Let T be a linear operator on the space of all m × n complex matrices such that the rank of T(A) is the rank of A whenever the rank of A is k. We show that there are nonsingular m × m and n × n matrices U and V respectively such that either T(A) =UAV for all m × n matrices A, or m =n and T(A) =UAtV for all m ×m matrices A.
The primary purpose in performing analysis is to obtain information on which information decisions can be made. This article will discuss the principles of the concept `moving from official to traceable methods', where traceable methods are validated methods which are not published as official/standard methods.
We show that preference-homogeneity and loss-aversion are necessary and sufficient for the value function to have the power form with identical powers for gains and losses and for the probability weighting functions for gains and losses to be identical.
For a fixed 2-f reference path, we demonstrate both theoretically and experimentally that based on the spatial correlation between two light fields, ghost imaging in spatial domain (GI) and Fourier-transform ghost interference (FRT) can be obtained by only increasing the transverse size of the Nike Free Run 5.0 Canada thermal source D. Both explanation of the transformation from GI to FRT and their Nike Free 5.0 Liberty potential applications are also discussed.