Related papers: Hyperfocused arcs in PG(2,32)
The Pascal Multimysticum is a system of points and lines constructed with a straight edge starting from six points on a conic. We show that the system contains 150 infinite ranges (and 150 infinite pencils) whose projective coordinates are…
Given a collection of points in the plane, classifying which subsets are collinear is a natural problem and is related to classical geometric constructions. We consider collections of points in a projective plane over a finite field such…
In this paper, we consider a number of results and seven conjectures on properly edge-coloured (PC) paths and cycles in edge-coloured multigraphs. We overview some known results and prove new ones. In particular, we consider a family of…
Trigonometric parallaxes of 16 nearby planetary nebulae are presented, including reduced errors for seven objects with previous initial results and results for six new objects. The median error in the parallax is 0.42 mas, and twelve…
In the course of an extensive campaign to measure radial velocities of galaxies in a selected sample of compact groups photometrically studied by de Carvalho et al. (1994), we report the discovery of a system very rich in starburst galaxies…
We present a novel approach to the age-old question of whether perfect cuboids exist. Our approach consists of two new computer search algorithms, arising from the analysis of "perfect plinths" reported by one of us recently, that are much…
We classify all finite linear spaces on at most 15 points admitting a blocking set. There are no such spaces on 11 or fewer points, one on 12 points, one on 13 points, two on 14 points, and five on 15 points. The proof makes extensive use…
A global superalgebra with 32 supercharges and all possible central extensions is studied in order to extract some general properties of duality and hidden dimensions in a theory that treats $p$-branes democratically. The maximal number of…
There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D.A. Gudkov. There are nine types of singular…
We show that there is no (75,32,10,16) strongly regular graph. The result is obtained by a mix of algebraic and computational approaches. The main idea is to build large enough induced structure and apply the star complement technique. Our…
We show that a rectifiable plane arc g has two parallel support lines and a triple of consecutive points g(r), g(s), g(t), r<s<t, so that g(s) lies on one line, while g(r) and g(t) lie on the other. If the arc is simple, such a pair of…
We establish that the minimum number of arcs required to partition the Galois projective plane $\text{PG}(2,q)$ is $q+1$. Furthermore, we determine the exact value for a fractional variant of this problem. We extend our analysis to affine…
We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent…
Probabilistic cross-identification has been successfully applied to a number of problems in astronomy from matching simple point sources to associating stars with unknown proper motions and even radio observations with realistic morphology.…
A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We prove that any connected subcubic graph with $n$ vertices and girth at least $5$ contains a uniquely restricted matching of…
Searching for lopsided/interacting objects among ~1500 isolated galaxies yields only eight strongly disturbed galaxies which may be explained as a result of their interaction with massive dark objects. We present results of spectral and…
Strongly lensed quadruply imaged quasars (quads) are extraordinary objects. They are very rare in the sky -- only a few tens are known to date -- and yet they provide unique information about a wide range of topics, including the expansion…
In the previous works of the authors, a step-by-step algorithm FOP which uses any fixed order of points in the projective plane $\mathrm{PG}(2,q)$ is proposed to construct small complete arcs. In each step, the algorithm adds to a current…
We show that almost complex circle actions with exactly three fixed points do not exist in dimension 8 and present an infinite series of 6-dimensional manifolds possessing an almost complex circle action with exactly two fixed points.
Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete…