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For Ξ∈$R$^{$n$},$t$∈$R$and$f$∈$S$($R$^{$n$}) define $$\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$$ . We determine the optimal regularity$s$_{0}such that $$\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$$ holds where$C$is independent of$f$∈$S$($R$^{$n$}) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11].

We show that the six-dimensional uplift of the five-dimensional Near-Horizon-Extremal-Kerr (NHEK) spacetime can be obtained from an AdS_3 X S^3 solution by a sequence of supergravity - but not string theory - dualities. We present three ways of viewing these pseudo-dualities: as a series of transformations in the STU model, as a combination of Melvin twists and T-dualities and, finally, as a sequence of two generalized spectral flows and a coordinate transformation. We then use these to find an infinite family of asymptotically flat embeddings of NHEK spacetimes in string theory, parameterized by the arbitrary values of the moduli at infinity. Our construction reveals the existence of non-perturbative deformations of asymptotically-NHEK spacetimes, which correspond to the bubbling of nontrivial cycles wrapped by flux, and paves the way for finding a microscopic field theory dual to NHEK which involves Melvin twists of the D1-D5 gauge theory. Our analysis also clarifies the meaning of the generalized spectral flow solution-generating techniques that have been recently employed in the literature.

We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293327, 2005) for each stationary point to be a solution, under suitable Cartesian <i>P</i>-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence results are obtained. Numerical results are reported for the second-order cone programs

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