In this work, we prove a version of the Sylvester–Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of $\Sigma ^{[3]}\Pi \Sigma \Pi ^{[2]}$ circuits. Specifically, we prove that, if a finite set of irreducible quadratic polynomials ${\mathcal {Q}}$ satisfies that for every two polynomials $Q_1,Q_2\in {\mathcal {Q}}$ there is a subset ${\mathcal {K}}\subset {\mathcal {Q}}$ such that $Q_1,Q_2 \notin {\mathcal {K}}$ and whenever $Q_1$ and $Q_2$ vanish, then $\prod _{i\in {\mathcal {K}}} Q_i$ vanishes, then the linear span of the polynomials in ${\mathcal {Q}}$ has dimension $O(1)$ . This extends the earlier result [21] that holds for the case $|{\mathcal {K}}| = 1$ .

We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.

We study local properties of the Bakry–Émery curvature function ${\mathcal{K}}_{G,x}:(0,\infty ]\rightarrow \mathbb{R}$ at a vertex $x$ of a graph $G$ systematically. Here ${\mathcal{K}}_{G,x}({\mathcal{N}})$ is defined as the optimal curvature lower bound ${\mathcal{K}}$ in the Bakry–Émery curvature-dimension inequality $CD({\mathcal{K}},{\mathcal{N}})$ that $x$ satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and $S^{1}$-out regularity, and relate the curvature functions of $G$ with various spectral properties of (weighted) graphs constructed from local structures of $G$. We prove that the curvature functions of the Cartesian product of two graphs $G_{1},G_{2}$ are equal to an abstract product of curvature functions of $G_{1},G_{2}$. We explore the curvature functions of Cayley graphs and many particular (families of) examples. We present various conjectures and construct an infinite increasing family of 6-regular graphs which satisfy $CD(0,\infty )$ but are not Cayley graphs.

Previous results about n-grids with acceptable colorings are extended here to n-indexed hyperspaces, which are structures ${\cal A} = \left( {A;{E_0},{E_1}, \ldots ,{E_{n - 1}}} \right)$, where each ${E_i}$ is an equivalence relation on A.

We characterize straightness of digital curves in the integer plane by means of difference operators. Earlier definitions of digital rectilinear segments have used, respectively, Rosenfeld’s chord property, word combinatorics, Reveillès’ double Diophantine inequalities, and the author’s refined hyperplanes. We prove that all these definitions are equivalent. We also characterize convexity of integer-valued functions on the integers with the help of difference operators.

By $\text{d(X,Y)}$ we denote the (multiplicative) Banach–Mazur distance between two normed spaces $X$ and $Y$. Let $X$ be an $n$-dimensional normed space with $\text{d(X,}\,l_{\infty }^{n}\text{)}\,\le \,\text{2}$, where $l_{\infty }^{n}$ stands for ${{\mathbb{R}}^{n}}$ endowed with the norm $\parallel ({{x}_{1}},\,.\,.\,.\,,\,{{x}_{n}}){{\parallel }_{\infty }}\,:=\,\max \{|{{x}_{1}}|,\,.\,.\,.\,,\,|{{x}_{n}}|\}$. Then every metric space $(S,\,\rho )$ of cardinality $n+1$ with norm $\rho $ satisfying the condition $\max D/\min D\,\le \,2/\,\text{d(}X,\,l_{\infty }^{n}\text{)}$ for $D\,:=\,\{\rho (a,\,b)\,:\,a,\,b\,\in \,S,\,a\,\ne \,b\}$ can be isometrically embedded into $X$.

Motivated by applications of Gabriel graphs and Yao graphs in wireless ad-hoc networks, we show that the maximum degree of a random Gabriel graph or Yao graph defined on n points drawn uniformly at random from a unit square grows as Θ (log n / log log n) in probability.

With an increase in size, configurations of points and lines in the plane usually become complicated and hard to analyze. The “floral” configurations we are introducing here represent a new type that makes accessible and visually intelligible even configurations of considerable size. This is achieved by combining a large degree of symmetry with a hierarchical construction. Depending on the details of the interdependence of these aspects, there are several subtypes that are described and investigated.

We describe a family of integrable lattice maps related to the known quad maps Q4. The integrability criterion we use is the vanishing of the algebraic entropy. The family has the advantage of being parametrized rationally: all its parameters are unconstrained.

The main goal of the paper is to find the discrete analogue of the Bianchi system in spaces of arbitrary dimension together with its geometric interpretation. We show that the proper geometric framework for such generalization is the language of dual quadrilateral lattices and of dual congruences. After introducing the notion of the dual Koenigs lattice in a projective space of arbitrary dimension, we define the discrete dual congruences and we present, as an important example, the normal dual discrete congruences. Finally, we introduce the dual Bianchi lattice as a dual Koenigs lattice allowing for a conjugate normal dual congruence, and we find its characterization in terms of a system of integrable difference equations.

The regular star-polyhedron $\left\{ 5,\,\frac{5}{2} \right\}$ is isomorphic to the abstract polyhedron $\left\{ 5,\,5|3 \right\}$, where the last entry “3” in its symbol denotes the size of a hole, given by the imposition of a certain extra relation on the group of the hyperbolic honeycomb $\left\{ 5,\,5 \right\}$. Here, analogous formulations are found for the groups of the regular 4-dimensional star-polytopes, and for those of the non-discrete regular 4-dimensional honeycombs. In all cases, the extra group relations to be imposed on the corresponding Coxeter groups are those arising from “deep holes”; thus the abstract description of $\left\{ 5,\,{{3}^{k}},\,\frac{5}{2} \right\}\,\text{is}\,\left\{ 5,\,{{3}^{k}},\,5|3 \right\}\,\text{for}\,k\,=\,1\,\text{or}\,\text{2}$. The non-discrete quasi-regular honeycombs in ${{\mathbb{E}}^{3}}$, on the other hand, are not determined in an analogous way.