The Elements of Mathematical Logic

Gradshteĭn, I. S.

doi:10.1016/b978-0-08-009832-6.50010-5

Cited by 2 publications

(2 citation statements)

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“…(where the last approximation comes from [33]) and where δ crit is the value of δ, in equation (17), that would create a black hole of critical mass, M crit . The comoving density of accreting black holes is n ≈ p acc /( …”

Section: Pbh Formation and Their Comoving Densitymentioning

confidence: 99%

Could supermassive black holes be quintessential primordial black holes?

2002

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There is growing observational evidence for a population of supermassive black holes (SMBHs) in galactic bulges. We examine in detail the conditions under which these black holes must have originated from primordial black holes (PBHs). We consider the merging and accretion history experienced by SMBHs to find that, whereas it is possible that they were formed by purely astrophysical processes, this is unlikely and most probably a populations of primordial progenitors is necessary. We identify the mass distribution and comoving density of this population and then propose a cosmological scenario producing PBHs with the right properties. Although this is not essential we consider PBHs produced at the end of a period of inflation with a blue spectrum of fluctuations. We constrain the value of the spectral tilt in order to obtain the required PBH comoving density. We then assume that PBHs grow by accreting quintessence showing that their mass scales like the horizon mass while the quintessence field itself is scaling. We find that if scaling is broken just before nucleosynthesis (as is the case with some attractive non-minimally coupled models) we obtain the appropriate PBH mass distribution. Hawking evaporation is negligible in most cases, but we also discuss situations in which the interplay of accretion and evaporation is relevant.

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Section: Pbh Formation and Their Comoving Densitymentioning

confidence: 99%

Could supermassive black holes be quintessential primordial black holes?

2002

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“…Extensively on these subjects, with a lot of examples, can be found in books [7,8] and also in a research [9].…”

Section: Introductionmentioning

confidence: 99%

Is the Converse Claim True and if It Is, How Is It Proven

Malek¹,

Stupel²,

Sigler³

2017

Jour. Math. Sci. Adv. Appl.

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Theorems and claims in Mathematics in general, and in geometry in particular, are an inseparable part of the material and of the mathematical education imparted to high school students. Proving claims is a central component in geometry classes. Students acquire knowledge on the properties of shapes and bodies and make use of these properties in proving different claims. For each claim, one can formulate the converse claim, which in some cases may be right and in others not. Teachers do not always make use of the opportunity to formulate with the students the converse claim and to examine its truth by ALIZA MALEK et al. 68 achieving a proof or finding an example to the contrary if it is not true. The present paper presents eight interesting mathematical claims in geometry, were for each claim and converse claim complete proofs are given using different mathematical tools, including indirect proofs. Teachers can make use of such and other examples to expose the students to the claim and the converse claim and to the methods for dealing with various methods of proof and of preventing mistakes.

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The Elements of Mathematical Logic

Cited by 2 publications

References 0 publications

Could supermassive black holes be quintessential primordial black holes?

Could supermassive black holes be quintessential primordial black holes?

Is the Converse Claim True and if It Is, How Is It Proven

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