10,830 research outputs found
Enumeration of rises and falls by position
AbstractLet Ï=(Ï1, Ï2,âŠ,Ïn) denote a permutation of Zn = {1, 2,âŠ, n}. The pair (Ïi, Ïi+1) is a rise if Ïi<Ïi+1 or a fall if Ïi>Ïi+1. Also a conventional rise is counted at the beginning of Ï and a conventional fall at the end. Let k be a fixed integer â„ 1. The rise Ïi,Ïi+1 is said to be in a in a j (mod k) position if i ⥠j (mod k); similarly for a fall. The conventional rise at the beginning is in a 0 (mod k) position, while the conventional fall at the end is in an n (mod k) position. Let PnâĄPn(r0,âŠ,rkâ1,Æ0,âŠ,Æ;kâ1) denote the number of permutations having ri rises i (mod k) positions and Æ;i falls in i (mod k) positions. A generating function for Pn is obtained. In particular, for k = 2 the generating function is quite explicit and also, for certain special cases when k = 4
Growth of the Brownian forest
Trees in Brownian excursions have been studied since the late 1980s. Forests
in excursions of Brownian motion above its past minimum are a natural extension
of this notion. In this paper we study a forest-valued Markov process which
describes the growth of the Brownian forest. The key result is a composition
rule for binary Galton--Watson forests with i.i.d. exponential branch lengths.
We give elementary proofs of this composition rule and explain how it is
intimately linked with Williams' decomposition for Brownian motion with drift.Comment: Published at http://dx.doi.org/10.1214/009117905000000422 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Average number of messages for distributed leader-finding in rings of processors
International audienceConsider a distributed system of n processors arranged on a ring. All processors are labeled with distinct identity-numbers, but are otherwise identical. In this paper, we make use of combinatorial enumeration methods in permutations and derive the one and the same exact asymptotic value, lJ2nH,,+O(n), of the expected number of messages in both probabilistic and deterministicbidirectional variants of Chang-Roberts distributed election algorithm. This confirms the result of Bodlaender and van Leeuwen (1986) that distributed Ieader finding is indeed strictly more efficient on bidirectional rings of processors than on unidirectional ones
On arithmetic and asymptotic properties of up-down numbers
Let , where , and let
denote the number of permutations of whose
up-down signature , for .
We prove that the set of all up-down numbers can be expressed by
a single universal polynomial , whose coefficients are products of
numbers from the Taylor series of the hyperbolic tangent function. We prove
that is a modified exponential, and deduce some remarkable congruence
properties for the set of all numbers , for fixed . We prove a
concise upper-bound for , which describes the asymptotic behaviour
of the up-down function in the limit .Comment: Recommended for publication in Discrete Mathematics subject to
revision
Exact average message complexity values for distributed election on bidirectional rings of processors
International audienceConsider a distributed system of n processors arranged on a ring. All processors are labeled with distinct identity-numbers, but are otherwise identical. In this paper, we make use of combinatorial enumeration methods in permutations and derive the one and the same exact asymptotic value, lJ2nH,,+O(n), of the expected number of messages in both probabilistic and deterministicbidirectional variants of Chang-Roberts distributed election algorithm. This confirms the result of Bodlaender and van Leeuwen (1986) that distributed Ieader finding is indeed strictly more efficient on bidirectional rings of processors than on unidirectional ones
The effects of grain shape and frustration in a granular column near jamming
We investigate the full phase diagram of a column of grains near jamming, as
a function of varying levels of frustration. Frustration is modelled by the
effect of two opposing fields on a grain, due respectively to grains above and
below it. The resulting four dynamical regimes (ballistic, logarithmic,
activated and glassy) are characterised by means of the jamming time of
zero-temperature dynamics, and of the statistics of attractors reached by the
latter. Shape effects are most pronounced in the cases of strong and weak
frustration, and essentially disappear around a mean-field point.Comment: 17 pages, 19 figure
Community, Comparisons and Subjective Well-being in a Divided Society
Using a South African data set, the paper poses six questions about the determinants of subjective well-being. Much of the paper is concerned with the role of relative concepts. We find that comparator income â measured as average income of others in the local residential cluster - enters the householdâs utility function positively but that income of more distant others (others in the district or province) enters negatively. The ordered probit equations indicate that, as well as comparator groups based on spatial proximity, race-based comparator groups are important in the racially divided South African society. It is also found that relative income is more important to happiness at higher levels of absolute income. Potential explanations of these results, and their implications, are considered.Subjective well-being; happiness; comparator groups; altruism; envy; relative deprivation; standard-setting; race; South Africa
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