An entry-wise combinatorial formula for the inverse of a modified Laplacian matrix of a connected signed graph G is provided. The approach employs the Moore–Penrose inverse of the full-rank matrix associated with G. Several properties of the entries of this inverse matrix are also derived, some of which reflect structural properties of the signed graph, including whether it is balanced.
Effervescent tablets enhance the delivery and acceptability of herbal formulations by improving stability, palatability, and bioavailability. In this study, Aegle marmelos (A. marmelos) dried leaf powder was formulated into effervescent tablets with ascorbic acid, zinc, and conventional excipients, and evaluated for pharmaceutical and therapeutic potential. Pre- and post-compression studies confirmed favorable micromeritic properties (Hausner ratio 1.14 ± 0.02; Carr's index 12.79 ± 1.43%) and compliance with pharmacopoeial specifications, including hardness (4.87 ± 0.09 kg/cm2), friability (<1%), pH (6.52 ± 0.05), moisture content (8.27 ± 0.63%), and rapid disintegration (195.3 ± 5.5 s). Network pharmacology identified key cancer-related targets (TNF, AKT1, EGFR, CASP3, MAPK3, HIF1A) linked to apoptosis and PI3K/AKT-MAPK pathways. Experimental prediction using the MTT assay in MCF-7 breast cancer cells revealed dose-dependent cytotoxicity with an IC50 of ∼200 μg/mL. The dual contribution of ascorbic acid and zinc provided synergistic antioxidant, immunomodulatory, and stability benefits. Taken together, these results suggest that A. marmelos effervescent tablets are feasible to formulate and may hold biological relevance, though further in vivo and mechanistic work is needed to confirm any functional or therapeutic benefit.
In order to model beam propagation and instability processes in multimode optical fibers with parabolic index profiles, this study explores the (2+1)-dimensional nonlinear Schrödinger equation with Kerr nonlinearity. Controlling nonlinear wave propagation and enhancing optical communication systems require an understanding of such dynamics. The Sardar-subequation method, Kumar-Malik approach, and modified generalized exponential rational function method are used to get analytical solutions of the model in order to solve this issue. Numerous nonlinear wave structures, such as periodic, bright, dark, singular, and Jacobi elliptic function solutions, are thus produced. To investigate the stability and creation of these structures, modulation instability is also examined. Two-dimensional, three-dimensional, and contour plots are used to show the physical behavior and interactions of chosen solutions. The results offer a greater understanding of the development and evolution of nonlinear waves in higher-dimensional optical medium and show rich dynamical aspects of the system. In contrast to other research, this work is innovative in that it simultaneously constructs many soliton profiles utilizing a variety of analytical methods and modulation instability analysis, providing a more thorough and cohesive framework.
Two recent modifications of ranked set sampling (RSS) pull in opposite directions: the truncation-based scheme (TBRSS) concentrates measurements on the extreme ranks, while the except-extreme scheme (EERSS) never measures them at all. This paper puts the two together. Starting from sets of size m+2, the true minimum and maximum are discarded, and truncation, governed by a coefficient k, then operates inside the remaining pool of ranks {2,…,m+1}; we call the result truncation-based except-extreme ranked set sampling (TB-EERSS). EERSS is the case k=0 and an except-extreme analogue of extreme RSS the case k=⌊m/2⌋. For symmetric parents the TB-EERSS mean estimator is unbiased and never less efficient than simple random sampling (SRS) at the same number of measured units, and its variance is given in closed form. Exact efficiencies are obtained for the uniform, normal and exponential distributions, and a Monte Carlo study covers the logistic, Laplace, Weibull and gamma cases. The construction matters most for heavy right tails: because the true extremes are never measured, TB-EERSS does not inherit the breakdown that TBRSS suffers under the exponential, and its efficiency stays above one for every k and peaks near k≈m/3. At the small set sizes m=3,4,5,6 it beats RSS in every case and TBRSS in nearly everyone, by a wide margin under skewness. Against EERSS the coefficient k acts as a tuning knob, larger for skewed parents and small for light symmetric ones. The same picture holds for the median. Imperfect ranking erodes the gains slowly: they fall as the correlation ρ with the ranking variable drops but stay above SRS throughout ρ∈[0.5,0.9], and TB-EERSS keeps its edge over RSS and TBRSS apart from the short-tailed uniform. Two real datasets confirm the findings.
The structural degradation induced by energetic α-particle irradiation remains a major challenge for direct-conversion nuclear batteries employing α-emitting radioisotopes also known as alphavoltaic battery. For this type of battery, semiconductors play a significant role in generating electricity; therefore, materials with high radiation tolerance, such as ZnO, are considered promising. In this work, the structural response of undoped ZnO and Al-doped ZnO (AZO) containing 5 wt% and 10 wt% Al was investigated following irradiation with Am-241 α-particle source. The Stopping and Range of Ions in Matter (SRIM) simulations were employed to evaluate α-particle transport, linear energy transfer (LET), and absorbed dose, while structural modifications were characterized using X-ray diffraction (XRD), Rietveld refinement, and scanning electron microscopy (SEM). The simulations indicate that Al incorporation slightly increases the α-particle stopping power, penetration range, and absorbed dose, resulting in enhanced energy deposition within the films. Despite the increased energy deposition, XRD and Rietveld analyses revealed that increasing Al concentration progressively reduced the crystallinity, crystallite size, and lattice quality while increasing the lattice microstrain. Following α-particle irradiation, all samples retained the hexagonal wurtzite structure, although a reduction in diffraction intensity and a slight increase in microstrain were observed. These results demonstrate that Al doping modifies the structural response of ZnO to α-particle irradiation by enhancing local lattice distortion while preserving the overall crystal structure, providing valuable insights into the design of radiation-tolerant oxide semiconductors for nuclear battery applications.
