Empty Triangles Research Articles

A sufficient condition for the existence of plane spanning trees on geometric graphs

Let P be a set of n⩾3 points in general position in the plane and let G be a geometric graph with vertex set P. If the number of empty triangles Δuvw in P for which the subgraph of G induced by {u,v,w} is not connected is at most n−3, then G contains a non-self intersecting spanning tree.

Monochromatic empty triangles in two-colored point sets

Improving a result of Aichholzer et al., we show that there exists a constant c>0 satisfying the following condition. Any two-colored set of n points in general position in the plane has at least cn4/3 triples of the same color such that the triangles spanned by them contain no element of the set in their interiors.

Erratum: Rapid and efficient protein digestion using trypsin‐coated magnetic nanoparticles under pressure cycles

Figure 2B in this paper is corrected as follows: (…) (B) Enzyme stabilities of CA-TR/NP (empty circles), EC-TR/NP (filled circles), and free trypsin (empty triangles) under recycled uses and shaking at 200 rpm at room temperature. The activity was measured by the hydrolysis of BAPNA at each time point, and the relative activity was calculated from the ratio of residual activity at each time point to the initial activity of each sample.

Empty monochromatic triangles

We consider a variation of a problem stated by Erdös and Guy in 1973 about the number of convex k-gons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As a main result we show that any bi-colored set of n points in R 2 in general position determines a super-linear number of empty monochromatic triangles, namely Ω ( n 5 / 4 ) .

Quality assurance for radiotherapy in prostate cancer: Point dose measurements in intensity modulated fields with large dose gradients

Purpose: We aimed to evaluate an optimization algorithm designed to find the most favorable points to position an ionization chamber (IC) for quality assurance dose measurements of patients treated for prostate cancer with intensity-modulated radiotherapy (IMRT) and fields up to 10 cm × 10 cm. Methods and Materials: Three cylindrical ICs (PTW, Freiburg, Germany) were used with volumes of 0.6 cc, 0.125 cc, and 0.015 cc. Dose measurements were made in a plastic phantom (PMMA) at 287 optimized points. An algorithm was designed to search for points with the lowest dose gradient. Measurements were made also at 39 nonoptimized points. Results were normalized to a reference homogeneous field introducing a dose ratio factor, which allowed us to compare measured vs. calculated values as percentile dose ratio factor deviations ΔF (%). A tolerance range of ΔF (%) of ±3% was considered. Results: Half of the ΔF (%) values obtained at nonoptimized points were outside the acceptable range. Values at optimized points were widely spread for the largest IC (i.e., 60% of the results outside the tolerance range), whereas for the two small-volume ICs, only 14.6% of the results were outside the tolerance interval. No differences were observed when comparing the two small ICs. Conclusions: The presented optimization algorithm is a useful tool to determine the best IC in-field position for optimal dose measurement conditions. A good agreement between calculated and measured doses can be obtained by positioning small volume chambers at carefully selected points in the field. Large chambers may be unreliable even in optimized points for IMRT fields ≤10 cm × 10 cm.

Tetanus Toxin Is Transported in a Novel Neuronal Compartment Characterized by a Specialized pH Regulation

Tetanus toxin binds specifically to motor neurons at the neuromuscular junction. There, it is internalized into vesicular carriers undergoing fast retrograde transport to the spinal cord. Despite the importance of this axonal transport pathway in health and disease, its molecular and biophysical characterization is presently lacking. We sought to fill this gap by determining the pH regulation of this compartment in living motor neurons using a chimera of the tetanus toxin binding fragment (TeNT HC) and a pH-sensitive variant of the green fluorescent protein (ratiometric pHluorin). We have demonstrated that moving retrograde carriers display a narrow range of neutral pH values, which is kept constant during transport. Stationary TeNT HC-positive organelles instead exhibit a wide spectrum of pH values, ranging from acidic to neutral. This distinct pH regulation is due to a differential targeting of the vacuolar (H+) ATPase, which is not present on moving TeNT HC compartments. Accordingly, inhibition of the vacuolar (H+) ATPase under conditions that completely abolish the intracellular accumulation of acidotrophic dyes does not affect axonal retrograde transport of TeNT HC. However, a functional vacuolar (H+) ATPase is required for early steps of TeNT HC trafficking following endocytosis, and it is localized to axonal vesicles containing TeNT HC. Altogether, these findings indicate that the vacuolar (H+ ATPase plays a specific role in early sorting events directing TeNT HC to axonal carriers but not in their subsequent progression along the retrograde transport route, which escapes acidification and targeting to degradative organelles.

Planar point sets with a small number of empty convex polygons

A subsetAof a finite setPof points in the plane is called anempty polygon, if each point ofAis a vertex of the convex hull ofAand the convex hull ofAcontains no other points ofP. We construct a set ofnpoints in general position in the plane with only ˜1.62n2empty triangles, ˜1.94n2empty quadrilaterals, ˜1.02n2empty pentagons, and ˜0.2n2empty hexagons.

Planar sets with few empty convex polygons

A configuration of n points in general position in the plane is described which has less than 1.684n 2 empty triangles, less than 2.132n 2 empty convex quadrilaterals, less than 1.229n 2 empty convex pentagons and less than 0.298n 2 empty convex hexagons. This improves the constants from a previous construction given by Valtr.

Spatial delaunay gibbs point processes

This paper studies two types of Delaunay Gibbs point processes originally introduced by Baddeley and Møller (1989) by combining stochastic geometry and computational geometry arguments. The energy function of these processes is for a given point pattern φ given by a sum of potentials associated to a subclass of the cliques, called “Delaunay-cliques”, given by the Delaunay graph correponding to the point pattern (empty set, singletons, Delaunay edges, and Delaunay triangles). This restriction is necessary in order that the Delaunay point process becomes Markov in the sense of Baddeley and Møller (1989). We demonstrate that the Markov property is satisfied when interactions are only permitted for “Delaunay–cliques” and to further establish a Hammersley-Clifford type theorem for the Delaunay Gibbs point processes. Furthermore local stability is studied and simulations are given

Empty triangles in drawings of the complete graph

If t( n) denotes the minimum number of empty triangles in all drawings of the complete graph of order n then t( n) ⩽ 2 n − 4 is proved and t(n) ⩾ 2n 3 is conjectured.

On Weiss' geometric characterization of the Rudvalis simple group

We give a computer-free proof for Weiss' theorem of the uniqueness of the Rudvalis simple group acting flag transitively on an extended generalized octagon with empty triangles.

Finding minimum area simple pentagons

Given a set P of n points in the plane, we want to find a simple, not necessarily convex, pentagon Q with vertices in P of minimum area. We present an algorithm for solving this problem in time O( nT( n)) and space O( n), where T( n) is the number of empty triangles in the set.

Searching for empty convex polygons

A key problem in computational geometry is the identification of subsets of a point set having particular properties. We study this problem for the properties of convexity and emptiness. We show that finding empty triangles is related to the problem of determining pairs of vertices that see each other in a star-shaped polygon. A linear-time algorithm for this problem which is of independent interest yields an optimal algorithm for finding all empty triangles. This result is then extended to an algorithm for finding empty convex r-gons (r> 3) and for determining a largest empty convex subset. Finally, extensions to higher dimensions are mentioned.

On empty triangles determined by points in the plane

On empty triangles determined by points in the plane