Competitive Boolean Function Evaluation: Beyond Monotonicity, and the Symmetric Case

We study the extremal competitive ratio of Boolean function evaluation. We provide the first non-trivial lower and upper bounds for classes of Boolean functions which are not included in the class of monotone Boolean functions. For the particular case of symmetric functions our bounds are matching and we exactly characterize the best possible competitiveness achievable by a deterministic algorithm. Our upper bound is obtained by a simple polynomial time algorithm.Comment: 15 pages, 1 figure, to appear in Discrete Applied Mathematic

Absorption by Branes and Schwinger Terms in the World Volume Theory

We study how coincident Dirichlet 3-branes absorb incident gravitons polarized along their world volume. We show that the absorption cross-section is determined by the central term in the correlator of two stress-energy tensors. The existence of a non-renormalization theorem for this central charge in four-dimensional N=4 supersymmetric Yang-Mills theories shows that the leading term at low energies in the absorption cross-section is not renormalized. This guarantees that the agreement of the cross-section with semiclassical supergravity, found in earlier work, survives all loop corrections. The connection between absorption of gravitons polarized along the brane and Schwinger terms in the stress-energy correlators of the world volume theory holds in general. We explore this connection to deduce some properties of the stress-energy tensor OPE's for 2-branes and 5-branes in 11 dimensions, as well as for 5-branes in 10 dimensions.Comment: 13 pages, latex, minor changes, version to appear in Physics Letters

Superconformal Ward Identities and their Solution

Superconformal Ward identities are derived for the the four point functions of chiral primary BPS operators for $\N=2,4$ superconformal symmetry in four dimensions. Manipulations of arbitrary tensorial fields are simplified by introducing a null vector so that the four point functions depend on two internal $R$-symmetry invariants as well as two conformal invariants. The solutions of these identities are interpreted in terms of the operator product expansion and are shown to accommodate long supermultiplets with free scale dimensions and also short and semi-short multiplets with protected dimensions. The decomposition into $R$-symmetry representations is achieved by an expansion in terms of two variable harmonic polynomials which can be expressed also in terms of Legendre polynomials. Crossing symmetry conditions on the four point functions are also discussed.Comment: 73 pages, plain Tex, uses harvmac, version 2, extra reference

All Inequalities for the Relative Entropy

The relative entropy of two n-party quantum states is an important quantity exhibiting, for example, the extent to which the two states are different. The relative entropy of the states formed by reducing two n-party to a smaller number $m$ of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy. Using techniques from convex geometry, we prove that monotonicity under restrictions is the only general inequality satisfied by relative entropies. In doing so we make a connection to secret sharing schemes with general access structures. A suprising outcome is that the structure of allowed relative entropy values of subsets of multiparty states is much simpler than the structure of allowed entropy values. And the structure of allowed relative entropy values (unlike that of entropies) is the same for classical probability distributions and quantum states.Comment: 15 pages, 3 embedded eps figure

Hidden tail chains and recurrence equations for dependence parameters associated with extremes of higher-order Markov chains

We derive some key extremal features for kth order Markov chains, which can be used to understand how the process moves between an extreme state and the body of the process. The chains are studied given that there is an exceedance of a threshold, as the threshold tends to the upper endpoint of the distribution. Unlike previous studies with k>1 we consider processes where standard limit theory describes each extreme event as a single observation without any information about the transition to and from the body of the distribution. The extremal properties of the Markov chain at lags up to k are determined by the kernel of the chain, through a joint initialisation distribution, with the subsequent values determined by the conditional independence structure through a transition behaviour. We study the extremal properties of each of these elements under weak assumptions for broad classes of extremal dependence structures. For chains with k>1, these transitions involve novel functions of the k previous states, in comparison to just the single value, when k=1. This leads to an increase in the complexity of determining the form of this class of functions, their properties and the method of their derivation in applications. We find that it is possible to find an affine normalization, dependent on the threshold excess, such that non-degenerate limiting behaviour of the process is assured for all lags. These normalization functions have an attractive structure that has parallels to the Yule-Walker equations. Furthermore, the limiting process is always linear in the innovations. We illustrate the results with the study of kth order stationary Markov chains based on widely studied families of copula dependence structures.Comment: 35 page