By Marston Morse, Stewart S. Cairns
This can be an advent to a mathematical thought that has revolutionized the equipment and goals of world research and differential topology and has unfolded new chances in mathematical physics and engineering. This publication discusses Bott's paintings on homotopy equivalence, Smale at the Poincare challenge, Milnor and Thom on cobordism. It additionally presents a finite-dimensional creation to Morse's serious pont concept and develops Eilenberg's singular homology concept on differentiable manifolds with none international triangulation of the manifolds.
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Additional info for Critical point theory in global analysis and differential topology, Volume 33: An introduction
1 1 2 1 3 5 6 4 – 34 53 – 2 3 68 – 18 T o subtract mixed numbers that have common denominators, follow these two simple steps: 1. Subtract the numerators. If the numerator of the minuend is smaller than that of the “subtrahend” (the number you are subtracting), subtract 1 from the whole number of the minuend and carry the numerator to the fraction. For example, in 1 4 1 4 5 the problem 7 5 – 6 5, 5 is smaller than 5. Subtract 1 from 7, change this 1 to 5, and 1 6 add this to the fraction. So the minuend 7 5 becomes 6 5, and the problem 6 4 6 4 2 becomes 6 5 – 6 5.
Reduce the fraction if necessary: 5 3 cannot be reduced. So 6 - 3 = 5 3. CIRCLES INSIDE CIRCLES 51 PartIII 11/28/00 9:22 AM Page 52 18 Subtraction Chip Trading Practice subtracting mixed numbers that have common denominators. M AT E R I A L S poker chips (yellow and blue, or any other two colors) or nickels and pennies pencil paper Procedure 1. To subtract mixed numbers that have common denominators, first change the “minuend” (the number you are subtracting from) to chips or coins. Use yellow chips or nickels to represent the whole number and blue chips or pennies to represent the numerator of the fraction.
ZERO 37 PartII 12/6/00 9:25 AM Page 38 PartIII 11/28/00 9:22 AM Page 39 III ADDING AND SUBTRACTING FRACTIONS N ow that you know fraction basics, it is time to learn how to add and subtract fractions. You’ll learn how to add and subtract fractions that have common denominators and fractions that have different denominators. You’ll add mixed numbers to mixed numbers, mixed numbers to proper fractions, and fractions to whole numbers. You’ll subtract all kinds of fractions, too. Along the way, you will make paper pizzas, trade poker chips, make pattern pictures, play tic-tac-toe, roll fraction dice, create fractions with dominoes, and play lots of different games.