Universality of the superpotential for d = 4 extremal black holes

We provide a general strategy to obtain the superpotential W for both BPS and non-BPS extremal black holes in N=2 four dimensional supergravities based on symmetric spaces. This extends the construction of W in terms of U-duality invariants that was presented in previous work on the $t^3$ model. As an application, we explicitly provide W and the solutions to the related gradient flows for the $st^2$ and stu models. The procedure is shown to hold also for the full N=8 theory. The role of flat directions in moduli space is clarified.Comment: 29 pages. v2: typos corrected, comments and references adde

Transfer Function Synthesis without Quantifier Elimination

Traditionally, transfer functions have been designed manually for each operation in a program, instruction by instruction. In such a setting, a transfer function describes the semantics of a single instruction, detailing how a given abstract input state is mapped to an abstract output state. The net effect of a sequence of instructions, a basic block, can then be calculated by composing the transfer functions of the constituent instructions. However, precision can be improved by applying a single transfer function that captures the semantics of the block as a whole. Since blocks are program-dependent, this approach necessitates automation. There has thus been growing interest in computing transfer functions automatically, most notably using techniques based on quantifier elimination. Although conceptually elegant, quantifier elimination inevitably induces a computational bottleneck, which limits the applicability of these methods to small blocks. This paper contributes a method for calculating transfer functions that finesses quantifier elimination altogether, and can thus be seen as a response to this problem. The practicality of the method is demonstrated by generating transfer functions for input and output states that are described by linear template constraints, which include intervals and octagons.Comment: 37 pages, extended version of ESOP 2011 pape

Invariants of solvable Lie algebras with triangular nilradicals and diagonal nilindependent elements

The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on Cartan's method of moving frames.Comment: 21 pages, enhanced and extended version. Section 2 reviews the method of finding invariants of Lie algebras that was proposed in arXiv:math-ph/0602046 and arXiv:math-ph/0606045. The computation is based on developing a specific technique given in arXiv:0704.0937. Results generalize ones of arXiv:0705.2394 to the case of arbitrary relevant number of nilindependent element

Invariants of Triangular Lie Algebras

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.Comment: LaTeX2e, 16 pages; misprints are corrected, some proofs are extende

What's Decidable About Sequences?

We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the quantifier-free fragment, based on an encoding into the first-order theory of concatenation; the procedure has PSPACE complexity. The quantifier-free fragment of the theory of sequences can express properties such as sortedness and injectivity, as well as Boolean combinations of periodic and arithmetic facts relating the elements of the sequence and their positions (e.g., "for all even i's, the element at position i has value i+3 or 2i"). The resulting expressive power is orthogonal to that of the most expressive decidable logics for arrays. Some examples demonstrate that the fragment is also suitable to reason about sequence-manipulating programs within the standard framework of axiomatic semantics.Comment: Fixed a few lapses in the Mergesort exampl

Group theory factors for Feynman diagrams

We present algorithms for the group independent reduction of group theory factors of Feynman diagrams. We also give formulas and values for a large number of group invariants in which the group theory factors are expressed. This includes formulas for various contractions of symmetric invariant tensors, formulas and algorithms for the computation of characters and generalized Dynkin indices and trace identities. Tables of all Dynkin indices for all exceptional algebras are presented, as well as all trace identities to order equal to the dual Coxeter number. Further results are available through efficient computer algorithms (see http://norma.nikhef.nl/~t58/ and http://norma.nikhef.nl/~t68/ ).Comment: Latex (using axodraw.sty), 47 page