A primal Barvinok algorithm based on irrational decompositions

We introduce variants of Barvinok's algorithm for counting lattice points in polyhedra. The new algorithms are based on irrational signed decomposition in the primal space and the construction of rational generating functions for cones with low index. We give computational results that show that the new algorithms are faster than the existing algorithms by a large factor.Comment: v3: New all-primal algorithm. v4: Extended introduction, updated computational results. To appear in SIAM Journal on Discrete Mathematic

Unique Minimal Liftings for Simplicial Polytopes

For a minimal inequality derived from a maximal lattice-free simplicial polytope in $\R^n$, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers $\R^n$. We then use this characterization to show that a minimal inequality derived from a maximal lattice-free simplex in $\R^n$ with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points.Comment: 15 page

Angle-resolved Auger spectrum of the N₂ molecule

Angle-resolved Auger electron spectra of N2 have been measured with good statistics at photon energies corresponding to the π* resonance and the σ* shape resonance, below and above the N 1s threshold, respectively. Angular anisotropy is observed in both cases, but disappears as expected far above threshold. Satellite Auger transitions also show some angular anisotropy close to the N 1s threshold. This is attributed to the creation and decay of conjugate shakeup initial states, which have non-ground-state symmetry

Software for cut-generating functions in the Gomory--Johnson model and beyond

We present software for investigations with cut generating functions in the Gomory-Johnson model and extensions, implemented in the computer algebra system SageMath.Comment: 8 pages, 3 figures; to appear in Proc. International Congress on Mathematical Software 201

Variation of Cross-Section Enhancement in Decay Spectra of CO under Resonant Raman Conditions

We have measured participator and spectator decay at several photon energies within the range of the lifetime-broadened first vibrational component of the C 1s→π* resonance in CO. From the branching ratios it is evident that the resonant enhancement is different for single-hole and two-hole–one-electron states: The maximum in the resonant intensity peaks at different photon energies. It now becomes necessary to calculate energy-dependent transition matrix elements within the lifetime-broadening range

Core Hole Double-Excitation and Atomiclike Auger Decay in N₂

Core hole decay spectra of the free N2 molecule show evidence for hitherto unobserved molecular resonances both below and above the K-shell photoionization threshold. Based on earlier calculations they are assigned to doubly excited neutral states which could not be seen below threshold in recent high resolution absorption spectra because of the more intense core-to-Rydberg excitations. By calculating the Auger spectrum of core-excited nitrogen atoms, we show that the features are atomiclike

Integer Polynomial Optimization in Fixed Dimension

We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope, we show an algorithm to compute lower and upper bounds for the optimal value. For polynomials that are non-negative over the polytope, these sequences of bounds lead to a fully polynomial-time approximation scheme for the optimization problem.Comment: In this revised version we include a stronger complexity bound on our algorithm. Our algorithm is in fact an FPTAS (fully polynomial-time approximation scheme) to maximize a non-negative integer polynomial over the lattice points of a polytop

The structure of the infinite models in integer programming

The infinite models in integer programming can be described as the convex hull of some points or as the intersection of halfspaces derived from valid functions. In this paper we study the relationships between these two descriptions. Our results have implications for corner polyhedra. One consequence is that nonnegative, continuous valid functions suffice to describe corner polyhedra (with or without rational data)

High-resolution C 1s photoelectron spectra of methane

The C 1s partial photoionization cross section and photoelectron angular distribution of methane (CH4) have been measured with high-energy resolution between threshold and 385 eV photon energy. From the analysis of the vibrational fine structure on the C 1s−1 photoelectron line a vibrational energy of 396±2 meV and an equilibrium bond length of 1.039(±0.001) Å for the CH+4 ion have been determined. The lifetime broadening was found to be 83(±10) meV. The weak feature in the photoabsorption cross section just above threshold does not influence the vibrational fine structure in a way typical for a shape resonance. We therefore suggest that it is due to doubly excited states of the type C (1s)−1(Val)−1(Ryd)1a(Ryd)1b, an assignment which is supported by recent Auger decay studies. Measurements of the shakeup structure revealed six satellite lines, one of which increases strongly in intensity at threshold, thus pointing to the existence of a conjugate shakeup process