Cheng Equation: A Revisit Through Symmetry Analysis

The symmetry analysis of the Cheng Equation is performed. The Cheng Equation is reduced to a first-order equation of either Abel's Equations, the analytic solution of which is given in terms of special functions. Moreover, for a particular symmetry the system is reduced to the Riccati Equation or to the linear nonhomogeneous equation of Euler type. Henceforth, the general solution of the Cheng Equation with the use of the Lie theory is discussed, as also the application of Lie symmetries in a generalized Cheng equation.Comment: 10 pages. Accepted for publication in Quaestiones Mathematicae journa

Effect of cation ratio on microstructure and optical absorbance of magnesium aluminate spinel

This work is pertaining to the synthesis of fine magnesium aluminate spinel (MgAl2O4) powders of varied trivalent:bivalent cation ratio along line of homogeneity of the solid solution (MgO. xAl(2)O(3), x = 1, 1.25, 1.50, 1.75, and 2) via gel combustion method. Magnesium- and aluminum- nitrate were used as the oxidants in combustion reaction fuelled by urea in combination with stoichiometric formaldehyde solution acting as reductant. Synthesized powders were characterized in terms of microscopic analysis and optical absorbance measurements. The cation ratio, through a change in gel structure influences the nature of crystallization of the product, while on the other hand does not affect grain shapes and sizes. Distinct enhancement in both absorption intensity and the corresponding estimated energy band gap has been observed against increasing excess than stiochiometric alumina concentrations. Evaluated optical band gaps were widened in proportion to the Al: Mg ratio which may be attributed to Burstein-Moss effect in consequence of substitutional insertion of introduced Al3+ ions in spinel lattices

Gastric Follicular Dendritic Cell Sarcoma: A Case Report of a Rare Entity

Follicular Dendritic cell sarcoma arises from the follicular dendritic cells present in the lymphnode.Though are commonly seen in head and neck area but are extremely rare in the abdomen. Less than eighty cases are reported in the indexed literature. We herein describe a case of follicular dendritic cell sarcoma arising from the stomach wall with infiltration into pancreas in an 85 year old patient

Detector imperfections in photon-pair source characterization

We analyze how imperfections in single-photon detectors impact the characterization of photon-pair sources. We perform exact calculations to reveal the effects of multi-pair emissions and of noisy, non-unit efficiency, non photon-number resolving detections on the Cauchy-Schwarz parameter, on the second order auto-correlation and cross-correlation functions, and on the visibilities of both Hong-Ou-Mandel and Bell-like interferences. We consider sources producing either two-mode squeezed states or states with a Poissonian photon distribution. The proposed formulas are useful in practice to determine the impacts of multi-pair emissions and dark counts in standard tests used in quantum optics.Comment: 9 pages, 11 figure

Insights into the structure and Ion transport of pectin-[BMIM][PF6] electrolytes

We investigate the effect of pectin on the structure and ion transport properties of the room-temperature ionic liquid electrolyte 1-n-butyl-3-methylimidazolium hexafluorophosphate ([BMIM][PF6]) using molecular dynamics simulations. We find that pectin induces intriguing structural changes in the electrolyte that disrupt large ionic aggregates and promote the formation of smaller ionic clusters, which is a promising finding for ionic conductivity. Due to pectin in [BMIM][PF6] electrolytes, the diffusion coefficient of cations and anions is observed to decrease by a factor of four for a loading of 25 wt. % of pectin in [BMIM][PF6] electrolyte. A strong correlation between the ionic diffusivities (D) and ion-pair relaxation timescales (τc) is observed such that D ∼ τc−0.75 for cations and D ∼ τc−0.82 for anions. The relaxation timescale exponents indicate that the ion transport mechanisms in pectin-[BMIM][PF6] electrolytes are slightly distinct from those found in neat [BMIM][PF6] electrolytes (⁠⁠). Since pectin marginally affects ionic diffusivities at the gain of smaller ionic aggregates and viscosity, our results suggest that pectin-ionic liquid electrolytes offer improved properties for battery applications, including ionic conductivity, mechanical stability, and biodegradability