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Local polynomials and the Montel theorem

On Wilson’s functional equations

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Regularity properties of measurable functions satisfying a multiplicative type functional equation almost everywhere

Improving regularity of solutions of a difference equation

Deriving meaningful scientific laws from abstract, “gedanken” type, axioms: five examples

Relations between generalized Aczél–Jabotinsky and Aczél–Jabotinsky differential equations

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Deriving meaningful scientific laws from abstract, “gedanken” type, axioms: five examples

Content Provider	Springer Nature Link
Author	Falmagne, Jean Claude
Copyright Year	2015
Abstract	A scientific law can be a faithful representation of the physical world only if its form is invariant with respect to changes in the unit or units. This is referred to as the ‘meaningfulness condition.’ This condition is powerful. If we require it, the mathematical form of a scientific or geometric law may be derivable from some abstract constraint, possibly verifiable by a thought experiment or a trivial argument. We discuss five examples of such abstract constraints in this paper: 1. the $${associativity equation: \qquad\qquad\qquad\qquad\quad F(F(x,y),z) \,= \,F(x,F(y,z));}$$ a s s o c i a t i v i t y e q u a t i o n : F ( F ( x , y ) , z ) = F ( x , F ( y , z ) ) ; 2. the $${quasi-permutability equation: \qquad\qquad\quad\ \ F(G(x,y),z) \,=\, F(G(x,z),y));}$$ q u a s i - p e r m u t a b i l i t y e q u a t i o n : F ( G ( x , y ) , z ) = F ( G ( x , z ) , y ) ) ; 3. the $${bisymmetry equation: \qquad\qquad\qquad\quad F(F(x,y),\!F(z,w))\! = \!F(F(x,z),F(y,w));}$$ b i s y m m e t r y e q u a t i o n : F ( F ( x , y ) , F ( z , w ) ) = F ( F ( x , z ) , F ( y , w ) ) ; 4. the $${translation equation: \qquad\qquad\qquad\qquad\quad\ \ F(F(x,y),z) = F(x, y+z);}$$ t r a n s l a t i o n e q u a t i o n : F ( F ( x , y ) , z ) = F ( x , y + z ) ; 5. the $${abstract Lorentz-FitzGerald contraction: \quad\ \ \ L(L(\ell ,v),w) = L(\ell , v\oplus w).}$$ a b s t r a c t L o r e n t z - F i t z G e r a l d c o n t r a c t i o n : L ( L ( ℓ , v ) , w ) = L ( ℓ , v ⊕ w ) . In each case, just one or a couple of meaningful mathematical representations are possible. In this paper, we derive the possible meaningful representations in five examples. These results are obtained under some general conditions, in addition to those listed in 1–5. Other meaningful representations may be possible under different additional conditions.
Ending Page	435
Page Count	43
Starting Page	393
File Format	PDF
ISSN	00019054
e-ISSN	14208903
Journal	Aequationes mathematicae
Issue Number	2
Volume Number	89
Language	English
Publisher	Springer Basel
Publisher Date	2015-03-17
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Research exposition (monographs, survey articles) Analysis Combinatorics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Discrete Mathematics and Combinatorics

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Deriving meaningful scientific laws from abstract, “gedanken” type, axioms: five examples

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