A model hyperbolic partial differential equation with singular convolution operators and infinitely smooth solutions is studied. It is shown that short pulses, including finite-bandwidth pulses, propagate with a delay with respect to the wavefront. For a two-parameter family of such equations Green's functions are obtained in a simple self-similar form. As an application, it is demonstrated that the Gurevich-Lopatnikov dispersion law for a thin-layered porous medium can be approximated by a hyperbolic equation with singular memory.
Corrado, Pilch and Warner found in 2001 the second 11-dimensional solution where the deformed geometry of S 7 in the lift contains S 2 × S 2 . We identify the gauge dual of this background with the theory described by Franco, Klebanov and Rodriguez-Gomez recently. It is the U(N) × U(N) × U(N) gauge theory with two SU (2) doublet chiral fields B 1 transforming in the (N, N, 1), B 2 transforming in the (1, N, N), C 1 in the (1, N, N) and C 2 in the (N, N, 1) as well as an adjoint field in the (1, adj, 1) representation. By adding the mass term for adjoint field , the detailed correspondence between fields of AdS 4 supergravity and composite operators of the IR field theory is determined. Moreover, we compute the spin-2 KK modes around a warped product of AdS 4 and a squashed and stretched seven-manifold. This background with global SU (2)×SU (2)×U(1) R symmetry is related to a U(N)×U(N)×U(N) N = 2 superconformal Chern-Simons matter theory with eighth-order superpotential and Chern-Simons level (1, 1, −2). The mass-squared in AdS 4 depends on SU (2) × SU (2) × U(1) R quantum numbers and the KK excitation number. The dimensions of spin-2 operators are found.
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