1,980 research outputs found
Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read
A Boolean function of n bits is balanced if it takes the value 1 with
probability 1/2. We exhibit a balanced Boolean function with a randomized
evaluation procedure (with probability 0 of making a mistake) so that on
uniformly random inputs, no input bit is read with probability more than
Theta(n^{-1/2} sqrt{log n}). We give a balanced monotone Boolean function for
which the corresponding probability is Theta(n^{-1/3} log n). We then show that
for any randomized algorithm for evaluating a balanced Boolean function, when
the input bits are uniformly random, there is some input bit that is read with
probability at least Theta(n^{-1/2}). For balanced monotone Boolean functions,
there is some input bit that is read with probability at least Theta(n^{-1/3}).Comment: 11 page
Tug-of-war and the infinity Laplacian
We prove that every bounded Lipschitz function F on a subset Y of a length
space X admits a tautest extension to X, i.e., a unique Lipschitz extension u
for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do
not intersect Y.
This was previously known only for bounded domains R^n, in which case u is
infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also
prove the first general uniqueness results for Delta_infty u = g on bounded
subsets of R^n (when g is uniformly continuous and bounded away from zero), and
analogous results for bounded length spaces.
The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be
the value of the following two-player zero-sum game, called tug-of-war: fix
x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner
chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y,
and player one's payoff is
F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i)
We show that the u^\epsilon converge uniformly to u as epsilon tends to zero.
Even for bounded domains in R^n, the game theoretic description of
infinity-harmonic functions yields new intuition and estimates; for instance,
we prove power law bounds for infinity-harmonic functions in the unit disk with
boundary values supported in a delta-neighborhood of a Cantor set on the unit
circle.Comment: 44 pages, 4 figure
Analysis of Coolant-flow Requirements for an Improved, Internal-strut-supported, Air-cooled Turbine-rotor Blade
An analytical evaluation of a new typ An analytical evaluation of a new type of air-cooled turbine-rotor-blade design, based on the principle of submerging the load-carrying element in cooling air within a thin high-temperature sheel, indicates that this principle of blade design permits the load carrying element to be operated at considerably lower temperature than that of the enveloping shell. Comparison with an air-cooled shell-supported air-cooled blade has greater potentiality to withstand increased stresses that can be anticipated in future engines
Finitary Coloring
Suppose that the vertices of are assigned random colors via a
finitary factor of independent identically distributed (iid) vertex-labels.
That is, the color of vertex is determined by a rule that examines the
labels within a finite (but random and perhaps unbounded) distance of ,
and the same rule applies at all vertices. We investigate the tail behavior of
if the coloring is required to be proper (that is, if adjacent vertices
must receive different colors). When , the optimal tail is given by a
power law for 3 colors, and a tower (iterated exponential) function for 4 or
more colors (and also for 3 or more colors when ). If proper coloring is
replaced with any shift of finite type in dimension 1, then, apart from trivial
cases, tower function behavior also applies.Comment: 35 pages, 3 figure
- …