A residual-set framework is introduced for analyzing additive prime conjectures, with particular emphasis on the Strong Goldbach Conjecture (SGC). For each even integer $E_n\ge 4$, the residual set $ℛ\left(E_n\right)=\{E_n-p\mid$ $p<E_n,p\in \mathbb{P}\}$ is defined, and the universal residual set ${ℛ}_{\mathbb{E}}={{{\displaystyle \cup }}^{\text{}}}_{E_n}ℛ\left(E_n\right)$ is constructed. It is shown that ${ℛ}_{\mathbb{E}}$ contains infinitely many primes. A nontrivial constructive lower bound is derived, establishing that the number of Goldbach partitions satisfies $G\left(E\right)\ge 2$ for all $E\ge 8$, and that the cumulative partition count satisfies ${\displaystyle \sum }_{E\le N}G\left(E\right)\gg \frac{N^2}{{\text{log}}^{4}N}$. An optimized deterministic algorithm is implemented to verify the SGC for even integers up to 16,000 digits. Each computed partition $E_n=p+q$ is validated using elliptic curve primality testing, and no exceptions are observed. Runtime variability observed in the empirical tests corresponds with known fluctuations in prime density and modular residue distribution. A recursive construction is formulated for generating Goldbach partitions, using residual descent and leveraging properties of the residual sets. The method extends naturally to Lemoine's Conjecture, asserting that every odd integer $n\ge 7$ can be expressed as $n=p+2q$, where $p,q\in \mathbb{P}$. A corresponding residual formulation is developed, and it is proven that at least two valid partitions exist for all $n\ge 9$. Comparative analysis with the Hardy-Littlewood and Chen estimates is provided to contextualize the cumulative growth rate. The residual-set methodology offers a deterministic, scalable, and structurally grounded approach to additive problems in prime number theory, supported by both theoretical results and large-scale computational evidence.

Objective: To investigate the status of the acoustic environment of a typical Chinese pediatric emergency care department in a time series and identify the relationship between noise levels and factors such as crowd density and movement. Methods: A descriptive study was designed based on a multi-loci time-series method. We measured three loci under three variable settings: the decibel value, observation volume, and emergency care volume. Results: The noise levels of the three loci were significantly higher than the internationally recommended levels, exceeding rate reached more than 86.3%. The 24-hour mean map of the three loci showed similar fluctuation patterns, all of which had two peaks at approximately 10:00 AM and 16:00 PM. Conclusions: The daytime and nighttime noise levels were well-fitted by cubic functions with different coefficients. It is suggested that crowd density and movement may play important roles in noise mean fluctuations, which can be optimized to ensure a satisfactory environment in a pediatric emergency care department.

In this paper, we found a new model for a compact star with charged anisotropic matter distribution considering the generalized Chaplygin equation of state. The Einstein-Maxwell field equations have been solved with a particular form of metric potential and electric field intensity. The plots show that physical variables such as radial pressure, energy density, charge density, anisotropy, radial speed sound, and the mass are fully well defined and are regular in the star's interior. We obtained models consistent with stellar objects such as GJ 832, LHS 43, SAO 81292, GJ 380, GJ 412, and SAO 62377.

Recently, we have studied drug-receptor interaction of the peptidic HIV-1 protease inhibitors based on polar and hydrophobic interactions. We have also studied pharmacokinetics of these inhibitors based on Lipinski’s rule of five and its extended form. After that there was a need to study intermolecular interactions. From literatures, drug-receptor interaction involves hydrogen bonds between acceptor and donor sites of drug and its receptor. These donor acceptor sites must be more than four to be dominant. As single intermolecular H-bond is relatively weak and unlikely to support this type of interaction. It is also clear from literature that this interaction contribute to the alignment of reacting species in proper three-dimensional space in such a position that strong and effective polar or hydrophobic or both interaction occurs to form drug-receptor adduct or enzyme inhibitor complex as appropriate. The strength of H-bonds formed between drug and receptor was judged by bond lengths, bond angles and bond orders. As well as, its nature (strong, moderate or weak) and its number, too. Along with H-bonding, we have also studied Van der Walls i.e. non-bonding type interaction. These non-bonding interactions were studied using charge transfer from donor to acceptor and this results transfer of electron flux from donor molecule (drug/receptor) towards acceptor (receptor/ drug). Thus, lowering of energy of the system under investigation will occur. For this resulted interaction energy was also studied that very clearly explain feasibility of interactions. As we know that all above phenomena are molecular properties and do not cover involvement of orbitals. To cover this we have also studied drug-receptor interaction involving molecular orbital. It was HOMO of one reacting molecule (B) that donates electron pair, electron cloud or electron density to LUMO of another reacting molecule (A) that accepts or accommodates this electron pair, electron cloud or electron density. The quantity of the electron flux from HOMO to LUMO was judged by the value of ∆E_LH. A lower value of this will support strong and effective drug-receptor interaction. Results of orbital based study have also been found to supports the results as abstracted from interaction energy.