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The spinal roots in the lumbar region, after exiting from the spinal cord, course downward in the spinal canal and are gradually displaced laterally until they angulate and exit at the intervertebral foramina Prior to entering the short foraminal canal, the spinal root lies in a shallow furrow along the inner surface of the pedicle called the lateral recess This is a common site of root entrapment by disc fragments and bony overgrowth The parts of the back that possess the greatest freedom of movement and hence are most frequently subject to injury are the lumbar, lumbosacral, and cervical In addition to bending, twisting, and other voluntary movements, many actions of the spine are re exive in nature and are the basis of posture.

To morph these equations, we can use the same sort of algebra that gets two-variable linear equations into SI form. Here are the processes, step-by-step. You have learned enough algebra so you can follow along without detailed explanations. 4x + 2y 3z = 5 2y 3z = 4x + 5 3z = 4x 2y + 5 z = (4/3)x (2/3)y + 5/3 z = ( 4/3)x + (2/3)y 5/3 2x 5y = z 1 2x 5y + 1 = z z = 2x 5y + 1 3x = 6y + 7z 3x + 6y = 7z (3/7)x + (6/7)y = z z = (3/7)x + (6/7)y These three equations represent relations that map pairs of variables (in this case x and y) into a single variable (in this case z). If we call the relations f, g, and h, then we can write: f (x, y) = ( 4/3)x + (2/3)y 5/3 g (x, y) = 2x 5y + 1 h (x, y) = (3/7)x + (6/7)y

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WMLScript ignores any spaces, tabs, and carriage returns that appear in your script, except those that are included in string literals. The WMLScript compiler recognizes the three following strings as different:

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Aging Changes in Spinal Structures Changes in the intervertebral discs and ligaments as a consequence of aging and perhaps a succession of minor traumas begin to occur as early as the rst part of the third decade Deposition of collagen and elastin and alterations of glycosaminoglycans combine to decrease the water content of the nucleus pulposus; concomitantly, the cartilagenous end plate becomes less vascular (Hassler) The dehydrated disc thins out and becomes more fragile Similar changes occur in the annulus of the disc, which frays to an increasing degree with the passage of time, permitting the nucleus pulposus to bulge and, sometimes with injury, to extrude This process can be observed by magnetic resonance imaging (MRI), which shows a gradual re-.

20 is devotedto one-stepmethodsfor solving initial-value ODEs. As the name suggests, one-step methodscomputea futurepredictiony;a1, basedonly on information at a single point r; and no other previous information.This is in contrastto multistepapproachesthat useinlbrmation from several previouspointsas the basisfor extrapolating to a newvalue. presented Chap.20 belongto With all but a rninorexception, one-step the methods in wh:rt are called, Runge-Kuttatechni4lle.s. Although the chapterrnight havebeenorganized to aroundthis theoretical notion, we haveoptedfor a more graphical.intuitive approach inThus, we begin the chapterwrth Euler's ntethod,which has a very troduce the rnethods. graphicalinterpretation. addition,because have alreadyintroduced we straightforward In Euler's methodin Chap. l, our emphasis hereis on quantifyingits truncationeror and describineits stabilitv.

Matrix multiplication does not commute: AB = BA. Matrix division is unde ned. Matrix multiplication does have the following useful formulas: (AB) C = (A + B) C = (ABC) = A (BC) AC + BC C B A . Associative property Distributive property (B.4) (B.5) (B.6)

(a) Plot the clata. (b) Fit a straightline to the data with linearregression. Add this line to the plot. (c) If the minimum safeaverage distance between bikes and passingcars is considered be 2 m, determine cor. to the points for the datawas takenat l5 time 13.22 Concentration minimum lanewidth. responding polyrnerization reaction: 11.25 In water-resourccs engineering,the sizing of reserxA*,tB - A,B:. voirs dependson accurateestimatesof water flow in the We assumethe reaction occurs via a complex mechanism river that is being impounded.For some rivers, long-term consistingof many steps.Severalmodelshavebeenhypoth- historicalrecordso1'such flow data are difficult to obtain. In esized,and the sum of the squares the residualshad been contrast,meteorolo-vical of data on precipitationis often avail-

FIGURE 11-3

To call an external file, you must specify the following: M L The URL of the WMLScript resource (in this example http://www.host.com/app/script) The resource s name (in this example UtilityScript)

The di erential equations for the error variables are found by subtracting eqns. (4.94-4.95) from eqns. (4.91-4.92) p v = = v a(t) k + ba (t) + a (t)

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