National Defence Academy (India)

In 1971, Reich (Some remarks concerning contraction mappings. Can Math Bull 14:121–124, 1971) unified the Banach contraction principle and the Kannan contraction theorem by introducing a new class of contractions, now known as Reich contractions, which ensure the existence and uniqueness of a fixed point in a metric space. His condition permits discontinuity in the domain of definition but requires the self-mapping to be continuous at the fixed point. In this paper, we extend Reich’s fixed point theorem to include both contractive and non-expansive mappings in metric spaces. This extension admits the possibility of multiple fixed points, even when a fixed point is itself a point of discontinuity. The resulting structured fixed-point sets and domains exhibit rich algebraic, geometric, and dynamical structures. Our theorem thus provides a broad generalization of several well-known results on contractive mappings. As a byproduct, we present a new solution–distinct from the known one–to the problem posed by Rhoades (Contractive definitions and continuity. Contemp Math 72:233–245, 1988) concerning the existence of contractive mappings that admit a fixed point which is also a point of discontinuity.

Background Semiconductor-based heterojunctions in the realm of photocatalysis has attracted significant consideration among scientific community as a promising strategy to advance environmentally resilient development for long-term generational benefit. This approach has been acknowledged as an effective solution to ameliorate the ongoing environmental crisis. Samarium vanadate (SmVO4), a fascinating rare earth metal orthovanadate, has been the focal point in the solar-driven applications due to optimal band positions and a narrow band gap, thereby broadening the absorption capacity within the visible light region. It also acts as a highly effective photocatalyst for many reactions, such as the dehydrogenation of alkanes, olefins, etc. Methods To address the broad spectrum of photocatalytic application demands, the bare photocatalyst is ineffective owing to its limitations, such as high recombination rate, agglomeration, low surface area, etc. Therefore, various strategies were explored and implemented by the researchers to mitigate the above-mentioned shortcomings and to improve the overall photocatalytic performance. To fill this gap, the review systematically elucidates the crystallographic, electronic and optical properties, followed by synthesis methods. Thereafter, modification strategy i.e., heterojunction formation via integrating bare SmVO4 with other semiconductors such as metal oxide, metal-free, metal halide, metal organic frameworks, insulating material and more was analysed. Significant findings This approach increases the surface area, chemical stability, facilitates the charge-segregation capability, offers more active sites and exhibits high quantum efficiency. The various photocatalytic applications, including antibiotic, dye, and heavy metal degradation via SmVO4-based heterojunctions were examined, aligning with the United Nations Sustainable Development Goals-6 i.e., clean water and sanitation. Finally, the article concluded with key issues and future directions related to SmVO4-based heterojunctions for sustainable living.

The merger of binary neutron stars (BNSs) is a remarkable astrophysical event where all four fundamental forces interplay dynamically across multiple stages, producing a rich spectrum of multi-messenger signals. These observations present a significant multiphysics modeling challenge but also offer a unique opportunity to probe the nature of gravity and the strong nuclear interaction under extreme conditions. The landmark detection of GW170817 provided essential constraints on the properties of non-rotating neutron stars (NSs), including their maximum mass (Mmax) and radius distribution, thereby informing the equation of state (EOS) of cold, dense nuclear matter. While the inspiral phase of such events has been extensively studied, the post-merger signal holds even greater potential to reveal the behavior of matter at supranuclear densities, particularly in scenarios involving a transition to deconfined quark matter. Motivated by the recent gravitational wave event GW190814 (2.5-2.67M circle dot), we revisit the modeling of high-mass compact stars to investigate their internal structure via a generalized polytropic EOS. This framework incorporates a modified energy density profile and is coupled with the Tolman-Oppenheimer-Volkoff (TOV) equations. We explore mass-radius (M-R) relationships within both general relativity (GR) and the minimal geometric deformation (MGD) approach. Specifically, we constrain the radii of four massive compact objects - PSR J1614-2230 (1.97 0.04-0.04 M circle dot), PSR J0952-0607 (2.35 0.17-0.17 M circle dot), GW190814 (2.5-2.67M circle dot), and GW200210 (2.83 0.47-0.42 M circle dot) -and demonstrate that our theoretical M-R curves are consistent with observational data. These findings provide meaningful constraints on the EOS and underscore the potential of alternative gravity models to accommodate ultra-massive compact stars within a physically consistent framework.

Ion exchange and exfoliation are widely used chemical processes for layered perovskites; however, their impact on vacuum-level-referenced electronic band alignment remains unclear. In this study, we investigate the crystal and electronic band structures of the Dion–Jacobson-type layered perovskite KCa2Nb3O10, its protonated form HCa2Nb3O10, and exfoliated Ca2Nb3O10 nanosheets. Synchrotron x-ray powder diffraction reveals that KCa2Nb3O10 crystallizes in the monoclinic space group P21/m, whereas HCa2Nb3O10 adopts a tetragonal structure with space group P42212, indicating enhanced structural symmetry upon protonation. Energy diagrams referenced to the vacuum level are experimentally determined for all three compounds, enabling direct comparison of valence-band positions and in-gap states induced by ion exchange and exfoliation. In HCa2Nb3O10, a hydrogen-bonded network is found to play a key role in controlling the formation and energy level of in-gap states. These results demonstrate that chemical processing provides an effective route for vacuum-level-referenced band tuning in layered perovskites, offering a practical guideline for band engineering in photocatalytic and related functional oxide materials.

In this study, the structural, electronic, magnetic, optical, and thermoelectric properties of FeHfTiX (X = Si, Ge, Sn) Heusler alloys were investigated using first-principles density functional theory (DFT) calculations. The results reveal that both compounds possess ferromagnetic ground states and are thermodynamically stable in their cubic phase. With a band gap of approximately 1.06 eV and 1.07 eV in the spin methods, we found that both behave in a half-metallic manner utilizing the Generalized Gradient Approximate (GGA) method. The half-metallic nature of these alloys encourages high spin polarization. According to the Slater-Pauling criterion, such alloys have an overall magnetic moment of 2.0 μB as well as a magnetic moment at half metallic properties. Optical properties were analysed to understand light–matter interaction and electronic transitions. Furthermore, the Seebeck coefficient and thermoelectric figure of merit (ZT) were evaluated using semi-classical Boltzmann transport theory, showing promising thermoelectric performance. The calculated Seebeck coefficients reach 121.8 μV/K, 100.6 μV/K, and 110.5 μV/K at 600 K, 750 K, and 900 K for FeHfTiX (X = Si, Ge). The corresponding ZT values of 0.9, 0.59, and 0.85 underscore their suitability for spintronic and thermoelectric applications. We calculate the Curie temperature of the FeHfTiX (X = Si, Ge) alloys, considering the exchange interactions. The Curie temperatures of both Heusler alloys are above 900 K, which is significantly higher than room temperature. FeHfTiX (X = Si, Ge) alloys show promise for spintronics applications, as indicated by theoretical simulation calculations.