By MacLane S., G. Birkhoff

This ebook goals to offer sleek algebra from first ideas, so that it will be

accessible to undergraduates or graduates, and this via combining ordinary

materials and the wanted algebraic manipulations with the final strategies

which make clear their that means and value.

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**Example text**

10) S1 = − log |S0′ | = − log(V ). 2 4 Therefore, to O(ε), the solution must be a linear combination of the two obtained above R √ R √ 1 x 1 x V (y) dy V (y) dy u(x) ∼ A1 V (x)−1/4 e ε + A2 V (x)−1/4 e− ε . 11). 12) uε (x, 0) = A(x)eiS(x)/ε . 14) where Aε (x, t) ∼ A0 (x, t) + εA1 (x, t) + · · · . 12), we obtain the leading and next order equations 1 ∂t S + |∇S|2 + V (x) = 0, 2 1 ∂t A0 + ∇S · ∇A0 + A0 ∆S = 0. 3. 17) where H(x, p) = |p|2 /2+V (x). The second equation is a transport equation for the leading order behavior of the amplitude function A0 .

8] J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publisher, London, 1990. [9] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill Book Company, New York, 1978. 23 24 BIBLIOGRAPHY [10] A. L. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Pub. , Amsterdam, 1978. B. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics, John Wiley, New York, 1987.

London, Ser. A 326, pp. 445–470, (1988). [20] H. Conrad and J. , vol. 42, no. 11, pp. 1025–1030, 2000. [21] I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. [22] P. Dirac, “Quantum mechanics of many-electron systems,” Proc. Royal Soc. London, vol. 123, no. 792, 714–733, 1929. J. Durlofsky, “Numerical calculation of equivalent grid block permeability tensors for heterogeneous poros-media,” Water. Resour. , vol. 27, pp. 699–708, 1991. [24] W. E and B. Engquist, “The heterogeneous multi-scale methods,” Comm.