Limits of Boolean Functions on $\mathbb{F}_p^n$

Hatami, Hamed; Hatami, Pooya; Hirst, James

doi:10.37236/4445

Cited by 7 publications

(22 citation statements)

References 4 publications

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“…In Section 8 we use this fact to rule out certain other natural candidates. In particular, we were interested in clarifying the relation between our results concerning limit domains and the results in [25]. The latter paper studies a convergence notion for sequences of Boolean functions (f n : F n p → {0, 1}) n∈N , for any prime p, and for p = 2 that notion is equivalent to the one studied in this paper (we detail this equivalence in Appendix E).…”

Section: )mentioning

confidence: 92%

“…The latter paper studies a convergence notion for sequences of Boolean functions (f n : F n p → {0, 1}) n∈N , for any prime p, and for p = 2 that notion is equivalent to the one studied in this paper (we detail this equivalence in Appendix E). However, in Subsection 8.1 we prove that the group G ∞ used to define the limit objects in [25] cannot be used as a limit domain in the sense of statement (iii) in Theorem 1.7.…”

Section: )mentioning

confidence: 95%

“…Variants of this notion have been considered previously (e.g. for Boolean functions in [25]); we shall define it in detail in the sequel (in Section 7) but let us briefly outline the notion here. Given a sequence (f n : F n 2 → B) n∈N for B a compact metric space, we can identify any given system of finitely many linear forms over F 2 as a set…”

Section: Introductionmentioning

confidence: 99%

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On $\mathbb{F}_2^ω$-affine-exchangeable probability measures

Candela¹,

González-Sánchez²,

Szegedy³

2022

Preprint

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For any standard Borel space B, let P(B) denote the space of Borel probability measures on B. In relation to a difficult problem of Aldous in exchangeability theory, and in connection with arithmetic combinatorics, Austin raised the question of describing the structure of affine-exchangeable probability measures on product spaces indexed by the vector space F ω 2 , i.e., the measures in P(B F ω 2 ) that are invariant under the coordinate permutations on B F ω 2 induced by all affine automorphisms of F ω 2 . We answer this question by describing the extreme points of the space of such affine-exchangeable measures. We prove that there is a single structure underlying every such measure, namely, a random infinite-dimensional cube (sampled using Haar measure adapted to a specific filtration) on a group that is a countable power of the 2-adic integers. Indeed, every extreme affineexchangeable measure in P(B F ω 2 ) is obtained from a P(B)-valued function on this group, by a vertex-wise composition with this random cube. The consequences of this result include a description of the convex set of affine-exchangeable measures in P(B F ω 2 ) equipped with the vague topology (when B is a compact metric space), showing that this convex set is a Bauer simplex. We also obtain a correspondence between affine-exchangeability and limits of convergent sequences of (compact-metric-space valued) functions on vector spaces F n 2 as n → ∞. Via this correspondence, we establish the above-mentioned group as a general limit domain valid for any such sequence.

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Section: )mentioning

confidence: 92%

Section: )mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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On $\mathbb{F}_2^ω$-affine-exchangeable probability measures

Candela¹,

González-Sánchez²,

Szegedy³

2022

Preprint

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“…An example of the output from one of the oscillatory time sweep tests performed in the gelling agent content (30%) is illustrated in Figure 2. In the test, G ′ and G ″ were measured versus time and their intersection point, where both are increasing, was taken as the gelation time 32 …”

Section: Preparation Of Oil‐based Gelmentioning

confidence: 99%

Experimental study on an oil‐based polymer gel for lost circulation control in high‐temperature fractured formation

Qiu

Bai

Sun

et al. 2021

J of Applied Polymer Sci

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Aiming at the lack of plugging agent for oil‐based drilling fluid leakage in fractured formation, an oil‐based gel‐plugging agent with high structural strength and strong thermal stability was synthesized based on monomer polymerization and crosslinking. According to the preparation experiment, when the total concentration of methyl methacrylate (MMA) and butyl acrylate (BA) is 30% and the ratio is 6:4, the oil‐based gel solution can be completely gelatinized from 80 to 200°C, which has high gel strength and controllable gelation time. The effects of total concentration of BA and MMA, the ratio of BA and MMA, synthesis temperature, oil‐based drilling fluid invasion, shear effect, and fiber concentration on gelling performance were systematically studied. Fourier transform infrared molecular structure analysis showed that the oil‐based gel contained benzene ring group and ester characteristic group. From the rheological properties of oil‐based gel with 30% gelling agent at 160°C, it could be seen that the elastic modulus G′ of 10 Hz is 232 kPa, which is much larger than the storage modulus G″ is 99 kPa, which reflects the solid properties and has good elasticity and high strength. The displacement experiment of sand filling pipe and dynamic and static plugging experiment of high temperature and high pressure shows that the oil‐based gel has high bearing capacity in fractures and sand filling pipe. The bearing pressure is 3.01 MPa in 6 mm fractures, and the bearing pressure gradient is 0.602 MPa/cm.

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“…Bhattacharyya et al [17] used the same proof technique to replace the condition that each L i is affine, with the condition that all the coefficients of L i are in {0, 1}. Finally, in the work of Hatami et al [46] the joint distribution of the matrix A B,L was fully characterized for nonclassical polynomials without any extra assumptions on the linear forms. Instead, it requires that the polynomials are homogeneous.…”

Section: Strong Equidistribution Of Regular Factorsmentioning

confidence: 99%

Higher-order Fourier Analysis and Applications

Hatami¹,

Hatami²,

Lovett³

2019

Self Cite

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Fourier analysis has been extremely useful in many areas of mathematics. In the last several decades, it has been used extensively in theoretical computer science. Higher-order Fourier analysis is an extension of the classical Fourier analysis, where one allows to generalize the "linear phases" to higher degree polynomials. It has emerged from the seminal proof of Gowers of Szemerédi's theorem with improved quantitative bounds, and has been developed since, chiefly by the number theory community. In parallel, it has found applications also in theoretical computer science, mostly in algebraic property testing, coding theory and complexity theory.The purpose of this book is to lay the foundations of higherorder Fourier analysis, aimed towards applications in theoretical computer science with a focus on algebraic property testing.

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Limits of Boolean Functions on $\mathbb{F}_p^n$

Cited by 7 publications

References 4 publications

On $\mathbb{F}_2^ω$-affine-exchangeable probability measures

On $\mathbb{F}_2^ω$-affine-exchangeable probability measures

Experimental study on an oil‐based polymer gel for lost circulation control in high‐temperature fractured formation

Higher-order Fourier Analysis and Applications

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