LII. The accumulation of chance effects and the Gaussian frequency distribution

Goddard, Linda

doi:10.1080/14786444508520925

Cited by 10 publications

(6 citation statements)

References 6 publications

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“…Thus, one way to compute the estimator (6) is to solve the LASSO problem (30) and use the resulting coefficients to construct the rectangular piecewise constant function (6). Note the strong similarity between the two expressions (29) and (32).…”

Section: Computational Feasibilitymentioning

confidence: 99%

“…) so that the first column of A is the vector of ones. Therefore in the lattice design setting, the optimization problems ( 28) and (30) for computing the two estimators f EM and f HK0,V can be rewritten as…”

Section: Special Case: the Equally-spaced Lattice Designmentioning

confidence: 99%

“…, x n and vectors β EM and β HK0,V in R p which are obtained by solving the NNLS problem (28) and the LASSO probimsart-generic ver. 2014/10/16 file: PaperPiecewiseConstant_arxiv21Dec2019.tex date: December 24, 2019 lem (30) respectively. Here I [z j ,1] denotes the indicator of the rectangle [z j , 1] (defined via (16)).…”

Section: Introductionmentioning

confidence: 99%

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Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation

Fang

Guntuboyina

Sen

2019

Preprint

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Section: Computational Feasibilitymentioning

confidence: 99%

Section: Special Case: the Equally-spaced Lattice Designmentioning

confidence: 99%

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Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation

Fang

Guntuboyina

Sen

2019

Preprint

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“…Remark

σ_{k}

obeys a simple asymptotic (see, for example, , or for more terms):

σ_{k} = \sqrt{\frac{6}{k π}} (1 + O false(1 / k false))

k \to \infty

. We may interpret

σ_{k}

combinatorially.…”

Section: Equations In More Than 3 Variablesmentioning

confidence: 99%

Maximising the number of solutions to a linear equation in a set of integers

Aaronson

2019

Bull. London Math. Soc.

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Given a linear equation of the form a1x1 + a2x2 + a3x3 = 0 with integer coefficients ai, we are interested in maximising the number of solutions to this equation in a set S ⊆ Z, for sets S of a given size.We prove that, for any choice of constants a1, a2 and a3, the maximum number of solutions is at least ( 1 12 + o(1))|S| 2 . Furthermore, we show that this is optimal, in the following sense. For any ε > 0, there are choices of a1, a2 and a3, for which any large set S of integers has at most ( 1 12 + ε)|S| 2 solutions. For equations in k 3 variables, we also show an analogous result. Set σ k = ∞ −∞ ( sin πx πx ) k dx. Then, for any choice of constants a1, . . . , a k , there are sets S with at least ( σ k k k−1 + o(1))|S| k−1 solutions to a1x1 + · · · + a k x k = 0. Moreover, there are choices of coefficients a1, . . . , a k for which any large set S must have no more than ( σ k k k−1 + ε)|S| k−1 solutions, for any ε > 0.

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“…The result (cf. [4]) is found by contour integration to be sm x. cos xt dx 2,+(2m 1) m set of polynomial arcs so joined t 2s that all deriwtives are continuous except those of order 2m 1 t 0, 2, 2m.…”

mentioning

confidence: 99%

Evaluation of a Fourier Transform

Mackenzie¹

1967

SIAM Rev.

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LII. The accumulation of chance effects and the Gaussian frequency distribution

Cited by 10 publications

References 6 publications

Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation

Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation

Maximising the number of solutions to a linear equation in a set of integers

Evaluation of a Fourier Transform

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