In the Euclidean plane, the Erdős-Mordell inequality indicates that the sum of distances of an interior point of a triangle T to its vertices is larger than or equal to twice the sum of distances to the sides of T. We extend this theorem to arbitrary (normed or) Minkowski planes, and we generalize in an analogous way some other related inequalities, e.g. referring to polygons. We also derive Minkowskian analogues of Erdős-Mordell inequalities for tetrahedra and n-dimensional simplices. Finally, some related inequalities are obtained which additionally involve total edge-lengths of simplices.
Thermal properties of Co2FeV3O11
have been reinvestigated. It has been proved that this compound does not exhibit
polymorphism. It melts incongruently at the temperature of 7705C
and the phase with lyonsite type structure is the solid product of this melting.
Phase relations in the whole subsolidus area of the CoO–V2O5–Fe2O3 system have been determined. The solidus area projection onto
the component concentration triangle plane of this system has been constructed
using the DTA and XRD methods. 15 subsidiary subsystems can be distinguished
in this system.
Authors:Nazli Yazici Gözütok and Bahadir Özgür Güler
In this study, we investigate suborbital graphs Gu,n of the normalizer ΓB (N) of Γ0 (N) in PSL(2, ℝ) for N = 2α3β where α = 1, 3, 5, 7, and β = 0 or 2. In these cases the normalizer becomes a triangle group and graphs arising from the action of the normalizer contain quadrilateral circuits. In order to obtain graphs, we first define an imprimitive action of ΓB (N) on using the group (N) and then obtain some properties of the graphs arising from this action.
Most accidents in roundabouts occur at the entries, partly due to poor visibility. To analyse this problem a sample roundabout with some visibility obstacles was measured by laser scanner and data were put into a 3D model. The fields of view were studied from the viewpoint heights of car and truck drivers as well. To assess the results, three indicators were defined: the hidden sight distance, the hidden sight triangle area and the angle of hide. These indicators were measured in the point cloud model. Through these indicators the adequacy of visibility conditions can be assessed.
Phenylacetone was chromatographed on alumina with mixture of toluene and 1,4-dioxane as a mobile phase in a thin-layer chromatography (TLC) system. All the chromatograms were scanned with a densitometer. It was found that for such a binary mobile phase, there is a range of stronger component concentration that gives both peaks of Gaussian shape and triangle-like shape as well. Non-Gaussian peaks of phenylacetone chromatographed on alumina with a mixture of toluene and 1,4-dioxane were observed in the case of relatively low concentration of the latter in the mobile phase. Appropriate model describing the observed retention process is suggested.
Műemlékes építész számára gyakran talány a műemlékben meglelni a régi alkotók méretezési vagy arányosítási módszereit. Számos eljárás ismeretes a középkori, kivált a reneszánsz művészetből, építészetből, például az aranymetszés vagy a Fibonacci-számsor. Ezek ókori művészek és papok általi használatáról nincs konkrét bizonyíték. Számos feltételezés viszont létezik, mint „szent geometria”.Elképzelésünk alapja: az egyiptomiak által használt szeked (seked) méretezés létezett rézsűk, rámpák, gátak tervezésére. Egy seked 2 háromszög hasonló az aranymetszés szerkesztő háromszögéhez. És feltételezzük, hogy körzővel ki tudták jelölni a P pontot a háromszög átfogóján. A P ponttal pedig háromféle arányosság szerkeszthető: a φ (fi) 0,618 arányszám, amit ismerünk a reneszánsz óta, s a leleményünk szerinti χ (khi) 0,523, valamint a ψ (pszi) 0,381. Mindhárom arányszám megtalálható az óegyiptomi monumentális, kultikus építészetben mint átló, tengely, kétoldali átló az alaprajzokban, és a hosszúság-magasság arány a metszetekben és homlokzatokon.A cikk végén Taposiris Magna sajátos rekonstrukciói szerepelnek.
In this paper it will be constructed an abstract geometry will be called a triple space, which is defined in general sense
by the closure theoretic definition of geometry “see ”. And it is proved that the category of triple spaces is isomorphic
to the category of Steiner triple systems. And hence it could be shown that the class of Steiner triple systems which satisfy
the geometric axiomI3,
Bialostocki, Dierker, and Voxman proved that for any n = p +2, there is an integer B(n; p) with the following property. Every set of B(n; p) points in general position in the plane has n points in convex position such that the number of points in the interior of their convex hull is 0 mod p. They conjectured that the same is true for all pairs n = 3, p =2. In this note, we show that every su&ciently large point set determining no triangle with more than one point in its interior has n elements that form the vertex set of an empty convex n-gon. As a consequence, we show that the above conjecture is true for all n =5p=6+O(1).
This paper describes a novel application of mobile robot technology to the construction of mobility for the frail blind. Trajectory control of an experimental mobile robot using proposed neural network based controller structure for frail blind persons are presented. The robot is a non-holonomic experimental mobile robot with all electronic equipments. A prescribed double triangle reference trajectory is employed to test the robot with proposed controller structure for following exact trajectory of blind disable persons. The gain parameters of the P controller are tuned with neural network. The results of the proposed adaptive neural network based P controller are improved that this controller has superior performance at adapting large disturbances on the mobile robot.
Authors:Eric Mielke, Dominique M. Roberge, and Arturo Macchi
Based on previous work studying complex microreactors, it was desired to further improve the mixing efficiency by varying the mixing unit design for fast liquid—liquid reactions. Different flow regimes were studied, including slug flow, parallel flow, and drop flow. The two-phase hydrolysis of 4-nitrophenyl acetate in sodium hydroxide solution was used to evaluate the overall volumetric mass transfer coefficients (Korga) as a function of the average rate of energy dissipation (ε) for each microreactor design and all flow regimes. The liquid—liquid systems investigated used n-butanol or toluene as the organic phase solvent and a 0.5-M NaOH aqueous solution. The use of surfactant was also investigated with the toluene—water system. All microreactor geometry designs were based on contraction—expansion repeating units with asymmetric obstacles to aid the breakup of slugs and desynchronize the recombination of split streams. The investigated designs were chosen to avoid the formation of the parallel flow regime, contrary to curvature-based mixing-unit designs. The microreactor design can then be optimized to reduce the ε required to reach drop flow, since Korga has been found to be constant at equal ε for a given solvent system in this flow regime, regardless of the reactor selection. Additionally, the “3/7th” scaleup rule was applied and confirmed with the LL-Triangle mixer. It was found that, for low interfacial-tension systems (i.e., n-butanol—water), the onset of drop flow occurred at a lower ε for the LL-Triangle mixer when compared with the Sickle or LL-Rhombus mixers.