On Exhaustion of Domains - Department of Mathematics, Statistics
... imbedding f : D → G such that f (D) ⊃ K. This is equivalent to the following: There exists a sequence fk : D → G of biholomorphic imbeddings such that the sets fk (D) → G as k → ∞, i.e., for any K G there exists a number N that for k ≥ N , fk (D) ⊃ K. Below we call such a sequence {fk } an exhaust ...
... imbedding f : D → G such that f (D) ⊃ K. This is equivalent to the following: There exists a sequence fk : D → G of biholomorphic imbeddings such that the sets fk (D) → G as k → ∞, i.e., for any K G there exists a number N that for k ≥ N , fk (D) ⊃ K. Below we call such a sequence {fk } an exhaust ...
Fibonacci Numbers
... as a sum of distinct Fibonacci numbers. We could begin by looking for the largest Fibonacci number that is less than 62, in this case 55, and writing 62 as 55 + 7. Since the difference between 62 and 55 (i.e. 7) must be smaller than 55, then if we know how to decompose it into a sum of distinct Fibo ...
... as a sum of distinct Fibonacci numbers. We could begin by looking for the largest Fibonacci number that is less than 62, in this case 55, and writing 62 as 55 + 7. Since the difference between 62 and 55 (i.e. 7) must be smaller than 55, then if we know how to decompose it into a sum of distinct Fibo ...
Square Roots - HSU Users Web Pages
... “Some real numbers, such as√2 , can't be expressed as a ratio of integers, and are therefore called irrational numbers. … Every [real] number has a decimal representation. ... Real numbers can be represented by points on a line... … Often we identify the point with its coordinate and think of a numb ...
... “Some real numbers, such as√2 , can't be expressed as a ratio of integers, and are therefore called irrational numbers. … Every [real] number has a decimal representation. ... Real numbers can be represented by points on a line... … Often we identify the point with its coordinate and think of a numb ...
Document
... Write a trace table for the Max algorithm and the input a = {7, 3, 8, 6, 4}, n = 5. Iteration ...
... Write a trace table for the Max algorithm and the input a = {7, 3, 8, 6, 4}, n = 5. Iteration ...
Linear independence of continued fractions
... where L is a positive real constant which does not depend on n. It follows 1 for every sufficiently large positive integer n. This and (11) that IMnl( 1 for every sufficiently large n, where Mn is an imply that 0 I MnI ...
... where L is a positive real constant which does not depend on n. It follows 1 for every sufficiently large positive integer n. This and (11) that IMnl( 1 for every sufficiently large n, where Mn is an imply that 0 I MnI ...
Contents 1 The Natural Numbers
... be found in the first two chapters of Stoll’s book. We shall use the result stated by the theorem, however. Theorem 1. Suppose that A is a given set with a fixed element a0 selected in A. Suppose further that we have functions hn : A → A, one for each n ∈ N. Then, there exists a unique function k : ...
... be found in the first two chapters of Stoll’s book. We shall use the result stated by the theorem, however. Theorem 1. Suppose that A is a given set with a fixed element a0 selected in A. Suppose further that we have functions hn : A → A, one for each n ∈ N. Then, there exists a unique function k : ...
The Sum of Two Squares
... where ϕ(m) is the Euler totient function, which counts the positive integers less than or equal to m but which are relatively prime to m. How is this related? Let a = 1 and m = 4. This grants that the primes satisfying p ≡ 1 (mod 4) are exactly 1/2 the total primes as ϕ(4) = 2. Also, the primes for ...
... where ϕ(m) is the Euler totient function, which counts the positive integers less than or equal to m but which are relatively prime to m. How is this related? Let a = 1 and m = 4. This grants that the primes satisfying p ≡ 1 (mod 4) are exactly 1/2 the total primes as ϕ(4) = 2. Also, the primes for ...
Full text
... To obtain a partial generalization of Theorem II of Horadam and the corresponding proposition of 5ubbaRao,we first define {U^} * a fundamental sequence of order r; we illustrate its fundamental nature by showing that any linear recursive sequence of order v can be expressed in terms of {U^}. We defi ...
... To obtain a partial generalization of Theorem II of Horadam and the corresponding proposition of 5ubbaRao,we first define {U^} * a fundamental sequence of order r; we illustrate its fundamental nature by showing that any linear recursive sequence of order v can be expressed in terms of {U^}. We defi ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.