The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t 0 = t is naturally understood where t 0 is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be the Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink-anti-kink and soliton-soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.

Using the generalized coherent states we argue that the path integral formulae for SU (2) and SU (1, 1) (in the discrete series) are WKB exact, if the starting point is expressed as the trace of e −iTĤ withĤ being given by a linear combination of generators. In our case, WKB approximation is achieved by taking a large "spin" limit: J, K → ∞. The result is obtained directly by knowing that the each coefficient vanishes under the J −1 (K −1 ) expansion and is examined by another method to be legitimated. We also point out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression leads us to a wrong result. Therefore a great care must be taken when some geometrical action would be adopted,

Background Validation and standardization of methodologies for microbial community measurements by high-throughput sequencing are needed to support human microbiome research and its industrialization. This study set out to establish standards-based solutions to improve the accuracy and reproducibility of metagenomics-based microbiome profiling of human fecal samples. Results In the first phase, we performed a head-to-head comparison of a wide range of protocols for DNA extraction and sequencing library construction using defined mock communities, to identify performant protocols and pinpoint sources of inaccuracy in quantification. In the second phase, we validated performant protocols with respect to their variability of measurement results within a single laboratory (that is, intermediate precision) as well as interlaboratory transferability and reproducibility through an industry-based collaborative study. We further ascertained the performance of our recommended protocols in the context of a community-wide interlaboratory study (that is, the MOSAIC Standards Challenge). Finally, we defined performance metrics to provide best practice guidance for improving measurement consistency across methods and laboratories. Conclusions The validated protocols and methodological guidance for DNA extraction and library construction provided in this study expand current best practices for metagenomic analyses of human fecal microbiota. Uptake of our protocols and guidelines will improve the accuracy and comparability of metagenomics-based studies of the human microbiome, thereby facilitating development and commercialization of human microbiome-based products.

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