Background Human studies on exercise, cognition, and apolipoprotein E (APOE) genotype show that ε4 carriers may benefit from regular physical activity. Methods We examined voluntary wheel-running, memory, and hippocampal plasticity in APOE ε3 and APOE ε4 transgenic mice at 10–12 months of age. Results Sedentary ε4 mice exhibited deficits in cognition on the radial-arm water maze (RAWM), a task dependent on the hippocampus. Six weeks of wheel-running in ε4 mice resulted in improvements on the RAWM to the level of ε3 mice. Hippocampal brain-derived neurotrophic factor (BDNF) levels were similar in ε3 and ε4 mice, and after exercise BDNF was similarly increased in both ε3 and ε4 mice. In sedentary ε4 mice, tyrosine kinase B (Trk B) receptors were reduced by 50%. Exercise restored Trk B in ε4 mice to the level of ε3 mice, and in ε4 mice, exercise dramatically increased synaptophysin, a marker of synaptic function. Conclusions Our results support the hypothesis that exercise can improve cognitive function, particularly in ε4 carriers.

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The Green’s function for the biharmonic equation in an infinite angular wedge is considered. The main result is that if the angle a is less than a 1 ≅ 0.812 π {a_1} \cong 0.812\pi , then the Green’s function does not remain positive; in fact it oscillates an infinite number of times near zero and near ∞ \infty . The method uses a number of transformations of the problem including the Fourier transform. The inversion of the Fourier transform is accomplished by means of the calculus of residues and depends on the zeros of a certain transcendental function. The distribution of these zeros in the complex plane gives rise to the determination of the angle a 1 {a_1} . A general expression for the asymptotic behavior of the solution near zero and near infinity is obtained. This result has the physical interpretation that if a thin elastic plate is deflected downward at a point, the resulting shape taken by the plate will have ripples which protrude above the initial plane of the plate.

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