Related papers: A note on the capacity of the binary perceptron
Let ${\mathcal D}(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and $\mathcal R(n) = {\mathcal D}(n)/n^{n/2}$ be the ratio of ${\mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove…
A power is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer and a square is a word of the form $uu$. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares…
Determining the energy scale of inflation is crucial to understand the nature of inflation in the early Universe. We place observational constraints on the energy scale of the observable part of the inflaton potential by combining the…
A strong version of Andrica's conjecture can be formulated as follows: Except for $p_n\in\{3,7,13,23,31,113\}$, that is $n\in\{2,4,6,9,11,30\}$, one has$\sqrt{p_{n+1}}-\sqrt{p_n} < \frac{1}{2}.$ While a proof is far out of reach I shall…
Heisenberg's uncertainty principle states that the position and momentum of a particle cannot be sharply determined simultaneously. Standard-deviation and entropic formulations capture the spread of the probability distribution but say…
A classic theorem of Kazhdan and Margulis states that for any semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices. H. C. Wang's subsequent quantitative analysis showed that the…
Cells adapt to changing environments by sensing ligand concentrations using specific receptors. The accuracy of sensing is ultimately limited by the finite number of ligand molecules bound by receptors. Previously derived physical limits to…
At the linear level, the gravitational wave (GW) spectrum predicted by inflation, and many of its alternatives, can have arbitrarily small amplitude and consequently an unconstrained tilt. However, at second order, tensor fluctuations are…
In physical-layer security, one of the most fundamental issues is the secrecy capacity. The objective of this paper is to determine the secrecy capacity for an indoor visible light communication system consisting of a transmitter, a…
New capacity upper bounds are presented for the discrete-time Poisson channel with no dark current and an average-power constraint. These bounds are a simple consequence of techniques developed for the seemingly unrelated problem of upper…
Ahlswede, Cai, and Zhang proved that, in the noise-free limit, the zero-undetected-error capacity is lower bounded by the Sperner capacity of the channel graph, and they conjectured equality. Here we derive an upper bound that proves the…
The assertion that there is an intrinsic excess of binaries with mass ratios q \simeq 1 - the twin hypothesis - is investigated. A strong version of this hypothesis (H_s), due to Lucy & Ricco (1979) and Tokovinin (2000), refers to a narrow…
Cosmological $\alpha$-attractors are a compelling class of inflationary models. They lead to universal predictions for large-scale observables, broadly independent from the functional form of the inflaton potential. In this work we derive…
Independently of any theory, the possibility that the large value of the Tevatron cross section claimed by CDF is correct suggests that the total cross section at the LHC may be large. Because of the experimental and theoretical…
We study the determination problem of the channel capacity for the discrete memoryless channels in the finite blocklength regime. We derive explicit lower and upper bounds of the capacity. We shall demonstrate that the information spectrum…
A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and…
We study the random binary symmetric perceptron problem, focusing on the behavior of rare high-margin solutions. While most solutions are isolated, we demonstrate that these rare solutions are part of clusters of extensive entropy,…
A lower bound is derived for the boundary entropy s = ln g of a 1+1d quantum critical system with boundary, under the conditions that the bulk conformal central charge c is >=1 and the most relevant bulk scaling dimension is >(c-1)/12. This…
R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings. We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the delta-vector of…
We search for a possible $\Lambda_{c} \bar{{\Sigma}}_{c}$ bound state, denoted as $H_{c}^{\pm}$, via the $ e^{+}e^{-} \to \pi^{+} \pi^{-} \Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ process for the first time. This analysis utilizes 207.8 and…