Related papers: The Average Sensitivity of Bounded-Depth Formulas
The sensitivity conjecture of Nisan and Szegedy [CC '94] asks whether for any Boolean function $f$, the maximum sensitivity $s(f)$, is polynomially related to its block sensitivity $bs(f)$, and hence to other major complexity measures.…
We prove that the pigeonhole upper bound $\lambda(s,m) \leq \binom{m}{2}(s+1)$ is asymptotically tight whenever $s/\!\sqrt{m} \to \infty$. In particular, $\lambda(s,m) \sim \binom{m}{2}\,s$ in this regime. As corollaries: $\lambda(n,n)/n^3…
We show that 4-point vector boson one-loop amplitudes, computed in ref.[1] in the RNS formalism, around vacuum configurations with open unoriented strings, preserving at least N=1 SUSY in D=4, satisfy the correct supersymmetry Ward…
The compact fourth-order finite-difference scheme for solving the 1d wave equation is studied. New error bounds of the fractional order $\mathcal{O}(h^{4(\lambda-1)/5})$ are proved in the mesh energy norm in terms of data, for two initial…
Microscopic variations in composition or structure can lead to nanoscale inhomogeneity in superconducting properties such as the magnetic penetration depth, but measurements of these properties are usually made on longer length scales. We…
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…
We investigate error bounds for numerical solutions of divergence structure linear elliptic PDEs on compact manifolds without boundary. Our focus is on a class of monotone finite difference approximations, which provide a strong form of…
A power law is postulated for both site and bond percolation thresholds. The formula writes $p_c=p_0[(d-1)(q-1)]^{-a}d^{\ b}$, where $d$ is the space dimension and $q$ the coordination number. All thresholds up to $d\rightarrow \infty$ are…
We study Frege proofs using depth-$d$ Boolean formulas for the Tseitin contradiction on $n \times n$ grids. We prove that if each line in the proof is of size $M$ then the number of lines is exponential in $n/(\log M)^{O(d)}$. This…
We derive bounds $ |\frac{d\psi(\alpha)}{d\alpha}| \leq 1 $, $ \frac{d(\frac{d\psi(\alpha)}{d\alpha}\psi(\alpha))}{d\alpha} \leq 1 $ on the GL (Gell-Mann--Low) function $\psi(\alpha)$ from the Kallen-Lehmann dispersion representation in…
Random measurements have been shown to induce a phase transition in an extended quantum system evolving under chaotic unitary dynamics, when the strength of measurements exceeds a threshold value. Below this threshold, a steady state with a…
We consider the precision $\Delta \varphi$ with which the parameter $\varphi$, appearing in the unitary map $U_\varphi = e^{ i \varphi \Lambda}$ acting on some type of probe system, can be estimated when there is a finite amount of prior…
Given a function $f$ on $\mathbb{F}_2^n$, we study the following problem. What is the largest affine subspace $\mathcal{U}$ such that when restricted to $\mathcal{U}$, all the non-trivial Fourier coefficients of $f$ are very small? For the…
Uniform stability of a learning algorithm is a classical notion of algorithmic stability introduced to derive high-probability bounds on the generalization error (Bousquet and Elisseeff, 2002). Specifically, for a loss function with range…
Estimating the second frequency moment of a stream up to $(1\pm\varepsilon)$ multiplicative error requires at most $O(\log n / \varepsilon^2)$ bits of space, due to a seminal result of Alon, Matias, and Szegedy. It is also known that at…
We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on $n$ vertices up to a $1+\epsilon$ error must…
We consider the Stochastic Boolean Function Evaluation (SBFE) problem where the task is to efficiently evaluate a known Boolean function $f$ on an unknown bit string $x$ of length $n$. We determine $f(x)$ by sequentially testing the…
In this paper we characterize the limiting behavior of the principal eigenvalue, $\s_1[-\D,\b,\O]$, of the boundary value problem \eqref{1.1} as the Lebesgue measure of the underlying domain, $\O$, tends to zero. Naturally, the domains $\O$…
We resolve the long-standing open problem of Boolean dynamic data structure hardness, proving an unconditional lower bound of $\Omega((\log n / \log\log n)^2)$ for the Multiphase Problem of Patrascu [STOC 2010] (instantiated with Inner…
We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq2$ and a domain $\Omega$ of boundary $\Gamma$ in $\mathbb {R}^d$. Let $\Gamma ^1$ and $\Gamma ^2$ be two disjoint open subsets of…