Curvature space curves

The cooling effect of the channel walls on flame parameters is effective for narrow channels. This influence is illustrated in Figure 6.1.3, in the form of the dead-space curve. When the walls are <4 mm apart, the dead space becomes rapidly wider. This is accompanied by falling laminar burning velocity and probably lowering of the local reaction temperature. For wider charmels, the propagation velocity w is proportional to the effective flame-front area, which can be readily calculated. On analysis of Figures 6.1.2b and 6.1.3, it is evident that the curvature of the flame is a function of... 

The most austere representation of a polymer backbone considers continuous space curves with a persistence in their tangent direction. The Porod-Kratky model [99,100] for a chain molecule incorporates the concept of constant curvature c0 everywhere on the chain skeleton c0 being dependent on the chemical structure of the polymer. It is frequently referred to as the wormlike chain, and detailed studies of this model have already appeared in the literature [101-103], In his model, Santos accounts for the polymer-like behavior of stream lines by enforcing this property of constant curvature. 

The curvature of a space curve, kc, is equal to the rate at which the tangent vector changes as the curve is traversed and is therefore given by the relation... 

Equations C.12 and C.13 correspond to the curvatures along space curves in independent directions when the derivative in Eq. C.8 is applied. The curvature formula corresponding to the choice of coordinates in Eq. C.ll is... 

One further measure of the bending of curves needs mention. A nonplanar space curve exhibits curvature (which is measured by the radius of the circle of best fit to the curve) and torsion. 

In this section, we introduce the computational method in the form used for the surfaces exhibited in Section IV, i.e., where the prescribed mean curvature of the computed surface is everywhere constant and the boundary conditions are determined by two dual periodic graphs. We also give generalizations of the method for the computation for a surface of prescribed—not necessarily constant—mean curvature, with prescribed contact angle against surface. Generalization to the computation of space curves of prescribed curvature or geodesic curvature is available (Anderson 1986). 

Two additional families of global descriptors are related to the curvature features of molecular space curves. Since molecular chains are sequences of bonded straight line segments, they are differentiable along the bonds but not at the nuclear positions. Shape descriptors can be derived assuming two... 

Local geometrical features are essential for understanding binding properties, catalytic behavior, and molecular recognition. Many of the descriptors used for global analysis can be adapted to study local features. For instance, mean and Gaussian curvature distributions of a surface, curvature and torsion of a molecular space curves, and the variation of the fractal index Df(r) over a molecular model serve this purpose. 

A useful assumption about a curve in an image is that it arises from a space curve that is as planar as possible and as uniformly curved as possible. One might also assume that the surface bounded by this space curve has the least possible surface curvature (it is a soap bubble surface) or that the curve is a line of curvature of the surface. Families of curves in an image can be used to suggest the shape of a surface very compellingly we take advantage of this when we plot perspective views of 3-D surfaces. 

Figure 2.12 Sketches of a wormlike chain, (a) Freely rotating Kuhn chain, (b) Space curve representation R(s) is the position vector of the segment at the contour variable s u(s) and 9u(s)/9s are the local tangent and curvature, respectively.

B. Rieger and L.J. van Vliet (2002) Curvature of n-dimensional space curves in grey-value images. IEEE Transactions on Image Processing 11(7), 738-745. 

One further effect of the formation of bands of electron energy in solids is that the effective mass of elecuons is dependent on the shape of the E-k curve. If dris is the parabolic shape of the classical free electron tlreoty, the effective mass is the same as tire mass of the free electron in space, but as tlris departs from the parabolic shape the effective mass varies, depending on the curvature of tire E-k curve. From the dehnition of E in terms of k, it follows that the mass is related to the second derivative of E widr respect to k tlrus... 

The competition between the polar and steric dipoles of molecules may also lead to internal frustration. In this case, the local energetically ideal configuration cannot be extended to the whole space, but tends to be accomodated by the appearance of a periodic array of defects. For example, the presence of the strong steric dipole at the head of a molecule forming bilayers will induce local curvature. As the size of the curved areas increases, an increase in the corresponding elastic energy makes energetically preferable the... 

Nearly two years ago, studying electrodynamics in curved space-time I found1 that Maxwell s equations impose on space-time a restriction which can be formulated by saying that a certain vector q determined by the curvature field must be the gradient of a scalar function, or... 

In the real world the stress tensor never vanishes and so requires a nonvanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in flat Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Associated with the misfit (mass) a strain field appears in the curved surface. 

A particular important property of silicon electrodes (semiconductors in general) is the sensitivity of the rate of electrochemical reactions to the radius of curvature of the surface. Since an electric field is present in the space charge layer near the surface of a semiconductor, the vector of the field varies with the radius of surface curvature. The surface concentration of charge carriers and the rate of carrier supply, which are determined by the field vector, are thus affected by surface curvature. The situation is different on a metal surface. There exists no such a field inside the metal near the surface and all sites on a metal surface, whether it is curved not, is identical in this aspect. 

You throw one of the balls to Sally. The curvature of our 3-D universe would be in the direction of the fourth dimension. Our straight lines would actually be curved, but in a direction unknown to us. This would be similar to a creature living on the two-space surface of a sphere. Lines that appeared straight to him would actually be curved. Parallel lines could actually intersect, just as longitude lines (which seem parallel at the equator) intersect at the poles. This curvature could be hard to detect if his, or our, universe were large compared to the local curvature. In other words, only if the radius of the hypersphere (whose hypersurface forms our 3-D space) were very small, could we notice it. ... 

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