40,748 research outputs found
Novel Lower Bounds on the Entropy Rate of Binary Hidden Markov Processes
Recently, Samorodnitsky proved a strengthened version of Mrs. Gerber's Lemma,
where the output entropy of a binary symmetric channel is bounded in terms of
the average entropy of the input projected on a random subset of coordinates.
Here, this result is applied for deriving novel lower bounds on the entropy
rate of binary hidden Markov processes. For symmetric underlying Markov
processes, our bound improves upon the best known bound in the very noisy
regime. The nonsymmetric case is also considered, and explicit bounds are
derived for Markov processes that satisfy the -RLL constraint
Isoperimetric properties of the mean curvature flow
In this paper we discuss a simple relation, which was previously missed,
between the high co-dimensional isoperimetric problem of finding a filling with
small volume to a given cycle, and extinction estimates for singular, high
co-dimensional, mean curvature flow. The utility of this viewpoint is first
exemplified by two results which, once casted in the light of this relation,
are almost self evident. The first is a genuine, 5-lines proof, for the
isoperimetric inequality for -cycles in , with a constant
differing from the optimal constant by a factor of only , as opposed
to a factor of produced by all of the other soft methods (like
Michael-Simon's or Gromov's). The second is a 3-lines proof of a lower bound
for extinction for arbitrary co-dimensional, singular, mean curvature flows
starting from cycles, generalizing the main result of a paper of Giga and
Yama-uchi. We then turn to use the above mentioned relation to prove a bound on
the parabolic Hausdorff measure of the space time track of high co-dimensional,
singular, mean curvature flow starting from a cycle, in terms of the mass of
that cycle. This bound is also reminiscent of a Michael-Simon Isoperimetric
inequality. To prove it, we are lead to study the geometric measure theory of
Euclidean rectifiable sets in parabolic space, and prove a co-area formula in
that setting. This formula, the proof of which occupies most this paper, may be
of independent interes
Religion, religious fervour, and universalist education
This paper is conceived from a secular perspective, and designed to address three elements identified in the call for papers: âPluralistic tendenciesâ, their counterpart of âexclusivist attitudesâ, and âcreating an ethos of inter-religious harmonyâ. I choose to tackle these aspects by (a) exploring the meaning of religion, (b) addressing a specific attitude often corresponding to religion, namely religious fervour, and (c) assessing the validity and instrumentality of facilitating a universalist education as a tool to defuse âmistrust and hatred among various faith-communitiesâ. The following paper is intended to serve only as a preliminary discussion guidance paper
- âŠ