In [4] R.Cowen considers a generalization of the resolution rule for hypergraphs and introduces a notion of satisfiability of families of sets of vertices via 2-colorings piercing elements of such families. He shows, for finite hypergraphs with no one-element edges that if the empty set is a consequence ofA by the resolution rule, thenA is not satisfiable. Alas the converse is true for a restricted class of hypergraphs only, and need not to be true in the general case. In this (...) paper we show that weakening slightly the notion of satisfiability, we get the equivalence of unsatisfiability and the derivability of the empty set for any hypergraph. Moreover, we show the compactness property of hypergraph satisfiability and state its equivalence to BPI, i.e. to the statement that in every Boolean algebra there exists an ultrafilter. (shrink)

In this paper we present a general method of solving Smullyan’s puzzles. We do this by showing how a puzzle is translated into Classical Propositional Calculus.

In the following we show that general property S considered by Cowen [1], Cowen and Kolany in [3] and earlier by Cowen in [2] and Kolany in [4] as hypergraph satisfiability, can be constructively reduced to (3, 2) · SAT , that is to satisfiability of (at most) triples with two-element forbidden sets. This is an analogue of the“classical” result on the reduction of SAT to 3 · SAT.

Progress in the medical diagnostic is relentlessly pushing the measurement technology as well with its intertwined mathematical models and solutions. Mathematics has applications to many problems that are vital to human health but not for all. In this article we describe how the mathematics of acoustocerebrography has become one of the most important applications of mathematics to the problems of brain monitoring as well we will show some algebraic problems which still have to be solved. Acoustocerebrography is a set of (...) techniques of visualizing the state of brain tissue and its changes with use of ultrasounds, which mainly rely on a relation between the tissue density and speed of propagation for ultrasound waves in this medium. Propagation speed or, equivalently, times of arriving for an ultrasound pulse, can be inferred from phase relations for various frequencies. Since, due to Kramers-Kronig relations, the propagation speeds depend significantly on the frequency of investigated waves, we consider multispectral wave packages of the form W = ∑Hh=1 Ah · sin, n = 0,..., N – 1 with appropriately chosen frequencies fh, h = 1,...,H, amplifications Ah, h = 1,...,H, start phases ψh, h = 1,..., H and sampling frequency F. In this paper we show some problems of algebraic and, to some extend, algorithmic nature which raise up in this topic. Like, for instance, the influence of relations between the signal length and frequency values on the error on estimated phases or on neutralizing alien frequencies. Another problem is finding appropriate initial phases for avoiding improper distributions of peaks in the resulting signal or finding a stable algorithm of phase unwinding which is resistant to sudden random disruptions. (shrink)