Many data sets are visualized effectively with area proportional Venn diagrams, where the area of the regions is in proportion to a defined specification. In particular, Venn diagrams with three intersecting curves are considered useful for visualizing data in many applications, including bioscience, ecology and medicine. To ease the understanding of such diagrams, using restricted ‘nice’ shape...

Modifying stress-strain curves is one of the important topics in mechanical engineering and materials science. Real stress-strain curves should be modified after necking point as stress becomes three-dimensional after creation of throat, and consequently, equivalent stress should be used instead of axial one. Also, distribution of triple stresses across throat section is not uniform anymore, an...

We show that a smooth unknotted curve in R3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r2. In the direction ...

A FISC, or family of intersecting simple closed curves, is a collection of simple closed curves in the plane with the properties that there is some open region common to the interiors of all the curves, and that every two curves intersect in finitely many points or arcs. Let F be a FISC with a set of open regions R. F is said to be area-proportional with respect to weight function ω : R → R if ...

the average approach and the ME/MC approach. The arrow shows the region where the motion artifact is severer. Fig. 1 Uptake curves reconstructed by RICS with the ME/MC approaches. (a) A representative image showing the tumor and control area (indicated by the black squares) from which uptake curves were calculated. Each of these two areas contained 900 pixels. (b) and (c) The “true” uptake curv...