A Very Special Congruency Problem

The following question appears in Chapter 7 of the Class VII NCERT Mathematics textbook (page 150): Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles are not congruent. Here, ‘parts’ refers to the three sides and three angles of the triangle; ‘congruent parts’ means that the corresponding parts of the two triangles are identically equal to each other. The questions that quickly come to mind are: “Why a rough sketch? Why not the exact figure?” “Is it possible to draw such pairs of triangles exactly?” We will find the answers to these questions in this article and prove some related results

Nanoengineered condenser surfaces for enhancing transport in thermal desalination by air gap membrane distillation

Thermal desalination is a technique that uses heat or thermal energy to desalinate water, unlike reverse osmosis. Membrane distillation (MD) is a type of thermal desalination technology having various configurations. Air gap membrane distillation (AGMD) is one of the more energy efficient MD configurations, being especially advantageous over other configurations at high salinity. However, the large mass transfer resistance of the air gap dramatically reduces the permeate flux, impairing performance. Higher condensation performance can be achieved by using a smaller air gap size, but typical film-wise condensation flow patterns flood the air gap at the optimal gap size (\u3c1 mm). Experiments show that dropwise and jumping-droplet condensation regimes, achieved using hydrophobic and superhydrophobic condensing surfaces respectively, can improve droplet shedding, allowing for thinner gap sizes. A systemlevel numerical model is used to demonstrate that these surfaces could thereby enable improved energy efficiency (2.1× increase of gained output ratio) while avoiding flooding at gap sizes as small as 0.2 mm. Superhydrophobic surfaces with directional jumping of droplets, specifically in the direction of gravity, are also tested and compared to droplets that jump normal to the condensing surface. Novel condensing surfaces that include a combination of the superhydrophobic and superhydrophilic patterns create flow regimes having pathways for faster permeate removal. Other condensing surfaces, including SLIPS (slippery liquidinfused porous surfaces) and laser-ablated superhydrophobic patterned surfaces are tested to the check the extent to which they improve the permeate removal rate while exhibiting different condensation regimes that merit further exploration

Measurement of higher cumulants of net-charge multiplicity distributions in Au++Au collisions at sNN=7.7−200\sqrt{s_{_{NN}}}=7.7-200 GeV

We report the measurement of cumulants ($C_n, n=1\ldots4$) of the net-charge distributions measured within pseudorapidity ($|\eta|<0.35$) in Au$+$Au collisions at $\sqrt{s_{_{NN}}}=7.7-200$ GeV with the PHENIX experiment at the Relativistic Heavy Ion Collider. The ratios of cumulants (e.g. $C_1/C_2$, $C_3/C_1$) of the net-charge distributions, which can be related to volume independent susceptibility ratios, are studied as a function of centrality and energy. These quantities are important to understand the quantum-chromodynamics phase diagram and possible existence of a critical end point. The measured values are very well described by expectation from negative binomial distributions. We do not observe any nonmonotonic behavior in the ratios of the cumulants as a function of collision energy. The measured values of $C_1/C_2 = \mu/\sigma^2$ and $C_3/C_1 = S\sigma^3/\mu$ can be directly compared to lattice quantum-chromodynamics calculations and thus allow extraction of both the chemical freeze-out temperature and the baryon chemical potential at each center-of-mass energy.Comment: 512 authors, 8 pages, 4 figures, 1 table. v2 is version accepted for publication in Phys. Rev. C as a Rapid Communication. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm

Nuclear matter effects on J/ψJ/\psi production in asymmetric Cu+Au collisions at sNN\sqrt{s_{_{NN}}} = 200 GeV

We report on $J/\psi$ production from asymmetric Cu+Au heavy-ion collisions at $\sqrt{s_{_{NN}}}$=200 GeV at the Relativistic Heavy Ion Collider at both forward (Cu-going direction) and backward (Au-going direction) rapidities. The nuclear modification of $J/\psi$ yields in Cu$+$Au collisions in the Au-going direction is found to be comparable to that in Au$+$Au collisions when plotted as a function of the number of participating nucleons. In the Cu-going direction, $J/\psi$ production shows a stronger suppression. This difference is comparable in magnitude and has the same sign as the difference expected from shadowing effects due to stronger low-$x$ gluon suppression in the larger Au nucleus. The relative suppression is opposite to that expected from hot nuclear matter dissociation, since a higher energy density is expected in the Au-going direction.Comment: 349 authors, 10 pages, 4 figures, and 4 tables. Submitted to Phys. Rev. C. For v2, fixed LaTeX error in 3rd-to-last sentence. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm

Cross Section and Transverse Single-Spin Asymmetry of η\eta Mesons in p↑+pp^{\uparrow}+p Collisions at s=200\sqrt{s}=200 GeV at Forward Rapidity

We present a measurement of the cross section and transverse single-spin asymmetry ($A_N$) for $\eta$ mesons at large pseudorapidity from $\sqrt{s}=200$~GeV $p^{\uparrow}+p$ collisions. The measured cross section for $0.5<p_T<5.0$~GeV/$c$ and $3.0<|\eta|<3.8$ is well described by a next-to-leading-order perturbative-quantum-chromodynamics calculation. The asymmetries $A_N$ have been measured as a function of Feynman-$x$ ($x_F$) from $0.2<|x_{F}|<0.7$, as well as transverse momentum ($p_T$) from $1.0<p_T<4.5$~GeV/$c$. The asymmetry averaged over positive $x_F$ is $\langle{A_{N}}\rangle=0.061{\pm}0.014$. The results are consistent with prior transverse single-spin measurements of forward $\eta$ and $\pi^{0}$ mesons at various energies in overlapping $x_F$ ranges. Comparison of different particle species can help to determine the origin of the large observed asymmetries in $p^{\uparrow}+p$ collisions.Comment: 484 authors, 13 pages, 11 figures, 4 tables, 2008 data. v2 is version accepted by Phys. Rev. D. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be)publicly available at http://www.phenix.bnl.gov/papers.htm