D4+˙T2D_4\dot{+} T^2 Mode Interactions and Hidden Rotational Symmetry

Bifurcation problems in which periodic boundary conditions or Neumann boundary conditions are imposed often involve partial differential equations that have Euclidean symmetry. As a result the normal form equations for the bifurcation may be constrained by the ``hidden'' Euclidean symmetry of the equations, even though this symmetry is broken by the boundary conditions. The effects of such hidden rotation symmetry on $D_4\dot{+} T^2$ mode interactions are studied by analyzing when a $D_4\dot{+} T^2$ symmetric normal form $\tilde{F}$ can be extended to a vector field ${\rm \cal F}$ with Euclidean symmetry. The fundamental case of binary mode interactions between two irreducible representations of $D_4\dot{+} T^2$ is treated in detail. Necessary and sufficient conditions are given that permit $\tilde{F}$ to be extended when the Euclidean group ${\rm \cal E}(2)$ acts irreducibly. When the Euclidean action is reducible, the rotations do not impose any constraints on the normal form of the binary mode interaction. In applications, this dependence on the representation of ${\rm \cal E}(2)$ implies that the effects of hidden rotations are not present if the critical eigenvalues are imaginary. Generalization of these results to more complicated mode interactions is discussed.Comment: 77 pages (Latex), 3 figures available in hard copy from the author ([email protected]); paper accepted for publication in Nonlinearit

Age-Related Changes in Hair Shaft Protein Profiling and Genetically Variant Peptides.

Recent reports highlight possible improvements in individual identification using proteomic information from human hair evidence. These reports have stimulated investigation of parameters that affect the utility of proteomic information. In addition to variables already studied relating to processing technique and anatomic origin of hair shafts, an important variable is hair ageing. Present work focuses on the effect of age on protein profiling and analysis of genetically variant peptides (GVPs). Hair protein profiles may be affected by developmental and physiological changes with age of the donor, exposure to different environmental conditions and intrinsic processes, including during storage. First, to explore whether general trends were evident in the population at different ages, hair samples were analyzed from groups of different subjects in their 20's, 40's and 60's. No significant differences were seen as a function of age, but consistent differences were evident between European American and African American hair profiles. Second, samples collected from single individuals at different ages were analyzed. Mostly, these showed few protein expression level differences over periods of 10 years or less, but samples from subjects at 44 and 65 year intervals were distinctly different in profile. The results indicate that use of protein profiling for personal identification, if practical, would be limited to decadal time intervals. Moreover, batch effects were clearly evident in samples processed by different staff. To investigate the contribution of storage (at room temperature) in affecting the outcomes, the same proteomic digests were analyzed for GVPs. In samples stored over 10 years, GVPs were reduced in number in parallel with the yield of identified proteins and unique peptides. However, a very different picture emerged with respect to personal identification. Numbers of GVPs sufficed to distinguish individuals despite the age differences of the samples. As a practical matter, three hair samples per person provided nearly the maximal number obtained from 5 or 6 samples. The random match probability (where the log increased in proportion to the number of GVPs) reached as high as 1 in 108. The data indicate that GVP results are dependent on the single nucleotide polymorphism profile of the donor genome, where environmental/processing factors affect only the yield, and thus are consistent despite the ages of the donors and samples and batchwise effects in processing. This conclusion is critical for application to casework where the samples may be in storage for long periods and used to match samples recently collected