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The Tur\'{a}n number $ex(n,H)$ of a graph $H$ is the maximum number of edges in any $H$-free graph on $n$ vertices. The triangular pyramid of $k$-layers, denoted by $TP_k$, is a generalization of a triangle. The Tur\'an problems of a…
Through the two specific problems, the 2D hidden linear function problem and the 1D magic square problem, Bravyi et al. have recently shown that there exists a separation between $\mathbf{QNC^0}$ and $\mathbf{NC^0}$, where $\mathbf{QNC^0}$…
The famous $n$-queens problem asks how many ways there are to place $n$ queens on an $n \times n$ chessboard so that no two queens can attack one another. The toroidal $n$-queens problem asks the same question where the board is considered…
We show the 3 by 3 magic square of squares problem equivalent to solving quartic polynomials with certain factorization constraints over an abelian extension of the rationals. We analyze a particular case in which said extension is assumed…
In card games, in casino games with multiple decks of cards and in cryptography, one is sometimes faced with the following problem: how can a human (as opposed to a computer) shuffle a large deck of cards? The procedure we study is to break…
In this paper, we study the Contiguous Art Gallery Problem, introduced by Thomas C. Shermer at the 2024 Canadian Conference on Computational Geometry, a variant of the classical art gallery problem from 1973 by Victor Klee. In the…
We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal…
This paper considers the triangle finding problem in the CONGEST model of distributed computing. Recent works by Izumi and Le Gall (PODC'17), Chang, Pettie and Zhang (SODA'19) and Chang and Saranurak (PODC'19) have successively reduced the…
We present a $9^k\cdot n^{O(1)}$-time algorithm for the proper circular-arc vertex deletion problem, resolving an open problem of van 't Hof and Villanger [Algorithmica 2013] and Crespelle et al. [arXiv:2001.06867]. Our structural study…
We study outer length billiards; our main results are as follows. We prove 3- and 4-periodic versions of the Ivrii conjecture. We show that, for every period $n\ge 3$, there exists a functional space of billiard tables that possess…
E. R. Lamken prove in [5] that there exists a partitioned balanced tournament design of side $n$, PBTD($n$), for $n$ a positive integer, $n \ge 5$, except possibly for $n \in \{9,11,15\}$. In this article, we show the existence of PBTD($n$)…
As a continuation of a previous paper (arXiv:2303.05769 [nlin.CD]), we introduce examples of H\'enon-type mappings that exhibit new horseshoe topologies in three and four dimensional spaces that are otherwise impossible in two dimensions.
The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion…
We present a finite-order system of recurrence relations for a permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k = 1, 2, and 3) as well as the method for deriving such recurrence…
Following the setup proposed by Jardim-Maciocia-Martinez in the case of the projective space, we study some numerical and actual Bridgeland walls for the (twisted) Chern character $v=(-R,0,D,0)$ in certain half-plane of stability…
We prove that nonlinear Schr\"odinger equations on the circle, without external parameters, admits plenty of almost periodic solutions. Indeed, we prove that arbitrarily close to most of the finite dimensional KAM tori constructed by…
In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body $K\subset \mathbb{R}^d$ has the property that the tangent cone of every non-smooth…
Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in…
The Dinitz conjecture states that, for each $n$ and for every collection of $n$-element sets $S_{ij}$, an $n\times n$ partial latin square can be found with the $(i,j)$\<th entry taken from $S_{ij}$. The analogous statement for $(n-1)\times…
In this paper, we study the solution to the 1-dimensional $\lambda$-self shrinkers and show that for certain $\lambda<0$, there are some closed, embedded solutions other than the circle.