Longchamp Canada Vancouver cause “metallisation” of tungsten oxide and
Let M = Mn,n(q) be the space of n × n matrices over the finite field GF(q), q a prime power. For each t ≤ n a subspace Φ(t) Longchamp Tote Toronto of dimension n2 − tn is constructed which contains no nonzero matrix of rank Longchamp Canada Vancouver strictly less than t + 1, and it is demonstrated that these subspaces are maximal with respect to this property. The conjugate-normal Toeplitz problem is the one of characterizing the matrices that are conjugate-normal and Toeplitz at the same time. Based on a result of Gu and Patton and our results related to the normal Hankel problem, we show that a complex matrix is conjugate-normal and Toeplitz if and only if it is in one of the seven classes explicitly described in our paper. Intense irradiation (25–40 A/cm2) at intermediate energy (400 keV) has been found to cause “metallisation” of tungsten oxide and titanium niobate crystal surfaces. The possible mechanisms and the likely consequences for high-resolution imaging of clean oxide surfaces are briefly discussed. In this note the method introduced by Kokubu, Oka, and Wang is improved and the unsolved problem in a paper of Baider and Sanders for the unique normal form of Bogdanov–Takens singularities is solved under a very general condition.