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

The structural transformation between manufacturing and services and the deline in the U.S. GDP volatility

For a single firm with a given volatility of total factor productivity at the gross output level (GTFP), the volatility of total factor productivity at the value added level (YTFP) increases with the share of intermediate goods in gross output. For a Cobb-Douglas production function in capital, labor and intermediate goods, YTFP volatility is equal to GTFP volatility divided by one minus the share of intermediate goods in gross output. In the U.S., this share is steadily around 0.6 for manufacturing and 0.38 for services during the 1960-2005 period. Thus, the same level of GTFP volatility in the two sectors implies a 55% larger YTFP volatility in manufacturing. This fact contributes to the higher measured YTFP volatility in manufacturing with respect to services. It follows that, as the services share in GDP increases from 0.53 in 1960 to 0.71 in 2005 in the U.S., GDP volatility is reduced. I construct a two-sector dynamic general equilibrium input-output model to quantify the role of the structural transformation between manufacturing and services in reducing the U.S. GDP volatility. Numerical results for the calibrated model economy suggest that the structural transformation can account for 32% of the GDP volatility reduction between the 1960-1983 and the 1984-2005 periods