Inversion of multidimensional Mellin transforms

Antipova, I. A.

doi:10.1070/rm2007v062n05abeh004459

Cited by 6 publications

(19 citation statements)

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“…In this section, we compute Mellin amplitudes in minimal models. 30 Minimal models were discovered in [77] and formulas for correlation functions were found in [78] and [79]. These theories provide a context where we can explicitly compute Mellin amplitudes in interacting non-perturbative CFT's.…”

Section: Mellin Amplitudes In Minimal Modelsmentioning

confidence: 98%

“…30) with α = α m,n and inserting two positive screening charges.Its Mellin amplitude can be obtained fom the general formulas (7.27) and(7.29). We can write the Mellin amplitude in the formM (γ 12 , γ 14 ) = C 0 [dξ 12 ] [dξ 15 ] [dξ 24 ] [dξ 34 ] [dξ 35 ]Γ(ξ 12 )Γ(ξ 15 ) (7.31) Γ(−2αα + +γ 12 −ξ 12 )Γ(ξ 24 )Γ(ξ 15 −ξ 24 −ξ 34 +1)Γ −2α 2 + +2αα + +γ 12 −ξ 34 +2 Γ(ξ 34 )Γ(−2αα + −ξ 12 −ξ 15 +ξ 34 −1)Γ(−2αα + +γ 14 −ξ 15 +ξ 24 +ξ 34 −1) Γ −2α 2 + +2αα + +γ 14 +ξ 12 +ξ 24 −ξ 35 +1 Γ 2α 2 + −ξ 15 +ξ 24 −ξ 35 −1 Γ ξ 12 −ξ 34 −ξ 35 +2αα + −2α 2 + +1 Γ(ξ 35 )Γ(−ξ 12 −ξ 24 +ξ 35 +1) Γ −γ 12 −γ 14 +ξ 15 −ξ 24 +ξ 35 +4α 2 + −1 Γ 2α 2 + +ξ 15 Γ(−4αα + −ξ 12 −ξ 15 +ξ 34 −1) Γ(−γ 12 −γ 14 −ξ 24 ) Γ 2α 2 + +ξ 35 Γ −4α 2 + +4αα + +ξ 12 −ξ 34 −ξ 35 +3…”

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“…We use mostly plus signature 30. By minimal models, we mean the 2d CFT's with finite number of Virasoro primaries at c < 1 and with diagonal partition function.…”

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confidence: 99%

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Nonperturbative Mellin amplitudes: existence, properties, applications

Penedones

Silva

Zhiboedov

2020

J. High Energ. Phys.

161

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We argue that nonperturbative CFT correlation functions admit a Mellin amplitude representation. Perturbative Mellin representation readily follows. We discuss the main properties of nonperturbative CFT Mellin amplitudes: subtractions, analyticity, unitarity, Polyakov conditions and polynomial boundedness at infinity. Mellin amplitudes are particularly simple for large N CFTs and 2D rational CFTs. We discuss these examples to illustrate our general discussion. We consider subtracted dispersion relations for Mellin amplitudes and use them to derive bootstrap bounds on CFTs. We combine crossing, dispersion relations and Polyakov conditions to write down a set of extremal functionals that act on the OPE data. We check these functionals using the known 3d Ising model OPE data and other known bootstrap constraints. We then apply them to holographic theories.

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Section: Mellin Amplitudes In Minimal Modelsmentioning

confidence: 98%

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Nonperturbative Mellin amplitudes: existence, properties, applications

Penedones

Silva

Zhiboedov

2020

J. High Energ. Phys.

161

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“…In analogy to the multivariate extensions to Fourier and Laplace transforms one can define a multivariate Mellin transform [40]. For x ∈ R D we define the multivariate Mellin transform as the integral transform…”

Section: Multivariate Mellin Transformmentioning

confidence: 99%

A Natural Representation of Functions for Exact Learning

Irwin¹

2021

Preprint

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We present a collection of mathematical tools and emphasise a fundamental representation of analytic functions. Connecting these concepts leads to a framework for `exact learning', where an unknown numeric distribution could in principle be assigned an exact mathematical description. This is a new perspective on machine learning with potential applications in all domains of the mathematical sciences and the generalised representations presented here have not yet been widely considered in the context of machine learning and data analysis. The moments of a multivariate function or distribution are extracted using a Mellin transform and the generalised form of the coefficients is trained assuming a highly generalised Mellin-Barnes integral representation. The functions use many fewer parameters than contemporary machine learning methods and any implementation that connects these concepts successfully will likely carry across to non-exact problems and provide approximate solutions. We compare the equations for the exact learning method with those for a neural network which leads to a new perspective on understanding what a neural network may be learning and how to interpret the parameters of those networks.

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“…For a proof of this theorem see [21,Appendix 4,. A treatment of the multidimensional case can also be found for example in [3].…”

Section: Mellin-barnes Integralsmentioning

confidence: 99%