This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The logician Wilfred Hodges and specialist of game theory used the sad story of the logician monk of the xiith century, Pierre Abélard, the “tychastic” player, nicknamed “\( \forall \)belard”, who was cruelly punished for seducing the future abbess of Argenteuil, Eloise (Héloïse), the “contingent regulator” nicknamed “\( \exists \)loise”. She was at that time his young student attracted by an inclination that logic does not know, but biology does. While working on logics, the young Pierre-Astrolabe was born as the fruit of their love. His punishment was the reverse retaliation of his “sin” and the transgression of ecclesiastical canons. Driven from their paradise by a singular punishment, \( \exists \)loise and \( \forall \)belard could have given way to “\( \forall \)dam” and “\( \exists \)va”, also punished for wanting to “know” and being the culprits for the sins of all of us. The author thanks Allen Mann for informing him that Wilfred Hodges had proposed in 1985 to name these two players in Building models by games, [150, Hodges], discovered in his book Building models by games Independence-Friendly Logic: A Game-Theoretic Approach [183, Mann, Sandu and Sevenster], in collaboration with Gabriel Sandu and Merlijn Sevenster. They made the link between a disembodied logic and (human) game theory.
- 2.
At this stage, mathematics are in the same situation than the yamonamö (see Footnote 19, p. 20) who have a name for the number two only, as we have a name only for maps, which are binary relations, whereas general relations are not currently used.
- 3.
We do not mention here the case of discrete time systems nor punctuated evolutions of impulse systems (see Sect. 12.3, p. 503 of Viability Theory. New Directions, [31, Aubin, Bayen and Saint-Pierre]).
- 4.
See Chap. 9, p. 319, of Viability Theory. New Directions, [31, Aubin, Bayen and Saint-Pierre], for the case of and Chap. 10, p. 375, for results specific to differential inclusions.
- 5.
Actually, the upper and lower stability assumptions involve the Painlevé–Kuratowski limits (see Definition 8.1.4, p. 213) and are defined respectively by
$$\begin{aligned} {\left\{ \begin{array}{l} \mathrm{{Limsup}}_{n \mapsto +\infty }\mathrm{{Capt}}_{\mathcal {S}}(K_{n},C_{n}) \; = \; \mathrm{{Capt}}_{\mathcal {S}}(\mathrm{{Limsup}}_{n \mapsto +\infty }K_{n},\mathrm{{Limsup}}_{n \mapsto +\infty } C_{n})\\ \mathrm{{Liminf}}_{n \mapsto +\infty }\mathrm{{Inv}}_{\mathcal {S}}(K_{n},C_{n}) \; = \; \mathrm{{Inv}}_{\mathcal {S}}(\mathrm{{Liminf}}_{n \mapsto +\infty }K_{n},\mathrm{{Liminf}}_{n \mapsto +\infty } C_{n})\\ \end{array} \right. }\quad {(8.22)} \end{aligned}$$ - 6.
See Sects. 6.5, p. 233 and 7.2, p. 248 of Viability Theory. New Directions, [31, Aubin, Bayen and Saint-Pierre] for more details on this important topic.
- 7.
See [199, Quincampoix].
- 8.
Later, Gottfried Leibniz characterized them in terms of limits of differential quotients. Limits were fuzzily defined until Cauchy proposed a rigorous definition, actually precise but too demanding. Since then, the history of differential calculus has been punctuated by revolutions relaxing Cauchy requirements by taking weaker and weaker concepts of limits to obtain more and more “differentiable” functions. Such attempts started with Giuseppe Peano in the case of functions, followed by Laurent Schwartz for distributions, later for set-valued maps (graphical derivatives) and maps from a metric space to another one, replacing the linear structure by a mutational one. This was the route ever since used with the success we all know. See Sect. 18.9, The Graal of the Ultimate Derivative, p. 765, of Viability Theory.New Directions, [31, Aubin, Bayen and Saint-Pierre].
- 9.
Parameterized their slope in this simplest case.
- 10.
See Sect. 18.9, p. 765 of Viability Theory. New Directions, [31, Aubin, Bayen and Saint-Pierre] for more details.
- 11.
[118, Dolecki and Greco], [140, Greco, Mazzucchi and Pagani]. We refer only to the books Applicazioni geometriche del calcolo infinitesimale, [195, Peano], and Formulario Mathematico, [196, Peano], by Giuseppe Peano.
- 12.
Or a zoo...
- 13.
See Set-Valued Analysis, [42, Aubin and Frankowska], Variational Analysis, [203, Rockafellar and Wets], Mutational and Morphological Analysis: Tools for Shape Regulation and Morphogenesis, [18, Aubin], and Mutational Analysis. A Joint Framework for Cauchy Problems in and Beyond Vector Spaces, [179, Lorenz].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Aubin, JP., Désilles, A. (2017). Mathematical Appendixes. In: Traffic Networks as Information Systems. Mathematical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54771-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-54771-3_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54770-6
Online ISBN: 978-3-642-54771-3
eBook Packages: EngineeringEngineering (R0)