110 research outputs found

    Fractality of certain quantum states

    Full text link
    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

    Get PDF
    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

    Get PDF
    We prove that if p1p\ge 1 and 0<rp0< r\le p then the sequence (mp+rm)rmp+r\binom{mp+r}{m}\frac{r}{mp+r}, m=0,1,2,...m=0,1,2,..., is positive definite, more precisely, is the moment sequence of a probability measure μ(p,r)\mu(p,r) with compact support contained in [0,+)[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=2x=2. We show that if p>1p>1 is a rational number, 0<rp0<r\le p, then μ(p,r)\mu(p,r) is absolutely continuous and its density Wp,r(x)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, Wp,r(x)W_{p,r}(x) turns out to be an elementary function

    Square root voting system, optimal threshold and \pi

    Full text link
    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
    corecore