Δ\Delta-Algebra and Scattering Amplitudes

In this paper we study an algebra that naturally combines two familiar operations in scattering amplitudes: computations of volumes of polytopes using triangulations and constructions of canonical forms from products of smaller ones. We mainly concentrate on the case of $G(2,n)$ as it controls both general MHV leading singularities and CHY integrands for a variety of theories. This commutative algebra has also appeared in the study of configuration spaces and we called it the $\Delta$-algebra. As a natural application, we generalize the well-known square move. This allows us to generate infinite families of new moves between non-planar on-shell diagrams. We call them sphere moves. Using the $\Delta$-algebra we derive familiar results, such as the KK and BCJ relations, and prove novel formulas for higher-order relations. Finally, we comment on generalizations to $G(k,n)$.Comment: 36+13 page

Point Line Cover: The Easy Kernel is Essentially Tight

The input to the NP-hard Point Line Cover problem (PLC) consists of a set $P$ of $n$ points on the plane and a positive integer $k$, and the question is whether there exists a set of at most $k$ lines which pass through all points in $P$. A simple polynomial-time reduction reduces any input to one with at most $k^2$ points. We show that this is essentially tight under standard assumptions. More precisely, unless the polynomial hierarchy collapses to its third level, there is no polynomial-time algorithm that reduces every instance $(P,k)$ of PLC to an equivalent instance with $O(k^{2-\epsilon})$ points, for any $\epsilon>0$. This answers, in the negative, an open problem posed by Lokshtanov (PhD Thesis, 2009). Our proof uses the machinery for deriving lower bounds on the size of kernels developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients: We first show, by reduction from Vertex Cover, that PLC---conditionally---has no kernel of total size $O(k^{2-\epsilon})$ bits. This does not directly imply the claimed lower bound on the number of points, since the best known polynomial-time encoding of a PLC instance with $n$ points requires $\omega(n^{2})$ bits. To get around this we build on work of Goodman et al. (STOC 1989) and devise an oracle communication protocol of cost $O(n\log n)$ for PLC; its main building block is a bound of $O(n^{O(n)})$ for the order types of $n$ points that are not necessarily in general position, and an explicit algorithm that enumerates all possible order types of n points. This protocol and the lower bound on total size together yield the stated lower bound on the number of points. While a number of essentially tight polynomial lower bounds on total sizes of kernels are known, our result is---to the best of our knowledge---the first to show a nontrivial lower bound for structural/secondary parameters

Scalable RDF Data Compression using X10

The Semantic Web comprises enormous volumes of semi-structured data elements. For interoperability, these elements are represented by long strings. Such representations are not efficient for the purposes of Semantic Web applications that perform computations over large volumes of information. A typical method for alleviating the impact of this problem is through the use of compression methods that produce more compact representations of the data. The use of dictionary encoding for this purpose is particularly prevalent in Semantic Web database systems. However, centralized implementations present performance bottlenecks, giving rise to the need for scalable, efficient distributed encoding schemes. In this paper, we describe an encoding implementation based on the asynchronous partitioned global address space (APGAS) parallel programming model. We evaluate performance on a cluster of up to 384 cores and datasets of up to 11 billion triples (1.9 TB). Compared to the state-of-art MapReduce algorithm, we demonstrate a speedup of 2.6-7.4x and excellent scalability. These results illustrate the strong potential of the APGAS model for efficient implementation of dictionary encoding and contributes to the engineering of larger scale Semantic Web applications

Inferring phylogenetic networks with maximum pseudolikelihood under incomplete lineage sorting

Phylogenetic networks are necessary to represent the tree of life expanded by edges to represent events such as horizontal gene transfers, hybridizations or gene flow. Not all species follow the paradigm of vertical inheritance of their genetic material. While a great deal of research has flourished into the inference of phylogenetic trees, statistical methods to infer phylogenetic networks are still limited and under development. The main disadvantage of existing methods is a lack of scalability. Here, we present a statistical method to infer phylogenetic networks from multi-locus genetic data in a pseudolikelihood framework. Our model accounts for incomplete lineage sorting through the coalescent model, and for horizontal inheritance of genes through reticulation nodes in the network. Computation of the pseudolikelihood is fast and simple, and it avoids the burdensome calculation of the full likelihood which can be intractable with many species. Moreover, estimation at the quartet-level has the added computational benefit that it is easily parallelizable. Simulation studies comparing our method to a full likelihood approach show that our pseudolikelihood approach is much faster without compromising accuracy. We applied our method to reconstruct the evolutionary relationships among swordtails and platyfishes ($Xiphophorus$: Poeciliidae), which is characterized by widespread hybridizations

Ab Initio Calculations of Medium-Mass Nuclei with Explicit Chiral 3N Interactions

We present the first ab initio coupled-cluster calculations of medium-mass nuclei with explicit chiral three-nucleon (3N) interactions. Using a spherical formulation of coupled cluster with singles and doubles excitations including explicit 3N contributions, we study ground states of 16,24-O, 40,48-Ca and 56-Ni. We employ chiral NN plus 3N interactions softened through a similarity renormalization group (SRG) transformation at the three-body level. We investigate the impact of all truncations and quantify the resulting uncertainties---this includes the contributions from triples excitations, the truncation of the set of three-body matrix elements, and the omission of SRG-induced four-body interactions. Furthermore, we assess the quality of a normal-ordering approximation of the 3N interaction beyond light nuclei. Our study points towards the predictive power of chiral Hamiltonians in the medium-mass regime.Comment: 6 pages, 3 figures, 2 table