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Clark et al.['The axiomatizability of topological prevarieties', Adv. Math. 218 (2008Math. 218 ( ), 1604Math. 218 ( -1653 have shown that, for k ≥ 2, there exists a Boolean topological graph that is k-colourable but not topologically k-colourable; that is, for every > 0, it cannot be coloured by a paintbrush of width . We generalize this result to show that, for k ≥ 2, there is a Boolean topological graph that is 2-colourable but not topologically k-colourable. This graph is an inverse limit of finite graphs which are shown to exist by an Erdős-style probabilistic argument of Hell and Nešetřil ['The core of a graph', Discrete Math. 109 (1992), 117-126]. We use the fact that there exists a Boolean topological graph that is 2-colourable but not k-colourable, and some other results (some new and some previously known), to answer the question of which finitely generated topological residual classes of graphs are axiomatizable by universal Horn sentences. A more general version of this question was raised in the above-mentioned paper by Clark et al., and has been investigated by various authors for other structures.2000 Mathematics subject classification: primary 05C80; secondary 08C15, 05C15, 57M15.

We say that a directed graph is pre-bipartite if its symmetric closure is bipartite. We will show that the class B of all pre-bipartite digraphs containing no cycles is a universal Horn class. Let U be a universal Horn class contained in B. We determine when it is possible to axiomatise, by first-order sentences, the class R CT (U fin ) of compact topological digraphs that are topologically residually in the class of finite members of U. We show that if R CT (U fin ) is axiomatisable by firstorder sentences, then it is axiomatisable by universal Horn sentences.

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<p>Improvements to skill (and other characteristics) of quantitative probabilistic rainfall forecasting for weather and hydrological purposes are a high priority at the Australian Bureau of Meteorology (the Bureau), highlighted by a series of major floods in eastern Australia in early 2022.  Post-processed ensemble numerical weather prediction (NWP) rainfall guidance is key to increasing forecast automation in routine conditions and providing better guidance for high impact rainfall events. </p><p>The Bureau's existing NWP post-processing system has markedly increased weather rainfall forecast skill in recent years. However, to enable further forecast improvements and greater integration of rainfall processing for weather and hydrological purposes, a more general approach with fewer calibration steps was required. To these ends, we have developed 'RainForests': a multi-ensemble rainfall processing system, utilising gradient boosted decision tree (GBDT) ensembles for forecast calibration.</p><p>RainForests is inspired by the ECPoint method of Hewson and Pillosu (2021). RainForests, like ECPoint, is a non-parametric and generally non-local method which uses decision trees to create situation-dependent error distributions for each input (analogous to an extension of Bayesian Joint Probabilities) that can be used to calibrate grid-scale rainfall guidance to point-scale.</p><p>Key features of RainForests:</p><ul><li>Uses a series of GBDT ensembles to construct the error distribution in place of a single manually trained decision tree. This produces robust outputs which are near-continuous relative to inputs, allows for rapid retraining on new data or addition of feature variables, and utilises open source GBDT software.</li> <li>Uses additive error (delta = obs - forecast) in place of the forecast error ratio. This allows for calibration when forecast rainfall is zero. Error distributions vary with forecast rainfall amount (and other predictors).</li> <li>Models are trained and verified using both rain gauge and gauge-calibrated radar data, each of which have their uncertainties, strengths and weaknesses. Basic Bureau and RainForests QC is applied to the data to reduce gross errors.</li> <li>Error distributions are pooled for each NWP model (e.g. ECMWF ensemble), from around 100 individual input ensemble and deterministic ('ensemble of one') members in total. Resulting calibrated probability distributions for each model are then blended in probability space.</li> </ul><p>Additionally, RainForests uses and contributes to capabilities in the Integrated Model post-PROcessing and VERification (IMPROVER) system being developed in collaboration with the UK Met Office.</p><p>Initial RainForests outputs have comparable skill to the Bureau's existing post-processing system.  Improvements are planned and will be reported on.  It is also planned to provide calibrated rainfall ensemble members, derived from RainForests outputs, to downstream applications.  This aims to support production of ensemble multiple-variable indices, hydrological applications and aggregation of rainfall forecasts in space and time.</p>

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class C of algebras letAssume that C is a class of semigroups possessing a nontrivial member with a neutral element and let H be the universal Horn class generated by G(C ). We prove that the Boolean core of H , i.e., the topological prevariety generated by finite members of H equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of H . We derive analogous results when C is a class of monoids or groups with a nontrivial member.