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We establish a structure theorem, analogous to the classical result of Milnor and Moore, for differential graded Hopf algebras: any differential Hopf algebra $H$ that is free as a coalgebra carries an underlying $B_\infty$ algebra structure that restricts to the subspace of primitives, and conversely $H$ may be recovered via a universal enveloping differential-2-associative algebra. This extends the work of Loday and Ronco [12] where the ungraded non-differential case was treated, and only the multibrace part of the $B_\infty$ structure was found. We show that the multibrace structure of [12] originates from a twisting of a quasi-trivial structure, extending the work of Markl [14] on the $A_\infty$ structure underlying any algebra with a square-zero endomorphism. In this framework it is also clear that the multibrace and $A_\infty$ structures are compatible, and provide an appropriate $B_\infty$ structure for the structure theorem., Comment: 18pp

The radius valley separating super-Earths from mini-Neptunes is a fundamental benchmark for theories of planet formation and evolution. Observations show that the location of the radius valley decreases with decreasing stellar mass and with increasing orbital period. Here, we build from our previous pebble-based formation model, which, combined with photoevaporation after disc dispersal, unveiled the radius valley as a separator between rocky- and water-worlds. We expand our models for a range of stellar masses spanning from 0.1 to 1.5 $M_\odot$. We find that the location of the radius valley is well described by a power-law in stellar mass as $R_{\rm valley} = 1.8197 \, M_{\star}^{\!0.14({+0.02}/{-0.01})}$, which is in excellent agreement with observations. We also find very good agreement with the dependence of the radius valley on orbital period, both for FGK- and M-dwarfs. Additionally, we note that the radius valley gets filled towards low stellar masses, particularly at 0.1-0.4 $M_\odot$, yielding a rather flat slope in $R_{\rm valley} - P_{\rm orb}$. This is the result of orbital migration occurring at lower planet mass for less massive stars, which allows for low-mass water-worlds to reach the inner regions of the system, blurring the separation in mass (and size) between rocky- and water-worlds. Furthermore, we find that for planetary equilibrium temperatures above 400 K, the water in the volatile layer exists fully in the form of steam, puffing the planet radius up compared to condensed-water worlds. This produces an increase in planet radii of $\sim 30\%$ at 1 $M_\oplus$, and of $\sim 15\%$ at 5 $M_\oplus$, compared to condensed-water-worlds. As with Sun-like stars, pebble accretion leaves its imprint on the overall exoplanet population as a depletion of planets with intermediate compositions, carving a valley in planet density for all spectral types (abridged)., Comment: Accepted for publication in A&A Letters. Results from simulations available at https://zenodo.org/records/10719523

For any set S, the free magmatic algebra spanned by card(S) binary products is the vector space spanned by the set of all planar rooted binary trees with the internal nodes colored by the elements of S, graded by the number of leaves of a tree. We show that it has a unique structure of coassociative coalgebra such that the coproduct satisfies the unital infinitesimal condition with each magmatic product, and prove an analog of Aguiar-Sottile formula in this context, describing the coproduct in terms of the Moebius basis for the Tamari order. The last result allows us to compute the subspace of primitive elements of any unital infinitesimal S-magmatic bialgebra. As an example, we construct a set of generators of the dual of Pilaud and Pons bialgebra of integer relations and compute an explicit basis of its subspace of primitive elements., Comment: 36 pages and 15 figures

The study of protoplanetary disc evolution and planet formation has mainly concentrated on solar (and low) mass stars since they host the majority of the confirmed exoplanets. Nevertheless, the numerous planets found orbiting stars up to $\sim3M_\odot$ has sparked interest in understanding how they form and how their hosting discs evolve. Our goal is to improve our knowledge on the gas disc evolution around intermediate mass stars for future planet formation studies. We study the long-term evolution of protoplanetary discs affected by viscous accretion, X-ray and FUV photoevaporation from the central star around stars between $1 - 3M_\odot$ considering the effects of stellar evolution. We explore different values of the viscosity parameter and the initial mass of the disc. We find that the evolutionary pathway of disc dispersal depends on the stellar mass. Our simulations reveal four distinct evolutionary pathways for the gas disc not reported before that are a consequence of stellar evolution, and which will likely impact dust evolution and planet formation. As the stellar mass grows from 1 to $\sim2M_\odot$, the disc evolution changes from the conventional inside-out clearing to a homogeneous disc evolution scenario where both inner and outer discs, formed after photoevaporation opened a gap, vanish over a similar timescale. As the stellar mass continues to increase, reaching $\sim 3M_\odot$, we have identified a distinct pathway that we refer to as revenant disc evolution, where the inner and outer discs reconnect after the gap opened. For the largest masses, we observe outside-in disc dispersal, in which the outer disc dissipates first due to the strong FUV photoevaporation. Revenant disc evolution stands out as it is capable of extending the disc lifespan. Otherwise, the disc dispersal time scale decreases with increasing stellar mass except for low viscosity discs., Comment: 20 pages, 13 figures. Resubmitted to A&A after minor revisions

The gas dissipation from a protoplanetary disk is one of the key processes affecting planet formation, and it is widely accepted that it happens on timescales of a few million years for disks around single stars. Over the last years, several protoplanetary disks have been discovered in multiple star systems, and despite the complex environment in which they find themselves, some of them seem to be quite old, a situation that may favor planet formation. A clear example of this is the disk around HD 98800 B, a binary in a hierarchical quadruple stellar system, which at a $\sim$10 Myr age seems to still be holding significant amounts of gas. Here we present a 1D+1D model to compute the vertical structure and gas evolution of circumbinary disks in hierarchical triple star systems considering different stellar and disk parameters. We show that tidal torques due to the inner binary together with the truncation of the disk due to the external companion strongly reduce the viscous accretion and expansion of the disk. Even allowing viscous accretion by tidal streams, disks in these kind of environments can survive for more than 10 Myr, depending on their properties, with photoevaporation being the main gas dissipation mechanism. We particularly apply our model to the circumbinary disk around HD 98800 B and confirm that its longevity, along with the current non-existence of a disk around the companion binary HD 98800 A, can be explained with our model and by this mechanism., Comment: 20 pages, 9 figures. Accepted for publication in ApJ

In about 6 Giga years our Sun will evolve into a red giant and finally end its life as a white dwarf. This stellar metamorphosis will occur to virtually all known host stars of exo-planetary systems and is therefore crucial for their final fate. It is clear that the innermost planets will be engulfed and evaporated during the giant phase and that planets located farther out will survive. However, the destiny of planets in-between, at ~ 1 - 10 au, has not yet been investigated with a multi-planet tidal treatment. We here combine for the first time multi-planet interactions, stellar evolution, and tidal effects in an N-body code to study the evolution of a Neptune-Jupiter planetary system. We report that the fate of the Neptune-mass planet, located closer to the star than the Jupiter-mass planet, can be very different from the fate of a single Neptune. The simultaneous effects of gravitational interactions, mass loss and tides can drive the planetary system towards mean motion resonances. Crossing these resonances affects particularly the eccentricity of the Neptune and thereby also its fate, which can be engulfment, collision with the Jupiter-mass planet, ejection from the system, or survival at a larger separation., Comment: Accepted for publication in ApJL. 10 pages, 5 figures

M. Carr and S. Devadoss introduced in [7] the notion of tubing on a finite simple graph $\Gamma$, in the context of configuration spaces on the Hilbert plane. To any finite simple graph $\Gamma$ they associated a finite partially ordered set, whose elements are the tubings of $\Gamma$ and whose geometric realization is a convex polytope ${\mathcal K}\Gamma$, the graph-associahedron. For the complete graphs they recovered permutahedra, for linear graphs they got Stasheff's associahedra, while for simple graph they obtained the standard simplexes. The goal of the present work is to give an \emph{algebraic} description of graph associahedra. We introduce a substitution operation on tubings, which allows us to describe the set of faces of graph-associahedra as a free object, spanned by the set of all connected simple graphs, under operations given via connected subgraphs. The boundary maps of graph-associahedra defines natural derivations in this context. Along the way, we introduce a topological interpretation of the graph tubings and our new operations. In the last section, we show that substitution of tubings may be understood in the context of M. Batanin and M. Markl's operadic categories.

Aims. We study the formation and water delivery of planets in the habitable zone (HZ) around solar-type stars. In particular, we study different dynamical environments that are defined by the most massive body in the system. Methods. First of all, a semi-analytical model was used to define the mass of the protoplanetary disks that produce each of the five dynamical scenarios of our research. Then, we made use of the same semi-analytical model to describe the evolution of embryos and planetesimals during the gaseous phase. Finally, we carried out N-body simulations of planetary accretion in order to analyze the formation and water delivery of planets in the HZ in the different dynamical environments. Results. Water worlds are efficiently formed in the HZ in different dynamical scenarios. In systems with a giant planet analog to Jupiter or Saturn around the snow line, super-Earths tend to migrate into the HZ from outside the snow line as a result of interactions with other embryos and accrete water only during the gaseous phase. In systems without giant planets, Earths and super-Earths with high water by mass contents can either be formed in situ in the HZ or migrate into it from outer regions, and water can be accreted during the gaseous phase and in collisions with water-rich embryos and planetesimals. Conclusions. The formation of planets in the HZ with very high water by mass contents seems to be a common process around Sun- like stars. Our research suggests that such planets are still very efficiently produced in different dynamical environments. Moreover, our study indicates that the formation of planets in the HZ with masses and water contents similar to those of Earth seems to be a rare process around solar-type stars in the systems under consideration., Comment: 18 pages. 19 figures

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs., Comment: Expanded for publication, previous title was "Polytopes and Hopf algebras of painted trees: Fan graphs and Stellohedra "

We introduce natural binary set-theoretical products on the set of all $m$-Dyck paths, which led us to define a non-symmetric algebraic operad $\Dy^m$, described on the vector space spanned by $m$-Dyck paths. Our construction is closely related to the $m$-Tamari lattice, so the products defining $\Dy^m$ are given by intervals in this lattice. For $m=1$, we recover the notion of dendriform algebra introduced by J.-L. Loday in \cite{Lod}, and there exists a natural operad morphism from the operad ${\mbox {\it Ass}}$ of associative algebras into the operad $\Dy^m$, consequently $\Dy ^m$ is a Hopf operad. We give a description of the coproduct in terms of $m$-Dyck paths in the last section. As an additional result, for any composition of $m+1\geq 2$ with $r+1$ parts, we get a functor from the category of $\Dy ^m$ algebras into the category of $\Dy ^r$ algebras., Comment: 40 pages

In general, most of the studies of terrestrial-type planet formation typically use ad hoc initial conditions. In this work we improved the initial conditions described in Ronco & de El\'ia (2014) starting with a semi-analytical model wich simulates the evolution of the protoplanetary disk during the gas phase. The results of the semi-analytical model are then used as initial conditions for the N-body simulations. We show that the planetary systems considered are not sensitive to the particular initial distribution of embryos and planetesimals and thus, the results are globally similar to those found in the previous work., Comment: 2 pages, 2 figures - Complex Planetary Systems - IAU Symposium No. 310, 2014

Several studies, observational and theoretical, suggest that planetary systems with only rocky planets should be the most common in the Universe. We study the diversity of planetary systems that might form around Sun-like stars in low-mass disks without giant planets. We focus on the formation process of terrestrial planets in the habitable zone (HZ) and analyze their water contents with the goal to determine systems of astrobiological interest. Besides, we study the formation of planets on wide orbits because they can be detected with the microlensing technique. N-body simulations of high resolution (embryos + planetesimals) are developed for a wide range of surface density profiles. The surface density profile combines a power law to the inside of the disk of the form r^{-gamma}, with an exponential decay to the outside. We adopt a disk of 0.03M_sun and values of gamma = 0.5, 1 and 1.5. All our simulations form planets in the HZ with different masses and final water contents depending on the 3 profiles. For gamma = 0.5, we produce 3 planets in the HZ with masses between 0.03 M_e to 0.1 M_e and water contents between 0.2 and 16 Earth oceans. For gamma = 1, 3 planets form in the HZ with masses between 0.18 M_e and 0.52 M_e and water contents from 34 to 167 Earth oceans. For gamma = 1.5, we find 4 planets in the HZ with masses from 0.66 M_e to 2.21 M_e and water contents between 192 and 2326 Earth oceans. This profile shows distinctive results because it is the only one of those studied here that leads to the formation of water worlds. Since planetary systems with gamma = 1 and 1.5 present planets in the HZ with suitable masses to retain a long-live atmosphere and to maintain plate tectonics, they seem to be the most outstanding candidates to be potentially habitable. Particularly, these systems form Earths and Super-Earths near the snow line which can be discovered by microlensing., Comment: Accepted for publication in Astronomy and Astrophysics - 13 pages, 11 figures

We unravel the algebraic structure which controls the various ways of computing the word ((xy)(zt)) and its siblings. We show that it gives rise to a new type of operads, that we call permutads. It turns out that this notion is equivalent to the notion of "shuffle algebra" introduced by the second author. It is also very close to the notion of "shuffle operad" introduced by V. Dotsenko and A. Khoroshkin. It can be seen as a noncommutative version of the notion of nonsymmetric operads. We show that the role of the associahedron in the theory of operads is played by the permutohedron in the theory of permutads., Comment: Same results, re-arranged and more details. 38 pages

We extend the definition of tridendriform bialgebra by introducing a weight q. The subspace of primitive elements of a q-tridendriform bialgebra is equipped with an associative product and a natural structure of brace algebra, related by a distributive law. This data is called q-Gerstenhaber-Voronov algebras. We prove the equivalence between the categories of connected q-tridendriform bialgebras and of q-Gerstenhaber-Voronov algebras. The space spanned by surjective maps, as well as the space spanned by parking functions, have natural structures of q-tridendriform bialgebras, denoted ST(q) and PQSym(q)*, in such a way that ST(q) is a sub-tridendriform bialgebra of PQSym(q)*. Finally we show that the bialgebra of M-permutations defined by T. Lam and P. Pylyavskyy may be endowed with a natural structure of q-tridendriform algebra which is a quotient of ST(q).

We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). The choice of such an isomorphism implies the existence a finer algebraic structure on the Hopf algebra and on the indecomposables (resp. the primitives). For instance a cofree-cocommutative right-sided combinatorial Hopf algebra is completely determined by its primitive part which is a pre-Lie algebra. The key example is the Connes-Kreimer Hopf algebra. The study of all these combinatorial Hopf algebra types gives rise to several good triples of operads. It involves the operads: dendriform, pre-Lie, brace, and variations of them., Comment: The second part, dealing with right-sided combinatorial Hopf algebras, has been completely modified in this new version

The goal of our work is to study the spaces of primitive elements of the Hopf algebras associated to the permutaedra and the associaedra. We introduce the notion of shuffle and preshuffle bialgebras, and compute the subpaces of primitive elements associated to these algebras. These spaces of primitive elements are free objects for some types of algebras which we describe in terms of generators and relations., Comment: 50 pages

We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n-2 operations of arity n. The dimension of the space of all the n-ary operations of this primitive operad turns out to be the Fine number F_{n-1}. In short, the triple of operads (As, Mag, MagFine) is good., Comment: 17 pages

We prove an analogue of the Poincare'-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a B-infini-algebra. We construct a universal enveloping functor U2 from B-infini-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U2(Prim H). We take advantage of the simple description of the free 2as-algebra in terms of planar trees to unravel the structure of the operad B-infini., Comment: Minor modifications

We extend the weak Bruhat order of a finite Coxeter group to the set of its coclasses, modulo parabolic standard subgroups. We use this order to describe associative algebra structures on the vector spaces spanned by the faces of permutahedra on one side, and on the vector space spanned by the planar rooted trees, on the other one., Comment: 30 pages

We show that the family of standard simplices and the family of Stasheff polytopes are dual to each other in the following sense. The chain modules of the standard simplices, resp. the Stasheff polytopes, assemble to give an operad. We show that these operads are dual of each other in the operadic sense. The main result of this paper is to show that they are both Koszul operads. As a consequence the generating series of the standard simplices and the generating series of the Stasheff polytopes are inverse to each other. The two operads give rise to new types of algebras with 3 generating operations, 11 relations, respectively 7 relations, that we call {\it associative trialgebras} and {\it dendriform trialgebras} respectively. The free dendriform trialgebra, which is based on planar trees, has an interesting Hopf algebra structure, which will be dealt with in another paper. Similarly the family of cubes gives rise to an operad which happens to be self-dual for Koszul duality., Comment: 29 pages

We show that the family of chain modules over the standard simplices can be equipped with an operad structure. Similarly, the family of cochain modules of the Stasheff polytopes can be equipped with an operad structure. We first show that these operads are Koszul dual to each other, and second that they are both Koszul operads. The algebras over the standard simplices operad, called associative trialgebras, are defined by 3 operations and 11 relations. The algebras over the Stasheff polytopes operad, called dendriform trialgebras, are defined by 3 operations and 7 relations., Comment: In french, english abridged version, 8 pages

In "Hopf algebra of the planar binary trees", Adv. Math. 139 (1998), no. 2, 293--309, we constructed by induction a graded associative product on the vector space generated by the planar binary trees (resp. the permutations). In the present paper we prove an explicit formula for this product in terms of a partial order on the set of planar binary trees (resp. the weak Bruhat order on the symmetric groups)., Comment: 19 pages