Fundamentals Of Taxation 2020 Edition 13th Edition Ana Cruz, Michael Deschamps, Frederick Niswander, Debra Prendergast, Dan Schisler - Solutions

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For the Pareto problem in Section 13.6, code the NSGA-II algorithm and see if you can replicate the results presented in the chapter to your own satisfaction.Next, increase the number of variables to four and five and rerun the algorithm.What is your expectation about the algorithm's run time until
Consider the problem of finding a personalized shoe for elite professional athletes. How could you use interactive evolutionary computation to help design such a shoe? Detail the steps and experiments you would conduct.
Imagine that you wanted to design a possible improvement to the winglets that are now ubiquitous on commercial airplane wingtips. Winglets are a relatively recent invention, having been developed in the 1970s at NASA. Think about a series of experiments you could conduct using evolvable hardware
Implement an ant colony optimization algorithm to demonstrate the double-bridge problem. The problem has ants that start on one side of a double bridge and must go across one of the two bridges to get to a food source. Start with one of the bridges at half the length of the other. How quickly does
Refer back to software you have written or use an available feed forward neural network simulator that employs sigmoid functions for the nodes and instead of using backpropagation, substitute a differential evolution algorithm to do the training. Test the results on data from
Then extend the function to be a function of n dimensions:f(x) = C1-1 x2 -20(2_ (cos (TEX;)- 1)Estimate the convergence rate in terms of number of generations to reach a target level of the function for an initialization that you choose as a function of n for n = 2, 3, 5, and 10.
Implement a particle swarm optimization algorithm to find the minimum of the function f(x, y) = x2 + y2 - 20[cos ("X) + cos (ny) - 2]as shown in Figure
Give an interpretation of the Sugeno and Choquet fuzzy integrals; compare and contrast them: you might want to illustrate with examples.
Use the Sugeno measure in Table 9.2 and redo Example 9.6 with the same input training data O. You will have to write the QP program (or use the Matlab or other package version). Do you recover the measure in Table 9.2?
Fill in all the details in Example 9.6, that is, hand-compute «, Z, D, A, and b.
Write the complete matrix A and vectors g and b in Eq. 9.25 for a set of information sources X that has four elements.
Using the notational convention in Eq. 9.21 with three sources of information, intuitively describe the fusion operation performed by the Choquet integral with the following measures:i. g= [0, 0, 0, 1, 1, 1, 1]ii. g= [1, 1, 1, 1, 1, 1, 1]iii. g = [1/3, 1/3, 1/3, 2/3, 2/3, 2/3, 1]iv. g=[0, 0, 0, 0,
Generate Eq. 9.18, that is, compute and simplify @/agk.
Verify that g in Eq. 9.13 is a fuzzy measure and that Cg(h) (Eq. 9.12) is the corresponding LOS operator.
Let h be the characteristic function for a subset A of X, that is, 1, if x E A h(x)=10, if x ( A . What is the integral (Sugeno or Choquet) of this h with respect to a given fuzzy measure g?
What can you conclude if Sg(h) = 0? Is it true that the function h must be identically 0?
Is the Sugeno/Choquet integral linear, that is, is Sg(a · h +b) = a . S.(h) + b, where a and b are real numbers in [0,1]? (Same equation substituting Cg for Sg).You can assume that the linear combinations remain in the interval [0,1], although that is not strictly required. Supply a justification
Show that if h1(x) ≤ h2(x) for all x € X, then Sg(h1) ≤ Sg (h2) (same for the Choquet integrals).
Show that if g (A) < g2(A) for all A C X, then Sg, (h) { Sg2 (h) (same for the Choquet integrals).
Show that if h(x)=a for all x, then Sg(h)=a, and C(h) =a.
Given a finite set X={x1, X2, ..., x1}, a measure g : 2" -> [0, 1], and a function h : X -> [0, 1], show that both the Sugeno and Choquet integrals of h with respect to g are generalized expected values, that is, they both lie between the minimum and maximum of the values h(x1), . .., h(x).
Given the measure Subset A g(A)Ø0{x1}0.5{x2}0.1{ X3}0.3{ X1, X2}0.7{X1, X3}0.9{X2, X3}0.5 X1.0 Calculate the Sugeno and Choquet fuzzy integrals for the following functions:i. h(x1)=0.3; h(x2)=0.5; h(x3)=0.9 ii. h(x1)=0.5; h(x2)=0.5; h(x3)=0.5
Show that Eq. 9.4 holds for finite sets. Hint: What are the level sets for the sorted fuzzy set {h(x(1)) ,..., h(x(m))}?
Let g'=0.4; g2=0.3; g3 =0.2 Calculate the complete Sugeno À-measure on X = {X1, X2, X3}.Subset A g(A)Ø{x1}(x2){X3}{X1, X2}{X1, X3}{X2, X3}{X1, X2,X3}
If X=0 in Eq. 9.2, show that g is a probability measure and that >" , gi = 1.
Show that Eq. 9.2 holds for X = {X1, X2, X3 } and densities {g1, g2, g3}.
Let F : X - [0, 1] be a normal fuzzy subset of X and let Pos : 2" -> [0, 1] be a function defined by 0, if A = фPos(A) =maxx EA(F(x)), if A + фA. Show that Pos is a fuzzy measure (called a possibility measure).B. Show that Pos(AUB) = Pos(A) V Pos(B) for all subsets A and B of X.C. Define a
Let X = {x1, . . . x"} and F : X - [0, 1] be a fuzzy subset of X.Define g(Ø) = 0 g(A) = max{1 -F(x)}, for all x € A Is g a fuzzy measure? Justify your answer. Consider conditions on the fuzzy set F.
Show that all probability measures are fuzzy measures, that is, if a set function g : 2* = > [0, 1] satisfies g(Ø) = 0 and g(X) = 1 and g(AUB) =g(A)+g(B), if AnB = p, then it must also satisfy g(A) ≤ g(B) if A C B.
a. Give a variation to Eq. 8.9 for a fuzzy k-NN and for a possibilistic k-NN.b. Can you think of other ways to assign soft labels to training points?
a. Use the final partition matrix from Problem 8.3 as fuzzy labels for two 2.5 and y2 = (0.5 classes. Compute the memberships for y1 = 2 using the fuzzy 2-NN and fuzzy 3-NN algorithms.b. Now suppose that X is a small piece of a larger training set, and the fuzzy labels for X are given as follows:X1
What are the similarities and differences between the hard (crisp) C-means(HCM), the fuzzy C-means (FCM), and the possibilistic C-means (PCM)? Be as specific as possible, but don't just write down the update equations.
Develop the necessary conditions for a minimum of the possibilistic C-means criterion function.
For the mathematically hardy, prove that Eq. 8.6 represents the necessary conditions on the covariance matrices when using cluster-specific Mahalanobis distances.
Prove that the necessary condition for the cluster centers in an FCM algorithm with Mahalanobis distances takes the same form as in Theorem8.1.
a. Apply the crisp 2-means algorithm using Euclidean distance to the data set:0 54 in in 45×11 NN with vi (1) = (5) and v2(1)= ( 3.75b. Now run the crisp 3-means beginning with the final two cluster centers from(a) and a third center that is the point of X furthest from its cluster center. Do you
Let X = (72), (=) (8). (0).(7). (2)} and let v1 = (3)and v2 = (3).Execute one complete iteration of the fuzzy 2-means algorithm with m =2 and one complete iteration of the possibilistic 2-means with m = 2 and each nis equal to 1.
What is the basic concept underlying all clustering procedures?
Right after Eq. 7.1, we mentioned how it is a challenge to define appropriate domains to describe the values (create fuzzy sets) for some linguistic variables.Take the example of HEALTH. How could you define a domain on which you could define values like good, average, poor, and so on, so that
You are to design a fuzzy rule-based system to balance a broomstick: a 2-input, 1-out control function. The inputs are the angle the broom makes with the vertical axis and the rate of change in that angle between instances. The output is the force you apply to your hand to bring the broom into
ª You are to design a Takagi-Sugeno fuzzy rule-based system to perform a 2-input, 1-output control function. Based on a considerable experiment in which a skilled human controlled the plant, it is decided that the linguistic variable for the first antecedent should have two linguistic values,
Let X= {a,b, c,d, e, f}; Y = {1, 2, 3, 4}; Z= {s, t}; and let A=0.0/a+0.0/b+1.0/c+1.0/d+0.0/e +1.0/f B=1.0/1+0.7/2+0.2/3+0.0/4 C =. 9/s+0.5/t Consider the rule:IF V is A THEN W is B A. Translate this rule using Zadeh's original translation formula (i.e ., the Lukasiewicz implication).B. Show the
Let X1 = {1, 2, 3}; X2 = {w, x, y, z}; Y = {a, b}; and let A1=(0.0,1.0,0.6)(fuzzy subset of X1)A2=(0.0,0.4,0.8, 1.0)(fuzzy subset of X2)B=(0.2,0.8)(fuzzy subset of Y)Consider the rule:IF V is A, THEN W is B A. Translate this rule using Correlation-Product encoding.B. Show the result of inference
If the exact fuzzy antecedent is applied to a fuzzy rule that is translated using Zadeh's original formula (i.e ., the Lukasiewicz implication), the result is not exactly the consequent fuzzy set. However, what nice property does this translation of modus ponens possess? Can you prove it (not just
Let X= {1, 2, 3}; Y = {a, b]; Z = {w, x, y, z}; and let A=1.0/1+0.4/2+0.1/3 B=0.2/a +0.8/b C=0.0/w+0.4/x+0.8/y+1.0/z Consider the rule:IF U is A THEN W is C A. Translate this rule using Zadeh's original translation formula (i.e ., the Lukasiewicz implication).B. Show the result of inference if the
Let Y ={-3, - 2, - 1, 0, 1, 2, 3}. Suppose that the output of three rules are as follows:B1 = (0 ,. 1 ,. 6, . 6, . 6, . 2, . 1)B2 = (0,0, 0 ,. 2 ,. 7 ,. 7 ,. 2)B3 = (0,0, 0,0,0, 1, 0)Compute the crisp output generated by a fuzzy control system, using centroid defuzzification.
In a fuzzy logic control application, explain what is meant by the terms"fuzzification" and "defuzzification." Why are they necessary?
Repeat Exercise 6.13 for (A - B) and (A x B).
For the fuzzy sets defined in Example 6.4(B), compute the «-cuts, "+(A + B), for several values of 0 < < < 1 and sketch the graphs of A, B, and (A + B).Does this match your intuition? Why or why not?(Hint:To get a nice graph, you can use the strong «-cut "+(A + B) for a =0, to define the support
For the fuzzy sets defined in Example 6.4 (A), generate all of the nonzero values of (A + B) and Max(A,B).
Let A be any fuzzy subset of X. Show that "A C BA if B< a
Suppose X = {a,b, c,d, e }. Let A be the fuzzy set with memberships (0.5, 0.4, 0.7, 0.8, 1) fora, b,c, d, ande, respectively. (A common way you might see in the literature to write this is A =0.5/a +0.4/b+0.7/c+0.8/d+ 1/e). List all nonempty «-cuts of A.
0, if x -1 Let A(x) =x+3, if-3≤x≤-2-x-1, if -23 B(x) =x-1, if 1 { x ≤2-x+3, if 2 < x < 3 A. Compute 0.7 A.B. Using the standard definitions, sketch a picture of Bc.0, if x < 2 or x>4 C. Now, let C(x) ={ x-2, if 2 ≤x ≤3 Compute BNC -x+4, if3
Consider the operators:(AUB)(x)=1A (A(x)+B(x)(ANB)(x)=0v(A(x)+B(x)-1)Ac(x) = 1- A(x)A. Show that intersection and complement satisfy the law of contradiction.B. Is it true that intersection is idempotent: ANA = A? (prove or give a counterexample)
Show that LEM and LOC do not hold for the standard fuzzy set theory definitions, that is, show that AUAc = X and ANAc = dare not true in general for fuzzy set theory. Note that X(x) = 1 and ((x) = 0 for all x E X. Hint: Think about how you show that a statement is not a theorem.
Show that DeMorgan's laws hold for the standard fuzzy set theory definitions, that is, that (AUB) = AenBe and (ANB)" = ACUBc.
Finish the proof of the distributive law in Example6.2.
Suppose X = {-3, - 2, - 1, 0, 1, 2, 3}. Let fuzzy subsets A and B of X be defined by their membership vectors A = (0.0, 0.3, 0.8, 1.0, 0.8, 0.3, 0.0) and B =(1.0, 0.9, 0.7, 0.5, 0.2, 0.0, 0.0)A. Using Zadeh's original definitions, compute AºAUB ANB B. What is AUB if AUB(x) = 1 A (A(x) + B(x))?
A. Derive the equation for the S-function of Example 6.1.Hint: There are two parabola pieces and for each piece you know two points, the value of the function at two of the parameters. But wait you say, a parabola has three coefficients, so you need at least three equations to find them. Think
Let X=[0, 10]. Define and draw the graphs of the Trap(x; 1,2,3,5), Trap(x;3,5,7,9), Trap(x; 8,9,9,10), Trap(x; - 2 ,- 1,3,5), and Trap(x; 8,9,11,12).
A trapezoid membership function is defined by four parameters a
Define the Lyapunov function related to the cellular neural network described in Section 5.3 as 1V =-; ELA(i, j; k,l)vyij(t)VyKI(t) +R ≥ "zij(t)(i.j) (k,I)(i,j)-E(i)_(k.1)B(i,j;k,I)Vyij(t)Vuk1-E(i.j) IVyij(t)Try to perform the stability analysis of the network.
Consider a Hopfield network made up two neurons. The synaptic weight matrix of the network is w= [2 ]The bias applied to each neuron is zero. The following are the four possible states of the network:x = [+1,+1]™x2 = [-1,+1]™x3 = [-1 ,- 1]™x4 = [+1, - 1]]1. Demonstrate that states x2 and x4
The discrete (time) gamma model of a neurodynamical system is described by the following pair of equations:>w(mx(m) (k) + Kj xj(k) =¢i
Given a nonlinear dynamic system as X2 -2x1 (x2 + x2)dx= =dt-X1 -2×2(x}+×2)study the stability of the origin, by making use of the following Lyapunov function:
Compute the equilibrium state of the following dynamic system:dx-6+1)-x1+x2 dt-×2(x1 + 1)Then, confirm the stability of the equilibrium state by choosing the Lyapunov function as V(x) = x'x
The Lyapunov function of a Hopfield network is written as V = -(7x2+12x1×2-2×2)2 Point out the matrix of synaptic weight of the network.
The input-output relationship of a Gaussian-based RBF network is defined by K||x(i) - w(k)||2 y(i) = >Wj(k) exp i = 1,2, ..., k 262(k)j=1 where w (k) is the center point of the jth Gaussian unit, the width o(k) is common to all the K units, and w;(k) is the linear weight assigned to the output of
Reconsider the XOR problem by using an RBF network with four hidden units, with each radial-basis function center being determined by each piece of input data.1. Construct the interpolation matrix $ for the resulting RBF network and then compute the inverse matrix -1.2. Calculate the linear weights
Define a pair of Gaussian hidden functions asØ;(x) = exp (-|x-G|2), j = 1,2 where the centers c1 and c2 are q= H| c= [8]00 1. Calculate the weight vector of the RBF network and explain why this kind of network can be utilized to deal with the XOR problem successfully.2. Study the necessity of
Here, we solve it using an RBF network, whose structure is given in Figure
Revisit the XOR problem described in Table
The multiquadrics (4.1) and the inverse multiquadrics (4.2) provide two possible choices for radial-basis functions. An RBF network using the inverse multiquadrics constructs local approximation to nonlinear input-output map-ping. However, the use of a multiquadrics represents a counterexample to
There is a remarkable fact that radial-basis functions that grow at infinity, such as multiquadrics, can be used to approximate a smooth input-output mapping with greater accuracy than those that yield a positive definite interpolation matrix. Try to explain the statement.
For the inverse multiquadrics (4.2) and the Gaussian functions (4.3), the cor-responding interpolation matrix $ is positive definite. However, the + related to the multiquadrics (4.1) is not positive definite. Prove these propositions.
Classify the two classes of input patterns depicted in Figure 3.16 by using backpropagation training algorithm.
Investigate the use of backpropagation learning algorithm employing a sigmoi-dal nonlinearity to achieve one-to-one mappings, as described below:1. f(x) ==, 1 ≤ x ≤100 2. f(x)=1gx, 1≤x≤10 3. f(x)= exp(=x), 1≤x≤ 10, 4. f(x) = sin x, 0≤x≤;For each mapping, do the following things.1.
Linearly nonseparable patterns as shown in Figure 3.15 have to be classified into two categories by using a layered network. Construct the separating planes in the pattern space and draw patterns in the image space. Calculate all weights andthreshold values of related units. Use the minimum number
Consider the network shown in Figure 3.14, with the initial weight and basis are chosen as W1 = 1, b1 = 1, w2 == 2, b2=1.The activation function ((.) is set the same as Exercise3.2.Assume the input and the desired responses of the network are x = 1, d = 1, respectively.1. Calculate the total
Use the backpropagation algorithm for computing a set of synaptic weights and bias levels for a neural network structured as in Figure 3.7 to solve the XOR problem. Assume the use of a logistic function for the nonlinearity.
Show that the network of Figure 3.13 solves the XOR problem by constructing decision regions and a truth table for the network.+1+3+4+3 Inputs Output+3+2=5+3+4 FIGURE 3.12 The network of Exercise3.3. +1 +1 -2 Inputs Output -1.5 +1 -0.5 FIGURE 3.13 The network of Exercise 3.4.
Figure 3.13 shows a neural network involving a single hidden neuron for solving the XOR problem. It can be viewed as an alternative to the network considered in Section
The network shown in Figure 3.12 has been trained to classify correctly a set of two-dimensional, two-class patterns. Identify the function performed by the classifier, assuming initially that the neurons have function 1 ф(v) = 1+ exp (-av)Draw the resulting separating lines between the two
Consider a two-layer network containing no hidden neurons. Assume that the network has q inputs and a single output neuron. Let x; denote the ith input signal and define the corresponding output asу= фWiXi=0 where w; is a threshold and 1ф(v) =1 + exp (-v)Show that this network implements a
Consider the functions x1 e -(2/2) dtф(х) ==V25 1-00 and 2 ф(x) = = tan~'(x)Explain why both the functions fit the requirement of a sigmoid function. What is the difference between the two functions?
Consider two one-dimensional, Gaussian-distributed classes X1 and X2 that have a common variance equal to 1. Their mean values are p1 =- 10, p2 = 10, respectively. These two classes are essentially linearly separable. Design a classifier that separates these two classes.
Show the single-layer perceptron can classify the patterns described in Table 2.3 successfully. However, a basic limitation of the perceptron isthat it cannot implement the EXCLUSIVE OR function. Explain the reason for this limitation. TABLE 2.3 Pattern Classification of Exercise 2.7 X1 X2 d 0 0 0
Table 2.2 describes the two classes of patterns in the two-dimensional plane.Classify the two classes by using the single-layer perceptron. Given the detailed iterative process of the training algorithm, draw the separating line on the x1, X2 plane.
Suppose that in the signal flow graph of the perceptron shown in Figure 2.7, the hard limiter is prescribed as the sigmoidal nonlinearity:ф(v) = tanh where v is the induced local field. The classification decisions made by the perceptron are defined as> observation vector x that belongs to class
Another activation function 10, if v 2 1112ф(v) =v, if - - 2 11, if v 2 5 is depicted in Figure 2.24 and is called the piecewise-linear function. Here, the amplification factor inside the linear region of operation is assumed to be unity. Based on this expression, consider the activation functions
The limiting values of the algebraic sigmoid function Vф(v) =V1 + v2 are -1 and +1. Show that the derivative of (v) with respect to v is given by dф _ ф®(v)=dv v3Calculate the value of the derivative at the origin. () 1 0.5 -0.5 0 0.5 FIGURE 2.24 The piecewise-linear function of Exercise 2.4.
Study how the graph of the logistic function is affected by varying the value of slope parameter a in Eq.2.60.Calculate the slope of the logistic function at origin. Then point out what the logistic function will be when the slope parameter approaches infinity.
Suppose the inputs applied to neuron j, which is built around the McCulloch-Pitts model, are 10, - 20, 4, - 2. The synaptic weights of connecting to the neuron are 0.8, 0.2, - 1, -0.9.Given that the externally applied bias is 0, compute the induced local field and the output of neuron j,
Which of the following statements is false?a. A corporation with average sales in excess of $26,000,000 must use the accrual method of accounting.b. The charitable contributions of a corporation may be limited.c. A corporation may be entitled to a deduction for dividends received from other
In each of the following independent cases, determine the amount of business interest expense deduction and disallowed interest expense carryforward, if any.a. Company A has ATI of $70,000 and business interest expense of $20,000.b. Company B has ATI of $90,000, business interest expense of
For each of the following cases, determine the amount of capital gain or loss to report in each year (after taking into account any applicable carrybacks) and the capital loss carryforward to 2019, if any. Assume that 2013 is the first year of operation for each corporation.? Capital Gain or Loss
Concerning the business interest expense deduction, which of the following statements is true?a. Disallowed interest expense can be carried forward indefinitely.b. The interest expense deduction cannot exceed 30% of adjusted taxable income.c. The interest expense limitation does not apply to a
A corporation has a fiscal year-end of July. If the corporation does not receive an automatic extension of time to file its return, the return will be due on the 15th of:a. December.b. November.c. October.d. September.
The Subchapter S status of a calendar year corporation is statutorily terminated on August 12, 2019. The Subchapter S status is deemed to be terminated on what date? What is the answer if the status were voluntarily revoked on that date?
A calendar year corporation properly files a Subchapter S election on January 10, 2019. On what date is the election effective? What if the election were filed on June 1, 2019?

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Fundamentals Of Taxation 2020 Edition 13th Edition Ana Cruz, Michael Deschamps, Frederick Niswander, Debra Prendergast, Dan Schisler - Solutions

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