Complementary vertices and adjacency testing in polytopes

Our main theoretical result is that, if a simple polytope has a pair of complementary vertices (i.e., two vertices with no facets in common), then it has at least two such pairs, which can be chosen to be disjoint. Using this result, we improve adjacency testing for vertices in both simple and non-simple polytopes: given a polytope in the standard form {x \in R^n | Ax = b and x \geq 0} and a list of its V vertices, we describe an O(n) test to identify whether any two given vertices are adjacent. For simple polytopes this test is perfect; for non-simple polytopes it may be indeterminate, and instead acts as a filter to identify non-adjacent pairs. Our test requires an O(n^2 V + n V^2) precomputation, which is acceptable in settings such as all-pairs adjacency testing. These results improve upon the more general O(nV) combinatorial and O(n^3) algebraic adjacency tests from the literature.Comment: 14 pages, 5 figures. v1: published in COCOON 2012. v2: full journal version, which strengthens and extends the results in Section 2 (see p1 of the paper for details

A Perturbative Approach to the Relativistic Harmonic Oscillator

A quantum realization of the Relativistic Harmonic Oscillator is realized in terms of the spatial variable $x$ and {\d\over \d x} (the minimal canonical representation). The eigenstates of the Hamiltonian operator are found (at lower order) by using a perturbation expansion in the constant $c^{-1}$. Unlike the Foldy-Wouthuysen transformed version of the relativistic hydrogen atom, conventional perturbation theory cannot be applied and a perturbation of the scalar product itself is required.Comment: 9 pages, latex, no figure

Drawing Planar Graphs with a Prescribed Inner Face

Given a plane graph $G$ (i.e., a planar graph with a fixed planar embedding) and a simple cycle $C$ in $G$ whose vertices are mapped to a convex polygon, we consider the question whether this drawing can be extended to a planar straight-line drawing of $G$. We characterize when this is possible in terms of simple necessary conditions, which we prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, it can be computed in the same running time

Dark Matter from Monogem

As a supernova shock expands into space, it may collide with dark matter particles, scattering them up to velocities more than an order of magnitude larger than typical dark matter velocities in the Milky Way. If a supernova remnant is close enough to Earth, and the appropriate age, this flux of high-velocity dark matter could be detectable in direct detection experiments, particularly if the dark matter interacts via a velocity-dependent operator. This could make it easier to detect light dark matter that would otherwise have too little energy to be detected. We show that the Monogem Ring supernova remnant is both close enough and the correct age to produce such a flux, and thus we produce novel direct detection constraints and sensitivities for future experiments.Comment: 8 Pages of Text, 3 Figure

Quantum correlations in the temporal CHSH scenario

We consider a temporal version of the CHSH scenario using projective measurements on a single quantum system. It is known that quantum correlations in this scenario are fundamentally more general than correlations obtainable with the assumptions of macroscopic realism and non-invasive measurements. In this work, we also educe some fundamental limitations of these quantum correlations. One result is that a set of correlators can appear in the temporal CHSH scenario if and only if it can appear in the usual spatial CHSH scenario. In particular, we derive the validity of the Tsirelson bound and the impossibility of PR-box behavior. The strength of possible signaling also turns out to be surprisingly limited, giving a maximal communication capacity of approximately 0.32 bits. We also find a temporal version of Hardy's nonlocality paradox with a maximal quantum value of 1/4.Comment: corrected versio

Anti-de Sitter boundary in Poincare coordinates

We study the space-time boundary of a Poincare patch of Anti-de Sitter (AdS) space. We map the Poincare AdS boundary to the global coordinate chart and show why this boundary is not equivalent to the global AdS boundary. The Poincare AdS boundary is shown to contain points of the bulk of the entire AdS space. The Euclidean AdS space is also discussed. In this case one can define a semi-global chart that divides the AdS space in the same way as the corresponding Euclidean Poincare chart.Comment: In this revised version we add a discussion of the physical consequences of the choice of a coordinate system for AdS space. We changed figure 1 and added more references. Version to be published in Gen. Relat. Grav

Prospectus, July 20, 1989

https://spark.parkland.edu/prospectus_1989/1015/thumbnail.jp

Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For inputs in general position the number of bounded faces is O(n^d). For any fixed d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a polynomial number of linear programs

A survey of maincrop potatoes I Estimates of yield 1948–50

The survey shows that objective estimates of the yield of maincrop potatoes can be obtained from small samples carefully selected and dug by hand. Samples taken from about 1000 fields gave estimates of the mean yield of all counties sampled with a standard error due to sampling of less than ± 0·2 ton/acre. The precision of the estimate could have been improved by a better distribution of samples among counties.The results point to underestimation on the part of the official estimates, in each of the 3 years, especially in the case of high yields in particular counties, and in particular years. The discrepancy between the official and the survey yields is of the order of 1¾ tons/acre, after all necessary corrections have been applied to the survey yields.The experience gained in the survey indicates that the method of sampling adopted provides an accurate and reliable method of estimating the yields of potatoes which could supplement, and, possibly, ultimately replace the present official estimates if more accurate estimates are required. A national scheme, properly designed, which would include all the potato-growing areas in due proportion should not be unduly expensive to operate. Estimates so obtained would not only be generally more accurate than those obtained by the present official method, but, perhaps more important, would indicate far more closely the fluctuation in yield from year to year