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The logarithmic potential is of great interest and relevance in the study of the dynamics of galaxies. Some small corrections to the work of Contopoulos & Seimenis who used the method of Prendergast to find periodic orbits and bifurcations within such a potential are presented. The solution of the orbital radial equation for the purely radial logarithmic potential is then considered using the precessing ellipse (p-ellipse) method pioneered by Struck. This differential orbital equation is a special case of the generalized Burgers equation. The apsidal angle is also determined, both numerically and analytically by means of the Lambert W and the polylogarithmic functions. The use of these functions in computing the gravitational lensing produced by logarithmic potentials is discussed.

and the Alzheimer's Disease Neuroimaging InitiativePurpose: To create a standardized, MRI-compatible, lifesized phantom of the brain ventricles to evaluate ventricle segmentation methods using T 1 -weighted MRI. An objective phantom is needed to test the many different segmentation programs currently used to measure ventricle volumes in patients with Alzheimer's disease. Materials and Methods:A ventricle model was constructed from polycarbonate using a digital mesh of the ventricles created from the 3 Tesla (T) MRI of a subject with Alzheimer's disease. The ventricle was placed in a brain mold and surrounded with material composed of 2% agar in water, 0.01% NaCl and 0.0375 mM gadopentetate dimeglumine to match the signal intensity properties of brain tissue in 3T T 1 -weighted MRI. The 3T T 1 -weighted images of the phantom were acquired and ventricle segmentation software was used to measure ventricle volume. Results:The images acquired of the phantom successfully replicated in vivo signal intensity differences between the ventricle and surrounding tissue in T 1 -weighted images and were robust to segmentation. The ventricle volume was quantified to 99% accuracy at 1-mm voxel size. Conclusion:The phantom represents a simple, realistic and objective method to test the accuracy of lateral ventricle segmentation methods and we project it can be extended to other anatomical structures.

We present analytical and numerical studies of the Fourier transform (FT) of the gravitational wave (GW) signal from a pulsar, taking into account the rotation and orbital motion of the Earth. We also briefly discuss the Zak–Gelfand integral transform and a special class of the generalized hypergeometric function of potential relevance. The Zak–Gelfand integral transform that arises in our analytic approach has also been useful for Schrödinger operators in periodic potentials in condensed matter physics (Bloch wavefunctions) and holds promise for the study of periodic GW signals for long integration times.

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