Solvability of

Group Solvability

SOLVABILITY OF FREE PRODUCTS, CAYLEY GRAPHS AND COMPLEXES

In this paper, we verify the solvability of free product of finite cyclic groups with topological methods. We use Cayley graphs and Everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. In particular, using these methods, we prove that the commutator subgroup of $mathbb{Z}_{m}*mathbb{Z}_{n}$ is free of rank $(m-1)(n-1)$ fo...

Solvability of Diophantine Equations

Attila Bérczes (University of Debrecen): On arithmetic properties of solutions of norm form equations. Abstract. Let α be an algebraic number of degree n and K := Q(α). Consider the norm form equation NK/Q(x0 + x1α+ x2α + . . .+ xn−1α) = b in x0, . . . , xn−1 ∈ Z. (1) Let H denote the solution set of (1). Arranging the elements of H in an |H| × n array H, one may ask at least two natural questi...

Solvability of Graph Inequalities

We investigate a new type of graph inequality (in the tradition of Cvetković, Simić, and Capobianco) which is based on the subgraph relation and which allows as terms fixed graphs, graph variables with specified vertices, and the operation of identifying vertices. We present a simple graph inequality which does not have a solution, and show that the solvability of inequalities with only one gra...

Solvability of Laplace’s Equation

We’ve shown that if a solution exists, it’s unique (though only up to an additive constant for the Neumann boundary condition; in this case there’s also a requirement on g). We’ve also shown other interesting properties possessed by any solution, e.g., the maximum principle and the mean value property. Existence is actually the toughest issue. On certain special domains like rectangles and circ...