Related papers: On The "Majority is Least Stable" Conjecture
What is the smallest number of random transpositions (meaning that we swap given pairs of elements with given probabilities) that we can make on an $n$-point set to ensure that each element is uniformly distributed -- in the sense that the…
A popular method for selecting the number of clusters is based on stability arguments: one chooses the number of clusters such that the corresponding clustering results are "most stable". In recent years, a series of papers has analyzed the…
It is shown that $3$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with nearly minimum total Gaussian surface area must be close to adjacent $120$ degree sectors, when $n\geq2$. These same results hold for any…
The Standard Simplex Conjecture of Isaksson and Mossel asks for the partition $\{A_{i}\}_{i=1}^{k}$ of $\mathbb{R}^{n}$ into $k\leq n+1$ pieces of equal Gaussian measure of optimal noise stability. That is, for $\rho>0$, we maximize $$…
We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by…
We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $\alpha$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars, each of…
For a dynamical system on n-dimensional projective space over a number field or a function field, we show that semi-stable reduction implies the minimality of the resultant. We use this to show that every such dynamical system over a number…
The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning…
It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal…
We study deviations by a group of agents in the three main types of matching markets: the house allocation, the marriage, and the roommates models. For a given instance, we call a matching $k$-stable if no other matching exists that is more…
We revisit the phenomenon of instability of solitons in the generalized Korteweg-de Vries equation, $u_t + \partial_x(u_{xx} + u^p) = 0$. It is known that solitons are unstable for nonlinearities $p \geq 5$, with the critical power $p=5$…
Algorithms for resolving majority cycles in preference aggregation have been studied extensively in computational social choice. Several sophisticated cycle-resolving methods, including Tideman's Ranked Pairs, Schulze's Beat Path, and…
The profiles of narrow lattice solitons are calculated analytically using perturbation analysis. A stability analysis shows that solitons centered at a lattice (potential) maximum are unstable, as they drift toward the nearest lattice…
The algebraic stability theorem for $\mathbb{R}$-persistence modules is a fundamental result in topological data analysis. We present a stability theorem for $n$-dimensional rectangle decomposable persistence modules up to a constant…
We give a new method to prove the existence, non-existence, multiplicity, orbital stability/instability of standing waves for NLS with partial confinement without the subcritical hypothesis, even in the reduction equation. Using this…
We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for…
We study majority dynamics on the binomial random graph $G(n,p)$ with $p = d/n$ and $d > \lambda n^{1/2}$, for some large $\lambda>0$. In this process, each vertex has a state in $\{-1,+1 \}$ and at each round every vertex adopts the state…
There is no trivial mathematics, there are only trivial mathematicians! A mathematician is trivial if he or she believes that there exists trivial mathematics. Being a non-trivial mathematician myself, I will describe ten different proofs…
We consider the problem of computing the closest stable/unstable non-negative matrix to a given real matrix. This problem is important in the study of linear dynamical systems, numerical methods, etc. The distance between matrices is…
In this paper, I characterize minimal stable voting rules and minimal self-stable constitutions (i.e., pairs of voting rules) for societies in which only power matters. To do so, I first let players' preference profiles over voting rules…