We prove that the lattice EqΩ of all equivalence relations on an infinite set Ω contains, as a 0, 1-sublattice, the 0-coproduct of two copies of itself, thus answering a question by G.M. Bergman. Hence, by using methods initiated by de Bruijn and further developed by Bergman, we obtain that EqΩ also contains, as a sublattice, the coproduct of 2 Ω copies of itself.

In these lectures we plan to present a survey of certain aspects of harmonic analysis on a Heisenberg nilmanifold Γ\Hn. Using Weil-Brezin-Zak transform we obtain an explicit decomposition of L(Γ\H) into irreducible subspaces invariant under the right regular representation of the Heisenberg group. We then study the Segal-Bargmann transform associated to the Laplacian on a nilmanifold and charac...

In this paper we consider a class of weighted integral operators on L2(0,∞) and show that they are unitarily equivalent to little Hankel operators between weighted Bergman spaces of the right half plane. We use two parameters α, β ∈ (−1,∞) and involve two weights to define Bergman spaces of the domain and range of the little Hankel operators. We obtained conditions for the Hankel integral opera...

We prove that the Bergman kernel function associated to a finitely connected domain Ω in the plane is given as a rational combination of only three basic functions of one complex variable: an Alhfors map, its derivative, and one other function whose existence is deduced by means of the field of meromorphic functions on the double of Ω. Because many other functions of conformal mapping and poten...

The synthesis and cyclization of acyclic quinoxaline, pyridine, and pyrimidine enediynes (1-3) are described. These compounds were prepared using palladium(0) coupling of trimethylsilyl acetylene to o-dihaloor o-halotriflic heteroarenes. All compounds were prepared in modest to good yields. The enediynes prepared were shown to undergo Bergman cyclization. Kinetics over a minimum of 3 half-lives...

Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Φ with corresponding oriented matroid MΦ, then B (MΦ) is dual ...

Philip J. Bergman*, Jedd D. Wolchok 1 Bright Heart Veterinary Centers Ludwig Center for Cancer Immunotherapy, Memorial Sloan-Kettering Cancer Center __________________________________________________________________________________ *Correspondence: Philip J. Bergman DVM, MS, PhD, Diplomate ACVIM-Oncology, Chief Medical Officer, Bright Heart Veterinary Centers, 80 Business Park Drive, Suite 110,...

We study the positive Bergman complex B(M) of an oriented matroid M , which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid M . The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid MI , the positive tropical variety associated to I is equal to the fan over B(MI). Our main result is that a certain “fine” ...