In this note, we outline a definition of propositional manifold and logical cohomology. An application is also considered for mathematics, two Boole algebras of mathematical propositions are non equivalent if their two cohomologies are not isomorphic.

Hosted by: The Claude Shannon Institute for Discrete Mathematics, Coding, Cryptography and Information Security, The Boole Centre for Research in Informatics, UCC School of Engineering, UCC School of Mathematical Sciences, UCC

We study a connection between Euler-MacLaurin Summation and Boole Summation suggested in an AMM note from 1960, which explains them as two cases in a general approach to approximation that also encompasses Taylor sums. Here we give additional details of the construction.

Let p be a prime number with p ≡ 1(mod 2). Throughout this paper, Zp,Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp. The p-adic norm | · |p is normalized as |p|p = 1 p . Let q be an indeterminate in Cp such that |1− q|p < p −1 p−1 . The q-extension of number x be defined as [x]q = 1−qx 1−q . Note that limq→1[...

Let Qρ,γ ⊃ 1 be arbitrary. In [34], the main result was the computation of bounded, contraconnected, Boole paths. We show that Λ ≥ e. Here, existence is trivially a concern. Now this could shed important light on a conjecture of Clifford.

The so-called “new machinery for computing” was early computers, and ACM considered it a machinery for “reasoning .. of information”. However, the computers are not the only models for information change. The exciting history of derivations (or so called syllogisms) go back till Aristotle, and gained so much importance in Medieval ages via the rise of logical discourse among religious scholars....

Three extensions and reinterpretations of nonclassical probabilities are reviewed. (i) We propose to generalize the probability axiom of quantum mechanics to self-adjoint positive operators of trace one. Furthermore, we discuss the Cartesian and polar decomposition of arbitrary normal operators and the possibility to operationalize the corresponding observables. Thereby we review and emphasize ...