R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

Aims. We examine the uncertainties in current planetary models and quantify their impact on the planet cooling histories and massradius relationships. Methods. These uncertainties include (i) the differences between the various equations of state used to characterize the heavy material thermodynamical properties, (ii) the distribution of heavy elements within planetary interiors, (iii) their chemical composition, and (iv) their thermal contribution to the planet evolution. Our models, which include a gaseous H/He envelope, are compared with models of solid, gasless Earth-like planets in order to examine the impact of a gaseous envelope on the cooling and the resulting radius. Results. We find that, for a fraction of heavy material larger than 20% of the planet mass, the distribution of the heavy elements in the planet’s interior substantially affects the evolution and thus the radius at a given age. For planets with large core mass fractions (>50%), such as the Neptune-mass transiting planet GJ 436b, the contribution of the gravitational and thermal energy from the core to the planet cooling history is not negligible, yielding a ∼10% effect on the radius after 1 Gyr. We show that the present mass and radius determinations of the massive planet Hat-P-2b require at least 200 M⊕ of heavy material in the interior, at the edge of what is currently predicted by the core-accretion model for planet formation. As an alternative avenue for massive planet formation, we suggest that this planet, and similarly HD 17156b, may have formed from collisions between one or several other massive planets. This would explain these planets unusually high density and high eccentricity. We show that if planets as massive as ∼25 MJ can form, as predicted by improved core-accretion models, deuterium is able to burn in the H/He layers above the core, even for core masses as high as ∼100 M⊕. Such a result highlights the confusion provided by a definition of a planet based on the deuterium-burning limit. Conclusions. We provide extensive grids of planetary evolution models from 10 M⊕ to 10 MJup, with various fractions of heavy elements. These models provide a reference for analyzing the transit discoveries expected from the CoRoT and Kepler missions and for inferring the internal composition of these objects.

/pdf/structure-and-evolution-of-super-earth-to-super-jupiter-2oyza71jys.pdf

Structure and evolution of super-Earth to super-Jupiter exoplanets - I. Heavy element enrichment in the interior

This research monograph integrates ideas from category theory, algebra and combinatorics. It is organised in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Benabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work. Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes. Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature. The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students. Titles in this series are co-published with the Centre de Recherches Mathematiques.|This research monograph integrates ideas from category theory, algebra and combinatorics. It is organised in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Benabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work. Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes. Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature. The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students. Titles in this series are co-published with the Centre de Recherches Mathematiques.

Monoidal Functors, Species, and Hopf Algebras

Bialgebroid actions on depth two extensions and duality

An introduction to the theory of groups

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups (H;G; α; β) is deformed using a combinatorial datum (σ; v; r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: G → H in order to obtain a new matched pair (H; (G; *); α′, β′) such that there exists a σ-invariant isomorphism of groups H
 α⋈β
 G ≅H
 α′⋈β′
 (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H
 α⋈β
 G ≅ H
 α′⋈β′
 G′ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed products of groups are given.

Schreier type theorems for bicrossed products

3.1 Separable functors and Frobenius functors


Frobenius functors


Sepanable functors and Maschke’s Theorem


Relative injective and projective objects


Separable functors of the second kind


Fully faithful functors






3.2 Restriction of scalars and the smash product


Application to the smash product


Application to Hopf algebras


Frobenius modules and $\Lambda$-separability






3.3 The functor forgetting the C-coaction


Relative separability






3.4 The functor forgetting the A-action


3.5 The general induction functor


T-generators


Separability of F


Separability of the adjoint of the induction functor

3. Frobenius and separable functors for entwined modules

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups $(H, G, \alpha, \beta)$ is deformed using a combinatorial datum $(\sigma, v, r)$ consisting of an automorphism $\sigma$ of $H$, a permutation $v$ of the set $G$ and a transition map $r: G\to H$ in order to obtain a new matched pair $\bigl(H, (G,*), \alpha', \beta' \bigl)$ such that there exist an $\sigma$-invariant isomorphism of groups $H {}_{\alpha} \bowtie_{\beta} G \cong H {}_{\alpha'} \bowtie_{\beta'} (G,*)$. Moreover, if we fix the group $H$ and the automorphism $\sigma \in \Aut(H)$ then any $\sigma$-invariant isomorphism $H {}_{\alpha} \bowtie_{\beta} G \cong H {}_{\alpha'} \bowtie_{\beta'} G'$ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed product of groups are given.

Rings with finite quotients

Separable functors applied to Doi Hopf Modules. Applications

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Papers

Schreier type theorems for bicrossed products

3. Frobenius and separable functors for entwined modules

Schreier type theorems for bicrossed products

Rings with finite quotients

Separable functors applied to Doi Hopf Modules. Applications