257 research outputs found
Subspace-Invariant AC Formulas
We consider the action of a linear subspace of on the set of
AC formulas with inputs labeled by literals in the set , where an element acts on formulas by
transposing the th pair of literals for all such that . A
formula is {\em -invariant} if it is fixed by this action. For example,
there is a well-known recursive construction of depth formulas of size
computing the -variable PARITY function; these
formulas are easily seen to be -invariant where is the subspace of
even-weight elements of . In this paper we establish a nearly
matching lower bound on the -invariant depth
formula size of PARITY. Quantitatively this improves the best known
lower bound for {\em unrestricted} depth
formulas, while avoiding the use of the switching lemma. More generally,
for any linear subspaces , we show that if a Boolean function is
-invariant and non-constant over , then its -invariant depth
formula size is at least where is the minimum Hamming
weight of a vector in
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
A Milnor-Wood inequality for complex hyperbolic lattices in quaternionic space
We prove a Milnor-Wood inequality for representations of the fundamental
group of a compact complex hyperbolic manifold in the group of isometries of
quaternionic hyperbolic space. Of special interest is the case of equality, and
its application to rigidity. We show that equality can only be achieved for
totally geodesic representations, thereby establishing a global rigidity
theorem for totally geodesic representations.Comment: 13 page
Geometry of invariant domains in complex semi-simple Lie groups
We investigate the joint action of two real forms of a semi-simple complex
Lie group S by left and right multiplication. After analyzing the orbit
structure, we study the CR structure of closed orbits. The main results are an
explicit formula of the Levi form of closed orbits and the determination of the
Levi cone of generic orbits. Finally, we apply these results to prove
q-completeness of certain invariant domains in S.Comment: 20 page
Ergodic properties of infinite extensions of area-preserving flows
We consider volume-preserving flows on , where is a closed connected surface of genus and
has the form , where is a locally Hamiltonian flow
of hyperbolic periodic type on and is a smooth real valued function on
. We investigate ergodic properties of these infinite measure-preserving
flows and prove that if belongs to a space of finite codimension in
, then the following dynamical dichotomy holds: if
there is a fixed point of on which does not
vanish, then is ergodic, otherwise, if
vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial
extension . The proof of this result exploits the
reduction of to a skew product automorphism over
an interval exchange transformation of periodic type. If there is a fixed point
of on which does not vanish, the reduction
yields cocycles with symmetric logarithmic singularities, for which we prove
ergodicity.Comment: 57 pages, 4 picture
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