13,908 research outputs found
Subspace-Invariant AC^0 Formulas
The n-variable PARITY function is computable (by a well-known recursive construction) by AC^0 formulas of depth d+1 and leaf size n2^{dn^{1/d}}. These formulas are seen to possess a certain symmetry: they are syntactically invariant under the subspace P of even-weight elements in {0,1}^n, which acts (as a group) on formulas by toggling negations on input literals. In this paper, we prove a 2^{d(n^{1/d}-1)} lower bound on the size of syntactically P-invariant depth d+1 formulas for PARITY. Quantitatively, this beats the best 2^{Omega(d(n^{1/d}-1))} lower bound in the non-invariant setting
The Weitzenb\"ock Machine
In this article we give a unified treatment of the construction of all
possible Weitzenb\"ock formulas for all irreducible, non--symmetric holonomy
groups. The resulting classification is two--fold, we construct explicitly a
basis of the space of Weitzenb\"ock formulas on the one hand and characterize
Weitzenb\"ock formulas as eigenvectors for an explicitly known matrix on the
other. Both classifications allow us to find tailor--suit Weitzenb\"ock
formulas for applications like eigenvalue estimates or Betti number estimates.Comment: 48 page
Yang-Mills theory and the Segal-Bargmann transform
We use a variant of the classical Segal-Bargmann transform to understand the
canonical quantization of Yang-Mills theory on a space-time cylinder. This
transform gives a rigorous way to make sense of the Hamiltonian on the
gauge-invariant subspace. Our results are a rigorous version of the widely
accepted notion that on the gauge-invariant subspace the Hamiltonian should
reduce to the Laplacian on the compact structure group. We show that the
infinite-dimensional classical Segal-Bargmann transform for the space of
connections, when restricted to the gauge-invariant subspace, becomes the
generalized Segal-Bargmann transform for the the structure group
Decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of Z_2-graded Lie algebras
We give uniform formulas for the branching rules of level 1 modules over
orthogonal affine Lie algebras for all conformal pairs associated to symmetric
spaces. We also provide a combinatorial intepretation of these formulas in
terms of certain abelian subalgebras of simple Lie algebras.Comment: Latex, 56 pages, revised version: minor corrections, Subsection 6.2
added. To appear in Advances in Mathematic
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