On All-loop Integrands of Scattering Amplitudes in Planar N=4 SYM

We study the relationship between the momentum twistor MHV vertex expansion of planar amplitudes in N=4 super-Yang-Mills and the all-loop generalization of the BCFW recursion relations. We demonstrate explicitly in several examples that the MHV vertex expressions for tree-level amplitudes and loop integrands satisfy the recursion relations. Furthermore, we introduce a rewriting of the MHV expansion in terms of sums over non-crossing partitions and show that this cyclically invariant formula satisfies the recursion relations for all numbers of legs and all loop orders.Comment: 34 pages, 17 figures; v2: Minor improvements to exposition and discussion, updated references, typos fixe

Multiple-channel generalization of Lellouch-Luscher formula

We generalize the Lellouch-Luscher formula, relating weak matrix elements in finite and infinite volumes, to the case of multiple strongly-coupled decay channels into two scalar particles. This is a necessary first step on the way to a lattice QCD calculation of weak decay rates for processes such as D -> pi pi and D -> KK. We also present a field theoretic derivation of the generalization of Luscher's finite volume quantization condition to multiple two-particle channels. We give fully explicit results for the case of two channels, including a form of the generalized Lellouch-Luscher formula expressed in terms of derivatives of the energies of finite volume states with respect to the box size. Our results hold for arbitrary total momentum and for degenerate or non-degenerate particles.Comment: 16 pages, 2 figures. v3: Added references, clarified relation to and corrected comments about previous work, and minor stylistic improvements. v4: Minor clarifications added, typos fixed, references updated---matches published versio

QRAT+: Generalizing QRAT by a More Powerful QBF Redundancy Property

The QRAT (quantified resolution asymmetric tautology) proof system simulates virtually all inference rules applied in state of the art quantified Boolean formula (QBF) reasoning tools. It consists of rules to rewrite a QBF by adding and deleting clauses and universal literals that have a certain redundancy property. To check for this redundancy property in QRAT, propositional unit propagation (UP) is applied to the quantifier free, i.e., propositional part of the QBF. We generalize the redundancy property in the QRAT system by QBF specific UP (QUP). QUP extends UP by the universal reduction operation to eliminate universal literals from clauses. We apply QUP to an abstraction of the QBF where certain universal quantifiers are converted into existential ones. This way, we obtain a generalization of QRAT we call QRAT+. The redundancy property in QRAT+ based on QUP is more powerful than the one in QRAT based on UP. We report on proof theoretical improvements and experimental results to illustrate the benefits of QRAT+ for QBF preprocessing.Comment: preprint of a paper to be published at IJCAR 2018, LNCS, Springer, including appendi

Lagrange Multipliers and Rayleigh Quotient Iteration in Constrained Type Equations

We generalize the Rayleigh quotient iteration to a class of functions called vector Lagrangians. The convergence theorem we obtained generalizes classical and nonlinear Rayleigh quotient iterations, as well as iterations for tensor eigenpairs and constrained optimization. In the latter case, our generalized Rayleigh quotient is an estimate of the Lagrange multiplier. We discuss two methods of solving the updating equation associated with the iteration. One method leads to a generalization of Riemannian Newton method for embedded manifolds in a Euclidean space while the other leads to a generalization of the classical Rayleigh quotient formula. Applying to tensor eigenpairs, we obtain both an improvements over the state-of-the-art algorithm, and a new quadratically convergent algorithm to compute all complex eigenpairs of sizes typical in applications. We also obtain a Rayleigh-Chebyshev iteration with cubic convergence rate, and give a clear criterion for RQI to have cubic convergence rate, giving a common framework for existing algorithms