This paper investigates the structure of general affine subspaces of ( )  L . For a d × d expansive matrix A, it shows that every affine subspace can be decomposed as an orthogonal sum of spaces each of which is generated by dilating some shift invariant space in this affine subspace, and every non-zero and non-reducing affine subspace is the orthogonal direct sum of a reducing subspace and a ...

an efficient and convenient protocol for synthesis of 3-hydroxy-3-indolyl-2-oxindoles and 3,3-diindolyl-2-oxindoles is presented here. the syntheses were achieved selectively under catalysis of liclo4 whereby the products were obtained purely.

We study essentially non-Abelian backgrounds in the five dimensional N=4 gauged SU(2)×U(1) supergravity. Static configurations that are invariant under either the SO(4) spatial rotations or with respect to the SO(3) rotations and translations along the fourth spatial coordinate are considered. By analyzing consistency conditions for the equations for supersymmetric Killing spinors we derive the...

We show that all pretzel knots satisfy the (purely) cosmetic surgery conjecture, i.e. Dehn surgeries with different slopes along a knot provide oriented three-manifolds.