Countable and Uncountable in Group Theory

A prominent, recurring feature of group theory has been the determination of groups (all of) whose subgroups possess some group theoretical property. For infinite groups in a suitable universe a number of different approaches have been used in this regard. For locally finite groups, for example, knowledge of the structure of the finite subgroups is often crucial. On the other hand the concept of "largeness" has also recently played an interesting role. Moving from this, I started to study how subgroups of uncountable cardinality affect an uncountable group. Let X be a group theoretical proper, let G be a group of uncountable cardinality and suppose that all its proper uncountable subgroups satisfy X. Is it true that all (proper) subgroups of G satisfy X? The thesis exploits this question, showing that, under some soluble conditions, the answer is often positive. Finally the thesis deals with countably recognizable properties, which has a strong relation with the previous question

On derived-indecomposable solutions of the Yang--Baxter equation

If $(X,r)$ is a finite non-degenerate set-theoretic solution of the Yang--Baxter equation, the additive group of the structure skew brace $G(X,r)$ is an $FC$-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an $FC$-group itself. If one additionally assumes that the derived solution of $(X,r)$ is indecomposable, then for every element $b$ of $G(X,r)$ there are finitely many elements of the form $b*c$ and $c*b$, with $c\in G(X,r)$. This naturally leads to the study of a brace-theoretic analogue of the class of $FC$-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation.Comment: 24 pages. Accepted for publication in Publicacions Matem\`atique

Scenario analysis of pollutants loads to European regional seas for the year 2020. Part II: Assessment of priority chemicals – an example with three pilot substances

In order to support the implementation of the Marine Strategy Framework Directive, DG Environment and the Joint Research Centre joined to carry out a study on the expected cumulative impact of existing EU environmental legislation on the quality of the marine environment, with specific reference to the case of aquatic discharges to the European seas. The report describes a few scenario analyses affecting emissions to the European regional seas up to 2020 for Lindane, Trifluralin and Perfluorooctane sulfonate (PFOS) taken as pilot substances. The scenarios developed are agreed with stakeholders at DG ENV following some preparatory meetings. The scenarios do not intend be exhaustive, but examples of what can be further achieved making use of the modelling and database development made in the different phases of the project. For Lindane, the model estimated European sea load of 745 tons per 1995, based on the official emission data provided by EMEP, appears to be reduced by 98.3% in 2005, ten years after the start of the EU regulations for γ-HCH. Besides, under the BAU scenario, a Lindane sea load of ca.12.5 tons per year should be expected. The trend and ban scenarios support, respectively, a reduction of the load to the European seas in 2020 by 74% and 95% when compared to the BAU estimate. Aimed at Trifluralin, according to the BAU scenario, an annual load of ca.61.7 tones is estimated in 2020. However, this is an overestimation, because the aggregated emission data of EUROSTAT for the agriculture use of the entire group of dinitroaniline herbicides in for EU25 have been considered as model input data. The complete ban scenario forecasts ca. 0.07 t/y and in practice eliminates the concern about loads of Trifluralin to European seas to a negligible level in a time-frame of one year due to degradation in soil. Considering PFOS under BAU scenario conditions the total sea load from all European contries is estimated to be 5.8 tons per year. The model forecasts approximately a half of this amount when a 50% reduction of emissions takes place.JRC.H.1-Water Resource

On the lattice of closed subgroups of a profinite group

The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behaviour of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained.Comment: 23 page

Assessment of Persistent Organic Pollutants load to marine environment by MAPPE-Global model focusing on European regional seas

MAPPE-Global modeling tool belongs to the group of the global box models for environmental fate and transport of POPs. The estimated error level of MAPPE-Global is about a factor of two, which suggests that the model could be considered as a tool for a screening or initial evaluation of chemical risk for POPs at global scale. The verified MAPPE-Global is applied in practical assessments of chemical loads to the European regional seas. The PCBs case study relates to a sum of 22 congeners and considers two emission scenarios: first - the current status relative to the 2010 year; second – a future projection corresponding to the 2020 year. The total amount of 22PCBs released to air equals to 101.03 t for year 2010 and to 50.2 t for year 2020, respectively. The Baseline scenario for Lindane assumes, for the reference year 2005, 86.6 t atmospheric emissions only from European sources and omits the impact of the long range atmospheric transport. The second scenario for Lindane aims to targeting the situation in the year 2020. This scenario (denoted as LRT) suggests no emissions to atmosphere from any European origin due to the banning of this substance but admits that Europe is affected by an unavoidable „import” of 5.4 t Lindane through trans-continental air transport. According to the MAPPE-Global model, the extended European area is exporting to the marine water 3.7 t of 22PCBs in 2010 and 1.9 t in 2020, respectively. In both scenarios, the most affected seas are the Mediterranean Sea (ca.35% from the total) followed by Northern (ca. 21.5%) and Black sea (ca. 19%). It was estimated that the European seas receive by atmospheric deposition about 7.9 t of 22PCBs in 2010 and ca. 4t in 2020. This is two times more when comparing to the entire riverine discharge of 22PCBs for these years. MAPPE-Global forecasts 10.1 t riverine sea load of Lindane under the Baseline scenario and 0.26t in the case of LRT meaning 97% reduction compared to the baseline option. The highest discharges are observed for Atlantic Ocean – in Baseline scenario 2.2t (21.8% from the total) and for LRT 0.06t (23%); and Mediterranean Sea - Baseline 3.5t (34.7%) and LRT 0.04t (15.4%). For the gamma-HCH, likewise for the PCBs, it is found that the atmospheric deposition over the European seas dominate the river input to the coastal zone. Under the Baseline scenario, the total air deposition (50.2t/y) is about 5 times higher than the riverine component of the sea load. Potentially the outcome of MAPPE-Global model could serve in the assessments of different policy options related to the EU Water Framework Directive (WFD) or Marine Strategy Framework Directive (MSFD) as well as to support the implementation of the European sea conventions as HELCOM (Baltic Sea), OSPAR (North-East Atlantic), MEDPOL (Mediterranean Sea) and BSC (Black Sea).JRC.H.1-Water Resource

Methods for Regional Integrated Assessment: High resolution gridded emission distribution in the LUISA Platform

This report illustrates the progresses made towards the inclusion of air quality related issues in the Land Use-based Integrated Sustainability Assessment (LUISA) platform. It focuses on the description of the methodology to derive high-resolution gridded-emission spatially geo-referenced layers from outputs and datasets integrated in LUISA. In the framework of the integration of the Regional Integrated Assessment Tool (RIAT model) and the Land Use Modelling Integrated Sustainability Assessment (LUISA) platform, we implemented the downscaling of atmospheric emission data from national level to very high spatial resolution (100m). The GAINS model (IIASA) provides the input emission data for different scenarios, up to year 2030, which are disaggregated based on 34 different surrogates. Each surrogate is calculated by means of the integration of several proxies derived by statistical datasets, ancillary models and GIS layers in the framework of the LUISA platform. The preliminary results for NOx, PM10 and NH3 (year 2010) are presented in this report together with their first assessment, based on existing emission maps at 7 and 10 Km resolution. Future steps for further refinements are also discussed.JRC.H.8-Sustainability Assessmen

Multimedia assessment of pollutant pathways in the environment: a Global scale model

The report describes the assumptions, equations and a few examples of preliminary applications of a global spatial steady-state box model entitled Multimedia Assessment of Pollutant Pathways in the Environment (MAPPE-Global). The model grounds on the concept of already developed European version of MAPPE chemical fate model. MAPPE-Global computes the removal rates of a substance with given physical-chemical properties in an evaluative environment for the Globe with a resolution of 1x1 degree considering atmosphere, land (natural and agriculture soils, forests, impervious surfaces, frozen territories), surface water (including lakes, inland wetlands and reservoirs) and oceans and seas. MAPPE-Global is able to consider chemical emissions in one or more of the environmental compartments and estimates chemical concentrations and fluxes accounting chemical partitioning (gas, liquid or solid), degradation, advective and diffusive transport. At this stage, MAPPE Global does not explicitly compute chemical transport in space, but only the fate of a substance at each location in space. However, the model estimates for each grid cell the mass fluxes of chemical that are available for transport inside or outside of the cell, in addition to concentrations from local emissions. Thus, MAPPE Global is developed specifically to respond questions as: • How will a chemical spread across different media in the different climatic and landscape settings? • How important is the variability of environmental processes in determining the fate of chemicals across the globe? In addition, the model enables estimating, for virtually any location in the world, representative parameters of the environmental removal rates that determine the fate of a contaminant. These rates may be used to feed a zero-dimensional time-dependent model that allows computing the main receptors of the chemical emissions. Besides, in order to evaluate the performance of the MAPPE-Global model a comparison with established models, such as Impact World and USEtox was made by crosschecking of the intermedia removal rate coefficients. Finally, MAPPE-Global was used to quantify for a set of 34 representative pollutants at global scale the range of variability of chemical removal rates for the different environmental compartments and to identify the fate patterns of flyers, swimmers, soil-bound and multimedia chemical substances.JRC.H.5-Rural, water and ecosystem resource

A note on right-nil and strong-nil skew braces

The aim of this short note is to completely answer Questions 2.34 and 2.35 of arXiv:1806.01127. In particular, we show that a finite strong-nil skew brace $B$ of abelian type need not be right-nilpotent, but that this is the case if~$B$ is of nilpotent type and $b\ast b=0$ for all $b\in B$ (our examples show that this is the best possible result).Comment: arXiv admin note: text overlap with arXiv:2310.07474. A typo is corrected in proof of Lemma 2.1 and a step of the proof is explained with more detai

Central nilpotency of left skew braces and solutions of the Yang-Baxter equation

Nipotency of skew braces is related to certain types of solutions of the Yang-Baxter equation. This paper delves into the study of centrally nilpotent skew braces. In particular, we study their torsion theory (Section 4.1) and we introduce an "index" for subbraces (Section 4.2), but we also show that the product of centrally nilpotent ideals need not be centrally nilpotent (Example B), a rather peculiar fact. To cope with these examples, we introduce a special type of nilpotent ideal, using which, we define a {\it good} Fitting ideal. Also, a Frattini ideal is defined and its relationship with the Fitting ideal is investigated. A key ingredient in our work is the characterisation of the commutator of ideals in terms of absorbing polynomials (Section 3); this solves Problem 3.4 of arXiv:2109.04389. Moreover, we provide an example (Example A) showing that the idealiser of a subbrace (as defined in arXiv:2205.01572v2) does not exist in general.Comment: 44 pages, definition of centralizer is modifie

A History of Group Theory through the Lives of Group Theorists: Sophus Lie - Part 1

We continue here our attempt of a systematic historical account of Group Theory inspected by means of the lives and the works of its main actors. The aim is to bring the interested reader through orig- inal correspondences, published and unpublished works, historical perspectives, diatribes and friendships. This issue contains the translation of a memory of Sophus Lie writ- ten by Ludwig Sylow. It was published in the 1899 issue of Archiv for Mathematik of Naturvidenskab soon after Lie’s death. We are grateful to Gunnar Traustason for his translation from Nor- wegian