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Pyridoxine, pyridoxal, and pyridoxamine are the vitamins collectively known as vitamin B6 Vitamin B6 is important in the metabolism of proteins and amino acids, helps regulate blood glucose levels, is necessary for the manufacture of hemoglobin, is important in normalizing nervous system function, and plays a key role in enabling the brain to transmit nerve impulses Vitamin B6 promotes healthy skin, teeth, gums, and blood vessels It also helps the body in the utilization of carbohydrates and essential fatty acids for energy, promotes the production of antibodies, and provides support for the overall healthy function of the immune system Vitamin B6 is one of the most critical of all the vitamins; it participates as a coenzyme in more than sixty enzymatic reactions Vitamin B12 (cyanocobalamin) is a very important vitamin that contains the trace mineral cobalt.

// Add match to matches table and log file. private void addMatch(String url) {

587

or, written out in expanded form, eq. (587) becomes y nT b0 x nT b1 x nT T b2 x nT 2T bp x nT pT a1 y nT T a2 y nT 2T aq y nT qT 588

// Add URL to matches table. DefaultTableModel model = (DefaultTableModel) table.getModel(); model.addRow(new Object[]{url}); // Add URL to matches log file. try { logFileWriter.println(url); } catch (Exception e) { showError("Unable to log match."); } }

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Equations (587) and (588) are called di erence equations because of the presence of the past history forms having the di erence notation nT kT . Let us now operate on eqs. (587) and (588) in such a way as to express things in terms of the z-transform. This can be done as follows. First note that, in the above equations, y nT and x nT denote sampled values of y t and x t for any particular value of n we might be interested in. However, to bring the ztransform into the picture we must summate the values of y nT and x nT over the entire range of n that there is, for n 0 to n ! 1 (in accordance with the basic de nition of eq. (573) in section 13.3). Therefore (so that we can apply eq. (573)) let us multiply both sides of eq. (588) by z n and then summate from n 0 to n ! 1; doing this, eq. (588) becomes X X X X x nT z n b1 x nT T z n bp x nT pT z n y nT z n b0 X X X a1 y nT T z n a2 y nT 2T z n aq y nT qT z n where all summations are understood to be from n 0 to n 1. Now note that (with same summation from n 0 to n 1) by eq. (573) X y nT z n Y z and by item (4), Table 2 X y nT kT z n Y z z k

and the same, of course, for x nT in place of y nT . Now apply the last two relations to the preceding equation; doing this gives the relationship Y z b0 X z b1 X z z 1 bp X z z p a1 Y z z 1 a2 Y z z 2 aq Y z z q Now, on the right-hand side of the last equation, factor out X z and Y z and then solve for the ratio of OUTPUT TO INPUT, that is, Y z =X z . Doing this, and using the notation of eq. (585), you should nd that b0 b1 z 1 b2 z 2 bp z p Y z H z X z 1 a1 z 1 a2 z 2 aq z q 589

This method first adds the URL to the matches table by retrieving the table s data model and calling its addRow( ) method. Notice that the addRow( ) method takes an Object array as input. In order to satisfy that requirement, the url String object is wrapped in an Object array. After adding the URL to the matches table, the URL is written to the log file with a call to logFileWriter.println( ). This call is wrapped in a try-catch block; and if an exception is thrown, the showError( ) method is called to alert the user that an error has occurred while trying to write to the log file.

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