Analysis of Incomplete Data and an Intrinsic-Dimension Helly Theorem

The analysis of incomplete data is a long-standing challenge in practical statistics. When, as is typical, data objects are represented by points in R^d , incomplete data objects correspond to affine subspaces (lines or Δ-flats).With this motivation we study the problem of finding the minimum intersection radius r(L) of a set of lines or Δ-flats L: the least r such that there is a ball of radius r intersecting every flat in L. Known algorithms for finding the minimum enclosing ball for a point set (or clustering by several balls) do not easily extend to higher dimensional flats, primarily because “distances” between flats do not satisfy the triangle inequality. In this paper we show how to restore geometry (i.e., a substitute for the triangle inequality) to the problem, through a new analog of Helly’s theorem. This “intrinsic-dimension” Helly theorem states: for any family L of Δ-dimensional convex sets in a Hilbert space, there exist Δ + 2 sets L' ⊆ L such that r(L) ≤ 2r(L'). Based upon this we present an algorithm that computes a (1+ε)-core set L' ⊆ L, |L'| = O(Δ^4/ε), such that the ball centered at a point c with radius (1 +ε)r(L') intersects every element of L. The running time of the algorithm is O(n^(Δ+1)dpoly(Δ/ε)). For the case of lines or line segments (Δ = 1), the (expected) running time of the algorithm can be improved to O(ndpoly(1/ε)).We note that the size of the core set depends only on the dimension of the input objects and is independent of the input size n and the dimension d of the ambient space

Extended self-energy functional approach for strongly-correlated lattice bosons in the superfluid phase

Among the various numerical techniques to study the physics of strongly correlated quantum many-body systems, the self-energy functional approach (SFA) has become increasingly important. In its previous form, however, SFA is not applicable to Bose-Einstein condensation or superfluidity. In this paper we show how to overcome this shortcoming. To this end we identify an appropriate quantity, which we term $D$, that represents the correlation correction of the condensate order parameter, as it does the self-energy for the Green's function. An appropriate functional is derived, which is stationary at the exact physical realizations of $D$ and of the self-energy. Its derivation is based on a functional-integral representation of the grand potential followed by an appropriate sequence of Legendre transformations. The approach is not perturbative and therefore applicable to a wide range of models with local interactions. We show that the variational cluster approach based on the extended self-energy functional is equivalent to the "pseudoparticle" approach introduced in Phys. Rev. B, 83, 134507 (2011). We present results for the superfluid density in the two-dimensional Bose-Hubbard model, which show a remarkable agreement with those of Quantum-Monte-Carlo calculations.Comment: 1 additional figure showing the region close to the tip of the Mott lobe, minor changes in the tex

Perceived Stress, Social Support and Survival: North Carolina Farm Operators and the Farm Crisis

The current farm crisis draws attention to the sources and consequences of the stress process among farm operators. Using panel data from statewide surveys of North Carolina farm operators collected during a period of economic and ecological crisis, the relationships among perceived stress, social support, and survival in agriculture are investigated. Analyses reveal that while the level of perceived stress has no relationship with survival, social support has a significant impact upon both social psychological (plans to remain in farming) and behavioral (continuing as a farm operator) dimensions of survival in agriculture. Perceived social support increased plans to remain in agriculture and increased the probability of an operator continuing farming. The results point to the importance of social support and have implications for policy intervention and programs