Combination probe for optically assisted ultrasonic velocity-change imaging aimed at detecting unstable blood vessel plaque

Optical Review - Blood vessel plaque with a large lipid core is at risk of becoming thrombus and is likely to induce acute heart disease. To prevent this, it is necessary to determine not only the...     

Generalized Joseph–Lundgren exponent and intersection properties for supercritical quasilinear elliptic equations

We study the solution \({u(r,\rho)}\) of the quasilinear elliptic problem

$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u'|^{\beta-1}u')'+|u|^{p-1}u=0, & 0 < r < \infty, \\u(0)=\rho > 0,\ u'(0)=0.\end{cases}$$

The usual Laplace, \({m}\)-Laplace, and \({k}\)-Hessian operators are included in the differential operator \({r^{-(\gamma-1)}(r^{\alpha}|u'|^{\beta-1}u')'}\). Under certain conditions on \({\alpha}\), \({\beta}\), \({\gamma}\), and \({p}\), the equation has a singular positive solution \({u^*(r)}\) and the solution \({u(r,\rho)}\) is positive for \({r\ge 0}\). We study the intersection numbers between \({u(r,\rho)}\) and \({u^*(r)}\) and between \({u(r,\rho_0)}\) and \({u(r,\rho_1)}\). A generalized Joseph–Lundgren exponent \({p^*_{JL}}\) plays a crucial role. The main technique is a phase plane analysis. In particular, we use two changes of variables which transform the equation into two autonomous systems.

     

A new resin glycoside from Ipomoea maxima

A new resin glycoside (1) was isolated from the aerial part of Ipomoea maxima, together with three known compounds, pescaprein XX (2), stoloniferin X (3), and stoloniferin IX (4). The structure of 1 was elucidated on the basis of 1D NMR spectroscopy, a fragmentation study by APCIMS, and HRESIMS analysis.