Related papers: A note on the capacity of the binary perceptron
The Twin Prime conjecture states that there are infinitely many pairs of distinct primes which differ by $2$. Until recently this conjecture had seemed to be far out of reach with current techniques. However, in April 2013, Yitang Zhang…
For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three applications. (1) An open question in…
We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a~function of the gap to…
In this paper we consider the classical spherical perceptron problem. This problem and its variants have been studied in a great detail in a broad literature ranging from statistical physics and neural networks to computer science and pure…
John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of binary words. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with $m$ free indices and of size $n$…
The asymptotic rate vs. distance problem is a long-standing fundamental problem in coding theory. The best upper bound to date was given in 1977 and has received since then numerous proofs and interpretations. Here we provide a new,…
We consider the problem of determining the trade-off between the rate and the block-length of polar codes for a given block error probability when we use the successive cancellation decoder. We take the sum of the Bhattacharyya parameters…
We compute the channel capacity of non-binary fingerprinting under the Marking Assumption, in the limit of large coalition size c. The solution for the binary case was found by Huang and Moulin. They showed that asymptotically, the capacity…
We consider the standard site percolation model on the $d$-dimensional lattice. A direct consequence of the proof of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman [Comm. Math. Phys. 111 (1987) 505-531] is that the…
In a recent breakthrough, Dimitrov solved the Schinzel-Zassenhaus Conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form $T^p+c$ where $p$ is a prime number and where the orbit of…
The binary perceptron problem asks us to find a sign vector in the intersection of independently chosen random halfspaces with intercept $-\kappa$. We analyze the performance of the canonical discrepancy minimization algorithms of…
For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an…
The Shannon capacity of a graph $G$ is defined as $c(G)=\sup_{d\geq 1}(\alpha(G^d))^{\frac{1}{d}},$ where $\alpha(G)$ is the independence number of $G$. The Shannon capacity of the cycle $C_5$ on $5$ vertices was determined by Lov\'{a}sz in…
We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $\epsilon>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-\epsilon}$ with finitely many…
We provide a two-sided inequality for the alpha-optimal partition value of a measurable space according to n nonatomic finite measures. The result extends and often improves Legut (1988) since the bounds are obtained considering several…
By studying its subgraphs, it is argued that the lower critical percolation threshold of the enhanced binary tree (EBT) is bounded as $p_{c1} < 0.355059$, while the upper threshold is bounded both from above and below by 1/2 according to…
The two-user broadcast channel (BC) with receivers connected by bidirectional cooperation links of finite capacities, known as conferencing decoders, is considered. A novel capacity region outer bound is established based on multiple…
Capacity bounds for waveform channels under square-law detection of time-limited complex-valued signals are derived. The upper bound is the capacity of the channel under (complex-valued) coherent detection. The lower bound is one bit less,…
We derive here a robust bound on the effective number of neutrinos from constraints on primordial nucleosynthesis yields of deuterium and helium. In particular, our results are based on very weak assumptions on the astrophysical…
The $abc$ conjecture predicts a highly non trivial upper bound for the height of an algebraic point in terms of its discriminant and its intersection with a fixed divisor of the projective line counted without multiplicity. We describe the…