4,959 research outputs found
Effects of Ionizing Radiation on Solid Rocket Motor Components
Problems which the solid propellant rocket engineer will encounter in designing for long-term storage in a radiation environment are discussed. A summary of present knowledge of the radiation environment is given. Mechanisms of radiation degradation and its effects on tensile properties of propellant binders are discussed qualitatively. Data from a program of irradiation of several propellants is presented. Properties of two of the propellants were changed significantly by doses of the order of 4 x 10(exp 6) rads
On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies
We present a method which shows that in \Eb the Busemann-Petty problem,
concerning central sections of centrally symmetric convex bodies, has a
positive answer. Together with other results, this settles the problem in each
dimension.Comment: 5 page
DEVELOPING COLLABORATION IN RURAL POLICY: LESSONS FROM A STATE RURAL DEVELOPMENT COUNCIL
Community/Rural/Urban Development,
Reverse and dual Loomis-Whitney-type inequalities
Various results are proved giving lower bounds for the th intrinsic volume
, , of a compact convex set in , in
terms of the th intrinsic volumes of its projections on the coordinate
hyperplanes (or its intersections with the coordinate hyperplanes). The bounds
are sharp when and . These are reverse (or dual, respectively)
forms of the Loomis-Whitney inequality and versions of it that apply to
intrinsic volumes. For the intrinsic volume , which corresponds to mean
width, the inequality obtained confirms a conjecture of Betke and McMullen made
in 1983
Operations between sets in geometry
An investigation is launched into the fundamental characteristics of
operations on and between sets, with a focus on compact convex sets and star
sets (compact sets star-shaped with respect to the origin) in -dimensional
Euclidean space . For example, it is proved that if , with three
trivial exceptions, an operation between origin-symmetric compact convex sets
is continuous in the Hausdorff metric, GL(n) covariant, and associative if and
only if it is addition for some . It is also
demonstrated that if , an operation * between compact convex sets is
continuous in the Hausdorff metric, GL(n) covariant, and has the identity
property (i.e., for all compact convex sets , where
denotes the origin) if and only if it is Minkowski addition. Some analogous
results for operations between star sets are obtained. An operation called
-addition is generalized and systematically studied for the first time.
Geometric-analytic formulas that characterize continuous and GL(n)-covariant
operations between compact convex sets in terms of -addition are
established. The term "polynomial volume" is introduced for the property of
operations * between compact convex or star sets that the volume of ,
, is a polynomial in the variables and . It is proved that if
, with three trivial exceptions, an operation between origin-symmetric
compact convex sets is continuous in the Hausdorff metric, GL(n) covariant,
associative, and has polynomial volume if and only if it is Minkowski addition
Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms
We propose strongly consistent algorithms for reconstructing the
characteristic function 1_K of an unknown convex body K in R^n from possibly
noisy measurements of the modulus of its Fourier transform \hat{1_K}. This
represents a complete theoretical solution to the Phase Retrieval Problem for
characteristic functions of convex bodies. The approach is via the closely
related problem of reconstructing K from noisy measurements of its covariogram,
the function giving the volume of the intersection of K with its translates. In
the many known situations in which the covariogram determines a convex body, up
to reflection in the origin and when the position of the body is fixed, our
algorithms use O(k^n) noisy covariogram measurements to construct a convex
polytope P_k that approximates K or its reflection -K in the origin. (By recent
uniqueness results, this applies to all planar convex bodies, all
three-dimensional convex polytopes, and all symmetric and most (in the sense of
Baire category) arbitrary convex bodies in all dimensions.) Two methods are
provided, and both are shown to be strongly consistent, in the sense that,
almost surely, the minimum of the Hausdorff distance between P_k and K or -K
tends to zero as k tends to infinity.Comment: Version accepted on the Journal of the American Mathematical Society.
With respect to version 1 the noise model has been greatly extended and an
appendix has been added, with a discussion of rates of convergence and
implementation issues. 56 pages, 4 figure
Convergence of algorithms for reconstructing convex bodies and directional measures
We investigate algorithms for reconstructing a convex body in from noisy measurements of its support function or its brightness
function in directions . The key idea of these algorithms is
to construct a convex polytope whose support function (or brightness
function) best approximates the given measurements in the directions
(in the least squares sense). The measurement errors are assumed
to be stochastically independent and Gaussian. It is shown that this procedure
is (strongly) consistent, meaning that, almost surely, tends to in
the Hausdorff metric as . Here some mild assumptions on the
sequence of directions are needed. Using results from the theory of
empirical processes, estimates of rates of convergence are derived, which are
first obtained in the metric and then transferred to the Hausdorff
metric. Along the way, a new estimate is obtained for the metric entropy of the
class of origin-symmetric zonoids contained in the unit ball. Similar results
are obtained for the convergence of an algorithm that reconstructs an
approximating measure to the directional measure of a stationary fiber process
from noisy measurements of its rose of intersections in directions
. Here the Dudley and Prohorov metrics are used. The methods are
linked to those employed for the support and brightness function algorithms via
the fact that the rose of intersections is the support function of a projection
body.Comment: Published at http://dx.doi.org/10.1214/009053606000000335 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Characterizing the dual mixed volume via additive functionals
Integral representations are obtained of positive additive functionals on
finite products of the space of continuous functions (or of bounded Borel
functions) on a compact Hausdorff space. These are shown to yield
characterizations of the dual mixed volume, the fundamental concept in the dual
Brunn-Minkowski theory. The characterizations are shown to be best possible in
the sense that none of the assumptions can be omitted. The results obtained are
in the spirit of a similar characterization of the mixed volume in the
classical Brunn-Minkowski theory, obtained recently by Milman and Schneider,
but the methods employed are completely different
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