On Stable Ideals

Kubota, Kazuji

doi:10.3836/tjm/1270151225

Cited by 3 publications

(6 citation statements)

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“…(i) The "soft state" ULXs have inner disk temperatures Tin which are too high for their implied high masses. Equivalently, when the BH mass is estimated from the Eddington argument, Rin with reasonable correction for a boundary condition (Kubota et al 1998) and spectral hardening factor (Shimura & Takahara 1995) falls much below the last stable orbit for a non-spinning BH, 6Rg, where Rg = GM/c 2 is the gravitational radius. This contrasts with soft-state BHBs, where Rin generally agrees with 6Rg.…”

Section: A Comparison Of the Ulx With Bhbmentioning

confidence: 99%

Another interpretation of the power-law-type spectrum of an ultraluminous compact X-ray source in IC 342

Kubota

Done

Makishima

2002

Monthly Notices of the Royal Astronomical Society

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The ultraluminous compact X‐ray sources (ULXs) generally show a curving spectrum in the 0.7–10 keV ASCA bandpass, which looks like a high‐temperature analogue of the disc‐dominated high/soft‐state spectra seen in Galactic black hole binaries (BHBs) at high mass accretion rates. Several ULXs have been seen to vary, and to make a transition at their lowest luminosity to a spectrum which looks more like a power law. These have been previously interpreted as the analogue of the power‐law‐dominated low/hard state in Galactic BHBs. However, the ULX luminosity at which the transition occurs must be at least 10–50 per cent of the Eddington limit assuming that their highest luminosity phase corresponds to the Eddington limit, while for the Galactic BHBs the high/soft–low/hard transition occurs at a few per cent of the Eddington limit. Here we show that the apparently power‐law spectrum in a ULX in IC 342 can be equally well fitted over the ASCA bandpass by a strongly Comptonized optically thick accretion disc with a maximum temperature of ∼1 keV. Recent work on the Galactic BHBs has increasingly shown that such components are common at high mass accretion rates, and that this often characterizes the very high (or anomalous) state. Thus we propose that the power‐law‐type ULX spectra are not to be identified with the low/hard state, but rather represent the Comptonization‐dominated very high/anomalous state in the Galactic BHBs.

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Section: A Comparison Of the Ulx With Bhbmentioning

confidence: 99%

Another interpretation of the power-law-type spectrum of an ultraluminous compact X-ray source in IC 342

Kubota

Done

Makishima

2002

Monthly Notices of the Royal Astronomical Society

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“…Obviously (ii) is equivalent to (iii) and (iii) implies (iv). (v) follows from (iv) by [5] Corollary 6 and Proposition 8. Assume (v).…”

Section: Introductionmentioning

confidence: 81%

Superficial Saturation

Watanabe¹

1986

Tokyo J. Math.

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“…We shall show in a forthcoming paper [3], an extension of this fact $t\mathfrak{c}$ any open ideal of an equi-dimensional Cohen-Macaulay semi-local ring.…”

mentioning

confidence: 93%

“…Again, we may assume that $I$ is an ideal of definition and that $A/M$ are infinite fields for all maximal ideals $M$ of $A$ . Let $x_{1},$ $x_{2},$ $\cdots,$ $x_{d}$ be a system of superficial parameters of $I$ , then $e_{0}(I)=$ $\swarrow(A/(x_{1}, x_{2}, \cdots, x_{d}))$ by (3) of Lemma 4 and by the fact that $x_{d}mod$ . $(x_{1}, \cdots, x_{d-1})$ is a transversal element of $I/(x_{1}, \cdots, x_{d-1})$ .…”

mentioning

confidence: 99%