Related papers: Semiarcs with a long secant in $\mathrm{PG}(2,q)$

In a projective plane $\Pi_q$ of order $q$, a non-empty point set ${\cal S}_t$ is a $t$-semiarc if the number of tangent lines to ${\cal S}_t$ at each of its points is $t$. If ${\cal S}_t$ is a $t$-semiarc in $\Pi_q$, $t<q$, then each line…

A $2$-semiarc is a pointset ${\mathcal S}_k$ with the property that the number of tangent lines to ${\mathcal S}_k$ at each of its points is two. Using some theoretical results and computer aided search, the complete classification of…

A blocking semioval is a set of points in a projective plane that is both a blocking set (i.e., every line meets the set, but the set contains no line) and a semioval (i.e., there is a unique tangent line at each point). The smallest size…

A k-arc in a Dearguesian projective plane whose secants meet some external line in k-1 points is said to be hyperfocused. Hyperfocused arcs are investigated in connection with a secret sharing scheme based on geometry due to Simmons. In…

The classification of all semiovals and blocking semiovals in $\mathrm{PG}(2,8)$ and in $\mathrm{PG}(2,9)$ of size less than $17$ is determined. Also, some information on the stabilizer groups and the intersection sizes with lines is given.

A $k$-arc in PG($2,q$) is a set of $k$ points no three of which are collinear. A hyperfocused $k$-arc is a $k$-arc in which the $k \choose 2$ secants meet some external line in exactly $k-1$ points. Hyperfocused $k$-arcs can be viewed as…

A set without tangents in $\PG(2,q)$ is a set of points S such that no line meets S in exactly one point. An exterior set of a conic $\mathcal{C}$ is a set of points $\E$ such that all secant lines of $\E$ are external lines of…

All sets of lines providing a partition of the set of internal points to a conic C in PG(2,q), q odd, are determined. There exist only three such linesets up to projectivities, namely the set of all nontangent lines to C through an external…

A semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities having real coefficients and is a union of finitely many maximally connected components. We consider the problem of deciding whether two…

The subject of this paper is the study of small complete arcs in $\mathrm{PG}(2,q)$, for $q$ odd, with at least $(q+1)/2$ points on a conic. We give a short comprehensive proof of the completeness problem left open by Segre in his seminal…

A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not…

A {\em generalized hyperfocused arc} $\mathcal H $ in $PG(2,q)$ is an arc of size $k$ with the property that the $k(k-1)/2$ secants can be blocked by a set of $k-1$ points not belonging to the arc. We show that if $q$ is a prime and…

We prove a determinantal formula for quantities related to the problem of enumeration of (semi-) meanders, namely the topologically inequivalent planar configurations of non-self-intersecting loops crossing a given (half-) line through a…

In this paper, we study the cardinality of the smallest set of lines of the finite projective spaces $\operatorname{PG}(n,q)$ such that every plane is incident with at least one line of the set. This is the first main open problem…

We devote this work to the functional equation $\psi \circ g = g\circ \psi$, where $\psi$ is an unknown function and $g$ is piecewise linear unimodal map, which is topologically conjugated to the tent map. We will call such $\psi$…

An algebra $A$ is said to be strongly semisimple if every principal congruence of $A$ is an intersection of maximal congruences. We give a geometrical characterisation of strongly semisimple MV-algebras in terms of Bouligand-Severi…

Blocking semiovals and the determination of their (minimum) sizes constitute one of the central research topics in finite projective geometry. In this article we introduce the concept of blocking set with the $r_\infty$-property in a finite…

In this paper we consider two pointsets in $\mathrm{PG}(2,q^n)$ arising from a linear set $L$ of rank $n$ contained in a line of $\mathrm{PG}(2,q^n)$: the first one is a linear blocking set of R\'edei type, the second one extends the…

In this paper we introduce a quotient structure on topological ternary semigroup by defining a congruence suitably. We have found conditions under which this quotient structure becomes a topological ternary semigroup. We have also obtained…

Semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities. In this paper, we consider the problem of deciding whether two given points in a semi-algebraic set are connected. We restrict to the case…