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References
Baiocchi, C., Gastaldi, F., and Tomarelli, F.: ‘Some existence results on noncoercive variational inequalities’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV 13 (1986), 617–659.
Barbu, V.: Optimal control of variational inequalities, Vol. 100 of Res. Notes Math., Pitman, 1984.
Brézis, H.: ‘Problèmes unilatéraux’, J. Math. Pures Appl. 51 (1972), 1–168.
Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland&Amer. Elsevier, 1973.
Duvaut, G., and Lions, J.L.: Les Inéquations en Mécanique et en Physique, Dunod, 1972.
Fichera, G.: ‘Problemi Elastostatici con Vincoli Unilaterali, il Problema di Signorini con Ambigue Condizioni al Contorno’, Mem. Accad. Naz. Lincei, VIII 7 (1964), 91–140.
Glowinski, R., Lions, J.L., and Trémoliéres, R.: Analyse numérique des inéquations variationnelles, Dunod, 1976.
Goeleven, D.: Noncoercive variational problems and related results, Vol. 357 of Res. Notes Math. Sci., Longman, 1996.
Haslinger, J., Hlavacek, I., and Necas, J.: ‘Numerical methods for unilateral problems’, in P.G. Giarlet and J.L. Lions (eds.): Solid Mechanics, Vol. IV of Handbook Numer.Anal., Elsevier, 1996, pp. 313–477.
Le, V.K., and Schmitt, K.: Global bifurcation in variationalinequalities, Springer, 1997.
Lions, J.L., and Stampacchia, G.: ‘Variational inequalities’, Commun. Pure Appl. Math. XX (1967), 493–519.
Moreau, J.J.: Fonctionnelles convexes. Sém. sur les Équations aux Dérivées Partielles, Collége de France, 1967.
Panagiotopoulos, P.D.: Inequality problems in mechanics and applications. Convex and nonconvex energy functions, Birkhäuser, 1985.
Begle, E.G.: ‘The Vietoris mappings theorem for bicompact spaces’, Ann. of Math. 51, no. 2 (1950), 534–550.
Borisovich, Yu.G.: ‘A modern appoach to the theory of topological characteristics of nonlinear operators II’: Global analysis: Studies and Applications IV, Vol. 1453 of Lecture Notes Math., Springer, 1990, pp. 21–49.
Borisovich, Yu.G., Bliznyakov, N.M., Fomenko, T.N., and Izrailevich, Y.A.: Introduction to differential and algebraic topology, Kluwer Acad. Publ., 1995.
Borisovich, Yu.G., Gelman, B.D., Myshkis, A.D., and Obukhovskii, V.V.: ‘Topological methods in the fixed-point theory of multi-valued maps’, Russian Math. Surveys 35, no. 1 (1980), 65–143. (Translated from the Russian.)
Borisovich, Yu.G., Gelman, B.D., Myshkis, A.D., and Obukhovskii, V.V.: ‘Multivalued mappings’, J. Soviet Math. 24 (1984), 719–791. (Translated from the Russian.)
Borisovich, Yu.G., Gelman, B.D., and Obukhovskii, V.V.: ‘Of some topological invariants of set-valued maps with nonconvex images’, Proc. Sem. Functional Analysis, Voronezh State Univ. 12 (1969), 85–95.
Bouvgin, D.G.: ‘Cones and Vietoris-Begle type theorems’, Trans. Amer. Math. Soc. 174 (1972), 155–183.
Eilenberg, S., and Montgomery, D.: ‘Fixed point theorems for multi-valued transformations’, Amer. J. Math. 68 (1946), 214–222.
Eilenberg, S., and Steenrod, N.: Foundations of algebraic topology, Princeton Univ. Press, 1952.
Górniewicz, L.: ‘On non-acyclic multi-valued mappings of subsets of Euclidean spaces’, Bull. Acad. Polon. Sci. 20, no. 5 (1972), 379–385.
Górniewicz, L.: ‘Homological methods in fixed-point theory of multi-valued maps’, Dissert. Math. CXXIX (1976), 1–71.
Granas, A., and Jaworowski, J.W.: ‘Some theorems on multi-valued maps of subsets of the Euclidean space’, Bull. Acad. Polon. Sci. 7, no. 5 (1959), 277–283.
Sklyarenko, E.G.: ‘Of some applications of theory of bundles in general topology’, Uspekhi Mat. Nauk 19, no. 6 (1964), 47–70. (In Russian.)
Spanier, E.H.: Algebraic topology, McGraw-Hill, 1966.
Antosik, P., and Swartz, C.: Matrix methods in analysis, Vol. 1113 of Lecture Notes Math., Springer, 1985.
Dunford, N., and Schwartz, J.T.: Linear operators, PartI, Interscience, 1958.
Hahn, H.: ‘Über Folgen linearer Operationen’, Monatsh.Math. Physik 32 (1922), 3–88.
Pap, E.: Null-additive set functions, Kluwer Acad. Publ.&Ister Sci., 1995.
Phillips, R.S.: ‘Integration in a convex linear topological space’, Trans. Amer. Math. Soc. 47 (1940), 114–145.
Rickart, C.E.: ‘Integration in a convex linear topological space’, Trans. Amer. Math. Soc. 52 (1942), 498–521.
Saks, S.: ‘Addition to the note on some functionals’, Trans. Amer. Math. Soc. 35 (1933), 967–974.
Vitali, G.: ‘Sull’ integrazione per serie’, Rend. Circ. Mat.Palermo 23 (1907), 137–155.
Behzad, M.: ‘Graphs and their chromatic numbers’, Doctoral Thesis Michigan State Univ. (1965).
Beineke, L.W.: ‘Derived graphs and digraphs’: Beiträge zur Graphentheorie, Teubner, 1968, pp. 17–33.
Chetwynd, A.G., and Hilton, A.J.W.: ‘1-factorizing regular graphs of high degree: an improved bound’, Discrete Math. 75 (1989), 103–112.
Chew, K.H.: ‘Total chromatic number of graphs of high maximum degree’, J. Combin. Math. Combin. Comput. 18 (1995), 245–254.
Chew, K.H.: ‘On Vizing’s theorem, adjacency lemma and fan argument generalized to multigraphs’, Discrete Math. 171 (1997), 283–286.
Choudom, S.A.: ‘Chromatic bound for a class of graphs’, Quart. J. Math. 28 (1977), 257–270.
Erdös, P., and Wilson, R.J.: ‘On the chromatic index of almost all graphs’, J. Combin. Th. 23 B (1977), 255–257.
Galvin, F.: ‘The list chromatic index of a bipartite multi-graph’, J. Combin. Th. B 68 (1995), 153–158.
Goldberg, M.K.: ‘Edge-coloring of multigraphs: recoloring technique’, J. Graph Theory 8 (1984), 123–137.
Hilton, A.J.W., and Hind, H.R.: ‘The total chromatic number of graphs having large maximum degree’, Discrete Math. 117 (1993), 127–140.
Holyer, I.: ‘The NP-completeness of edge-coloring’, SIAM J. Comput. 10 (1981), 718–720.
König, D.: ‘Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre’, Math. Ann. 77 (1916), 453–465.
Kostochka, A.V.: ‘List edge chromatic number of graphs with large girth’, Discrete Math. 101 (1992), 189–201.
Molloy, M., and Reed, B.: ‘A bound on the total chromatic number’, Combinatorica 18 (1998), 241–280.
Niessen, T., and Volkmann, L.: ‘Class 1 conditions depending on the minimum degree and the number of vertices of maximum degree’, J. Graph Theory 14 (1990), 225–246.
Randerath, H.: ‘The Vizing bound for the chromatic number based on forbidden pairs’, Doctoral Thesis RWTH Aachen (1998).
Shannon, C.E.: ‘A theorem on coloring the lines of a network’, J. Math. Phys. 28 (1949), 148–151.
Tait, P.G.: ‘On the colouring of maps’, Proc. R. Soc. Edinburgh 10 (1880), 501–503
Tait, P.G.: ‘On the colouring of maps’, Proc. R. Soc. Edinburgh 10 (1880), 729.
Vizing, V.G.: ‘On an estimate of the chromatic class of a p-graph’, Diskret. Anal. 3 (1964), 25–30. (In Russian.)
Vizing, V.G.: ‘Critical graphs with a given chromatic class’, Diskret. Anal. 5 (1965), 9–17. (In Russian.)
Vizing, V.G.: ‘Vertex colouring with given colours’, Diskret. Anal. 29 (1976), 3–10. (In Russian.)
Volkmann, L.: Fundamente der Graphentheorie, Springer, 1996.
Yap, H.P.: Total colourings of graphs, Vol. 1623 of Lecture Notes Math., Springer, 1996.
Carlitz, L.: ‘A degenerate Staudt-Clausen theorem’, Arch. Math. Phys. 7 (1956), 28–33.
Carlitz, L.: ‘A note on the Staudt-Clausen theorem’, Amer. Math. Monthly 64 (1957), 19–21.
Carlitz, L.: ‘Arithmetic properties of generalized Bernoulli numbers’, J. Reine Angew. Math. 202 (1959), 174–182.
Clarke, F.: ‘The universal von Staudt theorems’, Trans. Amer. Math. Soc. 315 (1989), 591–603.
Clarke, F., and Slavutskii, I.Sh.: ‘The integrality of the values of Bernoulli polynomials and of generalised Bernoulli numbers’, Bull. London Math. Soc. 29 (1997), 22–24.
Clausen, Th.: ‘Lehrsatz aus einer Abhandlung über die Bernoullischen Zahlen’, Astr. Nachr. 17 (1840), 351–352.
Girstmair, K.: ‘Ein v. Staudt-Clausenscher Satz für periodische Bernoulli-Zahlen’, Monatsh. Math. 104 (1987), 109–118.
Goss, D.: ‘Von Staudt for F q (T)’, Duke Math. J. 45 (1978), 887–910.
Hermite, Ch.: ‘Extrait d’une lettre à M. Borchardt (sur les nombres de Bernoulli)’, J. Reine Angew. Math. 81 (1876), 93–95.
Katz, N.: ‘The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers’, Math. Ann. 216 (1975), 1–4.
Lipschitz, R.: ‘Sur la représentation asymptotique de la valeur numérique ou de la partie entière des nombres de Bernoulli’, Bull. Sci. Math. (2) 10 (1886), 135–144.
Rado, R.: ‘A note on Bernoullian numbers’, J. London Math.Soc. 9 (1934), 88–90.
Staudt, K.G.C. von: ‘Beweis eines Lehrsatzes die Bernoulli’schen Zahlen betreffend’, J. Reine Angew. Math. 21 (1840), 372–374.
Staudt, K.G.C. von: De Numeris Bernoullianis, Erlangen, 1845.
Stern, M.A.: ‘Über eine Eigenschaft der Bernoulli’sehen Zahlen’, J. Reine Angew. Math. 81 (1876), 290–294.
Sun, Zhi-Hong: ‘Congruences for Bernoulli numbers and Bernoulli polynomials’, Discrete Math. 163 (1997), 153–163.
Vandiver, H.S.: ‘Simple explicit expressions for generalized Bernoulli numbers of the first order’, Duke Math. J. 8 (1941), 575–584.
Washington, L.C.: Introduction to cyclotomic fields, Springer, 1982, Second ed.: 1996.
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© 2000 Kluwer Academic Publishers and Elliott H. Lieb for “Lieb-Thirring inequalities” and “Thomas-Fermi theory”
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Hazewinkel, M. (2000). V. In: Hazewinkel, M. (eds) Encyclopaedia of Mathematics. Encyclopaedia of Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-1279-4_22
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