Linear transformations of monotone functions on the discrete cube

Nathan Keller; Haran Pilpel

doi:10.1016/j.disc.2008.12.018

Linear transformations of monotone functions on the discrete cube

Nathan Keller, Haran Pilpel

Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

For a function f : {0, 1}ⁿ → R and an invertible linear transformation L ∈ G L_n (2), we consider the function L f : {0, 1}ⁿ → R defined by L f (x) = f (L x). We raise two conjectures: First, we conjecture that if f is Boolean and monotone then I (L f) ≥ I (f), where I (f) is the total influence of f. Second, we conjecture that if both f and L (f) are monotone, then f = L (f) (up to a permutation of the coordinates). We prove the second conjecture in the case where L is upper triangular.

Original language	English
Pages (from-to)	4210-4214
Number of pages	5
Journal	Discrete Mathematics
Volume	309
Issue number	12
DOIs	https://doi.org/10.1016/j.disc.2008.12.018
State	Published - 28 Jun 2009
Externally published	Yes

Bibliographical note

Funding Information:
First author is supported by the Adams Fellowship Program of the Israeli Academy of Sciences and Humanities. Second author is supported by the Giora Yoel Yashinsky Memorial Scholarship.

Funding

First author is supported by the Adams Fellowship Program of the Israeli Academy of Sciences and Humanities. Second author is supported by the Giora Yoel Yashinsky Memorial Scholarship.

Funders	Funder number
Giora Yoel Yashinsky Memorial Scholarship
Israeli Academy of Sciences and Humanities

Keywords

Boolean functions
Discrete Fourier analysis
Fourier-Walsh expansion
Influences

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10.1016/j.disc.2008.12.018

Cite this

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KW - Discrete Fourier analysis

KW - Fourier-Walsh expansion

KW - Influences

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Linear transformations of monotone functions on the discrete cube

Abstract

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Keywords

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