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Mechanisms of Heterophase Polymerization
Published in Hugo Hernandez, Klaus Tauer, Heterophase Polymerization, 2021
The average value of a distribution corresponds to its first moment. For example, the average chain length (corresponding to the degree of polymerization) will be DP=M1(l)=E(l)=∑i=1NlipLli
Data Acquisition and Intelligent Diagnosis
Published in Diego Galar, Uday Kumar, Dammika Seneviratne, Robots, Drones, UAVs and UGVs for Operation and Maintenance, 2020
Diego Galar, Uday Kumar, Dammika Seneviratne
Kurtosis is the fourth standardized statistical moment from a distribution. It is generally defined as a measure that reflects the degree to which a distribution has a peak shape (Albendea, Madruga, Cobo, & Lopez-Higueral, 2010). In particular, kurtosis provides information about the height of the distribution in relation to the value of the standard deviation. Mathematically, it is defined as Equation (5.3) (Usamentiaga et al., 2014.): k4=E[(x−μ)4]σ4,
Torque and moments
Published in Paul Grimshaw, Michael Cole, Adrian Burden, Neil Fowler, Instant Notes in Sport and Exercise Biomechanics, 2019
A torque is defined as a twisting or turning moment. The term moment is the force acting at a perpendicular distance from an axis of rotation. Torque can therefore be calculated by multiplying the force applied by the perpendicular distance at which the force acts from the axis of rotation. Often the term torque is referred to as the moment of force. The moment of force is the tendency of a force to cause rotation about an axis. Torque is a vector quantity and as such it is expressed with both magnitude and direction. Within human movement or exercise science torques cause angular acceleration that result in the rotational movements of the limbs and segments. These rotational movements take place about axes of rotation. For example, the rotational movements created in the leg while kicking a soccer ball would occur about the ankle (the foot segment), the knee (lower leg segment) and the hip (upper leg segment) joints or axes of rotation. If an object is pushed with a force through its centre of mass it will move in a straight line (linear motion) in the same direction as the applied force. However, if an object is pushed with a force at a perpendicular distance away from its centre of mass it will both rotate (about an axis of rotation) and its centre of mass will translate (move in a straight line). Figure C3.1 and Figure C3.2 illustrate this in more detail.
Analysis of the fabric of undisturbed and pluviated silty sand under load over time
Published in European Journal of Environmental and Civil Engineering, 2021
Muhamad Yusa, Elisabeth T. Bowman, Misko Cubrinovski
The distance between particles as measured along a scan line generally produces a highly skewed distribution with many small gaps and few larger ones. In order to apply meaningful statistics, this type of distribution should be transformed to approach normality (Chatfield, 1983), as previously proposed by Bowman and Soga (2003). Parameters used in this analysis are therefore mean log of void distance, kurtosis of the log of void distance, and the ratio of variance of the log of void distance to the mean log void distance, also known as the index of dispersion. The mean void distance of a sample image is, as expected, somewhat related to the whole sample void ratio. Void ratio of the sample, e, is defined as the volume (or image area) of voids (white area) to that of the solids (black area), as illustrated in Figure 5. Kurtosis is defined as the fourth moment of a probability distribution and is a measure of the shape of a curve. High values of kurtosis can occur where there are large numbers of values concentrated in the extreme tails of the distribution, with a value of 3 indicating a normal distribution. The excess population kurtosis, β, used here, is defined as kurtosis minus 3, so a value of zero indicates a normal distribution. It is formulated for a sub-set of samples within a population as:
Numerical analysis of breakthrough curves and temporal moments for solute transport in triple-permeability porous medium
Published in ISH Journal of Hydraulic Engineering, 2020
Pramod Kumar Sharma, Muskan Mayank, Chandra Shekhar Prasad Ojha
Temporal moment analysis is a useful statistical technique for quantifying solute transport properties independently from an underlying mathematical model. Temporal moments describe the moments of the breakthrough curve of solute plume, i.e. the mean residence time, first time moment and variance of the residence time, and these moments are used to characterize the transport process of solute through porous media. For solute transport in a finite soil column, the zeroth time moment of the breakthrough curve is a measure of the total mass recovery from the soil column (Skopp 1984). The first time moment of the breakthrough curve is a measure of the average time solute molecules spent inside the column. The second time moment is a measure of the variance of the breakthrough curve. Variance describes solute spreading in the soil column, a process attributed to diffusion, hydrodynamic dispersion, time variation of the input concentration (Valocchi 1989), and macrodispersion (Gelhar and Axness 1983). The third time moment is a measure of the skewness of the concentration distribution. Skewness describes the degree of asymmetry of the solute distribution, a process attributed to the deviation of the flow process from flow ideality. The expressions of zeroth, first and second time moments (Valocchi 1985) are given below.
Performance statistics of broadcasting networks with receiver diversity and Fountain codes
Published in Journal of Information and Telecommunication, 2023
Lam-Thanh Tu, Tan N. Nguyen, Phuong T. Tran, Tran Trung Duy, Quang-Sang Nguyen
Moments are used to describe the characteristic of a distribution. Particularly, the first moment is the expected value of the distribution while the second moment measures how the distribution spreads out around the mean value. The third moment is called ‘Skewness’ which measures the symmetric of the distribution and the fourth moment is ‘kurtosis’ measuring whether the distribution is heavy tails or not. In this section, we address the raw moments of the needed time slot to dispatch the message to all users. The kth raw moments of the needed time slot under the θ diversity scheme denoted by is calculated as follows