On the minimal travel time needed to collect n items on a circle

Consider n items located randomly on a circle of length 1. The locations of the items are assumed to be independent and uniformly distributed on [0,1). A picker starts at point 0 and has to collect all n items by moving along the circle at unit speed in either direction. In this paper we study the minimal travel time of the picker. We obtain upper bounds and analyze the exact travel time distribution. Further, we derive closed-form limiting results when n tends to infinity. We determine the behavior of the limiting distribution in a positive neighborhood of zero. The limiting random variable is closely related to exponential functionals associated with a Poisson process. These functionals occur in many areas and have been intensively studied in recent literature

Remembering Wassily Hoeffding

Wasssily Hoeffding's terminal illness and untimely death in 1991 put an end to efforts that were made to interview him for Statistical Science. An account of his scientific work is given in Fisher and Sen [The Collected Works of Wassily Hoeffding (1994) Springer], but the present authors felt that the statistical community should also be told about the life of this remarkable man. He contributed much to statistical science, but will also live on in the memory of those who knew him as a kind and modest teacher and friend, whose courage and learning were matched by a wonderful sense of humor.Comment: Published in at http://dx.doi.org/10.1214/08-STS271 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Debt, Deficits and Inflation: An Application to the Public Finances of India

The paper studies the solvency of the Indian public sector and the eventual monetization and inflation implied by stabilization of the debt-GNP ratio without any changes in the primary deficit. The nonstationarity of the discounted public debt suggests that indefinite continuation of the pattern of behavior reflected in the historical discounted debt process is inconsistent with the maintenance of solvency. This message is reinforced by the recent behavior of the debt-GNP ratio and the ratio of primary surplus plus seigniorage to GNP. Our estimates of the base money demand function suggest that even maximal use of seigniorage will not be sufficient to restore solvency.

The graphs with all but two eigenvalues equal to −2-2 or 00

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to $-2$, or $0$, and determine which of these graphs are determined by their adjacency spectrum

Thermoelectric performance of multiphase XNiSn (X = Ti, Zr, Hf) half-Heusler alloys

Quantitative X-ray powder diffraction analysis demonstrates that mixing Ti, Zr and Hf on the ionic site in the half-Heusler structure, which is a common strategy to lower the lattice thermal conductivity in this important class of thermoelectric materials, leads to multiphase behaviour. For example, nominal Ti0.5Zr0.5NiSn has a distribution of Ti1−xZrxNiSn compositions between 0.24 ≤ x ≤ 0.70. Similar variations are observed for Zr0.50Hf0.5NiSn and Ti0.5Hf0.5NiSn. Electron microscopy and elemental mapping demonstrate that the main compositional variations occur over micrometre length scales. The thermoelectric power factors of the mixed phase samples are improved compared to the single phase end-members (e.g. S2/ρ = 1.8 mW m−1 K−2 for Ti0.5Zr0.5NiSn, compared to S2/ρ = 1.5 mW m−1 K−2 for TiNiSn), demonstrating that the multiphase behaviour is not detrimental to electronic transport. Thermal conductivity measurements for Ti0.5Zr0.5NiSn0.95 suggest that the dominant reduction comes from Ti/Zr mass and size difference phonon scattering with the multiphase behaviour a secondary effect

Regular graphs with maximal energy per vertex

We study the energy per vertex in regular graphs. For every k, we give an upper bound for the energy per vertex of a k-regular graph, and show that a graph attains the upper bound if and only if it is the disjoint union of incidence graphs of projective planes of order k-1 or, in case k=2, the disjoint union of triangles and hexagons. For every k, we also construct k-regular subgraphs of incidence graphs of projective planes for which the energy per vertex is close to the upper bound. In this way, we show that this upper bound is asymptotically tight

Notes on simplicial rook graphs

The simplicial rook graph ${\rm SR}(m,n)$ is the graph of which the vertices are the sequences of nonnegative integers of length $m$ summing to $n$, where two such sequences are adjacent when they differ in precisely two places. We show that ${\rm SR}(m,n)$ has integral eigenvalues, and smallest eigenvalue $s = \max (-n, -{m \choose 2})$, and that this graph has a large part of its spectrum in common with the Johnson graph $J(m+n-1,n)$. We determine the automorphism group and several other properties

Core Precession and Global Modes in Granular Bulk Flow

A transition from local to global shear zones is reported for granular flows in a modified Couette cell. The experimental geometry is a slowly rotating drum which has a stationary disc of radius R_s fixed at its bottom. Granular material, which fills this cell up to height H, forms a wide shear zone which emanates from the discontinuity at the stationary discs edge. For shallow layers (H/R_s < 0.55), the shear zone reaches the free surface, with the core of the material resting on the disc and remaining stationary. In contrast, for deep layers (H/R_s > 0.55), the shear zones meet below the surface and the core starts to precess. A change in the symmetry of the surface velocities reveals that this behavior is associated with a transition from a local to a global shear mode.Comment: 4 pages, 7 figures, submitte