INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 18 Contents
... 1.2. Last time. Goal: Rational maps of nonsingular curves to projective varieties can be extended to morphisms. We discussed reasons why you can’t extend P1 99K A1 , A2 99K P1 . Lemma. Let Y be a prevariety, and suppose P and Q are two points contained in a single affine open U , and OY,Q ⊂ OY,P (as ...
... 1.2. Last time. Goal: Rational maps of nonsingular curves to projective varieties can be extended to morphisms. We discussed reasons why you can’t extend P1 99K A1 , A2 99K P1 . Lemma. Let Y be a prevariety, and suppose P and Q are two points contained in a single affine open U , and OY,Q ⊂ OY,P (as ...
The only even prime is 2.
... Let p1 < p2 < p3 < · · · denote the prime numbers listed in increasing order. Given a finite subset S = {x1 , x2 , . . . , xn } ⊂ N, relabel the xi if necessary so that x1 < x2 · · · < xn . Then define f (S) = px1 1 px2 2 · · · pxnn . By uniqueness of the decomposition of a natural number into a pro ...
... Let p1 < p2 < p3 < · · · denote the prime numbers listed in increasing order. Given a finite subset S = {x1 , x2 , . . . , xn } ⊂ N, relabel the xi if necessary so that x1 < x2 · · · < xn . Then define f (S) = px1 1 px2 2 · · · pxnn . By uniqueness of the decomposition of a natural number into a pro ...
Operators and Expressions
... character at a time from the leftmost character of each string. The ASCII values of the characters from the two strings are compared. ...
... character at a time from the leftmost character of each string. The ASCII values of the characters from the two strings are compared. ...
EXERCISES: CHAPTER 12 Section 12.1 (Partitions of a set) 1
... (i) x3 y 2 z 4 in (x + y + z)9 (ii) xy 3 zt2 u in (x + y + z + t + u)8 . 3. Calculate p(8), the total number of partitions of 8, and verify that the number which have distinct parts is equal to the number whose parts are all odd. Can you explain this equality (which holds for any n, not just n = 8) ...
... (i) x3 y 2 z 4 in (x + y + z)9 (ii) xy 3 zt2 u in (x + y + z + t + u)8 . 3. Calculate p(8), the total number of partitions of 8, and verify that the number which have distinct parts is equal to the number whose parts are all odd. Can you explain this equality (which holds for any n, not just n = 8) ...