Abel's Version of Abel's Theorem
... on the curve. Euler—we are still in the 18th century here— showed that his formula applied in this more general case as well, but the sum of the integrals Z Z Z ω(x1 , y1 ) dx1 + ω(x2 , y2 ) dx2 + · · · + ω(xk , yk ) dxk while it is no longer constant, can be expressed in terms of elementary functio ...
... on the curve. Euler—we are still in the 18th century here— showed that his formula applied in this more general case as well, but the sum of the integrals Z Z Z ω(x1 , y1 ) dx1 + ω(x2 , y2 ) dx2 + · · · + ω(xk , yk ) dxk while it is no longer constant, can be expressed in terms of elementary functio ...
Lecture_Notes (reformatted)
... Note that the last 3 examples have the same answers (not obvious). Note the second and third examples have the same answer (obvious). Counting is an important tool in discrete math as we will see later. ...
... Note that the last 3 examples have the same answers (not obvious). Note the second and third examples have the same answer (obvious). Counting is an important tool in discrete math as we will see later. ...
Fibonacci sequence
... first month the rabbits produced no offspring, but each month thereafter produced one new pair of rabbits. If each new pair thus produced reproduces in the same manner, how many pairs of rabbits will there be at the end of one year? ...
... first month the rabbits produced no offspring, but each month thereafter produced one new pair of rabbits. If each new pair thus produced reproduces in the same manner, how many pairs of rabbits will there be at the end of one year? ...
NORMAL FAMILIES, ORDERS OF ZEROS, AND OMITTED VALUES
... normal function, then there exist sequences {zn } and {̺n } such that zn ∈ D , |zn | → 1 , ̺n > 0 , ̺n /(1 − |zn |) → 0 , and the sequence {gn (t) = f (zn + ̺n t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g . Theorem Z [11, Lemma, p. 814]. Let F be a fam ...
... normal function, then there exist sequences {zn } and {̺n } such that zn ∈ D , |zn | → 1 , ̺n > 0 , ̺n /(1 − |zn |) → 0 , and the sequence {gn (t) = f (zn + ̺n t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g . Theorem Z [11, Lemma, p. 814]. Let F be a fam ...
3.5 Arithmetic Sequences as Linear Functions
... 1) Use inductive reasoning in continuing number patterns 2) Write rules for arithmetic sequences 3) Relate arithmetic sequences to linear functions ...
... 1) Use inductive reasoning in continuing number patterns 2) Write rules for arithmetic sequences 3) Relate arithmetic sequences to linear functions ...
On certain positive integer sequences (**)
... by the maximal number of repetitions. Many other open probems on practical numbers and related questions have been raised by Erdös in [6]. ...
... by the maximal number of repetitions. Many other open probems on practical numbers and related questions have been raised by Erdös in [6]. ...
The Chinese Restaurant Approach to Integer
... three or fewer members of the set {30 , 31 , 32 , 33 , 34 } if we are allowed to use the same power more than once. For example, 5 can be represented, but 8 cannot. (1991 State Math Contest of North Carolina.) 12. How many integers can be expressed as a sum of two or more different members of the se ...
... three or fewer members of the set {30 , 31 , 32 , 33 , 34 } if we are allowed to use the same power more than once. For example, 5 can be represented, but 8 cannot. (1991 State Math Contest of North Carolina.) 12. How many integers can be expressed as a sum of two or more different members of the se ...
Conflicts in the Learning of Real Numbers and Limits
... To avoid such conflicts, the stress should be placed on the actuality of the limit process. We live in a world of limited accuracy, and to any desired degree of accuracy, if lim sn=s, then from some term onwards the terms are indistinguishable from the limit. In a very practical sense we soon reach ...
... To avoid such conflicts, the stress should be placed on the actuality of the limit process. We live in a world of limited accuracy, and to any desired degree of accuracy, if lim sn=s, then from some term onwards the terms are indistinguishable from the limit. In a very practical sense we soon reach ...
Sample Individual Questions
... (b) Let t = T N Y W R. Let k = 3t. The convex area bounded by x = 1, x = k, the x-axis and the line y = mx + 4 is 17. Determine the value of m. (c) Let t = T N Y W R. Let k = 15t. Let n represent the number of ways that k dollars can be changed into dimes and quarters, with at least one of each coin ...
... (b) Let t = T N Y W R. Let k = 3t. The convex area bounded by x = 1, x = k, the x-axis and the line y = mx + 4 is 17. Determine the value of m. (c) Let t = T N Y W R. Let k = 15t. Let n represent the number of ways that k dollars can be changed into dimes and quarters, with at least one of each coin ...
Linear and Quadratic Regression
... arithmetic Sequence: A sequence of numbers where the common difference occurs at level D1 ...
... arithmetic Sequence: A sequence of numbers where the common difference occurs at level D1 ...
1.7 Simplifying Expressions
... Use D-Prop to express a sum of 2 whole numbers with common factors as a sum of two whole numbers with no common factors. Apply properties of expressions to generate equivalent expressions. Identify when two expressions are equivalent. ...
... Use D-Prop to express a sum of 2 whole numbers with common factors as a sum of two whole numbers with no common factors. Apply properties of expressions to generate equivalent expressions. Identify when two expressions are equivalent. ...