Related papers: An Improved Dictatorship Test with Perfect Complet…
We present a regularity lemma for Boolean functions $f:\{-1,1\}^n \to \{-1,1\}$ based on noisy influence, a measure of how locally correlated $f$ is with each input bit. We provide an application of the regularity lemma to weaken the…
Coherent states are required to form a complete set of vectors in the Hilbert space by providing the resolution of identity. We study the completeness of coherent states for two different models in a noncommutative space associated with the…
The Ratio Test and the Root Test for absolute convergence/divergence of series of numbers $\sum_{n=0}^{\infty}a_n$ are frequently discussed and proved independently in Calculus courses. The Root Test is stronger (verifies convergence for…
In this paper we offer a complete methodology for sufficient dimension reduction called the test function (TF). TF provides a new family of methods for the estimation of the central subspace (CS) based on the introduction of a nonlinear…
Let $0 < \epsilon < 1/2$ be a noise parameter, and let $T_{\epsilon}$ be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$. We upper bound the entropy of $T_{\epsilon}…
We consider the basic statistical problem of detecting truncation of the uniform distribution on the Boolean hypercube by juntas. More concretely, we give upper and lower bounds on the problem of distinguishing between i.i.d. sample access…
Deciding termination is a fundamental problem in the analysis of probabilistic imperative programs. We consider the qualitative and quantitative probabilistic termination problems for an imperative programming model with discrete…
A proposed measure of voting power should satisfy two conditions to be plausible: first, it must be conceptually justified, capturing the intuitive meaning of what voting power is; second, it must satisfy reasonable postulates. This paper…
The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96] seeks to relate two fundamental measures of Boolean function complexity: it states that $H[f] \leq C Inf[f]$ holds for every Boolean function $f$, where $H[f]$…
A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)\chi_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq…
This paper considers elections in which voters choose one candidate each, independently according to known probability distributions. A candidate receiving a strict majority (absolute or relative, depending on the version) wins. After the…
We introduce a variant of de Bruijn words that we call perfect necklaces. Fix a finite alphabet. Recall that a word is a finite sequence of symbols in the alphabet and a circular word, or necklace, is the equivalence class of a word under…
Let $\mathsf{TH}_k$ denote the $k$-out-of-$n$ threshold function: given $n$ input Boolean variables, the output is $1$ if and only if at least $k$ of the inputs are $1$. We consider the problem of computing the $\mathsf{TH}_k$ function…
Condorcet's paradox is a fundamental result in social choice theory which states that there exist elections in which, no matter which candidate wins, a majority of voters prefer a different candidate. In fact, even if we can select any $k$…
A unary constraint (on the Boolean domain) is a function from {0,1} to the set of real numbers. A free use of auxiliary unary constraints given besides input instances has proven to be useful in establishing a complete classification of the…
Let $\sigma(n)$ be the sum of the positive divisors of $n$. A number $n$ is said to be 2-near perfect if $\sigma(n) = 2n +d_1 +d_2 $, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We give a complete description of those $n$…
A dynamical system is called contractive if any two solutions approach one another at an exponential rate. More precisely, the dynamics contracts lines at an exponential rate. This property implies highly ordered asymptotic behavior…
An operator set is functionally incomplete if it can not represent the full set $\lbrace \neg,\vee,\wedge,\rightarrow,\leftrightarrow\rbrace$. The verification for the functional incompleteness highly relies on constructive proofs. The…
A discrete function of $n$ variables is a mapping $g : X_1 \times \ldots \times X_n \rightarrow A$, where $X_1, \ldots, X_n$, and $A$ are arbitrary finite sets. Function $g$ is called {\em separable} if there exist $n$ functions $g_i : X_i…
This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterised by a finite set F of non-negative functions that…