dataesr/scientific-paragraphs-categorization huggingface dataset:
2018; Yan et al. 2019; Tröster et al. 2022 ). Given the central role that small-scale primary CMB data and CMB lensing will play in cosmology in the next decade  (Schmittfull & Seljak 2018; Fang et al. 2022) , improving our understanding of the CIB, and developing better models that allow us to quantify its impact on these observations, is of paramount importance. On the other hand, the CIB is in itself a treasure trove for both astrophysics and cosmology. First, it is a high-sensitivity tracer of the star-formation rate (SFR) density (i. e. the total mass of stars created per unit time and volume)  (Dole et al. 2006 ). Understanding star formation across time and galaxy type is a central piece in the puzzle of galaxy formation and evolution  (Tinsley 1980). Through observations of the infrared luminosity function at high redshifts  (Gruppioni et al.
Angular power spectra and covariances We estimate auto-and cross-power spectra from all datasets in this study using the so-called pseudoℓ or MASTER algorithm as implemented in NaMaster  (Alonso et al. 2019 ). The method is described briefly here, and further details can be found in  Hivon et al. (2002) ;  Alonso et al. (2019). In all cases studied here (galaxy overdensity and CIB intensity maps), the observed field ˜ ( n) is a masked version of the true underlying field ( n), i. e. ˜ ( n) = ( n) ( n), ( 32 ) where is the mask. can simply be a binary map defining the footprint over which has been observed, but in general it can be thought of as a local weight that can be designed to maximise the signal-to-noise ratio of the resulting power spectrum estimator. Consider two such fields, ˜ and ˜ , with masks and respectively.
Likelihood We derive parameter constraints from the power spectrum data in two different ways. First, as described in Section 4. 2, we derive model-independent tomographic measurements of SFR from the galaxy auto-correlations and their cross-correlation with the CIB. We will then use these measurements, as described in Section 4. 3, to place constraints on the free parameters of the halo models for the SFR described in Section 2. 1. 3. In both cases, we will make use of Gaussian likelihoods of the form -2 log (d|Θ) = (d -m(Θ)) C -1 d (d -m(Θ)) + const. , ( 36 ) where d is the data vector (power spectra or measurements of SFR ), C is the covariance of d, m is the theoretical model for d, and Θ denotes the set of parameters used to define m. The posterior parameter distribution is then given by the product of this likelihood and the parameter priors, which we discuss in Section 4.
5 respectively. The redshift 4 The data can be found at https://gitlab. com/qianjunhang/ desi-legacy-survey-cross-correlations. We thank the authors for the remarkable care with which these data were made publicly available. distribution of each bin is estimated directly from the data as a histogram of the measured spectroscopic redshifts. We construct maps of the galaxy overdensity from both samples using the methods described in  García-García et al. (2021). These include the construction of galaxy masks, the use of random catalogues for eBOSS, the correction of large-scale systematic fluctuations for DELS, and the estimation of the noise bias for both samples. Details can be found in Section 3 of  García-García et al. (2021). The impact of large-scale systematics in the galaxy autocorrelation is minimal given the scale cuts used for this analysis (see Section 4.
are able to recover the diagonal errors within ∼ 4 -5% (see top panel of Fig. 7 ). However, we find that the jackknife estimate recovers cross-correlation coefficients between power spectra involving different CIB frequencies that are consistently lower than the analytical prediction by ∼ 2% (see bottom panel of Fig. 7 ). Although this is a small difference, the tight correlation between different CIB frequencies makes it necessary to accurately characterise these offdiagonal covariance elements. We find that, otherwise, the goodness of fit to the full dataset combining all frequencies was unacceptable, even if the same model parameters were able to provide a good fit to the three different frequencies separately, which clearly points to an inconsistent covariance matrix. To correct for this, we therefore multiply all off-diagonal covariance matrix elements involving two different CIB frequencies by an overall factor of 0.
SFR We obtain tomographic measurements of the bias-weighted mean SFR density, SFR , by fitting the phenomenological model in Eq. 29 to the power spectrum data described in the previous section. The model consists on one galaxy bias parameter and one value of SFR for each redshift bin, as well as a free white-noise amplitude ( ) for each individual power spectrum. Although the white noise levels for different CIB frequencies should be tightly correlated, we choose to allow each of them to vary freely, in order to account for different levels of point-source subtraction, and non-Poissonian stochastic contributions in galaxy clustering. The full model therefore has a total of 6 × bin = 36 free parameters ( SFR , and one white noise amplitude for the four different power spectra in each galaxy redshift bin).
Note, however, that since the white noise amplitudes are linear parameters, it is possible to marginalise over them analytically, significantly reducing the number of degrees of freedom we have to explore to 2 × bin = 12. This is achieved by modifying the inverse covariance matrix used in the Gaussian likelihood of Eq. 36 as  (Rybicki & Press 1992 ) C -1 d = C -1 d -C -1 d T T C -1 d T -1 T C -1 d , (37) MNRAS 000, 1-18 (  2022 ) The star-formation history from CIB correlations 9  where T is a matrix containing, as columns, the templates corresponding to the different linear parameters of the model (in this case, a constant vector for each of the different power spectra included in the data vector). We used flat, uninformative priors on all parameters. Figure  8  shows the resulting measurements of SFR.
We recover constraints that are almost equal to our fiducial measurements, and the model is able to reproduce the data for each redshift bin individually, with 2 -values above 0. 1. These results are shown as grey points in Fig. 8. The model used here relies on having an accurate description of the mean infrared spectral energy distribution ( eff in Eq. 13), and inaccuracies in it would lead to inconsistent measurements of SFR for different frequencies. To test for this, we repeat our measurement of SFR three more times, using only one of the three CIB frequencies at a time. The results are presented in Table  A3 , and as orange/red/brown points in Fig. 8  for the 353/545/857 GHz channels respectively. Overall we find good agreement between the three frequencies. The 353 GHz channel achieves consistently larger uncertainties than the other two, and the recovered values of SFR are also consistently higher, although compatible with 0.
Of potential concern is the fact that the eBOSS and DELS samples correspond to different galaxy types, which likely reside in significantly different environments and have different infrared properties. As emphasized in Section 1, these differences should be irrelevant in large-scale limit of the halo model, as long as both samples trace the same underlying large-scale structure. Nevertheless, potential inaccuracies in the halo model could invalidate this as- sumption. To explore this possibility, we have re-analysed the four DELS and two eBOSS samples separately within the context of Model B, and compared the constraints resulting from each dataset. Since in this case we work with significantly fewer data points, we simplified Model B by fixing the mass parameters to the best-fit values from  Moster et al.
(2018)  ( 0 = 11. 339, = 0. 692), and varied only ( 0 , ). The results are shown in Fig. 12 , together with the constraints from the full dataset. We find that the two different galaxy samples recover results that are in good agreement with each other and with the combined constraints. It is therefore unlikely that the differences in the astrophysical origin of both samples lead to a significant bias in the results presented here. These caveats aside, the fact that our measurements of SFR are compatible with direct measurements of the SFR density is an interesting and non-trivial result, beyond the fact that both constraints come from independent datasets. The contribution of haloes of different masses to SFR is weighted by their linear bias, and thus SFR is more sensitive than SFR to star formation in the most massive haloes.
These include accurately describing the scale dependence of the mean and variance of the galaxy overdensity in haloes, the covariance between galaxy overdensity and SFR density in haloes (or, equivalently, characterising star formation in the target galaxy sample), the transition between the 1-halo and 2-halo-dominated regimes, and the different contributions (Poisson or otherwise) to the shot-noise component of all auto-and crosscorrelations involved. We find that this phenomenological model provides a good fit to the data on the scales probed. We thus obtained measurements for SFR at six different redshifts, reported in Table  2. The measurements are robust with respect to the choice of scale cuts, as well as the combination of datasets used (i. e. the choice of redshift bins and frequency maps).
In terms of systematics, the presence of other sources of extragalactic contamination in the CIB maps, which would correlate with the galaxy distribution, could bias the measured values of SFR. The most harmful contaminants in this case are thermal Sunyaev-Zel'dovich and emission from radio point sources. The fact that we recover compatible results from the three Planck frequencies reassures that any residual contamination is likely negligible within the range of scales and frequencies explored here. This may not be the case if a more ambitious analysis was carried out, using smallerscale data or other frequencies, and a more thorough quantification of the impact from possible sources of contamination would have been necessary. On the galaxy side, Galactic extinction may cause a non-cosmological modulation in the observed number counts of galaxies.
On the CIB side, high-resolution data from experiments such as CCAT-prime  (Stacey et al. 2018)  or the Simons Observatory  (Ade et al. 2019)  will improve the quality, sensitivity, and frequency coverage of current CIB observations. On the galaxy side, future deep photometric surveys such as the Rubin Observatory Legacy Survey of Space and Time (LSST,  LSST Science Collaboration et al. 2009 ) will enable us to trace the large-scale structure tomographically to redshifts ∼ 3, significantly improving our ability to trace the evolution of star formation at earlier times. Likewise, spectroscopic surveys such as DESI  (Levi et al. 2019)  will provide denser quasar samples, enabling the same type of constraints with better control over redshift uncertainties. With a better understanding of the star-formation history, it will also be possible to use the CIB itself as a cosmological tool, allowing us to map the Universe's structures on very large scales with a sensitivity higher than will be achievable by future CMB lensing experiments.
ACKNOWLEDGEMENTS We would like to thank  Jens Chluba ,  Martin Rey ,  Aprajita Verma , and  Ziang Yan  for useful discussions. BJ is supported by the  ENS Paris-Saclay. CGG is supported by  European Research Council  Grant No:  693024  and the  Beecroft Trust. DA is supported by the  Science and Technology Facilities Council  through an  Ernest Rutherford Fellowship , grant reference  ST/P004474. JRZ and NK are supported by an  STFC doctoral studentship. NK also acknowledges funding from the  University of Oxford, Department of Physics. We also made extensive use of computational resources at the  University of Oxford Department of Physics , funded by the  John Fell Oxford University Press Research Fund. We made extensive use of the numpy (Oliphant 2006;Van Der Walt et al. 2011), scipy (Virtanen et al.
The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Duág (Kitt Peak), a mountain with particular significance to the  Tohono O'odham Nation. NOIRLab is operated by the  Association of Universities for Research in Astronomy (AURA)  under a cooperative agreement with the  National Science Foundation. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the  Dark Energy Survey  (DES) collaboration. Funding for the DES Projects has been provided by the  U. S. Department of Energy , the  U. S. National Science Foundation , the  Ministry of Science and Education of Spain , the  Science and Technology Facilities Council of the United Kingdom , the  Higher Education Funding Council for England , the  National Center for Supercomputing Applications  at the  University of Illinois at Urbana-Champaign , the  Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics  at the  Ohio State University , the  Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos ,  Fundacao Carlos Chagas Filho de Amparo ,  Financiadora de Estudos e Projetos ,  Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro ,  Conselho Nacional de Desenvolvimento Cientifico e Tecnologico  and the  Ministerio da Ciencia, Tecnologia e Inovacao , the  Deutsche Forschungsgemeinschaft  and the  Collaborating Institutions  in the  Dark Energy Survey.
BASS is a key project of the  Telescope Access Program  (TAP), which has been funded by the  National Astronomical Observatories of China , the  Chinese Academy of Sciences  (the  Strategic Priority Research Program "The   Emergence of Cosmological Structures " Grant #  XDB09000000 ), and the  Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the  External Cooperation Program of Chinese Academy of Sciences  (Grant #  114A11KYSB20160057 ), and  Chinese National Natural Science Foundation  (Grant #  11433005 ). The  Legacy Survey  team makes use of data products from the Near-Earth Object Wide-field  Infrared Survey Explorer (NE-OWISE ), which is a project of the  Jet Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by the  National Aeronautics and Space Administration.
This tells us that if genus is a measure of complexity it has properties that interestingly distinguish it from measures of length. Because we will discuss structural proof rules later, we will use Statman's term of global structure to refer to the overall structure of the graph or proof. So genus appears to be a good choice for measuring the global structural complexity of graphs. But, for genus to measure the global structure of the proof, the proof graphs must encode the global structure of the proof. This is not the case for logical flow graphs, as we will now show.
On Coq's side Support for hardware floating-point arithmetic in Coq was inspired by two libraries: the Flocq library  [7] , which gathers an IEEE-754-compliant executable formalization, formerly in the module IEEE754. Binary, 2  and the formalization of primitive 63-bit integers  [10] , now part of the standard library as module UInt63. 3  The former provides a precise specification in the form of a reference implementation of floating-point operators, while the latter guided the implementation methodology. Flocq defines an inductive type full float that represents signed zeros, signed infinities, unbounded floating-point numbers with any integer mantissa m and exponent e, and NaNs (Not-a-Number) with a payload m. The payload distinguishes the different binary encodings of NaNs. 4 Variant full_float := | F754_zero (s : bool) | F754_infinity (s : bool) | F754_nan (s : bool) (m : positive) | F754_finite (s : bool) (m : positive) (e : Z).
Our aim was to extract a subset of this Flocq theory sufficient to completely specify floating-point numbers and operators, so that Coq does not depend on Flocq at compilation time. We started by porting this definition, just removing the NaN payloads because they are not fully specified by the IEEE-754 standard, including the distinction between signaling and quiet NaNs, which can lead to implementation discrepancies between hardware vendors. Thus, ignoring these payloads in Coq is paramount to guarantee the portability of computations and proofs performed with the same Coq script on different processors. Values of type spec float are not normalized, since no bound is enforced on mantissas and exponents. But in practice, all the operators will make sure that the exponent e is bounded, and that the mantissa m contains 53 bits for normalized numbers, and at most 52 for subnormal numbers, thus ensuring that the number belongs to the binary64 format.
This property is denoted by the valid binary predicate. Note that this definition actually matches Flocq's original formalization of the IEEE-754 standard  [11]. But Flocq was later extended to make it suitable for the semantics used in the CompCert C compiler  [12]. Indeed, even if nothing can be specified about NaN payloads, they can still be distinguished by a C program and thus need to exist. As part of this work, the original formalization was added back to Flocq, in the module IEEE754. BinarySingleNaN. Next, following the same methodology as that of the Uint63 formalization, we have declared an abstract type and some arithmetic operations: Primitive float := #float64_type. Primitive add := #float64_add. (* and so on *) Here, the Primitive vernacular amounts to introducing Parameters (i.
On Flocq's side Comprehensive formalizations of floating-point arithmetic exist for several proof assistants, e. g. , HOL Light  [14]  and PVS  [15, 16]. In the case of Coq, the largest formalization is provided by the Flocq library  [7]. A whole hierarchy of formats is provided, ranging from real numbers with bounded mantissas but unbounded exponents to computable numbers with all the floating-point special values: signed zeros, infinities, and NaNs. Along with these formats and the links between them, the library contains many classical results about roundings and error-free transformations. When verifying properties of floating-point algorithms, two families of formats are commonly encountered: Numbers with an unbounded exponent range, i. e. , without underflow nor overflow. Although unrealistic, this model is attractive for its simplicity and commonly used for error bounds  [17].
Reduction engines As explained in Section 2. 1, Coq provides an abstract type float as well as operations over it, and Coq's version of λ-calculus is extended with dedicated reduction rules for these operations. This means that these rules have to be implemented in the software. Unfortunately, Coq supports various engines, each one with its own implementation of the rules of Coq's calculus. First, there is the conversion engine, which is responsible of checking that two λ-terms are equivalent according to the rules of the calculus. The conversion engine works great in general, but it falls short when performing a proof by computational reflection. Indeed, in that case, the archetypal goal to prove is f (x) = true for some given x. This is a proof by reflexivity, which means that Coq has to check that the term f (x) = true is equivalent to the term true = true.
Since true is already in normal form, this amounts to computing the normal form of f (x). But the conversion engine tries hard to never normalize terms, as the size of the normal form explodes in the general case. So, it is unable to handle this use case efficiently. That is why Coq provides two reduction engines that are solely designed to compute normal forms. The first one, invoked by the vm compute tactic, compiles the λ-term to bytecode and then evaluates it using an interpreter derived from the bytecode interpreter for the OCaml language  [19]. The second one, invoked by the native compute tactic, follows a similar approach. It turns the λ-term into an OCaml function, compiles it to machine code using the OCaml compiler, and then loads it and executes it  [20]. For all three engines, the underlying runtime comes from the OCaml language.
A similar approach is used for multiplication, division, and so on. Unfortunately, while rounding to nearest is readily available in floating-point units, this is not the case for directed rounding. And even if it was, there is no foolproof way of performing floating-point operations with directed rounding in programming languages such as OCaml and C. For instance, as of writing this article, the GCC compiler still does not handle dynamically changing the rounding mode in a safe way, let alone an efficient one. 7  So, to ensure portability, we had to stop at rounding to nearest. But since interval arithmetic is such an ubiquitous paradigm when it comes to proving properties about real numbers, we have provided two more primitives to ease its implementation: predecessor and successor. Indeed, interval arithmetic does not really care whether the bounds of the enclosures are correctly rounded toward -∞ and +∞.
CoqInterval Since version 4. 0, the CoqInterval library uses hardware floating-point numbers by default. This implementation of interval arithmetic lives alongside the original implementation, which emulates floating-point numbers using arbitrary-precision integers bigZ from the bignums library  [25]. Two such integers are used, one for the mantissa and the other one for the exponent. A special value NaN is provided, but infinities are not (thanks to the unbounded exponent, neither overflow nor underflow are possible). Every operation is performed at a given target precision, i. e. , number of figures in the resulting mantissa. Let us consider addition as an illustration. The original interface of the operation was as follows in the FloatOps module type. Parameter add : rounding_mode -> precision -> type -> type -> type.
In practice, the result is close to the middle of the interval, but this property is left unspecified as it is not used in correctness proofs. Exact operations (addition, subtraction, and multiplication), using as much precision as needed to exactly encode their result, were used mostly to compute some constants (typically integers). So, the latter were replaced with tight intervals containing the constants. With this new coarser FloatOps interface, it was relatively easy to implement hardware floating-point numbers in CoqInterval. Regarding the implementation, 9700 lines were added to the library (including 3300 for the hardware-based implementation of the interface FloatOps) while 4300 were removed. The use of hardware floating-point numbers provides a speedup of about one order of magnitude, as illustrated by the following proof of 8 0 sin(t + exp t)dt -0.
3474 ≤ 0. 1. From Coq Require Import Reals. From Coquelicot Require Import Coquelicot. From Interval Require Import Tactic. 3. 60 GHz, with 16 GB of RAM. All the benchmarks have been executed sequentially (namely, without the -j option of make). Our benchmarks rely on a large set of dependencies that can take about an hour to compile. For greater convenience, we devised some Docker images containing the benchmark environments, based on Debian 11, opam 2 (the OCaml package manager), and OCaml 4. 07. 1. The source code of all the benchmarks as well as guidelines to run them are gathered at the following URL: https://github. com/validsdp/ benchs-primitive-floats.