 the different simulations are indicated by different line styles. The shaded regions indicate the half-mass range M/2 for the corresponding mass bins. This analysis reveals: (1) The correlation coefficient peaks as expected for haloes of mass M on scales around M/2; and (2) the correlation with the large scale density field on several h −1 Mpc is significant. that survive until z = 0 will preferentially form in regions that are underdense on these scales in the initial density field. They will then migrate into regions of higher density over time. We determine the evolution of the background density using first-order Lagrangian perturbation theory. Given the eigenvalues Λi of the initial tidal field on scales of R b,0, the evolution of δ5 at the positions of the haloes is given before shell-crossing by in the Zel'dovich approximation (Zel'dovich 1970). For each halo in the N -body simulation, we thus obtain an environmental density at z = 0 and the redshift of collapseẑ form by knowing a priori its final mass and its Lagrangian position. In Figure A2 we show the median formation redshiftẑ form (in grey) as a function of density δ5 for three mass bins. 6 The corresponding relation is shown also for the quantities directly obtained from the N -body simulation (in black). We find very good agreement with the N -body data suggesting that this simple model suffices to predict the correlation between formation redshift and environment seen in the simulations if the correct final mass is known.
Replacing the actual halo masses with those predicted at the Lagrangian halo locations by the Extended Press-Schechter (EPS) formalism is sufficient to completely erase any dependence of the formation redshift on environmental density. The spherical collapse model (on which the EPS method is based) tends to overestimate halo masses, particularly for those haloes that are located within 6 We have checked that results are not influenced by the small fraction of haloes that already experienced shell-crossing on the 5 h −1 Mpc scale. Figure A2. Halo formation time as a function of environmental density on scales 5 h −1 Mpc obtained from the N -body simulations (grey) and obtained from the initial conditions data using a combination of two spherical collapse models for the evolution of the background and the peak (black), the mass of the halo is taken from the N -body simulation. Stripped haloes are not considered. Solid lines indicate the median, dashed lines the ±1σ spread around the median. strongly sheared flows. In Figure A3 we show how the formation redshift z form depends on the ratio between the mass MEPS and the mass M, where z form and M are determined from the N -body simulations. We find that the mismatch increases from a median ratio of about 1.5 at low formation redshifts to about 3 at high formation redshifts for the considered masses (cf. Figure 3 in Wang et al. 2007). When we exclude the mass-stripped haloes (grey lines) the scatter is significantly reduced. The ratio between the EPS-predicted mass and the mass found in the N -body simulations correlates with the largest eigenvalue λ3 of the rate-of-strain tensor as strongly as with the halo formation redshift. For haloes at z = 0 with masses between 2 and 4 × 10 10 h −1 M⊙ the Spearman rank correlation coefficient between MEPS/M and the shear eigenvalue λ3 is 0.29. It is 0.27 between MEPS/M and z form. This provides further evidence that tidally suppressed accretion is responsible for the existence of the assembly bias. The proximity to a massive halo is reflected in the correlation coefficient with density obtained with the top-hat filter at z=0. We find that the correlation coefficient with MEPS/M is 0.22 for density determined on scales of 5 h −1 Mpc, 0.31 for 2 h −1 Mpc and 0.41 for 0.5 h −1 Mpc.

A2 Lagrangian assembly bias
It is interesting to verify whether the assembly bias is already present in the spatial distribution of protohaloes in the linear density field. The Lagrangian bias is given by bL = bE − 1 if bE is the Eulerian bias (e.g. Mo & White 1996). For M < M * haloes bE < 1. Thus, if b E,old > bE,young it follows that 0 b L,old > bE,young and thus the old haloes should have a lower Lagrangian clustering amplitude than the young ones.
To verify this, we compute the two-point correlation of the Lagrangian patches of old and young haloes (selected at z = 0) at Figure A3. The ratio of the mass determined in the initial conditions using the EPS model to the mass measured in the N -body simulation as a function of halo formation time. Solid lines indicate the median, dashed lines the 16th and 84th percentiles. Shown is the relation for haloes that are not undergoing tidal mass-loss (grey) and for all haloes (black) in three mass bins. Figure A4. Two-point correlation functions of the Lagrangian centres of mass in the initial density field (z = 78.5) for haloes with masses 2 − 4 × 10 10 h −1 M ⊙ at z = 0 (upper panel) and 2 − 4 × 10 11 h −1 M ⊙ (lower panel). The solid line shows the correlation function for all haloes, while open circles and filled squares correspond to the 10 per cent of these haloes with lowest and highest formation redshifts, respectively. the initial redshift of our simulations. Figure A4 shows the correlation function of the protohaloes that will form objects with masses between 2 and 4 × 10 10 h −1 M⊙ at z = 0. The corresponding correlations at z = 0 have been presented in Figure 1. At z = 0 the oldest haloes cluster more strongly than the youngest ones. For the Lagrangian patches, as expected, the opposite is true: the oldest (youngest) haloes are significantly less (more) clustered than the total. This indicates that, already in the linear density field, the environments surrounding the Lagrangian patches of young and old haloes have different properties. In particular, we find that both young and old haloes at these masses form in initially underdense (i.e. with a median δ5 < 0) regions. However, the Lagrangian patches of young protohaloes reside (on average) in more underdense regions which, in a Gaussian random field, are more strongly clustered. Using the mean squared overdensities at the halo centres as a proxy for their clustering amplitude (i.e. assuming a linear bias model independent of halo age for the halo density), B =˙δ 2 5¸1 /2, we find B old /B all = 0.8 and Byoung/B all = 1.2. On the other hand, with time, the protohaloes will collapse and flow towards denser regions thus changing their clustering properties. At z = 0 we find B old /B all = 1.8 and Byoung/B all = 0.33. The reversal of the assembly bias is then a simple result of the evolution of the halo environmental density. Older haloes have thus always resided in more overdense regions than younger haloes.<?php

declare(strict_types=1);

use App\Models\Listing;

/** @var Listing[] $listings */

?>
<div class="panel-outer">
    <p class="ml-2 mb-1 text-grey-dark text-sm">Latest</p>
    <div class="panel-inner">
        <div class="panel-heading">
            <div class="flex text-green-darker">
                <div class="w-full md:w-3/5">
                    <p class="panel-heading-text">Job Title</p>
                </div>
                <div class="hidden md:flex justify-between md:w-2/5 pl-4 lg:pl-6">
                    <p class="panel-heading-text">Category</p>
                    <p class="panel-heading-text">Location</p>
                </div>
            </div>
        </div>
        <div class="flex flex-col justify-center text-green-darker flex-wrap lg:flex-no-wrap px-4 pt-3 pb-1">
            @foreach($listings as $listing)
                <div class="text-center md:text-left flex justify-between items-center my-3">
                    <div class="w-full md:w-3/5">
                        <a class="font-semibold text-left text-green-darker no-underline"
                           href="{{ route('region_category_listing.show', ['region' => $listing->region, 'category' => $listing->category, 'listing' => $listing]) }}">
                            {{ $listing->title }}
                        </a>
                        <small
                            class="md:my-2 hidden md:block lg:inline-block">{{ str_limit($listing->body, 60) }}</small>
                    </div>
                    <div class="hidden md:flex justify-between items-center md:w-2/5 pl-4 lg:pl-6">
                        <a class="text-sm text-green-darker"
                           href="{{ route('region_category_listing.index', ['region' => $listing->region, 'category' => $listing->category]) }}">{{ $listing