Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion

We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decomposition that was obtained from the Baker-Campbell-Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer can be obtained from a certain Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the even-odd decomposition of combinatorial Hopf algebra characters as well as to the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.Comment: accepted for publication in Comm. in Math. Phy

Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications

Abstract. Let A be an n by N real-valued matrix with n < N; we count the number of k-faces fk(AQ) when Q is either the standard N-dimensional hypercube IN or else the positive orthant RN +. To state results simply, consider a proportional-growth asymptotic, where for fixed δ, ρ in (0, 1), we have a sequence of matrices An,Nn and of integers kn with n/Nn → δ, kn/n → ρ as n → ∞. If each matrix An,Nn has its columns in general position, then fk(AIN)/fk(I N) tends to zero or one depending on whether ρ> min(0, 2 − δ−1) or ρ < min(0, 2 − δ−1). Also, if each An,Nn is a random draw from a distribution which is invariant under right multiplication by signed permutations, then fk(ARN +)/fk(RN +) tends almost surely to zero or one depending on whether ρ> min(0, 2 − δ−1) or ρ < min(0, 2 − δ−1). We make a variety of contrasts to related work on projections of the simplex and/or cross-polytope. These geometric face-counting results have implications for signal processing, information theory, inverse problems, and optimization. Indeed, face counting is related to conditions for uniqueness of solutions of underdetermine

Anomaly in the K^0_S Sigma^+ photoproduction cross section off the proton at the K* threshold

The $\gamma + p \rightarrow K^0 + \Sigma^+$ photoproduction reaction is investigated in the energy region from threshold to $E_\gamma = 2250$\,MeV. The differential cross section exhibits increasing forward-peaking with energy, but only up to the $K^*$ threshold. Beyond, it suddenly returns to a flat distribution with the forward cross section dropping by a factor of four. In the total cross section a pronounced structure is observed between the $K^*\Lambda$ and $K^*\Sigma$ thresholds. It is speculated whether this signals the turnover of the reaction mechanism from t-channel exchange below the $K^*$ production threshold to an s-channel mechanism associated with the formation of a dynamically generated $K^*$-hyperon intermediate state.Comment: 14 pages, 7 figure

Linearly polarised photon beams at ELSA and measurement of the beam asymmetry in pi^0-photoproduction off the proton

At the electron accelerator ELSA a linearly polarised tagged photon beam is produced by coherent bremsstrahlung off a diamond crystal. Orientation and energy range of the linear polarisation can be deliberately chosen by accurate positioning of the crystal with a goniometer. The degree of polarisation is determined by the form of the scattered electron spectrum. Good agreement between experiment and expectations on basis of the experimental conditions is obtained. Polarisation degrees of P = 40% are typically achieved at half of the primary electron energy. The determination of P is confirmed by measuring the beam asymmetry, \Sigma, in pi^0 photoproduction and a comparison of the results to independent measurements using laser backscattering.Comment: 9 pages, 10 figures, submitted to EPJ