SummaryWe provide a visual proof, using successive partitions of a triangle, that the sum of the alternating geometric series with common ratio −13 is 34.
SummaryA rational triangle is a triangle where all side lengths are positive rational numbers. We prove an equivalence between finding rational triangles with a 120° angle and finding pairs of ...incongruent isosceles rational triangles with both a common perimeter and a common rational area.
SummaryWe extend a result from triangles to quadrilaterals and prove without words that circles externally tangent to a side and tangent to the extended adjacent sides can be used to bisect the ...perimeter of the quadrilateral.
The application of finding the nearest public facility using 2 methods to measure the distance between 2 points, i.e. the Euclidean Method and the Haversine Formula. Euclidean is a heuristic function ...obtained based on direct distance without obstacles such as to get the value of the length of a diagonal line on a triangle. Whereas Haversine is an equation that looks for the distance of an arc between two points on longitude and latitude. The results of the calculation of the average distance Euclidean deviations with an average value of data 2.539764, and Haversine 2.536912. This shows that the comparison of the measurements of the distance between Euclidean and Haversine has a difference of 0.002852 or the percentage of the distance between the two methods is 99.89 percent. Of the two methods, which yield values almost by measurements on Google maps is Haversine. For Euclidean, it is used to measure the distance between two points on a flat plane so that the results have differences when compared to the Haversine formula.
We describe and explain the fluxionality of B(13)(+). The chemical bonding analysis shows that the inner triangle of B(13)(+) is bound to the peripheral ring by delocalized bonds only, allowing a ...quasi-free rotation of the inner ring.
A new beryllium-free zincoborate, BaZn3(BO3)2F2, with a KBBF-type structure has been synthesized for the first time. The electrostatic force of interaction in BaZn3(BO3)2F2 provides better linkage in ...neighboring ZnBO3F∞ single layers. BaZn3(BO3)2F2 is the first case of borates with both ZnO3F tetrahedra and ZnO6 octahedra, enriching the structural chemistry of borate system. All the coplanar BO3 triangles align in the same direction with a high density, which endows BaZn3(BO3)2F2 with a large birefringence of cal. 0.076 at 1064 nm. This work is of great significance to design beryllium-free borates with a KBBF-type structure.
We study the problem of finding a maximum-cardinality set of r-cliques in an undirected graph of fixed maximum degree Δ, subject to the cliques in that set being either vertex disjoint or edge ...disjoint. It is known for r=3 that the vertex-disjoint (edge-disjoint) problem is solvable in linear time if Δ=3 (Δ=4) but APX-hard if Δ≥4 (Δ≥5).
We generalise these results to an arbitrary but fixed r≥3, and provide a complete complexity classification for both the vertex- and edge-disjoint variants in graphs of maximum degree Δ.
Specifically, we show that the vertex-disjoint problem is solvable in linear time if Δ<3r/2−1, solvable in polynomial time if Δ<5r/3−1, and APX-hard if Δ≥⌈5r/3⌉−1. We also show that if r≥6 then the above implications also hold for the edge-disjoint problem. If r≤5, then the edge-disjoint problem is solvable in linear time if Δ<3r/2−1, solvable in polynomial time if Δ≤2r−2, and APX-hard if Δ>2r−2.
We point out that the recently increased value of the angle Formula omitted in the unitarity triangle (UT), determined in tree-level decays to be Formula omitted by the LHCb collaboration, combined ...with the most recent value of Formula omitted implies an enhancement of Formula omitted over the data in the ballpark of Formula omitted. This is larger by roughly a factor of two than the enhancement of Formula omitted that is independent of Formula omitted. This disparity of enhancements is problematic for models with constrained minimal flavour violation (CMFV) and also for Formula omitted models. In view of the prospects of measuring Formula omitted with the precision of Formula omitted by Belle II and LHCb in the coming years, we propose to use the angles Formula omitted and Formula omitted together with Formula omitted and Formula omitted as the fundamental parameters of the CKM matrix until Formula omitted from tree-level decays will be known precisely. Displaying Formula omitted as functions of Formula omitted clearly demonstrates the tension between the value of Formula omitted from tree-level decays, free from new physics (NP) contributions, and Formula omitted calculated in CMFV and Formula omitted models and thus exhibits the presence of NP contributions to Formula omitted beyond these frameworks. We calculate the values of Formula omitted and Formula omitted as functions of Formula omitted and Formula omitted and discuss the implications of our results for Formula omitted and rare K and B decays. We also briefly discuss a future strategy in which Formula omitted, possibly affected by NP, is replaced by Formula omitted.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
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![Emerging [Formula omitted]-...](http://api.summon.serialssolutions.com/2.0.0/image/issn/XR4PS5YY4K/1434-6044/small "Emerging [Formula omitted]-anomaly from tree-level determinations of [Formula omitted] and the angle [Formula omitted]")