The Radon Transform on $\mathbbZ_n^k$

DeDeo, Michelle; Velasquez, Elinor

doi:10.1137/s0895480103430764

Cited by 4 publications

(2 citation statements)

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“…Discovering whether or not the dark energy is a cosmological constant is the primary goal of the study of dark energy. If dark energy is shown not to be a cos-mological constant, then the next important issues are whether or not w < −1, which can be theoretically problematic [25] (however, see [26,27,28,29,30] for an alternative viewpoint), and whether or not w is changing with time. While a quintessence with constant w ≈ −1 is very difficult to distinguish quantitatively from a cosmological constant, the qualitative difference for fundamental physics is enormous, so it is critical that the maximum effort be made to reduce the uncertainty in |w +1| and its time derivative.…”

Section: Introductionmentioning

confidence: 99%

Dynamical dark energy: Current constraints and forecasts

2005

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We consider how well the dark energy equation of state $w$ as a function of red shift $z$ will be measured using current and anticipated experiments. We use a procedure which takes fair account of the uncertainties in the functional dependence of $w$ on $z$, as well as the parameter degeneracies, and avoids the use of strong prior constraints. We apply the procedure to current data from WMAP, SDSS, and the supernova searches, and obtain results that are consistent with other analyses using different combinations of data sets. The effects of systematic experimental errors and variations in the analysis technique are discussed. Next, we use the same procedure to forecast the dark energy constraints achieveable by the end of the decade, assuming 8 years of WMAP data and realistic projections for ground-based measurements of supernovae and weak lensing. We find the $2 \sigma$ constraints on the current value of $w$ to be $\Delta w_0 (2 \sigma) = 0.20$, and on $dw/dz$ (between $z=0$ and $z=1$) to be $\Delta w_1 (2 \sigma)=0.37$. Finally, we compare these limits to other projections in the literature. Most show only a modest improvement; others show a more substantial improvement, but there are serious concerns about systematics. The remaining uncertainty still allows a significant span of competing dark energy models. Most likely, new kinds of measurements, or experiments more sophisticated than those currently planned, are needed to reveal the true nature of dark energy.Comment: 24 pages, 20 figures. Added SN systematic uncertainties, extended discussio

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Section: Introductionmentioning

confidence: 99%

Dynamical dark energy: Current constraints and forecasts

2005

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“…When X is S n , the symmetric group, and Y is unit balls in the Cayley metric, the transform is one to one if and only if n is in {1, 2, 4, 5, 6, 8, 10, 12}. Further work on inversion formulas for functions on finite symmetric spaces is found in Velasquez [47] and for functions on the torus Z k n in Dedeo and Velasquez [13]. Fill [24] discusses invertibility when the Radon transform of f at x averages over a set of translates of f (x) which has applications to directional data and time series.…”

Section: Uniqueness Of Radon Transformsmentioning

confidence: 99%

Projection pursuit for discrete data

Diaconis¹,

Salzman²

2008

Institute of Mathematical Statistics Collections

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This paper develops projection pursuit for discrete data using the discrete Radon transform. Discrete projection pursuit is presented as an exploratory method for finding informative low dimensional views of data such as binary vectors, rankings, phylogenetic trees or graphs. We show that for most data sets, most projections are close to uniform. Thus, informative summaries are ones deviating from uniformity. Syllabic data from several of Plato's great works is used to illustrate the methods. Along with some basic distribution theory, an automated procedure for computing informative projections is introduced.Comment: Published in at http://dx.doi.org/10.1214/193940307000000482 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

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Cubes and the Radon Transform

Stanley

2013

Undergraduate Texts in Mathematics

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The Radon Transform on $\mathbbZ_n^k$

Cited by 4 publications

References 4 publications

Dynamical dark energy: Current constraints and forecasts

Dynamical dark energy: Current constraints and forecasts

Projection pursuit for discrete data

Cubes and the Radon Transform

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