We present 10 mainsequence ALPINE galaxies (log (M/M⊙) = 9.2−11.1 and ${\rm SFR}=23190\, {\rm M_{\odot }\, yr^{1}}$) at z ∼ 4.5 with optical [O ii] measurements from Keck/MOSFIRE spectroscopy and Subaru/MOIRCS narrowband imaging. This is the largest such multiwavelength sample at these redshifts, combining various measurements in the ultraviolet, optical, and farinfrared including [C ii]158 $\mu$m line emission and dust continuum from ALMA and H α emission from Spitzer photometry. For the first time, this unique sample allows us to analyse the relation between [O ii] and total starformation rate (SFR) and the interstellar medium (ISM) properties via [O ii]/[C ii] and [O ii]/H α luminosity ratios at z ∼ 4.5. The [O ii]−SFR relation at z ∼ 4.5 cannot be described using standard local descriptions, but is consistent with a metaldependent relation assuming metallicities around $50{{\ \rm per\ cent}}$ solar. To explain the measured dustcorrected luminosity ratios of $\log (L_{\rm [OII]}/L_{\rm [CII]}) \sim 0.98^{+0.21}_{0.22}$ and $\log (L_{\rm [OII]}/L_{\rm H\alpha }) \sim 0.22^{+0.13}_{0.15}$ for our sample, ionization parameters log (U) < −2 and electron densities $\log (\rm n_e / {\rm [cm^{3}]}) \sim 2.53$ are required. The former is consistent with galaxies at z ∼ 2−3, however lower than at z > 6. The latter may be slightly higher than expected given the galaxies’ specific SFR. The analysis of this pilot sample suggests that typical log (M/M⊙) > 9 galaxies at z ∼ 4.5 to have broadly similar ISM properties as their descendants at z ∼ 2 and suggest a strong evolution of ISM properties since the epoch of reionization at z > 6.
We investigate the asymptotics of the total number of simple $(4a+1)$knots with Alexander polynomial of the form $mt^2 +(12m) t + m$ for some nonzero $m \in [X, X]$. Using Kearton and Levine’s classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, this count is the same as the total number of ${\mathbb{Z}}[1/m]$equivalence classes of binary quadratic forms of discriminant $14m$, for $m$ running through the same range. Our heuristics, based on the Cohen–Lenstra heuristics, suggest that this total is asymptotic to $X^{3/2}/\log X$ and the largest contribution comes from the values of $m$ that are positive primes. Using sieve methods, we prove that the contribution to the total coming from $m$ positive prime is bounded above by $O(X^{3/2}/\log X)$ and that the total itself is $o(X^{3/2})$.
more » « less NSFPAR ID:
 10205612
 Publisher / Repository:
 Oxford University Press
 Date Published:
 Journal Name:
 International Mathematics Research Notices
 ISSN:
 10737928
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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ABSTRACT 
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J = Li/B = 2), has been measured over a wide temperature range by high‐energy X‐ray diffraction, and compared to that of its glass and borate liquids of other compositions. The results indicate a gradual increase in tetrahedral boron fraction from 3(1)% to 6(1)% during cooling fromT = 1271(15) to 721(8) K, consistent with the largerN _{4} = 10(1)% found for the glass, and literature^{11}B nuclear magnetic resonance measurements. van't Hoff analysis based on a simple boron isomerization reaction BØ_{3}O^{2}^{–}⇌ BØO_{2}^{2–}yields ΔH = 13(1) kJ mol^{–1}and ΔS = 40(1) J mol^{–1} K^{–1}for the boron coordination change from 4 to 3, which are, respectively, smaller and larger than found for singly charged isomers forJ ≤ 1. With these, we extend our model forN _{4}(J ,T ), nonbridging oxygen fractionf _{nbr}(J ,T ), configurational heat capacity , and entropyS ^{conf}(J ,T ) contributions up toJ = 3. A maximum is revealed in atJ = 1, and shown semi‐quantitatively to lead to a corresponding maximum in fragility contribution, akin to that observed in the total fragilities by temperature‐modulated differential scanning calorimetry. Lithium is bound to 4.6(2) oxygen in the pyroborate liquid, with 2.7(1) bonds centered around 1.946(8) Å and 1.9(1) around 2.42(1) Å. In the glass,n _{LiO}= 5.4(4), the increase being due to an increase in the number of short Li–O bonds. 
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