Related papers: On unbalanced Boolean functions with best correlat…

We consider a multiscale Boolean percolation on $\mathbb R^d$ with radius distribution $\mu$ on $[1,+\infty)$, $d\ge 2$. The model is defined by superposing the original Boolean percolation model with radius distribution $\mu$ with a…

We show that Boolean functions expressible as monotone disjunctive normal forms are PAC-evolvable under a uniform distribution on the Boolean cube if the hypothesis size is allowed to remain fixed. We further show that this result is…

Suppose $X$ is a uniformly distributed $n$-dimensional binary vector and $Y$ is obtained by passing $X$ through a binary symmetric channel with crossover probability $\alpha$. A recent conjecture by Courtade and Kumar postulates that…

Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts the number of partitions of $n$ in which every even part is less than each odd part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which…

We prove that there is a constant $C\leq 6.614$ such that every Boolean function of degree at most $d$ (as a polynomial over $\mathbb{R}$) is a $C\cdot 2^d$-junta, i.e. it depends on at most $C\cdot 2^d$ variables. This improves the $d\cdot…

It is proved that the number of 9-regular partitions of n is divisible by 3 when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An infinite family of congruences mod 3 holds in other progressions modulo powers of 4…

We give a combinatorial proof of the result of Kahn, Kalai, and Linial, which states that every balanced boolean function on the $n$-dimensional boolean cube has a variable with influence of at least Omega(\frac{log n}{n}). The methods of…

We initiate the study of a new model of query complexity of Boolean functions where, in addition to 0 and 1, the oracle can answer queries with ``unknown''. The query algorithm is expected to output the function value if it can be…

For the Hamming graph $H(n,q)$, where a $q$ is a constant prime power and $n$ grows, we construct perfect colorings without non-essential arguments such that $n$ depends exponentially on the off-diagonal part of the quotient matrix. In…

Estimates for initial coefficients of Taylor-Maclaurin series of bi-univalent functions belonging to certain classes defined by subordination are obtained. Our estimates improve upon the earlier known estimates for second and third…

In this paper, we study the uniform measure for the self-conjugate partitions. We derive the $q$-difference equation which is satisfied by the $n$-point correlation function related to the uniform measure. As applications, we give explicit…

We prove a lower bound of $\tilde{\Omega}(n^{1/3})$ for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is monotone or far from monotone. This…

In this paper some cryptographic properties of Boolean functions, including weight, balancedness and nonlinearity, are studied, particularly focusing on splitting functions and cubic Boolean functions. Moreover, we present some quantities…

We provide the first-ever calculation of the number of inequivalent monotone Boolean functions of 9 variables, which is equal to 789,204,635,842,035,040,527,740,846,300,252,680.

For any positive regularity parameter $\beta < \frac 12$, we construct non-conservative weak solutions of the 3D incompressible Euler equations which lie in $H^{\beta}$ uniformly in time. In particular, we construct solutions which have an…

The 93 minions of Boolean functions stable under left composition with the clone of self-dual monotone functions are described. As an easy consequence, all $(C_1,C_2)$-stable classes of Boolean functions are determined for an arbitrary…

We give an algorithm for learning symmetric k-juntas (boolean functions of $n$ boolean variables which depend only on an unknown set of $k$ of these variables) in the PAC model under the uniform distribution, which runs in time n^{O(k/\log…

Let $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In analogy with Ramanujan's work on the partition function, Lin recently proved in \cite{Lin} that $p_{-3}(11n+7)\equiv0\pmod{11}$ for every integer $n$. Such…

We study the binary perceptron, a random constraint satisfaction problem that asks to find a Boolean vector in the intersection of independently chosen random halfspaces. A striking feature of this model is that at every positive constraint…

Let $\prec$ be the product order on $\mathbb{R}^k$ and assume that $X_1,X_2,\ldots,X_n$ ($n\geq3$) are i.i.d. random vectors distributed uniformly in the unit hypercube $[0,1]^k$. Let $S$ be the (random) set of vectors in $\mathbb{R}^k$…