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Linear algebra is now rife with results that concern inequalities. Some of these results appear in Chapter 12. Additionally, the simplex method is the main technique for solving linear systems some of whose decision variables are required to be nonnegative. The simplex method actually solves a pair of linear programs – the one under attack and its dual. That it does so is an important – and largely unanticipated – facet of linear algebra whose implications are probed in Chapter 12. Duality is an important addition to the mathematician’s tool kit; it facilitates the proof of many theorems, as is evident in nearly every issue of the journals, Mathematics of Operations Research and Mathematical Programming.

By the summer of 1946, Dantzig was back in Washington as the lead mathematician in a group whose assignment was to mechanize the planning problems faced by the Air Force. By the spring of 1947, Dantzig had observed that a variety of Air Force planning problems could be posed as linear programs. By the summer of 1947 he had developed the simplex method. These and a string of subsequent accomplishments have cemented his stature as the preeminent figure in linear programming. Tjalling C. Koopmans Tjalling C.

If the text you wish to place in a cell could be misinterpreted, begin with an apostrophe, which will not appear. 1, select cell A7, type ‘= mean, and hit the Enter key. The leading apostrophe tells Excel that what follows is text, not a function. 1, cell A8 displays the fraction 1/3. Making that happen looks easy. But suppose you select cell A8, type 1/3 and then press the Enter key. ” Excel has decided that you wish to put a date in cell A8. And Excel will interpret everything that you subsequently enter into cell A8 as a date.

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