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... Saposhenko [9]» Prodinger & Tichy [11] and later together with Kirschenhofer [7], [8] considered that problem in particular for trees. They introduced the notion of the Fibonacci number of a graph for the number of independent sets in it because the case of paths yields the Fibonacci numbers. We wil ...
... Saposhenko [9]» Prodinger & Tichy [11] and later together with Kirschenhofer [7], [8] considered that problem in particular for trees. They introduced the notion of the Fibonacci number of a graph for the number of independent sets in it because the case of paths yields the Fibonacci numbers. We wil ...
Reasoning without Contradiction
... result. But I think that this is untrue and that a case can be made that reasoning without contradiction has some very beneficial consequences. Limitations of space prevent me giving more than a sketch, but I set out the argument in such a way that the reader can easily identify where, if anywhere, ...
... result. But I think that this is untrue and that a case can be made that reasoning without contradiction has some very beneficial consequences. Limitations of space prevent me giving more than a sketch, but I set out the argument in such a way that the reader can easily identify where, if anywhere, ...
Classical BI - UCL Computer Science
... Accordingly, the contexts Γ on the left-hand side of the sequents in the rules above are not sets or sequences, as in standard sequent calculi, but rather bunches: trees whose leaves are formulas and whose internal nodes are either ‘;’ or ‘,’ denoting respectively additive and multiplicative combina ...
... Accordingly, the contexts Γ on the left-hand side of the sequents in the rules above are not sets or sequences, as in standard sequent calculi, but rather bunches: trees whose leaves are formulas and whose internal nodes are either ‘;’ or ‘,’ denoting respectively additive and multiplicative combina ...
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... 19 - 30, he brought out the fact that the last (units) digit of the sequence is p e r i odic with period 60, and that the last two digits are similarly periodic with period 300. Setting up an IBM 1620 he further found that the last three digits repeat every 1,500 times, the last four every 15,000, t ...
... 19 - 30, he brought out the fact that the last (units) digit of the sequence is p e r i odic with period 60, and that the last two digits are similarly periodic with period 300. Setting up an IBM 1620 he further found that the last three digits repeat every 1,500 times, the last four every 15,000, t ...
CS 103X: Discrete Structures Homework Assignment 3 — Solutions
... So if d | a, then |a| is either a, which is trivially divisible by d, or −a, which by the above reasoning is also divisible by d. Similarly, if d | |a|, then a is either |a| or −|a|, both of which are divisible by d as above. This proves the result. Lemma 2. gcd(a, b) = gcd(|a|, |b|) = gcd(|b|, |a|) ...
... So if d | a, then |a| is either a, which is trivially divisible by d, or −a, which by the above reasoning is also divisible by d. Similarly, if d | |a|, then a is either |a| or −|a|, both of which are divisible by d as above. This proves the result. Lemma 2. gcd(a, b) = gcd(|a|, |b|) = gcd(|b|, |a|) ...
You`re a mathematician! Oh! I never was much good at maths
... mathematician and the maths that I do is closer to philosophy than' to engineering. Consequently, somewhere along the way GOOD usually asks, "What is all .this good for?" I then explain that I do what I do because I enjoy it and don't seek or expect any applications. It comes as. a surprise to· me ( ...
... mathematician and the maths that I do is closer to philosophy than' to engineering. Consequently, somewhere along the way GOOD usually asks, "What is all .this good for?" I then explain that I do what I do because I enjoy it and don't seek or expect any applications. It comes as. a surprise to· me ( ...
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... identities 7fn = fn+4 + fn−4 and 7 = φ4 + φ−4 . 2.1. Termination. Before turning to the interpretation of the algorithm, it is worth saying a few words about termination. The algorithm alternates between making a sequence of shifts and outputting the next term. Bergman [2] shows that numbers can be ...
... identities 7fn = fn+4 + fn−4 and 7 = φ4 + φ−4 . 2.1. Termination. Before turning to the interpretation of the algorithm, it is worth saying a few words about termination. The algorithm alternates between making a sequence of shifts and outputting the next term. Bergman [2] shows that numbers can be ...
Sperner`s Lemma and its application
... This turns out be link with a very deep result in topology: Hopf’s index theorem. In dimension one, the question is the same as how many times a function will come cross the x-coordinated if we know f (a) < 0 and f (b) > 0. Then if counted by the order of the zero points, we know it will always be o ...
... This turns out be link with a very deep result in topology: Hopf’s index theorem. In dimension one, the question is the same as how many times a function will come cross the x-coordinated if we know f (a) < 0 and f (b) > 0. Then if counted by the order of the zero points, we know it will always be o ...
MATHEMATICAL STATEMENTS AND PROOFS In this note we
... 1. Notation; sets and sequences We will assume that the reader is familiar with the basic notation connected with sets. Thus if A, B are sets then “a ∈ A” means that a is an element in A; “A ∪ B” is the union of A and B; “A ∩ B” is the intersection of A and B; and “A − B” is the difference set A min ...
... 1. Notation; sets and sequences We will assume that the reader is familiar with the basic notation connected with sets. Thus if A, B are sets then “a ∈ A” means that a is an element in A; “A ∪ B” is the union of A and B; “A ∩ B” is the intersection of A and B; and “A − B” is the difference set A min ...
Arithmetic Polygons
... Corollary 5.1. There are no arithmetic heptagons. It is tempting to conjecture that the same results hold for any equiangular polygon with an odd number of sides. However, interleaving (for instance) the edges of three equilateral triangles of different edge lengths, two of them rotated by ±20◦ with ...
... Corollary 5.1. There are no arithmetic heptagons. It is tempting to conjecture that the same results hold for any equiangular polygon with an odd number of sides. However, interleaving (for instance) the edges of three equilateral triangles of different edge lengths, two of them rotated by ±20◦ with ...
Absolutely Abnormal Numbers - Mathematical Association of America
... A number is normal to the base b if it is simply normal to each of the bases b, b2, b3..... This is equivalent to demanding that for any finite string a.a2... ak Of base-b digits, the limiting frequency of occurrences of this string in the b-ary expansion of a (analogous to (2)) exists and equals 1/ ...
... A number is normal to the base b if it is simply normal to each of the bases b, b2, b3..... This is equivalent to demanding that for any finite string a.a2... ak Of base-b digits, the limiting frequency of occurrences of this string in the b-ary expansion of a (analogous to (2)) exists and equals 1/ ...
Congruent Number Problem 1 Congruent number problem
... As an exercise, I suggest you extend this table up to all square free n < 100. I wonder if someone has noticed something miraculous in this table? In fact, for n square free and n ≤ 23 it verifies the following: Deep Conjecture. (Birch-Swinnerton-Dyer-Tunnell) Let N be any odd square free positive i ...
... As an exercise, I suggest you extend this table up to all square free n < 100. I wonder if someone has noticed something miraculous in this table? In fact, for n square free and n ≤ 23 it verifies the following: Deep Conjecture. (Birch-Swinnerton-Dyer-Tunnell) Let N be any odd square free positive i ...
Probabilistic proofs of existence of rare events, Springer Lecture
... k ≥ 1 there are tournaments in which for every set of k players there is one who beats them all. The proof given in [Er] actually shows that for every fixed k if the number n of players is sufficiently large then almost all tournaments with n players satisfy this property, i.e., the probability that ...
... k ≥ 1 there are tournaments in which for every set of k players there is one who beats them all. The proof given in [Er] actually shows that for every fixed k if the number n of players is sufficiently large then almost all tournaments with n players satisfy this property, i.e., the probability that ...
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.