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This finishes the proof of the theorem. 7. Uniformization and large antichains First, let us explain why large uncountable antichains appear even in the non-special tree [MATH] of Theorem 6.2 . Fix a ladder system [MATH] , and consider the colourings [MATH] for [MATH] so that [MATH] is constant 1 on [MATH] if [MATH] , ... |
We can arrange the [MATH] -sequence [MATH] so that [MATH] for [MATH] . So, as in the proof of Theorem 6.2 , we can build all the uniformizations [MATH] for the colouring [MATH] using the same club [MATH] |
Let [MATH] be the domain of [MATH] . Since [MATH] for [MATH] , we must have [MATH] and [MATH] agrees with [MATH] here. Now, [MATH] implies that |
[EQUATION] whenever [MATH] . In turn, any selection [MATH] defines an uncountable antichain [MATH] of [MATH] meeting a club set of levels. |
The antichain above appeared because the map that assigned the uniformization to the colouring was continuous: if two colourings agreed up to some level, their corrensponding uniformizations agreed up to that level as well. However, the fact alone that the above defined colourings all have [MATH] -uniformizations does ... |
Theorem 7.1 Suppose that [MATH] is a family of [MATH] many ladder system colourings. Then if [MATH] holds then there is a Suslin tree [MATH] so that any [MATH] has a full [MATH] -uniformization. |
One can similarly show, without any extra assumptions beyond ZFC, that for any family [MATH] of [MATH] many ladder system colourings, there is a (necessarily special) Aronszajn tree [MATH] so that any [MATH] has a full [MATH] -uniformization. |
Proof. Let [MATH] where [MATH] and [MATH] . Let [MATH] denote the diamond sequence. We aim to construct a Suslin tree [MATH] and the uniformizations [MATH] for [MATH] simultaneously for each [MATH] . So, by induction on [MATH] , we construct [MATH] and [MATH] so that |
(1) [MATH] is a countable, downward closed and normal subtree of [MATH] (2) [MATH] is a uniformization of [MATH] (3) for all [MATH] [MATH] is an end-extension of [MATH] and [MATH] is an end-extension of [MATH] |
(4) if [MATH] is limit and [MATH] is a maximal antichain of [MATH] then any [MATH] extends some element of [MATH] (5) for any [MATH] [MATH] [MATH] and [MATH] there are infinitely many [MATH] above [MATH] so that |
[EQUATION] for all [MATH] The latter condition is a simultaneous version of the richness condition that we introduced before the proof of Theorem 6.1 . It is clear that if we succeed in building these objects then [MATH] is a Suslin tree and [MATH] is a full [MATH] -uniformization of [MATH] |
In limit steps, we can take unions and all the assumptions are preserved. As usual, the non-trivial step in our construction is when we are given [MATH] for some limit [MATH] along with [MATH] for [MATH] with the above conditions. We need to define [MATH] , which is the next level of the tree, extend all the maps [MATH... |
We first construct a map [MATH] that will serve as the restriction of [MATH] to [MATH] Claim 7.2 There is a map [MATH] that uniformizes [MATH] and so that for any [MATH] [MATH] [MATH] and [MATH] , there are infinitely many [MATH] above [MATH] so that |
[EQUATION] for all [MATH] Proof. It is straightforward to check that the forcing [MATH] defined in the proof of Lemma 6.4 can be used to provide this map [MATH] , by choosing the filter [MATH] to be generic enough over the maps [MATH] |
Our next goal is to define [MATH] : first, list [MATH] as [MATH] , each node infinitely often. We will inductively construct cofinal branches [MATH] above [MATH] so that |
(a) [MATH] for all [MATH] (b) [MATH] extends some element of [MATH] if [MATH] is a maximal antichain in [MATH] , and (c) for any [MATH] , and almost all [MATH] |
[EQUATION] If we succeed then we let [MATH] and [MATH] . The last condition ensures that for any [MATH] [MATH] uniformizes [MATH] as well and not just [MATH] |
Suppose that [MATH] is already defined for [MATH] . Let [MATH] extend [MATH] so that [MATH] and [MATH] extends some element of [MATH] if [MATH] is a maximal antichain in [MATH] . List [MATH] as [MATH] . We now build a sequence [MATH] for [MATH] in [MATH] so that |
(i) [MATH] is the [MATH] th element of [MATH] (ii) if [MATH] then for any [MATH] such that [MATH] [EQUATION] Clearly, the branch [MATH] will satisfy our requirements. |
Given [MATH] , we describe the construction of [MATH] . Consider the finite set [EQUATION] and let [MATH] denote the increasing enumeration. We need to find an increasing sequence in [MATH] |
[EQUATION] so that [MATH] and if [MATH] for some [MATH] then [MATH] for [MATH] . Given [MATH] , we simply apply the simultaneous richness property ( ) of the colourings [MATH] to find [MATH] (this is done by setting [MATH] [MATH] and [MATH] ). |
Now that we defined these branches and the level [MATH] , our last job is to define the maps [MATH] (extending [MATH] ) on the new level [MATH] for [MATH] . The only requirement to keep in mind is the simultaneous richness condition ( ) with [MATH] . One can do this by a simple induction of length [MATH] enumerating al... |
Claim 7.3 Suppose that [MATH] is a countable elementary submodel so that [MATH] , and let [MATH] be mutually [MATH] -generic Cohen-functions from [MATH] to [MATH] . If we let [MATH] for [MATH] then [MATH] satisfies condition ( ). |
In any case, we leave the proof to the reader. With this, the [MATH] th step of the induction is done and hence we finished our contruction of the tree [MATH] with the full [MATH] -uniformizations. The fact that [MATH] is Suslin follows form condition ( ). |
8. Closing remarks and open problems Hopefully, our exposition convinced the reader that there is a diverse theory behind Definition 1.1 well worth studying in detail. At the same time, we believe that our proofs demonstrated rather different applications of a wide spectrum of guessing principles, starting from the wea... |
Remarks on Kurepa trees Kurepa tree is an [MATH] -tree with at least [MATH] branches. One can construct Kurepa trees [MATH] from [MATH] or using strong diamond assumptions |
. Moreover, it is also possible to exclude Aronszajn subtrees in these construction. Let us also mention a rather flexible forcing approach due to Todorcevic |
to achieve the same result (with the extra feature that complete binary subtrees are also avoided). Finally, we mention 37 , Lemma 5.8] where it is proved that any ccc forcing preserves that an [MATH] -tree contains no Aronszajn subtrees. |
Now, what can we say about [MATH] -uniformizations for a Kurepa tree [MATH] ? First, let us look at models of CH or the weak diamond [MATH] . Now, for any ladder system [MATH] , there is a colouring [MATH] without an [MATH] -uniformization. So, if [MATH] holds then any [MATH] -uniformization [MATH] of this particular [... |
Problem 8.1 Is it consistent with CH that there is a Kurepa tree [MATH] with [MATH] so that for any Aronszajn subtree [MATH] of [MATH] [MATH] fails. |
Now, if [MATH] and [MATH] has no Aronszajn subtrees then actually [MATH] holds (since the domain of any [MATH] -uniformization includes an [MATH] -branch). In particular, CH must fail. Note that the above cited results from |
imply that even [MATH] (and so [MATH] for all ) is consistent with the existence of Kurepa trees with no Aronszajn subtrees. Various problems |
First, it is natural to ask how crucial it was in Theorem 2.3 that all antichains are countable. Problem 8.2 Suppose CH holds. Does [MATH] for some [MATH] imply the existence of stationary antichains in [MATH] |
Upon reading Theorem 7.1 , it is also natural to consider the smallest size [MATH] of a family [MATH] of ladder system colourings such that there is no single Aronszajn [MATH] such that any [MATH] has a full [MATH] -uniformization. Note that under [MATH] , no such family exists since all ladder system colourings have a... |
Problem 8.3 Is it consistent that [MATH] exists and is bigger than [MATH] In Section , we cited 15 , Theorem 6.2] stating that a Suslin tree always forces [MATH] . It seems unclear if that argument can be extended to show the existence of non [MATH] -uniformizable colourings. |
Problem 8.4 Suppose [MATH] is a Suslin tree and [MATH] and [MATH] are ( [MATH] -names for) a ladder system and Aronszajn tree. Does [MATH] force [MATH] or even [MATH] |
In , we presented a model of CH so that [MATH] holds for some Aronszajn trees [MATH] but fails for others (e.g., for any Suslin tree in that model). It would be nice to solve the following. |
Problem 8.5 Given two sufficiently different trees [MATH] and [MATH] , can we force [MATH] together with the failure of [MATH] Another natural question is whether the number of colours plays a crucial role in the unformization property for trees. |
Problem 8.6 Suppose that [MATH] is an Aronszajn tree and is a ladder system. Does [MATH] imply [MATH] In the classical setting of [MATH] -uniformizations, the answer is no: in fact, if a ladder system [MATH] is only defined on a stationary, co-stationary set [MATH] then the conjunction of CH plus [MATH] and [MATH] are ... |
The following question is inspired by Corollary 6.3 and concerns the classical [MATH] -uniformization theory. Problem 8.7 Is it consistent that [MATH] holds but [MATH] fails for some ladder system [MATH] |
We conjecture that the answer is yes, and one might start by looking at the and where similar questions are studied. Let us point out that we focused solely on ladder systems ranging over all countable limit ordinals. It would be very interesting to see how the tree uniformization theory changes if one restricts the la... |
Problem 8.8 Suppose that is a stationary, co-stationary ladder system. Does CH imply that whenever [MATH] is a Suslin tree then there is a (monochromatic) 2-colouring of with no [MATH] -uniformization? |
Finally, it would be interesting to look at ladder systems and trees on [MATH] , and to study uniformizations there. This was initiated in 27 , Appendix 3] for the classical theory. Whether the uniformization property here also gives consequences on minimal linear orders of size [MATH] is an interesting question for fu... |
# Source: arxiv 1806.03944 # Title: Supervised Machine Learning for Analysing Spectra of Exoplanetary Atmospheres # Sections: all # Downloaded: 2026-03-03T05:16:39.489030+00:00 |
Supervised Machine Learning for Analysing Spectra of Exoplanetary Atmospheres Pablo Márquez-Neila 1,2 , Chloe Fisher Raphael Sznitman , Kevin Heng 2,3 |
1: University of Bern, ARTORG Center for Biomedical Engineering, Murtenstrasse 50, CH-3008, Bern, Switzerland 2: University of Bern, Center for Space and Habitability, Gesellschaftsstrasse 6, CH-3012, Bern, Switzerland |
3: Corresponding author: kevin.heng@csh.unibe.ch The use of machine learning is becoming ubiquitous in astronomy , but remains rare in the study of the atmospheres of exoplanets. Given the spectrum of an exoplanetary atmosphere, a multi-parameter space is swept through in real time to find the best-fit model |
. Known as “atmospheric retrieval”, it is a technique that originates from the Earth and planetary sciences . Such methods are very time-consuming and by necessity there is a compromise between physical and chemical realism versus computational feasibility. Machine learning has previously been used to determine which m... |
. Here, we report an adaptation of the “random forest” method of supervised machine learning , trained on a pre-computed grid of atmospheric models, which retrieves full posterior distributions of the abundances of molecules and the cloud opacity. The use of a pre-computed grid allows a large part of the computational ... |
. We obtain results consistent with the standard nested-sampling retrieval method. Additionally, we can estimate the sensitivity of the measured spectrum to constraining the model parameters and we can quantify the information content of the spectrum. Our method can be straightforwardly applied using more sophisticated... |
We use the previously analysed Hubble Space Telescope Wide Field Camera 3 (WFC3) transmission spectrum of the hot Jupiter WASP-12b, where the volume mixing ratio of water was inferred to be [MATH] to [MATH] and the temperature [MATH] |
. Transmission spectra measure the wavelength-dependent obscuration of starlight by a transiting exoplanet, which encodes signatures of absorption by molecules and clouds in the exoplanetary atmosphere. The choice of this spectrum was to ensure continuity between previous studies |
and because we expect WFC3 to be the workhorse for measuring exo-atmospheric spectra for the immediate future. We implement the random forest method |
, which is a supervised form of machine learning. It combines the use of a decision tree and bootstrapping with replacement, and may be used on both discrete and continuous training sets. A decision tree is a way of splitting a training set into subsets based on common characteristics of its members |
. The splitting is performed so as to maximize the gain in information entropy . Since decision trees are sensitive to slight changes in the training set, they are suitable for use with the bootstrapping method, which constructs the decision tree by randomly drawing from the training set |
The training set consists of 80,000 synthetic WFC3 transmission spectra, each described by 5 parameters: the temperature ( [MATH] ), volume mixing ratios (relative abundances by number) of water ( [MATH] ), ammonia ( [MATH] ) and hydrogen cyanide ( [MATH] ), and a constant cloud opacity ( [MATH] ). Given that these 5 p... |
. For each spectrum, the values of the 5 parameters are randomly generated either from a log-uniform (volume mixing ratios and cloud opacities) or uniform (temperature) distribution. In addition to adopting the same wavelength range and 13 bins of the measured WASP-12b WFC spectrum |
, we assume a noise floor of 50 parts per million (ppm) on the transit depth. In a general machine-learning situation, each member of a training set is associated with a number of characteristics known as “features” (in the jargon of machine learning), e.g., color, height, type of terrain. For a spectrum, the features ... |
Upon setting up the regression tree, we use it in tandem with a bootstrapping method. To train each regression tree, we randomly draw from the 80,000 synthetic spectra in the training set. Upon each draw, the drawn synthetic spectrum is placed back into the training set, allowing for it to be drawn more than once. Each... |
Figure shows the posterior distributions of the temperature, cloud opacity and volume mixing ratios of water, ammonia and hydrogen cyanide. The retrieved water volume mixing ratio ( [MATH] ) and temperature ( [MATH] K) values are broadly consistent with the previous analysis |
. A non-zero cloud opacity ( [MATH] ) is necessary to flatten the spectral continuum blueward of the 1.4 [MATH] m water feature. The degeneracies between the temperature, molecular abundances and cloud opacity are consistent with physical intuition. As the temperature increases linearly, the molecular opacities increas... |
The retrieved volume mixing ratios of ammonia and hydrogen cyanide are several orders of magnitude lower than that of water: [MATH] [MATH] . Running a pair of nested-sampling retrievals shows that the Bayes factor |
between a model with water only versus one with all three molecules is 0.6 (with the former having the higher Bayesian evidence), implying that there is a lack of evidence for strongly favouring one model over the other. Essentially, there is no evidence for claiming the detection of either hydrogen cyanide or ammonia. |
As a consistency check, Figure shows the posterior distributions of parameters from our nested-sampling retrieval . The retrieved parameter values from the nested-sampling retrieval are [MATH] K, [MATH] [MATH] [MATH] [MATH] . It is worth noting that the interpretation of transmission spectra suffers from a “normalizati... |
. To break this normalization degeneracy requires that one specifies a unique relationship between a reference transit radius ( [MATH] ) and reference pressure ( [MATH] ), which cannot be directly inferred from the WFC3 data alone. In practice, what this means is that instead of the volume mixing ratio of molecules ( [... |
Having demonstrated that we can use supervised machine learning to perform atmospheric retrieval, we now push beyond the regular analysis. First, we would like to check the values of the 5 parameters predicted by the random forest method versus “ground truth” values. For the latter, we generate another 20,000 WFC3 synt... |
This comparison between the predicted versus real parameter values provides a rough estimate of the minimum values of the parameters that the retrieval is sensitive to, given the noise model assumed (a constant 50 ppm in our case). For example, the linear trend between the predicted versus real values of the volume mix... |
Second, we can use our approach to analyse the information content of the measured WFC3 transmission spectrum. While information content analysis has been previously considered |
, we offer a complementary analysis and show that this is a natural outcome of the random forest method, which is called the “feature importance” analysis. Figure shows the relative weight of each of the 13 data points in the WFC3 transmission spectrum towards determining the value of each parameter. Physical intuition... |
There are straightforward extensions of random-forest retrieval for which no conceptual obstacles exist. We have demonstrated the method on a spectrum with 13 data points, but the random forest method has been shown to work well even for 1000–10,000 data points |
. This property implies that random-forest retrieval is applicable to future James Webb Space Telescope (JWST) spectra spanning a broader range of wavelengths with [MATH] –1000 data points |
. The information content analysis may be used to influence observational campaigns and the design of spectrographs, depending on the intended scientific goal. |
Another straightforward extension is to train a random forest once and apply it to an ensemble of spectra. In the current study, we picked a specific object (WASP-12b) to demonstrate our method. There is no conceptual obstacle to making model grids where the surface gravity is allowed to vary. The random forest is trai... |
. It is conceivable that one may use model grids produced by different research groups to perform retrievals, even if the computer codes used to generate these grids are proprietary. |
For the current study, we have showed that more sophisticated models are not necessary to analyse the WFC3 spectrum of WASP-12b. However, there is nothing that prevents one from considering more sophisticated models. For example, using the non-isothermal model of |
in tandem with the non-grey cloud model of would add 4 more parameters to the retrieval. A longstanding shortcoming of atmospheric retrieval, which is the non-self-consistency of the physics and chemistry in the models, may now be obviated using random-forest retrieval. |
Data Availability: The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. |
Code Availability: The code used to generate all random-forest retrievals can be accessed at Correspondence and request for materials should be made to K.H. We acknowledge partial financial support from the Center for Space and Habitability (P.M.N. and K.H.), the University of Bern International 2021 Ph.D Fellowship (C... |
P.M.N. led the development of computer codes used for this study, performed the machine-learning related calculations, participated in the experimental design and made the majority of the figures. C.F. computed the grid of atmospheric models used as the training set, participated in the experimental design and performe... |
Methods For the physics input, we choose to use a previously validated analytical formula to convert the temperatures, molecular opacities and relative abundances of molecules into transit radii |
. The simplicity of this forward model allows us to straightforwardly diagnose problems and understand trends in the posterior distributions. We use the simplest incarnation of this formula, which assumes that the atmosphere is isothermal, isobaric and hosts a grey cloud. Using the nested sampling method |
, we have performed regular retrievals, which indicate that non-isothermal behavior and non-grey clouds are not necessary to explain the data given its current level of quality and sophistication. We include the opacities of water (H O), hydrogen cyanide (HCN) and ammonia (NH ), computed using the ExoMol spectroscopic ... |
as input and in the standard way, meaning that the opacities are products of the integrated line strength and line shape, and the line shapes are assumed to be truncated Voigt profiles |
For each model, we randomly pick values of the parameters over the following ranges: [MATH] –2900 K, [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] cm -1 . The surface gravity of WASP-12b is taken to be 977 cm s -2 |
. The spectroscopic database used to construct the NH opacities does not exist for temperatures above 1600 K . For computational reasons, we set the NH opacity to be zero and the volume mixing ratio to be small ( [MATH] ) if the temperature exceeds this threshold. Fortunately, ammonia is expected to be a minor species ... |
The “features” are the 13 values of the transit radius, across wavelength, associated with each transmission spectrum. One may visualize 13 columns, each with 80,000 values of the transit radius. One then visualizes a 13-dimensional space, where each dimension is marked by a set of numerical thresholds. Boundaries in t... |
. The implementations of the random forest method and [MATH] metric are from the open-source scikit.learn library in the Python programming language. |
It has been previously shown that the random forest method is capable of handling systems with 1000–10,000 features and tree depths of several tens to hundreds |
. Our current problem has 13 features and the regression trees have, on average, about 19,000 nodes and depths of 14. To check the robustness of our results with respect to our implementation of the random forest method, we examine retrieval outcomes with different numbers of regression trees. Like before, we train on ... |
We also ran the same mock retrievals for model grids where the atmosphere contains water only versus one that contains hydrogen cyanide and ammonia (without water), as shown in Figure . In the former case, the retrievals return [MATH] and [MATH] even when neither molecule is present in the mock spectra, which is consis... |
As a final test and precursor for future studies, we generated mock JWST-like data in the NIRSpec range of wavelengths (0.8 to 5.0 [MATH] m) at a resolution of 100 (not shown). Despite the increase in the number of features (data points) from 13 to 181, the time needed to train the random forest on 80,000 mock spectra ... |
To determine the spectral resolution used for our opacities, we ran retrievals with resolutions of 1, 2, 5 and 10 cm -1 assuming an isothermal atmosphere containing grey clouds and all three molecules. Retrieval practitioners typically use a spectral resolution of 1 cm -1 for their opacities |
, although it is not uncommon for workers to not state the spectral resolution used. For these 4 resolutions, the retrievals are shown in Figure . The corresponding retrieved parameter values are tabulated. Based on this resolution test, we adopt 5 cm -1 as our spectral resolution for the opacities. |
We assume pressure broadening to be negligible. Since the inferred atmospheric temperature does not fall well below 1000 K and the volume mixing ratios are typically much smaller than unity, this is not an unreasonable assumption |
. Operationally, to implement this assumption we assume a pressure of 1 mbar when computing the opacities. As is accepted practice |
, our ignorance of the physics of pressure broadening forces us to truncate the Voigt profile at some distance from line center. We have made an ad hoc choice of 100 cm -1 , but since pressure broadening is assumed to be negligible this has little to no effect on the outcome. |
To check our assumption of a constant/grey cloud opacity, we ran another retrieval calculation with the non-grey cloud model of . The Bayes factor for the pair of models with grey versus non-grey clouds is 0.6 (with the former having a higher Bayesian evidence), which implies there is no evidence for the data favouring... |
. In fact, we note that the model with non-grey clouds and water only has the same Bayesian evidence as one with grey clouds and all three molecules. Similarly, the Bayes factor for a pair of models with isothermal versus non-isothermal atmospheres (both with water only) is 0.7, implying a lack of evidence for non-isot... |
# Source: arxiv 1806.04034 # Title: The Influence of One Strategic Agent on the Core of Stable Matchings # Sections: all # Downloaded: 2026-03-02T09:22:56.152303+00:00 |
The Influence of One Strategic Agent on the Core of Stable Matchings Abstract In this work, we analyze the influence of a single strategic agent on the quality of the other agents’ matchings in a matching market. We consider a stable matching problem with [MATH] men and [MATH] women when preferences are drawn uniformly... |
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