Related papers: Direction-Critical Configurations in Noncentral Ge…
We establish the existence of finitely many sign-changing solutions to the Lane-Emden system $$-\Delta u=|v|^{q-2}v,\quad -\Delta v=|u|^{p-2}u \quad \text{ in }\mathbb{R}^N, \ \ N\geq 4,$$ where the exponents $p$ and $q$ lie on the critical…
We show that a homeomorphism of Euclidean space is quasiconformal if and only if at each point there exists a sequence of uncentered open sets with bounded eccentricity shrinking to that point whose images also have bounded eccentricity.…
We identify the asymptotic probability of a configuration model $\mathrm{CM}_n(\boldsymbol{d})$ to produce a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well…
The maximum number of non-crossing straight-line perfect matchings that a set of $n$ points in the plane can have is known to be $O(10.0438^n)$ and $\Omega^*(3^n)$. The lower bound, due to Garc\'ia, Noy, and Tejel (2000) is attained by the…
Given a set of $n$ points in $R^2$, the Szemer\'edi-Trotter theorem establishes that the number of lines which can be incident to at least $k > 1$ of these points is $O(n^2/k^3 + n/k)$. J.\ Solymosi conjectured that if one requires the…
The union-closed sets conjecture (sometimes referred to as Frankl's conjecture) states that every finite, nontrivial union-closed family of sets has an element that is in at least half of its members. Although the conjecture is known to be…
A graph $G$ is \textit{asymmetric} if its automorphism group of vertices is trivial. Asymmetric graphs were introduced by Erd\H{o}s and R\'{e}nyi in 1963. They showed that the probability of a graph on $n$ vertices being asymmetric tends to…
We show that every set $\mathcal{P}$ of $n$ non-collinear points in the plane contains a point incident to at least $\lceil\frac{n}{3}\rceil+1$ of the lines determined by $\mathcal{P}$.
We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central…
We consider the nonconvex optimization problem associated with the decomposition of a real symmetric tensor into a sum of rank-one terms. Use is made of the rich symmetry structure to construct infinite families of critical points…
A. Chermak has recently proved that to each saturated fusion system over a finite $p$-group, there is a unique associated centric linking system. B. Oliver extended Chermak's proof by showing that all the higher cohomological obstruction…
We show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\sqrt{\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a…
In the present paper, as a continuation of our preceding paper [10], we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a viewpoint of discrete geometric analysis…
The Frankl or Union-Closed Sets conjecture states that for any finite union-closed family of sets $\mathcal{F}$ containing some nonempty set, there is some element $i$ in the ground set $U(\mathcal F) := \bigcup_{S \in \mathcal{F}} S$ of…
We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be…
Nonsymmorphic symmetries can enforce band connectivity that obstructs a single-band Wannier description. We show that a fractional translation $\mathcal{L}$ connecting distinct high-symmetry Wyckoff positions generically renders the Wannier…
We study perimeters of connecting cycles for concentric circles. More precisely, we are interested in characterization of those connecting cycles which are critical points of perimeter considered as a function on the product of given…
We investigate multi-field multicritical scalar theories using CFT constraints on two- and three-point functions combined with the Schwinger-Dyson equation. This is done in general and without assuming any symmetry for the models, which we…
We classify the possible discrete (finite) symmetries of two--dimensional critical models described by unitary minimal conformally invariant theories. We find that all but six models have the group Z_2 as maximal symmetry. Among the six…
Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic,…