent, all the results of this chapter are standard parts of the functional equation literature. Specific references are given in due place. These results provide the mathematical foundations on which the meaningfulness axiom will operate.
Jean-Claude Falmagne, Christopher Doble
5. Defining Meaningfulness
Abstract
We turn to the core condition of this book. One of our goals here is to axiomatize a particular type of invariance that must hold for all scientific laws that are expressed in ratio scale units. The consequence of this axiomatization should be that the form of an expression representing a scientific law should not be altered by changing the units of the variables. The next definition, which generalizes that used by Falmagne (2004) applies to n-codes regarded as functions of n real, ratio scale variables.
Jean-Claude Falmagne, Christopher Doble
6. Propagating Axioms via Meaningfulness
Abstract
The meaningfulness condition introduced in Definition 5.2.1 and Equation (5.6) has a remarkable property. In some cases, a condition imposed on a single code of a meaningful collection may be automatically transported to all the codes in the collection. This applies to many properties, such as solvability, quasi-permutability, symmetry, differentiability, and others.
Jean-Claude Falmagne, Christopher Doble
7. Meaningful Representations of Scientific Codes
Abstract
In this chapter, which recalls some results of Falmagne (2015), we derive some exemplary consequences of the meaningfulness condition paired with some abstract axioms, in particular: associativity, quasi-permutability, bisymmetry, translatability, and quasi-permutability, the latter in the context of LF-systems.
Jean-Claude Falmagne, Christopher Doble
8. Order Invariance under Transformations
Abstract
So far in this book, we have investigated the consequences of some abstract axioms such as transitivity or permutability, combined with the meaningfulness condition, on the mathematical form of the codes. This chapter is in the same spirit, with the abstract axioms replaced by ‘order-invariance’ axioms. The next equation gives an example.
Jean-Claude Falmagne, Christopher Doble
9. Dimensional Invariance and Dimensional Analysis
Abstract
The notion of invariance is a fundamental one in mathematics and the sciences. This chapter is a brief introduction to two of its subtopics, dimensional invariance and dimensional analysis, which are relevant to the subject of this book. Specifically, dimensional invariance is closely related to, but technically different from, meaningfulness.
Jean-Claude Falmagne, Christopher Doble
10. Open Problems
Abstract
In the vein of Theorem 7.4.1, derive the meaningful collection, satisfying the translation equation, whose initial code is
$$F(x,\ y)={{\left( {{x}^{\frac{1}{\vartheta }}}+{{c}^{\frac{1}{\vartheta }}y} \right)}^{\vartheta }}.$$
Jean-Claude Falmagne, Christopher Doble
Backmatter
Metadaten
Titel
On Meaningful Scientific Laws
verfasst von
Jean-Claude Falmagne
Christopher Doble
Copyright-Jahr
2015
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-46098-6
Print ISBN
978-3-662-46097-9
DOI
https://doi.org/10.1007/978-3-662-46098-6