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Taylor dispersion or Taylor diffusion is an apparent or effective diffusion of some scalar field arising on the large scale due to the presence of a strong, confined, zero-mean shear flow on the small scale. Essentially, the shear acts to smear out the concentration distribution in the direction of the flow, enhancing the rate at which it ...
However, sometimes we are faced with a stochastic differential equation for a more complex process , in which the process appears on both sides of the differential equation. That is, say d Y t = a 1 ( Y t , t ) d t + a 2 ( Y t , t ) d B t , {\displaystyle dY_{t}=a_{1}(Y_{t},t)\ dt+a_{2}(Y_{t},t)\ dB_{t},} for some functions a 1 {\displaystyle a ...
The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. For x > 1, Taylor polynomials of higher degree provide worse approximations. The Taylor approximations for ln(1 + x) (black). For x > 1, the approximations diverge. Pictured is an accurate approximation of sin x around the point x = 0. The ...
where: k 1 is the rate coefficient for the reaction that consumes A and B; k −1 is the rate coefficient for the backwards reaction, which consumes P and Q and produces A and B. The constants k 1 and k −1 are related to the equilibrium coefficient for the reaction (K) by the following relationship (set v=0 in balance):
That is, r −1 ≪ r 2, so that r 1 − r 2 ≈ 0. But the overall rate of reaction is the rate of formation of final product (here CO 2), so that r = r 2 ≈ r 1. That is, the overall rate is determined by the rate of the first step, and (almost) all molecules that react at the first step continue to the fast second step.
Reverse accumulation traverses the chain rule from outside to inside, or in the case of the computational graph in Figure 3, from top to bottom. The example function is scalar-valued, and thus there is only one seed for the derivative computation, and only one sweep of the computational graph is needed to calculate the (two-component) gradient.
Taylor Swift has broken Elvis Presley’s longstanding record for the most weeks spent atop the Billboard 200 album chart by a solo artist. She set a new mark of 68 total weeks, as “1989 (Taylor ...
In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time. Because science and engineering often relate quantities to each other, the methods of related rates have broad ...