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

Existence of Probability Measures With Given Marginals

We show that if f is a probability density on Rn wrt Lebesgue measure (or any absolutely continuous measure) and 0 ≤ f ≤ 1, then there is another density g with only the values 0 and 1 and with the same (n−1)-dimensional marginals in any finite number of directions. This sharpens, unifies and extends the results of Lorentz and of Kellerer. Given a pair of independent random variables 0 ≤ X, Y ≤ 1, we further study functions 0 ≤ ϕ ≤ 1 such that Z = ϕ (X,Y) satisfies E(Z|X) = X and E(Z|Y) = Y. If there is a solution then there also is a nondecreasing solution ϕ(x,y). These results are applied to tomography and baseball

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

Inequalities and Positive-Definite Functions Arising From a Problem in Multidimensional Scaling

We solve the following variational problem: Find the maximum of E ∥ X−Y ∥ subject to E ∥ X ∥2 ≤ 1, where X and Y are i.i.d. random n-vectors, and ∥⋅∥ is the usual Euclidean norm on Rn. This problem arose from an investigation into multidimensional scaling, a data analytic method for visualizing proximity data. We show that the optimal X is unique and is (1) uniform on the surface of the unit sphere, for dimensions n ≥ 3, (2) circularly symmetric with a scaled version of the radial density ρ/(1−ρ2)1/2, 0 ≤ ρ ≤1, for n=2, and (3) uniform on an interval centered at the origin, for n=1 (Plackett\u27s theorem). By proving spherical symmetry of the solution, a reduction to a radial problem is achieved. The solution is then found using the Wiener-Hopf technique for (real) n \u3c 3. The results are reminiscent of classical potential theory, but they cannot be reduced to it. Along the way, we obtain results of independent interest: for any i.i.d. random n-vectors X and Y,E ∥ X−Y ∥ ≤ E ∥ X+Y ∥. Further, the kernel Kp, β(x,y) = ∥ x+y ∥βp− ∥x−y∥βp, x, y∈Rn and ∥ x ∥ p=(∑|xi|p)1/p, is positive-definite, that is, it is the covariance of a random field, Kp,β(x,y) = E [ Z(x)Z(y) ] for some real-valued random process Z(x), for 1 ≤ p ≤ 2 and 0 \u3c β ≤ p ≤ 2 (but not for β \u3ep or p\u3e2 in general). Although this is an easy consequence of known results, it appears to be new in a strict sense. In the radial problem, the average distance D(r1,r2) between two spheres of radii r1 and r2 is used as a kernel. We derive properties of D(r1,r2), including nonnegative definiteness on signed measures of zero integral

Review of measures of worksite environmental and policy supports for physical activity and healthy eating

INTRODUCTION: Obesity prevention strategies are needed that target multiple settings, including the worksite. The objective of this study was to assess the state of science concerning available measures of worksite environmental and policy supports for physical activity (PA) and healthy eating (HE). METHODS: We searched multiple databases for instruments used to assess worksite environments and policies. Two commonly cited instruments developed by state public health departments were also included. Studies that were published from 1991 through 2013 in peer-reviewed publications and gray literature that discussed the development or use of these instruments were analyzed. Instrument administration mode and measurement properties were documented. Items were classified by general health topic, 5 domains of general worksite strategy, and 19 subdomains of worksite strategy specific to PA or HE. Characteristics of worksite measures were described including measurement properties, length, and administration mode, as well as frequencies of items by domain and subdomain. RESULTS: Seventeen instruments met inclusion criteria (9 employee surveys, 5 manager surveys, 1 observational assessment, and 2 studies that used multiple administration modes). Fourteen instruments included reliability testing. More items were related to PA than HE. Most instruments (n = 10) lacked items in the internal social environment domain. The most common PA subdomains were exercise facilities and lockers/showers; the most common HE subdomain was healthy options/vending. CONCLUSION: This review highlights gaps in measurement of the worksite social environment. The findings provide a useful resource for researchers and practitioners and should inform future instrument development

Hormonal responses to cholinergic input are different in humans with and without type 2 diabetes mellitus

<div><p>Peripheral muscarinic acetylcholine receptors regulate insulin and glucagon release in rodents but their importance for similar roles in humans is unclear. Bethanechol, an acetylcholine analogue that does not cross the blood-brain barrier, was used to examine the role of peripheral muscarinic signaling on glucose homeostasis in humans with normal glucose tolerance (NGT; n = 10), impaired glucose tolerance (IGT; n = 11), and type 2 diabetes mellitus (T2DM; n = 9). Subjects received four liquid meal tolerance tests, each with a different dose of oral bethanechol (0, 50, 100, or 150 mg) given 60 min before a meal containing acetaminophen. Plasma pancreatic polypeptide (PP), glucose-dependent insulinotropic polypeptide (GIP), glucagon-like peptide-1 (GLP-1), glucose, glucagon, C-peptide, and acetaminophen concentrations were measured. Insulin secretion rates (ISRs) were calculated from C-peptide levels. Acetaminophen and PP concentrations were surrogate markers for gastric emptying and cholinergic input to islets. The 150 mg dose of bethanechol increased the PP response 2-fold only in the IGT group, amplified GLP-1 release in the IGT and T2DM groups, and augmented the GIP response only in the NGT group. However, bethanechol did not alter ISRs or plasma glucose, glucagon, or acetaminophen concentrations in any group. Prior studies showed infusion of xenin-25, an intestinal peptide, delays gastric emptying and reduces GLP-1 release but not ISRs when normalized to plasma glucose levels. Analysis of archived plasma samples from this study showed xenin-25 amplified postprandial PP responses ~4-fold in subjects with NGT, IGT, and T2DM. Thus, increasing postprandial cholinergic input to islets augments insulin secretion in mice but not humans.</p><p><b><i>Trial Registration</i>:</b> ClinicalTrials.gov <a href="https://clinicaltrials.gov/ct2/show/NCT01434901?term=NCT01434901&rank=1" target="_blank">NCT01434901</a></p></div