Some Algebraic and Algorithmic Problems in Acoustocerebrography

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 ([4, 1]) is a set of techniques of visualizing the state of (human) 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 (n) = ∑Hh=1 Ah · sin(2π ·fh · n/F + ψh), 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

A general method of solving Smullyan’s puzzles

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

Representation theorems for hypergraph satisfiability

Given a set of propositions, one can consider its inconsistency hypergraph. Then the satisfiability of sets of clauses with respect to that hypergraph (see [1], [6]) turns out to be the usual satisfiability. The problem is which hypergraphs can be obtained from sets of formulas as inconsistency hypergraphs. In the present paper it is shown that this can be done for all hypergraphs with countably many vertices and pairwise incomparable edges. Then, a general method of transforming the combinatorial problems into the satisfiability problem is shown

Consequence Operators Based on

Four consequence operators based on hypergraph satisfiability are defi-ned. Their properties are explored and interconnections are displayed. Finally their relation to the case of the Classical Propositional Calculus is shown. 1. Preliminaries. Let us recall some definitions and facts which can also be found in Cowen [1,2] and Kolany [3]. A hypergraph is a structure G = (V, E), where V is a set and E is a family of nonempty subsets of V. The elements of V will be called vertices, and the elements of the set E, edges of the hypergraph G. Sets of vertices will sometimes be called clauses. A hypergraph is compact iff every edge contains a finite one. A hypergraph is locally finite iff every vertex belongs to a finite number of edges only. A hypergraph is edge disjoint iff its edges are pairwise disjoint. Notice that a graph is a hypergraph with at most two-element edges. Here, the vertices of a fixed hypergraph G will be interpreted as some elementary propositions and the edges of G will be inconsistent sets of them. This interpretation leads to the following generalizatio