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We show that a Boolean degree $d$ function on the slice $binom{[n]}{k} = { (x_1,ldots,x_n) in {0,1} : sum_{i=1}^n x_i = k }$ is a junta, assuming that $k,n-k$ are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube. Moreover, we show that the maximum number of coordinates that a Boolean degree $d$ function can depend on is the same on the slice and the hypercube.
We show that if $fcolon S_n to {0,1}$ is $epsilon$-close to linear in $L_2$ and $mathbb{E}[f] leq 1/2$ then $f$ is $O(epsilon)$-close to a union of mostly disjoint cosets, and moreover this is sharp: any such union is close to linear. This constitute
Classification of Non-linear Boolean functions is a long-standing problem in the area of theoretical computer science. In this paper, effort has been made to achieve a systematic classification of all n-variable Boolean functions, where only one affi
Let $G$ be a finite, undirected $d$-regular graph and $A(G)$ its normalized adjacency matrix, with eigenvalues $1 = lambda_1(A)geq dots ge lambda_n ge -1$. It is a classical fact that $lambda_n = -1$ if and only if $G$ is bipartite. Our main result p
Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of
For a graph $G=(V,E)$, $kin mathbb{N}$, and a complex number $w$ the partition function of the univariate Potts model is defined as [ {bf Z}(G;k,w):=sum_{phi:Vto [k]}prod_{substack{uvin E phi(u)=phi(v)}}w, ] where $[k]:={1,ldots,k}$. In this paper w