Semiclassical methods provide important tools for approximating solutions in quantum mechanics. In several cases these are intriguingly exact rather than approximate, as has been shown by direct calculations on particular systems. this paper we prove that the long-conjectured exactness of supersymmetry-based semiclassical quantization condition broken supersymmetry is a consequence additive sha...

let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...

We briefly review differential forms on manifolds. We prove homotopy invariance of cohomology, the Poincaré lemma and exactness of the Mayer–Vietoris sequence. We then compute the cohomology of some simple examples. Finally, we prove Poincaré duality for orientable manifolds.

Zijian Diao Mathematics Department, Ohio University Eastern Campus, St Clairsville, OH 43950, USA∗ Abstract It is well-known that when searching one out of four, the original Grover’s search algorithm is exact; that is, it succeeds with certainty. It is natural to ask the inverse question: If we are not searching one out of four, is Grover’s algorithm definitely not exact? In this article we gi...