Narrow Results

The main objective of this research is to analyze some geometrical properties of a quaternary four-point interpolatory subdivision scheme ($Q_4$-scheme) with a shape parameter and then find the precise range of the shape parameter of the $Q_4$-scheme to generate fractal and convexity of limit curves. That allows the curve to change easily and be more flexible without altering its control points. Therefore, by taking the various values of tension parameters, the curve still preserves its characteristics and geometrical configuration. These geometric modeling examples show that our techniques can be easily performed, and they can also provide us with an alternative strong strategy for the modeling of complex figures.

Consider a function $F(X,Y)$ of pairs of positive matrices with values in the positive matrices such that whenever $X$ and $Y$ commute $F(X,Y)=X^p{Y}^q.$ Our first main result gives conditions on $F$ such that $\mathrm{\text{Tr}}[X \mathrm{\text{log}}(F(Z,Y))]\le \mathrm{\text{Tr}}[X(p \mathrm{\text{log}} X+q \mathrm{\text{log}} Y)]$ for all $X,Y,Z$ such that $\mathrm{\text{Tr}}Z=\mathrm{\text{Tr}}X$. (Note that $Z$ is absent from the right side of the inequality.) We give several examples of functions $F$ to which the theorem applies. Our theorem allows us to give simple proofs of the well-known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables $X,Y,Z$ instead of just $X,Y$ alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy $D(X\parallel Y)=\mathrm{\text{Tr}}[X(\mathrm{\text{log}} X- \mathrm{\text{log}} Y])$, and two others, the Donald relative entropy $D_D(X\parallel Y)$, and the Belavkin–Stasewski relative entropy $D_{\mathrm{\text{BS}}}(X\parallel Y)$. They are known to satisfy $D_D(X\parallel Y)\le D(X\parallel Y)\le D_{\mathrm{\text{BS}}}(X\parallel Y)$. We prove that the Donald relative entropy provides the sharp upper bound, independent of $Z$ on $\mathrm{\text{Tr}}[X \mathrm{\text{log}}(F(Z,Y))]$ in a number of cases in which $F(Z,Y)$ is homogeneous of degree $1$ in $Z$ and $-1$ in $Y$. We also investigate the Legendre transforms in $X$ of $D_D(X\parallel Y)$ and $D_{\mathrm{\text{BS}}}(X\parallel Y)$, and show how our results for these lead to new refinements of the Golden–Thompson inequality.

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.