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We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,...,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,...,X_m$, modulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,...,Y_r$.
The subject of this textbook is the analysis of Boolean functions. Roughly speaking, this refers to studying Boolean functions $f : {0,1}^n to {0,1}$ via their Fourier expansion and other analytic means. Boolean functions are perhaps the most basic o
We apply down operators in the affine nilCoxeter algebra to yield explicit combinatorial expansions for certain families of non-commutative k-Schur functions. This yields a combinatorial interpretation for a new family of k-Littlewood-Richardson coefficients.
The seminal result of Kahn, Kalai and Linial shows that a coalition of $O(frac{n}{log n})$ players can bias the outcome of any Boolean function ${0,1}^n to {0,1}$ with respect to the uniform measure. We extend their result to arbitrary product measur
A famous conjecture of Gyarfas and Sumner states for any tree $T$ and integer $k$, if the chromatic number of a graph is large enough, either the graph contains a clique of size $k$ or it contains $T$ as an induced subgraph. We discuss some results a
Let $(S,mathcal L)$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $mathcal L$ of degree $d > 35$. In this paper we prove that $K^2_Sgeq -d(d-6)$. The bound is sharp, and $K^2_S=-d(d-6)$ if and only if $d