Distribution of Summands in Generalized
... For any f : N0 → N0 , there exists a sequence of natural numbers {an }∞ n=0 such that every positive integer has a unique f -decomposition using {an }. Construction of {an }. Let a0 = 1 and an = an−1 + an−1−f (n−1) (assume an = 1 when n < 0). The proof of existence is by induction. Consider any x ∈ ...
... For any f : N0 → N0 , there exists a sequence of natural numbers {an }∞ n=0 such that every positive integer has a unique f -decomposition using {an }. Construction of {an }. Let a0 = 1 and an = an−1 + an−1−f (n−1) (assume an = 1 when n < 0). The proof of existence is by induction. Consider any x ∈ ...
Mathematical Reasoning: Writing and Proof
... Important Features of the Book Following are some of the important features of this text that will help with the transition from calculus to upper-level mathematics courses. 1. Emphasis on Writing in Mathematics Issues dealing with writing mathematical exposition are addressed throughout the book. G ...
... Important Features of the Book Following are some of the important features of this text that will help with the transition from calculus to upper-level mathematics courses. 1. Emphasis on Writing in Mathematics Issues dealing with writing mathematical exposition are addressed throughout the book. G ...
Modal Logic for Artificial Intelligence
... Some rules are very simple: if you can prove ϕ and you can prove ψ, then you can also prove their conjunction ϕ ∧ ψ. Other rules are more complicated. For example, the only way to ‘eliminate’ the disjunction ϕ ∨ ψ is by proving, first that ϕ ∨ ψ, and second, that some conclusion χ can be proven both ...
... Some rules are very simple: if you can prove ϕ and you can prove ψ, then you can also prove their conjunction ϕ ∧ ψ. Other rules are more complicated. For example, the only way to ‘eliminate’ the disjunction ϕ ∨ ψ is by proving, first that ϕ ∨ ψ, and second, that some conclusion χ can be proven both ...
- ESAIM: Proceedings
... for some non-negative random variable W . [We recall that if (ξn ) is a sub-martingale with supn E(ξn+ ) < ∞, then ξn converges almost surely to a finite random variable. Apply this to (−Wn ).] By Fatou’s lemma, E(W ) ≤ lim inf n→∞ E(Wn ) = 1. It is, however, possible that W = 0. So it is important ...
... for some non-negative random variable W . [We recall that if (ξn ) is a sub-martingale with supn E(ξn+ ) < ∞, then ξn converges almost surely to a finite random variable. Apply this to (−Wn ).] By Fatou’s lemma, E(W ) ≤ lim inf n→∞ E(Wn ) = 1. It is, however, possible that W = 0. So it is important ...
Braid Index, Genus and Crossing Number of Links
... of deficiency zero so there are infinitely many such links. In this paper, we will prove that the number of deficiency zero links actually grows exponentially with respect to the crossing number of the links. Another part of this paper is devoted to exploring further the relationships among the cros ...
... of deficiency zero so there are infinitely many such links. In this paper, we will prove that the number of deficiency zero links actually grows exponentially with respect to the crossing number of the links. Another part of this paper is devoted to exploring further the relationships among the cros ...
Complex varieties and the analytic topology
... Exercise 1.3. (a) A regular function X → C gives a continuous map Xan → C, where C is equipped with the analytic topology. (b) If also Y ⊆ Am C is an affine complex variety, and X → Y is a morphism, then the induced map Xan → Yan is continuous. (c) The analytic topology on X is an isomorphism invari ...
... Exercise 1.3. (a) A regular function X → C gives a continuous map Xan → C, where C is equipped with the analytic topology. (b) If also Y ⊆ Am C is an affine complex variety, and X → Y is a morphism, then the induced map Xan → Yan is continuous. (c) The analytic topology on X is an isomorphism invari ...
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... granted to members of The Fibonacci Association for noncommercial reproduction of a limited quantity of individual articles (in whole or in part) provided complete reference is made to the source. Annual domestic Fibonacci Association membership dues, which include a subscription to THE FIBONACCI QU ...
... granted to members of The Fibonacci Association for noncommercial reproduction of a limited quantity of individual articles (in whole or in part) provided complete reference is made to the source. Annual domestic Fibonacci Association membership dues, which include a subscription to THE FIBONACCI QU ...
SECTION C Properties of Prime Numbers
... It means that we can choose any positive integer n, 5 for example, and somewhere else on the real number line we will find 5 composite numbers sat between 2 primes. In the case of 5 we find that further along the number line, 5 composite numbers 24, 25, 26, 27 and 28 sit between the prime numbers 23 ...
... It means that we can choose any positive integer n, 5 for example, and somewhere else on the real number line we will find 5 composite numbers sat between 2 primes. In the case of 5 we find that further along the number line, 5 composite numbers 24, 25, 26, 27 and 28 sit between the prime numbers 23 ...
Logical Methods in Computer Science Vol. 8(4:19)2012, pp. 1–28 Submitted Oct. 27, 2011
... to estimate the degree of similarity of two systems in terms of their behaviours. The metric theory for Markov processes was initiated by Desharnais et al. [DGJP04] and greatly developed and explored by van Breugel, Worrell and others [vBW01, vB+03]. One way to do this is to relax the satisfiability ...
... to estimate the degree of similarity of two systems in terms of their behaviours. The metric theory for Markov processes was initiated by Desharnais et al. [DGJP04] and greatly developed and explored by van Breugel, Worrell and others [vBW01, vB+03]. One way to do this is to relax the satisfiability ...
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.