Gilbreath's Sequences and Proof of Conditions for Gilbreath's Conjecture

The conjecture attributed to Norman L. Gilbreath, but formulated by Francois Proth in the second half of the 1800s, concerns an interesting property of the ordered sequence of prime numbers $P$. Gilbreath’s conjecture stated that, computing the absolute value of differences of consecutive primes on ordered sequence of prime numbers, and if this calculation is done for the terms in the new sequence and so on, every sequence will starts with 1. In this paper is defined the concept of Gilbreath’s sequence, Gilbreath’s triangle and Gilbreath’s equation. On the basis of the results obtained from the proof of properties, an inductive proof is produced thanks to which it is possible to establish the necessary condition to state that the Gilbreath's conjecture is true.

Topological generation results for free unitary and orthogonal groups

We show that for every [Formula: see text] the free unitary group [Formula: see text] is topologically generated by its classical counterpart [Formula: see text] and the lower-rank [Formula: see text]. This allows for a uniform inductive proof that a number of finiteness properties, known to hold for all [Formula: see text], also hold at [Formula: see text]. Specifically, all discrete quantum duals [Formula: see text] and [Formula: see text] are residually finite, and hence also have the Kirchberg factorization property and are hyperlinear. As another consequence, [Formula: see text] are topologically generated by [Formula: see text] and their maximal tori [Formula: see text] (dual to the free groups on [Formula: see text] generators) and similarly, [Formula: see text] are topologically generated by [Formula: see text] and their tori [Formula: see text].

Understanding Mathematical Induction by Writing Analogies

Mathematical induction has some notoriety as a difficult mathematical proof technique, especially for beginning students. In this note, I describe a writing assignment in which students are asked to develop, describe in detail, critique, defend, and finally extend their own analogies for mathematical induction. By putting the work of explanation into the students' hands, this assignment requires them to engage in detail with the necessary parts of an inductive proof. Students select their subject for the analogy, allowing them to connect abstract mathematics to their lived experiences. The process of peer review helps students recognize and remedy several of the most common errors in writing an inductive proof. All of this takes place in the context of a creative assignment, outside the work of writing formal inductive proofs.

Data Conversion Method between a Natural Number and a Binary Tree for an Inductive Proof and Its Application

This paper presents modeling ofa binary tree that represents a natural numberand gives an inductive proof for its propertiesusing theorem provers.We define a function for converting data from a natural numberinto a binary treeand give an inductive proof for its well-definedness.We formalize this method, develop a computational model based on it,and apply it to an electronic cash protocol.We also define the payment function on the binary treeand go on to prove the divisibility of electronic cashusing the theorem provers Isabelle/HOL and Coq, respectively.Furthermore, we discuss the effectiveness of this method.

Newtonian and Non-Newtonian Elements in Hume

For the last forty years, Hume's Newtonianism has been a debated topic in Hume scholarship. The crux of the matter can be formulated by the following question: Is Hume a Newtonian philosopher? Debates concerning this question have produced two lines of interpretation. I shall call them ‘traditional’ and ‘critical’ interpretations. The traditional interpretation asserts that there are many Newtonian elements in Hume, whereas the critical interpretation seriously questions this. In this article, I consider the main points made by both lines of interpretations and offer further arguments that contribute to this debate. I shall first argue, in favor of the traditional interpretation, that Hume is sympathetic to many prominently Newtonian themes in natural philosophy such as experimentalism, criticality of hypotheses, inductive proof, and criticality of Leibnizian principles of sufficient reason and intelligibility. Second, I shall argue, in accordance with the critical interpretation, that in many cases Hume is not a Newtonian philosopher: His conceptions regarding space and time, vacuum, reality of forces, specifics about causation, and the status of mechanism differ markedly from Newton's related conceptions. The outcome of the article is that there are both Newtonian and non/anti-Newtonian elements in Hume.

W-types in homotopy-type theory – CORRIGENDUM

In the article below, Theorem 3.4 requires the additional assumption that A is Kan as well. Indeed, the inductive proof as given only shows that if W(f)<α is a Kan complex, then W(f)<α+1 → A is a Kan fibration.