where B ∈ R n×n was introduced and investigated in [1]. The unique solvability of AVEs (2) was given in [2]. The algorithm which can compute all solutions of AVEs (2) (A, B square) with several steps, was proposed in [3]. Using the relationship between the **absolute** **value** **equations** and second order cone, a generalized Newton algorithm was introduced in [4] for solving AVEs (2). When the matrix B is reversible, AVEs (2) can be converted into AVEs (1), so some scholars have begun to study AVEs (1) instead of AVEs (2). AVEs (1) was investigated in theoretical detail in [5], when the smallest singular **value** of A is not less than 1, AVEs (1) is equivalent to the generalized LCP, the standard LCP and the bilinear program, based on the LCP, Mangasarian introduced the suﬃcient conditions of the exis- tence of unique solution, 2 n solutions and nonnegative solutions and the nonexistence of

**absolute** **value** **equations** is also given. By utilizing an equivalence relation to the unconstrained optimization problem, we propose a modiﬁed HS conjugate gradient method to solve the transformed unconstrained optimization problem. Only under mild conditions, the global convergence of the given method is also established. Finally, the numerical results show the eﬃciency of the proposed method.

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In this paper, the nonlinear MHSS-like method has been established for solving the weakly **absolute** **value** **equations** (AVE). In the proposed method, two real linear subsystems with symmetric positive deﬁnite matrices αI + W and αI +T are solved at each step. In contrast, in the nonlinear HSS-like method a shifted skew-Hermitian linear subsystem with the matrix αI + iT is solved at each iteration. By using a smoothing approximate function, the local convergence of the proposed method has been analyzed. Numerical experiments have shown that the nonlinear MHSS-like method is feasible, robust, and eﬃcient.

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In this paper, we give a smoothing neural network algorithm for **absolute** **value** **equations** (AVE). By using smoothing function, we reformulate the AVE as a differentiable unconstrained optimiza- tion and we establish a steep descent method to solve it. We prove the stability and the equili- brium state of the neural network to be a solution of the AVE. The numerical tests show the effi- cient of the proposed algorithm.

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transformational approach extends the initial conceptual understanding to realizing that the distance between two values is the **absolute** **value** of their difference - that the distance from 𝑥 to b is |𝑥 − 𝑏|” (Ellis and Bryson, 2011). To find the distance between two numbers on the number line involves finding the difference between them, which is why subtraction is used. Students need to first understand this concept before moving on. Then students can start connecting the symbolic to the verbal phrase. For example, for the symbolic statement |𝑥 − 𝑎| = 𝑏 the verbal translation is “𝑥 is b units from a in both directions.” A contextual example might be my friend lives at 12 Fairhill Drive, and I live 5 houses away (house numbers change by units of 1). Where would my house be located? One would realize that you could move 5 units from 12 in the positive or negative direction. Students can make a visual representation of this on a number line. A teacher can then have can have students write a

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0, 1, 2, ... of the inner HSS iteration steps are often problem-dependent and difficult to be determined in actual computations [ 23 ] . Moreover, the iteration vector can not be updated timely. Thus, to avoid these defect and still preserve the advantages of the Picard-HSS iterative method, based on the HSS (4) and the nonlinear fixed- point **equations**

The presence of insects can also be influenced by soil pH. Soil pH values affect the diversity index, because pH that is too acidic or too alkaline can result in insect mortality. According to [17] that the soil's pH **value** affects the number of insect species, because a pH that is too acidic or too alkaline can result in death in insects because there are some insects that cannot survive at a certain pH. The acidity (pH) of the soil is a limiting factor for the life of the organism. PH conditions that are too acidic or basic will cause organisms to experience imperfect life or even experience death. The average pH at the observation field is 7 pH measures which are still within tolerance limits that can allow insects to live and breed on the surface of the soil. According to [18] the optimum pH that is tolerated by insects ranges from 5-7.

The explanation for no significant association between ROT and signed abnormal accrual variables may be due to the fact that positive accruals and negative accruals are offset against each other. Myers et al. (2003) argue that regulators are not solely concerned with the dispersion in accruals, but they are also concerned about the distortion in earnings due to inappropriate income-increasing or income- decreasing accruals. Earnings can either be managed upward (income-increasing) or downward (income-decreasing) on terms favorable to management. Myers et al. (2003) and Chi et al. (2009) also separate **absolute** abnormal accruals into positive and negative accruals. Following these studies, we identify positive and negative abnormal accruals to test whether new auditors restrict extreme income-increasing and/or decreasing activities. Previous studies posit that ordinary least squares (OLS) estimates can be considered biased in a truncated sample; therefore, we estimate a ML (maximum likelihood) truncated regression, consistent with previous studies (Greene, 2000; Myers et al., 2003; Chi et al., 2009). In untabulated results, we find mixed results. Specifically, for income-increasing accruals from DAMJ, the coefficient for ROT is significantly positive (0.006, z = 2.69) for FROT versus PROT comparison, suggesting that the FROT sample do not constrain extremely positive accruals compared to the PROT sample. Second, for income-decreasing accruals from DAMJ, the coefficient for ROT is insignificant (0.001, z = 0.20) for the FROT versus PROT comparison, suggesting that the audit quality of the FROT sample is indistinguishable from that of itself in prior years. All the coefficients for ROT for the FROT versus VROT comparison appear to be insignificant, again suggesting that there is no evidence supporting that the mandatory rotation regime enhances audit quality. The results from the DAKO partitions are consistent with above findings.

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As you study from this book, you’ll often be asked to use computer software and graphics. Exercises with this intent are marked as C (computer or calculator required), C/G (computer and/or graphics required), or L (laboratory work requiring software and/or graphics). Often you may not completely understand how the software does what it does. This is similar to the situation most people are in when they drive automobiles or watch television, and it doesn’t decrease the **value** of using modern technology as an aid to learning. Just be careful that you use the technology as a supplement to thought rather than a substitute for it.

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was made on the logarithm of the ratio. This is because the derived equation has a physical interpretation, i.e. exponential dependence on magnitude and decay due to anelastic effects. Also, directly using the ratios of maximum acceleration assumes that the uncertainty associated with this ratio is the same for all levels of ground motion [17, pp. 237–238]. This assumption must be false because otherwise us- ing the standard deviation associated with the equation, to derive predicted ratios for percentiles less than 50%, would lead to the prediction of negative PGA (by definition a positive quantity). Also working directly on the untransformed ratio violates the requirement of the standard least-squares method that the residuals be homoscedastic, i.e. that the residuals are similarly distributed with respect to the predicted **value** and the independent parameters.

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Spectroscopy (UATR-FT-IR). The selected excipients were microcrystalline cellulose, lactose monohydrate, magnesium stearate and talc. Equal proportion of active substance and excipients (w/w) was utilized in the interaction study. The **absolute** **value** of the difference between the melting point peak of active substances and the one corresponding for the active substances in the analysed mixture, as well the **absolute** **value** of the difference between the enthalpy of the pure active ingredient melting peak and that of its melting peak in the different analysed mixtures were chosen as indexes of the drug-excipient interaction degree. All the results obtained through thermal analysis were also sustained by FT-IR spectroscopy.

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Abstract. We apply the method of Hamilton shooting to obtain the well- posedness of boundary **value** problems for certain Hamiltonian systems and some estimates for their solutions. The examples of Hamiltonian functions covered by the method include elliptic polynomials and exponentially growing functions. As a consequence we prove global existence, smoothness and almost everywhere uniqueness of **absolute** minimizers in the corresponding problem of calculus of variations and hence construct the global field of extremals.

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The finite element results of the normal stress in the function of the moment are shown on Figure 10. Results are shown on a diagram (Figure 11). In **absolute** **value**, as an effect of the increasing load moment, normal stress values also increase on the tooth surfaces of the driven gear in **absolute** **value** (Figure 11).

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A Several different measures for PI (a measure of inflation that affects uncertainty) were used to reflect different beliefs about how inflation causes uncertainty. The different measures of PI that were used was (a) inflation and the **absolute** **value**, (b) inflation squared (c) the **absolute** deviation of inflation from a four-year average and (d) the **absolute** deviation of inflation from an ARIMA model of inflation. The first two measures reflected the idea that inflation generated uncertainty about prices whereas the last two measures reflected the belief that this was only unexpected inflation that generated uncertainty about prices.

(6) A life is at a neutral well-being level if and only if it is at the same well-being level as a life which is at each time at a neutral level of temporal well-being.28 One might wonder how a life could fail to be neutral if it’s always at a neutral level of temporal well-being. It seems that a life cannot be good or bad for the person living unless it is at some time at a good or bad level of temporal well-being. And one might wonder how a life could be undistinguished for the person living it if it is at all times at a neutral level of temporal well-being. Where would the undistinguishedness come from? The most plausible answer is, I think, that an atemporal component of any life is to be alive at least some point in time and that component is undistinguished for the person.29 And, since being alive at least at some point in time is a component of any possible life, it has no effect on the comparative evaluations of lives. Hence it doesn’t conflict with the full comparability between lives. It can, however, have an effect on the **absolute** personal **value** of a life. It can outweigh certain amounts of personal goodness from moments at good levels of temporal well-being and against certain amounts of personal badness from moments at bad level of temporal well-being. And it makes a life that’s always at a neutral level of temporal well-being overall undistinguished for the person living it. The contribution of the neutral temporal well-being in the life will be overall neutral. But, in combination with the undistinguished atemporal component, the life will be overall undistinguished.

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Also, the number of director’s influences board decisions and corporate performance (Jensen, 1993). According to prior research, board size is negatively correlated with firm **value** (Yermack, 1996). However, Coles, Daniel, and Naveen (2008) insist to require a large board of directors that companies operating in various fields have difficulties in consultation and supervision.

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**absolute** **value** symbols could be positive or negative. The **absolute** **value** represents the distance a number is from zero on a number line. **Absolute** **value** equa- tions can be solved by graphing them or by writing them as a compound sentence and solving alge- braically. To solve algebraically, write the expression inside the symbol as equal to the given **value** and then equal to the opposite of the given **value**. Solve each equation. Write both solutions inside one set of brackets.

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In this note we study a function which frequently appears in partial di ff eren- tial **equations**. We prove that this function is absolutely continuous, hence it can be written as a definite integral. As a result we obtain some estimates regarding solutions of the Hamilton-Jacobi systems.

1. It’s possible to describe the solutions of ⎜ x ⎟ + 2 < 5 and ⎜ x ⎟ + 2 > 5 using inequalities that don’t involve **absolute** **value**. For instance, you can write the solutions of ⎜ x ⎟ + 2 < 5 as x > -3 and x < 3. Notice that the word and is used because x must be both greater than -3 and less than 3. How would you write the solutions of ⎜ x ⎟ + 2 > 5? Explain.

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“To avoid any confusion between the scarcity and the exchange **value**, it is essential to note again that the exchange **value** is real or objective, it exists in the things; whereas the scarcity is in us, it is subjective or personal. There is not such a thing as the scarcity of the commodity (A) or the commodity (B). Consequently, there is not anything as a ratio of the scarcity of (A) to the scarcity of (B)… There exist only the scarcities of the commodity (A) and the commodity (B) for the carriers of (1), (2), (3)…of these two commodities, and the common ratios of the scarcities of (A) to the scarcities of (B)…for these carriers. It is only in conjunction with this or that individual that we can…define the scarcity as the derivative of the effective utility with respect to the quantity owned.” ([1873]1993, p. 45–46, emphases added).

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