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Unformatted text preview: The correct answer is (b) . square Solution of problem 1.5: The intersection of B with any one of the four dimensional subspaces is given by one constraint so it depends on three parameters. Since we have a two parameter family of four dimensional subspaces, the points in B depend on a tota; pf 3 + 2 = 5 parameters. This shows that B is 5 dimensional. This argument can be made also in terms of formulas. We can solve the constraint for x 3 in terms of the other variables. Then a general point in the slice of B by the subspace satisfying x 1 = a , x 5 = b is given by ( a, x 2 , x 2 2 2 x 3 4 + 5 x 3 6 + ( a + 7 b ) 2 , x 4 , b, x 6 ) . In other words, the general point of B is described by the five param eters a , b , x 2 , x 4 , x 6 , and so B is five dimensional. Therefore the correct answer is (f) . square Solution of problem 1.6: Cleraly D can be distorted into O by rounding off the corners on the left side of D . Thus, the correct answer must be (b) . We can also easily rule out the other answers directly. Indeed, if we delete a point from the bottom part of one of the legs of A , we will get two connected pieces. But removing any point from either D or O we get a single piece. Therefore A can not be distorted into either D or O . Similarly, removing a point from M results in two pieces, and so M can not be distorted into either D or O . Finally, if we delete the top apex of A we will get one piece, whereas when we remove any point from M we will get two pieces. So A can not be distorted into M . The correct answer is (b) . square 14 Solution of problem 1.7: If we cut the two hole pretzel along the indi cated circle we will obtain two connected pieces each with one hole and one boundary. Thus the correct choice is (d) . square Solution of problem 1.8: (1) A connected planar graph has Euler char acteristic 2. So if the graph has 54 vertices, 177 edges, and F faces, then it follows that 54 177 + F = 2, or that F = 125. Therefore (1) is True. (2) A planar graph with 7 vertices, 4 edges, and 7 faces has Euler characteristic 7 4 + 7 = 10. By the Euler characteristic theorem such a graph must have 10 1 = 9 connected components. Therefore (2) is False. (3) If a planar graph has 2 connected pieces then it has Euler charac teristic 3. If the graph also has 12 edges, 7 vertices, and F faces, then 3 = 7 12 + F . In other words, we must have that F = 8. Therefore (3) is True. The correct answer is (b) . square Solution of problem 1.9: Since the pond can sustain 300 fish and the ini tial population is 1200 fish, it follows that the initial population density is P 1 = 1200 300 = 4 . Similarly, the population density after two years is P 2 = 600 300 = 2 . The Verhulst model gives the relation P 2 P 1 = 1 + c (1 P 1 ) , 15 or equivalently 2 4 = 1 3 c....
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 Spring '08
 schneps
 Natural Numbers, Rational number, Irrational number, Verhulst

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