The QLV model with Yeoh strain energy

density function shows remarkable agreement with the experimental results as it adequately captures the behavior of all four filler concentrations and of the polymer.

The

density function. In this section we give a proof of Theorem 1.1 with the help of the Laplace transform of [[tau]r.sup.([??])].

The time required to terminate client's service in the bank follows an exponential distribution, with an average of one minute, let us write a

density function that represents the time required for terminating client's service, and then calculate the probability of terminating client's service in less than a minute.

In a clutch pack with free separator plates, the instantaneous value of the total clearance, [y.sub.1+2+3], can also change in time as the clutch plates move axially Lastly, to calculate a time-averaged drag torque considering the changes in [y.sub.1+2+3] described by the probability

density function f ([y.sub.1+2+3]), the expected value of the drag torque for the whole clutch pack can be written as follows:

The DQI can be converted to probability

density function as input parameters of uncertainty analysis, specifically shown as follows:

We also apply the same result to the difference of the probability

density function in Figure 2.

So, the Gaussian mixture probability

density function can be characterized by the expected [u.sub.i], the variance [[sigma].sub.i.sup.2], and the mixing weighting factors of the Gaussian probability density of all components:

The kernel density function-based method is a positivity-preserving representation and can be regarded as the development of parameter-fitting method, which approximates the PSD by a superposition of weighted kernel

density functions [10].

The

density function exhibits different kinds of shapes.

Here, [h.sub.j+1](x) is the exp-EE

density function with power parameter j+1, and is given by (for j[greater than or equal to] 0) [mathematical expression not reproducible] By interchanging [mathematical expression not reproducible] in the last equation, where Equation 21:

Different vertical velocity distribution laws are used to describe stream-wise velocity structure in coastal currents (Soulsby, 1980; Anwar, 1996, 1998; De Serio and Mossa, 2014), in tidal channels (Soulsby and Dyer, 1981), and on the continental shelf (Soulsby, 1983; De Serio and Mossa, 2010).Vertical distribution of suspended sediment concentration (SSC) in rivers, estuaries, and coastal currents have been studied by many researchers through the years (Rouse, 1937; Taylor and Dyer, 1977; Li uet al., 2014).This work derives general formulas to describe the mean velocity and SSC distributions of turbulent shear flow to fit the whole cross section from the bottom boundary to the upper boundary, using a probability

density function method.

With the new data, the team built a statistical description of theoretically possible plant shapes by studying the plant's branch

density function. The branch

density function depicts the likelihood of finding a branch at any point in the space surrounding a plant.