On a conjecture of Wilf about the Frobenius number

Given coprime positive integers $a_1 < ...< a_d$, the Frobenius number $F$ is the largest integer which is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote the number of all non-representable positive integers: Wilf conjectured that $d \geq \frac{F+1}{F+1-g}$. We prove that for every fixed value of $\lceil \frac{a_1}{d} \rceil$ the conjecture holds for all values of $a_1$ which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional local rings and a question on the equality $d = \frac{F+1}{F+1-g}$

Oscillating solutions for prescribed mean curvature equations: Euclidean and Lorentz-Minkowski cases

This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if $N=1$, while they are radial symmetric and decay to zero at infinity with their derivatives, if $N\ge 2$.Comment: 11 page

Somewhere over the... what?

In order to defend his controversial claim that observation is unaided perception, Bas van Fraassen, the originator of constructive empiricism, suggested that, for all we know, the images produced by a microscope could be in a situation analogous to that of the rainbows, which are ‘images of nothing’. He added that reflections in the water, rainbows, and the like are ‘public hallucinations’, but it is not clear whether this constitutes an ontological category apart or an empty set. In this paper an argument will be put forward to the effect that rainbows can be thought of as events, that is, as part of a subcategory of entities that van Fraassen has always considered legitimate phenomena. I argue that rainbows are actually not images in the relevant (representational) sense and that there is no need to ontologically inflate the category of entities in order to account for them, which would run counter to the empiricist principle of parsimony