Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bass, F., and Fuks, I.: Wave scattering from statistically rough surfaces, Pergamon, 1979.
Colton, D., and Kress, R.: Integral equations methods in scattering theory, Wiley, 1983.
Kupradze, V.: Randwertaufgaben der Schwingungstheorie und Integralgleichungen, DVW, 1956.
Leis, R.: Initial boundary value problems in mathematical physics, New York, 1986.
Marchenko, V., and Khruslov, E.: Boundary value problems in domains with granulated boundary, Nauk. Dumka, Kiev, 1974. (In Russian.)
Maz’ja, V.: Sobolev spaces, Springer, 1985.
Ramm, A.G.: ‘Spectral properties of the Schroedinger operator in some domains with infinite boundaries’, Soviet Math. Dokl. 152 (1963), 282–285.
Ramm, A.G.: ‘Reconstruction of the domain shape from the scattering amplitude’, Radiotech. i Electron. 11 (1965), 2068–2070.
Ramm, A.G.: ‘Approximate formulas for tensor polarizability and capacitance of bodies of arbitrary shape and its applications’, Soviet Phys. Dokl. 195 (1970), 1303–1306.
Ramm, A.G.: ‘Electromagnetic wave scattering by small bodies of an arbitrary shape’, in V. VARADAN (ed.): Acoustic, Electromagnetic and Elastic Scattering: Focus on T-Matrix Approach, Pergamon, 1980, pp. 537–546.
Ramm, A.G.: Theory and applications of some new classes of integral equations, Springer, 1980.
Ramm, A.G.: Iterative methods for calculating the static fields and wave scattering by small bodies, Springer, 1982.
Ramm, A.G.: ‘On inverse diffraction problem’, J. Math. Anal. Appl. 103 (1984), 139–147.
Ramm, A.G.: Scattering by obstacles, Reidel, 1986.
Ramm, A.G.: ‘Necessary and sufficient condition for a scattering amplitude to correspond to a spherically symmetric scatterer’, Appl. Math. Lett. 2 (1989), 263–265.
Ramm, A.G.: Multidimensional inverse scattering problems, Longman/Wiley, 1992.
Ramm, A.G.: ‘Stability estimates in inverse scattering’, Acta Applic. Math. 28,no. 1 (1992), 1–42.
Ramm, A.G.: ‘New method for proving uniqueness theorems for obstacle inverse scattering problems’, Appl. Math. Lett. 6,no. 6 (1993), 19–22.
Ramm, A.G.: ‘Stability estimates for obstacle scattering’, J. Math. Anal. Appl. 188,no. 3 (1994), 743–751.
Ramm, A.G.: ‘Stability of the solution to inverse obstacle scattering problem’, J. Inverse Ill-Posed Probl. 2,no. 3 (1994), 269–275.
Ramm, A.G.: ‘Uniqueness theorems for inverse obstacle scattering problems in Lipschitz domains’, Applic. Anal. 59 (1995), 377–383.
Ramm, A.G., Pang, P., and Yan, G.: ‘A uniqueness result for the inverse transmission problem’, Internat. J. Appl. Math. 2,no. 5 (2000), 625–634.
Ramm, A.G., and Sammartino, M.: ‘Existence and uniqueness of the scattering solutions in the exterior of rough domains’, in A.G. RAMM, P.N. SHIVAKUMAR, and A.V. STRAUSS (eds.): Operator Theory and Its Applications, Vol. 25 of Fields Inst. Commun., Amer. Math. Soc, 2000, pp. 457–472.
Ursell, F.: ‘On the exterior problems of acoustics’, Proc. Cambridge Philos. Soc. 74 (1973), 117–125, See also: 84 (1978), 545–548.
Vekua, I.: ‘Metaharmonic functions’, Trudy Tbil. Math. Inst. 12 (1943), 105–174. (In Russian.)
References
Golod, E.S., and Shafarevich, I.R.: ‘On the class-field tower’, Izv. Akad. Nauk. SSSR 28 (1964), 261–272. (In Russian.)
Martinet, J.: ‘Tours de corps de classes et estimations de discriminants’, Invent. Math. 44 (1978), 65–73.
Martinet, J.: ‘Petits discriminants’, Ann. Inst. Fourier (Grenoble) 29, no. fasc.l (1979), 159–170.
Martinet, J.: ‘Petits discriminants des corps de nombres’: Journ. Arithm. 1980, Cambridge Univ. Press, 1982, pp. 151–193.
Minkowski, H.: ‘Théorèmes arithmétiques’, C.R. Acad. Sci. Paris 112 (1891), 209–212.
Mulholland, H.P.: ‘On the product of n complex homogeneous linear forms’, J. London Math. Soc. 35 (1960), 241–250.
Odlyzko, A.: ‘Some analytic estimates of class numbers and discriminants’, Invent. Math. 29 (1975), 275–286.
Odlyzko, A.: ‘Lower bounds for discriminants of number fields’, Acta Arith. 29 (1976), 275–297.
Odlyzko, A.: ‘Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results’, Sém. de Théorie des Nombres, Bordeaux 2 (1990), 119–141.
Poitou, G.: ‘Sur les petits discriminants’, Sém. Delange-Pisot-Poitou 18,no. 6 (1976/77).
Poitou, G.: ‘Minoration de discriminants (d’aprés A.M. Odlyzko)’: Sém. Bourbaki (1975/76), Vol. 567 of Lecture Notes in Mathematics, Springer, 1977, pp. 136–153.
Stark, H.M.: ‘Some effective cases of the Brauer-Siegel theorem’, Invent. Math. 23 (1974), 135–152.
Stark, H.M.: ‘The analytic theory of numbers’, Bull. Amer. Math. Soc. 81 (1975), 961–972.
References
Gell-Mann, M.: ‘Symmetries of baryons and mesons’, Phys. Rev. 125 (1962), 1067–1084.
Okubo, S.: ‘Deformation of the Lie-admissible pseudo-octonion algebra into the octonion algebra’, Hadronic J. 1 (1978), 1383–1431.
Okubo, S.: ‘A generalization of Hurwitz theorem and flexible Lie-admissible algebras’, Hadronic J. 3 (1978), 1–52.
Okubo, S.: ‘Octonion as traceless 3 × 3 matrices via a flexible Lie-admissible algebra’, Hadronic J. 1 (1978), 1432–1465.
Okubo, S.: ‘Pseudo-quaternion and psuedo-octonion algebras’, Hadronic J. 1 (1978), 1250–1278.
Okubo, S.: Introduction to octonion and other non-associative algebras in physics, Cambridge Univ. Press, 1995.
Okubo, S., and Myung, H.C.: ‘Some new classes of division algebras’, J. Algebra 67 (1980), 479–490.
References
Aihara, S.I., and Bagchi, A.: ‘Nonlinear smoothing for random fields’, Stochastic Processes Appl. 55 (1995), 143–158.
Dembo, A., and Zeitouni, O.: ‘A maximum a posteriori estimator for trajectories of diffusion processes’, Stochastics 20 (1987), 221–246.
Fujita, T., and Kotani, S.: ‘The Onsager-Machlup function for diffusion processes’, J. Math. Kyoto Univ. 22 (1982), 115–130.
Ikeda, N., and Watanabe, S.: Stochastic differential equations and diffusion processes, second ed., North-Holland, 1989.
Onsager, L., and Machlup, S.: ‘Fluctuations and irreversible proceses, MI’, Phys. Rev. 91 (1953), 1505–1512; 1512-1515.
Stratonovich, R.L.: ‘On the probability functional of diffusion processes’, Selected Transl. Math. Statist. Prob. 10 (1971), 273–286.
References
Alpay, D., Dijksma, A., Rovnyak, J., and SNOO, H.S.V. DE: Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Vol. 96 f Oper. Th. Adv. Appl, Birkhäuser, 1997.
Arov, D.Z., and Grossman, L.Z.: ‘Scattering matrices in the theory of extensions of isometric operators’, Math. Nachr. 157 (1992), 105–123.
Ball, J.A., and Trent, T.T.: ‘Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables’, J. Fund. Anal. 157 (1998), 1–61.
Bart, H., Gohberg, I., and Kaashoek, M.A.: Minimal factorization of matrix and operator functions, Vol. 1 of Oper. Th. Adv. Appl, Birkhäuser, 1979.
Branges, L. de, and Rovnyak, J.: Square summable power series, Holt, Rinehart & Winston, 1966.
Brodskii, M.S.: Triangular and Jordan representations of linear operators, Vol. 32 of Transl. Math. Monographs, Amer. Math. Soc, 1971. (Translated from the Russian.)
Brodskii, M.S.: ‘Unitary operator colligations and their characteristic functions’, Russian Math. Surveys 33,no. 4 (1978), 159–191. (Uspekhi Mat. Nauk. 33, no. 4 (202) (1978), 141–168.)
Brodskii, M.S., and Livšic, M.S.: ‘Spectral analysis of non-selfadjoint operators and intermediate systems’, Amer. Math. Soc. Transl. 13,no. 2 (1960), 265–346. (Uspekhi Mat. Nauk. 13, no. 1 (79) (1958), 3–85.)
Kuzhel, A.: Characteristic functions and models of nonselfadjoint operators, Kluwer Acad. Publ., 1996.
Livšic, M.S.: ‘On the spectral decomposition of linear nonselfadjoint operators’, Amer. Math. Soc. Transl. 5,no. 2 (1957), 67–114. (Mat. Sb. 34, no. 76 (1954), 145–199.)
Livšic, M.S.: Operators, oscillations, waves, Vol. 34 of Transl. Math. Monographs, Amer. Math. Soc, 1973. (Translated from the Russian.)
Livšic, M.S., Kravitsky, N., Markus, A.S., and Vinnikov, V.: Theory of commuting nonselfadjoint operators, Kluwer Acad. Publ., 1995.
Livšic, M.S., and Yantsevich, A.A.: Operator colligations in Hilbert spaces, Winston, 1979. (Translated from the Russian.)
Nlkolski, N., and Vasyunin, V.: ‘Elements of spectral theory in terms of the free function model I. Basic constructions’: Holomorphic spaces (Berkeley, CA, 1995), Cambridge Univ. Press, 1998, pp. 211–302.
Pavlov, B.S.: ‘Spectral analysis of a singular Schrödinger operator in terms of a functional model’: Partial Differential Equations VIII, Springer, 1995, pp. 89–153.
Sakhnovich, L.A.: Interpolation theory and its applications, Kluwer Acad. Publ., 1997.
Sz.-Nagy, B., and Foiaş, C.: Harmonic analysis of operators on Hilbert space, North-Holland, 1970.
Štrauss, A.V.: ‘Characteristic functions of linear operators’, Amer. Math. Soc. Transl. 40,no. 2 (1964), 1–37. (Izv. Akad. Nauk. SSSR Ser. Mat. 24 (1960), 43–74.)
Tsekanovskii, E.R., and Shmulyan, Yu.L.: ‘The theory of bi-extensions of operators on rigged Hilbert spaces. Unbounded operator colligations and characteristic functions’, Russian Math. Surveys 32,no. 5 (1977), 73–131. (Uspekhi Mat. Nauk. 32, no. 5 (1977), 69–124.)
References
Livšic, M.S.: ‘Operator waves in Hilbert space and related partial differential equations’, Integral Eq. Oper. Th. 2,no. 1 (1979), 25–47.
Livšic, M.S.: ‘A method for constructing triangular canonical models of commuting operators based on connections with algebraic curves’, Integral Eq. Oper. Th. 3,no. 4 (1980), 489–507.
Livšic, M.S., Kravitsky, N., Markus, A.S., and Vinnikov, V.: Theory of commuting nonselfadjoint operators, Kluwer Acad. Publ., 1995.
Vinnikov, V.: ‘Commuting operators and function theory on a Riemann surface’, in S. Axler, J. McCarthy, and D. Sarason (eds.): Holomorphic Spaces and Their Operators, Vol. 33 of Math. Sci. Res. Inst. Publ, Cambridge Univ. Press, 1998, pp. 445–476.
References
Bingham, N.H., and Kiesel, R.: Risk-neutral valuation: The pricing and hedging of financial derivatives, Springer, 1998.
Björk, T.: Arbitrage theory in continuous time, Oxford Univ. Press, 1998.
Black, F., and Scholes, M.: ‘The pricing of options and corporate liabilities’, J. Political Economy 81 (1973), 637–659.
Elliott, R.J., and Kopp, E.: Mathematics of financial markets, Springer, 1999.
Karatzas, I., and Shreve, S.E.: Methods of mathematical finance, Springer, 1998.
Kwok, Y.-K.: Mathematical models of financial derivatives, Springer, 1997.
Lamberton, D., and Lapeyre, B.: Introduction to stochastic calculus applied to finance, Chapman and Hall, 1996.
Luenberger, D.G.: Investment science, Oxford Univ. Press, 1997.
Merton, R.C.: ‘Theory of rational option pricing’, Bell J. Economics and Management Sci. 4 (1973), 141–183.
Musiela, M., and Rutkowski, M.: Martingale methods in financial modeling. Theory and applications, Springer, 1997.
Nielsen, L.T.: Pricing and hedging of derivative securities, Oxford Univ. Press, 1999.
Pliska, S.R.: Introduction to mathematical finance. Discrete time models, Blackwell, 1997.
Shiryaev, A.N.: Essentials of stochastic finance, World Sci., 1999.
Wilmott, P.: Derivatives. The theory and practice of financial engineering, Wiley, 1998.
References
Ramm, A.G.: ‘Recovery of the potential from I-function’, Math. Rept. Acad. Sci. Canada 9 (1987), 177–182.
Ramm, A.G.: ‘Inverse scattering problem with part of the fixed-energy phase shifts’, Comm. Math. Phys. 207,no. 1 (1999), 231–247.
Ramm, A.G.: ‘Property C for ODE and applications to inverse scattering’, Z. Angew. Anal. 18,no. 2 (1999), 331–348.
Ramm, A.G.: ‘Property C for ODE and applications to inverse problems’, in A.G. Ramm, P.N. Shivakumar, and A.V. Strauss (eds.): Operator Theory And Its Applications, Vol. 25 of Fields Inst. Commun., Amer. Math. Soc, 2000, pp. 15–75.
Editor information
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this chapter
Cite this chapter
(2001). O. In: Hazewinkel, M. (eds) Encyclopaedia of Mathematics, Supplement III. Encyclopaedia of Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-0-306-48373-8_15
Download citation
DOI: https://doi.org/10.1007/978-0-306-48373-8_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-0198-7
Online ISBN: 978-0-306-48373-8
eBook Packages: Springer Book Archive