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A split system $\mathcal S$ on a finite set $X$, $|X|\ge3$, is a set of bipartitions or splits of $X$ which contains all splits of the form $\{x,X-\{x\}\}$, $x \in X$. To any such split system $\mathcal S$ we can associate the Buneman graph $\mathcal B(\mathcal S)$ which is essentially a median graph with leaf-set $X$ that displays the splits in $\mathcal S$. In this paper, we consider properties of injective split systems, that is, split systems $\mathcal S$ with the property that $\mathrm{med}_{\mathcal B(\mathcal S)}(Y) \neq \mathrm{med}_{\mathrm B(\mathcal S)}(Y')$ for any 3-subsets $Y,Y'$ in $X$, where $\mathrm {med}_{\mathcal B(\mathcal S)}(Y)$ denotes the median in $\mathcal B(\mathcal S)$ of the three elements in $Y$ considered as leaves in $\mathcal B(\mathcal S)$. In particular, we show that for any set $X$ there always exists an injective split system on $X$, and we also give a characterization for when a split system is injective. We also consider how complex the Buneman graph $\mathcal B(\mathcal S)$ needs to become in order for a split system $\mathcal S$ on $X$ to be injective. We do this by introducing a quantity for $|X|$ which we call the injective dimension for $|X|$, as well as two related quantities, called the injective 2-split and the rooted-injective dimension. We derive some upper and lower bounds for all three of these dimensions and also prove that some of these bounds are tight. An underlying motivation for studying injective split systems is that they can be used to obtain a natural generalization of symbolic tree maps. An important consequence of our results is that any three-way symbolic map on $X$ can be represented using Buneman graphs., Comment: 22 pages, 3 figures

Introgression is an evolutionary process in which genes or other types of genetic material are introduced into a genome. It is an important evolutionary process that can, for example, play a fundamental role in speciation. Recently the concept of an overlaid species forest was introduced to represent introgression histories. Basically this approach takes a putative gene history in the form of a phylogenetic gene tree and tries to overlay this onto a forest which usually consists of a collection of lineage trees for the species of interest. The result is a network called an overlaid species forest in which genes jump or introgress between lineages. In this paper we study properties of overlaid species forests, showing that they have various connections with models for lateral gene transfer, maximum parsimony, and unfolding of phylogenetic networks. In particular, we show that a certain algorithm called OSF-B UILDER for constructing overlaid species forests is guaranteed to a produce a special type of overlaid species forest with a minimum number introgressions, as well as providing some characterizations for networks that can arise from overlaid species forests. We expect that these results will be useful in developing new methods for representing introgression histories, a growing area of interest in phylogenetics., Comment: 18 pages, 5 figures

The tight-span of a finite metric space is a polytopal complex that has appeared in several areas of mathematics. In this paper we determine the polytopal structure of the tight-span of a totally split decomposable (finite) metric. Totally split-decomposable metrics are a generalization of tree-metrics and have importance within phylogenetics. In previous work, we showed that the cells of the tight-span of such a metric are zonotopes that are polytope isomorphic to either hypercubes or rhombic dodecahedra. Here, we extend these results and show that the tight-spanof a totally split-decomposable metric can be broken up into a canonical collection of polytopal complexes whose polytopal structures can be directly determined from the metric. This allows us to also completely determine the polytopal structure of the tight-span of a totally split-decomposable metric in a very direct way.We anticipate that our improved understanding of this structure may ultimately lead to improved techniques for phylogenetic inference.

The mass of galaxy clusters is not a direct observable, nonetheless it is commonly used to probe cosmological models. Based on the combination of all main cluster observables, that is, the X-ray emission, the thermal Sunyaev-Zel'dovich (SZ) signal, the velocity dispersion of the cluster galaxies, and gravitational lensing, the gravitational potential of galaxy clusters can be jointly reconstructed. We derive the two main ingredients required for this joint reconstruction: the potentials individually reconstructed from the observables and their covariance matrices, which act as a weight in the joint reconstruction. We show here the method to derive these quantities. The result of the joint reconstruction applied to a real cluster will be discussed in a forthcoming paper. We apply the Richardson-Lucy deprojection algorithm to data on a two-dimensional (2D) grid. We first test the 2D deprojection algorithm on a $\beta$-profile. Assuming hydrostatic equilibrium, we further reconstruct the gravitational potential of a simulated galaxy cluster based on synthetic SZ and X-ray data. We then reconstruct the projected gravitational potential of the massive and dynamically active cluster Abell 2142, based on the X-ray observations collected with XMM-Newton and the SZ observations from the Planck satellite. Finally, we compute the covariance matrix of the projected reconstructed potential of the cluster Abell 2142 based on the X-ray measurements collected with XMM-Newton. The gravitational potentials of the simulated cluster recovered from synthetic X-ray and SZ data are consistent, even though the potential reconstructed from X-rays shows larger deviations from the true potential. Regarding Abell 2142, the projected gravitational cluster potentials recovered from SZ and X-ray data reproduce well the projected potential inferred from gravitational-lensing observations. (abridged), Comment: accepted for publication in the journal A&A

We report the detection of the secondary eclipse of HAT-P-11 b, a Neptune-sized planet orbiting an active K4 dwarf. Using all available short-cadence data of the Kepler mission, we derive refined planetary ephemeris increasing their precision by more than an order of magnitude. Our simultaneous primary and secondary transit modeling results in improved transit and orbital parameters. In particular, the precise timing of the secondary eclipse allows to pin down the orbital eccentricity to $0.26459_{-0.00048}^{+0.00069}$. The secondary eclipse depth of $6.09_{-1.11}^{+1.12}$ ppm corresponds to a $5.5\sigma$ detection and results in a geometric albedo of $0.39\pm0.07$ for HAT-P-11 b, close to Neptune's value, which may indicate further resemblances between these two bodies. Due to the substantial orbital eccentricity, the planetary equilibrium temperature is expected to change significantly with orbital position and ought to vary between $630^\circ$ K and $950^\circ$ K, depending on the details of heat redistribution in the atmosphere of HAT-P-11 b., Comment: Accepted by A&A, 27/10/2016

Reconstructing the evolutionary past of a family of genes is an important aspect of many genomic studies. To help with this, simple operations on a set of sequences called orthology relations may be employed. In addition to being interesting from a practical point of view they are also attractive from a theoretical perspective in that e. g. a characterization is known for when such a relation is representable by a certain type of phylogenetic tree. For an orthology relation inferred from real biological data it is however generally too much to hope for that it satisfies that characterization. Rather than trying to correct the data in some way or another which has its own drawbacks, as an alternative, we propose to represent an orthology relation $\delta$ in terms of a structure more general than a phylogenetic tree called a phylogenetic network. To compute such a network in the form of a level-1 representation for $\delta$, we introduce the novel {\sc Network-Popping} algorithm which has several attractive properties. In addition, we characterize orthology relations $\delta$ on some set $X$ that have a level-1 representation in terms of eight natural properties for $\delta$ as well as in terms for level-1 representations of orthology relations on certain subsets of $X$., Comment: 27 pages, 7 Figures

Phylogenetic networks have gained prominence over the years due to their ability to represent complex non-treelike evolutionary events such as recombination or hybridization. Popular combinatorial objects used to construct them are triplet systems and cluster systems, the motivation being that any network $N$ induces a triplet system $\mathcal R(N)$ and a softwired cluster system $\mathcal S(N)$. Since in real-world studies it cannot be guaranteed that all triplets/softwired clusters induced by a network are available it is of particular interest to understand whether subsets of $\mathcal R(N)$ or $\mathcal S(N)$ allow one to uniquely reconstruct the underlying network $N$. Here we show that even within the highly restricted yet biologically interesting space of level-1 phylogenetic networks it is not always possible to uniquely reconstruct a level-1 network $N$ even when all triplets in $\mathcal R(N)$ or all clusters in $\mathcal S(N)$ are available. On the positive side, we introduce a reasonably large subclass of level-1 networks the members of which are uniquely determined by their induced triplet/softwired cluster systems. Along the way, we also establish various enumerative results, both positive and negative, including results which show that certain special subclasses of level-1 networks $N$ can be uniquely reconstructed from proper subsets of $\mathcal R(N)$ and $\mathcal S(N)$. We anticipate these results to be of use in the design of, for example, algorithms for phylogenetic network inference.

Transit timing variations (TTVs) of exoplanets are normally interpreted as the consequence of gravitational interaction with additional bodies in the system. However, TTVs can also be caused by deformations of the system transits by starspots, which might thus pose a serious complication in their interpretation. We therefore simulate transit light curves deformed by spot-crossing events for different properties of the stellar surface and the planet, such as starspot position, limb darkening, planetary period, and impact parameter. Mid-transit times determined from these simulations can be significantly shifted with respect to the input values; these shifts cannot be larger than ~1% of the transit duration and depend most strongly on the longitudinal position of the spot during the transit and the transit duration. Consequently, TTVs with amplitudes larger than the above limit are very unlikely to be caused by starspots. We also investigate whether TTVs from sequences of consecutive transits with spot-crossing anomalies can be misinterpreted as the result of an additional body in the system. We use the Generalized Lomb-Scargle periodogram to search for periods in TTVs and conclude that low amplitude TTVs with statistically significant periods around active stars are the most problematic cases. In those cases where the photometric precision is high enough to inspect the transit shapes for deformations, it should be possible to identify TTVs caused by starspots, however, especially for cases with low transit signal to noise light curves (TSNR $\lesssim$ 15) it becomes quite difficult to reliably decide whether these periods come from starspots, physical companions in the system or if they are random noise artifacts., Comment: 9 pages, 11 figures

HD 181068 is the only compact, triply eclipsing, hierarchical triple system containing a giant star known to date. With its central, highly-active G-type giant orbited by a close pair of main-sequence dwarfs, the system is ideal to study tidal interactions. We carried out a multiwavelength study to characterize the magnetic activity of the HD 181068 system. To this end, we obtained in- and out-of-eclipse X-ray snapshots with XMM-Newton and an optical spectrum, which we analyzed along with the Kepler light-curve. The primary giant shows strong quiescent X-ray emission at a level of 2e31 ergs, an S-index of 0.41 +/- 0.01, and marked white-light flares releasing up to 6e38 erg in the Kepler-band. During the second X-ray observation, we found a three-times elevated -- yet decaying -- level of X-ray emission, which might be due to an X-ray flare. The high level of magnetic activity is compatible with the previously reported absence of solar-like oscillations in the giant, whose atmosphere, however, undergoes tidally-induced oscillations imposed by the changing configuration of the dwarf-binary. We found that the driving force exciting these oscillations is comparable to the disturbances produced by a typical hot Jupiter, making the system a potential test bed to study the effects of tidal interactions also present in planetary systems., Comment: Accepted for publication in A&A

The photon-ion merged-beams technique has been employed at the new Photon-Ion spectrometer at PETRA III (PIPE) for measuring multiple photoionization of Xe$^{q+}$ (q=1-5) ions. Total ionization cross sections have been obtained on an absolute scale for the dominant ionization reactions of the type h\nu + Xe$^{q+}$ $\to$ Xe$^{r+}$ + (q-r) e$^-$ with product charge states q+2 $\le$ r $\le$ q+5. Prominent ionization features are observed in the photon-energy range 650-750 eV, which are associated with excitation or ionization of an inner-shell 3d electron. Single-configuration Dirac-Fock calculations agree quantitatively with the experimental cross sections for non-resonant photoabsorption, but fail to reproduce all details of the measured ionization resonance structures., Comment: 20 pages, 12 figures, submitted for publication to J. Phys. B

Given a finite connected simple graph $G=(V,E)$ with vertex set $V$ and edge set $E\subseteq \binom{V}{2}$, we will show that $1.$ the (necessarily unique) smallest block graph with vertex set $V$ whose edge set contains $E$ is uniquely determined by the $V$-indexed family ${\bf P}_G:=\big(\pi_0(G^{(v)})\big)_{v \in V}$ of the various partitions $\pi_0(G^{(v)})$ of the set $V$ into the set of connected components of the graph $G^{(v)}:=(V,\{e\in E: v\notin e\})$, $2.$ the edge set of this block graph coincides with set of all $2$-subsets $\{u,v\}$ of $V$ for which $u$ and $v$ are, for all $w\in V-\{u,v\}$, contained in the same connected component of $G^{(w)}$, $3.$ and an arbitrary $V$-indexed family ${\bf P}p=({\bf p}_v)_{v \in V}$ of partitions $\pi_v$ of the set $V$ is of the form ${\bf P}p={\bf P}p_G$ for some connected simple graph $G=(V,E)$ with vertex set $V$ as above if and only if, for any two distinct elements $u,v\in V$, the union of the set in ${\bf p}_v$ that contains $u$ and the set in ${\bf p}_u$ that contains $v$ coincides with the set $V$, and $\{v\}\in {\bf p}_v$ holds for all $v \in V$. As well as being of inherent interest to the theory of block graphs, these facts are also useful in the analysis of compatible decompositions and block realizations of finite metric spaces.

A simple expression for the zeros of Weierstrass' function is given which follows from a formula for relativistic orbits., Comment: 5 pages

Phylogenetic networks are a generalization of phylogenetic trees that are used in biology to represent reticulate or non-treelike evolution. Recently, several algorithms have been developed which aim to construct phylogenetic networks from biological data using {\em triplets}, i.e. binary phylogenetic trees on 3-element subsets of a given set of species. However, a fundamental problem with this approach is that the triplets displayed by a phylogenetic network do not necessary uniquely determine or {\em encode} the network. Here we propose an alternative approach to encoding and constructing phylogenetic networks, which uses phylogenetic networks on 3-element subsets of a set, or {\em trinets}, rather than triplets. More specifically, we show that for a special, well-studied type of phylogenetic network called a 1-nested network, the trinets displayed by a 1-nested network always encode the network. We also present an efficient algorithm for deciding whether a {\em dense} set of trinets (i.e. one that contains a trinet on every 3-element subset of a set) can be displayed by a 1-nested network or not and, if so, constructs that network. In addition, we discuss some potential new directions that this new approach opens up for constructing and comparing phylogenetic networks.

CoRoT-2 is one of the most unusual planetary systems known to date. Its host star is exceptionally active, showing a pronounced, regular pattern of optical variability caused by magnetic activity. The transiting hot Jupiter, CoRoT-2b, shows one of the largest known radius anomalies. We analyze the properties and activity of CoRoT-2A in the optical and X-ray regime by means of a high-quality UVES spectrum and a 15 ks Chandra exposure both obtained during planetary transits. The UVES data are analyzed using various complementary methods of high-resolution stellar spectroscopy. We characterize the photosphere of the host star by deriving accurate stellar parameters such as effective temperature, surface gravity, and abundances. Signatures of stellar activity, Li abundance, and interstellar absorption are investigated to provide constraints on the age and distance of CoRoT-2. Furthermore, our UVES data confirm the presence of a late-type stellar companion to CoRoT-2A that is gravitationally bound to the system. The Chandra data provide a clear detection of coronal X-ray emission from CoRoT-2A, for which we obtain an X-ray luminosity of 1.9e29 erg/s. The potential stellar companion remains undetected in X-rays. Our results indicate that the distance to the CoRoT-2 system is approximately 270 pc, and the most likely age lies between 100 and 300 Ma. Our X-ray observations show that the planet is immersed in an intense field of high-energy radiation. Surprisingly, CoRoT-2A's likely coeval stellar companion, which we find to be of late-K spectral type, remains X-ray dark. Yet, as a potential third body in the system, the companion could account for CoRoT-2b's slightly eccentric orbit., Comment: accepted for publication by A&A

A classical result, fundamental to evolutionary biology, states that an edge-weighted tree $T$ with leaf set $X$, positive edge weights, and no vertices of degree 2 can be uniquely reconstructed from the set of leaf-to-leaf distances between any two elements of $X$. In biology, $X$ corresponds to a set of taxa (e.g. extant species), the tree $T$ describes their phylogenetic relationships, the edges correspond to earlier species evolving for a time until splitting in two or more species by some speciation/bifurcation event, and their length corresponds to the genetic change accumulating over that time in such a species. In this paper, we investigate which subsets of $\binom{X}{2}$ suffice to determine (`lasso') a tree from the leaf-to-leaf distances induced by that tree. The question is particularly topical since reliable estimates of genetic distance - even (if not in particular) by modern mass-sequencing methods - are, in general, available only for certain combinations of taxa.

Given a set $\Sg$ of bipartitions of some finite set $X$ of cardinality at least 2, one can associate to $\Sg$ a canonical $X$-labeled graph $\B(\Sg)$, called the Buneman graph. This graph has several interesting mathematical properties - for example, it is a median network and therefore an isometric subgraph of a hypercube. It is commonly used as a tool in studies of DNA sequences gathered from populations. In this paper, we present some results concerning the {\em cut vertices} of $\B(\Sg)$, i.e., vertices whose removal disconnect the graph, as well as its {\em blocks} or 2-{\em connected components} - results that yield, in particular, an intriguing generalization of the well-known fact that $\B(\Sg)$ is a tree if and only if any two splits in $\Sg$ are compatible.

The lightcurve of CoRoT-2 shows substantial rotational modulation and deformations of the planet's transit profiles caused by starspots. We consistently model the entire lightcurve, including both rotational modulation and transits, stretching over approximately 30 stellar rotations and 79 transits. The spot distribution and its evolution on the noneclipsed and eclipsed surface sections are presented and analyzed, making use of the high resolution achievable under the transit path. We measure the average surface brightness on the eclipsed section to be (5\pm1) % lower than on the noneclipsed section. Adopting a solar spot contrast, the spot coverage on the entire surface reaches up to 19 % and a maximum of almost 40 % on the eclipsed section. Features under the transit path, i.e. close to the equator, rotate with a period close to 4.55 days. Significantly higher rotation periods are found for features on the noneclipsed section indicating a differential rotation of $\Delta \Omega > 0.1$. Spotted and unspotted regions in both surface sections concentrate on preferred longitudes separated by roughly 180 deg., Comment: Paper accepted by A&A 17/02/2010. For a better resolution paper please visit my homepage: http://www.hs.uni-hamburg.de/EN/Ins/Per/Huber/index.html

The theory of the tight span, a cell complex that can be associated to every metric $D$, offers a unifying view on existing approaches for analyzing distance data, in particular for decomposing a metric $D$ into a sum of simpler metrics as well as for representing it by certain specific edge-weighted graphs, often referred to as realizations of $D$. Many of these approaches involve the explicit or implicit computation of the so-called cutpoints of (the tight span of) $D$, such as the algorithm for computing the "building blocks" of optimal realizations of $D$ recently presented by A. Hertz and S. Varone. The main result of this paper is an algorithm for computing the set of these cutpoints for a metric $D$ on a finite set with $n$ elements in $O(n^3)$ time. As a direct consequence, this improves the run time of the aforementioned $O(n^6)$-algorithm by Hertz and Varone by ``three orders of magnitude''., Comment: 17 pages, 1 eps-figure

We analyze the surface structure of the planet host star CoRoT-2a using a consistent model for both the `global' (i.e., rotationally modulated) lightcurve and the transit lightcurves, using data provided by the CoRoT mission. Selecting a time interval covering two stellar rotations and six transits of the planetary companion CoRoT-2b, we adopt a `strip' model of the surface to reproduce the photometric modulation inside and outside the transits simultaneously. Our reconstructions show that it is possible to achieve appropriate fits for the entire sub-interval using a low-resolution surface model with 36 strips. The surface reconstructions indicate that the brightness on the eclipsed section of the stellar surface is (6 +/- 1) % lower than the average brightness of the remaining surface. This result suggests a concentration of stellar activity in a band around the stellar equator similar to the behavior observed on the Sun., Comment: accepted by A&A on 12/09/2009

The detection of planets around very low-mass stars with the radial velocity method is hampered by the fact that these stars are very faint at optical wavelengths where the most high-precision spectrometers operate. We investigate the precision that can be achieved in radial velocity measurements of low mass stars in the near infrared (nIR) Y-, J-, and H-bands, and we compare it to the precision achievable in the optical. For early-M stars, radial velocity measurements in the nIR offer no or only marginal advantage in comparison to optical measurements. Although they emit more flux in the nIR, the richness of spectral features in the optical outweighs the flux difference. We find that nIR measurement can be as precise than optical measurements in stars of spectral type ~M4, and from there the nIR gains in precision towards cooler objects. We studied potential calibration strategies in the nIR finding that a stable spectrograph with a ThAr calibration can offer enough wavelength stability for m/s precision. Furthermore, we simulate the wavelength-dependent influence of activity (cool spots) on radial velocity measurements from optical to nIR wavelengths. Our spot simulations reveal that the radial velocity jitter does not decrease as dramatically towards longer wavelengths as often thought. The jitter strongly depends on the details of the spots, i.e., on spot temperature and the spectral appearance of the spot. Forthcoming nIR spectrographs will allow the search for planets with a particular advantage in mid- and late-M stars. Activity will remain an issue, but simultaneous observations at optical and nIR wavelengths can provide strong constraints on spot properties in active stars., Comment: accepted by ApJ, v2 accepted revision with new precision calculations, abstract abrideg

We analyze variations in the transit lightcurves of CoRoT-2b, a massive hot Jupiter orbiting a highly active G star. We use one transit lightcurve to eclipse-map a photospheric spot occulted by the planet. In this case study we determine the size and longitude of the eclipsed portion of the starspot and systematically study the corresponding uncertainties. We determine a spot radius between $4.5\degr$ and $10.5\degr$ on the stellar surface and the spot longitude with a precision of about $\pm 1$ degree. Given the well-known transit geometry of the CoRoT-2 system, this implies a reliable detection of spots on latitudes typically covered by sunspots; also regarding its size the modelled spot is comparable to large spot groups on the Sun. We discuss the future potential of eclipse mapping by planetary transits for the high-resolution analysis of stellar surface features., Comment: accepted by Astronomy and Astrophysics

Lightcurves have long been used to study stellar activity and have more recently become a major tool in the field of exoplanet research. We discuss the various ways in which stellar activity can influence transit lightcurves, and study the effects using the outstanding photometric data of the CoRoT-2 exoplanet system. We report a relation between the `global' lightcurve and the transit profiles which turn out to be shallower during high spot coverage on the stellar surface. Furthermore, our analysis reveals a color dependence of the transit lightcurve compatible with a wavelength-dependent limb darkening law as observed on the Sun. Taking into account activity-related effects, we re-determine the orbit inclination and planetary radius and find the planet to be approximately 3% larger than reported previously. Our findings also show that exoplanet research cannot generally ignore the effects of stellar activity., Comment: submitted to A&A (to be accepted soon)

The young active G-dwarf star V889 Herculis (HD 171488) shows pronounced spots in Doppler images as well as large variations in photometry and radial velocity (RV) measurements. However, the lifetime and evolution of its active regions are not well known. We study the existence and stability of active regions on the star's surface using complementary data and methods. Furthermore, we analyze the correlation of spot-induced RV variations and Doppler images. Photometry and high-resolution spectroscopy are used to examine stellar activity. A CLEAN-like Doppler Imaging (DI) algorithm is used to derive surface reconstructions. We study high-precision RV curves to determine their modulation due to stellar activity in analogy to the Rossiter-McLaughlin effect. To this end we develop a measure for the shift of a line's center and compare it to RV measurements. We show that large spotted regions exist on V889 Her for more than one year remaining similar in their large scale structure and position. This applies to several time periods of our observations, which cover more than a decade. Furthermore we use DI line profile reconstructions to identify influences of long-lasting starspots on RV measurements. In this way we verify the RV curve's agreement with our Doppler images. Based on long-term RV data we confirm V889 Her's rotation period of $1.3371 \pm 0.0002$ days., Comment: Accepted for publication in A&A

A contemporary and fundamental problem faced by many evolutionary biologists is how to puzzle together a collection $\mathcal P$ of partial trees (leaf-labelled trees whose leaves are bijectively labelled by species or, more generally, taxa, each supported by e. g. a gene) into an overall parental structure that displays all trees in $\mathcal P$. This already difficult problem is complicated by the fact that the trees in $\mathcal P$ regularly support conflicting phylogenetic relationships and are not on the same but only overlapping taxa sets. A desirable requirement on the sought after parental structure therefore is that it can accommodate the observed conflicts. Phylogenetic networks are a popular tool capable of doing precisely this. However, not much is known about how to construct such networks from partial trees, a notable exception being the $Z$-closure super-network approach and the recently introduced $Q$-imputation approach. Here, we propose the usage of closure rules to obtain such a network. In particular, we introduce the novel $Y$-closure rule and show that this rule on its own or in combination with one of Meacham's closure rules (which we call the $M$-rule) has some very desirable theoretical properties. In addition, we use the $M$- and $Y$-rule to explore the dependency of Rivera et al.'s ``ring of life'' on the fact that the underpinning phylogenetic trees are all on the same data set. Our analysis culminates in the presentation of a collection of induced subtrees from which this ring can be reconstructed., Comment: 25 pages