On the equivalence of extremal Teichmüller mapping

Fan, Jishan; Chen, Jixiu

doi:10.1007/s11425-008-0092-1

Cited by 5 publications

(6 citation statements)

References 16 publications

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“…Recently, Fan and Chen [9] showed that there exists some μ ∈ M (Δ) such that [μ] T is a Strebel point while [μ] B is not an infinitesimal Strebel point, and vice versa. In this section, we give a rather simple approach to this result.…”

Section: Two Examplesmentioning

confidence: 99%

“…This mapping is due to Reich (see [13]) and approximates the famous Teichmüller shift mapping (see [14]) when t is small. It also has been used in various situations (see [9,12]). Set…”

Section: Two Examplesmentioning

confidence: 99%

“…When μ ∞ is small so that Remark. As was asked by Fan and Chen [9] , a natural question is whether for any Beltrami coefficient μ ∈ M (Δ), there exist two Beltrami coefficients ν 1 , ν 2 in M (Δ) such that [ν 1 ] T = [μ] T , [ν 1 ] B is an infinitesimal Strebel point, and [ν 2 ] B = [μ] B , [ν 2 ] T is a Strebel point. Corollary 5.1 says that when μ ∞ is small, the answer is affirmative.…”

Section: Consequently [λ T ] B Is An Infinitesimal Strebel Point In mentioning

confidence: 99%

“…When μ is extremal, the answer is definitely affirmative. However, when μ is not extremal, Fan and Chen [9] gave a negative approach to the question when X is the unit disk.…”

Section: Introductionmentioning

confidence: 99%

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On Strebel points

Shen

2009

Sci. China Ser. A-Math.

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Section: Two Examplesmentioning

confidence: 99%

“…This mapping is due to Reich (see [13]) and approximates the famous Teichmüller shift mapping (see [14]) when t is small. It also has been used in various situations (see [9,12]). Set…”

Section: Two Examplesmentioning

confidence: 99%

Section: Consequently [λ T ] B Is An Infinitesimal Strebel Point In mentioning

confidence: 99%

“…When μ is extremal, the answer is definitely affirmative. However, when μ is not extremal, Fan and Chen [9] gave a negative approach to the question when X is the unit disk.…”

Section: Introductionmentioning

confidence: 99%

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On Strebel points

Shen

2009

Sci. China Ser. A-Math.

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“…If μ itself is of constant modulus, the answer is a fortiori. Recently, Fan and Chen[3] gave a negative answer to the above problem in virtue of the method used in this paper if μ need not be extremal. §6.…”

mentioning

confidence: 94%

Existence of extremal Beltrami coefficients with nonconstant modulus

Yao¹

2010

Nagoya Mathematical Journal

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Abstract. Suppose that [μ] T (Δ)is a point of the universal Teichmüller space T (Δ). In 1998, Božin, Lakic, Marković, and Mateljević showed that there exists μ such that μ is uniquely extremal in [μ] T (Δ) and has a nonconstant modulus. It is a natural problem whether there is always an extremal Beltrami coefficient of constant modulus in [μ] T (Δ) if [μ] T (Δ) admits infinitely many extremal Beltrami coefficients; the purpose of this paper is to show that the answer is negative. An infinitesimal version is also obtained. Extremal sets of extremal Beltrami coefficients are considered, and an open problem is proposed. The key tool of our argument is Reich's construction theorem.

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Absorption, fluorescence and resonance Rayleigh scattering spectra of hydrophobic hydrogen bonding of eosin Y/Triton X-100 nanoparticles and their analytical applications

Kong

Liu

et al. 2010

Sci. China Chem.

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In a weak acidic medium (pH 2.4-2.8), eosin Y molecules (H 2 L) could replace water molecules to associate with Triton X-100 to form hydrophobic hydrogen bonding complexes. These complexes could further aggregate to form nanoparticles through the squeezing action of the water phase and Van Der Waals force, resulting in changes in the absorption spectrum and fluorescence quenching of EY as well as the significant enhancement of resonance Rayleigh scattering. This enables the sensitive determination of Triton X-100 using the fading spectrophotometry, fluorescence quenching method and RRS method. Among them, the RRS method shows the highest sensitivity with a detection limit of 20.6 ng mL 1 for Triton X-100. The optimum experimental conditions and factors that affect the absorption, fluorescence and RRS spectra were tested. The effects of coexisting substances were investigated and the results showed good selectivity. Based on these results, new spectrophotometric methods, fluorescence quenching method and RRS method for the determination of Triton X-100, were established. The hydrogen bonding association of eosin Y with Triton X-100 and the formation of nanoparticles as well as their effects on related spectral characteristics were discussed utilizing infrared, transmission electron microscope technique and quantum chemical method. eosin Y, Triton X-100, resonance Rayleigh scattering, fluorescence quenching, absorption spectrum

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On the equivalence of extremal Teichmüller mapping

Cited by 5 publications

References 16 publications

On Strebel points

On Strebel points

Existence of extremal Beltrami coefficients with nonconstant modulus

Absorption, fluorescence and resonance Rayleigh scattering spectra of hydrophobic hydrogen bonding of eosin Y/Triton X-100 nanoparticles and their analytical applications

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