Toric Ideals of Lattice Path Matroids and Polymatroids

We show that the toric ideal of a lattice path polymatroid is generated by quadrics corresponding to symmetric exchanges, and give a monomial order under which these quadrics form a Gr\"obner basis. We then obtain an analogous result for lattice path matroids.Comment: 9 pages, 4 figure

Reflections on God and Evil in the Krishna Bhakti Theology of Caitanya

The story that forms the short prologue at the start of Valmiki’s Rāmāyana expresses something of how the bhakta deals with tragic loss and senseless violence. In the briefest of words, the story goes something like this: As Valmiki is performing his morning ablutions in the Tamasa River, he witnesses two cranes sweetly cooing at one another lovingly on a branch in a nearby tree. Suddenly, a hunter comes along and shoots an arrow through the chest of the male crane, who immediately drops to the ground. The female, herself alighting on the ground, sees her beloved mate writhing in agony with his mortal wound, and thus she begins to weep, drowning herself in the tears of her sorrow. Valmiki, upon seeing this senseless killing, in his anger curses the hunter from that day on to wander all the days of his life homelessly. Who can mortally wound such an innocent loving creature? What kind of person does this? What kind of senseless act is this? Where is God in all this? Questions such as these are easily elicited from those who hear this story. We shall return to this story following a brief sketch of the relationship between God and evil in a bhakti theology

Computer Analysis of Dielectric Waveguides: A Finite-Difference Method

A method for computing the modes of dielectric guiding structures based on finite differences is described. The numerical computation program is efficient and can be applied to a wide range of problems. We report here solutions for circular and rectangular dielectric waveguides and compare our solutions with those obtained by other methods. Limitations in the commonly used approximate formulas developed by Marcatili are discussed

Borel generators

We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to ordinary generators, this enables one to do manual computations much more easily. Moreover, this perspective allows us to find new connections to combinatorics involving Catalan numbers and their generalizations. We conclude with a surprising result relating the Betti numbers of certain principal Borel ideals to the number of pointed pseudo-triangulations of particular planar point sets.Comment: 23 pages, 2 figures; very minor changes in v2. To appear in J. Algebr