相关论文: On (not) computing the Mobius function using bound…

For a prime p and base b, the digit function delta(r) = floor(br/p) partitions the residues {1, ..., p-1} into b contiguous bins. The collision count C(g) records how many residues share a bin with their image under multiplication by g. We…

It is shown that monotone Boolean functions on the Boolean cube capture the expected number of primes, under he usual identification by binary expansion. This answers a question posed by G.Kalai.

In this short note we show that functions in the modulation space $\mathscr{F}W=\{ f: \sum_{j\in\mathbb{Z}^n}\| \hat{f}(\cdot+2\pi j)\|_{L_\infty([-\pi,\pi]^n)}<\infty \}$ enjoy similar recovery properties as band-limited functions. If…

In this note we compare two measures of the complexity of a class $\mathcal F$ of Boolean functions studied in (unconditional) pseudorandomness: $\mathcal F$'s ability to distinguish between biased and uniform coins (the coin problem), and…

Our work addresses the well-known open problem of distributed computing of bilinear functions of two correlated sources ${\bf A}$ and ${\bf B}$. In a setting with two nodes, with the first node having access to ${\bf A}$ and the second to…

Let $\mathcal{P}$ be the set of the primes. We consider a class of random multiplicative functions $f$ supported on the squarefree integers, such that $\{f(p)\}_{p\in\mathcal{P}}$ form a sequence of $\pm1$ valued independent random…

In this paper we examine the problem of computing majority function $\mathrm{MAJ}_n$ on $n$ bits by depth-two formula, where each gate is a majority function on at most $k$ inputs. We present such formula that gives the first nontrivial…

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold $M$. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the…

Let $({\mathcal X},d,\mu)$ be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, the authors establish some equivalent characterizations for the boundedness of fractional…

We demonstrate that the unbounded fan-out gate is very powerful. Constant-depth polynomial-size quantum circuits with bounded fan-in and unbounded fan-out over a fixed basis (denoted by QNCf^0) can approximate with polynomially small error…

Let $\{b_H(t),t\in\mathbb{R}\}$ be the fractional Brownian motion with parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of the type \[X(t)=c+\int_0^t\sigma\bigl(X(u)\bigr)\mathrm {d}b_H(u)+\int _0^t\mu\bigl(X(u)\bigr)\mathrm…

We establish nontrivial bounds for bilinear sums involving the M\"obius function evaluated over solutions to a broad class of equations. Several of our results may be regarded as M\"obius-function analogues of the ternary Goldbach problem.…

We present optimized quantum circuits for GF$(2^m)$ multiplication and division operations, which are essential computing primitives in various quantum algorithms. Our ancilla-free GF multiplication circuit has the gate count complexity of…

A new class of test functions for black box optimization is introduced. Affine OneMax (AOM) functions are defined as compositions of OneMax and invertible affine maps on bit vectors. The black box complexity of the class is upper bounded by…

Obtaining a non-trivial (super-linear) lower bound for computation of the Fourier transform in the linear circuit model has been a long standing open problem. All lower bounds so far have made strong restrictions on the computational model.…

It is shown that the minimal depth of an optimal prefix circuit (i.e., a zero-deficiency circuit) on $N$ inputs with fanout bounded by $k$ is ${\log_{\alpha_k} N \pm O(1)}$, where $\alpha_k$ is the unique positive root of the polynomial…

The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose…

We show how to distinguish circuits with $\log k$ negations (a.k.a $k$-monotone functions) from uniformly random functions in $\exp\left(\tilde{O}\left(n^{1/3}k^{2/3}\right)\right)$ time using random samples. The previous best…

Let $\mathbb{F}\subset \mathbb{K}$ be fields with characteristic zero, $n$ be a positive integer and $\kappa\in \mathbb{K}$. In this paper, we determine those monomials $f\colon \mathbb{F}\to \mathbb{K}$ of degree $n$ for which \[ f(x^{2})=…

We characterize the power of constant-depth Boolean circuits in generating uniform symmetric distributions. Let $f\colon\{0,1\}^m\to\{0,1\}^n$ be a Boolean function where each output bit of $f$ depends only on $O(1)$ input bits. Assume the…