110 research outputs found

### Fractality of certain quantum states

We prove the theorem announced in Phys. Rev. Lett. {\bf 85}:5022, 2001
concerning the existence and properties of fractal states for the Schr\"odinger
equation in the infinite one-dimensional well.Comment: Latex2e with svjour clas

### Penrose voting system and optimal quota

Systems of indirect voting based on the principle of qualified majority can
be analysed using the methods of game theory. In particular, this applies to
the voting system in the Council of the European Union, which was recently a
subject of a vivid political discussion. The a priori voting power of a voter
measures his potential influence over the decisions of the voting body under a
given decision rule. We investigate a system based on the law of Penrose, in
which each representative in the voting body receives the number of votes (the
voting weight) proportional to the square root of the population he or she
represents. Here we demonstrate that for a generic distribution of the
population there exists an optimal quota for which the voting power of any
state is proportional to its weight. The optimal quota is shown to decrease
with the number of voting countries.Comment: 12 pages, 2 figure

### Densities of the Raney distributions

We prove that if $p\ge 1$ and $0< r\le p$ then the sequence
$\binom{mp+r}{m}\frac{r}{mp+r}$, $m=0,1,2,...$, is positive definite, more
precisely, is the moment sequence of a probability measure $\mu(p,r)$ with
compact support contained in $[0,+\infty)$. This family of measures encompasses
the multiplicative free powers of the Marchenko-Pastur distribution as well as
the Wigner's semicircle distribution centered at $x=2$. We show that if $p>1$
is a rational number, $0<r\le p$, then $\mu(p,r)$ is absolutely continuous and
its density $W_{p,r}(x)$ can be expressed in terms of the Meijer and the
generalized hypergeometric functions. In some cases, including the
multiplicative free square and the multiplicative free square root of the
Marchenko-Pastur measure, $W_{p,r}(x)$ turns out to be an elementary function

### Square root voting system, optimal threshold and \pi

The problem of designing an optimal weighted voting system for the two-tier
voting, applicable in the case of the Council of Ministers of the European
Union (EU), is investigated. Various arguments in favour of the square root
voting system, where the voting weights of member states are proportional to
the square root of their population are discussed and a link between this
solution and the random walk in the one-dimensional lattice is established. It
is known that the voting power of every member state is approximately equal to
its voting weight, if the threshold q for the qualified majority in the voting
body is optimally chosen. We analyze the square root voting system for a
generic 'union' of M states and derive in this case an explicit approximate
formula for the level of the optimal threshold: q \simeq 1/2+1/\sqrt{{\pi} M}.
The prefactor 1/\sqrt{{\pi}} appears here as a result of averaging over the
ensemble of unions with random populations.Comment: revised version, 21 pages in late

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