Skip to main content
SpringerLink
Log in

W

  • Chapter
Encyclopaedia of Mathematics

Part of the book series: Encyclopaedia of Mathematics ((ENMA))

  • 1040 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Awata, H., Fukuma, M., Matsuo, Y., and Odake, S.: ‘Character and determinant formulae of quasifinite representation of the W1+∞ algebra’, Coram. Math. Phys. 172 (1995), 377–400.

    MathSciNet  MATH  Google Scholar 

  2. Awata, H., Fukuma, M., Matsuo, Y., and Odake, S.: ‘Representation theory of the W1+∞ algebra’, Prog. Theor. Phys. Proc. Suppl. 118 (1995), 343–373.

    MathSciNet  Google Scholar 

  3. Frenkel, E., Kac, V., Radul, A., and Wang, W.: ‘W1+∞ and W(gl N ) with central charge N’, Comm. Math. Phys. 170 (1995), 337–357.

    MathSciNet  MATH  Google Scholar 

  4. Kac, V.G., and Peterson, D.H.: ‘Spin and wedge representations of infinite-dimensional Lie algebras and groups’, Proc. Nat. Acad. Sci. USA 78 (1981), 3308–3312.

    MathSciNet  MATH  Google Scholar 

  5. Kac, V., and Radul, A.: ‘Quasifinite highest weight modules over the Lie algebra of differential operators on the circle’, Comm. Math. Phys. 157 (1993), 429–457.

    MathSciNet  MATH  Google Scholar 

  6. Krichever, I.M., and Novikov, S.P.: ‘Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons’, Funkts. Anal. Appl. 21, no. 2 (1987), 46–63.

    MathSciNet  Google Scholar 

  7. Radul, A.O.: ‘Lie algebras of differential operators, their central extensions and W algebras’, Funkts. Anal. Appl. 25, no. 1 (1991), 33–49.

    MathSciNet  Google Scholar 

  8. Schlichenmaier, M.: ‘Differential operator algebras on compact Riemann surfaces’, in H.-D. Doebner, V.K. Dobrev, and A.G Ushveridze (eds.): Generalized Symmetries in Physics, Clausthal 1993, World Sci., 1994, pp. 425–435.

    Google Scholar 

References

  1. Bickel, P.J., and Freedman, D.A.: ‘Some asymptotic theory for the bootstrap’, Ann. Statist. 9 (1981), 1196–1217.

    MathSciNet  MATH  Google Scholar 

  2. Dall’Aglio, G.: ‘Sugli estremi dei momenti delle funzioni di ripartizione doppia’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3, no. 1 (1956), 33–74.

    Google Scholar 

  3. Dobrushin, R.L.: ‘Prescribing a system of random variables by conditional distributions’, Theor. Probab. Appl. 15 (1970), 458–486.

    MATH  Google Scholar 

  4. Fréchet, M.: ‘Les tableaux de corréllation dont les marges sont données’, Ann. Univ. Lyon Ser. A 20 (1957), 13–31.

    Google Scholar 

  5. Gray, R.M., Neuhoff, D.L., and Shields, P.C.: ‘A generalization to Ornstein’s d-distance with applications to information theory’, Ann. Probab. 3 (1975), 315–328.

    MathSciNet  MATH  Google Scholar 

  6. Kantorovich, L.V.: ‘On one effective method of solving certain classes of extremal problems’, Dokl. Akad. Nauk USSR 28 (1940), 212–215.

    Google Scholar 

  7. Kantorovich, L.V., and Rubinstein, G.Sh.: ‘On the space of completely additive functions’, Vestnik Leningrad Univ., Ser. Mat. Mekh. i Astron. 13, no. 7 (1958), 52–59. (In Russian.)

    MATH  Google Scholar 

  8. Mallows, C.L.: ‘A note on asymptotic joint normality’, Ann. Math. Stat. 43 (1972), 508–515.

    MathSciNet  MATH  Google Scholar 

  9. Salvemini, T.: ‘Sul calcolo degli indici di concordanza tra due caratteri quantitative’, Atti della VI Riunione della Soc. Ital. di Statistica, Roma (1943).

    Google Scholar 

  10. Wasserstein, L.N.: ‘Markov processes over denumerable products of spaces describing large systems of automata’, Probl. Inform. Transmission 5 (1969), 47–52.

    Google Scholar 

References

  1. Amir, D., and Lindenstrauss, J.: ‘The structure of weakly compact sets in Banach spaces’, Ann. of Math. 88 (1968), 35–46.

    MathSciNet  MATH  Google Scholar 

  2. Benyamini, Y., Rudin, M.E., and Wage, M.: ‘Continuous images of weakly compact subsets of Banach spaces’, Pacific J. Math. 70 (1977), 309–324.

    MathSciNet  MATH  Google Scholar 

  3. Davis, W.J., Figiel, T., Johnson, W.B., and Pelczynski, A.: ‘Factoring weakly compact operators’, J. Funct. Anal. 17 (1974), 311–327.

    MathSciNet  MATH  Google Scholar 

  4. Deville, R., Godefroy, G., and Zizler, V.: Smoothness and renormings in Banach spaces, Longman, 1993.

    MATH  Google Scholar 

  5. Diestel, J.: Geometry of Banach spaces: Selected topics. Vol. 485 of Lecture Notes Math., Springer, 1975.

    MATH  Google Scholar 

  6. Fabian, M.: Gâteaux differentiability of convex functions and topology, Wiley-Interscience, 1997.

    MATH  Google Scholar 

  7. Fabian, M., and Godefroy, G.: ‘The dual of every Asplund space admits a projectional resolution of the identity’, Studia Math. 91 (1988), 141–151.

    MathSciNet  MATH  Google Scholar 

  8. Gul’ko, S.P.: ‘On the structure of spaces of continuous functions and their complete paracompactness’, Russian Math. Surveys 34, no. 6 (1979), 36–44.

    MathSciNet  MATH  Google Scholar 

  9. John, K., and Zizler, V.: ‘Smoothness and its equivalents in weakly compactly generated Banach spaces’, J. Funct. Anal. 15 (1974), 1–11.

    MathSciNet  Google Scholar 

  10. Johnson, W.B., and Lindenstrauss, J.: ‘Some remarks on weakly compactly generated Banach spaces’, Israel J. Math. 17 (1974), 219–230

    MathSciNet  MATH  Google Scholar 

  11. Johnson, W.B., and Lindenstrauss, J.: ‘Some remarks on weakly compactly generated Banach spaces’, Corrigendum: 32 (1979), 382–383.

    MathSciNet  MATH  Google Scholar 

  12. Lindenstrauss, J.: ‘On nonseparable reflexive Banach spaces’, Bull. Amer. Math. Soc. 72 (1966), 967–970.

    MathSciNet  MATH  Google Scholar 

  13. Lindenstrauss, J.: ‘Weakly compact sets: their topological properties and the Banach spaces they generate’, in R.D. Anderson (ed.): Symp. Infinite Dimensional Topol., Vol. 69 of Math. Studies, 1972, pp. 235–273.

    Google Scholar 

  14. Negrepontis, S.: ‘Banach spaces and topology’, in K. Kunen and J.E. Vaughan (eds.): Handbook of set-theoretic topology, Elsevier Sci., 1984, pp. 1045–1142.

    Google Scholar 

  15. Orihuela, J., and Valdivia, M.: ‘Projective generators and resolutions of identity in Banach spaces’, Rev. Mat. Univ. Complutense Madr. 2 (1989), 179–199.

    MathSciNet  Google Scholar 

  16. Rosenthal, H.P.: ‘The heredity problem for weakly compactly generated Banach spaces’, Compositio Math. 28 (1974), 83–111.

    MathSciNet  MATH  Google Scholar 

  17. Stegall, Ch.: ‘A proof of the theorem of Amir and Lindenstrauss’, Israel J. Math. 68 (1989), 185–192.

    MathSciNet  MATH  Google Scholar 

  18. Talagrand, M.: ‘Sur une conjecture de H.H. Corson’, Bull. Sci. Math. 99 (1975), 211–212.

    MathSciNet  Google Scholar 

  19. Troyanski, S.L.: ‘On locally uniformly convex and differen-tiable norms in certain non-separable Banach spaces’, Studia Math. 37 (1971), 173–180.

    MathSciNet  MATH  Google Scholar 

  20. Vašák, L.: ‘On one generalization of weakly compactly generated Banach spaces’, Studia Math. 70 (1981), 11–19.

    MathSciNet  MATH  Google Scholar 

References

  1. Drezner, Z.: Facility location, a survey of applications and methods, Springer, 1995.

    Google Scholar 

  2. Francis, R.L., and White, J.A.: Facility layout and location: an analytical approach, Prentice- Hall, 1974.

    Google Scholar 

  3. Kaplan, W., and Yang, W.H.: ‘Duality theorem for a generalized Fermat-Weber problem’, Math. Progr. 7, no. 6 (1997), 285–297.

    MathSciNet  Google Scholar 

  4. Kuhn, H.W.: ‘On a pair of dual nonlinear programs’, in J. Abadie (ed.): Nonlinear programming, Wiley, 1967, pp. 39–54.

    Google Scholar 

  5. Love, R.F., Morris, J.G., and Wesolovsky, CO.: Facilities location, models and methods, North-Holland, 1988.

    MATH  Google Scholar 

  6. Plastria, F.: ‘Continuous location problems’, in Z. Drezner (ed.): Facility location, a survey of applications and methods, Springer, 1995, pp. 225–262.

    Google Scholar 

  7. Weber, A.: Theory of the location of industries, Univ. Chicago Press, 1957. (Translated from the German.)

    Google Scholar 

  8. Weiszfeld, E.: ‘Sur le point lequel la somme des distances de n points donnés est minimum’, Tôhoku Math. J. 43 (1937), 355–386.

    Google Scholar 

  9. Witzgall, C: ‘Optimal location of a central facility: mathematical models and concepts’, Nat. Bureau Standards Report 8388 (1964).

    Google Scholar 

References

  1. Albrecht, E.: ‘Several variable spectral theory in the non-commutative case’: Spectral Theory, Vol. 8 of Banach Centre Publ., PWN, 1982, pp. 9–30.

    Google Scholar 

  2. Anderson, R.F.V.: ‘The Weyl functional calculus’, J. Fund. Anal. 4 (1969), 240–267.

    MATH  Google Scholar 

  3. Anderson, R.F.V.: ‘On the Weyl functional calculus’, J. Funct. Anal. 6 (1970), 110–115.

    MATH  Google Scholar 

  4. Anderson, R.F.V.: ‘The multiplicative Weyl functional calculus’, J. Funct. Anal. 9 (1972), 423–440.

    MATH  Google Scholar 

  5. Atiyah, M., Bott, R., and GÅrding, L.: ‘Lacunas for hyperbolic differential operators with constant coefficients I’, Acta Math. 124 (1970), 109–189.

    MathSciNet  MATH  Google Scholar 

  6. Atiyah, M., Bott, R., and GÅrding, L.: ‘Lacunas for hyperbolic differential operators with constant coefficients II’, Acta Math. 131 (1973), 145–206.

    MathSciNet  MATH  Google Scholar 

  7. Bazer, J., and Yen, D.H.Y.: ‘The Riemann matrix of (2+1)-dimensional symmetric hyperbolic systems’, Commun. Pure Appl. Math. 20 (1967), 329–363.

    MathSciNet  MATH  Google Scholar 

  8. Bazer, J., and Yen, D.H.Y.: ‘Lacunas of the Riemann matrix of symmetric-hyperbolic systems in two space variables’, Commun. Pure Appl. Math. 22 (1969), 279–333.

    MathSciNet  MATH  Google Scholar 

  9. Feynman, R.P.: ‘An operator calculus having applications in quantum electrodynamics’, Phys. Rev. 84 (1951), 108–128.

    MathSciNet  MATH  Google Scholar 

  10. Folland, G.B.: Harmonic analysis in phase space, Princeton Univ. Press, 1989.

    MATH  Google Scholar 

  11. Gotay, M.J., Grundling, H.B., and Tuynman, G.M.: ‘Obstruction results in quantization theory’, J. Nonlinear Sci. 6 (1996), 469–498.

    MathSciNet  MATH  Google Scholar 

  12. Groenewold, H.J.: ‘On the principles of elementary quantum mechanics’, Physica 12 (1946), 405–460.

    MathSciNet  MATH  Google Scholar 

  13. Hörmander, L.: ‘The Weyl calculus of pseudodifferential operators’, Commun. Pure Appl. Math. 32 (1979), 359–443.

    MATH  Google Scholar 

  14. Hörmander, L.: The analysis of linear partial differential operators, Vol. III, Springer, 1985.

    Google Scholar 

  15. Jefferies, B., and McIntosh, A.: ‘The Weyl calculus and Clifford analysis’, Bull. Austral. Math. Soc. 57 (1998), 329–341.

    MathSciNet  MATH  Google Scholar 

  16. Kohn, J.J., and Nirenberg, L.: ‘An algebra of pseudodifferential operators’, Commun. Pure Appl. Math. 18 (1965), 269–305.

    MathSciNet  MATH  Google Scholar 

  17. Maslov, V.P.: Operational methods, Mir, 1976.

    MATH  Google Scholar 

  18. Moyal, J.E.: ‘Quantum mechanics as a statistical theory’, Proc. Cambridge Philos. Soc. 45 (1949), 99–124.

    MathSciNet  MATH  Google Scholar 

  19. Nazaikinskii, V.E., Shatalov, V.E., and Sternin, B.Yu.: Methods of noncommutative analysis, Vol. 22 of Studies Math., W. de Gruyter, 1996.

    Google Scholar 

  20. Nelson, E.: ‘A functional calculus for non-commuting operators’, in F.E. Browder (ed.): Functional Analysis and Related Fields: Proc. Conf. in Honor of Professor Marshal Stone (Univ. Chicago, May (1968), Springer, 1970, pp. 172–187.

    Google Scholar 

  21. Petrovsky, I.: ‘On the diffusion of waves and lacunas for hyperbolic equations’, Mat. Sb. 17 (1945), 289–368. (In Russian.)

    Google Scholar 

  22. Pool, J.C.T.: ‘Mathematical aspects of the Weyl correspondence’, J. Math. Phys. 7 (1966), 66–76.

    MathSciNet  MATH  Google Scholar 

  23. Taylor, M.E.: ‘Functions of several self-adjoint operators’, Proc. Amer. Math. Soc. 19 (1968), 91–98.

    MathSciNet  MATH  Google Scholar 

  24. Taylor, M.E.: Pseudodifferential operators, Princeton Univ. Press, 1981.

    MATH  Google Scholar 

  25. Treves, F.: Introduction to pseudodifferential and Fourier integral operators, Vol. I, Plenum, 1980.

    MATH  Google Scholar 

  26. Vassiliev, V.A.: Ramified integrals, singularities and lacunas, Kluwer Acad. Publ., 1995.

    MATH  Google Scholar 

  27. Weyl, H.: The theory of groups and quantum mechanics, Methuen, 1931, Reprint: Dover, 1950.

    Google Scholar 

References

  1. Gadella, M.: ‘Moyal formulation of quantum mechanics’, Fortschr. Phys. 43 (1995), 229.

    MathSciNet  MATH  Google Scholar 

  2. Gracia-Bondia, J.M., and Varilly, J.C.: ‘Algebras of distributions suitable for phase space quantum mechanics’, J. Math. Phys. 29 (1988), 869.

    MathSciNet  MATH  Google Scholar 

  3. Gracia-Bondia, J.M., and Varilly, J.C.: ‘The Moyal representation of spin’, Ann. Phys. (NY) 190 (1989), 107.

    MathSciNet  MATH  Google Scholar 

  4. Grossmann, A.: ‘Parity operator and quantization of ô functions’, Comm. Math. Phys. 48 (1976), 191.

    MathSciNet  MATH  Google Scholar 

  5. Royer, A.: ‘Wigner function as the expectation value of a parity operator’, Phys. Rev. A 15 (1977), 449.

    MathSciNet  Google Scholar 

  6. Varilly, J.C.: ‘The Stratonovich-Weyl correspondence: a general approach to Wigner functions’, BIBOS preprint 345 Univ. Bielefeld, Germany (1988).

    Google Scholar 

  7. Weyl, H.: The theory of groups and quantum mechanics, Dover, 1931.

    MATH  Google Scholar 

References

  1. Andersen, H.H., Jantzen, J.C., and Soergel, W.: ‘Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: Independence of p’, Astérisque 220 (1994), 1–321.

    Google Scholar 

  2. Ariki, S.: ‘On the decomposition numbers of the Hecke algebra of G(m, l, n)’ J. Math. Kyoto Univ. 36, no. 4 (1996), 789–808.

    MathSciNet  MATH  Google Scholar 

  3. Beilinson, A.A., and Bernstein, I.N.: ‘Localisation de g-modules’, CR. Acad. Sci. Paris Ser. I Math. 292, no. 1 (1981), 15–18.

    MathSciNet  MATH  Google Scholar 

  4. Brylinski, J.L., and Kashiwara, M.: ‘Kazhdan Lusztig conjecture and holonomic systems’, Invent. Math. 64, no. 3 (1981), 387–410.

    MathSciNet  MATH  Google Scholar 

  5. Carter, R., and Lusztig, G.: ‘On the modular representations of the general linear and symmetric groups’, Math. Z. 136 (1974), 193–242.

    MathSciNet  MATH  Google Scholar 

  6. Dipper, R., and James, G.: ‘The g-Schur algebra’, Proc. London Math. Soc. 59 (1989), 23–50.

    MathSciNet  MATH  Google Scholar 

  7. Dipper, R., and James, G.: ‘ç-Tensor space and ç-Weyl modules’, Trans. Amer. Math. Soc. 327 (1991), 251–282.

    MathSciNet  MATH  Google Scholar 

  8. Green, J.A.: Polynomial representations of GL n , Vol. 830 of Lecture Notes Math., Springer, 1980.

    Google Scholar 

  9. Kashiwara, M., and Tanisaki, T.: ‘Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. I,II’, Duke Math. J. 77–84 (1995–1996), 21–62; 771–81.

    Google Scholar 

  10. Kazhdan, D., and Lusztig, G.: ‘Representations of Coxeter groups and Hecke algebras’, Invent. Math. 53, no. 2 (1979), 165–184.

    MathSciNet  MATH  Google Scholar 

  11. Kazhdan, D., and Lusztig, G.: ‘Affine Lie algebras and quantum groups’, Duke Math. J. 62 (1991), Also: Internat. Math. Res. Notices 2 (1991), 21–29.

    MathSciNet  Google Scholar 

  12. Kazhdan, D., and Lusztig, G.: ‘Tensor structures arising from affine Lie algebras I-III, III-IV’, J. Amer. Math. Soc. 6–7 (1993/94), 905–1011; 335–453.

    MathSciNet  Google Scholar 

  13. Lascoux, A., Leclerc, B., and Thibon, J.-Y.: ‘Hecke algebras at roots of unity and crystal bases of quantum affine algebras’, Comm. Math. Phys. 181, no. 1 (1996), 205–263.

    MathSciNet  MATH  Google Scholar 

  14. Schur, I.: ‘Uber die rationalen Darstellungen der allgemeinen linearen Gruppe (1927)’: J. Schur, Gesammelte Abhandlungen III, Springer, 1973, pp. 68–85.

    Google Scholar 

  15. Schur, I.: ‘Uber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen (1901)’: J. Schur, Gesammelte Abhandlungen I, Springer, 1973, pp. 1–70.

    Google Scholar 

  16. Soergel, W.: ‘Charakterformeln für Kipp-Moduln über Kac-Moody-Algebren’, Represent. Theory 1 (1997), 115–132, Electronic.

    MathSciNet  MATH  Google Scholar 

  17. Soergel, W.: ‘Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules’, Represent. Theory 1 (1997), 83–114.

    MathSciNet  MATH  Google Scholar 

  18. Varagnolo, M., and Vasserot, E.: ‘Canonical bases and Lusztig conjecture for quantized sl(N) at roots of unity’, Preprint (1998), math.QA/9803023.

    Google Scholar 

  19. Weyl, H.: The classical groups, their invariants and representations, Princeton Univ. Press, 1966.

    Google Scholar 

References

  1. Moór, A.: ‘Otsukische Übertragung mit rekurrenter Maßtensor’, Acta Sci. Math. 40 (1978), 129–142.

    MATH  Google Scholar 

  2. Moor, A.: ‘Über verschiedene geodätische Abweichungen in Weyl-Otsukischen Räumen’, Publ. Math. Debrecen 28 (1981), 247–258.

    MathSciNet  MATH  Google Scholar 

  3. Moör, A.: ‘Über die Begründung von Finsler-Otschukischen Räumen und ihre Dualität’, Tensor N.S. 37 (1982), 121–129.

    MathSciNet  Google Scholar 

  4. Moör, A.: ‘Über Transformationsgruppen in Weyl-Otsukischen Räumen’, Publ. Math. Debrecen 29 (1982), 241–250.

    MathSciNet  Google Scholar 

  5. Moör, A.: ‘Über spezielle Finsler-Otsukische Räume’, Publ. Math. Debrecen 31 (1984), 185–196.

    Google Scholar 

  6. Otsuki, T.: ‘On general connections. I’, Math. J. Okayama Univ. 9 (1959–60), 99–164.

    MathSciNet  Google Scholar 

  7. Otsuki, T.: ‘On metric general connections’, Proc. Japan Acad. 37 (1961), 183–188.

    MathSciNet  MATH  Google Scholar 

References

  1. Beals, R., and Fefferman, C: ‘On local solvability of linear partial differential equations’, Ann. oj Math. 97 (1973), 482–498.

    MathSciNet  MATH  Google Scholar 

  2. Bony, J.-M.: ‘Second microlocalization and propagation of singularities for semi-linear hyperbolic equations’, in K. Mizohata (ed.): Hyperbolic Equations and Related Topics, Kinokuniya, 1986, pp. 11–49.

    Google Scholar 

  3. Bony, J.-M., and Chemin, J.-Y.: ‘Espaces fonctionnels associés au calcul de Weyl-Hörmander’, Bull. Soc. Math. France 122 (1994), 77–118.

    MathSciNet  MATH  Google Scholar 

  4. Bony, J.-M., and Lerner, N.: ‘Quantification asymtotique et microlocalisations d’ordre supérieur’, Ann. Sci. Ecole Norm. Sup. 22 (1989), 377–483.

    MathSciNet  MATH  Google Scholar 

  5. Boulkhemair, A.: ‘Remarque sur la quantification de Weyl pour la classe de symboles S0 1,1’ CR. Acad. Sci. Paris 321, no. 8 (1995), 1017–1022.

    MathSciNet  MATH  Google Scholar 

  6. Fefferman, C, and Phong, D.H.: ‘On positivity of pseudo-differential operators’, Proc. Nat. Acad. Sci. USA 75 (1978), 4673–4674.

    MathSciNet  MATH  Google Scholar 

  7. Hörmander, L.: ‘The Weyl calculus of pseudo-differential operators’, Commun. Pure Appl. Math. 32 (1979), 359–443.

    MATH  Google Scholar 

  8. Hörmander, L.: The analysis of linear partial differential operators, Springer, 1985.

    Google Scholar 

  9. Segal, I.: ‘Transforms for operators and asymptotic automorphisms over a locally compact abelian group’, Math. Scand. 13 (1963), 31–43.

    MathSciNet  MATH  Google Scholar 

  10. Unterberger, A.: ‘Oscillateur harmonique et opérateurs pseudo-différentiels’, Ann. Inst. Fourier 29, no. 3 (1979), 201–221.

    MathSciNet  MATH  Google Scholar 

  11. Weil, A.: ‘Sur certains groupes d’opérateurs unitaires’, Acta Math. 111 (1964), 143–211.

    MathSciNet  MATH  Google Scholar 

  12. Weyl, H.: Gruppentheorie und Quantenmechanik, S. Hirzel, 1928.

    MATH  Google Scholar 

References

  1. Eisenhart, L.P.: Riemannian geometry, Princeton Univ. Press, 1966.

    Google Scholar 

  2. Petrov, A.Z.: Einstein spaces, Pergamon, 1969.

    MATH  Google Scholar 

  3. Schouten, J.A.: ‘Ueber die konforme Abbildung n-dimensionaler Mannigfaltigkeiten mit quadratischer Massbestimmung auf eine Mannigfaltigkeit mit Euklidischer Massbestimmung’, Math. Z. 11 (1921), 58–88.

    MathSciNet  Google Scholar 

  4. Weyl, H.: ‘Reine Infinitesimalgeometrie’, Math. Z. 2 (1918), 384–411.

    MathSciNet  Google Scholar 

References

  1. Gustafson, K.: ‘Necessary and sufficient conditions for Weyl’s theorem’, Michigan Math. J. 19 (1972), 71–81.

    MathSciNet  MATH  Google Scholar 

  2. Kato, T.: Perturbation theory for linear operators, Springer, 1976.

    MATH  Google Scholar 

  3. Reed, M., and Simon, B.: Methods in modern mathematical physics IV. Analysis of operators, Acad. Press, 1978.

    Google Scholar 

  4. Weyl, H.: ‘Über quadratische Formen, deren Differenz vollstetig ist’, Rend. Circ. Mat. Palermo 27 (1909), 373–392.

    Google Scholar 

References

  • Eklof, P.C., and Mekler, A.H.: Almost free modules: settheoretic methods, North-Holland, 1990.

    Google Scholar 

  • Eklof, P.C., and Shelah, S.: ‘On Whitehead modules’, J. Algebra 142 (1991), 492–510.

    MathSciNet  MATH  Google Scholar 

  • Eklof, P.C., and Shelah, S.: ‘A combinatorial principle equivalent to the existence of non-free Whitehead groups’, Contemp. Math. 171 (1994), 79–98.

    MathSciNet  Google Scholar 

  • Eklof, P.C., and Trlifaj, J.: ‘How to make Ext vanish’, preprint.

    Google Scholar 

  • Shelah, S.: ‘Infinite abelian groups, Whitehead problem and some constructions’, Israel J. Math. 18 (1974), 243–256.

    MathSciNet  MATH  Google Scholar 

  • Shelah, S.: ‘A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals’, Israel J. Math. 21 (1975), 319–349.

    MathSciNet  MATH  Google Scholar 

  • Trlifaj, J.: Associative rings and the Whitehead property of modules, Vol. 63 of Algebra Berichte, R. Fischer, 1990.

    MATH  Google Scholar 

  • Trlifaj, J.: ‘Whitehead test modules’, Trans. Amer. Math. Soc. 348 (1996), 1521–1554.

    MathSciNet  MATH  Google Scholar 

References

  1. Illanes, A.: ‘The space of Whitney decompositions’, Ann. Inst. Mat. Univ. Nac. Autonoma México 28 (1988), 47–61.

    MathSciNet  MATH  Google Scholar 

  2. Illanes, A., and Nadler Jr., S.B.: Hyperspaces, fundamentals and recent advances, Vol. 216 of Monogr. Textbooks Pure Appl. Math., M. Dekker, 1999.

    MATH  Google Scholar 

References

  1. Baer, R.: ‘Der Kern, eine charakteristische Untergruppe’, Compositio Math. 1 (1934), 254–283.

    MathSciNet  MATH  Google Scholar 

  2. Beidleman, J.C., Dixon, M.R., and Robinson, D.J.S.: ‘The generalized Wielandt subgroup of a group’, Canad. J. Math. 47, no. 2 (1995), 246–261.

    MathSciNet  MATH  Google Scholar 

  3. Brandl, R., Franciosi, S., and Giovanni, F. de: ‘On the Wielandt subgroup of infinite soluble groups’, Glasgow Math. J. 32 (1990), 121–125.

    MathSciNet  MATH  Google Scholar 

  4. Bryce, R.A., and Cossey, J.: ‘The Wielandt subgroup of a finite soluble group’, J. London Math. Soc. (2) 40 (1989), 244–256.

    MathSciNet  MATH  Google Scholar 

  5. Camina, A.R.: ‘The Wielandt length of finite groups’, J. Algebra 15 (1970), 142–148.

    MathSciNet  MATH  Google Scholar 

  6. Casolo, C: ‘Soluble groups with finite Wielandt length’, Glasgow Math. J. 31 (1989), 329–334.

    MathSciNet  MATH  Google Scholar 

  7. Cossey, J.: ‘The Wielandt subgroup of a polycyclic group’, Glasgow Math. J. 33 (1991), 231–234.

    MathSciNet  MATH  Google Scholar 

  8. Cossey, J.: ‘Finite groups generated by subnormal T-subgroups’, Glasgow Math. J. 37 (1995), 363–371.

    MathSciNet  MATH  Google Scholar 

  9. Giovanni, F. de, and Franciosi, S.: ‘Groups in which every infinite subnormal subgroup is normal’, J. Algebra 96 (1985), 566–580.

    MathSciNet  MATH  Google Scholar 

  10. Heineken, H.: ‘Groups with restrictions on their infinite subnormal subgroups’, Proc. Edinburgh Math. Soc. 31 (1988), 231–241.

    MathSciNet  MATH  Google Scholar 

  11. Kegel, O.H.: ‘Über den Normalisator von subnormalen und erreichbaren Untergruppen’, Math. Ann. 163 (1966), 248–258.

    MathSciNet  MATH  Google Scholar 

  12. Lennox, J.C., and Stonehewer, S.E.: Subnormal subgroups of groups, Oxford, 1987.

    MATH  Google Scholar 

  13. Ormerod, E.: ‘The Wielandt subgroup of a metacyclic p-group’, Bull. Austral. Math. Soc. 42 (1990), 499–510.

    MathSciNet  MATH  Google Scholar 

  14. Robinson, D.J.S.: ‘Groups in which normality is a transitive relation’, Proc. Cambridge Philos. Soc. 60 (1964), 21–38.

    MathSciNet  MATH  Google Scholar 

  15. Robinson, D.J.S.: ‘On the theory of subnormal subgroups’, Math. Z. 89 (1965), 30–51.

    MathSciNet  MATH  Google Scholar 

  16. Roseblade, J.E.: ‘On certain subnormal coalition classes’, J. Algebra 1 (1964), 132–138.

    MathSciNet  MATH  Google Scholar 

  17. Schenkman, E.: ‘On the norm of a group’, Illinois J. Math. 4 (1960), 150–152.

    MathSciNet  MATH  Google Scholar 

  18. Wielandt, H.: ‘Über den Normalisator der subnormalen Untergruppen’, Math. Z. 69 (1958), 463–465.

    MathSciNet  MATH  Google Scholar 

References

  1. Adler, R.J.: ‘The uniform dimension of the level sets of a Brownian sheet’, Ann. of Probab. 6 (1978), 509–518.

    MATH  Google Scholar 

  2. Adler, R.J.: The geometry of random fields, Wiley, 1981.

    MATH  Google Scholar 

  3. Chentsov, N.N.: ‘Wiener random fields depending on several parameters’, Dokl. Akad. Nauk SSSR 106 (1956), 607–609.

    MathSciNet  MATH  Google Scholar 

  4. Chentsov, N.N.: ‘A multiparametric Brownian motion Levy and generalized white noise’, Theory Probab. Appl. 2 (1957), 281–282.

    MATH  Google Scholar 

  5. Csörgö, M., and Révész, P.: Strong approximations inprobability and statistics, Akad. Kiado, 1981.

    MATH  Google Scholar 

  6. Dalang, R.C., and Mountford, T.: ‘Nondifferentiability of curves on the Brownian sheet’, Ann. of Probab. 24 (1996), 182–195.

    MathSciNet  MATH  Google Scholar 

  7. Dalang, R.C., and Russo, F.: ‘A prediction problem for the Brownian sheet’, J. Multivariate Anal. 26 (1988), 16–47.

    MathSciNet  MATH  Google Scholar 

  8. Dalang, R.C., and Walsh, J.B.: ‘The sharp Markov property of the Brownian sheet and related processes’, Acta Math. 168 (1992), 153–218.

    MathSciNet  MATH  Google Scholar 

  9. Dalang, R.C., and Walsh, J.B.: ‘Geography of the level sets of the Brownian sheet’, Probab. Th. Rel. Fields 96 (1993), 153–176.

    MathSciNet  MATH  Google Scholar 

  10. Dalang, R.C., and Walsh, J.B.: ‘The structure of a Brownian bubble’, Probab. Th. Rel. Fields 96 (1993), 475–501.

    MathSciNet  MATH  Google Scholar 

  11. Dudley, R.M.: ‘Sample functions of the Gaussian process’, Ann. of Probab. 1 (1973), 66–103.

    MathSciNet  MATH  Google Scholar 

  12. Goodman, V.: ‘Distribution estimates for functionals of the two-parameter Wiener process’, Ann. of Probab. 4 (1976), 977–982.

    MATH  Google Scholar 

  13. Kendall, W.: ‘Contours of Brownian processes with several-dimensional time’, Z. Wahrscheinlichkeitsth. verw. Gebiete 52 (1980), 269–276.

    MathSciNet  Google Scholar 

  14. Kitagava, T.: ‘Analysis of variance applied to function spaces’, Mem. Fac. Sci. Kyushu Univ. Ser. A 6 (1951), 41–53.

    Google Scholar 

  15. Levy, P.: Processes stochastiques et mouvement brownien, Gauthier-Villars, 1948.

    Google Scholar 

  16. McKean Jr., H.P.: ‘Brownian motion with a several-dimensional time’, Theory Probab. Appl. 8 (1963), 335–354.

    MathSciNet  MATH  Google Scholar 

  17. Molchan, G.M.: ‘Some problems for Levy’s Brownian motion’, Theory Probab. Appl. 12 (1967), 682–690.

    MATH  Google Scholar 

  18. Orey, S., and Pruitt, W.: ‘Sample functions of the N-parameter Wiener process’, Ann. of Probab. 1 (1973), 138–163.

    MathSciNet  MATH  Google Scholar 

  19. Paranjape, S.R., and Park, C: ‘Distribution of the supre-mum of the two-parameter Yeh-Wiener process on the boundary’, J. Appl. Probab. 10 (1973), 875–880.

    MathSciNet  MATH  Google Scholar 

  20. Paranjape, S.R., and Park, C: ‘Laws of iterated logarithm of multiparameter Wiener process’, J. Multivariate Anal. 3 (1973), 132–136.

    MathSciNet  MATH  Google Scholar 

  21. Piterbarg, V.I.: Asymptotic methods in the theory of Gaussian processes and fields, Amer. Math. Soc, 1996.

    MATH  Google Scholar 

  22. Rosanov, Yu.A.: Markov random fields, Springer, 1982.

    Google Scholar 

  23. Walsh, J.B.: ‘Propagation of singularities in the Brownian sheet’, Ann. of Probab. Ann. 10 (1982), 279–288.

    MATH  Google Scholar 

  24. Yadrenko, M.I.: Spectral theory of random fields, Optim. Software, 1983.

    MATH  Google Scholar 

References

  1. Brezzi, F., and Markowich, P.: ‘The three-dimensional Wigner-Poisson problem: existence, uniqueness and approximation’, Math. Meth. Appl. Sci. 14 (1991), 35.

    MathSciNet  MATH  Google Scholar 

  2. Folland, G.B.: Harmonic analysis in phase space, Princeton Univ. Press, 1989.

    MATH  Google Scholar 

  3. Illner, R., Lange, H., and Zweifel, P.F.: ‘Global existence and asymptotic behaviour of solutions of the Wigner-Poisson and Schrödinger-Poisson systems’, Math. Meth. Appl. Sci. 17 (1994), 349–376.

    MathSciNet  MATH  Google Scholar 

  4. Neumann, J. von: Mathematical foundations of quantum mechanics, Princeton Univ. Press, 1955.

    MATH  Google Scholar 

  5. Neunzert, H.: ‘The nuclear Vlasov equation: methods and results that can (not) be taken over from the ‘classical’ case’, Il Nuovo Cimento 87A (1985), 151–161.

    MathSciNet  Google Scholar 

  6. Weyl, H.: The theory of groups and quantum mechanics, Dover, 1950.

    Google Scholar 

  7. Wigner, E.: ‘On the quantum correction for thermodynamic equilibrium’, Phys. Rev. 40 (1932), 749–759.

    MATH  Google Scholar 

  8. Zweifel, P.F.: ‘The Wigner transform and the Wigner-Poisson system’, Trans. Theor. Stat. Phys. 22 (1993), 459–484.

    MathSciNet  MATH  Google Scholar 

  9. Zweifel, P.F., and Toomire, B.: ‘Quantum transport theory’, Trans. Theor. Stat. Phys. 27 (1998), 347–359.

    MathSciNet  MATH  Google Scholar 

References

  1. Brass, H.: Quadraturverfahren, Vandenhoeck&Ruprecht, 1977.

    MATH  Google Scholar 

  2. Davis, P.J., and Rabinowitz, P.: Methods of numerical integration, 2 ed., Acad. Press, 1984.

    MATH  Google Scholar 

  3. Engels, H.: Numerical quadrature and cubature, Acad. Press, 1980.

    MATH  Google Scholar 

  4. Wilf, H.S.: ‘Exactness conditions in numerical quadrature’, Numer. Math. 6 (1964), 315–319.

    MathSciNet  MATH  Google Scholar 

References

  1. Agaian, S.S.: Hadamard matrices and their applications, Vol. 1168 of Lecture Notes Math., Springer, 1985.

    MATH  Google Scholar 

  2. Baumert, L.D., and Hall Jr., M.: ‘A new construction for Hadamard matrices’, Bull. Amer. Math. Soc. 71 (1965), 169–170.

    MathSciNet  MATH  Google Scholar 

  3. Djokovic, D.Z.: ‘Williamson matrices of order An for n = 33,35,39’, Discrete Math. 115 (1993), 267–271.

    MathSciNet  MATH  Google Scholar 

  4. Geramita, A.V., and Seberry, J.: Orthogonal designs: Quadratic forms and Hadamard matrices, M. Dekker, 1979.

    MATH  Google Scholar 

  5. Goethals, J.M., and Seidel, J.J.: ‘A skew-Hadamard matrix of order 36’, J. Austral. Math. Soc. A 11 (1970), 343–344.

    MathSciNet  MATH  Google Scholar 

  6. Plotkin, M.: ‘Decomposition of Hadamard matrices’, J. Combin. Th. A 2 (1972), 127–130.

    MathSciNet  Google Scholar 

  7. Seberry, J., and Yamada, M.: ‘Hadamard matrices, sequences and block designs’, in J.H. Dinitz and D.R. Stinson (eds.): Contemporary Design Theory: A Collection of Surveys, Wiley, 1992, pp. 431–560.

    Google Scholar 

  8. Wallis, J.S.: ‘Construction of Williamson type matrices’, Linear and Multilinear Algebra 3 (1975), 197–207.

    MathSciNet  Google Scholar 

  9. Wallis, W.D., Street, A.P., and Wallis, J.S.: Combinatorics: Room squares, sum-free sets and Hadamard matrices, Vol. 292 of Lecture Notes Math., Springer, 1972.

    Google Scholar 

  10. Williamson, J.: ‘Hadamard’s determinant theorem and the sum of four squares’, Duke Math. J. 11 (1944), 65–81.

    MathSciNet  MATH  Google Scholar 

  11. Xia, M.Y.: ‘An infinite class of supplementary difference sets and Williamson matrices’, J. Combin. Th. A 58 (1991), 310–317.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CWI, P.O.Box 94079, 1090GB, Amsterdam, The Netherlands

    Prof. Dr. Michiel Hazewinkel

Authors
  1. Prof. Dr. Michiel Hazewinkel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. CWI, P.O.Box 94079, 1090GB, Amsterdam, The Netherlands

    Michiel Hazewinkel

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Kluwer Academic Publishers and Elliott H. Lieb for “Lieb-Thirring inequalities” and “Thomas-Fermi theory”

About this chapter

Cite this chapter

Hazewinkel, M. (2000). W. In: Hazewinkel, M. (eds) Encyclopaedia of Mathematics. Encyclopaedia of Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-1279-4_23

Download citation

  • DOI: https://doi.org/10.1007/978-94-015-1279-4_23

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5378-7

  • Online ISBN: 978-94-015-1279-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation