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We present a new open‐source tool for magnetospheric computations, that is modeling of cosmic ray (CR) propagation in the geomagnetosphere, named "Oulu—Open‐source geomagneToSphere prOpagation tool" (OTSO), available on GitHub (https://github.com/NLarsen15/OTSO) and Zenodo (https://doi.org/10.5281/zenodo.7516233). A tool of this nature is required in order to interpret experiments and study phenomena within the CR research field. Within this work, OTSO is applied to the investigation of several ground‐level enhancement events. Here, we demonstrated several applications of OTSO, namely the computation of asymptotic directions of selected CR stations, effective rigidity cut‐off across the globe at various conditions within the design, and general properties, including the magnetospheric models employed. Comparison and validation of OTSO with older widely used tools such as MAGNETOCOSMICS was performed, and good agreement was achieved. An application of OTSO for providing the necessary background for the analysis of two notable ground‐level enhancements is demonstrated and their spectral and angular characteristics are presented. Plain Language Summary: Cosmic rays (CRs) are hazardous to humans and are constantly bombarding the Earth. Some impacts of these CRs are increased radiation doses for aviation crew and their passengers and damaging spacecraft. Luckily the Earth's magnetic field deflects most of these harmful particles away, mitigating most of the potential damage. The interaction between the Earth's magnetic field and incoming CRs is complex but needs to be properly understood and accurately modeled for the study of specific phenomena relevant to the CR research field. This work presents a new open‐source tool for magnetospheric computations named the "Oulu—Open‐source geomagneToSphere prOpagation tool" (OTSO). This tool can compute the trajectory of CRs within the Earth's magnetic field under various magnetospheric conditions. Within this work, comparison and validation of OTSO with an older widely used tool (MAGNETOCOSMICS) was performed with a good agreement observed. OTSO was also used to successfully aid in the analysis of two ground‐level enhancement events. OTSO demonstrates its usefulness as a tool to aid in the study of CRs in the magnetosphere, and its open‐source nature allows for further development by the research community. Key Points: We present a new open‐source tool for geomagnetospheric computations, named "Oulu—Open‐source geomagneToSphere prOpagation tool" (OTSO)The new tool shows a good agreement with other tools that preform a similar function (e.g., MAGNETOCOSMICS)Ground level enhancement (GLE) 66 and 71 were both successfully analyzed using the tool proving that it can preform comprehensive GLE analyses for future studies [ABSTRACT FROM AUTHOR]

We study the speed of convergence to the asymptotic cone for a finitely generated nilpotent group endowed with a word metric. The first result on this theme is given by Burago who showed that an abelian group endowed with a word metric converges to the normed space with the speed O 1 n in the sense of Gromov–Hausdorff distance. Later Krat showed the same statement for the Heisenberg group, and Breuillard and Le Donne constructed an example, the direct product of the Z and the Heisenberg group with a specific word metric, whose speed of convergence is precisely O 1 n . For 2-step nilpotent groups, we show that if the asymptotic cone is non-singular, then the speed of convergence is O 1 n for any choice of generating set. Our argument can be applied to every nilpotent Lie group with a left-invariant sub-Finsler metric. In terms of sub-Finsler geometry, the condition being non-singular is equivalent to the strongly bracket generating condition, and also to absence of abnormal curves. [ABSTRACT FROM AUTHOR]

We study optimization problems with interval objective functions. We focus on set-type solution notions defined using the Kulisch–Miranker order between intervals. We obtain bounds for the asymptotic cones of level, colevel and solution sets that allow us to deduce coercivity properties and coercive existence results. Finally, we obtain various noncoercive existence results. Our results are easy to check since they are given in terms of the asymptotic cone of the constraint set and the asymptotic functions of the end point functions. This work extends, unifies and sheds new light on the theory of these problems. [ABSTRACT FROM AUTHOR]

We use asymptotic analysis for dealing with quasiconvex optimization problems in reflexive Banach spaces. We study generalized asymptotic (recession) cones for nonconvex and nonclosed sets and its respective generalized asymptotic functions. We prove that the generalized asymptotic functions defined in previous works directly through closed formulae can also be generated from the generalized asymptotic cones. We establish three characterizations results for the nonemptiness and compactness of the solution set for noncoercive quasiconvex minimization problems using different asymptotic functions. Finally, we present a sufficient condition for the nonemptiness and boundedness of the solution set for quasiconvex pseudomonotone equilibrium problems. [ABSTRACT FROM AUTHOR]

We prove that the R. Thompson groups F, T, V have linear divergence functions. [ABSTRACT FROM AUTHOR]

Given a symplectic surface (Σ , ω) of genus g ≥ 4 , we show that the free group with two generators embeds into every asymptotic cone of (Ham (Σ , ω) , d H ) , where d H is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds. [ABSTRACT FROM AUTHOR]

We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion-free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other. As Lipschitz injections need not be bi-Lipschitz embeddings, this is a strengthening of a classical result of Pansu in the context of groups of the same homogeneous dimension. [ABSTRACT FROM AUTHOR]

First, we show that , the class of functions defined as the point-wise minimum of n+1 affine functions defined on the Euclidean space , has the peaking property. Then we characterize -convex functions by means of asymptotic cones and asymptotic functions. Finally, some conditions in order to guarantee the nonemptiness of subdifferentials of -convex functions are established. [ABSTRACT FROM AUTHOR]

We introduce a new asymptotic function, which is mainly adapted to quasiconvex functions. We establish several properties and calculus rules for this concept and compare it to previous notions of generalized asymptotic functions. Finally, we apply our new definition to quasiconvex optimization problems: we characterize the boundedness of the function, and the nonemptiness and compactness of the set of minimizers. We also provide a sufficient condition for the closedness of the image of a nonempty closed and convex set via a vector-valued function. [ABSTRACT FROM AUTHOR]

In this article, we prove that if a finitely presented group has an asymptotic cone which is tree-graded with respect to a precise set of pieces then it is relatively hyperbolic. This answers a question of Mark Sapir and generalizes a result of Kapovich and Kleiner to relatively hyperbolic groups. [ABSTRACT FROM AUTHOR]

We compute the divergence of the finitely generated group SLn(𝒪𝒮), where 𝒮 is a finite set of valuations of a function field and 𝒪𝒮is the corresponding ring of 𝒮-integer points. As an application, we deduce that all its asymptotic cones are without cut-points. [ABSTRACT FROM AUTHOR]

We show that a closed, connected, oriented, Riemannian n-manifold, admitting a branched cover of bounded length distortion from Rn, has a virtually Abelian fundamental group. [ABSTRACT FROM AUTHOR]

Let Γ be a finitely generated group and G be a noncompact semisimple connected real Lie group with finite center. We consider the space $${\fancyscript X(\Gamma, G)}$$ of conjugacy classes of semisimple representations of Γ into G, which is the maximal Hausdorff quotient of $${{\rm Hom}(\Gamma, G)/G}$$ . We define the translation vector of $${g \in G}$$, with value in a Weyl chamber, as a natural refinement of the translation length of g in the symmetric space associated with G. We construct a compactification of $${\fancyscript X(\Gamma, G)}$$ , induced by the marked translation vector spectrum, generalizing Thurston's compactification of the Teichmüller space. We show that the boundary points are projectivized marked translation vector spectra of actions of Γ on affine buildings with no global fixed point. An analoguous result holds for any reductive group G over a local field. [ABSTRACT FROM AUTHOR]

We prove an analog of the base change functor of Λ-trees in the setting of generalized affine buildings. The proof is mainly based on local and global combinatorics of the associated spherical buildings. As an application we obtain that the class of generalized affine buildings is closed under taking ultracones and asymptotic cones. Other applications involve a complex of groups decompositions and fixed point theorems for certain classes of generalized affine buildings. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study two problems concerning Assouad–Nagata dimension: [(1)] Is there a metric space of positive asymptotic Assouad–Nagata dimension such that all of its asymptotic cones are of Assouad–Nagata dimension zero? (Dydak and Higes, 2008 ). [(2)] Suppose G is a locally finite group with a proper left invariant metric . If , is infinite? (Brodskiy et al., preprint ). The first question is answered positively. We provide examples of metric spaces of positive (even infinite) Assouad–Nagata dimension such that all of its asymptotic cones are ultrametric. The metric spaces can be groups with proper left invariant metrics. The second question has a negative solution. We show that for each n there exists a locally finite group of Assouad–Nagata dimension n. As a consequence this solves for non-finitely generated countable groups the question about the existence of metric spaces of finite asymptotic dimension whose asymptotic Assouad–Nagata dimension is larger but finite. [ABSTRACT FROM AUTHOR]

Let Mg denote the moduli space of compact Riemann surfaces of genus g and let Ag be the moduli space of principally polarized abelian varieties of dimension g. Let J : Mg → Ag be the map which associates to a Riemann surface its Jacobian. The map J is injective, and the image Jg ≔ J(Mg) is contained in a proper subvariety of Ag when g ≥ 4. The classical and long-studied Schottky problem is to characterize the Jacobian locus Jg in Ag. In this paper we address a large scale version of this problem posed by Farb and called the coarse Schottky problem: What does Jg look like "from far away", or how "dense" is Jg in the sense of coarse geometry? The large scale geometry of Ag is encoded in its asymptotic cone, Cone∞(Ag), which is a Euclidean simplicial cone of real dimension g. Ourmain result asserts that the Jacobian locus Jg is "coarsely dense" in Ag, which implies that the subset of Cone∞(Ag) determined by Jg actually coincides with this cone. The proof shows that the Jacobian locus of hyperelliptic curves is coarsely dense in Ag as well. We also study the boundary points of the Jacobian locus Jg in Ag and in the Baily-Borel and the Borel-Serre compactification. We show that for large genus g the set of boundary points of Jg in these compactifications is "small". [ABSTRACT FROM AUTHOR]

As a consequence of an abstract theorem proved elsewhere, a vector Weierstrass theorem for the existence of a weakly efficient solution without any convexity assumption is established. By using the notion (recently introduced in an earlier paper) of semistrict quasiconvexity for vector functions and assuming additional structure on the space, new existence results encompassing many results appearing in the literature are derived. Also, when the cone defining the preference relation satisfies some mild assumptions (but including the polyhedral and icecream cones), various characterizations for the nonemptiness and compactness of the weakly efficient solution set to convex vector optimization problems are given. Similar results for a class of nonconvex problems on the real line are established as well. [ABSTRACT FROM AUTHOR]