Proper 1-immersions of graphs triangulating the plane

In this paper we study what planar graphs are “rigid” enough such that they can not be drawn on the plane with few (1, 2, or 3) crossings per edge. A graph drawn on the plane is k$k$-immersed in the plane if each edge is crossed by at most k$k$ other edges. By a proper  k$k$-immersion of a graph we mean a k$k$-immersion of the graph in the plane such that there is at least one crossing point. We give a characterization (in terms of forbidden subgraphs) of 4-connected graphs which triangulate the plane and have a proper 1-immersion. We show that every planar graph has a proper 3-immersion.     

Another design for sundials

Based on Kepler's laws we calculate the position of the sun as a function of the date, the local time, the latitude and longitude and use the resulting formula to construct a portable sundial which works without being adjusted to the cardinal directions (i.e. there is no need for an additional compass).     

Sample size for the Z test and its confidence interval

The statistical power of a significance test is closely related to the length of the confidence interval (i.e. estimate precision). In the case of a Z test, the length of the confidence interval can be expressed as a function of the statistical power.     

Modular forms and effective Diophantine approximation

After the work of G. Frey, it is known that an appropriate bound for the Faltings height of elliptic curves in terms of the conductor (Frey?s height conjecture) would give a version of the ABC conjecture. In this paper we prove a partial result towards Frey?s height conjecture which applies to all elliptic curves over $\mathbb{Q}$, not only Frey curves. Our bound is completely effective and the technique is based in the theory of modular forms. As a consequence, we prove effective explicit bounds towards the ABC conjecture of similar strength to what can be obtained by linear forms in logarithms, without using the latter technique. The main application is a new effective proof of the finiteness of solutions to the S-unit equation (that is, S-integral points of ${\mathbb{P}}^1?\{0,1,\infty \}$), with a completely explicit and effective bound, without using any variant of Baker?s theory or the Thue–Bombieri method.     

Central Units,Class Sums and Characters of the Symmetric Group

In the search for central units of a group algebra, we look at the class sums of the group algebra of the symmetric group S _n in characteristic zero, and we show that they are units in very special instances.