Diophantine Representation of the Fibonacci Numbers
... The Fibonacci numbers were important in MatijaseviFs proof, because the sequence of Fibonacci numbers grows exponentially. Martin Davis, Julia Robinson and Hilary Putnam [3] had nearly solved Hilbert's tenth problem in 1961, when they succeeded in proving that the stated result would follow from the ...
... The Fibonacci numbers were important in MatijaseviFs proof, because the sequence of Fibonacci numbers grows exponentially. Martin Davis, Julia Robinson and Hilary Putnam [3] had nearly solved Hilbert's tenth problem in 1961, when they succeeded in proving that the stated result would follow from the ...
Ramsey Theory, Integer Partitions and a New Proof of the Erd˝os
... Organization: The rest of the paper is organized as follows. In Section 2 we mainly focus on 3-uniform hypergraphs. We first give a new proof of EST by showing that N3 (2, n) ≤ P1 (n) + 1. We will then move on to prove the more general bound N3 (q, n) ≤ Pq−1 (n) + 1. The proof of the general bound w ...
... Organization: The rest of the paper is organized as follows. In Section 2 we mainly focus on 3-uniform hypergraphs. We first give a new proof of EST by showing that N3 (2, n) ≤ P1 (n) + 1. We will then move on to prove the more general bound N3 (q, n) ≤ Pq−1 (n) + 1. The proof of the general bound w ...
Oliver Johnson and Christina Goldschmidt 1. Introduction
... (|c(i)|, i ≥ 0) is a unimodal sequence and Welsh [21] conjectured, more strongly, that it is log-concave. Hoggar proved Theorem 1.4 in a partial attempt to resolve this conjecture. It remains unproven, but progress towards it is reviewed in Brenti [5]. In Section 3, we show how a version of Theorem ...
... (|c(i)|, i ≥ 0) is a unimodal sequence and Welsh [21] conjectured, more strongly, that it is log-concave. Hoggar proved Theorem 1.4 in a partial attempt to resolve this conjecture. It remains unproven, but progress towards it is reviewed in Brenti [5]. In Section 3, we show how a version of Theorem ...
STANDARD COMPLETENESS THEOREM FOR ΠMTL 1
... (1) if x, y ∈ F , then x ∗ y ∈ F , (2) if x ∈ F , x ≤ y, then y ∈ F . LEMMA 2.6. For any filter F in a ΠMTL-algebra L, let us define the following equivalence relation in L: x ∼F y iff x → y ∈ F and y → x ∈ F . Then ∼F is a congruence and the quotient L/F is a ΠMTL-algebra. We will denote the equiva ...
... (1) if x, y ∈ F , then x ∗ y ∈ F , (2) if x ∈ F , x ≤ y, then y ∈ F . LEMMA 2.6. For any filter F in a ΠMTL-algebra L, let us define the following equivalence relation in L: x ∼F y iff x → y ∈ F and y → x ∈ F . Then ∼F is a congruence and the quotient L/F is a ΠMTL-algebra. We will denote the equiva ...
Techniques for proving the completeness of a proof system
... Sometimes it is easier to show the truth of a formula than to derive the formula. The completeness result shows that nothing is missing in a proof system. The completeness result formalizes what a proof system achieves. With a completeness result, a paper about a proof system has more chances to get ...
... Sometimes it is easier to show the truth of a formula than to derive the formula. The completeness result shows that nothing is missing in a proof system. The completeness result formalizes what a proof system achieves. With a completeness result, a paper about a proof system has more chances to get ...
4-3: Alternating Series, and the Alternating Series Theorem
... • Definition: A series is called Pan alternating series if the terms alternate in sign. That is, an alternating series is a series of the form (−1)k+1 ak where ak > 0 for all k. • The series above is thus an example of an alternating series, and is called the alternating harmonic series. • The idea ...
... • Definition: A series is called Pan alternating series if the terms alternate in sign. That is, an alternating series is a series of the form (−1)k+1 ak where ak > 0 for all k. • The series above is thus an example of an alternating series, and is called the alternating harmonic series. • The idea ...
![[Part 1]](http://s1.studyres.com/store/data/008795780_1-2d32093eb8955eecb4220b99bc38a981-300x300.png)
On the rational approximation to the binary Thue–Morse–Mahler
... Section 13.4 of [1]). Since the irrationality exponent of ξt,b is equal to 2 (see [3]), the transcendence of ξt,b cannot be proved by applying Roth’s theorem. In the present note, we focus on the so-called Thue–Morse constant ξt := ξt,2 = 0.412454 . . . Open Problem 9 on page 403 of [1] asks whether ...
... Section 13.4 of [1]). Since the irrationality exponent of ξt,b is equal to 2 (see [3]), the transcendence of ξt,b cannot be proved by applying Roth’s theorem. In the present note, we focus on the so-called Thue–Morse constant ξt := ξt,2 = 0.412454 . . . Open Problem 9 on page 403 of [1] asks whether ...
Effectively Polynomial Simulations
... boolean formulas, R(f, m) such that when m is at least is said to be weakly automatizable. the size of the shortest B-proof of f , R(f, m) has an A When we say “automatizable” in future, we mean proof of size polynomial in |f | + m. If there also exists a polynomial-time function (again polytime in ...
... boolean formulas, R(f, m) such that when m is at least is said to be weakly automatizable. the size of the shortest B-proof of f , R(f, m) has an A When we say “automatizable” in future, we mean proof of size polynomial in |f | + m. If there also exists a polynomial-time function (again polytime in ...
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.