The Green-function transform and wave propagation

Fourier methods well known in signal processing are applied to three-dimensional wave propagation problems. The Fourier transform of the Green function, when written explicitly in terms of a real-valued spatial frequency, consists of homogeneous and inhomogeneous components. Both parts are necessary to result in a pure out-going wave that satisfies causality. The homogeneous component consists only of propagating waves, but the inhomogeneous component contains both evanescent and propagating terms. Thus we make a distinction between inhomogenous waves and evanescent waves. The evanescent component is completely contained in the region of the inhomogeneous component outside the k-space sphere. Further, propagating waves in the Weyl expansion contain both homogeneous and inhomogeneous components. The connection between the Whittaker and Weyl expansions is discussed. A list of relevant spherically symmetric Fourier transforms is given

A Derivation of Moment Evolution Equations for Linear Open Quantum Systems

Given a linear open quantum system which is described by a Lindblad master equation, we detail the calculation of the moment evolution equations from this master equation. We stress that the moment evolution equations are well-known, but their explicit derivation from the master equation cannot be found in the literature to the best of our knowledge, and so we provide this derivation for the interested reader

Task-Related, Low-Frequency Task-Residual, and Resting State Activity in the Default Mode Network Brain Regions

The hypothesis of a default mode network (DMN) of brain function is based on observations of task-independent decreases of brain activity during effort as participants are engaged in tasks in contrast to resting. On the other hand, studies also showed that DMN regions activate rather than deactivate in response to task-related events. Thus, does DMN “deactivate” during effort as compared to resting? We hypothesized that, with high-frequency event-related signals removed, the task-residual activities of the DMN would decrease as compared to resting. We addressed this hypothesis with two approaches. First, we examined DMN activities during resting, task residuals, and task conditions in the stop signal task using independent component analysis (ICA). Second, we compared the fractional amplitude of low-frequency fluctuation (fALFF) signals of DMN in resting, task residuals, and task data. In the results of ICA of 76 subjects, the precuneus and posterior cingulate cortex (PCC) showed increased activation during task as compared to resting and task residuals, indicating DMN responses to task events. Precuneus but not the PCC showed decreased activity during task residual as compared to resting. The latter finding was mirrored by fALFF, which is decreased in the precuneus during task residuals, as compared to resting and task. These results suggested that the low-frequency blood oxygen level-dependent signals of the precuneus may represent a useful index of effort during cognitive performance