Splitting Proofs for Interpolation

We study interpolant extraction from local first-order refutations. We present a new theoretical perspective on interpolation based on clearly separating the condition on logical strength of the formula from the requirement on the com- mon signature. This allows us to highlight the space of all interpolants that can be extracted from a refutation as a space of simple choices on how to split the refuta- tion into two parts. We use this new insight to develop an algorithm for extracting interpolants which are linear in the size of the input refutation and can be further optimized using metrics such as number of non-logical symbols or quantifiers. We implemented the new algorithm in first-order theorem prover VAMPIRE and evaluated it on a large number of examples coming from the first-order proving community. Our experiments give practical evidence that our work improves the state-of-the-art in first-order interpolation.Comment: 26th Conference on Automated Deduction, 201

The Gap Function Phi(k,w) for a Two-leg t-J Ladder and the Pairing Interaction

The gap function phi(k,omega), determined from a Lanczos calculation for a doped 2-leg t-J ladder, is used to provide insight into the spatial and temporal structure of the pairing interaction. It implies that this interaction is a local near-neighbor coupling which is retarded. The onset frequency of the interaction is set by the energy of an S=1 magnon-hole-pair and it is spread out over a frequency region of order the bandwith

Star cluster ecology IVa: Dissection of an open star cluster---photometry

The evolution of star clusters is studied using N-body simulations in which the evolution of single stars and binaries are taken self-consistently into account. Initial conditions are chosen to represent relatively young Galactic open clusters, such as the Pleiades, Praesepe and the Hyades. The calculations include a realistic mass function, primordial binaries and the external potential of the parent Galaxy. Our model clusters are generally significantly flattened in the Galactic tidal field, and dissolve before deep core collapse occurs. The binary fraction decreases initially due to the destruction of soft binaries, but increases later because lower mass single stars escape more easily than the more massive binaries. At late times, the cluster core is quite rich in giants and white dwarfs. There is no evidence for preferential evaporation of old white dwarfs, on the contrary the formed white dwarfs are likely to remain in the cluster. Stars tend to escape from the cluster through the first and second Lagrange points, in the direction of and away from the Galactic center. Mass segregation manifests itself in our models well within an initial relaxation time. As expected, giants and white dwarfs are much more strongly affected by mass segregation than main-sequence stars. Open clusters are dynamically rather inactive. However, the combined effect of stellar mass loss and evaporation of stars from the cluster potential drives its dissolution on a much shorter timescale than if these effects are neglected. The often-used argument that a star cluster is barely older than its relaxation time and therefore cannot be dynamically evolved is clearly in error for the majority of star clusters.Comment: reduced abstract, 33 pages (three separate color .jpg figures), submitted to MNRA

Binary--single-star scattering VI. Automatic Determination of Interaction Cross Sections

Scattering encounters between binaries and single stars play a central role in determining the dynamical evolution of a star cluster. In addition, three-body scattering can give rise to many interesting exceptional objects: merging can produce blue stragglers; exchange can produce binaries containing millisecond pulsars in environments quite different from those in which the pulsars were spun up; various types of X-ray binaries can be formed, and their activity can be either shut off or triggered as a result of triple interactions. To date, all published results on three-body scattering have relied on human guidance for determining the correct parameter range for the envelope within which to perform Monte--Carlo scattering experiments. In this paper, we describe the first fully automatic determination of cross sections and reaction rates for binary--single-star scattering. Rather than relying on human inspection of pilot calculations, we have constructed a feedback system that ensures near-optimal coverage of parameter space while guaranteeing completeness. We illustrate our approach with a particular example, in which we describe the results of a three-body encounter between three main-sequence stars of different masses. We provide total cross sections, as well as branching ratios for the various different types of two-body mergers, three-body mergers, and exchanges, both non-resonant and resonant. The companion paper in this series, Paper VII, provides a full survey of unequal-mass three-body scattering for hard binaries in the point-mass limit.Comment: 20 pages, TeX + 5 ps-figures, to appear in Ap