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For an arbitrary coherent state |x 0 , x 0 [0, 1], it is interesting to evaluate the lower symbol of Ax p This gives x 0 |Ax p |x 0 =

Prosser, F.P., and D.E. Winkel. The Art ofDigita1 Design. New Jersey: Prentice Hall. KersHaw, John D. Digital Electronics. Boston: PWS-Kent. Stout, D.F., and M. Kaufman. Handbook o Microcircuit Design and f Applications. New York: McCraw-Hill. Roth, Charles H. Jr. Fundurnentals offlogic Design. Eagan, MN: West Publishing Company. Taub, H., and D. Schilling. Digital Integrated Circuits. New York: McGraw-Hill.

u x0 u 1 ,

(1430)



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Industry has less room for improvement than the residential sector, which represents 5 percent of United States energy usage In the industrialized world, industrial energy ef ciency has been pursued vigorously since the oil crises in the 1970s and is re ected by the lower levels of energy intensity in most countries that are members of the Organization of Economic Cooperation and Development (OECD) As GDP in these countries grows their use of energy becomes more ef cient In fact, since 1970 the costs for producing every dollar of national income have shrunk by 49 percent in the United States84 Yet there are still technologies available to save energy in the industrial sector, and the government could support industry in its energy savings efforts by undertaking demonstration projects and energy audits Although energy intensity in emerging market countries is slowly decreasing, those countries still have a long way to go before achieving the standards of the industrialized world (see Figure 38) Considering the stakes for the entire world if energy demand goes out of control, wealthy countries have a real interest in helping their less developed neighbors to increase their energy ef ciency Technology and knowledge sharing, a reduction of tariff barriers for energy-effective technologies, and high domestic taxes for inef cient technologies coming out of the developing world can produce the right incentives to accelerate a less energy consuming pattern in the developing world You might nd it surprising





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the two possible values being precisely the eigenvalues of the above matrix Note the average values of the position operator: x 0 |Ax |x 0 = 1/4 if 0 u x 0 u 1/2 and 3/4 if 1/2 u x 0 u 1 Clearly, like in the N = 2 case in the previous section, all operators A f commute, since they are linear combinations of the identity matrix and the Pauli matrix 1 The procedure is easily generalized to higher dimensions Let us add to the previous set { 1 , 2 } other elements of the Haar basis, say, up to the scale J: { 1 (x), 2 (x), 3 (x) = , s (x) = 2

7414 (Schrnitttrigger inverter)

2 2 (2x), 4 (x) =

2 2 (2x 1) , (1431)

Thousand Btu per 2000 Dollar of GDP 25 20 15 10 5 Other Non-OECD 0 1980 1995 2005 2015 2030 OECD Non-OECD Europe and Eurasia History Projections

2 (2x k), N (x) = 2 J/2 2 (2x 2 J + 1)} ,

2 mr1 (1 ) 1

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where, at given j = 1, 2, , J, the integer k assumes its values in the range 0 u k u 2 j 1 The total number of elements of this orthonormal system is N = 2 J+1 N 2 J+1 The normalization function is equal to N (x) = , and this n=1 = | n (x)| = 2 clearly diverges at the limit J Then, it is remarkable if not expected that spectral values as well as average values of the position operator are given by x 0 |Ax |x 0 = (2k + 1)/2 J+1 for k/2 J u x 0 u (k + 1)/2 J where 0 u k u 2 J 1 Hence, our quantization scheme yields a dyadic discretization of the localization in the unit interval

Now we choose another orthonormal system, namely, the rst two elements of the trigonometric Fourier basis, (1432) 1 (x) = 1(x), 2 (x) = 2 sin 2 x Then we have

Energy Intensity by Region, 1980 2030

N (x) =

1.0E 4

(1433)

(1434)

that countries still subsidize energy (especially fossil fuel), but they do McKinsey estimates that average fuel ef ciency in countries with fuel subsidies are half of what they are in countries without subsidies Cut them out, and the world theoretically can save three million barrels of oil a day85

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