ﻻ يوجد ملخص باللغة العربية
Let $mathcal{H}$ denote a collection of subsets of ${1,2,ldots,n}$, and assign independent random variables uniformly distributed over $[0,1]$ to the $n$ elements. Declare an element $p$-present if its corresponding value is at most $p$. In this paper, we quantify how much the observation of the $r$-present ($r>p$) set of elements affects the probability that the set of $p$-present elements is contained in $mathcal{H}$. In the context of percolation, we find that this question is closely linked to the near-critical regime. As a consequence, we show that for every $r>1/2$, bond percolation on the subgraph of the square lattice given by the set of $r$-present edges is almost surely noise sensitive at criticality, thus generalizing a result due to Benjamini, Kalai and Schramm.
We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first for which the critical probability p_c e 1/2. Our proof
Let $T_{epsilon}$ be the noise operator acting on Boolean functions $f:{0, 1}^nto {0, 1}$, where $epsilonin[0, 1/2]$ is the noise parameter. Given $alpha>1$ and fixed mean $mathbb{E} f$, which Boolean function $f$ has the largest $alpha$-th moment $m
We determine the size of $k$-core in a large class of dense graph sequences. Let $G_n$ be a sequence of undirected, $n$-vertex graphs with edge weights ${a^n_{i,j}}_{i,j in [n]}$ that converges to a kernel $W:[0,1]^2to [0,+infty)$ in the cut metric.
We consider a controlled second order differential equation which is partially observed with an additional fractional noise. we study the asymptotic (for large observation time) design problem of the input and give an efficient estimator of the unkno
We consider Boolean functions f:{-1,1}^n->{-1,1} that are close to a sum of independent functions on mutually exclusive subsets of the variables. We prove that any such function is close to just a single function on a single subset. We also conside