Binomial Difference Ideal and Toric Difference Variety

In this paper, the concepts of binomial difference ideals and toric difference varieties are defined and their properties are proved. Two canonical representations for Laurent binomial difference ideals are given using the reduced Groebner basis of Z[x]-lattices and regular and coherent difference ascending chains, respectively. Criteria for a Laurent binomial difference ideal to be reflexive, prime, well-mixed, perfect, and toric are given in terms of their support lattices which are Z[x]-lattices. The reflexive, well-mixed, and perfect closures of a Laurent binomial difference ideal are shown to be binomial. Four equivalent definitions for toric difference varieties are presented. Finally, algorithms are given to check whether a given Laurent binomial difference ideal I is reflexive, prime, well-mixed, perfect, or toric, and in the negative case, to compute the reflexive, well-mixed, and perfect closures of I. An algorithm is given to decompose a finitely generated perfect binomial difference ideal as the intersection of reflexive prime binomial difference ideals.Comment: 72 page

A generalized Macdonald operator

We present an explicit difference operator diagonalized by the Macdonald polynomials associated with an (arbitrary) admissible pair of irreducible reduced crystallographic root systems. By the duality symmetry, this gives rise to an explicit Pieri formula for the Macdonald polynomials in question. The simplest examples of our construction recover Macdonald's celebrated difference operators and associated Pieri formulas pertaining to the minuscule and quasi-minuscule weights. As further by-products, explicit expansions and Littlewood-Richardson type formulas are obtained for the Macdonald polynomials associated with a special class of small weights.Comment: 11 pages. To appear in Int. Math. Res. Not. IMR

KOI 1224, a Fourth Bloated Hot White Dwarf Companion Found With Kepler

We present an analysis and interpretation of the Kepler binary system KOI 1224. This is the fourth binary found with Kepler that consists of a thermally bloated, hot white dwarf in a close orbit with a more or less normal star of spectral class A or F. As we show, KOI 1224 contains a white dwarf with Teff = 14400 +/- 1100 K, mass = 0.20 +/- 0.02 Msun, and radius = 0.103 +/- 0.004 Rsun, and an F-star companion of mass = 1.59 +/- 0.07 Msun that is somewhat beyond its terminal-age main sequence. The orbital period is quite short at 2.69802 days. The ingredients that are used in the analysis are the Kepler binary light curve, including the detection of the Doppler boosting effect; the NUV and FUV fluxes from the Galex images of this object; an estimate of the spectral type of the F-star companion; and evolutionary models of the companion designed to match its effective temperature and mean density. The light curve is modelled with a new code named Icarus which we describe in detail. Its features include the full treatment of orbital phase-resolved spectroscopy, Doppler boosting, irradiation effects and transits/eclipses, which are particularly suited to irradiated eclipsing binaries. We interpret the KOI 1224 system in terms of its likely evolutionary history. We infer that this type of system, containing a bloated hot white dwarf, is the direct descendant of an Algol-type binary. In spite of this basic understanding of the origin of KOI 1224, we discuss a number of problems associated with producing this type of system with this short of an short orbital period.Comment: 14 pages, 8 figures, 2 tables, submitted to Ap

On Congruence Compact Monoids

A universal algebra is called congruence compact if every family of congruence classes with the finite intersection property has a non-empty intersection. This paper determines the structure of all right congruence compact monoids S for which Green’s relations J and H coincide. The results are thus sufficiently general to describe, in particular, all congruence compact commutative monoids and all right congruence compact Clifford inverse monoids