Line Transversals Research Articles

Distribution of endemic Garra culiciphaga (Pellegrin, 1927) in the Ceyhan River System, some diagnostic characteristics, habitats and conservation requirements

In this study, the distribution and some diagnostic features of the endemic Garra culiciphaga belonging to the Cyprinidae family in the Ceyhan River systems were determined. Accordingly, the average total length of G. culiciphaga individuals is 49.58 mm, and their average weight is 1.32 g. Line lateral scales are 30-33, line transversal 8/7, and pharyngeal teeth are 3.3.5-5.3.3. G. culiciphaga has a very limited distribution in the Ceyhan River basin and has been identified from the Kumaşır Lake, Aksu Stream and Akçasu Stream. However, G. culiciphaga individuals were detected in May 2015 in Aksu stream could not encountered in the following year (April 2016) due to habitat loss.

Line Transversals to Disjoint Disks

Line Transversals to Disjoint Disks

Line Transversals in Families of Connected Sets in the Plane

We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of Eckhoff [Discrete Comput. Geom., 9 (1993), pp. 203--214], who proved that, under the same condition, there are four lines intersecting all the sets. In fact, we prove a colorful version of this result under weakened conditions on the sets. Three sets $A,B,C$ form a tight triple if $\textrm{conv}(A\cup B)\cap \textrm{conv}(A\cup C)\cap \textrm{conv}(B\cap C)\neq \emptyset.$ This notion was first introduced by Holmsen, who showed that if $\mathcal{F}$ is a family of compact convex sets in the plane in which every three sets form a tight triple, then there is a line intersecting at least $\frac{1}{8}|\mathcal{F}|$ members of $\mathcal{F}$. Here we prove that if $\mathcal{F}_1,\dots,\mathcal{F}_6$ are families of compact connected sets in the plane such that every three sets, chosen from three distinct families $\mathcal{F}_i$, form a tight triple, then there exists $1\le j\le 6$ and three lines intersecting every member of $\mathcal{F}_j$. In particular, this improves $\frac{1}{8}$ to $\frac{1}{3}$ in Holmsen's result.

Cooperation research between electrode line transversal differential protection and bridge differential protection in HVDC project

On July 14, 2017, the electrode line occurred ground fault near the ZhaoQing converter station of GaoZhao HVDC, the Delta Bridge Differential Protection(87CBD) of the valve area protection acted before the Electrode Line Transversal Differential Protection (60EL) of electrode area protection, which leads to the pole blocking. In view of the exposed problem between the 87CBD and the 60EL about the cooperation of the protection delays, the paper studies the feasibility of the delays adjustment. Overall considering from the aspects from the 87CBD delay, the 60EL export time, the 60EL measurement error, the cooperation of the 60EL and the restart blocking time, and so on, the optimization is proposed which improves the 60EL delay for 400ms, and improves the restart blocking time for 400ms. Simulation show that the proposed measures are feasible and effective, to ensure that when the electrode line fault occurred near the converter station, the pole blocking can avoid by the DC restart sequence of the 60EL.

Evaluating the damage degree of cracking in facades using infrared thermography

The study of facade cracking is of great importance in the investigation of pathologies and degradation, as well as in the rehabilitation processes of buildings. Cracking is one of the major defects in facades; so assessing damage severity is necessary to define the best maintenance and rehabilitation strategies. The infrared thermography has been increasingly used as an inspection technique to identify and map facade defects and their severity. The use of hygrothermal simulation to understand the dynamics of the heat flux allows to observe important references that can help in the interpretation of the thermogram under different heating or cooling conditions. This study investigates the use of the quantitative passive thermography to measure the damage degree of cracks in a building facade. The crack depth was evaluated by the Delta-T values investigated in the three cracked regions. For the width, the variation of the studied temperature in a line transversal to the crack was used. The analysis was performed at a distance of 33 m from the facade. The conclusions corroborated the importance of the hygrothermal simulation as a pre-thermography technique, which made it possible to limit the period of investigation focusing on the phenomena of interest. It is also observed that it is possible to obtain quantitative values of the degree of damage by thermographic investigation. The measurement of the damage degree, in turn, brings important contribution to the application of the inspection technique.

The Katchalski–Lewis transversal problem for regular polygons

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family $${\mathcal{F}}$$ has property $${T-m}$$ , if there exists a subfamily $${\mathcal{G} \subset \mathcal{F}}$$ with $${|\mathcal{F} - \mathcal{G}| \le m}$$ admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if $${\mathcal{F}}$$ is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which $${\mathcal{F}}$$ has property $${T-m}$$ . In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon $${(n \ge 5)}$$ . We find out that, for $${5 \le n \le 34}$$ , the family has property $${T - 3}$$ ; for $${n \ge 35}$$ , the family has property $${T - 2}$$ .

A conjecture on line transversals to five unit discs

In this paper, we prove the following: Let $$\mathcal {F}=\{x_1+B,x_2+B,x_3+B,x_4+B,x_5+B\}$$ be a family of translates of the unit diameter disc B such that every three members of $${\mathcal {F}}$$ are intersected by some straight line, then there is a line intersecting every member of the family $${\mathcal {F}}'=\{x_1+\varphi B,x_2+\varphi B,x_3+ \varphi B,x_4+ \varphi B,x_5+ \varphi B\}$$ , where $$\varphi =\frac{1+\sqrt{5}}{2}$$ .

Raman scattering measurements of mixing and finite-rate chemistry in a supersonic reacting flow over a piloted, ramped cavity

UV Raman scattering is applied to measure fuel/air mixing and combustion of a Mach-2 air stream flowing over a step-ramp cavity fueled with 70% methane/30% hydrogen. Average and RMS fluctuations of temperature and major species profiles as well as scatter plots of simultaneous temperature and chemical scalars are determined from single-shot Raman scattering measurements along a 6-mm line transverse to the main supersonic flow. In the fuel-rich regions of the pilot cavity, the 248-nm KrF laser induces broadband fluorescence interference that reduces the number of analyzable Raman spectra; nonetheless, on the whole, a significant fraction of the spectra were reducible. In the cavity, hydrogen fuel reacts quickly resulting in a uniform water concentration in the recirculation zone. Methane reacts slowly to carbon monoxide/carbon dioxide in the cavity, leading to non-uniform concentrations of these species. Mean and instantaneous mixture fraction data inside the shear layer were indicative of oxygen transport across the shear layer. Temperature, water, and oxygen fluctuations are fairly constant throughout the combustor due to recirculation/turbulent transport across the shear layer and the slow reaction of methane.

The colored Hadwiger transversal theorem in ℝ d

Hadwiger's transversal theorem gives necessary and suffcient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful version appeared due to Arocha, Bracho and Montejano. We show that it is possible to combine both results to obtain a colored version of Hadwiger's theorem in higher dimensions. The proofs differ from the previous ones and use a variant of the Borsuk-Ulam theorem. To be precise, we prove the following. Let F be a family of convex sets in źd in bijection with a set P of points in źdź1. Assume that there is a coloring of F with suffciently many colors such that any colorful Radon partition of points in P corresponds to a colorful Radon partition of sets in F. Then some monochromatic subfamily of F has a hyperplane transversal.

Allowable Interval Sequences and Line Transversals in the Plane

Given a family F of n pairwise disjoint compact convex sets in the plane with non-empty interiors, let T(k) denote the property that every subfamily of F of size k has a line transversal, and T the property that the entire family has a line transversal. We illustrate the applicability of allowable interval sequences to problems involving line transversals in the plane by proving two new results and generalizing three old ones. Two of the old results are Klee’s assertion that if F is totally separated then T(3) implies T, and the following variation of Hadwiger’s Transversal Theorem proved by Wenger and (independently) Tverberg: If F is ordered and each four sets of F have some transversal which respects the order on F, then there is a transversal for all of F which respects this order. The third old result (a consequence of an observation made by Kramer) and the first of the new results (which partially settles a 2008 conjecture of Eckhoff) deal with fractional transversals and share the following general form: If F has property T(k) and meets certain other conditions, then there exists a transversal of some m sets in F, with k<m<n. The second new result establishes a link between transversal properties and separation properties of certain families of convex sets.

Lower bounds to helly numbers of line transversals to disjoint congruent balls

A line l is a transversal to a family F of convex objects in ℝ d if it intersects every member of F. In this paper we show that for every integer d ⩾ 3 there exists a family of 2d−1 pairwise disjoint unit balls in ℝ d with the property that every subfamily of size 2d − 2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957. Crucial to the proof is the notion of a pinned transversal, which means an isolated point in the space of transversals. Here we investigate minimal pinning configurations and construct a family F of 2d−1 disjoint unit balls in ℝ d with the following properties: (i) The space of transversals to F is a single point and (ii) the space of transversals to any proper subfamily of F is a connected set with non-empty interior.

Line transversals to blown up closed balls

The number λd(k) is defined as the minimum λ > 0 such that the following holds: For any finite family \({\mathcal {F}=\{B_1,B_2, \ldots , B_n\}}\) of closed balls in \({{\mathbb{R}}^d}\) such that every k elements of \({\mathcal {F}}\) have a common line transversal, the elements of the blown up family \({\lambda\mathcal {F}=\{\lambda B_1,\lambda B_2, \ldots , \lambda B_n\}}\) have a common line transversal. In this paper we show that \({\lambda_d(d+1)\leq4, \lambda_2(4)\leq 2\sqrt 2}\) and λ2(3) < 2.88.

Lines Pinning Lines

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show that any minimal pinning of a line by convex polytopes such that no face of a polytope is coplanar with the line has size at most eight. If, in addition, the polytopes are disjoint, then it has size at most six. We completely characterize configurations of disjoint polytopes that form minimal pinnings of a line.

Pinning a Line by Balls or Ovaloids in ℝ3

We show that if a line l is an isolated line transversal to a finite family $\mathcal{F}$ of (possibly intersecting) balls in ℝ3 and no two balls are externally tangent on l, then there is a subfamily $\mathcal{G}\subseteq\mathcal{F}$ of size at most 12 such that l is an isolated line transversal to $\mathcal{G}$. We generalize this result to families of semialgebraic ovaloids.

Line Transversals of Convex Polyhedra in $\mathbb{R}^3$

We establish a bound of $O(n^2k^{1+\varepsilon})$, for any $\varepsilon>0$, on the combinatorial complexity of the set $\mathcal{T}$ of line transversals of a collection $\mathcal{P}$ of k convex polyhedra in $\mathbb{R}^3$ with a total of n facets, and we present a randomized algorithm which computes the boundary of $\mathcal{T}$ in comparable expected time. Thus, when $k\ll n$, the new bounds on the complexity (and construction cost) of $\mathcal{T}$ improve upon the previously best known bounds, which are nearly cubic in n. To obtain the above result, we study the set $\mathcal{T}_{\ell_0}$ of line transversals which emanate from a fixed line $\ell_0$, establish an almost tight bound of $O(nk^{1+\varepsilon})$ on the complexity of $\mathcal{T}_{\ell_0}$, and provide a randomized algorithm which computes $\mathcal{T}_{\ell_0}$ in comparable expected time. Slightly improved combinatorial bounds for the complexity of $\mathcal{T}_{\ell_0}$ and comparable improvements in the cost of constructing this set are established for two special cases, both assuming that the polyhedra of $\mathcal{P}$ are pairwise disjoint: the case where $\ell_0$ is disjoint from the polyhedra of $\mathcal{P}$, and the case where the polyhedra of $\mathcal{P}$ are unbounded in a direction parallel to $\ell_0$. Our result is related to the problem of bounding the number of geometric permutations of a collection $\mathcal{C}$ of k pairwise-disjoint convex sets in $\mathbb{R}^3$, namely, the number of distinct orders in which the line transversals of $\mathcal{C}$ visit its members. We obtain a new partial result on this problem.

Lower Bounds for Pinning Lines by Balls (Extended Abstract)

Abstract It is known that if n ⩾ 2 d pairwise disjoint balls in R d have a unique line l intersecting them in a given order ≺, one can always remove a ball so that l remains the only line intersecting the balls in the order induced by ≺. We show that the constant 2d is best possible, in any dimension, and derive lower bounds on Helly numbers for sets of line transversals to disjoint balls in arbitrary dimension.

On the degree of standard geometric predicates for line transversals in 3D

In this paper we study various geometric predicates for determining the existence of and categorizing the configurations of lines in 3D that are transversal to lines or segments. We compute the degrees of standard procedures of evaluating these predicates. The degrees of some of these procedures are surprisingly high (up to 168), which may explain why computing line transversals with finite-precision floating-point arithmetic is prone to error. Our results suggest the need to explore alternatives to the standard methods of computing these quantities.

Line Transversals in Large T(3)- and T(4)-Families of Congruent Discs

A family of closed discs is said to have property T(k) if to every subset of at most k discs there belongs a common line transversal. A family of discs is said to be d-disjoint, d≥1, if the mutual distance between the centers of the discs is larger than d. It is known that a d-disjoint T(3)-family ℱ of unit diameter discs has a line transversal if $d=\sqrt{2}$. Similarly, a d-disjoint T(4)-family has a line transversal if $d=2/\sqrt{3}$. Both results are sharp in d, i.e., they do not hold for smaller values of d. The main result of this paper is that while the above lower bounds on d cannot be relaxed in general, some reduction of d can be compensated by imposing a proper d-dependent lower bound on the size of the family in both cases.

Common transversals in the plane: The fractional perspective

A fresh look is taken at the fractional Helly theorem for line transversals to families of convex sets in the plane. This theorem was first proved in 1980 by Katchalski and Liu [M. Katchalski, A. Liu, Symmetric twins and common transversals, Pacific J. Math. 86 (1980) 513–515]. It asserts that for every integer k ≥ 3 , there exists a real number ρ ( k ) ∈ ( 0 , 1 ) such that the following holds: If K is a family of n compact convex sets in the plane, and any k or fewer members of K have a line transversal, then some subfamily of K of size at least ρ ( k ) n has a line transversal. A lower bound on ρ ( k ) is obtained which is stronger than the one obtained in [M. Katchalski, A. Liu, Symmetric twins and common transversals, Pacific J. Math. 86 (1980) 513–515]. Also, examples are given to show that a conjecture of Katchalski concerning the value of ρ ( 3 ) , if true, is the best possible.

On the Algorithmic Aspects of Discrete and Lexicographic Helly-Type Theorems and the Discrete LP-Type Model

Helly's theorem says that, if every $d+1$ elements of a given finite set of convex objects in $\mathbb{R}^d$ have a common point, there is a point common to all of the objects in the set. In discrete Helly theorems the common point should belong to an a priori given set. In lexicographic Helly theorems the common point should not be lexicographically greater than a given point. Using discrete and lexicographic Helly theorems we get linear time solutions for various optimization problems. For this, we introduce the DLP-type (discrete linear programming-type) model, and provide new algorithms that solve in randomized linear time fixed-dimensional DLP-type problems. For variable-dimensional DLP-type problems, our algorithms run in time subexponential in the combinatorial dimension. Finally, we use our results in order to solve in randomized linear time problems such as the discrete $p$-center on the real line, the discrete weighted 1-center problem in $\mathbb{R}^d$ with either $l_1$ or $l_\infty$ norm, the standard (continuous) problem of finding a line transversal for a totally separable set of planar convex objects, a discrete version of the problem of finding a line transversal for a set of axis-parallel planar rectangles, and the (planar) lexicographic rectilinear $p$-center problem for $p=1,2,3$. These are the first known linear time algorithms for these problems. Moreover, we use our algorithms to solve in randomized subexponential time various problems in game theory, improving upon the best known algorithms for these problems.