Σ 1 Formulae in Σ L

Explicit Formulae for Cocycles of Holomorphic Vector Fields with values in λ Densities

The continuous cohomology of Lie algebras of C-vector fields has been studied by I. M. Gelfand, D. B. Fuks, R. Bott, A. Haefliger and G. Segal in some outstanding papers [4], [9], [1]. B. L. Feigin [2] and N. Kawazumi [11], whose work is continued in [18], studied Gelfand-Fuks cohomology of Lie algebras of holomorphic vector fields Hol(Σ) on an open Riemann surface. Kawazumi calculated the coho...

An Explicit ∂-integration Formula for Weighted Homogeneous Varieties Ii, Forms of Higher Degree

Let Σ be a weighted homogeneous (singular) subvariety of C . The main objective of this paper is to present a class of explicit integral formulae for solving the ∂-equation ω = ∂λ on the regular part of Σ, where ω is a ∂-closed (0, q)-form with compact support and degree q ≥ 1. Particular cases of these formulae yield L-bounded solution operators for 1 ≤ p ≤ ∞ if Σ is a homogeneous and pure dim...

σ-Normality of Topological Spaces

Quantum Theory of Light

PLEASE read over the entire examination before you start. DO ALL QUESTIONS and show all your work in submitted material to be eligible for partial credit. Useful formulae: cos(2x)=cos 2 (x)-sin 2 (x) sin(2x)=2sin(x)cos(x) The Pauli spin matrices: σ z = −       1 0 0 1 ; σ x =       0 1 1 0 ; σ y i i = −       0 0 ; σ σ σ ± = ± 1 2 () x y i

On Semantic Generalizations of the Bernays-Schönfinkel-Ramsey Class with Finite or Co-finite Spectra

Motivated by model-theoretic properties of the BernaysSchönfinkel-Ramsey (BSR) class, we present a family of semantic classes of FO formulae with finite or co-finite spectra over a relational vocabulary Σ. A class in this family is denoted EBSΣ(σ), where σ ⊆ Σ. Formulae in EBSΣ(σ) are preserved under substructures modulo a bounded core and modulo re-interpretation of predicates in Σ \ σ. We stu...