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Results are obtained on the scattering theory for the Schrödinger equation $$i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$$ in spaces$L$^{$r$}($R$;$L$^{$q$}($R$^{$d$})) for a certain range of$r, q$, the so-called space-time scattering. In the linear case (i.e.$F$≡)) the relation with usual configuration space scattering is established.

We describe$T$-equivariant Schubert calculus on$G$($k$,$n$),$T$being an$n$-dimensional torus, through derivations on the exterior algebra of a free$A$-module of rank$n$, where$A$is the$T$-equivariant cohomology of a point. In particular,$T$-equivariant Pieri’s formulas will be determined, answering a question raised by Lakshmibai, Raghavan and Sankaran (Equivariant Giambelli and determinantal restriction formulas for the Grassmannian,$Pure Appl. Math. Quart.$$2$(2006), 699–717).

We construct balanced metrics on the family of non-Khler Calabi-Yau threefolds that are obtained by smoothing after contracting ( 1, 1) -rational curves on a Khler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to the connected sum of ( 1, 1) copies of ( 1, 1) .

We consider the optimization approach to the acousto-electric imaging problem. Assuming that the electric conductivity distribution is a small perturbation of a constant, we investigate the least-squares estimate analytically using (multiple) Fourier series, and confirm the widely observed fact that acousto-electric imaging has high resolution and is statistically stable. We also analyze the case of partial data and the case of limited-view data, in which some singularities of the conductivity can still be imaged.