Sets Uniquely Determined by Projections on Axes I. Continuous Case

This paper studies sets S in Rn which are uniquely reconstructible from their hyperplane integral projections Pi(xi ;S) = ∬ . . . ∫ΧS ( {x1, . . . ,xi, . . . ,xn) dx1 . . . dxi - 1 dxi + 1 . . .dxn onto the n coordinate axes of Rn. It is shown that any additive set S = {x = (x1, . . .,xn) : ∑i = 1n fi(xi)≧0}, where each fi(xi) is a bounded measurable function, is uniquely reconstructible. In particular, balls are uniquely reconstructible. It is shown that in R2 all uniquely reconstructible sets are additive. For n≧3, Kemperman has shown that there are uniquely reconstructible sets in Rn of bounded measure that are not additive. It is also noted for n≧3 that neither of the properties of being additive and being a set of uniqueness is closed under monotone pointwise limits. A necessary condition for S to be a set of uniqueness is that S contain no bad configuration. A bad configuration is two finite sets of points T1 in Int(S) and T2 in Int(Sc), where Sc=Rn - S, such that T1 and T2 have the same number of points in any hyperplane xi = c for 1≦ i ≦n, and all c ∈ R2. We show that this necessary condition is sufficient for uniqueness for open sets S in R2. The results show that prior information about a density f in R2 to be reconstructed in tomography (namely if f is known to have only values 0 and 1) can sometimes reduce the problem of reconstructing f to knowing only two projections of f. Thus even meager prior information can in principle be of enormous value in tomography

Some Problems in Probabilistic Tomography

Given probability distributions F1 , F2 , . . ., Fk on R and distinct directions θ1, . . ., θk, one may ask whether there is a probability measure μ on R2 such that the marginal of μ in direction θj is Fj, j = 1, . . ., k. For example for k = 3 we ask what the marginal of μ at 45° can be if the x and y marginals are each say standard normal? In probabilistic language, if X and Y are each standard normal with an arbitrary joint distribution, what can the distribution of X + Y or X - Y be? This type of question is familiar to probabilists and is also familiar (except perhaps in that μ is positive) to tomographers, but is difficult to answer in special cases. The set of distributions for Z = X - Y is a convex and compact set, C, which contains the single point mass Z ≡ 0 since X ≡ Y, standard normal, is possible. We show that Z can be 3-valued, Z=0, ±a for any a, each with positive probability, but Z cannot have any (genuine) two-point distribution. Using numerical linear programming we present convincing evidence that Z can be uniform on the interval [-ε, ε] for ε small and give estimates for the largest such ε. The set of all extreme points of C seems impossible to determine explicitly. We also consider the more basic question of finding the extreme measures on the unit square with uniform marginals on both coordinates, and show that not every such measure has a support which has only one point on each horizontal or vertical line, which seems surprising

The long-term effects of invasive signal crayfish (Pacifastacus leniusculus) on instream macroinvertebrate communities

Non-native species represent a significant threat to indigenous biodiversity and ecosystem functioning worldwide. It is widely acknowledged that invasive crayfish species may be instrumental in modifying benthic invertebrate community structure, but there is limited knowledge regarding the temporal and spatial extent of these effects within lotic ecosystems. This study investigates the long term changes to benthic macroinvertebrate community composition following the invasion of signal crayfish, Pacifastacus leniusculus, into English rivers. Data from long-term monitoring sites on 7 rivers invaded by crayfish and 7 rivers where signal crayfish were absent throughout the record (control sites) were used to examine how invertebrate community composition and populations of individual taxa changed as a result of invasion. Following the detection of non-native crayfish, significant shifts in invertebrate community composition were observed at invaded sites compared to control sites. This pattern was strongest during autumn months but was also evident during spring surveys. The observed shifts in community composition following invasion were associated with reductions in the occurrence of ubiquitous Hirudinea species (Glossiphonia complanata and Erpobdella octoculata), Gastropoda (Radix spp.), Ephemeroptera (Caenis spp.), and Trichoptera (Hydropsyche spp.); although variations in specific taxa affected were evident between regions and seasons. Changes in community structure were persistent over time with no evidence of recovery, suggesting that crayfish invasions represent significant perturbations leading to permanent changes in benthic communities. The results provide fundamental knowledge regarding non-native crayfish invasions of lotic ecosystems required for the development of future management strategies

The Hilbertian Tensor Norm and Entangled Two-Prover Games

We study tensor norms over Banach spaces and their relations to quantum information theory, in particular their connection with two-prover games. We consider a version of the Hilbertian tensor norm $\gamma_2$ and its dual $\gamma_2^*$ that allow us to consider games with arbitrary output alphabet sizes. We establish direct-product theorems and prove a generalized Grothendieck inequality for these tensor norms. Furthermore, we investigate the connection between the Hilbertian tensor norm and the set of quantum probability distributions, and show two applications to quantum information theory: firstly, we give an alternative proof of the perfect parallel repetition theorem for entangled XOR games; and secondly, we prove a new upper bound on the ratio between the entangled and the classical value of two-prover games.Comment: 33 pages, some of the results have been obtained independently in arXiv:1007.3043v2, v2: an error in Theorem 4 has been corrected; Section 6 rewritten, v3: completely rewritten in order to improve readability; title changed; references added; published versio

Attacker Control and Impact for Confidentiality and Integrity

Language-based information flow methods offer a principled way to enforce strong security properties, but enforcing noninterference is too inflexible for realistic applications. Security-typed languages have therefore introduced declassification mechanisms for relaxing confidentiality policies, and endorsement mechanisms for relaxing integrity policies. However, a continuing challenge has been to define what security is guaranteed when such mechanisms are used. This paper presents a new semantic framework for expressing security policies for declassification and endorsement in a language-based setting. The key insight is that security can be characterized in terms of the influence that declassification and endorsement allow to the attacker. The new framework introduces two notions of security to describe the influence of the attacker. Attacker control defines what the attacker is able to learn from observable effects of this code; attacker impact captures the attacker's influence on trusted locations. This approach yields novel security conditions for checked endorsements and robust integrity. The framework is flexible enough to recover and to improve on the previously introduced notions of robustness and qualified robustness. Further, the new security conditions can be soundly enforced by a security type system. The applicability and enforcement of the new policies is illustrated through various examples, including data sanitization and authentication