We present 10 main-sequence ALPINE galaxies (log (M/M⊙) = 9.2−11.1 and ${\rm SFR}=23-190\, {\rm M_{\odot }\, yr^{-1}}$) at z ∼ 4.5 with optical [O ii] measurements from Keck/MOSFIRE spectroscopy and Subaru/MOIRCS narrow-band imaging. This is the largest such multiwavelength sample at these redshifts, combining various measurements in the ultraviolet, optical, and far-infrared 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 star-formation 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 metal-dependent relation assuming metallicities around $50{{\ \rm per\ cent}}$ solar. To explain the measured dust-corrected 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 $\log (\rm n_e / {\rm [cm^{-3}]}) \sim 2.5-3$ 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 $mt^2 +(1-2m) t + m$ 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 $1-4m$, 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- NSF-PAR ID:
- 10205612
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
ABSTRACT -
more » « less
Abstract The structure of liquid lithium pyroborate, Li4B2O5(
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 literature11B nuclear magnetic resonance measurements. van't Hoff analysis based on a simple boron isomerization reaction BØ3O2–⇌ BØO22–yields ΔH = 13(1) kJ mol–1and ΔS = 40(1) J mol–1 K–1for 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. -
more » « less
Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow/circulation
X on a directed graphG into weighted source-to-sink paths whose weighted sum equalsX . We show that, for acyclic graphs, considering thewidth of the graph (the minimum number of paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the version of the problem that uses only non-negative weights, we identify and characterise a new class ofwidth-stable graphs, for which a popular heuristic is aO (logVal (X ))-approximation (Val (X ) being the total flow ofX ), and strengthen its worst-case approximation ratio from\(\Omega (\sqrt {m})\) m /logm ) for sparse graphs, wherem is the number of edges in the graph. We also study a new problem on graphs with cycles, Minimum Cost Circulation Decomposition (MCCD), and show that it generalises MFD through a simple reduction. For the version allowing also negative weights, we give a (⌈ log ‖ X ‖ ⌉ +1)-approximation (‖X ‖ being the maximum absolute value ofX on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary circulations (‖X ‖ ≤ 1), using a generalised notion of width for this problem. Finally, we disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [2018 ], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version. -
more » « less
Abstract We present the KODIAQ-Z survey aimed to characterize the cool, photoionized gas at 2.2 ≲
z ≲ 3.6 in 202 Hi -selected absorbers with 14.6 ≤z ≲ 3.6 can be metal-rich (−1.6 ≲ [X/H] ≲ − 0.2) as seen in damped Lyα absorbers (DLAs); it can also be very metal-poor ([X/H] < − 2.4) or even pristine ([X/H] < − 3.8), which is not observed in DLAs but is common in the IGM. Forv < 500 km s−1imply that the metals are poorly mixed inz ≲ 3.6. We find that the mean metallicity increases withN Hi , consistent with what is found inz < 1 gas. The metallicity of gas in this column density regime has increased by a factor ∼8 from 2.2 ≲z ≲ 3.6 toz < 1, but the contribution of thez < 1 is a quarter of that at 2.2 ≲z ≲ 3.6. We show that FOGGIE cosmological zoom-in simulations have a similar evolution of [X/H] withN Hi , which is not observed in lower-resolution simulations. In these simulations, very metal-poor absorbers with [X/H] < − 2.4 atz ∼ 2–3 are tracers of inflows, while higher-metallicity absorbers are a mixture of inflows and outflows. -
Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness.more » « less
