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With a stretching exponent β between 0 and 1, the graph of log f versus t is characteristically stretched, hence the name of the function. The compressed exponential function (with β > 1) has less practical importance, with the notable exception of β = 2, which gives the normal distribution.
In this example, when the rectangle is horizontal, the stresses are given by [] = []. The corresponding Mohr's circle representation is shown at the bottom. The magnitude of the principal stresses are the abscissas of the points C {\displaystyle C} and E {\displaystyle E} (Figure 6) where the circle intersects the σ n {\displaystyle \sigma ...
More generally, a set of eigenfunctions is also found by noting that the differentiation rules imply that the ordinary differential equation [() + ()] = with constant and () being a non-constant even function remains invariant in form when applying the Fourier transform to both sides of the equation. The simplest example is provided by ...
The stretch factor is important in the theory of geometric spanners, weighted graphs that approximate the Euclidean distances between a set of points in the Euclidean plane. In this case, the embedded metric S is a finite metric space, whose distances are shortest path lengths in a graph, and the metric T into which S is embedded is the ...
It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator. The action of the map is defined geometrically by squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a horseshoe.
This strain is usually uniaxial, meaning it stretches or compresses the lattice in one direction while keeping other dimensions constant or periodic. Calculate stresses: For each strained configuration, run a DFT calculation to compute the resulting stress tensor. This involves solving the Kohn-Sham equations to find the ground state electron ...
The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude is given by φ): The only normal cylindrical projections that preserve area have a north-south compression precisely the reciprocal of east-west stretching (cos φ). This divides north-south distances by a factor equal to ...
Examples of the application of the logistic S-curve to the response of crop yield (wheat) to both the soil salinity and depth to water table in the soil are shown in modeling crop response in agriculture. In artificial neural networks, sometimes non-smooth functions are used instead for efficiency; these are known as hard sigmoids.