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We solve the regularity problem for Milnor's infinite dimensional Lie groups in the $C^0$-topological context, and provide necessary and sufficient regularity conditions for the (standard) $C^k$-topological setting. We prove that the evolution map is $C^0$-continuous on its domain $\textit{iff}\hspace{1pt}$ the Lie group $G$ is locally $\mu$-convex. We furthermore show that if the evolution map is defined on all smooth curves, then $G$ is Mackey complete. Under the assumption that $G$ is locally $\mu$-convex, we show that each $C^k$-curve for $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$ is integrable (contained in the domain of the evolution map) $\textit{iff}\hspace{1pt}$ $G$ is Mackey complete and $\mathrm{k}$-confined. The latter condition states that each $C^k$-curve in the Lie algebra $\mathfrak{g}$ of $G$ can be uniformly approximated by a special type of sequence that consists of piecewise integrable curves. A similar result is proven for the case $k\equiv 0$; and, we provide several mild conditions that ensure that $G$ is $\mathrm{k}$-confined for each $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$. We finally discuss the differentiation of parameter-dependent integrals in the (standard) $C^k$-topological context. In particular, we show that if the evolution map is defined and continuous on $C^k([0,1],\mathfrak{g})$ for $k\in \mathbb{N}\sqcup\{\infty\}$, then it is smooth thereon $\textit{iff}\hspace{1pt}$ it is differentiable at zero $\textit{iff}\hspace{1pt}$ $\mathfrak{g}$ is $\hspace{0.2pt}$ Mackey$\hspace{1pt}/ \hspace{1pt}$integral$\hspace{1pt}$ complete for $k\in \mathbb{N}_{\geq 1}\sqcup\{\infty\}\hspace{1pt}/\hspace{1pt}k\equiv 0$. This result is obtained by calculating the directional derivatives explicitly, recovering the standard formulas that hold, e.g., in the Banach case., Comment: 72 pages. Version as published in Communications in Analysis and Geometry

We generalize a classical extension result by Seeley in the context of Bastiani’s differential calculus to infinite dimensions. The construction follows Seeley’s original approach, but is significantly more involved as not only $C^k$ -maps (for ) on (subsets of) half spaces are extended, but also continuous extensions of their differentials to some given piece of boundary of the domains under consideration. A further feature of the generalization is that we construct families of extension operators (instead of only one single extension operator) that fulfill certain compatibility (and continuity) conditions. Various applications are discussed as well.

A significant number of bryophyte species are thought to have transcontinental geographic ranges, often with multiple disjunct distribution areas. One of these cases is Epipterygium tozeri (Mniaceae), with a Holarctic distribution and disjunct ranges in western North America, the Mediterranean, Japan and central Asia. Collections from different geographic regions were lumped into E. tozeri based on morphology, but a molecular confirmation was lacking so far. Here, we tested species concepts in the genus Epipterygium, with a special focus on the E. tozeri species complex, combining morphological and DNA sequence data for the nuclear ribosomal ITS region and two plastid loci (trnG intron, trnT‐psbD spacer). In a second step, we reconstructed the historical biogeography of the genus. We found that Epipterygium most likely originated in Asia or North/Central America and that the alleged single widespread species E. tozeri with disjunct ranges is in fact a group of genetically and morphologically distinct taxa, including four overlooked species, for which we provide descriptions: E. atlanticum sp. nov., E. biauritum sp. nov., E. oreophilum sp. nov., and E. yunnanense sp. nov. The biogeographical history of these species is best explained by a step‐wise parallel colonization of the Eurasian and American continents followed by in‐situ speciation., JP was funded by the MINECO through the Juan de la Cierva Program‐Incorporation (IJCI‐2014‐19691) and Ramón y Cajal Program (RYC‐2016‐20506), and Marie Sklodowska‐Curie COFUND, Researchers’ Night and Individual Fellowships Global (MSCA grant agreement No 747238, “UNISLAND”). We thank R. Medina for comments on the manuscript. Open access funding enabled and organized by Projekt DEAL.

Cities have been shown to be biodiverse, but it is unclear what fraction of a regional species pool can live within city borders and how this differs between taxa. Among animals, most research has focused on a few well-studied taxa, such as birds or butterflies. For other species, progress is limited by the paucity of data. We used species occurrence data for 11 taxa and 23 German cities from the Global Biodiversity Information Facility (GBIF) and the different German states, in a 50-km buffer around the city centre, to investigate what proportion of species of the regional species pools also occur in cities. While data could be obtained for all cities from GBIF, state databases only provided data for a subset of cities. Sample coverage of data from GBIF was higher across all taxa than of the state databases. For each database and taxon, we analysed (i) all cities where the number of occurrences of a taxon was >50 and (ii) only those cities where additionally sample coverage was >0.85. Across all taxa studied on average, 44.9 ± 7.2% (GBIF) and 40.8 ± 9.6% (German states) of the species of the regional species pool were also found in cities. When all cities were considered together, more than 76% of all species occurred within city borders. Our results show that German cities harbour a large part of the regional diversity of different taxa when city borders rather than the city centre is considered. This opens up ample opportunities for conservation and for fostering human–nature relationships.

SummaryDue to climate warming, recently deglaciated glacier forefields create virtually uninhabited substrates waiting for initial colonization of bacteria, fungi and plants and serve as an ideal ecosystem for studying transformations in community composition and diversity over time and the interactions between taxonomic groups.In this study, we investigated the composition and diversity of bacteria, and fungi, plants and environmental factors (pH, temperature, plot age and soil nutrients) along a 1.5km glacier forefield. We used random forest analysis to detect how well the composition and diversity of taxonomic groups and environmental factors can be mutually predicted.Community composition and diversity of taxonomic groups predicted each other more accurately than environmental factors predicted the taxonomic groups; within the taxonomic groups bacteria and fungi predicted each other best and the taxa’s composition was better predicted than diversity indices. Additionally, accuracy of prediction among taxonomic groups and environmental factors considerably varied along the successional gradient.Although our results are no direct indication of interactions between the taxa investigated and the environmental conditions, the accurate predictions among bacteria, fungi, and plants do provide insights into the concerted community assembly of different taxa in response to changing environments along a successional gradient.

Community assembly is a result of dispersal, abiotic and biotic characteristics of the habitat as well as stochasticity. A direct comparison between the assembly of microbial and ‘macrobial’ organisms is hampered by the sampling of these communities in different studies, at different sites, or on different scales. In a glacier forefield in the Austrian Alps, we recorded the soil and plant microbiome (bacteria and fungi) and plants that occurred in the same landscape and in close proximity in the same plots. We tested five predictions deduced from assembly processes and revealed deviating patterns of assembly in these community types. In short, microbes appeared to be less dispersal limited than plants, soil microbes and plants strongly responded to abiotic factors whereas the leaf microbiome was plant species-specific and well buffered from environmental conditions. The observed differences in community assembly processes may be attributed to the organisms’ dispersal abilities, the exposure of the habitats to airborne propagules, and habitat characteristics. The finding that assembly is conditional to the characteristics of the organisms, the habitat, and the spatial scale under consideration is thus central for our understanding about the establishment and the maintenance of biodiversity.

Community assembly is a result of dispersal, abiotic and biotic characteristics of the habitat as well as stochasticity. A direct comparison between the assembly of microbial and ‘macrobial’ organisms is hampered by the sampling of these communities in different studies, at different sites or on different scales. In a glacier forefield in the Austrian Alps, we recorded the soil and plant microbiome (bacteria and fungi) and plants that occurred in the same landscape and in close proximity in the same plots. We tested five predictions deduced from assembly processes and revealed deviating patterns of assembly in these community types. In short, microbes appeared to be less dispersal limited than plants and soil microbes, and plants strongly responded to abiotic factors whereas the leaf microbiome was plant species specific and well buffered from environmental conditions. The observed differences in community assembly processes may be attributed to the organisms’ dispersal abilities, the exposure of the habitats to airborne propagules and habitat characteristics. The finding that assembly is conditional to the characteristics of the organisms, the habitat and the spatial scale under consideration is thus central for our understanding about the establishment and the maintenance of biodiversity., A comparison between bacterial and fungal communities associated with leaves or soil and plant communities suggests different assembly processes shaped by characteristics of the organisms and the habitat.

Research on ecological successions and community assembly shaped our understanding of community establishment, co-existence, and diversity. Although both lines of research address the same processes such as dispersal, species sorting, and biotic interactions, they lack unifying concepts. However, recent theoretical advances proposed to integrate both research lines and thus provided hypotheses on how communities assemble over time and form complex ecological systems. This framework predicts a sequence of stochastic and niche-based processes along successional gradients. Shifts in these assembly processes are assumed to occur abruptly once abiotic and biotic factors dominate over dispersal as main driver of community assembly. Considering the multidiversity composed of five organismal groups including plants, animals, and microbes, we empirically show that stochastic dispersal-dominated community assembly is replaced by environmental filters and biotic interactions after around 60 years of succession in a glacier forefield. The niche-based character of later successional processes is further supported by a pronounced decline in multi-beta-diversity after the shift in assembly processes. Our results support recent theories and provide new insights into the emergence of multidiverse and complex ecosystems. Our study will stimulate updates of concepts of community assembly considering multiple taxa with unique and complementary ecological roles and help to bridge the gap between research on successions and community assembly.

We investigate the Lax equation in the context of infinite-dimensional Lie algebras. Explicit solutions are discussed in the sequentially complete asymptotic estimate context, and an integral expansion (sums of iterated Riemann integrals over nested commutators with correction term) is derived for the situation that the Lie algebra is inherited by an infinite-dimensional Lie group in Milnor's sense. In the context of Banach Lie groups (and Lie groups with suitable regularity properties), we generalize the Baker-Campbell-Dynkin-Hausdorff formula to the product integral (with additional nilpotency assumption in the non-Banach case). We combine this formula with the results obtained for the Lax equation to derive an explicit representation of the product integral in terms of the exponential map. An important ingredient in the non-Banach case is an integral transformation that we introduce. This transformation maps continuous Lie algebra-valued curves to smooth ones and leaves the product integral invariant. This transformation is also used to prove a regularity statement in the asymptotic estimate context., Comment: 71 pages. Version as published in Annals of Global Analysis and Geometry

We show that the standard representation of homogeneous isotropic loop quantum cosmology (LQC) is the GNS-representation that corresponds to the unique state on the reduced quantum holonomy-flux $^*$-algebra that is invariant under residual diffeomorphisms $-$ both when the standard algebra is used as well as when one uses the extended algebra proposed by Fleischhack. More precisely, we find that in both situations the GNS-Hilbert spaces coincide, and that in the Fleischhack case the additional algebra elements are just mapped to zero operators. In order for the residual diffeomorphisms to have a well-defined action on the quantum algebra, we have let them act on the fiducial cell as well as on the dynamical variables, thereby recovering covariance. Consistency with Ashtekar and Campiglia in the Bianchi I case is also shown., Comment: 9 pages; Bianchi I case added in Section V; gamma factors corrected in equations (23) and (24); characters defined at the end of Section II; technical oversight corrected in Section IV; Acknowledgements updated; entry added to the bibliography

Given a group $G$ and an abelian $C^*$-algebra $\mathfrak{A}$, the antihomomorphisms $\Theta\colon G\rightarrow \mathrm{Aut}(\mathfrak{A})$ are in one-to-one with those left actions $\Phi\colon G\times \mathrm{Spec}(\mathfrak{A})\rightarrow \mathrm{Spec}(\mathfrak{A})$ whose translation maps $\Phi_g$ are continuous; whereby continuities of $\Theta$ and $\Phi$ turn out to be equivalent if $\mathfrak{A}$ is unital. In particular, a left action $\phi\colon G \times X\rightarrow X$ can be uniquely extended to the spectrum of a $C^*$-subalgebra $\mathfrak{A}$ of the bounded functions on $X$ if $\phi_g^*(\mathfrak{A})\subseteq \mathfrak{A}$ holds for each $g\in G$. In the present paper, we apply this to the framework of loop quantum gravity. We show that, on the level of the configuration spaces, quantization and reduction in general do not commute, i.e., that the symmetry-reduced quantum configuration space is (strictly) larger than the quantized configuration space of the reduced classical theory. Here, the quantum-reduced space has the advantage to be completely characterized by a simple algebraic relation, whereby the quantized reduced classical space is usually hard to compute., Comment: 33 pages. Revised version: Proof of Theorem 4.8 simplified; comments added to Sect. 1 and Sect. 5

We solve the differentiability problem for the evolution map in Milnor's infinite dimensional setting. We first show that the evolution map of each $C^k$-semiregular Lie group $G$ (for $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$) admits a particular kind of sequentially continuity $-$ called Mackey k-continuity. We then prove that this continuity property is strong enough to ensure differentiability of the evolution map. In particular, this drops any continuity presumptions made in this context so far. Remarkably, Mackey k-continuity arises directly from the regularity problem itself, which makes it particular among the continuity conditions traditionally considered. As an application of the introduced notions, we discuss the strong Trotter property in the sequentially-, and the Mackey continuous context. We furthermore conclude that if the Lie algebra of $G$ is a Fr\'{e}chet space, then $G$ is $C^k$-semiregular (for $k\in \mathbb{N}\sqcup\{\infty\}$) if and only if $G$ is $C^k$-regular., Comment: 40 pages. Version as published at Forum Mathematicum (up to additional generalization of Theorem 2 to nets)

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the asymptotic estimate context. Specifically, let $G$ be a Lie group with asymptotic estimate Lie algebra $\mathfrak{g}$, and denote its evolution map by $\mathrm{evol}\colon \mathrm{D}\equiv \mathrm{dom}[\mathrm{evol}]\rightarrow G$, i.e., $\mathrm{D}\subseteq C^0([0,1],\mathfrak{g})$. We show that $\mathrm{evol}$ is $C^\infty$-continuous on $\mathrm{D}\cap C^\infty([0,1],\mathfrak{g})$ if and only if $\mathrm{evol}$ is $C^0$-continuous on $\mathrm{D}\cap C^0([0,1],\mathfrak{g})$. We furthermore show that $G$ is k-confined for $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$ if $G$ is constricted. (The latter condition is slightly less restrictive than to be asymptotic estimate.) Results obtained in a previous paper then imply that an asymptotic estimate Lie group $G$ is $C^\infty$-regular if and only if it is Mackey complete, locally $\mu$-convex, and has Mackey complete Lie algebra - In this case, $G$ is $C^k$-regular for each $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$ (with ``smoothness restrictions'' for $k\equiv\mathrm{lip}$), as well as $C^0$-regular if $G$ is even sequentially complete with integral complete Lie algebra., Comment: 27 pages. Version as published at Indagationes Mathematicae (title refined; presentation improved; proof of Lemma 9 revised). arXiv admin note: text overlap with arXiv:1711.03508

Recent research has highlighted the significance of cities for biodiversity, making them important places for conservation in their own right. Current conservation approaches in cities are mostly defensive. Thus, they focus on remnant pockets of natural areas or try to protect particular species that occur in the built environment. These approaches are vulnerable to further urban development and do not create habitats. An alternative strategy is to make wildlife an integral part of urban development and thereby create a new habitat in the built-up area. Here we address the challenge of choosing target species for such wildlife-inclusive urban design. The starting point of our conceptual framework is the regional species pool, which can be obtained from geo-referenced species data. The existing habitat types on and around the development site and dispersal barriers limit the species numbers to the local species potential. In the next step, the site’s potential for each species is analyzed—how can it be upgraded to host species given the planned development and the life-cycle of the species? For the final choice of target species, traits related to the human–animal interaction are considered. We suggest that stakeholders will be involved in the final species selection. Our approach differs from existing practice, such as expert choice of priority species, by (1) representing an open process where many species are potential targets of conservation, (2) the involvement of stakeholders in a participatory way. Our approach can also be used at larger spatial scales such as quarters or entire cities.

In a paper by Ashtekar and Campiglia, invariance under volume preserving residual diffeomorphisms has been used to single out the standard representation of the reduced holonomy-flux algebra in homogeneous loop quantum cosmology (LQC). In this paper, we use invariance under all residual diffeomorphisms to single out the standard kinematical Hilbert space of homogeneous isotropic LQC for both the standard configuration space $\mathbb{R}_{\mathrm{Bohr}}$, as well as for the Fleischhack one $\mathbb{R} \sqcup \mathbb{R}_{\mathrm{Bohr}}$. We first determine the scale invariant Radon measures on these spaces, and then show that the Haar measure on $\mathbb{R}_{\mathrm{Bohr}}$ is the only such measure for which the momentum operator is hermitian w.r.t. the corresponding inner product. In particular, the measure is forced to be identically zero on $\mathbb{R}$ in the Fleischhack case, so that for both approaches, the standard kinematical LQC-Hilbert space is singled out., 7 pages, contribution to CQG focus issues on applications of loop quantum gravity to cosmology; Citations made consistent, minor typos corrected

In a previous article, analytic 1-submanifolds had been classified w.r.t. their symmetry under a given regular and separately analytic Lie group action on an analytic manifold. It was shown that such an analytic 1-submanifold is either free or (via the exponential map) analytically diffeomorphic to the unit circle or an interval. In this paper, we show that each free analytic 1-submanifold is discretely generated by the symmetry group, i.e., naturally decomposes into countably many symmetry free segments that are mutually and uniquely related by the Lie group action. This is proven under the assumption that the action is non-contractive (which is less restrictive than regular and separately analytic)., Comment: 52 pages. Presentation improved

Analytic curves are classified w.r.t. their symmetry under a regular and separately analytic Lie group action on an analytic manifold. We show that an analytic curve is either exponential or splits into countably many analytic immersive curves, each of them discretely generated by the symmetry group (i.e., each such curve naturally decomposes into countably many symmetry free subcurves that are mutually and uniquely related by the Lie group action). We additionally extend the classification result to the analytic 1-submanifold case. Specifically, we show that an analytic 1-submanifold is either free or (exponential, i.e.) analytically diffeomorphic (via the exponential map) to the unit circle or an interval. The corresponding decomposition results in the free case are outlined in this paper, but proven in a separate one., Comment: 77 pages. Version as published in Differential Geometry and its Applications (up to additional presentational improvements)

In a paper of Ashtekar and Campiglia, residual diffeomorphisms have been used to single out the standard representation of the reduced holonomy-flux algebra in homogeneous loop quantum cosmology (LQC). We show that, in the homogeneous isotropic case, unitarity of the translations w.r.t. the extended $\mathbb{R}$-action (exponentiated reduced fluxes in the standard approach) singles out the Bohr measure on both the standard quantum configuration space $\mathbb{R}_\mathrm{Bohr}$ as well as on the Fleischhack one. Thus, in both situation, the same condition singles out the standard kinematical Hilbert space of LQC., 4 pages

Given a principal fibre bundle with structure group $S$, and a fibre transitive Lie group $G$ of automorphisms thereon, Wang's theorem identifies the invariant connections with certain linear maps $\psi\colon \mathfrak{g}\rightarrow \mathfrak{s}$. In the present paper, we prove an extension of this theorem which applies to the general situation where $G$ acts non-transitively on the base manifold. We consider several special cases of the general theorem, including the result of Harnad, Shnider and Vinet which applies to the situation where $G$ admits only one orbit type. Along the way, we give applications to loop quantum gravity.

Projective structures have successfully been used for the construction of measures in the framework of loop quantum gravity. In the present paper, we establish such structures for the configuration space $\mathbb{R}\sqcup \mathbb{R}_{\mathbb{Bohr}}$, recently introduced in the context of homogeneous isotropic loop quantum cosmology. In contrast to the traditional space $\mathbb{R}_{\mathbb{Bohr}}$, the first one is canonically embedded into the quantum configuration space of the full theory. In particular, for the embedding of states into a corresponding symmetric sector of loop quantum gravity, this is advantageous. However, in contrast to the traditional space, there is no Haar measure on $\mathbb{R}\sqcup \mathbb{R}_{\mathbb{Bohr}}$ defining a canonical kinematical $L^2$-Hilbert space on which operators can be represented. The introduced projective structures allow to construct a family of natural measures on $\mathbb{R}\sqcup \mathbb{R}_{\mathbb{Bohr}}$ whose corresponding $L^2$-Hilbert spaces we finally investigate., Comment: 28 pages