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Anderson, R.L., and Ibragimov, N.H.: Lie-Bäcklund transformations in applications, SIAM (Soc. Industrial Applied Math.), 1979.
Rogers, C, and Shadwick, W.F.: Bäcklund transformations and their applications, Acad., 1982.
Steeb, W.-H.: Continuous symmetries, Lie algebras, differential equations and computer algebra, World Sci., 1996.
Steeb, W.-H.: Problems and solutions in theoretical and mathematical physics: Advanced problems, Vol. II, World Sci., 1996.
Steeb, W.-H., and Euler, N.: Nonlinear evolution equations and Painlevé test, World Sci., 1988.
Arcones, M.A.: ‘The Bahadur-Kiefer representation for U-quantiles’, Ann. Statist. 24 (1996), 1400–1422.
Arcones, M.A.: ‘The Bahadur-Kiefer representation of the two-dimensional spatial medians’, Ann. Inst. Statist. Math. 50 (1998), 71–86.
Bahadur, R.R.: ‘A note on quantiles in large samples’, Ann. Math. Stat. 37 (1966), 577–580.
Beirlant, J., Deheuvels, P., Einmahl, J.H.J., and Mason, D.M.: ‘Bahadur-Kiefer theorems for uniform spacings processes’, Theory Probab. Appl. 36 (1992), 647–669.
Beirlant, J., and Einmahl, J.H.J.: ‘Bahadur-Kiefer theorems for the product-limit process’, J. Multivariate Anal. 35 (1990), 276–294.
Deheuvels, P.: ‘Pointwise Bahadur-Kiefer-type theorems II’: Nonparametric statistics and related topics (Ottawa, 1991), North-Holland, 1992, pp. 331–345.
Deheuvels, P., and Mason, D.M.: ‘Bahadur-Kiefer-type processes’, Ann. of Probab. 18 (1990), 669–697.
Einmahl, J.H.J.: ‘A short and elementary proof of the main Bahadur-Kiefer theorem’, Ann. of Probab. 24 (1996), 526–531.
He, X., and Shao, Q.-M.: ‘A general Bahadur representation of M-estimators and its application to linear regression with nonstochastic designs’, Ann. Statist. 24 (1996), 2608–2630.
Hesse, C.H.: ‘A Bahadur-Kiefer type representation for a large class of stationary, possibly infinite variance, linear processes’, Ann. Statist. 18 (1990), 1188–1202.
Kiefer, J.C.: ‘On Bahadur’s representation of sample quantiles’, Ann. Math. Stat.38 (1967), 1323–1342.
Kiefer, J.C.: ‘Deviations between the sample quantile process and the sample df, in M. Puri (ed.): Non-parametric Techniques in Statistical Inference, Cambridge Univ. Press, 1970, pp. 299–319.
Shorack, G.R., and Wellner, J.A.: Empirical processes with applications to statistics, Wiley, 1986.
Balian, R.: ‘Un principe d’incertitude fort en théorie du signal ou en mécanique quantique’, C.R. Acad. Sci. Paris 292 (1981), 1357–1362.
Battle, G.: ‘Heisenberg proof of the Balian-Low theorem’, Lett. Math. Phys. 15 (1988), 175–177.
Battle, G.: ‘Phase space localization theorem for on-delettes’, J. Math. Phys. 30 (1989), 2195–2196.
Benedetto, J., Heil, C., and Walnut, D.: ‘Differentiation and the Balian-Low Theorem’, J. Fourier Anal. Appl. 1 (1995), 355–402.
Daubechies, I.: ‘The wavelet transform, time-frequency localization and signal analysis’, IEEE Trans. Inform. Th. 39 (1990), 961–1005.
Daubechies, I., and Janssen, A.J.E.M.: ‘Two theorems on lattice expansions’, IEEE Trans. Inform. Th. 39 (1993), 3–6.
Feichtinger, H., and Gröchenig, K.: ‘Gabor frames and time—frequency distributions’, J. Funct. Anal. 146 (1997), 464–495.
Low, F.: ‘Complete sets of wave packets’, in C. Detar et al. (eds.): A Passion for Physics: Essays in Honor of Geoffrey Chew, World Sci., 1985, pp. 17–22.
Ramanathan, J., and Steger, T.: ‘Incompleteness of Sparse Coherent States’, Appl. Comput. Harm. Anal. 2 (1995), 148–153.
Rieffel, M.: ‘Von Neumann algebras associated with pairs of lattices in Lie groups’, Math. Ann. 257 (1981), 403–418.
Abramovich, Y.A., and Wojtaszczyk, P.: ‘On the uniqueness of order in the spaces ℓ p and L P [0,1]’, Mat. Zametki 18 (1975), 313–325.
Aliprantis, C.D., and Burkinshaw, O.: Positive operators, Acad. Press, 1995.
Alspach, D., Enflo, P., and Odell, E.: ‘On the structure of separable ℒ p spaces, (1 < p < ∞)’, Studia Math. 60 (1977), 79–90.
Bennett, C., and Sharpley, R.: Interpolation of operators, Acad. Press, 1988.
Calderón, A.P.: ‘Intermediate spaces and interpolation, the complex method’, Studia Math. 24 (1964), 113–190.
Casazza, P.G., Kalton, N.J., Kutzarova, D., and Mastylo, M.: ‘Complex interpolation and comple-mentably minimal spaces’, in N. Kalton, E. Saab, and S. Montgomery-Smith (eds.): Interaction between Functional Analysis, Harmonic Analysis, and Probability (Proc. Conf. Univ. Missouri 1994), Vol. 175 of Lecture Notes Pure Appl. Math., M. Dekker, 1996, pp. 135–143.
Johnson, W.B., B. Maurey, V. Schechtmannn, and Tzafriri, L.: ‘Symmetric structures in Banach spaces’, Memoirs Amer. Math. Soc. 217 (1979).
Kalton, N.J.: ‘Lattice structures on Banach spaces’, Memoirs Amer. Math. Soc. 493 (1993).
Kalton, N.J.: ‘The basic sequence problem’, Studia Math. 116 (1995), 167–187.
Kantorovich, L.V., and Akilov, G.P.: Functional analysis, Pergamon, 1998.
Krein, S.G., Petunin, Yu.I., and Semenov, E.M.: Interpolation of linear operators, Amer. Math. Soc, 1982. (Translated from the Russian.)
Lindenstrauss, J., and Tzafriri, L.: Classical Banach spaces: Function spaces, Vol. 2, Springer, 1979.
Lozanovskiĭ, G.A.: ‘On some Banach lattices’, Sib. Math. J. 10 (1969), 419–430.
Luxemburg, W.A.J., and Zaanen, A.C.: Riesz spaces, Vol. 2, North-Holland, 1983.
Odell, E., and Schlumprecht, T.: ‘The distortion problem’, Acta Math. 173 (1994), 258–281.
Pisier, G.: ‘Some applications of the complex interpolation method to Banach lattices’, J. Anal. Math. 35 (1979), 264–281.
Aron, R., Cole, B., and Gamelin, T.: ‘Spectra of algebras of analytic functions on a Banach space’, J. Reine Angew. Math. 415 (1991), 51–93.
Aron, R., Galindo, P., Garcia, D., and Maestre, M.: ‘Regularity and algebras of analytic functions in infinite dimensions’, Trans. Amer. Math. Soc. 384, no. 2 (1996), 543–559.
Dineen, S.: Complex analysis in localy convex spaces, North-Holland, 1981.
Dineen, S.: Complex analysis on infinite dimensional spaces, Springer, 1999.
Farmer, J.: ‘Fibers over the sphere of a uniformly convex Banach space’, Michigan Math. J. 45, no. 2 (1998), 211–226.
Gamelin, T.: ‘Analytic functions on Banach spaces’: Complex Potential Theory (Montreal 1993), Vol. 439 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., 1994, pp. 187–233.
Mujica, J.: Complex analysis in Banach spaces, North-Holland, 1986.
Stensones, B.: ‘A proof of the Michael conjecture’, preprint (1999).
Bauer, K.W.: ‘Über eine der Differentialgleichung (math = 0 zugeordnete Funktionentheorie’, Bonner Math. Schriften 23 (1965).
Bauer, K.W.: ‘Über die Lösungen der elliptischen Differentialgleichung (math), J. Reine Angew. Math. 221 (1966), 48–84;
Bauer, K.W.: ‘Über die Lösungen der elliptischen Differentialgleichung (math), J. Reine Angew. Math. 221 (1966), 176–196.
Bauer, K.W., and Ruscheweyh, S.: Differential operators for partial differential equations and function theoretic applications, Vol. 791 of Lecture Notes Math., Springer, 1980.
Berglez, P.: ‘Darstellung und funktionentheoretische Eigenschaften von Lösungen partieller Differentialgleichungen’, Habilitationsschrift Techn. Univ. Graz (1988).
Heersink, R.: ‘Über Lösungsdarstellungen und funktionentheoretische Methoden bei elliptischen Differentialgleichungen’, Ber. Math. Statist. Sektion Forschungszentrum Graz 67 (1976).
Heersink, R.: ‘Zur Charakterisierung spezieller Lösungsdarstellungen für elliptische Gleichungen’, Österr. Akad. d. Wiss., Abt.II 192, no. 4–7 (1983), 267–293.
Kracht, M., and Kreyszig, E.: Methods of complex analysis in partial differential equations with applications, Wiley, 1988.
Peschl, E.: ‘Les invariants différentiels non holomorphes et leur role dans la théorie de fonctions’, Rend. Sem. Mat. Messina 1 (1955), 100–108.
Peschl, E.: ‘Über die Verwendung von Differentialinvarianten bei gewissen Funktionenfamilien und die Übertragung einer darauf gegründeten Methode auf partielle Differentialgleichungen vom elliptischen Typus’, Ann. Acad. Sci. Fenn., Ser. A I Math. 336, no. 6 (1963).
Alfsen, E.M.: Compact convex sets and boundary integrals, Springer, 1971.
Altomare, F., and Campiti, M.: Korovkin type approximation theory and its applications, W. de Gruyter, 1994.
Bauer, H.: ‘Schilowsche Rand und Dirichletsches Problem’, Ann. Inst. Fourier 11 (1961), 89–136.
Borgwardt, K.H.: The simplex method : A probabilistic approach, Springer, 1987.
Calvin, J.M.: ‘Average performance of a class of adaptive algorithms for global optimization’, Ann. Appl. Probab. 7 (1997), 711–730.
Diaconis, P.: ‘Bayesian numerical analysis’, in S.S. Gupta and J.O. Berger (eds.): Statistical Decision Theory and Related Topics IV, Vol. 1, Springer, 1988, pp. 163–175.
Kadane, J.B., and Wasilkowski, G.W.: ‘Average case e-complexity in computer science: a Bayesian view’, in J.M. Bernardo (ed.): Bayesian Statistics, North-Holland, 1988, pp. 361–374.
Novak, E., Ritter, K., and Wozniakowski, H.: ‘Average case optimality of a hybrid secant-bisection method’, Math. Comp. 64 (1995), 1517–1539.
Ritter, K.: Average case analysis of numerical problems, Lecture Notes Math. Springer, 2000.
Shub, M., and Smale, S.: ‘Complexity of Bezout’s theorem V: polynomial time’, Theoret. Comput. Sci. 133 (1994), 141–164.
Wasilkowski, G.W.: ‘Average case complexity of multivariate integration and function approximation: an overview’, J. Complexity 12 (1996), 257–272.
Bazilevich, I.E.: ‘On a class of integrability by quadratures of the equation of Loewner-Kufarev’, Mat. Sb. 37 (1955), 471–476.
Duren, P.L.: Univalent functions, Vol. 259 of Grundl. Math. Wissenschaft., Springer, 1983.
Eenigenburg, P.J., Miller, S.S., Mocanu, P.T., and Reade, M.O.: ‘On a subclass of Bazilevic functions’, Proc. Amer. Math. Soc. 45 (1974), 88–92.
Keogh, F.R., and Miller, S.S.: ‘On the coefficients of Bazilevic functions’, Proc. Amer. Math. Soc. 30 (1971), 492–496.
Miller, S.S.: ‘The Hardy class of a Bazilevic function and its derivative’, Proc. Amer. Math. Soc. 30 (1971), 125–132.
Mocanu, P.T., Reade, M.O., and Zlotkiewicz, E.J.: ‘On Bazilevic functions’, Proc. Amer. Math. Soc. 39 (1973), 173–174.
Nunokawa, M.: ‘On the Bazilevic analytic functions’, Sci. Rep. Fac. Edu. Gunma Univ. 21 (1972), 9–13.
Pommerenke, Ch.: Univalent functions, Vanden-hoeck&Ruprecht, 1975.
Sheil-Small, T.: ‘On Bazilevic functions’, Quart. J. Math. 23 (1972), 135–142.
Singh, R.: ‘On Bazilevic functions’, Proc. Amer. Math. Soc. 38 (1973), 261–271.
Thomas, D.K.: ‘On Bazilevic functions’, Trans. Amer. Math. Soc. 132 (1968), 353–361.
Zamorski, J.: ‘On Bazilevic schlicht functions’, Ann. Polon. Math. 12 (1962), 83–90.
Cercignani, C., Gerasimenko, V., and Petrina, D.: Many-particle dynamics and kinetic equations, Kluwer Acad. Publ., 1997.
Cercignani, C., Illner, R., and Pulvirenti, M.: The mathematical theory of dilute gases, Springer, 1994.
Petrina, D.: Mathematical foundations of quantum statistical mechanics, Kluwer Acad. Publ., 1995.
Petrina, D., Gerasimenko, V., and Malyshev, P.: Mathematical foundations of classical statistical mechanics. Continuous systems, Gordon&Breach, 1989.
Spohn, H.: Large scale dynamics of interacting particles, Springer, 1991.
Hale, J.: Theory of functional differential equations, Springer, 1977.
Samoilenko, A., and Perestyuk, N.: Impulsive differential equations, World Sci., 1995. (Translated from the Russian.)
Berger, M.: ‘Une borne inférieure pour le volume d’une variété riemannienes en fonction du rayon d’injectivité’, Ann. Inst. Fourier (Grenoble) 30 (1980), 259–265.
Chavel, I.: Riemannian geometry: A modem introduction, Cambridge Univ. Press, 1995.
Aleman, A., Richter, S., and Sundberg, C.: ‘Beurling’s theorem for the Bergman space’, Acta Math. 177 (1996), 275–310.
Apostol, C., Bercovici, H., Foias, C., and Pearcy, C.: ‘Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra’, J. Funct. Anal. 63 (1985), 369–404.
Axler, S.: ‘Bergman spaces and their operators’, in J.B. Conway and B.B. Morrel (eds.): Surveys of Some Recent Results in Operator Theory I, Vol. 171 of Res. Notes Math., Pitman, 1988, p. 1–50.
Axler, S., McCarthy, J., and Sarason, D. (eds.): Holomorphic Spaces, Cambridge Univ. Press, 1998.
Axler, S., and Zheng, D.: ‘Compact operators via the Berezin transform’, Indiana J. Math. 49 (1998), 311.
Bell, S.: The Cauchy transform, potential theory, and con-formal mapping, Studies Adv. Math. CRC, 1992.
Bergman, S.: The kernel function and conformai mapping, Vol. 5 of Math. Surveys, Amer. Math. Soc, 1950.
Carleson, L.: Selected problems on exceptional sets, v. Nostrand, 1967.
Cowen, C., and MacCluer, B.: Composition operators on spaces of analytic functions, Studies Adv. Math. CRC, 1995.
Duren, P., Khavinson, D., Shapiro, H., and Sundberg, C.: ‘Invariant subspaces in Bergman spaces and the bihar-monic equation’, Michigan Math. J. 41 (1994), 247–259.
Havin, V.P.: ‘Approximation in the mean by analytic functions’, Soviet Math. Dokl. 9 (1968), 245–248.
Hedberg, L.: ‘Non linear potentials and approximation in the mean by analytic functions’, Math. Z. 129 (1972), 299–319.
Hedenmalm, H.: ‘A factorization theorem for square area integrable functions’, J. Reine Angew. Math. 422 (1991), 45–68.
Hedenmalm, H.: ‘An invariant subspace of the Bergman space having the codimension two property’, J. Reine Angew. Math. 443 (1993), 1–9.
Korenblum, B.: ‘An extension of the Nevanlinna theory’, Acta Math. 135 (1975), 187–219.
Korenblum, B.: ‘A Beurling type theorem’, Acta Math. 138 (1977), 265–293.
Korenblum, B.: ‘Outer functions and cyclic elements in Bergman spaces’, J. Funct. Anal. 115 (1993), 104–118.
Li, H., and Luecking, D.: ‘BMO on strongly pseudocon-vex domains: Hankel operators, duality and ∂-estimates’, Trans. Amer. Math. Soc. 346 (1994), 661–691.
Seip, K.: ‘Beurling type density theorems in the unit disc’, Invent. Math. 113 (1993), 21–39.
Seip, K.: ‘On a theorem of Korenblum’, Ark. Mat. 32 (1994), 237–243.
Seip, K.: ‘On Korenblum’s density condition for the zero sets of Ap, α’, J. Anal. Math. 67 (1995), 307–322.
Zhu, K.: ‘Operator theory in function spaces’, Pure Appl. Math. 139 (1990).
Berlekamp, E.R.: Algebraic coding theory, McGraw-Hill, 1968.
Kailath, T.: ‘Encounters with the Berlekamp-Massey algorithm’, in R.E. Blahut, D.J. Costello Jr., U. Maureer, and T. Mittelholzer (eds.): Communications and Cryptography, Two Sides of One Tapestry, Kluwer Acad. Publ., 1994.
Massey, J.L.: ‘Shift register synthesis and BCH decoding’, IEEE Trans. Inform. Th. IT-19 (1969), 122–127.
McEliece, R.J.: The theory of information and coding, Vol. 3 of Encycl. Math. Appl, Addison-Wesley, 1977.
Sugiyama, Y., Kasahara, S., Hirasawa, S., and Namekawa, T.: ‘A method for solving key equation for decoding Goppa codes’, Inform. Control 27 (1975), 87–99.
Welch, L.R., and Schultz, R.A.: ‘Continued fractions and Berlekamp’s algorithm’, IEEE Trans. Inform. Th. IT-25 (1979), 19–27.
Chaudhuri, A., and Mukerjee, R.: Randomized response, M. Dekker, 1988.
Ferguson, T.S.: Mathematical statistics: a decision theoretic approach, Acad. Press, 1967.
Lehmann, E.L.: Theory of point estimation, Wiley, 1983.
Bernstein, S.N.: ‘Mathematical problems in modern biology’, Science in the Ukraine 1 (1922), 14–19. (In Russian.)
Bernstein, S.N.: ‘Solution of a mathematical problem related to the theory of inheritance’, Uchen. Zap. Nauch. Issl. Kafedr. Ukrain. 1 (1924), 83–115. (In Russian.)
Gonzales, S., Gutiérrez, J.C., and Martinez, C.: ‘The Bernstein problem in dimension 5’, J. Algebra 177 (1995), 676–697.
Gutiérrez, J.C.: ‘The Bernstein problem in dimension 6’, J. Algebra 185 (1996), 420–439.
Hardy, G.H.: ‘Mendelian proportions in a mixed population’, Science 28, no. 706 (1908), 49–50.
Lyubich, Y.I.: ‘Basic concepts and theorems of evolutionary genetics for free populations’, Russian Math. Surveys 26, no. 5 (1971), 51–123.
Lyubich, Y.I.: ‘Analogues to the Hardy-Weinberg Law’, Genetics 9, no. 10 (1973), 139–144. (In Russian.)
Lyubich, Y.I.: ‘Two-level Bernstein populations’, Math. USSR Sb. 24, no. 1 (1974), 593–615.
Lyubich, Y.I.: ‘Quasilinear Bernstein populations’, Teor. Funct. Funct. Anal. Appl. 26 (1976), 79–84.
Lyubich, Y.I.: ‘Proper Bernstein populations’, Probl. Inform. Transmiss. Jan. (1978), 228–235.
Lyubich, Y.I.: ‘A topological approach to a problem in mathematical genetics’, Russian Math. Surveys 34, no. 6 (1979), 60–66.
Lyubich, Y.I.: Mathematical structures in population genetics, Springer, 1992.
Lyubich, Y.I.: ‘A new advance in the Bernstein problem in mathematical genetics’, Preprint Inst. Math. Sci., SUNY Stony Brook 9 (1996), 1–33.
Adams, D.R., and Hedberg, L.I.: Function spaces and potential theory, Springer, 1996.
Triebel, H.: Theory of function spaces II, Birkhäuser, 1992.
Barwise, J.: ‘Infinitary logic and admissible sets’, Doctoral Diss. Stanford (1967).
Barwise, J.: ‘Infinitary logic and admissible sets’, J. Symbolic Logic 34 (1969), 226–252.
Barwise, J., and Feferman, S. (eds.): Model-theoretic logics, Springer, 1985.
Beth, E.W.: ‘On Padoa’s method in the theory of definition’, Indag. Math. 15 (1953), 330–339.
Craig, W.: ‘Satisfaction for n-th order languages defined in n-th order languages’, J. Symbolic Logic 30 (1965), 13–25.
Ebbinghaus, H.-D., and Flum, J.: Finite model theory, Springer, 1995.
Gostanian, R., and Hrbacek, K.: ‘On the failure of the weak Beth property’, Proc. Amer. Math. Soc. 58 (1976), 245–249.
Kolaitis, P.: ‘Implicit definability on finite structures and unambiguous computations’: Proc. 5th IEEE Symp. on Logic in Computer Science, 1990, pp. 168–180.
Lopez-Escobar, E.G.K.: ‘An interpolation theorem for de-numerably long sentences’, Funct. Math. 57 (1965), 253–272.
Mekler, A.H., and Shelah, S.: ‘Stationary logic and its friends I’, Notre Dame J. Formal Logic 26 (1985), 129–138.
Schütte, K.: ‘Der Interpolationssatz der intuitionistischen Prädikatenlogik’, Math. Ann. 148 (1962), 192–200.
Dahlberg, J., and Trubowitz, E.: ‘A remark on two dimensional periodic potentials’, Comment. Math. Helvetici 57 (1982), 130–134.
Eastham, M.S. P.: The spectral theory of periodic differential equations, Scottish Acad. Press, 1973.
Helffer, B., and Mohamed, A.: ‘Asymptotic of the density of states for the Schrödinger operator with periodic electric potential’, Duke Math. J. 92, no. 1 (1998), 1–60.
Karpeshina, Y.E.: Perturbation theory for the Schrödinger operator with a periodic potential, Vol. 1663 of Lecture Notes Math., Springer, 1977.
Kuchment, P.: Floquet theory for partial differential equations, Vol. 60 of Oper. Th. Adv. Appl., Birkhäuser, 1993.
Mohamed, A.: ‘Asymptotic of the density of states for Schrödinger operator with periodic electro-magnetic potential’, J. Math. Phys. 38, no. 8 (1997), 4023–4051.
Shubin, M.: ‘The spectral theory and the index of almost periodic coefficients’, Russian Math. Surveys 34, no. 2 (1979), 109–157.
Skriganov, M.M.: ‘Proof of the Bethe-Sommerfeld conjecture in dimension two’, Soviet Math. Dokl. 20, no. 5 (1979), 956–959.
Skriganov, M.M.: ‘The spectrum band structure of the three dimensional Schrödinger operator with periodic potential’, Invent. Math. 80 (1985), 107–121.
Skriganov, M.M.: ‘Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators’, Proc. Steklov Inst. Math., no. 2 (1987).
Sommerfeld, A., and Bethe, H.: Electronentheorie der Metalle, second ed., Handbuch Physik. Springer, 1933.
Abrahamse, M.B., and Douglas, R.G.: ‘A class of subnormal operators related to multiply connected domains’, Adv. Math. 19 (1976), 1–43.
Ball, J.A., Gohberg, I., and Rodman, L.: Interpolation of rational matrix functions, Vol. 45 of Oper. Th. Adv. Appi, Birkhäuser, 1990.
Ball, J.A., and Helton, J.W.: ‘A Beurling-Lax theorem for the Lie group U(m, n) which contains most classical interpolation’, J. Operator Th. 9 (1983), 632–658.
Ball, J.A., and Helton, J.W.: ‘Shift invariant manifolds and nonlinear analytic function theory’, Integral Eq. Operator Th. 11 (1988), 615–725.
Bercovici, H.: Operator theory and arithmetic in H ∞, Vol. 26 of Math. Surveys Monogr., Amer. Math. Soc, 1988.
Beurling, A.: ‘On two problems concerning linear transformations in Hilbert space’, Acta Math. 81 (1949), 239–255.
Branges, L. De, and Rovnyak, J.: Square summable powerseries, Holt, Rinehart&Winston, 1966.
Brodskii, M.S.: Triangular and Jordan representations of linear operators, Vol. 32 of Transl. Math. Monogr., Amer. Math. Soc, 1971.
Gohberg, I. (ed.): I. Schur Methods in Operator Theory and Signal Processing, Vol. 18 of Oper. Th. Adv. Appl., Birkhäuser, 1986.
Gohberg, I. (ed.): Time-Variant Systems and Interpolation, Vol. 56 of Oper. Th. Adv. Appl, Birkhäuser, 1992.
Halmos, P.R.: ‘Shifts on Hilbert spaces’, J. Reine Angew. Math. 208 (1961), 102–112.
Hedenmalm, H.: ‘A factorization theorem for square area-integrable analytic functions’, J. Reine Angew. Math. 422 (1991), 45–68.
Helton, J.W.: Operator theory, analytic functions, matrices, and electrical engineering, Vol. 68 of Conf. Board Math. Sci., Amer. Math. Soc, 1987.
Lax, P.D.: ‘Translation invariant subspaces’, Acta Math. 101 (1959), 163–178.
Potapov, V.P.: ‘The multiplicative structure of J-contractive matrix functions’, Amer. Math. Soc. Transl. 15, no. 2 (1960), 131–243.
Regalia, P.A.: Adaptive HR filtering and signal processing and control, M. Dekker, 1995.
Richter, S.: ‘A representation theorem for cyclic analytic two-isometries’, Trans. Amer. Math. Soc. 328 (1991), 325–349.
Sz.-Nagy, B., and Foias, C.: Harmonic analysis of operators on Hilbert space, North-Holland, 1970.
Bernstein, I.N., Gelfand, I.M., and Gelfand, S.I.: ‘Structure of representations generated by vectors of highest weight’, Funkts. Anal. Prilozh. 5, no. 1 (1971), 1–9.
Bernstein, I.N., Gelfand, I.M., and Gelfand, S.I.: ‘Differential operators on the base affine space and a study of g-modules’, in I.M. Gelfand (ed.): Lie groups and their representations, Proc. Summer School on Group Representations, Janos Bolyai Math. Soc.&Wiley, 1975, pp. 39–64.
Bernstein, I.N., Gelfand, I.M., and Gelfand, S.I.: ‘A certain category of g-modules’, Funkts. Anal. Prilozh. 10, no. 2 (1976), 1–8.
Rocha-Caridi, A.: ‘Splitting criteria for g-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite dimensional, irreducible g-module’, Trans. Amer. Math. Soc. 262, no. 2 (1980), 335–366.
Rocha-Caridi, A., and Wallach, N.R.: ‘Projective modules over graded Lie algebras’, Math. Z. 180 (1982), 151–177.
Rocha-Caridi, A., and Wallach, N.R.: ‘Highest weight modules over graded Lie algebras: Resolutions, filtrations and character formulas’, Trans. Amer. Math. Soc. 277, no. 1 (1983), 133–162.
Bhatnagar, P.L., Gross, E.P., and Krook, M.: ‘A model for collision processes in gases’, Phys. Rev. 94 (1954), 511.
Bouchut, F.: ‘Construction of BGK models with a family of kinetic entropies for a given system of conservation laws’, J-Statist. Phys. 95 (1999), 113–170.
Bremer, Y.: ‘Averaged multivalued solutions for scalar conservation laws’, SIAM J. Numer. Anal. 21 (1984), 1013–1037.
Brenier, Y., and Corrías, L.: ‘A kinetic formulation for multi-branch entropy solutions of scalar conservation laws’, Ann. Inst. H. Poincaré Anal. Non Lin. 15 (1998), 169–190.
Brenier, Y., Corrías, L., and Natalini, R.: ‘A relaxation approximation to a moment hierarchy of conservation laws with kinetic formulation’, preprint (1998).
Cercignani, C., Illner, R., and Pulvirenti, M.: The mathematical theory of dilute gases, Vol. 106, Springer, 1994.
Chen, G.Q., Levermore, C.D., and Liu, T.-P.: ‘Hyperbolic conservation laws with stiff relaxation terms and entropy’, Commun. Pure Appl. Math. 47 (1994), 787–830.
Giga, Y., and Miyakawa, T.: ‘A kinetic construction of global solutions of first order quasilinear equations’, Duke Math. J. 50 (1983), 505–515.
Jin, S., and Xin, Z.-P.: ‘The relaxation schemes for systems of conservation laws in arbitrary space dimensions’, Commun. Pure Appl. Math. 48 (1995), 235–276.
Levermore, CD.: ‘Moment closure hierarchies for kinetic theories’, J. Statist. Phys. 83 (1996), 1021–1065.
Lions, P.-L., Perthame, B., and Tadmor, E.: ‘A kinetic formulation of multidimensional scalar conservation laws and related equations’, J. Amer. Math. Soc. 7 (1994), 169–191.
Natalini, R.: ‘A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws’, J. Diff. Eq. 148 (1998), 292–317.
Perthame, B.: ‘Global existence to the BGK model of Boltz-mann equation’, J. Diff. Eq. 82 (1989), 191–205.
Perthame, B.: ‘Boltzmann type schemes for gas dynamics and the entropy property’, SIAM J. Numer. Anal. 27 (1990), 1405–1421.
Perthame, B., and Pulvirenti, M.: ‘Weighted L∞ bounds and uniqueness for the Boltzmann BGK model’, Arch. Rat. Mech. Anal. 125 (1993), 289–295.
Perthame, B., and Tadmor, E.: ‘A kinetic equation with kinetic entropy functions for scalar conservation laws’, Comm. Math. Phys. 136 (1991), 501–517.
Truesdell, C., and Muncaster, R.G.: Fundamentals of Maxwell’s kinetic theory of a simple monatomic gas, treated as a branch of rational mechanics, Vol. 83 of Pure Appl. Math., Acad. Press, 1980.
Bénabou, J.: Introduction to bicategories, Vol. 47 of Lecture Notes Math., Springer, 1967, pp. 1–77.
Bénabou, J.: ‘Les distributeurs’, Sem. Math. Pure Univ. Catholique de Louvain 33 (1973).
Betti, R., Carboni, A., and Walters, R.: ‘Variation through enrichment’, J. Pure Appl. Algebra 29 (1983), 109–127.
Bird, G.J., Kelly, G.M., Power, A.J., and Street, R.: ‘Flexible limits for 2-categories’, J. Pure Appl. Algebra 61 (1989), 1–27.
Brown, R.: ‘Higher dimensional group theory’, in R. Brown and T.L. Thickstun (eds.): Low dimensional topology, Vol. 48 of Lecture Notes London Math. Soc, Cam bridge Univ. Press, 1982, pp. 215–238.
Carboni, A., Johnson, S., Street, R., and Verity, D.: ‘Modulated bicategories’, J. Pure Appl. Algebra 94 (1994), 229–282.
Ehresmann, C.: Catégories et structures, Dunod, 1965.
Eilenberg, S., and Kelly, G.M.: ‘Closed categories’, Proc. Conf. Categorical Algebra, La Jolla. Springer, 1966, pp. 421–562.
Gabriel, P., and Zisman, M.: Calculus of fractions and homotopy theory, Vol. 35 of Ergebn. Math. Grenzgeb., Springer, 1967.
Giraud, J.: Cohomologie non abélienne, Springer, 1971.
Godement, R.: Topologie algébrique et théorie des faisceaux, Hermann, 1964.
Gordon, R., Power, A.J., and Street, R.: Coherence for tricategories, Vol. 117 of Memoirs, Amer. Math. Soc, 1995, p. 558.
Gray, J.W.: Report on the meeting of the Midwest Category Seminar in Zurich, Vol. 195 of Lecture Notes Math., Springer, 1971, pp. 248–255.
Gray, J.W.: Formal category theory: adjointness for 2- categories, Vol. 391 of Lecture Notes Math., Springer, 1974.
Hakim, M.: Topos annelés et schémas relatifs, Vol. 64 of Ergebn. Math. Grenzgeb., Springer, 1972.
Joyal, A., and Street, R.: ‘The geometry of tensor calculus I’, Adv. Math. 88 (1991), 55–112.
Kelly, G.M.: Adjunction for enriched categories, Vol. 106 of Lecture Notes Math., Springer, 1969, pp. 166–177.
Kelly, G.M.: An abstract approach to coherence, Vol. 281 of Lecture Notes Math., Springer, 1972, pp. 106–147.
Kelly, G.M., and Street, R.: Review of the elements of 2-categories, Vol. 420 of Lecture Notes Math., Springer, 1974, pp. 75–103.
Lawvere, F.W.: ‘The category of categories as a foundation for mathematics’: Proc. Conf. Categorical Algebra, La Jolla, Springer, 1966, pp. 1–20.
Lawvere, F.W.: ‘Metric spaces, generalised logic, and closed categories’, Rend. Sem. Mat. Fis. Milano 43 (1974), 135–166.
MacLane, S.: Categories for the working mathematician, Vol. 5 of Graduate Texts Math., Springer, 1971.
MacLane, S., and Paré, R.: ‘Coherence for bicategories and indexed categories’, J. Pure Appl. Algebra 37 (1985), 59–80.
Pitts, A.: ‘Applications of sup-lattice enriched category theory to sheaf theory’, Proc. London Math. Soc. (3)57 (1988), 433–480.
Rosebrugh, R.D., and Wood, R.J.: ‘Proarrows and cofibrations’, J. Pure Appl. Algebra 53 (1988), 271–296.
Street, R.: ‘The formal theory of monads’, J. Pure Appl. Algebra 2 (1972), 149–168.
Street, R.: Elementary cosmoi 1, Vol. 420 of Lecture Notes Math., Springer, 1974, pp. 134–180.
Street, R.: ‘Limits indexed by category-valued 2-functors’, J. Pure Appl. Algebra 8 (1976), 149–181.
Street, R.: ‘Fibrations in bicategories’, Cah. Topol. Géom. Diff. 2153–56 (1980; 1987)
Street, R.: ‘Fibrations in bicategories’, Cah. Topol. Géom. Diff. 21; 111–160 (1980; 1987)
Street, R.: ‘Fibrations in bicategories’, Cah. Topol. Géom. Diff. 28 (1980; 1987), 53–56;
Street, R.: ‘Fibrations in bicategories’, Cah. Topol. Géom. Diff. 28. (1980; 1987) 111–160
Street, R.: ‘Cauchy characterization of enriched categories’, Rend. Sem. Mat. Fis. Milano 51 (1981), 217–233.
Street, R.: ‘Conspectus of variable categories’, J. Pure Appl. Algebra 21 (1981), 307–338.
Street, R.: Characterization of bicategories of stacks, Vol. 962 of Lecture Notes Math., Springer, 1982, pp. 282–291.
Street, R.: ‘Two dimensional sheaf theory’, J. Pure Appl. Algebra 23 (1982), 251–270.
Street, R.: ‘Enriched categories and cohomology’, Quaest. Math. 6 (1983), 265–283.
Street, R.: ‘Higher categories, strings, cubes and simplex equations’, Appl. Categorical Struct. 3 (1995), 29–77
Street, R.: ‘Higher categories, strings, cubes and simplex equations’, Appl. Categorical Struct. 3 (1995)and 303.
Street, R.: ‘Categorical structures’, in M. Hazewinkel (ed.): Handbook of Algebra, Vol. I, Elsevier, 1996, pp. 529–577.
Street, R., and Walters, R.F.C.: ‘Yoneda structures on 2-categories’, J. Algebra 50 (1978), 350–379.
Walters, R.F.C.: ‘Sheaves on sites as Cauchy-complete categories’, J. Pure Appl. Algebra 24 (1982), 95–102.
Birkhoff, G.D.: ‘Singular points of ordinary linear differential equations’, Trans. Amer. Math. Soc. 10 (1909), 436–470.
Böttcher, A., and Silbermann, B.: Analysis of Toeplitz operators, Springer, 1990.
Clancey, K.F., and Gohberg, I.Z.: Factorization of matrix functions and singular integral operators, Birkhäuser, 1981.
Freed, D.: ‘The geometry of loop groups’, J. Diff. Geom. 28 (1988), 223–276.
Gakhov, F.D.: Boundary value problems, 3rd ed., Nauka, 1977.
Gohberg, I.Z., and Krein, M.G.: ‘Systems of integral equations on a half-line with kernels depending on the difference of the arguments’, Transl. Amer. Math. Soc. 14 (1960), 217–284.
Grothendieck, A.: ‘Sur la classification des fibres holomorphes sur la sphère de Riemann’, Amer. J. Math. 79 (1957), 121–138.
Hazewinkel, M., and Martin, C.F.: ‘Representations of the symmetric groups, the specialization order, systems, and Grassmann manifolds’, Enseign. Math. 29 (1983), 53–87.
Khimshiashvili, G.: ‘On the Riemann-Hilbert problem for a compact Lie group’, Dokl. Akad. Nauk SSSR 310 (1990), 1055–1058. (In Russian.)
Pressley, A., and Segal, G.: Loop groups, Clarendon Press, 1986.
Segal, G., and Wilson, G.: ‘Loop groups and equations of KdV type’, Publ. Math. IHES 61 (1985), 5–65.
Vekua, N.P.: Systems of singular integral equations, Nauka, 1970. (In Russian.)
Zhang, S.: ‘Factorizations of invertible operators and K-theory of C*-algebras’, Bull. Amer. Math. Soc. 28 (1993), 75–83.
Birkhoff, G.D.: ‘Singular points of ordinary linear differential equations’, Trans. Amer. Math. Soc. 10 (1909), 436–470.
Bojarski, B.: ‘On the stability of Hilbert problem for holo-morphic vector’, Bull. Acad. Sci. Georgian SSR 21 (1958), 391–398.
Disney, S.: ‘The exponents of loops on the complex general linear group’, Topology 12 (1973), 297–315.
Freed, D.: ‘The geometry of loop groups’, J. Diff. Geom. 28 (1988), 223–276.
Gohberg, I.Z., and Krein, M.G.: ‘Systems of integral equations on a half-line with kernels depending on the difference of the arguments’, Transl. Amer. Math. Soc. 14 (1960), 217–284.
Grothendieck, A.: ‘Sur la classification des fibres holomorphes sur la sphère de Riemann’, Amer. J. Math. 79 (1957), 121–138.
Khimshiashvili, G.: ‘Lie groups and transmission problems on Riemann surfaces’, Contemp. Math. 131 (1992), 164–178.
Pressley, A., and Segal, G.: Loop groups, Clarendon Press, 1986.
Duistermaat, J.J., and Grünbaum, F.A.: ‘Differential equations in the spectral parameter’, Comm. Math. Phys. 103 (1986), 177–240.
Grünbaum, F.A.: ‘Some nonlinear evolution equations and related topics arising in medical imaging’, Phys. D 18 (1986), 308–311.
Harnad, J., and Kasman, A. (eds.): The bispectral problem (Montreal, PQ, 1997), CRM Proc. Lecture Notes. Amer. Math. Soc, 1998.
Wilson, G.: ‘Bispectral commutative ordinary differential operators’, J. Reine Angew. Math. 442 (1993), 177–204.
Zubelli, J.P., and Magri, F.: ‘Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV,’ Comm. Math. Phys. 141, no. 2 (1991), 329–351.
Asmussen, S.: Applied probability and queues, Wiley, 1987.
Blackwell, D.: ‘A renewal theorem’, Duke Math. J. 15 (1948), 145–150.
Feller, W.: An introduction to probability theory and its applications, 3rd ed., Vol. 1, Wiley, 1968.
Feller, W.: An introduction to probability theory and its applications, 2nd ed., Vol. 2, Wiley, 1970.
Lindvall, T.: Lectures on the coupling method, 2nd ed., Vol. II, Wiley, 1992.
Fischer, S.D.: Function thory on planar domains, Wiley, 1983.
Garnett, J.B.: Bounded analytic functions, Acad. Press, 1981.
Voichick, M., and Zalcman, L.: ‘Inner and outer functions on Riemann Surfaces’, Proc. Amer. Math. Soc. 16 (1965), 1200–1204.
Bliedtner, J., and Hansen, W.: ‘Cones of hyperharmonic functions’, Math. Z. 151 (1976), 71–87.
Bliedtner, J., and Hansen, W.: Balayage spaces: An analytic and probabilistic approach to balayage, Universitext. Springer, 1986.
Boboc, N., Bucur, Gh., and Cornea, A.: Order and convexity in potential theory: H-cones, Vol. 853 of Lecture Notes Math., Springer, 1981.
Constantinescu, C., and Cornea, A.: Potential theory on harmonic spaces, Springer, 1972.
Lukeš, J., Malý, J., and Zajíček, L.: Fine topology methods in real analysis and potential theory, Vol. 1189 of Lecture Notes Math., Springer, 1986.
Aguirre, F., and Conca, C.: ‘Eigenfrequencies of a tube bundle immersed in a fluid’, Appl. Math. Optim. 18 (1988), 1–38.
Allaire, G., and Conca, C.: ‘Bloch-wave homogenization for a spectral problem in fluid-solid structures’, Arch. Rat. Mech. Anal. 135 (1996), 197–257.
Allaire, G., and Conca, C.: ‘Boundary layers in the homogenization of a spectral problem in fluid-solid structures’, SIAM J. Math. Anal. 29 (1998), 343–379.
Bensoussan, A., Lions, J.L., and Papanicolaou, G.: Asymptotic analysis in periodic structures, North-Holland, 1978.
Bloch, F.: ‘Über die Quantenmechanik der Electronen im Kristallgitern’, Z. Phys. 52 (1928), 555–600.
Brillouin, L.: Propagation of waves in periodic structures, Dover, 1953.
Conca, C., Planchard, J., and Vanninathan, M.: Fluids and periodic structures, Wiley&Masson, 1995.
Conca, C., and Vanninathan, M.: ‘A spectral problem arising in fluid-solid structures’, Comput. Meth. Appl. Mech. Eng. 69 (1988), 215–242.
Conca, C., and Vanninathan, M.: ‘Homogenization of periodic structures via Bloch decomposition’, SIAM J. Appl. Math. 57 (1997), 1639–1659.
Cracknell, A.P., and Wong, K.C.: The Fermi surface, Clarendon Press, 1973.
Eastham, M.: The spectral theory of periodic differential equations, Scottish Acad. Press, 1973.
Floquet, G.: ‘Sur les équations différentielles linéaires à coefficients périodiques’, Ann. Ecole Norm. Ser. 2 12 (1883), 47–89.
Gelfand, I.M.: ‘Entwicklung nach Eigenfunktionen einer Gleichung mit periodischer Koeffizienten’, Dokl. Akad. Nauk SSSR 73 (1950), 1117–1120.
Odeh, F., and Keller, J.B.: ‘Partial differential equations with periodic coefficients and Bloch waves in crystals’, J. Math. Phys. 5 (1964), 1499–1504.
Reed, M., and Simon, B.: Methods of modern mathematical physics, Acad. Press, 1978.
Santosa, F., and Symes, W.W.: ‘A dispersive effective medium for wave propagation in periodic composites’, SIAM J. Appl. Math. 51 (1991), 984–1005.
Sanchez-Hubert, J., and Sánchez-Palencia, E.: Vibration and coupling of continuous systems, Springer, 1989.
Titchmarsh, E.C.: Eigen]’unctions expansions Part II, Clarendon Press, 1958.
Wilcox, C.: ‘Theory of Bloch waves’, J. Anal. Math. 33 (1978), 146–167.
Ziman, J.M.: Principles of the theory of solids, Cambridge Univ. Press, 1972.
Bochner, S.: ‘Summation of multiple Fourier series by spherical means’, Trans. Amer. Math. Soc. 40 (1936), 175–207.
Fefferman, C.: ‘A note on spherical summation multipliers’, Israel J. Math. 15 (1973), 44–52.
Golubov, B.I.: ‘On Gibb’s phenomenon for Riesz spherical means of multiple Fourier integrals and Fourier series’, Anal. Math. 4 (1978), 269–287.
Levitan, B.M.: ‘Ueber die Summierung mehrfacher Fouri-erreihen und Fourierintegrale’, Dokl. Akad. Nauk SSSR 102 (1955), 1073–1076.
Sogge, C.: ‘On the convergence of Riesz means on compact manifolds’, Ann. of Math. 126 (1987), 439–447.
Stein, E.M.: Harmonic analysis, Princeton Univ. Press, 1993.
Thangavelu, S.: Lectures on H ermite and Laguerre expansions, Princeton Univ. Press, 1993.
Ando, T.: ‘Banachverbände und positive Projektionen’, Math. Z. 109 (1969), 121–130.
Bohnenblust, H.F.: ‘An axiomatic characterization of L p -spaces’, Duke Math. J. 6 (1940), 627–640.
Kakutani, S.: ‘Concrete representation of abstract L p -spaces and the mean ergodic theorem’, Ann. of Math. 42 (1941), 523–537.
Meyer-Nieberg, P.: Banach lattices, Springer, 1991.
Zippin, M.: ‘On perfectly homogeneous bases in Banach spaces’, Israel J. Math. 4 A (1966), 265–272.
Boas, H.P.: ‘Bohr’s power series theorem in several variables’, Proc. Amer. Math. Soc. 125 (1997), 2975–2979.
Caratheodory, C.: Theory of functions of a complex variable, Vol. 1, Chelsea, 1983, pp. Sects. 274–275.
Aizenberg, L.: ‘Multidimensional analogues of Bohr’s theorem on power series’, Proc. Amer. Math. Soc. 128 (2000).
Aizenberg, L., Aytuna, A., and Djakov, P.: ‘An abstract approach to Bohr phenomenon’, Proc. Amer. Math. Soc. (to appear).
Aizenberg, L., Aytuna, A., and Djakov, P.: ‘Generalization of Bohr’s theorem for arbitrary bases in spaces of holo-morphic functions of several variables’, J. Anal. Appl. (to appear).
Boas, H.P., and Khavinson, D.: ‘Bohr’s power series theorem in several variables’, Proc. Amer. Math. Soc. 125 (1997), 2975–2979.
Bohr, H.: ‘A theorem concerning power series’, Proc. London Math. Soc. 13, no. 2 (1914), 1–5.
Cercignani, C.: ‘On the Boltzmann equation for rigid spheres’, Transp. Theory Stat. Phys. 2 (1972), 211–225.
Cercignani, C., Gerasimenko, V., and Petrina, D.: Many-particle dynamics and kinetic equations, Kluwer Acad. Publ., 1997.
Gerasimenko, V., and Petrina, D.: ‘Mathematical problems of statistical mechanics of a hard-sphere system’, Russian Math. Surveys 45, no. 3 (1990), 159–211.
Grad, H.: ‘Principles of the kinetic theory of gases’: Handbuch Physik, Vol. 12, Springer, 1958, pp. 205–294.
Lanford, O.E.: Time evolution of large classical dynamical system, Vol. 38 of Lecture Notes Physics, Springer, 1975, pp. 1–111.
Baxter, R.J.: Exactly solved models in statistical mechanics, Acad. Press, 1992.
Drinfel’d, V.G.: ‘Hopf algebras and the quantum Yang-Baxter equation’, Soviet Math. Dokl. 32 (1985), 254–258. (Translated from the Russian.)
Jimbo, M.: ‘A q-difference analogue of U q g and the Yang-Baxter equation’, Lett. Math. Phys. 10 (1985), 63–69.
Jimbo, M. (ed.): Yang-Baxter equation in integrable systems, World Sci., 1990.
Kubo, R., et al.: Statistical physics, Vol. 1–2, Springer, 1985.
Reif, F.: Statistical and thermal physics, McGraw-Hill, 1965.
Tolman, R.C.: The principles of statistical mechanics, Oxford Univ. Press, 1938, Reprint: 1980.
Wadati, M., Deguchi, T., and Akutsu, Y.: ‘Exactly solvable models and knot theory’, Physics Reports 180 (1989), 247–332.
Yang, C.N., and Ge, M.L. (eds.): Braid group, knot theory and statistical mechanics, Vol. 1–2, World Sci., 1989; 1994.
Ajtai, M.: ‘Ej — formulae on finite structures’, Ann. Pure Appl. Logic 24 (1983), 1–48.
Ajtai, M., Komlós, J., and Szemerédi, E.: ‘An O(nlogn) sorting network’, Combinatorica 3 (1983), 1–19.
Alon, N., and Boppana, R.: ‘The monotone circuit complexity of Boolean functions’, Combinatorica 7, no. 1 (1987), 1–22.
Boppana, R., and Sipser, M.: ‘Complexity of finite functions’, in J. Van Leeuwen (ed.): Handbook of Theoretical Computer Science, Vol. A, 1990, pp. 758–804; Chap.14.
Furst, M., Saxe, J.B., and Sipser, M.: ‘Parity, circuits and the polynomial-time hierarchy’, Math. Systems Theory 17 (1984), 13–27.
Hastad, J.: ‘Almost optimal lower bounds for small depth circuits’, in S. Micali (ed.): Randomness and Computation, Vol. 5 of Adv. Comput. Res., JAI Press, 1989, pp. 143–170.
Kushilevitz, E., and Nisan, N.: Communication complexity, Cambridge Univ. Press, 1996.
Lupanov, O.B.: ‘A method of circuit synthesis’, Izv. V.U.Z. (Radiofizika) 1, no. 1 (1958), 120–140. (In Russian.)
Razborov, A.A.: ‘Lower bounds on the monotone complexity of some Boolean functions’, Soviet Math. Dokl. 31 (1985), 354–357.
Savage, J.E.: ‘Computational work and time on finite machines’, J. ACM 19, no. 4 (1972), 660–674.
Schonhage, A., and Strassen, V.: ‘Schnelle Multiplikation grosser Zahlen’, Computing 7 (1971), 281–292.
Shannon, C.E.: ‘The synthesis of two-terminal switching circuits’, Bell Systems Techn. J. 28, no. 1 (1949), 59–98.
Spira, P.M.: ‘On time-hardware complexity of tradeoffs for Boolean functions’: Proc. 4th Hawaii Symp. System Sciences, North Hollywood&Western Periodicals, 1971, pp. 525–527.
Wegener, I.: The complexity of Boolean functions, Wiley&Teubner, 1987.
Yao, Y.: ‘Separating the polynomial-time hierarchy by oracles’: Proc. 26th Ann. IEEE Symp. Found. Comput. Sci., 1985, pp. 1–10.
Athreya, K.B.: ‘Bootstrap of the mean in the infinite variance case’, Ann. Statist. 15 (1987), 724–731.
Davison, A.C., and Hinkley, D.V.: Bootstrap methods and their application, Cambridge Univ. Press, 1997.
Efron, B.: ‘Bootstrap methods: another look at the jack-knife’, Ann. Statist. 7 (1979), 1–26.
Efron, B., and Tibshirani, R.J.: An introduction to the bootstrap, Chapman&Hall, 1993.
Giné, E.: ‘Lectures on some aspects of the bootstrap’, in P. Bernard (ed.): Ecole d’Eté de Probab. Saint Flour XXVI-1996, Vol. 1665 of Lecture Notes Math., Springer, 1997.
Götze, F., and Künsch, H.R.: ‘Second order correctness of the blockwise bootstrap for stationary observations’, Ann. Statist. 24 (1996), 1914–1933.
Hall, P.: The bootstrap and Edgeworth expansion, Springer, 1992.
Mammen, E.: When does bootstrap work? Asymptotic results and simulations, Vol. 77 of Lecture Notes Statist., Springer, 1992.
Putter, H., and Zwet, W.R. van: ‘Resampling: consistency of substitution estimators’, Ann. Statist. 24 (1996), 2297–2318.
Shao, J., and Tu, D.: The jackknife and bootstrap, Springer, 1995.
Bartsch, T.: ‘On the existence of Borsuk-Ulam theorems’, Topology 31 (1992), 533–543.
Borsuk, K.: ‘Drei Sätze über die n-dimensionale Sphäre’, Funct. Math. 20 (1933), 177–190.
Krein, M.G., Krasnosel’skii, M.A., and Mil’man, D.P.: ‘On the defect numbers of linear operators in a Banach space and some geometrical questions’, Sb. Trud. Inst. Mat. Akad. Nauk Ukrain. SSR 11 (1948), 97–112. (In Russian.)
Steinlein, H.: ‘Borsuk’s antipodal theorem and its generalizations and applications: a survey. Méthodes topologiques en analyse non linéaire’: Sém. Math. Super. Montréal, Sém. Sci. OTAN (NATO Adv. Study Inst), Vol. 95, 1985, pp. 166–235.
Bott, R.: ‘Homogeneous vector bundles’, Ann. of Math. 66 (1957), 203–248.
Demazure, M.: ‘A very simple proof of Bott’s theorem’, Invent. Math. 33 (1976).
Wallach, N.R.: Harmonic analysis on homogeneous spaces, M. Dekker, 1973.
Boyer, R.S., Kaufmann, M., and Moore, J.S.: ‘The Boyer-Moore theorem prover and its interactive enhancement’, Cornput. Math. Appl. 29, no. 2 (1995), 27–62.
Boyer, R.S., and Moore, J.S.: A computational logic, Acad. Press, 1979.
Boyer, R.S., and Moore, J.S.: A computational logic handbook, Acad. Press, 1988.
FTP, http://ftp.cs.utexas.edu/pub/boyer/nqthm/index.html (1998).
Hunt, W.: FM8501: A verified microprocessor, Vol. 795 of Lecture Notes Computer Sci., Springer, 1994.
Kaufmann, M., and Moore, J.S.: ‘An industrial strength theorem prover for a logic based on common Lisp’, IEEE Trans. Software Engineering 23, no. 4 (1997), 203–213.
Shankar, N.: Metamathematics: Machines, and Goedel’s proof, Cambridge Univ. Press, 1994.
Joyal, A., and Street, R.: ‘Braided monoidal categories’, Math. Reports Macquarie Univ. 86008 (1986).
MacLane, S.: Categories for the working mathematician, Vol. 5 of GTM, Springer, 1974.
Majid, S.: Foundations of quantum group theory, Cambridge Univ. Press, 1995.
Majid, S.: ‘Examples of braided groups and braided matrices’, J. Math. Phys. 32 (1991), 3246–3253.
Majid, S.: Algebras and Hopf algebras in braided categories, Vol. 158 of Lecture Notes Pure Appl. Math., M. Dekker, 1994, pp. 55–105.
Majid, S.: Foundations of quantum group theory, Cambridge Univ. Press, 1995.
Majid, S.: ‘Double bosonisation and the construction of U q (g)’, Math. Proc. Cambridge Philos. Soc. 125 (1999), 151–192.
Alperin, J.L.: Local representation theory, Cambridge Univ. Press, 1986.
Alperin, J.L.: ‘Weights for finite groups’, in P. Fong (ed.): Representations of Finite Groups, Vol. 47 of Proc. Symp. Pure Math., Amer. Math. Soc, 1987, pp. 369–379.
Broué, M.: ‘Isométries parfaites, types de blocs, catégories dérivées’, Astérisque 181–182 (1990), 61–92.
Curtis, C., and Reiner, I.: Methods of representation theory, Vol. II, Wiley, 1987.
Feit, W.: The representation theory of finite groups, North-Holland, 1982.
Nagao, H., and Tsushima, Y.: Representation of finite groups, Acad. Press, 1987.
Berger, T.R., and Knörr, R.: ‘On Brauer’s height 0 conjecture’, Nagoya Math. J. 109 (1988), 109–116.
Feit, W.: The representation theory of finite groups, North-Holland, 1982.
Gluck, D., and Wolf, T.R.: ‘Brauer’s height conjecture for p-solvable groups’, Trans. Amer. Math. Soc. 282, no. 1 (1984), 137–152.
Brauer, R.: ‘Zur Darstellungstheorie der Gruppen endlicher Ordnung II’, Math. Z. 72 (1959), 22–46.
Curtis, C., and Reiner, I.: Methods of representation theory, Vol. II, Wiley, 1987.
Feit, W.: The representation theory of finite groups, North-Holland, 1982.
Nagao, H., and Tsushima, Y.: Representation of finite groups, Acad. Press, 1987.
Alperin, J.L.: Local representation theory, Cambridge Univ. Press, 1986.
Curtis, C., and Reiner, I.: Methods of representation theory, Vol. II, Wiley, 1987.
Nagao, H., and Tsushima, Y.: Representation of finite groups, Acad. Press, 1987.
Bredon, G.E.: Equivariant cohomology theories, Vol. 34 of Lecture Notes Math., Springer, 1967.
Dress, A.W.M.: ‘Contributions to the theory of induced representations’: Algebraic K-theory, II (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Vol. 342 of Lecture Notes Math., Springer, 1973, pp. 183–240.
Lück, W.: Tranformation groups and algebraic K-theory, Vol. 1408 of Lecture Notes Math., Springer, 1989.
May, J.P., et al.: Equivariant homotopy and cohomology theory, Vol. 91 of Regional Conf. Ser. Math., Amer. Math. Soc, 1996.
Moerdijk, I., and Svensson, J.A.: ‘The equivariant Serre spectral sequence’, Proc. Amer. Math. Soc. 118 (1993), 263–278.
Tom Dieck, T.: Transformation groups and representation theory, Vol. 766 of Lecture Notes Math., Springer, 1979.
Brooks, J., and Jewett, R.: ‘On finitely additive vector measures’, Proc. Nat. Acad. Sci. USA 67 (1970), 1294–1298.
Constantinescu, C.: ‘Some properties of spaces of measures’, Suppl. Atti Sem. Mat. Fis. Univ. Modena 35 (1991), 1–286.
D’Andrea, A.B., and Lucia, P. de: ‘The Brooks-Jewett theorem on an orthomodular lattice’, J. Math. Anal. Appl. 154 (1991), 507–522.
Drewnowski, L.: ‘Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym theorems’, Bull. Acad. Polon. Sci. 20 (1972), 725–731.
Pap, E.: Null-additive set functions, Kluwer Acad. Publ.& Ister Sci., 1995.
Weber, H.: ‘Compactness in spaces of group-valued contents, the Vitali-Hahn-Saks theorem and the Nikodym’s bounded-ness theorem’, Rocky Mtn. J. Math. 16 (1986), 253–275.
Blumenthal, R.M., and Getoor, R.K.: Markov processes and potential theory, Acad. Press, 1968.
Borodin, A.N., and Salminen, P.: Handbook of Brownian motion: Facts and formulae, Birkhäuser, 1996.
Ikeda, N., and Watanabe, S.: Stochastic differential equations and diffusion processes, North-Holland&Kodansha, 1981.
Itô, K., and McKean, H.P.: Diffusion processes and their sample paths, Springer, 1974.
Knight, F.: ‘Random walks and a sojourn density process of Brownian motion’, Trans. Amer. Math. Soc. 109 (1963), 56–86.
Lévy, P.: ‘Sur certains processus stochastiques homogénes’, Compositio Math. 7 (1939), 283–339.
Lévy, P.: Processus stochastiques et mouvement brownien, Gauthier-Villars, 1948.
McKean, H.P.: ‘Brownian local time’, Adv. Math. 15 (1975), 91–111.
Perkins, E.: ‘The exact Hausdorff measure of the level sets of Brownian motion’, Z. Wahrscheinlichkeitsth. verw. Gebiete 58 (1981), 373–388.
Ray, D.B.: ‘Sojourn times of a diffusion process III’, J. Math. 7 (1963), 615–630.
Taylor, S.J., and Wendel, J.G.: ‘The exact Hausdorff measure of the zero set of a stable process’, Z. Wahrscheinlichkeitsth. verw. Gebiete 6 (1966), 170–180.
Trotter, H.F.: ‘A property of Brownian motion paths. III’, J. Math. 2 (1958), 425–433.
Armijo, L.: ‘Minimization of functions having Lipschitz-continuous first partial derivatives’, Pacific J. Math. 16 (1966), 1–3.
Bathe, K.J., and Cimento, A.P.: ‘Some practical procedures for the solution of nonlinear finite element equations’, Comput. Meth. Appl. Mech. Eng. 22 (1980), 59–85.
Broyden, C.G.: ‘A new double-rank minimization algorithm. Notices Amer. Math. Soc. 16 (1969), 670.
Broyden, C.G., Dennis, J.E., and Moré, J.J.: ‘On the local and superlinear convergence of quasi-Newton methods’, J. Inst. Math. Appl 12 (1973), 223–246.
Byrd, R.H., and Nocedal, J.: ‘A tool for the analysis of quasi-Newton methods with application to unconstrained minimization’, SIAM J. Numer. Anal. 26 (1989), 727–739.
Byrd, R.H., Nocedal, J., and Schnabel, R.B.: ‘Representation of quasi-Newton matrices and their use in limited memory methods’, Math. Progr. 63 (1994), 129–156.
Byrd, R.H., Nocedal, J., and Yuan, Y.: ‘Global convergence of a class of quasi-Newton methods on convex problems’, SIAM J. Numer. Anal. 24 (1987), 1171–1190.
Dennis, J.E., and Schnabel, R.B.: Numerical Methods for Nonlinear Equations and Unconstrained Optimization, No. 16 in Classics in Applied Math. SIAM (Soc. Industrial Applied Math.), 1996.
Fletcher, R.: ‘A new approach to variable metric methods’, Comput. J. 13 (1970), 317–322.
Goldfarb, D.: ‘A family of variable metric methods derived by variational means’, Math. Comp. 24 (1970), 23–26.
Kelley, C.T.: Iterative methods for optimization, Vol. 18 of Frontiers in Appl. Math., SIAM (Soc. Industrial Applied Math.), 1999.
Matthies, H., and Strang, G.: ‘The solution of nonlinear finite element equations’, Internat. J. Numerical Methods Eng. 14 (1979), 1613–1626.
Nazareth, J.L.: ‘Conjugate gradient methods less dependent on conjugacy’, SIAM Review 28 (1986), 501–512.
Nocedal, J.: ‘Updating quasi-Newton matrices with limited storage’, Math. Comp. 35 (1980), 773–782.
Powell, M.J.D.: ‘Some global convergence properties of a variable metric algorithm without exact line searches’, Nonlinear Programming, in R. Cottle and C. Lemke (eds.). Amer. Math. Soc., 1976, pp. 53–72.
Shanno, D.F.: ‘Conditioning of quasi-Newton methods for function minimization, Math. Comp. 24 (1970), 647–657.
Werner, J.: ‘Über die globale konvergenz von Variable-Metric Verfahren mit nichtexakter Schrittweitenbestimmung’, Numer. Math. 31 (1978), 321–334.
Broyden, C.G.: ‘A class of methods for solving nonlinear simultaneous equations’, Math. Comp. 19 (1965), 577–593.
Broyden, C.G., Dennis, J.E., and Moré, J.J.: ‘On the local and superlinear convergence of quasi-Newton methods’, J. Inst. Math. Appl. 12 (1973), 223–246.
Decker, D.W., Keller, H.B., and Kelley, C.T.: ‘Convergence rates for Newton’s method at singular points’, SIAM J. Numer. Anal. 20 (1983), 296–314.
Decker, D.W., and Kelley, C.T.: ‘Sublinear convergence of the chord method at singular points’, Numer. Math. 42 (1983), 147–154.
Decker, D.W., and Kelley, C.T.: ‘Broyden’s method for a class of problems having singular Jacobian at the root’, SIAM J. Numer. Anal. 22 (1985), 566–574.
Dennis, J.E., and Moré, J.J.: ‘Quasi-Newton methods, methods, motivation and theory’, SIAM Review 19 (1977), 46–89.
Dennis, J.E., and Schnabel, R.B.: Numerical Methods for Nonlinear Equations and Unconstrained Optimization, No. 16 in Classics in Applied Math. SIAM (Soc. Industrial Applied Math.), 1996.
Deuflhard, P., Freund, R.W., and Walter, A.: ‘Fast Secant Methods for the Iterative Solution of Large Nonsymmetric Linear Systems’, Impact of Computing in Science and Engineering 2 (1990), 244–276.
Engelman, M.S., Strang, G., and Bathe, K.J.: ‘The application of quasi-Newton methods in fluid mechanics’, Internat. J. Numerical Methods Eng. 17 (1981), 707–718.
Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations, No. 16 in Frontiers in Appl. Math. SIAM (Soc. Industrial Applied Math.), 1995.
Kelley, C.T., and Sachs, E.W.: ‘A new proof of superlinear convergence for Broyden’s method in Hilbert space’, SIAM J. Optim. 1 (1991), 146–150.
Ortega, J.M., and Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables, Acad. Press, 1970.
Sherman, J., and Morrison, W.J.: ‘Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix (abstract)’, Ann. Math. Stat. 20 (1949), 621.
Sherman, J., and Morrison, W.J.: ‘Adjustment of an inverse matrix corresponding to a change in one element of a given matrix’, Ann. Math. Stat. 21 (1950), 124–127.
Bukhvalov, A.V.: ‘Integral representations of linear operators’, J. Soviet Math. 8 (1978), 129–137.
Dunford, N., and Pettis, J.B.: ‘Linear operators on sum-mable functions’, Trans. Amer. Math. Soc. 47 (1940), 323–392.
Meyer-Nieberg, P.: Banach lattices, Springer, 1991.
Schep, A.R.: ‘Kernel operators’, PhD Thesis Univ. Leiden (1977).
Zaanen, A.C.: Riesz spaces, Vol. II, North-Holland, 1983.
Burnside, W.: Theory of groups of finite order, Cambridge Univ. Press, 1897.
Burnside, W.: Theory of groups of finite order, second, much changed ed., Cambridge Univ. Press, 1911, Reprinted: Dover, 1955.
Frobenius, G.: ‘Über die Congruenz nach einem aus zwei endlichen Gruppen gebildeten Doppelmodm’, J. Reine Angew. Math. 101 (1887), 273–299, Also: Gesammelte Abh. II (1968), Springer, 304–330.
Neumann, Peter M.: ‘A lemma that is not Burnside’s’, Math. Scientist 4 (1979), 133–141.
Neumann, Peter M., Stoy, G.A., and Thompson, E.C.: Groups and geometry, Clarendon Press, 1994.
Pólya, G.: ‘Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen’, Acta Math. 68 (1937), 145–254.
Redfield, J.H.: ‘The theory of group-reduced distributions’, Amer. J. Math. 49 (1927), 433–455.
Wright, E.M.: ‘Burnside’s lemma: a historical note’, J. Combin. Th. B 30 (1981), 89–90.
Ballmann, W., Gromov, M., and Schroeder, V.: Manifolds of nonpositive curvature, Vol. 61 of Progr. Math., Birkhäuser, 1985.
Busemann, H.: The geometry of geodesics, Acad. Press, 1955.
Cheeger, J., and Gromoll, D.: ‘The splitting theorem for manifolds of nonnegative Ricci curvature’, J. Diff. Geom. 6 (1971/72), 119–128.
Cheeger, J., and Gromoll, D.: ‘On the structure of complete manifolds of nonnegative curvature’, Ann. of Math. (2) 96 (1972), 413–443.
Docquier, F., and Grauert, H.: ‘Leisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten’, Math. Ann. 140 (1960), 94–123.
Eberlein, P., and O’Neill, B.: ‘Visibility manifolds’, Pacific J. Math. 46 (1973), 45–109.
Greene, R.E., and Wu, H.: ‘On Kahler manifolds of positive bisectional curvature and a theorem of Hartogs’, Abh. Math. Sem. Univ. Hamburg 47 (1978), 171–185, Special issue dedicated to the seventieth birthday of Erich Käler.
Gromov, M.: Structures métriques pour les variétés rie-manniennes, Vol. 1 of Textes Mathématiques [Mathematical Texts], CEDIC, 1981, Edited by J. Lafontaine and P. Pansu.
Heintze, E., and Imhof, H.-C.: ‘Geometry of horospheres’, J. Diff. Geom. 12, no. 4 (1977), 481–491 (1978).
Innami, N.: ‘Differentiability of Busemann functions and total excess’, Math. Z. 180, no. 2 (1982), 235–247.
Innami, N.: ‘On the terminal points of co-rays and rays’, Arch. Math. (Basel) 45, no. 5 (1985), 468–470.
Kasue, A.: ‘A compactification of a manifold with asymptotically nonnegative curvature’, Ann. Sci. Ecole Norm. Sup. 4 21, no. 4 (1988), 593–622.
Shen, Z.: ‘On complete manifolds of nonnegative kth-Ricci curvature’, Trans. Amer. Math. Soc. 338, no. 1 (1993), 289–310.
Shiohama, K.: ‘Busemann functions and total curvature’, Invent. Math. 53, no. 3 (1979), 281–297.
Shiohama, K.: ‘The role of total curvature on complete non-compact Riemannian 2-manifolds’, Illinois J. Math. 28, no. 4 (1984), 597–620.
Shiohama, K.: ‘Topology of complete noncompact manifolds’, Geometry of Geodesies and Related Topics (Tokyo, 1982), Vol. 3 of Adv. Stud. Pure Math. North-Holland, 1984, pp. 423–450.
Siu, Y.T., and Yau, S.T.: ‘Complete Kahler manifolds with nonpositive curvature of faster than quadratic decay’, Ann. of Math. (2) 105, no. 2 (1977), 225–264.
Wu, H.: ‘An elementary method in the study of nonnegative curvature’, Acta Math. 142, no. 1–2 (1979), 57–78.
Buser, P.: ‘Über den ersten Eigenwert des Laplace-Operators auf kompakten Flächen’, Comment. Math. Helvetici 54 (1979), 477–493.
Buser, P.: ‘A note on the isoperimetric constant’, Ann. Sci. Ecole Norm. Sup. 15 (1982), 213–230.
Chavel, I.: Riemannian geometry: A modern introduction, Cambridge Univ. Press, 1995.
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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). B. In: Hazewinkel, M. (eds) Encyclopaedia of Mathematics. Encyclopaedia of Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-1279-4_2
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