Diagonalization
... Proof: (by contradiction) Let R denote the set of all real numbers, and suppose that R is countable. Then by definition it is either finite or countably infinite. Clearly, it is not finite, therefore it must be countably infinite. By definition, since it is countably infinite it has the same cardina ...
... Proof: (by contradiction) Let R denote the set of all real numbers, and suppose that R is countable. Then by definition it is either finite or countably infinite. Clearly, it is not finite, therefore it must be countably infinite. By definition, since it is countably infinite it has the same cardina ...
Chapter One Functions and Their Graphs
... Write an exponential equation in logarithmic form, and a logarithmic equation in exponential form. Graph logarithmic functions and transformations of logarithmic functions. ...
... Write an exponential equation in logarithmic form, and a logarithmic equation in exponential form. Graph logarithmic functions and transformations of logarithmic functions. ...
BASIC NOTIONS AND RESULTS IN TOPOLOGY 1. Metric spaces A
... Exercise 2. Show that C(X), ρ) is a complete metric space and that a sequence converges in this space if and only if it converges uniformly on X. A bounded set in (C(X), ρ) is sometimes called a uniformly bounded family of functions. A subset F ⊂ C(X) is called equicontinuous at x ∈ X if for every ε ...
... Exercise 2. Show that C(X), ρ) is a complete metric space and that a sequence converges in this space if and only if it converges uniformly on X. A bounded set in (C(X), ρ) is sometimes called a uniformly bounded family of functions. A subset F ⊂ C(X) is called equicontinuous at x ∈ X if for every ε ...
4. Hailstone Sequence (Wondrous Numbers)
... answer. Notice that for any initial value of ‘n’, the sequence always ends with “…., 8, 4, 2, 1”. 5. After the iterations are completed, display the number of iterations that the function took to “converge”. 6. Implement the program for 10 different seed values, note the iteration steps required and ...
... answer. Notice that for any initial value of ‘n’, the sequence always ends with “…., 8, 4, 2, 1”. 5. After the iterations are completed, display the number of iterations that the function took to “converge”. 6. Implement the program for 10 different seed values, note the iteration steps required and ...
1.1 Patterns and Inductive Reasoning
... The product of two positive numbers is greater than either number. ...
... The product of two positive numbers is greater than either number. ...
Number sequence
... The focus in this activity is the idea of proof and reasoning and the use of algebra to argue generalities. This question can be asked at grade 8 level too with no adjustment necessary. ...
... The focus in this activity is the idea of proof and reasoning and the use of algebra to argue generalities. This question can be asked at grade 8 level too with no adjustment necessary. ...
82 Eighty-Two LXXXII
... The number 82 is the forty-second even number and the fifty-ninth composite number. As a product of primes: 82 = 2 41. The number 82 has four divisors: 1, 2, 41, 82. The number 82 is the sixty-third deficient number: s(82) = 1 + 2 + 41 = 44 < 82. As a sum of four or fewer squares: 82 = 12 + 92 = 32 ...
... The number 82 is the forty-second even number and the fifty-ninth composite number. As a product of primes: 82 = 2 41. The number 82 has four divisors: 1, 2, 41, 82. The number 82 is the sixty-third deficient number: s(82) = 1 + 2 + 41 = 44 < 82. As a sum of four or fewer squares: 82 = 12 + 92 = 32 ...
Calculus for the Natural Sciences
... Fibonacci numbers. The idea is that you start with 1 (pair of) rabbit(s) the zeroth month. The first month you still have 1 pair. But then in the second month you have 1+1 = 2 pairs, the third you have 1 + 2 = 3 pairs, the fourth, 2 + 3 = 5 pairs, etc... The pattern is that if you have an pairs in t ...
... Fibonacci numbers. The idea is that you start with 1 (pair of) rabbit(s) the zeroth month. The first month you still have 1 pair. But then in the second month you have 1+1 = 2 pairs, the third you have 1 + 2 = 3 pairs, the fourth, 2 + 3 = 5 pairs, etc... The pattern is that if you have an pairs in t ...
9.3. Itô`s formula. First I stated the theorem. Then I did a simple
... The key point is Lévy’s theorem on quadratic variation. 9.3.1. statement of the theorem. Theorem 9.19 (Itô). Suppose that f (x) is a C 2 (twice continuously differentiable) function and Wt is standard Brownian motion. Then ! t ...
... The key point is Lévy’s theorem on quadratic variation. 9.3.1. statement of the theorem. Theorem 9.19 (Itô). Suppose that f (x) is a C 2 (twice continuously differentiable) function and Wt is standard Brownian motion. Then ! t ...
Geometric Sequence WS - Algebra with Mrs. Jett!
... Write an explicit formula for the following situations and use it to answer the questions. 10) The end of a spring is pulled as far as it will go and then is released. On the first bounce back it extends 54 cm. On its second bounce back it extends 18 cm. On its third bounce back it extends 6 cm. Ho ...
... Write an explicit formula for the following situations and use it to answer the questions. 10) The end of a spring is pulled as far as it will go and then is released. On the first bounce back it extends 54 cm. On its second bounce back it extends 18 cm. On its third bounce back it extends 6 cm. Ho ...
8.3 The number e
... decimal representation does not terminate or follow a repeating pattern. • The previous sequence of e can also be represented: • As n gets larger (n→∞), (1+1/n)n gets closer and closer to ...
... decimal representation does not terminate or follow a repeating pattern. • The previous sequence of e can also be represented: • As n gets larger (n→∞), (1+1/n)n gets closer and closer to ...
