Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows \map(t,x) in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups $M=K$ for $G=K\times K$, $H={\rm diag} G$. The derivation of these soliton hierarchies utilizes a moving parallel frame and connection 1-form along the curve flows, related to the Klein geometry of the Lie group $G\supset H$ where $H$ is the local frame structure group. The soliton equations arise in explicit form from the induced flow on the frame components of the principal normal vector N=\covder{x}\mapder{x} along each curve, and display invariance under the equivalence subgroup in $H$ that preserves the unit tangent vector T=\mapder{x} in the framing at any point $x$ on a curve. Their bi-Hamiltonian integrability structure is shown to be geometrically encoded in the Cartan structure equations for torsion and curvature of the parallel frame and its connection 1-form in the tangent space T_\map M of the curve flow. The hierarchies include group-invariant versions of sine-Gordon (SG) and modified Korteweg-de Vries (mKdV) soliton equations that are found to be universally given by curve flows describing non-stretching wave maps and mKdV analogs of non-stretching Schrodinger maps on $G/H$. These results provide a geometric interpretation and explicit bi-Hamiltonian formulation for many known multicomponent soliton equations. Moreover, all examples of group-invariant (multicomponent) soliton equations given by the present geometric framework can be constructed in an explicit fashion based on Cartan's classification of symmetric spaces.Comment: Published version, with a clarification to Theorem 4.5 and a correction to the Hamiltonian flow in Proposition 5.1

Multiplex PCRを用いた簡便で感度の高い溺死診断法の開発

For diagnosing death due to drowning, the method of acid digestion of diatoms is widely used to detect plankton in the organs of the corpse. However, the method is limited by its　being complex, hazardous, time-consuming, and insufficiently sensitive. We therefore, developed a novel simple method to diagnose death due to drowning, and determined the location of drowning by detecting genes of representative bacteria in the environment. To procure all the information in one step, the multiplex PCR method was designed. For the diagnosis of drowning, the genes of upper respiratory indigenous bacteria, Streptococcus salivarius and Streptococcus sanguinis were used as indicators. For detection of the location of drowning, Aeromonas hydrophila and Microcystis aeruginosa were used as indicators of freshwater, and Vibrio harveyi as an indicator of seawater. A set of primers was designed for multiplex PCR. to amplify all the bacterial genes simultaneously. Using this method, 47 cases of drowning were examined, and the causes and locations of death were diagnosed.博士（医学）・乙第1428号・平成31年3月15

Dendritic retraction, but not atrophy, is consistent in amyotrophic lateral sclerosis-comparison between Onuf’s neurons and other sacral motor neurons-

BACKGROUND: Fundamental cytological changes of amyotrophic lateral sclerosis (ALS) were looked for by comparing relatively preserved Onuf’s nucleus (ON) and severely affected neighboring motor neuron groups (dorsolateral alpha motoneurons (DL) and other anterior horn neurons (OAH)). The second sacral segments from 11 ALS patients and 5 controls were initially quadruple-labeled for phosphorylated and non-phosphorylated TAR DNA-binding protein of 43 kDa (TDP43), and p62 with DAPI to identify TDP43-related changes. After digital recording of these fluorescence data encompassing the entire specimen at a high resolution, the same sections were stained with Klüver-Barrera method to obtain their exact bright-field counterparts. This novel approach facilitated exact identification of ON. Furthermore, this cell to cell comparison enabled to correlate quantitative indices of the neuronal cell bodies: perimeter, area and circularity index (CI) i.e. the ratio of (perimeter/2π) divided by the square root of (area/π), which decreases with dendritic retraction, overall number of neurons and inclusions. RESULTS: In addition to known preservation of ON neuron number relative to DL and OAH, size reduction of ON neurons was not significant even in the advanced stage. Significant size reduction in DL was counteracted in the presence of TDP43-positive inclusions. Early increase of neuronal size in OAH was further enhanced by the presence of TDP43-positive inclusions. Even with these heterogeneous cytopathological changes, a decrease in CI was consistent in all groups at an early phase and was correlated with neuronal loss. CONCLUSIONS: Among variable cytological changes of ALS, a decrease in CI is a consistent early feature shared between non-atrophic ON neurons and other anterior horn neurons with either decreased (DL) or even increased (OAH) size and profounder neuronal loss. This decrease in CI, representative of dendritic retraction, is fundamental to ALS pathogenesis, not necessarily linked to cell size and pathological inclusions

Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin-vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy. A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2+1 dimensional integrable SO(3)-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, and constants of motion, describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and mKdV soliton equations. Based on the well-known equivalence between the Heisenberg model and the Schrodinger map equation in 1+1 dimensions, a geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is given in terms of dynamical maps into the 2-sphere. In particular, this formulation yields a new integrable generalization of the Schrodinger map equation in 2+1 dimensions as well as a mKdV analog of this map equation corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.Comment: Published version with typos corrected. Significantly expanded version of a talk given by the first author at the 2008 BIRS workshop on "Geometric Flows in Mathematics and Physics

Pressure-induced phase transition of Bi2Te3 into the bcc structure

The pressure-induced phase transition of bismuth telluride, Bi2Te3, has been studied by synchrotron x-ray diffraction measurements at room temperature using a diamond-anvil cell (DAC) with loading pressures up to 29.8 GPa. We found a high-pressure body-centered cubic (bcc) phase in Bi2Te3 at 25.2 GPa, which is denoted as phase IV, and this phase apperars above 14.5 GPa. Upon releasing the pressure from 29.8 GPa, the diffraction pattern changes with pressure hysteresis. The original rhombohedral phase is recovered at 2.43 GPa. The bcc structure can explain the phase IV peaks. We assumed that the structural model of phase IV is analogous to a substitutional binary alloy; the Bi and Te atoms are distributed in the bcc-lattice sites with space group Im-3m. The results of Rietveld analysis based on this model agree well with both the experimental data and calculated results. Therefore, the structure of phase IV in Bi2Te3 can be explained by a solid solution with a bcc lattice in the Bi-Te (60 atomic% tellurium) binary system.Comment: 12 pages, 5 figure

Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons

Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N-connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces $G/SO(n)$ provided with an N-connection structure and an adapted metric, for which we elaborate a complete classification, and by generalized Lagrange spaces with constant Hessian. In this approach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo) Riemannian metrics in gravity. The results yield horizontal/ vertical pairs of vector sine-Gordon equations and vector mKdV equations, with the corresponding geometric curve flows in the hierarchies described in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrodinger maps on a tangent bundle.Comment: latex 2e 50 pages, the manuscript is a Lagrange-Finsler generalization of the solitonic Riemannian formalism from math-ph/0608024, v3 modified following requests of Editor/Referee of J. Geom. Phys., new references and discussion provided in Conclusio

食用油の選択による脂肪酸バランスの改善と栄養教育の必要性

The effects of cooking oil on the fatty acid balance of the meats by deep-frying were investigated by measuring of fatty acid content and composition using gas chromatography. On the deep-fry using three kinds of cooking oil of which fatty acid composition is different, the SFA (saturated fatty acid) eluted from the meat (pork and chicken), and the characteristic fatty acids [MUFA (monounsaturated fatty acid), PUFA (polyunsaturated fatty acid)] in each cooking oil adhered to the meat. The adhesion of cooking oil increased, as the initial fatty acid content of meat increased. In the case of perilla oil, the adhesion ratio of the oil was the highest, and the n-6/n-3 PUFA ratio was the lowest. These results suggest that a fatty acid balance of diet could be easily improved, by appropriate selection of the cooking oil. The importance of the nutrition education on oil and cooking skill is also suggested