Towards historical roots of necessary conditions of optimality: Regula of Peano

• Autorzy: Dolecki S. , Greco G. H.
• Języki publikacji: EN

Abstrakty

EN

At the end of 19th century Peano discerned vector spaces, differentiability, convex sets, limits of families of sets, tangent cones, and many other concepts, in a modern perfect form. He applied these notions to solve numerous problems. The theorem on necessary conditions of optimality (Regula) is one of these. The formal language of logic that he developed, enabled him to perceive mathematics with great precision and depth. Actually he built mathematics axiomatically based exclusively on logical and set-theoretic primitive terms and properties, which was a revolutionary turning point in the development of mathematics.

Słowa kluczowe

EN

differentiability; convex set; Kuratowski limit of sets; tangent cones; condition of optimality; mean value theorem; Peano remainder; Peano's regula; strict derivability; Peano's Formulario Mathematico; Peano's Geometric Calculus; Grassmann's Ausdehnungslehre

PL

różniczkowalność; zbiór wypukły

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2007
• Tom: Vol. 36, no 3
• Strony: 491--518
• Opis fizyczny: Bibliogr. 87 poz., rys.

Twórcy

autor

Dolecki S.

autor

Greco G. H.

• Mathematical Institute of Burgundy, Burgundy University, B.P. 47 870, 21078 Dijon, France,

Bibliografia

• ACOSTA, E.G. and DELGADO, C.G. (1994) Frechetvs. Caratheodory. Amer. Math. Monthly, 101, 332-338. http://www.jstor.org.
• ACKER, F. and DICKSTEIN, F. (1983) Uma introdução a análise convexa. 14° Colóquio Brasileiro de Matemática, Poços de Caldas.
• ASCOLI, G. (1933) Sugli spazi lineari metrici e le loro varietả. Ann. Mat. Pura Appl. 10, 33-81, 203-232. http://www.springerlink.com
• ASCOLI, G. (1952) Sopra un’estensione di una formula asintotica di Laplace agli integrali multipli. Rend. Sem. Mat. Pad. 21, 209-227. http://www.numdam.org
• ASCOLI, G. (1955) I motivi fondamentali dell’opera di Giuseppe Peano. In: A. Terracini, ed., In memoria di Giuseppe Peano, Liceo Sc. Cuneo, 23-30.
• AUBIN, J.-P. (2000) Applied Functional Analysis. Wiley.
• AUBIN, J.-P. and FRANKOWSKA, H. (1990) Set-Valued Analysis. Birkhäuser.
• BANACH, S. (1951) Mechanics. Polish Mathematical Society. http: //matwbn.icm.edu.pl/kstresc.php?tom=24&wyd=10&jez=
• BIRKHOFF, G. (1973) A Source-book in Classical Analysis. Harvard Univ. Press.
• BOREL, E. (1903) Quelques remarques sur les ensembles de droites et de plans. Bull. Soc. Math. France 31, 272-275. http://www. numdam. org/item?id=BSMF. 1903__31-2T2_0
• BORWEIN, J. M. and LEWIS, A. S. (2000) Convex Analysis and Nonlinear Optimization. Springer-Verlag.
• BOULIGAND, G. (1924) Leçons de la Géometric Vectorielle. Vuibert, Paris.
• BOULIGAND, G. (1932) Introduction a la geométrie infinitésimale directe. Gauthier-Villars, Paris.
• BURALI-FORTI, C. (1895) Sur quelques propriete des ensembles d’ensembles et leurs applications a la limited’un ensemble variable. Math. Annalen 47, 20-32. http://gdzdoc.sub.uni-goettingen.de/sub/digbib/loader?did=D77340
• BURALI-FORTI, C. (1899) Sur l’égalité et sur l’introduction des eléments dérivés dans la sciences. Enseignement Math. 1, 246-261. http://retro.seals.ch/cntmng?type=pdf&aid=cl:43121lfesubp=lores
• CARATHÉODORY, C. (1964) Theory of functions of a complex variable, vol. 1. Chelsea Publ. Company, New York.
• CASSINA, U. (1926-27) Limiti delle funzioni plurivoche. Atti R. Ace. Scienze Torino 62, 4-21.
• DARBOUX, G. (1880) Sur le théorème fondamental de la géométrie projective. (Extrait d’une lettre a M. Klein) Math. Annalen 17, 55-61. http://dz-srvl.sub.uni-goettingen.de/sub/digbib/loader?did=D26213
• DINI, U. (1878) Fondamenti per la teorica delle funzioni di variabili reali. Nistri e C. Pisa.
• FEDERER, H. (1959) Curvature measures. Trans. Amer. Math. Soc. 93, 418-491.
• FÉLIX, L. (1957) L’aspect moderns des mathématiques. Librairie Blanchard, Paris.
• FRÉCHET, M. (191 la) Sur la notion de différentielle. C.R.A.Sc. Paris 152, 845-847, 27 March 1911. http://gallica.bnf .fr/ark:/12148/bpt6k3105c
• FRÉCHET, M. (191 la) Sur la notion de différentielle. C.R.A.Sc. Paris 152, 1950-1951, 18 April 1911. http://gallica.bnf .fr/ark:/12148/bpt6k3105c
• FRÉCHET, M. (1937) Sur la notion de différentielle. J. Math. Pures Appl. 16, 233-250.
• GENOCCHI, A. (1884) Calcolo differenziale e principii di calcolo integrate pubblicato con aggiunte dal Dr. Giuseppe Peano. Fratelli Bocca, Torino. http://historical.library.Cornell.edu/cgi-bin/cul.math/ docviewer?did=02840002&seq=l
• GRASSMANN, H.G. (1847) Geometrische Analyse. Leipzig.
• GRASSMANN, H.G. (1894-1911) Gesammelte Werke. Teubner, Leipzig, http://quod.lib.umich.edu/cgi/t/text/ text-idx?c=umhistmath&idno=ABW0785
• GRASSMANN, H.G. (2000) Extension Theory. American Mathematical Society.
• GRECO, G.H. AND PAGANI, E.M. (2007) Reworking on Affine Exterior Algebra of Grassmann: Peano and its School. Preprint.
• GRECO, G.H. (2007) Reworking on Derivation of Measures: Cauchy and Peano. Forthcoming.
• HANCOCK, H. (1903) Lectures on the Theory of Maxima and Minima of Functions of Several Variables. Weierstrass’ Theory. Cincinnati, University Press, http://historical.library.Cornell.edu/cgi-bin/cul.math/ docviewer?did=02120001&seq=l
• HANCOCK, H. (1917) Theory of maxima and minima. Dover Publications, 1960. http://www.archive.org/details/theorymaxima00hancrich
• HAUSDORFF, F. (1914) Grundzüge der Mengelehre. Chelsea Publishing Co, New York.
• HAUSDORFF, F. (1927) Mengelehre. Berlin.
• HIRRIART-URRUTY, J.-B. and LEMARECHAL, C. (1996) Convex Analysis and Minimization Algorithms. Springer-Verlag, Berlin.
• JACOBI, C. (1881-1891) Gesammelte Werke. Reimer, Berlin. http://quod.lib.umich.edu/cgi/t/text/ text-idx?c=umhistmath& idno=ABR8803
• JANISZEWSKI, Z. (1911) Les continus irréductibles entre deux points (Thèse, 1911). In: Oeuvres Choisies, P.W.N. (Polish Scientific Publishers), 1962, 31-125.
• JORDAN, C. (1893-96) Cours d’Analyse de l’École Polytechnique. 2nd edition, Gauthier-Villars, Paris.
• KENNEDY, H.C. (1980) Life and Works of Giuseppe Peano. D.Reidel, Dordrecht.
• KUHN, S. (1991) The Derivative ả la Carathéodory. Amer. Math Monthly, 98,40-44. http://vraw.jstor.org
• KURATOWSKI, K. (1928) Sur les décompositions semi-continues d’espaces métriques compacts. Fund. Math. 11, 167-85.
• KURATOWSKI, K. (1948) Topologie, vol. I. 4th edition, Monografie Matematyczne, Warszawa. http : //matwbn . icm . edu . pl/kstresc . php?tom=20&wyd=10& j ez= MANHEIM, J.H. (1964) The Genesis of Point Set Topology. Pergamon Press, Oxford.
• MAWHIN, J. (1997) Analyse. Fondaments, techniques, évolution. De Boeck, Bruxelles.
• MAZUR, S. (1933) Über konvexe Mengen in lineare normierte Räumen. Studia Math. 4, 70-84. http://matwbn.icm.edu.pl/ksiazki/sm/sm4/sm4113.pdf
• MAY, K.O. (1973) Bibliography and Research Manual of the History of Mathematics. Toronto.
• MINKOWSKI, H. (1896) Geometric der Zahlen. Teubner, Leipzig. http://gallica.bnf .fr/ark:/12148/bpt6k99643x and bpt6k3102f
• PALLASCHKE, D. and ROLEWICZ, S. (1997) Foundations of Mathematical Optimization. Convex Analysis without Linearity. Kluwer, Dordrecht.
• PEANO, G. (1884a) Extrait d’une lettre. Nouvelles Annales de Mathématiques 3, 45-47.
• PEANO, G. (1884b) Réponse a Ph. Gilbert. Nouvelles Annales de Mathématiques 3, 252-256.
• PEANO, G. (1887) Applicazioni geometriche del calcolo infinitesimale. Fratelli Bocca, Torino, http://historical.library.cornell.edu/cgi-bin/ cul.math/docviewer?did=00610002&seq=l
• PEANO, G. (1888a) Calcolo geometrico secondo Ausdehnungslehre di H. Grassmann. Fratelli Bocca, Torino.
• PEANO. G. (1888) Intégration par séries des équations différentielles linéaires. Mathernatische Annalen 32, 450-456. http://gdzdoc.sub.uni-goettingen.de/sub/digbib/ loader?ht=VIEW&did=D29534
• PEANO, G. (1889) / principii di geometria logicamente esposti. Fratelli Bocca Editori, Torino, http://quod.lib.umich.edu/cgi/t/text/ text-idx?c=umhistmath&idno=ABV4128
• PEANO, G. (1890a) Sulla definizione dell’area di una superficie. Rend. Ace. Lincei 6, 54-57.
• PEANO, G. (1890b) Démonstration de l’intégrabilité des équations différentielles ordinaires. Mathernatische Annalen 37, 182-228. http : //gdzdoc . sub . uni-goettingen . de/sub/digbib/ loader?ht=VIEW&did=D27538
• PEANO, G. (1891) Sulla formula di Taylor. Atti R. Accad. Scienze Torino 27, 40-46.
• PEANO, G. (1892) Sur la définition de la dérivée. Mathesis 2, 12-14.
• PEANO, G. (1893) Lezioni di analisi infintesimale. 2 vol., Candeletti, Torino.
• PEANO, G. (1894a) Sur les systèmes linéaires. Monatshefte für Mathematik und Physik 5, 136.
• PEANO, G. (1894b) Sur la définition de la limite d’une fonction. Exercice de la logique mathématique, Amer. J. Math. 17, 37-68.
• PEANO, G. (1895-96) Saggio di calcolo geometrico. Atti R. Accad. Scienze Torino 31, 952-975.
• PEANO, G. (1897-98) Analisi della teoria del vettori. Atti R. Ace. Scienze Torino 33, 513-534.
• PEANO, G. (1903) Formulaire Mathématique (tome IV). Fratelli Bocca, Torino.
• PEANO, G. (1908) Formulario Mathematico (Editio V). Fratelli Bocca, Torino.
• PEANO, G. (1957-9) Opere scelte. Edizioni Cremonese, Roma. PEANO, G. (1960) Formulario Mathematico. Edizioni Cremonese, Roma 1960 (reprint of Peano, 1908).
• PEANO, G. (1973) Selected works (H.C. Kennedy, ed.). University of Toronto Press.
• PEANO, G. (2002) Opera Omnia. 1 CD-ROM (S. Boero, ed.). Dipartimento di Matematica, Universitả, Torino.
• PIERPONT, J. (1905) The theory of functions of real variables, vol. I. Ginn and Co., Boston.
• POINCARE, H. (1908) L’avenir des mathématiques. Atti del IV Congresso Internazionale dei Matematici (Roma, 6-11 Aprile 1908), I, 167-182, Ace. Lincei, Roma 1909.
• REITBERGER, H. (2002) Leopold Vietoris (1891-2002). Notices AMS 49, 1232-6. http://www.ams.org/notices/200210/fea-vietoris.pdf
• ROCKAFELLAR, R.T. and WETS, R. (1997) Variational Analysis. Springer-Verlag.
• SAKS, S. (1937) Theory of the Integral. Hafner, New York. http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=
• STOLZ, O. (1893) Grundzüge der Differenzial-und Integralrechnung. Teubner, Leipzig.
• SVETIC, R.E. and VOLKMER, H. (1998) On the ultimate Peano derivative. J. Math. Anal. Appl. 218, 439-452.
• VASILESCO, F. (1925) Essai sur les fonctions multiformes de variables réelles (Thèse). Gauthier-Villars, Paris.
• VIETORIS, F. (1922) Bereiche zweiter Ordnung. Monatshefte für Mathematik und Physik 32, 258-80.
• WEIL, C. (1995) The Peano notion of higher order differentiation. Math. Japonica 42, 587-600.
• WEYL, H. (1918) Raum-Zeit-Materie. Springer, Berlin. (Engl. transl. Space, Time, Matter, Dover Publ. 1952).
• WHITNEY, H. (1972) Complex Analytic Varieties. Addison-Wesley Publ. Co, Reading.
• WILKOSZ, W. (1921) Sul concetto del differenziale esatto. Fundamenta Math. 2, 140-144. http://matwbn.icm.edu.pl/ksiazki/fm/fm2/fm2118.pdf
• YOUNG, W.H. (1910) The fundamental theorems of differential calculus. Cambridge Univ. Press, http://www.archive.org/details/ TheFundamentalTheoremsOfTheDifferentialCalculusNo11
• ZADDACH, A. (1988) Algebra de Grassmann y Geometría Proyectiva. Universidad de Tarapacá, Facultad de Ciencias.
• ZADDACH, A. (1994) Grassmanns Algebra in der Geometric mit Seitenblicken auf verwandte Strukturen. BI-Wissenschaftsverlag, Mannheim.
• ZERMELO, E. (1904) Beweis, dass jede Menge wohlgeordnet werden kann. Mathematische Annalen 59, 514-516. http://gdzdoc.sub.uni-goettingen.de/sub/digbib/loader?did=D28526
• ZORETTI, L. (1905) Sur les fonctions analytiques uniformes qui possèdent un ensemble parfait discontinu de points singuliers. J. Math. Pures Appl. 1, 1-51. http://gallica.bnf.fr/document?0=N107470
• ZORETTI, L. (1909) Un théorème de la théorie des ensembles. Bull. Soc. Math. France 37, 116-9. http://www.numdam.org/item?id=BSMF_1909— 37__116_0
• ZORETTI, L. (1912) Sur les ensembles de points. Encyclopédie des Sciences Mathematiques, II (vol. I), 113-170. http://gallica.bnf.fr/ark:/12148/bpt6k2025807