For unitary operators $U_0,U$ in Hilbert spaces ${\mathcal H}_0,{\mathcal H}$ and identification operator $J:{\mathcal H}_0\to{\mathcal H}$, we present results on the derivation of a Mourre estimate for $U$ starting from $U_0$ existence completeness wave triple $(U,U_0,J)$. As an application, determine spectral scattering properties class anisotropic quantum walks homogenous trees odd degree wi...

In this paper, we review two properties of completeness known as the Bourbaki-completeness and cofinal in setting metric spaces. These notions came from new classes generalized Cauchy sequences appearing when characterizing so-called Bourbaki-boundedness a similar way that characterize totally boundedness. For clustering Bourbaki–Cauchy cofinally sequences, have respectively what is call The to...

We show that computation of the SPERNER problem is PPA-complete on Möbius band under proper boundary conditions, settling a long term open problem. Further, same computational complexity results extend to other discrete fixed points band, such as Brouwer point problem, DPZP and simple version Tucker well projective plane Klein bottle. expect it opens up new route for further studies related com...

We have used the Keck 10m telescope to count objects as a function of image size in two high galactic latitude fields covering 1.5 arcmin and reaching 50% completeness depths of K=24 and J=24.5 for stellar sources. Our counts extend ∼1 magnitude deeper in K than those of surveys with other telescopes; complement other Keck surveys in the K band that provide counts at comparable or shallower dep...