The Triangle Closure is a Polyhedron

Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen, Louveaux and Weismantel [ An analysis of mixed integer linear sets based on lattice point free convex sets, Math. Oper. Res. 35 (2010), 233--256] and Averkov [On finitely generated closures in the theory of cutting planes, Discrete Optimization 9 (2012), no. 4, 209--215], some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron. In this paper, we show that when the number of integer variables $m=2$ the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data. The techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet-defining due to Cornu\'ejols and Margot [On the facets of mixed integer programs with two integer variables and two constraints, Mathematical Programming 120 (2009), 429--456] and obtain polynomial complexity results about the mixed integer hull.Comment: 39 pages; made self-contained by merging material from arXiv:1107.5068v

Note on the Complexity of the Mixed-Integer Hull of a Polyhedron

We study the complexity of computing the mixed-integer hull $\operatorname{conv}(P\cap\mathbb{Z}^n\times\mathbb{R}^d)$ of a polyhedron $P$. Given an inequality description, with one integer variable, the mixed-integer hull can have exponentially many vertices and facets in $d$. For $n,d$ fixed, we give an algorithm to find the mixed integer hull in polynomial time. Given $P=\operatorname{conv}(V)$ and $n$ fixed, we compute a vertex description of the mixed-integer hull in polynomial time and give bounds on the number of vertices of the mixed integer hull

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for the Gomory-Johnson model, in particular for testing extremality of piecewise linear functions whose breakpoints are rational numbers with huge denominators.Comment: 67 pages, 21 figures; v2: changes to sections 10.2-10.3, improved figures; v3: additional figures and minor updates, add reference to IPCO abstract. CC-BY-S

Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in $\mathbb{R}^2$ can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials. Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in $\mathbb{R}^2$ is solvable in polynomial time. We show that this holds for polynomials that can be translated into homogeneous polynomials, even when the translation vector is unknown. We demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input, thus requiring a compact representation of solutions for a general polynomial time algorithm for the unbounded case

SPICE Module for the Satellite Orbit Analysis Program (SOAP)

A SPICE module for the Satellite Orbit Analysis Program (SOAP) precisely represents complex motion and maneuvers in an interactive, 3D animated environment with support for user-defined quantitative outputs. (SPICE stands for Spacecraft, Planet, Instrument, Camera-matrix, and Events). This module enables the SOAP software to exploit NASA mission ephemeris represented in the JPL Ancillary Information Facility (NAIF) SPICE formats. Ephemeris types supported include position, velocity, and orientation for spacecraft and planetary bodies including the Sun, planets, natural satellites, comets, and asteroids. Entire missions can now be imported into SOAP for 3D visualization, playback, and analysis. The SOAP analysis and display features can now leverage detailed mission files to offer the analyst both a numerically correct and aesthetically pleasing combination of results that can be varied to study many hypothetical scenarios. The software provides a modeling and simulation environment that can encompass a broad variety of problems using orbital prediction. For example, ground coverage analysis, communications analysis, power and thermal analysis, and 3D visualization that provide the user with insight into complex geometric relations are included. The SOAP SPICE module allows distributed science and engineering teams to share common mission models of known pedigree, which greatly reduces duplication of effort and the potential for error. The use of the software spans all phases of the space system lifecycle, from the study of future concepts to operations and anomaly analysis. It allows SOAP software to correctly position and orient all of the principal bodies of the Solar System within a single simulation session along with multiple spacecraft trajectories and the orientation of mission payloads. In addition to the 3D visualization, the user can define numeric variables and x-y plots to quantitatively assess metrics of interest

Infrared astronomy

The decade of 1990's presents an opportunity to address fundamental astrophysical issues through observations at IR wavelengths made possible by technological and scientific advances during the last decade. The major elements of recommended program are: the Space Infrared Telescope Facility (SIRTF), the Stratospheric Observatory For Infrared Astronomy (SOFIA) and the IR Optimized 8-m Telescope (IRO), a detector and instrumentation program, the SubMilliMeter Mission (SMMM), the 2 Microns All Sky Survey (2MASS), a sound infrastructure, and technology development programs. Also presented are: perspective, science opportunities, technical overview, project recommendations, future directions, and infrastructure

4. The School Develops

Between 1947 and 1953, when M.P. Catherwood left the deanship to become New York’s industrial commissioner, the ILR School developed into a full fledged enterprise. These pages attempt to capture some of the excitement of this period of the school’s history, which was characterized by vigor, growth, and innovation. Includes: Alumni Recall Their Lives as Students; The Faculty Were Giants; Alice Cook: Lifelong Scholar, Consummate Teacher; Frances Perkins; Visits and Visitors; Tenth Anniversary: Reflection and Change; The Emergence of Departments at ILR; Development of International Programs and Outreach

Resolved Magnetic Field Mapping of a Molecular Cloud Using GPIPS

We present the first resolved map of plane-of-sky magnetic field strength for a quiescent molecular cloud. GRSMC 45.60+0.30 subtends 40 x 10 pc at a distance of 1.88 kpc, masses 16,000 M_sun, and exhibits no star formation. Near-infrared background starlight polarizations were obtained for the Galactic Plane Infrared Polarization Survey using the 1.8 m Perkins telescope and the Mimir instrument. The cloud area of 0.78 deg2 contains 2684 significant starlight polarizations for Two Micron All Sky Survey matched stars brighter than 12.5 mag in the H band. Polarizations are generally aligned with the cloud's major axis, showing an average position angle dispersion of 15 \pm 2{\deg} and polarization of 1.8 \pm 0.6%. The polarizations were combined with Galactic Ring Survey 13CO spectroscopy and the Chandrasekhar-Fermi method to estimate plane-of-sky magnetic field strengths, with an angular resolution of 100 arcsec. The average plane-of-sky magnetic field strength across the cloud is 5.40 \pm 0.04 {\mu}G. The magnetic field strength map exhibits seven enhancements or "magnetic cores." These cores show an average magnetic field strength of 8.3 \pm 0.9 {\mu}G, radius of 1.2 \pm 0.2 pc, intercore spacing of 5.7 \pm 0.9 pc, and exclusively subcritical mass-to-flux ratios, implying their magnetic fields continue to suppress star formation. The magnetic field strength shows a power-law dependence on gas volume density, with slope 0.75 \pm 0.02 for n_{H_2} >=10 cm-3. This power-law index is identical to those in studies at higher densities, but disagrees with predictions for the densities probed here.Comment: 11 pages, 15 figures, published in ApJ (2012, 755, 130