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(Hint: The problem is simpli ed by using permutation matrices.) 11.7. Consider the following de nition: A, B Cn n are simultaneously diagonalizable if there is a similarity matrix S Cn n such that S 1 AS, and S 1 BS are both diagonal matrices. Show that if A, B Cn n are simultaneously diagonalizable, then they commute (i.e., AB = BA). 11.8. Prove the following theorem. Let A, B Cn n be diagonalizable. Therefore, A and B commute iff they are simultaneously diagonalizable. 11.9. Matrix A Cn n is a square root of B Cn n if A2 = B. Show that every diagonalizable matrix in Cn n has a square root. 11.10. Prove the following (Bauer Fike) theorem (which says something about how perturbations of matrices affect their eigenvalues). If is an eigenvalue of A + E Rn n , and T 1 AT = D = diag( 0 , 1 , . . . , n 1 ), then

It is clear, the, that the intensity variations in the image plane show the presence of aberrations, provided that a modi cation is done over the Fourier spectrum of pupil function in the image plane, ltering with the knife edge, in order to obtain 0 Wimag x; y 6 0. The contrast of the image intensity variations over the uniform background is g 4p 0 W x; y l imag 8:48

measured supply of fresh air. Samples of wellmixed air are drawn off for analysis and the differences in O2 and CO2 concentrations in the ingoing and outgoing air are calculated. Depending on the protocol and equipment used, the data can be manipulated to yield BMR, thermogenesis, exercise and physical activity and macronutrient oxidation. Whole-body chambers are comfortably furnished, and subjects are not impeded by close tting apparatus; most include a bed, chair, desk, television and exercise equipment (cycle ergometer, step block, treadmill), and some also have an inbuilt toilet and showers. Food and drink and other necessary items, and urine and faecal samples, are passed in and out of the chamber through separate air-lock hatches. Some chambers also including a hermetically-sealed hatch for collecting blood samples from the subject s arm. Measurement periods are typically 24 or 36 hours, but can be up to 14 days. The technique is most valuable for assessing within-subject responses, or the effects of manipulations of diet or activity, or changes in physiological state such as pregnancy, weight loss or gain. Wholebody calorimetry is not appropriate for those who need close care (e.g. infants, young children, or frail, ill or other vulnerable people). Whole-body calorimetry enables very accurate and precise measurement to be made of 24-hr energy expenditure and its components, with a within-subject coef cient of variation of TEE measurement of 1%. It cannot mimic the

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Figure 2.15. Activation of agarose for af nity chromatography. Important methods for activation of agarose are; (i) CNBr, (ii) epoxy, (iii) carbonyldiimidazole, (iv) dichlorotriazine, (v) trecyl.

Photoshop provides three paint tools: pencil, paintbrush, and airbrush. You also get six edit tools: smudge, blur, sharpen, burn, dodge, and sponge. Figure 5-1 shows all the tools along with the keyboard shortcuts for selecting them.

Click Player Settings on the Preference tab, and you will see the windows displayed in Figure 3-21.

Identification of the damping time constant for synchrotron oscillations is almost as easy. Suppose the deviation from the synchronous energy of a particle is A E . In traversing an infinitesimal element of a turn, A E will change according to

9.4 TUNABLE OUTPUT OF HIGH-POWER DIODE LASERS USING A GRATING IN THE EXTERNAL CAVITY In this section we demonstrate how a laser diode array can be forced to oscillate in a single spatial mode with high temporal coherence. The laser diode array is driven with a drive current far above threshold, and the wavelength is widely tunable. Single longitudinal mode operation of a GaAlAs laser diode array operated with a drive current below threshold has been demonstrated by coupling it to an external cavity formed by a diffraction grating and a photorefractive phase conjugate mirror [71]. This system has proven to be widely tunable with respect to the frequency due to the presence of the diffractive grating. If operated far above threshold, this system is, in general, nonstable due to nonlinear processes in the photorefractive phase conjugate mirror (PCM), and self-induced frequency scanning can take place. In

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By default, CHAMFER trims the original lines that it chamfers. If you want to keep the full original lines when you add the chamfer line, choose the Trim option and choose No Trim. Use the Multiple option to continue the prompts and chamfer several corners in one command. The new Undo option lets you undo your last chamfer and try again.

ALTER TABLE emp ADD (home_zip VARCHAR2(10) NOT NULL);

OC : optical carry; STS : synchronous transfer signal;