Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we obtain the exact formulas for Ollivier’s Ricci-curvature for bipartite graphs and for the graphs with girth at least 5. These are the first formulas for Ricci-curvature that hold for a wide class of graphs, and e...

Bilayer actuators of gold and polypyrrole doped with dodecylbenzenesulfonate, PPy(DBS), were microfabricated with lengths and thicknesses that varied over two orders of magnitude. The actuators were connected to rigid plates. The curvature of the bilayers was examined as a function of PPy and Au layer thicknesses to enable a determination of the actuation strain and Young’s modulus of the PPy(D...

We prove an equality for the curvature function of a simple and closed curve on the plane. This equality leads to another proof of the four-vertex theorem in differential geometry. While examining the regularity assumption on the curve for the equality, we make comments on the relation between the boundary regularity of a Riemannmapping and two important subjects, the Schauder theory and the st...

The [Formula: see text] and [Formula: see text]-approximation schemes are introduced to approximate the popular generalized Cornu spirals with the help of the parametric rational cubic trigonometric Bézier curves. The [Formula: see text]-approximation scheme has two free parameters whereas [Formula: see text]-approximation scheme has four free parameters. To approximate the generalized Cornu sp...

We consider here how to separate multidimensional signals into two categories, such that the binary decision transmits the maximum possible information about those signals. Our motivation comes from the nervous system, where neurons process multidimensional signals into a binary sequence of responses (spikes). In a small noise limit, we derive a general equation for the decision boundary that l...

– We prove the convexity of the set which is delimited by the free boundary corresponding to a quasi-linear elliptic equation in a 2-dimensional convex domain. The method relies on the study of the curvature of the level lines at the points which realize the maximum of the normal derivative at a given level, for analytic solutions of fully nonlinear elliptic equations. The method also provides ...

Elasticity theory calculations predict the number N of depressions that appear at the surface of a spherical thin shell submitted to an external isotropic pressure. Using a model that mainly considers curvature deformations, we show that N depends on the relative volume variation and on an adimensional parameter that takes into account both the relative spontaneous curvature and the relative th...

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems canno...

This paper is to estimate the optimum plot size with the shape for field research experiments on paddy yield trial considering the effect of plot size on variability in yield of crop as well as studying the coefficients of variation of different plot sizes and shapes of plots. The maximum curvature technique and comparable variance methods were exercised to estimate optimum plot size and shape ...

An improved acoustic design for a hermetic compressor shell has been developed. The entire shell'surface is described by a single mathematical expression which insures that there are no discontinuities in either the surface or its curvature. The curvature of the improved housing is maximized within the geometric constraints of the size of the interior pumping mechanism and the specified maximum...