Fibonacci numbers, alternating parity sequences and
... (b) If B has only one inner entry, N = 1, and so FB is a triangle. (c) Suppose B has exactly two inner entries, b1 1 = b2 2 = 1, 1 < 2 . We have two cases: (i) 2 − 1 is even; then the two singletons, (1 ), (2 ), are the only a.p. subsequences of . So FB has four vertices. (ii) 2 − 1 is ...
... (b) If B has only one inner entry, N = 1, and so FB is a triangle. (c) Suppose B has exactly two inner entries, b1 1 = b2 2 = 1, 1 < 2 . We have two cases: (i) 2 − 1 is even; then the two singletons, (1 ), (2 ), are the only a.p. subsequences of . So FB has four vertices. (ii) 2 − 1 is ...
Mathematical Induction - Cambridge Computer Lab
... To be valid, this argument cannot go on forever: it requires an ordering; and it must terminate (see lecture 7); this is called a well-founded partial ordering. ...
... To be valid, this argument cannot go on forever: it requires an ordering; and it must terminate (see lecture 7); this is called a well-founded partial ordering. ...
ppt - Carnegie Mellon School of Computer Science
... Trying to establish that: 8n¸1 Sn Assume “Induction Hypothesis”: Sk (for any particular k¸ 1) 1+3+5+…+ (2k-1) = k2 Add (2k+1) to both sides. 1+3+5+…+ (2k-1)+(2k+1) = k2 +(2k+1) Sum of first k+1 odd numbers = (k+1)2 CONCLUSE: Sk+1 ...
... Trying to establish that: 8n¸1 Sn Assume “Induction Hypothesis”: Sk (for any particular k¸ 1) 1+3+5+…+ (2k-1) = k2 Add (2k+1) to both sides. 1+3+5+…+ (2k-1)+(2k+1) = k2 +(2k+1) Sum of first k+1 odd numbers = (k+1)2 CONCLUSE: Sk+1 ...
A Computationally-Discovered Simplification of the Ontological
... syntax. But for reasons of space, we shall not discuss all the details concerning these derivations in what follows.4 Our goal in this subsection is simply to show how these logical axioms and theorems are to be represented in prover9 syntax. Unfortunately, prover9, like other first-order automated ...
... syntax. But for reasons of space, we shall not discuss all the details concerning these derivations in what follows.4 Our goal in this subsection is simply to show how these logical axioms and theorems are to be represented in prover9 syntax. Unfortunately, prover9, like other first-order automated ...
The Coinductive Formulation of Common Knowledge
... for you to determine the colour of your dot, don’t wear a hat. Then you will all come back to this room and you will be able to see the other students’ hat choices. If some students will have made the correct hat choice, they will graduate. Those who make a mistaken choice will be expelled from the ...
... for you to determine the colour of your dot, don’t wear a hat. Then you will all come back to this room and you will be able to see the other students’ hat choices. If some students will have made the correct hat choice, they will graduate. Those who make a mistaken choice will be expelled from the ...
A Taste of Categorical Logic — Tutorial Notes
... Figure 2: Typing rules for logical connectives. Note that these are not introduction and elimination rules for connectives. These merely state that some things are propositions, i.e., of type Prop Notice that we did not include an equality predicate. This is just for brevity. In higher-order logic e ...
... Figure 2: Typing rules for logical connectives. Note that these are not introduction and elimination rules for connectives. These merely state that some things are propositions, i.e., of type Prop Notice that we did not include an equality predicate. This is just for brevity. In higher-order logic e ...
Partitions of numbers (concluded):
... Polya theory (see Ch. 14 of Brualdi, or handout) (Re-!)-definition: A permutation is a one-to-one and onto function from a finite set S to itself. Example: The six permutations of the set {1,2,3} are the functions x |1|2|3 x |1|2|3 x |1|2|3 ----------------- ----------------- ----------------g(x) | ...
... Polya theory (see Ch. 14 of Brualdi, or handout) (Re-!)-definition: A permutation is a one-to-one and onto function from a finite set S to itself. Example: The six permutations of the set {1,2,3} are the functions x |1|2|3 x |1|2|3 x |1|2|3 ----------------- ----------------- ----------------g(x) | ...
The application of a new mean value theorem to the fractional parts
... We note that the preceding argument is really intended as a demonstration that non-trivial estimates for Us can be obtained. The methods of Sections 4 and 5 are most effective when k is small, and under such circumstances the estimates of [11] supersede those of Theorem 2.1. Nonetheless, the argumen ...
... We note that the preceding argument is really intended as a demonstration that non-trivial estimates for Us can be obtained. The methods of Sections 4 and 5 are most effective when k is small, and under such circumstances the estimates of [11] supersede those of Theorem 2.1. Nonetheless, the argumen ...
Document
... A combinatory proof is a proof that uses counting arguments to prove a theorem, rather than some other method such as algebraic techniques. ...
... A combinatory proof is a proof that uses counting arguments to prove a theorem, rather than some other method such as algebraic techniques. ...
PDF
... Lemma 1. Let S be a subset of C that contains a nonzero complex number and α ∈ C. Then α is constructible from S if and only if there exists a finite sequence α1 , . . . , αn ∈ C such that α1 is immediately constructible from S, α2 is immediately constructible from S∪{α1 }, . . . , and α is immediat ...
... Lemma 1. Let S be a subset of C that contains a nonzero complex number and α ∈ C. Then α is constructible from S if and only if there exists a finite sequence α1 , . . . , αn ∈ C such that α1 is immediately constructible from S, α2 is immediately constructible from S∪{α1 }, . . . , and α is immediat ...
patterns in continued fraction expansions
... n can be used to find solutions to Pell’s equation, x2 ny 2 1 . For more information on Pell’s Equation and continued fractions, refer to [2]. Furthermore, continued fractions can be put to use in the factorization of large integers [5, p.246]. We can also make use of continued fractions to help ...
... n can be used to find solutions to Pell’s equation, x2 ny 2 1 . For more information on Pell’s Equation and continued fractions, refer to [2]. Furthermore, continued fractions can be put to use in the factorization of large integers [5, p.246]. We can also make use of continued fractions to help ...
New Generalized Cyclotomy and Its Applications
... generalized cyclotomic numbers are related to Waring’s problem [6], difference sets [3, 16, 17], sequences [4, 5, 10], coding theory [14, 15], and cryptography [7]. Classical cyclotomy was dealt to a good extent by Gauss in his ‘‘Disquisitiones Arithmeticae’’ [9], where he introduced the so called G ...
... generalized cyclotomic numbers are related to Waring’s problem [6], difference sets [3, 16, 17], sequences [4, 5, 10], coding theory [14, 15], and cryptography [7]. Classical cyclotomy was dealt to a good extent by Gauss in his ‘‘Disquisitiones Arithmeticae’’ [9], where he introduced the so called G ...
Proofs of a Trigonometric Inequality
... Trigonometric inequalities are very important in many mathematical areas. Because of its wide and profound application, it has become a popular research interest. Lohwarter mentioned in his book the following two inequalities (see [2], p.5 and p.78): ...
... Trigonometric inequalities are very important in many mathematical areas. Because of its wide and profound application, it has become a popular research interest. Lohwarter mentioned in his book the following two inequalities (see [2], p.5 and p.78): ...
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.