Related papers: On The "Majority is Least Stable" Conjecture

One of the interesting problems in arithmetic dynamics is to study the stability of polynomials over a field. In this paper, we study the stability of $f(z)=z^d+\frac{1}{c}$ for $d\geq 2$, $c\in{\mathbb{Z}\setminus\{0\}}$. We show that for…

We study in this paper stability estimates for the fault inverse problem. In this problem, faults are assumed to be planar open surfaces in a half space elastic medium with known Lam\'e coefficients. A traction free condition is imposed on…

We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies several…

Algorithmic stability is a central concept in statistics and learning theory that measures how sensitive an algorithm's output is to small changes in the training data. Stability plays a crucial role in understanding generalization,…

We show that, in the most general $N$-component theory with symmetry O(n_1)+O(n_2), N=n_1+n_2\geq 3, the O(N)-symmetric fixed point has (at least) three unstable directions: the temperature, the quadratic anisotropy, and the spin-4 quartic…

Super-stability and strong stability are properties of a matching in the stable matching problem with ties. In this paper, we introduce a common generalization of super-stability and strong stability, which we call non-uniform stability.…

We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs…

Estimation of structure, such as in variable selection, graphical modelling or cluster analysis is notoriously difficult, especially for high-dimensional data. We introduce stability selection. It is based on subsampling in combination with…

Some argue that political stability is best served through a two-party system. This study refutes this. The author mathematically defines the stability and rigidity of electoral systems comprised of any quantity of electors and parties. In…

Let $(M,g)$ be a compact riemannian manifold of dimension $n\geq 5$. We are interested in the stability of a slighly subcritical Paneitz-Branson type equation on $M$. Assuming that there exists a positive nondegenerate solution of the…

Given a simple graph $G$, the {\it irregularity strength} of $G$, denoted $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ for which each vertex weight $f^V(v):=…

Let $k$ and $\ell$ be positive integers. We prove that if $1 \leq \ell \leq o_k(k^{6/5})$, then in every large enough graph $G$, the fraction of $k$-vertex subsets that induce exactly $\ell$ edges is at most $1/e + o_k(1)$. Together with a…

Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly…

Let $M$ be an $n\times n$ matrix with iid subgaussian entries with mean $0$ and variance $1$ and let $\sigma_n(M)$ denote the least singular value of $M$. We prove that \[\mathbb{P}\big( \sigma_{n}(M) \leq \varepsilon n^{-1/2} \big) =…

We verify that for all $n \geq 3$ and $2 \leq k \leq n+1$, the standard $k$-bubble clusters, conjectured to be minimizing total perimeter in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$, are stable -- an infinitesimal regular…

We prove that the upper bounds for the consistency strength of certain instances of mutual stationarity considered by Liu-Shelah~\cite{MR1469093} are close to optimal. We also consider some related and, as it turns out, stronger properties.

We describe a new algorithm for verifying the Collatz conjecture for all n < 2^N for some fixed N. The algorithm takes less than twice as long to verify convergence for all n < 2^{N+1} as it does to verify convergence for all n < 2^N. We…

We report the finding of the new upper bound on the lowest positive integer $x$ for which the Mertens conjecture \begin{equation*} \left| \sum_{1 \leq n \leq x} \mu(n) \right| < \sqrt{x} \end{equation*} fails to hold: $x < \exp(1.017 \times…

Given a natural number N, one may ask what configuration of N points on the two-sphere minimizes the discrete generalized Coulomb energy. If one applies a gradient-based numerical optimization to this problem, one encounters many…

It is a classical fact that every $n$-element set of positive reals has at least $\binom{n+1}{2}+1$ distinct subset sums, with equality exactly for homogeneous arithmetic progressions (when $n\geq 4$). We establish stability versions of…