Analytic Functions of Several Complex Variables

Gunning, Robert C.; Rossi, Hugo

doi:10.1090/chel/368

Cited by 193 publications

(227 citation statements)

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“…See, for example, Gunning and Rossi [9]. Since the conjugate complex coordinates z i andz i correspond to x i and y i in a one-to-one manner, each quantity is evaluated in the original real coordinate if necessary.…”

Section: Fisher Metric On the Ar Model Manifoldmentioning

confidence: 99%

Superharmonic priors for autoregressive models

Tanaka

2017

Info. Geo.

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Tanaka and Komaki (Sankhya Ser A Indian Stat Inst 73-A: 2011) proposed superharmonic priors in Bayesian time series analysis as alternative to the famous Jeffreys prior. By definition the existence of superharmonic priors on a specific time series model with finite-dimensional parameter is equivalent to that of positive nonconstant superharmonic functions on the corresponding Riemannian manifold endowed with the Fisher metric. In the autoregressive models, whose Fisher metric and its inverse have quite messy forms, we obtain superharmonic priors in an explicit manner. To derive this result, we developed a systematic way of dealing with symmetric polynomials, which are related to Schur functions.

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Section: Fisher Metric On the Ar Model Manifoldmentioning

confidence: 99%

Superharmonic priors for autoregressive models

Tanaka

2017

Info. Geo.

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“…ǫ in an open neighborhood of ǫ = 0 [35]. We can therefore develop a perturbative series around ǫ = 0.…”

Section: The Harmonic Case: the Squarementioning

confidence: 99%

“…Thus except at most a finite number of points we can apply Weierstrass preparation theorem [35,31] (113) with u(β, ǫ), v(β c , ǫ) units and c k , g k analytic functions of ǫ, vanishing at ǫ 0 .…”

Section: The Harmonic Case: the Squarementioning

confidence: 99%

Accessory parameters for Liouville theory on the torus

Menotti

2012

J. High Energ. Phys.

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We give an implicit equation for the accessory parameter on the torus which is the necessary and sufficient condition to obtain the monodromy of the conformal factor. It is shown that the perturbative series for the accessory parameter in the coupling constant converges in a finite disk and give a rigorous lower bound for the radius of convergence. We work out explicitly the perturbative result to second order in the coupling for the accessory parameter and to third order for the one-point function. Modular invariance is discussed and exploited. At the non perturbative level it is shown that the accessory parameter is a continuous function of the coupling in the whole physical region and that it is analytic except at most a finite number of points. We also prove that the accessory parameter as a function of the modulus of the torus is continuous and real-analytic except at most for a zero measure set. Three soluble cases in which the solution can be expressed in terms of hypergeometric functions are explicitly treated.

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“…In the following holomorphic refer to the complex case and analytic to the real case. For a further reading about complex analytic spaces we suggest [19] while we remit the reader to [16,29] for the theory of real analytic spaces. …”

Section: Preliminaries On Real and Complex Analytic Spacesmentioning

confidence: 99%

On the irreducible components of globally defined semianalytic sets

Fernando

2016

Math. Z.

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In this work we present the concept of amenable C-semianalytic subset of a real analytic manifold M and study the main properties of this type of sets. Amenable C-semianalytic sets can be understood as globally defined semianalytic sets with a neat behavior with respect to Zariski closure. This fact allows us to develop a natural definition of irreducibility and the corresponding theory of irreducible components for amenable Csemianalytic sets. These concepts generalize the parallel ones for: complex algebraic and analytic sets, C-analytic sets, Nash sets and semialgebraic sets.

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Analytic Functions of Several Complex Variables

Cited by 193 publications

References 0 publications

Superharmonic priors for autoregressive models

Superharmonic priors for autoregressive models

Accessory parameters for Liouville theory on the torus

On the irreducible components of globally defined semianalytic sets

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