References
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- Chandrasekhar , S. 1960 . “Radiative Transfer”, , Second revised edition , N.Y. : Dover Publ. . the first edition appeared in 1950
- Van der Mee , C. V.M. 1981 . “Semigroup and Factorization Methods in Transport Theory” , Mathematical Centre Tract 146 Amsterdam
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- Hangelbroek , R. J. June 23 1980 . June 23 , announced this result during the Vienna Workshop on Transport Theory
- Zaanen , A. C. 1967 . “Integration” , Amsterdam : North-Holland . This book also contains general information about Bochner integrals, The vector-valued integrals appearing in this article are Bochner integrals. By strong measurability we mean measurability with respect to Lebesgue measure as defined in Section VI. 31 of
- For finite τ only the singularity of the Kernel H (.) B at x = 0 and not the one at x = +∞ has to be accounted for. Therefore, the equivalence proof does not change essentially if T is taken to be unbounded.
- We restrict ourselves to neutron transport in non-multiplying media.
- The function is known as the exponential integral function.
- Gohberg , I. C. and Feldman , I. A. 1971 . “Convolution Equations and Projection Methods for their Solution” , A.M.S. Transl. Monographs Providence, R.I.
- Gohberg , I. C. and Semençul , A. A. 1972 . Matem. Issled. , 7 : 201
- Dym , H. and McKean , H. P. 1976 . “Gaussian Processes, Function Theory and the Inverse Spectral Theorem” , 234 – 235 . N.Y. : Academic Press . The identity (3.9a) can also be derived from Theorems 7.2 and 7.3 in H. Dym and I. Gohberg, Integral Equations and Operator Theory, 1980, 3, 143
- Gohberg , I. C. and Heinig , G. 1975 . Rev. Roum. Math. Pures et Appl. , 20 : 55 The uniqueness of the solution ψ of Eq. (3.1) follows from its existence, because on the spaces of strongly measurable LP -functions on (0,τ) with values in a Hilbert space the integral operator is compact (1<p<+∞). The compactness of this operator follows from (the infinite dimensional analogue of) Lemma 1.1 in
- Of course, for the BGK model one has c=1. The constant c is still included to have available a solution of the analogous problem in neutron transport theory.
- Dunford , N. and Schwarz , J. T. 1958 . “Linear Operators” , New York/London : I, Interscience . For information about this Banach space we refer to
- Krein , M. G. 1958 . Uspehi Matem. Nauk , 13 ( 5 ) : 3
- Krein , M. G. 1976 . Matem. Issled. , 42 : 47
- Busbridge , I. W. 1967 . Astrophys. J. , 149 : 195 The constraints are easily found by requiring that the right-hand side of (5.13a) has a double zero at μ=∞
- Bart , H. , Gohberg , I. and Kaashoek , M. A. 1982 . “ Wiener-Hopf integral equations, Toeplitz matrices and linear systems ” . In “Toeplitz Matrices, Toeplitz Operators and Related Topics” , Edited by: Gohberg , I. vol. 3 , Birkhäuser Verlag . Operator Theory: Advances and Applications
- Bart , H. , Gohberg , I. and Kaashoek , M. A. “Convolution equations and linear systems” to appear in: Integral Equations and Operator Theory