Related papers: Capacity threshold for the Ising perceptron

In this work, novel upper and lower bounds for the capacity of channels with arbitrary constraints on the support of the channel input symbols are derived. As an immediate practical application, the case of multiple-input multiple-output…

We consider the state dependent channels with full state information with at the sender. For this state dependent channel, the channel capacity was determined by Gel'fand and Pinsker. In this paper, we study the correct probability of…

We show that the constraints which follow from the {\it Trans-Planckian Censorship Conjecture} for inflationary cosmology can be strengthened if the pre-inflationary universe was dominated by radiation. The resulting upper bound on the…

There are several examples in the literature showing that compactness-like properties of a cardinal $\kappa$ cause poor behavior of some generic ultrapowers which have critical point $\kappa$ (Burke \cite{MR1472122} when $\kappa$ is a…

We derive upper bounds on the fluctuations of a class of random surfaces of the $\nabla \phi$-type with convex interaction potentials. The Brascamp-Lieb concentration inequality provides an upper bound on these fluctuations for uniformly…

Let $(\cx,\,d,\,\mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors show that for the maximal Calder\'on-Zygmund operator associated with a…

Let $(R, m)$ be a $d$-dimensional Cohen-Macaulay local ring. In this note we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a $m$-primary ideal $I\subset R$ that improves all known upper…

We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss…

In this paper, we solve the optimal constant problem in the setting of Ohsawa's generalized $L^{2}$ extension theorem. As applications, we prove a conjecture of Ohsawa and the extended Suita conjecture, we also establish some relations…

We consider the additive white Gaussian noise channels. We prove that the error probability of decoding tends to one exponentially for rates above the capacity and derive the optimal exponent function. We shall demonstrate that the…

This work considers a Poisson noise channel with an amplitude constraint. It is well-known that the capacity-achieving input distribution for this channel is discrete with finitely many points. We sharpen this result by introducing upper…

We develop a least-squares method for computing the analytic capacity of compact plane sets with piecewise-analytic boundary. The method furnishes rigorous upper and lower bounds which converge to the true value of the capacity. Several…

We use the theory of resolutions for a given Hilbert function to investigate the multiplicity conjectures of Huneke and Srinivasan and Herzog and Srinivasan. To prove the conjectures for all modules with a particular Hilbert function, we…

This paper deals with probabilistic upper bounds for the error in functional estimation defined on some interpolation and extrapolation designs, when the function to estimate is supposed to be analytic. The error pertaining to the estimate…

In this paper we introduce capacitary analogues of the Hardy-Littlewood maximal function, \begin{align*} \mathcal{M}_C(f)(x):= \sup_{r>0} \frac{1}{C(B(x,r))} \int_{B(x,r)} |f|\;dC, \end{align*} for $C=$ the Hausdorff content or a Riesz…

In 1965, Vizing conjectured that the independence ratio of edge-chromatic critical graphs is at most $\frac{1}{2}$. We prove that for every $\epsilon > 0$ this conjecture is equivalent to its restriction on a specific set of edge-chromatic…

In this note, we establish a functional central limit theorem for the capacity of the range for a class of $\alpha$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5\alpha/2$. Using similar methods, we also prove an…

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Engbers and Galvin asked how large $i_t(G)$ could be in graphs with minimum degree at least $\delta$. They further conjectured that when $n\geq 2\delta$ and $t\geq…

For any $\alpha\in(0,d)$, we construct Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $\alpha$ such that the associated natural measure $\mu$ obeys the restriction estimate $\| \widehat{f d\mu} \|_{p} \leq C_p \| f \|_{L^2(\mu)}$ for…

We consider the critical FK-Ising measure $\phi_{\beta_c}$ on $\mathbb Z^d$ with $d\geq 3$. We construct the measure $\phi^\infty:=\lim_{|x|\rightarrow \infty}\phi_{\beta_c}[\:\cdot\: |\: 0\leftrightarrow x]$ and prove it satisfies…