In Accord With Physical Laws And Mathematical Equations.pdf

The Iterative Solution of Fully Implicit Discretizations of Three-Dimensional Transport Models

B.P. Sommeijer , in Parallel Computational Fluid Dynamics 1998, 1999

2 DEFINITION OF THE MATHEMATICAL MODEL

The mathematical equations describing the transport process in three dimensions are given by the system of advection-diffusion-reaction equations [ 7]

(1) $\begin{array}{l} \frac{\partial c_{i}}{\partial t} = ‐ \frac{\partial ({uc}_{i})}{\partial x} ‐ \frac{\partial (v c_{i})}{\partial y} ‐ \frac{\partial (w c_{i})}{\partial z} + \frac{\partial}{\partial x} (ε_{x} \frac{\partial c_{i}}{\partial x}) + \frac{\partial}{\partial y} (ε_{y} \frac{\partial c_{i}}{\partial y}) + \frac{\partial}{\partial z} (ε_{z} \frac{\partial c_{i}}{\partial z}) + \\ g_{i} (t, x, y, z, c_{1}, c_{2}, \dots, c_{m}), i = 1, \dots, m, \end{array}$

where c_i, are the unknown concentrations of the contaminants. The local fluid velocities u, v, w (in x, y, z directions, respectively) are considered to be known for this transport model. In practice they are computed by a hydrodynamical model, either in advance on the whole time interval (and stored on file), or concurrently with the transport model (for example, one time step in advance). The terms g_i describe sinks and sources (emissions) and the bio-chemical interactions of the species and therefore depend on the concentrations C_i. Hence, the mutual coupling of the equations in the system (1) is only due to these functions g_i. Finally, the diffusion coefficients ε_x, ε_y, and ε_z are assumed to be given functions.

The physical domain in space is bounded by vertical, closed boundary planes, by the water elevation surface, and by the bottom profile. On these boundaries, Dirichlet, Neumann or mixed boundary conditions have to be prescribed. Supplementing this with an initial condition, the concentrations c, can be computed in space and time.

Along the lines described in [8], the physical domain is covered by a set of N Cartesian grid points with mesh sizes Δx, Δy, and Δz. Next, the advection terms in (1) are discretized by a third-order upwind-biased scheme (see [9]), and the diffusion terms are discretized second-order, using the standard symmetric three-point formulas. This results in the semidiscrete initial value problem (IVP)

(2a) $\frac{d C (t)}{dt} = F (t, C (t)) : = H (t, C (t)) + G (t, C (t)), C (t_{0}) = C_{0},$

which is of dimension mN. Here, C contains the m concentrations c_i at all N grid points and C ₀ defines the initial values. The linear function H(t,C(t)) represents the advection-diffusion terms, and the generally nonlinear function G(t,C(t)) contains the reaction terms and emissions from sources. From (1) we see that H(t,C(t)) can be written as

(2b) $H (t, C (t)) = (X (t) + Y (t) + Z (t)) C (t),$

where the matrices X, Y and Z originate from the discretization of the derivatives in the various spatial directions. These matrices are block-diagonal, with m (identical) blocks of dimension N. In the next section we will discuss suitable time integration methods for the IVP (2).

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Groundwater Flow Within a Fracture, Matrix Rock and Leaky Aquifers: Fractal Geometry

Abdon Atangana , in Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, 2018

11.5.2 Problem Formulation

Groundwater representations portray the subsurface movement and transportation processes by means of mathematical equations supported by positive simplifying suppositions. These statements characteristically engage the bearing of flow, geometry of the aquifer, and the heterogeneity or anisotropy of sediments or bedrock within the aquifer. These are geological formations through which the groundwater flow is changing in time and space. The simplest oversimplification of groundwater flow equation, which while we're on the subject is also in accord with factual physics of the observable fact, is to assume that water level is not in a steady but transient state. In 1935, Theis [183] was the first to extend a modus operandi for unsteady-state stream that brought in the time factor and the storativity. He distinguished that at a time when a well trenchant a wide-ranging confined aquifer is pumped at a constant rate, the pressure of the discharge extends externally with time. The rate of decline of head, multiplied by the storativity and summed over the area of influence, equals the discharge. The unsteady-state (or Theis) equation, which was derived from the analogy between the flow of groundwater and the conduction of heat, is perhaps the most widely used partial differential equation in groundwater investigations:

(11.39) $U D_{t} H (s, t) = T D_{s s} H (s, t) + \frac{1}{s} D_{s} H (s, t) .$

The above equation is classified under parabolic equations. However, very few geological formations are completely impermeable to fluids. Leakage of the water could thus occur, should a confined aquifer be over- or under-lain by another aquifer. The behavior of such an aquifer, often referred to as a leaky or semi-confined aquifer, needs thus not be the same as that of a confined aquifer. Although the nature of a semi-confined aquifer differs from that of a true aquifer, it is still possible to use the basic principles of confined flow to arrive at the governing equation for such an aquifer. This is in particular true in those situations where the confining layer between the two aquifers is not too thick and the flow is mainly in the vertical direction. According to Hantush and Jacob [172,173], the drawdown due to pumping a leaky aquifer can be described by the following equation:

(11.40) $U D_{t} u (s, t) = T D_{s s} u (s, t) + \frac{1}{s} D_{s} u (s, t) + \frac{u (s, t)}{λ^{2}},$

where u is the drawdown or change in the level of water, U is the specific storativity of the aquifer, and T is the transmissivity:

$λ^{2} = \frac{B}{C^{'}},$

with C and $C^{'}$ the hydraulic conductivities of the main and confining layers, respectively; B and d the thicknesses of the main and confining layers, respectively; and Θ the discharge rate of the pumping. This partial differential equation describing the movement of water through the geological formation during the pumping is subjected to the following initial and boundary conditions:

$u (s, 0) = H_{0}, \lim_{s \to \infty} H (s, t) = H_{0}, Θ = \frac{2 π^{n / 2}}{Γ (n / 2)} s_{b}^{n - 1} C d^{3 - n} \partial_{u} (s_{b}, t) .$

However, when we consider the diffusion process in the porous medium, if the medium structure or external field changes with time, in this situation, the ordinary integer order and constant-order fractional diffusion equation model cannot be used to properly characterize such a phenomenon [180,184]. This is the case of the groundwater flow in the deformable aquifer, the medium through which the flow changes with time and space [165,166]. Note that Hantush equation cannot handle this case. One of the purposes of this work is therefore devoted to the discussion underpinning the description of water flowing through a deformable leaky aquifer, on one hand. In order to include explicitly the variability of the medium through which the flow takes place, the standard version of the partial derivative with respect to time is replaced here with variable-order (VO) fractional to obtain

(11.41) $U D_{t}^{α (x, t)} u (s, t) = T D_{s s} u (s, t) + \frac{1}{s} D_{s} u (s, t) + \frac{u (s, t)}{λ^{2}}, 0 < α (x, t) \leq 1 .$

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Groundwater Recharge Model With Fractional Differentiation

Abdon Atangana , in Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, 2018

13.3 Analysis of Uncertainties Within the Scope of Statistics

In this section, we devote the work to the study underpinning the uncertainties on parameters used in the mathematical equation as indicated in [220]. The aim of this study is to avoid all the errors obtained from field measurements [220]. It is important to note that data collected from field observations may differ from one time to another, therefore to get rid of uncertainties, we make use of statistical formula to evaluate uncertainties associated to the collected data. Harmonic mean: The mean is a typical value found within a data set over time series, and can often be seen as the operating point of a physical system generating the series of data [220]. Furthermore, the harmonic mean is a kind of mean used when the numbers are defined in relation to some unit; when a sample contains extreme values; and when more stability is needed regarding outliers. It is given as

(13.11) $H_{x} = \frac{n}{\sum_{i = 1}^{n} \frac{1}{x_{i}}}$

where $H_{x}$ is harmonic mean, n is a number of data points, $x_{i}$ is the ith sample point. Figs. 13.1 and 13.2 show the harmonic mean of the hydraulic head as a function of time for the coefficient drainage Dr and the storativity.

Standard deviation SD is a measure of dispersion round the mean value, for a generated time series of data. In other words, it gives insight to uncertainty and error based on how concentrated or scattered the samples are from the mean value. SD is given by the square root of the variance as follows:

(13.12) $S = \sqrt{\frac{\sum_{i = 1}^{n} {(X_{i} - \overline{X})}^{2}}{n - 1}}$

where s is the standard deviation, $X_{i}$ is a data point, and $\overline{X}$ is the mean X (see Figs. 13.3 and 13.4). Skewness is higher-order statistical attribute of a time series. It is essentially used for a measure of symmetry of the probability density function of the amplitude of a time series and the assessment of the departure from normality of the data. To expand, when the number of large and small amplitude values is equal within a time series of data, there is a value of zero skewness. Subsequently, a skewed distribution positive or negative has a mean and median that are not identical. Furthermore, skewness can be quantified using skewness mathematical formula to define the extent to which the time series distribution differs from a normal distribution [220]:

(13.13) $G = \frac{1}{n s^{3}} \sum_{i = 1}^{n} {(X_{i} - \overline{X})}^{3}$

where g is the skewness, $Y_{i}$ a data point, and $\overline{Y}$ is the mean.

Kurtosis is a measure of the peakedness of a distribution, or in other words how 'heavy-tailed' or 'light-tailed' the data is relative to a normal distribution. To expand, when a data set has a high kurtosis, it is associated with heavy tails, or outliers. Alternatively, when the measure of kurtosis is low, it is associated with a lack of outliers.

(13.14) $g = \frac{1}{n s^{4}} \sum_{i = 1}^{n} {(Y_{i} - \overline{Y})}^{4}$

where g is the kurtosis.

The harmonic mean indicates that the typical hydraulic head values occurring within the generated time series data as a result of change in S and change DR are in the ranges of 0–4000 m and 0–1650 m, respectively. These are given in Figs. 3 and 4, respectively; this can be found in the work done by Atangana and Jessica (see [220]). The SD appears to decrease for the S distribution and increase for the DR distribution over time as depicted in Figs. 5 and 10, respectively; this can be found in the work done by Atangana and Jessica (see [220]). This essentially means that DR has a greater deviation from the mean, and therefore a greater extent of uncertainty and error. Moreover, the SD for DR becomes significantly high after approximately 30 days, which means that the vulnerability to error is greater after 30 days. Furthermore, the graphs generated for skewness (Figs. 7 and 8; this can be found in the work done by Atangana and Jessica [220]) both indicate the data are not normal for both S and DR distributions. Moreover, when skewness is positive, it yields larger error, and so a greater uncertainty in the data. This is significant for the DR distribution, and more so after about 30 days. This supports what was previously said regarding DR having a greater extent of uncertainty and error. As mentioned, a higher kurtosis is associated with a presence of outliers. As a result, the distribution generated for S (Fig. 9; this can be found in the work done by Atangana and Jessica [220]) indicates that a shorter time period in hydraulic head changes would have less uncertainty than a longer period of time. This is because as time increases for this parameter, the value of kurtosis increases (presence of outliers increases). This means that as time increases, the error associated with S increases as well. On the other hand, the value of kurtosis is high already at a much earlier time for the distribution given for DR (Fig. 10; this can be found in the work done by Atangana and Jessica [220]). Ultimately, this means that both S and DR yield data that is not normal, but DR is the parameter associated with greater error and uncertainty. This statistical analysis infers that although both S and DR have associated error and so also uncertainty, it is DR that yields greater amount of error. This makes sense when looking at the initial plots (Figs. 1 and 2; this can be found in the work done by Atangana and Jessica [220]) because DR variation depicted an even greater change in hydraulic head over time, in comparison to S variation. With that being said, it is imperative to correctly distinguish between aquifer systems when using literature as a means for a value of S, if no fieldwork can be done. Even in the case where field work or laboratory work can be done to estimate S, critical consideration should be given to the reliability of the estimates as they have a tendency to be disputed. On the other hand, the aforementioned also suggests that incorrect DR estimates would yield inadequate changes in hydraulic head; and this would ultimately yield unreliable recharge estimates. More importantly, this means that sufficient consideration should be given to estimates of transmissivity, flow type, and flow path length, as all this information is needed to estimate DR. These considerations should be done with much thought for a groundwater system associated with significant heterogeneity, because this phenomenon causes one aquifer system's hydraulic properties to differ significantly from an adjacent aquifer system, and/or even within the same aquifer system. These considerations should be made because an inaccurate S or DR estimate would yield significant error in the eventual recharge estimate (this can be found in the work done by Atangana and Jessica [220]).

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Foreword 2 for 1st Edition

Dean Abbott , in Handbook of Statistical Analysis and Data Mining Applications (Second Edition), 2018

While it would be easier for everyone if data mining were merely a matter of finding and applying the correct mathematical equation or approach for any given problem, the reality is that both "art" and "science" are necessary. The "art" in data mining requires experience: when one has seen and overcome the difficulties in finding solutions from among the many possible approaches, one can apply newfound wisdom to the next project. However, this process takes considerable time, and particularly for data mining novices, the iterative process inevitable in data mining can lead to discouragement when a "textbook" approach doesn't yield a good solution.

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Robust control strategy for HBV treatment: Considering parametric and nonparametric uncertainties

Omid Aghajanzadeh , ... Ali Falsafi , in Control Applications for Biomedical Engineering Systems, 2020

2 HBV mathematical model

In the recent years, the methods of using antiviral drugs for the treatment of viral diseases like HBV has been studied via the mathematical equations of their dynamics ( Sheikhan and Ghoreishi, 2013a). Accordingly, control methods can be employed in order to design the treatment strategies for these kinds of antiviral therapies.

According to the literature, a recently proposed and validated nonlinear mathematical model frequently utilized to study the HBV infection is (Hattaf et al., 2009a):

(1) $\begin{array}{l} \frac{d x}{d t} & = λ - d x - (1 - u_{1}) β x v \end{array}$

(2) $\begin{array}{l} \frac{d y}{d t} & = (1 - u_{1}) β x v - δ y \end{array}$

(3) $\begin{array}{l} \frac{d v}{d t} & = (1 - u_{2}) p y - c v \end{array}$

in which, x, y, and v represent the uninfected cells, infected cells, and virion numbers, respectively. u ₁ and u ₂ are control inputs expressing the rate of drug usage. In addition, d, δ, c, p, β, and λ are model parameters which are positive constants and described in details in Table 1. The third column of the table consists of the values of parameters extracted from the research conducted by Sheikhan and Ghoreishi (2013a) and Aghajanzadeh et al. (2017). It is worth mentioning that in Eqs. (1)–(3) uninfected cells are created at a rate λ, die at a rate dx, and become infected at a rate βvx; infected cells are created at a rate βvx and die at a rate δy; virion are created from infected cells at a rate py and are removed at a rate cv (Wang and Wang, 2007).

Table 1. Parameters definition and values

Parameter	Definition	Value
d	Death rate of target cells	0.0038 day⁻¹ mL⁻¹
δ	Death rate of infected cells	0.0125 day⁻¹
c	Clearance rate of free virions	0.067 day⁻¹
p	Rate of production of virions per infected cell	842.0948 day⁻¹
β	Rate of infection of new target cells	1.981 × 10⁻¹³ day⁻¹ mL⁻¹
λ	Rate of production of new target cells	2.5251 × 10⁵ day⁻¹ mL⁻¹

Eqs. (1)–(3) are differential equations originally proposed by Nowak et al. (1996) according to some experimental study through the examination of reasonable numbers of patients during their treatment intervals. In the first study, 45 patients were treated for 28 days; in the following experiment, 50 patients were treated for 24 weeks which were explained by Nowak et al. (1996). u ₁ and u ₂, as the control inputs, were added to this model by Hattaf et al. (2009a), which represent the efficiency of two different treatment mechanisms of the HBV. These mechanisms block new infection and inhibit viral production. Ciupe et al. (2007) have investigated a nonlinear mathematical model of HBV and also studied its drug therapy using two drug therapy controls. Yosyingyong and Viriyapong (2018) studied the efficiency of drug treatment in preventing new infections and efficiency of drug treatment in inhibiting viral production. In this study, numerical simulations are established to show the role of optimal therapy in controlling viral replication.

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MINDWARE

AUGUST STERN , in Quantum Theoretic Machines, 2000

What is the mathematical structure of intelligent thought?

This is a critical question one has to answer meaningfully and in technical terms prior to being able to put human thought into precise mathematical equations. In science one canhardly finds a more complex and intractable problem. Despite various attempts at deciphering the code of the thinking brain, the fundamental mathematical structure of the intelligent thought has remained a mystery.

The enormous complexity of this problem has defeated some and inspired others. John von Neumann was led to believe, although later abandoned the idea, that the laws of the cognitive brain are not mathematical in nature and therefore noncomputable. A century earlier August DeMorgan and George Boole had offered a heuristic, but surprisingly effective, mathematical model of logic, in which the abstract logical operations AND, OR and NOT were identified as the algebraic operations over binary numbers. Their names are immortalized in the DeMorgan duality laws and Boolean logic algebra. However, in the ad hoc system of DeMorgan and Boole the conversion from logic to algebra and the connection between logical and algebraic operations do not stem from fundamental first principles. Nor is there a universal mathematical operation in the system: subtraction, product and addition are required, although these can be reduced to two operations making use of dualities linking AND and OR. Although there are universal and functionally complete monoconnectives NAND and NOR, these do not have a fundamental mathematical realization.

Boolean logic algebra conveys a sense of simplicity and beauty. George Boole entitled his landmark work The Laws of Thoughts, but they are rather the 'thoughts' of a computer than the thoughts of a creative mind. As far as the intelligent brain is concerned, Boolean logic, despite its many attractive features, cannot be considered fundamental. In a quest for the mathematics of thoughts we have to begin from square one. Our goal is to identify and to describe the universal thinking operation mathematically. Moreover, assuming that logical thinking obeys some mathematical laws, we hope to determine these laws through logical considerations alone. In persuing this goal we want to rely on the properties of thinking as such, and not on any other insight or conjecture which mathematical intuition may offer.

We now show that the syllogism rule hides within itself the fundamental mathematical principle of thinking. The logical string

$(i \to n) \land (m \to j)$

can be contracted to

$(i \to j)$

given the condition that

$n = m .$

Treating the logical symbols as tensor indices, and lowering and raising the indices in accordance with the tensor summation rule we can write:

$\to_{k}^{i} \to_{j}^{k} = \to_{j}^{i}$

When the covariant and contravariant indices coincide, the syllogism inference becomes computable. Introducing the summation sign explicitly, we get the composition

$z_{i j} = \sum_{k} x_{i k} y_{k j}$

which one recognizes as the usual rule for matrix multiplication: the product (i, n)·(m, j) exists if and only if n = m. Expressing the syllogism law in tensor form, we rediscover from the empirical result of logic the familiar rule of matrix multiplication. The transitivity of implication leads to the matrix principle at the foundation of logical thought. We have thus established the following principle of major importance:

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Numerical Analysis

John N. Shoosmith , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

I.A The Numerical Approach to Problem Solution

The means by which physical situations and processes are described, analyzed, designed, and simulated is through mathematics. Natural laws are stated in terms of mathematical equations, and the behavior of systems that obey those laws is described by their solutions.

Unfortunately, the mathematics of many of the processes we would like to study quickly becomes intractable when approached by conventional means. For example, analytical solutions to most nonlinear systems of equations simply cannot be found. The best that can be done, in the traditional sense, is to attempt a series expansion of the solution.

Today, there is another approach: the problem statement and variables of interest can be approximated numerically. Analysis and problem solution can then be performed through numerical computation with the aid of high-speed digital computers. To be sure, something is lost when it becomes necessary to resort to numerical methods, because characteristics of the solution that are immediately apparent from inspection of analytical expressions may be obscured in listings of numbers; however, a numerical solution is certainly better than no solution, and sometimes its nature can be revealed by repeating the process with small changes in the data. Also, computer generated graphs or images can often provide a sufficiently accurate visual interpretation of the solution to aid in its understanding.

The numerical approach is illustrated for a very simple problem in Fig. 1. The problem is posed in physical terms in (a), and its mathematical formulation is given in (b). In this situation we know the solution, but in more complex cases, of course, we may not.

The next step is to select or develop a numerical method. Here, we have chosen the Euler method for the solution of initial value problems involving ordinary differential equations, again because of its simplicity. Numerical methods can be thought of as operators that accept numbers as input (in this case the initial velocity V ₀, the problem parameters D and M, and the discretization parameter h) and produce other numbers as output (the successive values of time and velocity).

The final stage is to produce an algorithm, a step-by-step implementation of the method. Algorithms are thought of rather like flow charts and are usually described in an unambiguous way by means of an algorithmic or even a computer-programming language. Algorithms are recipes that could conceivably be followed by a person with pencil and paper; however, it is usual to convert them to computer programs, which can then be executed on a suitable computer.

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Radioactive Tracers

P. Brisset , J. Thereska , in Encyclopedia of Condensed Matter Physics, 2005

RTD Modeling

Experimental RTD is the basic information for further treatment. Throughout its modeling, the optimal parameters for process simulation and control could be determined. Modeling is realized generally by mathematical equations involving empirical or fundamental parameters, such as axial dispersion coefficients or arrangement of ideal mixers.

Evaluation of the dynamic parameters of continuous flows in vessels by optimizing the experimental RTD with the theoretical model (or RTD) is almost a common approach in field experiments (parametric approach or gray box principle). The fitting coefficient is found by using the least-squares method. Always knowing some features of the reactor performance, parametric modeling can be used to find the dynamic parameters.

Two classes of classical well-known models are mostly used: N (of equal size) fully mixed tanks in series and axial dispersion model with Peclet number Pe as a parameter of axial dispersion. In practice, however, the above ideal conditions are rarely achieved and the situation is usually somewhere between the two.

The axial dispersion model is used when the material that passes through a vessel moves along the longitudinal direction by advection as it tends to mix in the transverse section. The differential equation representing the unidirectional dispersion model is

$\frac{\partial C (x, t)}{\partial t} = D \frac{\partial^{2} C (x, t)}{\partial x^{2}} - u \frac{\partial C (x, t)}{\partial x}$

where C (x, t) is the concentration at a distance x at time t, D is the axial dispersion coefficient, and u is the mean velocity of advective transport.

For an instantaneous and planar injection at $t = 0$ and $x = 0$ , the solution is

$C (x, t) = \frac{M}{A \sqrt{4} π Dt} e^{- {(x - ut)}^{2} / 4 Dt}$

where M is the mass of a tracer injected into the cross section at the inlet.

The model parameter normally used as an index of mixing is the nondimensional Peclet number, $P e = u x / D$ ( $P e = infinite$ for plug flow whereas $P e = 0$ for completely mixed flow).

Ideal stirred tanks connected in series model are frequently used to describe the systems where it is assumed that an injected tracer is immediately (in comparison to the flow rate) mixed with the entire volume of the system as a result of either mechanical mixing or some circulation (Figure 3).

In such a case, the concentration of the tracer at the inlet and the outlet is equal. Then the time–concentration function for the outlet is

$\frac{{d C}_{0} (t)}{d t} = \frac{1}{\bar{t}} [C_{0} (t) - C_{i} (t)]$

As the C _i(t) function is usually a Dirac pulse $δ (t)$ , normalized C ₀(t) represents the RTD which in the time domain is equivalent to

$E (t) = \frac{1}{\bar{t}} exp (- \frac{1}{\bar{t}})$

It is common, in practice, to present the system as an arrangement of perfect mixers connected in series. For such a model, E(t) is

$E (t) = \frac{1}{(k - 1)!} \frac{1}{t_{0}} {(\frac{t}{t_{0}})}^{k - 1} exp (- \frac{t}{\bar{t}})$

where t ₀ is the MRT for a single mixer, k is the number of mixers. The total MRT is then $\bar{t} = k t_{0}$ .

In order to compare E(t) curves for different flow conditions and mixing efficiency, normalization to dimensionless time θ is performed: $θ = t / \bar{t}$ . Then the equation takes the form

$E (t) = \frac{k^{k}}{(k - 1)!} θ^{k - 1} exp (- k θ)$

where k is infinite for plug flow and is equal to 1 for completely mixed flow.

For continuous process vessels with high dispersion, the best model is the cascade of mixers-in-series. When dispersion is low, either the axial dispersion or the cascade of mixers follows the material transfer well. In the latter case, both models are equivalent (Villermaux): $P e = 2^{*} (N - 1)$ .

The cascade of mixers in series model describes quite well all the simple flows with partial dispersion. Moreover, the cascade of mixers-in-series model offers the possibility to build up more complicated models, combining the mixer units in various arrangements as well as adding into them several cells or zones with different flow regimes, that is, plug flow, stagnant zone, dead volume, bypass, recirculation, etc.

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An Introduction into the Intelligent Laboratory of the Economy–Energy–Electricity–Environment (ILE4)

Zhaoguang Hu , ... et al., in An Exploration into China's Economic Development and Electricity Demand by the Year 2050, 2014

Computable General Equilibrium Model

CGE model is a model that turns the abstract GE theory to a real economy situation. The basic principle of CGE model is shown in Figure 2.6 . Using a single group of mathematical equations, the model describes the equilibrium relationship between supply and demand on various markets. In this group of equations, there are exogenous variables (generally showing shocks received by the economic system) and endogenous variables (generally showing quantities and prices of goods within the economic system). Changes in the exogenous variables that influence any part of the economy can spread to the entire system. This leads to universal changes in the quantity and prices of key goods and factors. These changes can cause the entire economic system to shift from one state of equilibrium to another. The CGE model can calculate a set of figures and prices when the demand and supply reach equilibrium during the transition period. ⁷

In Figure 2.6, production activity, subjects, goods, essential factors, etc. are abstracted to eight types of accounts. These are shown in the boxes. The arrows between the boxes show the cash flow between accounts. The arrows which point into an account represent income going into that account, while the arrows that point out of an account represent expenses paid to another account. This figure describes the economic cycle; production leads to income, income leads to demand, and demand leads to production.

The accounts within the CGE work as follows: the production account shows production activity within the economic system. The factor account contains production factors, consisting of capital and the labor force. The enterprise account contains corporations, including agricultural, industrial, service, and other industries. The resident account contains residents (both urban and rural). The government account contains branches of the government, including both central and regional governments. The merchandise account portrays the production of various goods, including agricultural, industrial, and service goods. The foreign account represents countries other than China. China, on the one hand, wants to export goods, and, on the other hand, wants to import some foreign goods. The savings account contains deposits, which include the deposits of: residents, governments, corporations, and foreign entities. This account also contains the savings which have been shifted to investment.

Residents acting as owners of labor and corporations acting as owners of capital, provide the factor inputs into production. Production uses primary factor and intermediate input to produce goods. In this process, value is added which becomes profit within the factor account. In the next step, this factor profit is distributed to proprietors. Labor income goes to the residents, while capital profits go to corporations. Corporations also distribute a part of their income to residents. Governments act as agents of economic regulation. Government income comes from production taxes, sales taxes, resident and corporate income taxes, and import duties. Government income is sometimes used in transfer payments to businesses and residents. Both residents and governments use their income for the consumption of goods. Besides consumption, the remaining income is used for savings. Corporations, besides taxes and payments to residents, also use their income for savings. Goods that are not produced domestically come from foreign imports, and those that are not sold domestically can be sold internationally. A foreign account surplus, resulting from international trade, makes up one part of total deposits. Deposits are used to purchase capital goods by means of investment and become new capital factor inputs in the next cycle of the production process.

ILE4 developed computable general equilibrium electricity (CGE-E) model which is capable of performing scenario analyses on economic development. It is also capable of simulating the influence of exchange rates, taxation, and other policies on GDP, tertiary production growth, and national and regional industrial electricity supply and demand.

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Basic Presence/Absence Situation

Darryl I. MacKenzie , ... James E. Hines , in Occupancy Estimation and Modeling (Second Edition), 2018

8.2 An Implicit Dynamics Model

One approach to modeling detection/nondetection data from multiple seasons is to effectively apply a single season model to the data collected in each of the T seasons. Under this approach, occupancy in one season is considered to be a random process in the sense that the occupancy status of a unit in the previous season has no effect on the probability of occupancy at the units in the current season. Regardless of the underlying processes of change in occupancy, only the resulting pattern or level of occupancy each season is modeled. Here, let $ψ_{t}$ be the probability a unit is occupied in season t, and $p_{t, j}$ be the probability of detecting the species in the jth survey of a unit during season t (given the species was present at the unit in season t). Using the model-based approach of MacKenzie et al. (2002) (as detailed in Chapter 4), the observed data likelihood for season t would be

${ODL}_{t} (ψ_{t}, p_{t} | h_{t, 1}, h_{t, 2}, \dots, h_{t, s}) = \prod_{i = 1}^{s} P r (h_{t, i} | ψ_{t}, p_{t}),$

with the observed data likelihood evaluated for the full T seasons being the product of the seasonal likelihoods, i.e.,

$ODL (ψ, p | h_{1}, h_{2}, \dots, h_{s}) = \prod_{t = 1}^{T} {ODL}_{t} (ψ_{t}, p_{t} | h_{t, 1}, h_{t, 2}, \dots, h_{t, s}) .$

This same model can also be developed directly from the detection histories using the same techniques as in the previous chapters: taking a verbal description of the detection histories and translating them into a mathematical equation. Consider again the detection history $h_{i} = 110 000 010$ . A verbal description of these data would be:

In season 1:: the unit was occupied with the species being detected in the first and second surveys, but not in the third.
In season 2:: the unit was either occupied with the species not being detected in any of the 3 surveys, or the unit was unoccupied.
In season 3:: the unit was occupied with the species being detected in the second survey, but not in the first or third surveys.

Translating these statements into mathematical equations using the defined model parameters we have:

Season 1:: $ψ_{1} p_{1, 1} p_{1, 2} (1 - p_{1, 3})$ ,
Season 2:: $ψ_{2} (1 - p_{2, 1}) (1 - p_{2, 2}) (1 - p_{2, 3}) + (1 - ψ_{2})$ ,
Season 3:: $ψ_{3} (1 - p_{3, 1}) p_{3, 2} (1 - p_{3, 3})$ .

Therefore the probability of observing the entire detection history would be:

(8.1) $\begin{matrix} P r (h_{i} = 110 000 010 | ψ, p) & = ψ_{1} p_{1, 1} p_{1, 2} (1 - p_{1, 3}) \\ \times [ψ_{2} (1 - p_{2, 1}) (1 - p_{2, 2}) (1 - p_{2, 3}) + (1 - ψ_{2})] \\ \times ψ_{3} (1 - p_{3, 1}) p_{3, 2} (1 - p_{3, 3}) . \end{matrix}$

This procedure can be used to obtain the probability statement for each of the s observed detection histories, and the observed data likelihood would be calculated as

$ODL (ψ, p | h_{1}, h_{2}, \dots, h_{s}) = \prod_{i = 1}^{s} P r (h_{i} | ψ, p) .$

Expressed in terms of the underlying random variables, the implicit dynamics model would be:

$z_{t, i} \sim Bernoulli (ψ_{t}),$

$h_{t, i j} | z_{t, i} \sim Bernoulli (z_{t, i} p_{t, j}),$

which could be used to construct the complete data likelihood.

As in Chapter 4, this model can be easily generalized so that the probabilities of occupancy and detection are functions of covariates, and to allow for missing observations. Models can also be considered where there is some structural relationship among probabilities in different seasons. For example, Field et al. (2005) modeled a systematic decline in occupancy over time by defining seasonal occupancy probabilities with a linear trend on the logit scale, i.e., $logit (ψ_{t}) = β_{0} + β_{1} t$ .

Finally, we note that although the above modeling may appear to be relatively phenomenological, in the sense that vital rates (probabilities of local extinction and colonization) governing the dynamic process do not appear explicitly in this model, it actually makes fairly restrictive assumptions about these vital rates. In Section 10.4, we show that the implicit dynamics model is based on the assumption that the probability of the species not going locally extinct at a previously occupied unit is equal to the probability of colonization of a previously unoccupied unit. In the next section, we discuss a more general explicit model of occupancy dynamics, from which the above implicit dynamics model can be obtained as a special case.

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