A Panorama of Positivity. I: Dimension Free

Belton, Alexander C. R.; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai

doi:10.1007/978-3-030-14640-5_5

Cited by 14 publications

(12 citation statements)

References 122 publications

(215 reference statements)

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“…22 The Grunsky inequalities feature in other fascinating areas of mathematical physics as recently discussed in [40]. Positivity in mathematics via moment problems, as we will discuss in this section, appears in several areas; for a recent survey, see [41].…”

Section: Toeplitz Determinant Conditionsmentioning

confidence: 97%

QFT, EFT and GFT

Raman

Sinha

2021

J. High Energ. Phys.

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We explore the correspondence between geometric function theory (GFT) and quantum field theory (QFT). The crossing symmetric dispersion relation provides the necessary tool to examine the connection between GFT, QFT, and effective field theories (EFTs), enabling us to connect with the crossing-symmetric EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of Typically Real functions are summarized and shown to be of enormous use in bounding Wilson coefficients in the context of 2-2 scattering. We prove that two-sided bounds on Wilson coefficients are guaranteed to exist quite generally for the fully crossing symmetric situation. Numerical implementation of the GFT constraints (Bieberbach-Rogosinski inequalities) is straightforward and allows a systematic exploration. A comparison of our findings obtained using GFT techniques and other results in the literature is made. We study both the three-channel as well as the two-channel crossing-symmetric cases, the latter having some crucial differences. We also consider bound state poles as well as massless poles in EFTs. Finally, we consider nonlinear constraints arising from the positivity of certain Toeplitz determinants, which occur in the trigonometric moment problem.

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Section: Toeplitz Determinant Conditionsmentioning

confidence: 97%

QFT, EFT and GFT

Raman

Sinha

2021

J. High Energ. Phys.

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“…Much has been made of the Schoenberg theorem in recent years, see [1,2] for an extensive survey, including several multi-variable generalizations [3]. Our present goal will be to characterize products which Date: July 11, 2019.…”

Section: The Classical Casementioning

confidence: 99%

Noncommutative Schur-type products and their Schoenberg theorem

Pascoe

2019

Preprint

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Schoenberg showed that a function f :is also positive semi-definite must be analytic and have Taylor series coefficients nonnegative at the origin. The Schoenberg theorem is essentially a theorem about the functional calculus arising from the Schur product, the entrywise product of matrices. Two important properties of the Schur product are that the product of two rank one matrices is rank one, and the product of two positive semi-definite matrices is positive semi-definite. We classify all products which satisfy these two properties and show that these generalized Schur products satisfy a Schoenberg type theorem.

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“…These two classes of kernels will be denoted by CN D q (X) and SCN D q (X), respectively. Examples of kernels in CN D 1 (X) and SCN D 1 (X) can be found in [4] while connections between the classes P D 1 (X) and CN D 1 (X) are described in [3,4,10].…”

Section: Introductionmentioning

confidence: 99%

Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians

Menegatto

Oliveira

2021

Constructive Mathematical Analysis

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We introduce a method to construct general multivariate positive definite kernels on a nonempty set XX that employs a prescribed bounded completely monotone function and special multivariate functions on XX. The method is consistent with a generalized version of Aitken's integral formula for Gaussians. In the case in which XX is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation. In addition, it can be interpreted as an abstract multivariate version of the well-established Gneiting's model for constructing space-time covariances commonly highly cited in the literature. Many parametric models discussed in statistics can be interpreted as particular cases of the method.

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A Panorama of Positivity. I: Dimension Free

Cited by 14 publications

References 122 publications

QFT, EFT and GFT

QFT, EFT and GFT

Noncommutative Schur-type products and their Schoenberg theorem

Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians

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