Detailed Abundances of 15 Stars in the Metal-Poor Globular Cluster NGC 4833

We have observed 15 red giant stars in the relatively massive, metal-poor globular cluster NGC 4833 using the Magellan Inamori Kyocera Echelle spectrograph at Magellan. We calculate stellar parameters for each star and perform a standard abundance analysis to derive abundances of 43 species of 39 elements, including 20 elements heavier than the iron group. We derive = -2.25 +/- 0.02 from Fe I lines and = -2.19 +/- 0.013 from Fe II lines. We confirm earlier results that found no internal metallicity spread in NGC 4833, and there are no significant star-to-star abundance dispersions among any elements in the iron group (19 <= Z <= 30). We recover the usual abundance variations among the light elements C, N, O, Na, Mg, Al, and possibly Si. The heavy-element distribution reflects enrichment by r-process nucleosynthesis ([Eu/Fe] = +0.36 +/- 0.03), as found in many other metal-poor globular clusters. We investigate small star-to-star variations found among the neutron-capture elements, and we conclude that these are probably not real variations. Upper limits on the Th abundance, log epsilon (Th/Eu) < -0.47 +/- 0.09, indicate that NGC 4833, like other globular clusters where Th has been studied, did not experience a so-called "actinide boost."Comment: Accepted for publication in MNRAS. Version 2 adds final publication referenc

Differential and Functional Identities for the Elliptic Trilogarithm

When written in terms of $\vartheta$-functions, the classical Frobenius-Stickelberger pseudo-addition formula takes a very simple form. Generalizations of this functional identity are studied, where the functions involved are derivatives (including derivatives with respect to the modular parameter) of the elliptic trilogarithm function introduced by Beilinson and Levin. A differential identity satisfied by this function is also derived. These generalized Frobenius-Stickelberger identities play a fundamental role in the development of elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde equations of associativity, with the simplest case reducing to the above mentioned differential identity

A construction of Multidimensional Dubrovin-Novikov Brackets

A method for the construction of classes of examples of multi-dimensional, multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This is based on an extension of the original construction of Gelfand and Dorfman which gave examples of Novikov algebras in terms of structures defined from commutative, associative algebras. Given such an algebra, the construction involves only linear algebra

Generalized Legendre transformations and symmetries of the WDVV equations

The Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class - the so-called Legendre transformations - were introduced by Dubrovin. They are a discrete set of symmetries between the stronger concept of a Frobenius manifold, and are generated by certain flat vector fields. In this paper this construction is generalized to the case where the vector field (called here the Legendre field) is non-flat but satisfies a certain set of defining equations. One application of this more general theory is to generate the induced symmetry between almost-dual Frobenius manifolds whose underlying Frobenius manifolds are related by a Legendre transformation. This also provides a map between rational and trigonometric solutions of the WDVV equations.Comment: 23 page

Mammalian Septins Nomenclature

There are 10 known mammalian septin genes, some of which produce multiple splice variants. The current nomenclature for the genes and gene products is very confusing, with several different names having been given to the same gene product and distinct names given to splice variants of the same gene. Moreover, some names are based on those of yeast or Drosophila septins that are not the closest homologues. Therefore, we suggest that the mammalian septin field adopt a common nomenclature system, based on that adopted by the Mouse Genomic Nomenclature Committee and accepted by the Human Genome Organization Gene Nomenclature Committee. The human and mouse septin genes will be named SEPT1–SEPT10 and Sept1–Sept10, respectively. Splice variants will be designated by an underscore followed by a lowercase “v” and a number, e.g., SEPT4_v1

Compatible metrics on a manifold and non-local bi-Hamiltonian structures

Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local bi-Hamiltonian structure, and with additional quasi-homogeneity conditions one obtains the structure of a Frobenius manifold. With appropriate curvature conditions one may define a curved pencil of compatible metrics and these give rise to an associated non-local bi-Hamiltonian structure. Specific examples include the F-manifolds of Hertling and Manin equipped with an invariant metric. In this paper the geometry supporting such compatible metrics is studied and interpreted in terms of a multiplication on the cotangent bundle. With additional quasi-homogeneity assumptions one arrives at a so-called weak \F-manifold - a curved version of a Frobenius manifold (which is not, in general, an F-manifold). A submanifold theory is also developed.Comment: 17 page