Related papers: The Krzy\.z Conjecture and an Entropy Conjecture

After appropriate normalizations an embedded disk whose second fundamental form has large norm contains a multi-valued graph, provided the L^P norm of the mean curvature is sufficiently small. This generalizes to non-minimal surfaces a well…

Recently Bousso conjectured the entropy crossing a certain light-like hypersurface is bounded by the surface area. We point out a number of difficulties with this conjecture.

We define a notion of entropy for an infinite family $\mathcal{C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{C}$ under every ergodic transformation if and only if…

A number of recent works have sought to generalize the Kolmogorov-Sinai entropy of probability-preserving transformations to the setting of Markov operators acting on the integrable functions on a probability space $(X,\mu)$. These have…

We study the root distribution of a sequence of polynomials $\{P_n(z)\}_{n=0}^{\infty}$ with the rational generating function $$ \sum_{n=0}^{\infty} P_n(z)t^n= \frac{1}{1+ B(z)t^\ell +A(z)t^k}$$ for $(k,\ell)=(3,2)$ and $(4,3)$ where $A(z)$…

Using only continuous partitions of unity, we provide equivalent definitions for the metric, topological and topological tail entropies and pressures of a continuous self-map of a compact set, as well as their conditional versions. A tail…

We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1}…

We study the evolution of the roots of a polynomial of degree $N$, when the polynomial itself is evolving according to the heat flow. We propose a general conjecture for the large-$N$ limit of this evolution. Specifically, we propose (1)…

The slope conjecture gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this note we propose a generalization of the slope…

We prove a generalization of the Shapiro-Shapiro conjecture on Wronskians of polynomials, allowing the Wronskian to have complex conjugate roots. We decompose the real Schubert cell according to the number of real roots of the Wronski map,…

Entropy rate is a real valued functional on the space of discrete random sources which lacks a closed formula even for subclasses of sources which have intuitive parameterizations. A good way to overcome this problem is to examine its…

We exploit the idea to use the maximal-entropy method, successfully tested in information theory and statistical thermodynamics, to determine approximating function's coefficients and squared errors' weights simultaneously as output of one…

Using the recent proof of the polynomial Freiman-Ruzsa conjecture over $\mathbb{F}_p^n$ by Gowers, Green, Manners, and Tao, we prove a version of the polynomial Freiman-Ruzsa conjecture over function fields. In particular, we prove that if…

We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromov's…

A celebrated conjecture of Tuza states that in any finite graph the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge-disjoint triangles. For an $r$-uniform hypergraph ($r$-graph) $G$, let…

Let f be holomorphically continuable over the complex plane except for finitely many branch points contained in the unit disk. We prove that best rational approximants to f of degree n, in the L^2-sense on the unit circle, have poles that…

We prove that the Buchweitz-Greuel-Schreyer Conjecture on the minimal rank of a matrix factorization holds for a generic polynomial of given degree and strength. The proof introduces a notion of the secondary strength of a polynomial, and…

Inspired by the work of Karamata, we consider an extremization problem associated with the probability of intersecting two random chords inside a circle of radius $r, \, r \in (0,1]$, where the endpoints of the chords are drawn according to…

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \| f(z) \|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty…

For $r > 0$ and integers $t \ge n > 0$, we consider the following problem: maximize the amplitude $|x_t|$ at time $t$, over all complex solutions $x = (x_0, x_1, \dots)$ of arbitrary homogeneous linear difference equations of order $n$ with…