Global asymptotics of the Meixner polynomials

Wang, X. S.; Wong, R.

doi:10.1142/9789814656054_0059

Cited by 9 publications

(21 citation statements)

References 21 publications

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“…This analysis is very similar to the steepest descent analysis for the Meixner polynomials which was carried out by Wang and Wong [12], although they considered the parameter β in (3.10) to be fixed, while we allow it to grow with k. In this paper we take a different approach and compare the normalizing constants h k with the Meixner normalizing constants h M k , for which we have the exact formulae (3.12). In order to compare them, it is convenient to also introduce the Riemann-Hilbert problem for the Meixner polynomials.…”

Section: Riemann Hilbert Approach: Interpolation Problemmentioning

confidence: 95%

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Six-vertex model with partial domain wall boundary conditions: Ferroelectric phase

Bleher

Liechty

2015

Journal of Mathematical Physics

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Abstract. We obtain an asymptotic formula for the partition function of the six-vertex model with partial domain wall boundary conditions in the ferroelectric phase region. The proof is based on a formula for the partition function involving the determinant of a matrix of mixed Vandermonde/Hankel type. This determinant can be expressed in terms of a system of discrete orthogonal polynomials, which can then be evaluated asymptotically by comparison with the Meixner polynomials.

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Section: Riemann Hilbert Approach: Interpolation Problemmentioning

confidence: 95%

“…This implies the relation between partition functions, 12) and between Gibbs measures, µ(σ; w 1 , w 2 , w 3 , w 4 , w 5 , w 6 ) = µ(σ; ae −η , ae η , be −η , be η , c, c).…”

Section: Conservation Laws and Reduction Of Parametersmentioning

confidence: 99%

Six-vertex model with partial domain wall boundary conditions: Ferroelectric phase

Bleher

Liechty

2015

Journal of Mathematical Physics

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“…We now introduce a local transformation of the Riemann-Hilbert problem close to the endpoints e ±iα which allows for uniform estimates close to these points. The basic ideas behind this transformation can be found in [25]. Introduce the function This function has the following properties:…”

Section: The Parametrix At the Void-saturated Region End Pointsmentioning

confidence: 99%

The Fourier extension method and discrete orthogonal polynomials on an arc of the circle

Geronimo

Liechty

2020

Advances in Mathematics

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The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is defined. When the function being approximated is known at only finitely many points, the approximation is constructed as a projection based on this discrete set of points. In this paper we address the issue of estimating the absolute error in the approximation. The error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle, and these polynomials are then evaluated asymptotically using Riemann-Hilbert methods.

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“…The most parsimonious mechanism by which PTPσ likely regulates glutamate release is through the direct interaction of PTPσ with liprin-α via its intracellular D2 domain. Liprin-α3 functions as an upstream AZ scaffolding factor in AZ assembly, further interacting with RIMs and other presynaptic proteins, including ELKS, mDiaphanous, CASK, and GIT1 (Han et al, 2019, Südhof, 2012, Wong et al, 2018. Because PTPσ requires intracellular complexes for presynaptic assembly (Han et al, 2018), PTPσ may mediate various presynaptic functions by coordinating the interactions of liprin-α with other presynaptic proteins.…”

Section: Ptpσ Requires Various Molecular Mechanisms To Regulate Excitmentioning

confidence: 99%

Presynaptic PTPσ organizes neurotransmitter release machinery at excitatory synapses

Han

Lee

Lim

et al. 2020

Preprint

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Leukocyte common antigen-related receptor tyrosine phosphatases (LAR-RPTPs) are evolutionarily conserved presynaptic organizers. The synaptic role of vertebrate LAR-RPTPs in vivo, however, remains unclear. This study systematically analyzed the effects of genetic deletions of LAR-RPTP genes by generating single conditional knockout (cKO) mice targeting PTPσ and PTPδ. Although the numbers of synapses were reduced in cultured neurons deficient in individual PTPs, abnormalities in synaptic transmission, synaptic ultrastructures, and vesicle localization were observed only in PTPσ-deficient neurons. Strikingly, loss of presynaptic PTPσ reduced neurotransmitter release prominently at excitatory synapses, concomitant with drastic reductions in excitatory innervations onto postsynaptic target areas in vivo. However, postsynaptic PTPσ deletion had no effect on excitatory synaptic strength. Furthermore, conditional deletion of PTPσ in ventral CA1 specifically altered anxiety-like behaviors. Taken together, these results demonstrate that PTPσ is a bona fide presynaptic adhesion molecule that controls neurotransmitter release and excitatory inputs.11 100 nm from the presynaptic AZ membrane, was significantly increased in PTPσ-deficient presynaptic terminals (Fig 2E). In contrast, the number of tethered vesicles, defined as vesicular structures docked at presynaptic AZ membranes, was only marginally increased, an increase that was not statistically significant (Fig 2F). Surprisingly, the AZ length was increased by ~30% (Fig 2A and 2B), similar to the doubled AZ size in Caenorhabditis elegans mutants lacking ptp-3A, an orthologue of the type IIa RPTP gene (Ackley et al, 2005, Han et al, 2019. Consistent with this, a corresponding increase (~15%) in PSD length was observed (Fig 2A and 2C). This phenotype was not observed in PTPδ-deficient presynaptic terminals, nor were there any alterations in other anatomical parameters (Fig 2H-2M). Quantitative immunoblot analysis of PTPσand PTPδ-deficient neurons showed comparable expression of presynaptic AZ and PSD proteins, although the level of RIM1/2 was significantly lower in PTPδ-deficient neurons (Fig 2G, 2N, Appendix Fig 3A and 3B). These results suggest that PTPσ and PTPδ are not functionally redundant, and that PTPσ, but not PTPδ, is crucial in controlling the structural organization of both presynaptic AZs and PSDs.

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Global asymptotics of the Meixner polynomials

Cited by 9 publications

References 21 publications

Six-vertex model with partial domain wall boundary conditions: Ferroelectric phase

Six-vertex model with partial domain wall boundary conditions: Ferroelectric phase

The Fourier extension method and discrete orthogonal polynomials on an arc of the circle

Presynaptic PTPσ organizes neurotransmitter release machinery at excitatory synapses

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