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We have shown in Paragraph 7.5.1 that the finite element approximation has to have continuous tangential components on all mesh edges in order to conform to the space H(cur1, n). The lowest-order approximations that satisfy this requirement are with continuous and constant tangential components on the edges. Let us stay on the reference domain Kt first. The two-dimensional space [ P o ( K t ) ] 2too small to generate three linearly independent is constant tangential components on the edges of K t . Therefore we need to take one higher degree polynomial from the space [P' ( K t ) I 2 . Hence the lowest-order element ( K t ,P, 9) on the reference triangular domain Kt is equipped with the polynomial space

7. If needed, strip off the manually-applied yellow formatting from each of the hyperlinks, allowing the new white color to show. a. Triple-click the Protect Yourself button to select all of its text. b. Press Ctrl+Spacebar to remove the manual formatting. c. Repeat steps 7a and 7b for each button. 8. Format the buttons to make them more attractive. a. Select all of the buttons. Hold down the Shift key as you click each one. b. Choose Drawing Tools Format Shape Effects Shadow, and click the rst Shadow setting (top left).

The simulation results:

9 1M

with P$ = I - PH, Pi = I - Ps and where Ps and PH denote the orthogonal projectors onto the column spans of S and H , respectively, i.e., PS = S ( S * S ) - l S * and PH = H(H*H)-'H*. (c) Assume instead that z is modeled as a zero-mean random variable with covariance matrix II-' = Exz' > 0. Show that the linear least-mean-squares estimator of s = Hx is now given by L = PE - S(S*P2LS)-1S*P>'] y, where P>L = I - Pg and Pg = Compute the resulting minimum mean-square error (rn.rn.s.e.) and compare it with that of part (b).

= Pwl(1+ 1) =

During the print process, a laser printer produces ozone gas. Most laser printers have an ozone filter that captures the ozone gas as well as toner dust and paper dust. The ozone filter should be replaced or at least cleaned in accordance with the manufacturer s instructions, which are usually included in the owner s manual for the printer.

gains. Individual gains are classi ed as short-term and long-term and accumulated in separate accounts. Gains on futures and commodities are tallied separately from gains on securities. These gains, called Section 1256 gains, are taxed at a blended rate as if 60 percent of the results was long-term and 40 percent was short-term. The fund must report these gains to investors, who in turn include this income on their tax forms. The mechanics of this tax reporting are described in 10. Expenses Hedge funds pay commissions to execute trades. The fund is also charged management and incentive fees that are included as expenses on the income statement. The interest paid to nance a leveraged position is a typical expense for a hedge fund. Note that a corporation does not deduct the dividends it declares and pays to its shareholders, but a hedge fund includes payments paid for dividends declared by other companies if the hedge fund carries a short position in a security that pays a dividend (this is the scenario posed by question 6.13). The hedge fund must make substitute dividend and interest payments on all short positions in stocks and bonds, which are reported as expenses. Hedge funds accumulate short-term losses, long-term losses, and Section 1256 losses. These losses are included on the hedge fund income statement. The losses are also reported to investors, who include the amounts on their individual or corporate income tax forms. The losses reduce income and usually reduce the taxes investors must pay. 10 includes a brief description of these tax rules related to gains and losses.

V(I)

(8.95)