Commuting Flows and Conservation Laws for Noncommutative Lax Hierarchies

We discuss commuting flows and conservation laws for Lax hierarchies on noncommutative spaces in the framework of the Sato theory. On commutative spaces, the Sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the Lax hierarchies in terms of pseudo-differential operators. Noncommutative extension of the Sato theory has been already studied by the author and Kouichi Toda, and the existence of various noncommutative Lax hierarchies are guaranteed. In the present paper, we present conservation laws for the noncommutative Lax hierarchies with both space-space and space-time noncommutativities and prove the existence of infinite number of conserved densities. We also give the explicit representations of them in terms of Lax operators. Our results include noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera, modified KdV equations and so on.Comment: 22 pages, LaTeX, v2: typos corrected, references added, version to appear in JM

Methods for Scarless, Selection-Free Generation of Human Cells and Allele-Specific Functional Analysis of Disease-Associated SNPs and Variants of Uncertain Significance.

With the continued emergence of risk loci from Genome-Wide Association studies and variants of uncertain significance identified from patient sequencing, better methods are required to translate these human genetic findings into improvements in public health. Here we combine CRISPR/Cas9 gene editing with an innovative high-throughput genotyping pipeline utilizing KASP (Kompetitive Allele-Specific PCR) genotyping technology to create scarless isogenic cell models of cancer variants in ~1 month. We successfully modeled two novel variants previously identified by our lab in the PALB2 gene in HEK239 cells, resulting in isogenic cells representing all three genotypes for both variants. We also modeled a known functional risk SNP of colorectal cancer, rs6983267, in HCT-116 cells. Cells with extremely low levels of gene editing could still be identified and isolated using this approach. We also introduce a novel molecular assay, ChIPnQASO (Chromatin Immunoprecipitation and Quantitative Allele-Specific Occupation), which uses the same technology to reveal allele-specific function of these variants at the DNA-protein interaction level. We demonstrated preferential binding of the transcription factor TCF7L2 to the rs6983267 risk allele over the non-risk. Our pipeline provides a platform for functional variant discovery and validation that is accessible and broadly applicable for the progression of efforts towards precision medicine

The Sato Grassmannian and the CH hierarchy

We discuss how the Camassa-Holm hierarchy can be framed within the geometry of the Sato Grassmannian.Comment: 10 pages, no figure

Birkhoff strata of the Grassmannian Gr(2)\mathrm{^{(2)}}: Algebraic curves

Algebraic varieties and curves arising in Birkhoff strata of the Sato Grassmannian Gr${^{(2)}}$ are studied. It is shown that the big cell $\Sigma_0$ contains the tower of families of the normal rational curves of all odd orders. Strata $\Sigma_{2n}$, $n=1,2,3,...$ contain hyperelliptic curves of genus $n$ and their coordinate rings. Strata $\Sigma_{2n+1}$, $n=0,1,2,3,...$ contain $(2m+1,2m+3)-$plane curves for $n=2m,2m-1$ $(m \geq 2)$ and $(3,4)$ and $(3,5)$ curves in $\Sigma_3$, $\Sigma_5$ respectively. Curves in the strata $\Sigma_{2n+1}$ have zero genus.Comment: 14 pages, no figures, improved some definitions, typos correcte

Heating in current carrying molecular junctions

A framework for estimating heating and expected temperature rise in current carrying molecular junctions is described. Our approach is based on applying the Redfield approximation to a tight binding model for the molecular bridge supplemented by coupling to a phonon bath. This model, used previously to study thermal relaxation effects on electron transfer and conduction in molecular junctions, is extended and used to evaluate the fraction of available energy, i.e. of the potential drop, that is released as heat on the molecular bridge. Classical heat conduction theory is then applied to estimate the expected temperature rise. For a reasonable choice of molecular parameters and for junctions carrying currents in the nA range, we find the temperature rise to be a modest few degrees. It is argued, however, that using classical theory to describe heat transport away from the junction may underestimate the heating effect.Comment: 29 pages, 16 figures. J. Chem. Phys., in pres

Transition amplitudes and sewing properties for bosons on the Riemann sphere

We consider scalar quantum fields on the sphere, both massive and massless. In the massive case we show that the correlation functions define amplitudes which are trace class operators between tensor products of a fixed Hilbert space. We also establish certain sewing properties between these operators. In the massless case we consider exponential fields and have a conformal field theory. In this case the amplitudes are only bilinear forms but still we establish sewing properties. Our results are obtained in a functional integral framework.Comment: 33 page