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.

In this article, our investigation to study squeezing and statistical properties of the light by a two mode sub harmonic generator coupled to thermal reservoir via a single port-mirror. The equation of motion answers are then used to calculate the mean photon number, photon number variance, and quadrature variance for two mode cavity light. However, we have found that the degree of squeezing is indeed affected by the present of thermal light. The mean photon number of the system under consideration increases with increasing $\stackrel{-}{\mathrm{n}}$.

This study addressed the frequency of occurrence of the epistemic modal elements that the Ex-president prefers to express in the selected State of the Nation Addresses of Ghana. The study adopted a qualitative approach. Purposively, the two addresses selected for the study are State of the Nation Address: 16th February 2001 and State of the Nation Address: 14th February 2008. The selected addresses are a good starting point for determining tendencies in the meaning expressed by the selected modal categories. The focus is on the semantics of the epistemic category of modality and other related variables within the contexts in which they are used. The data were analysed based on specific epistemic modal meanings and the modal elements expressed. It was observed that there were more expressions of certainty during the 2008 State of the Nation Address than the first one delivered by the Ex-president. Additionally, it is observed that the selection of a particular modal element in a particular context is primarily influenced by the interplay of the speaker and the participants involved. Therefore, an epistemic modal element used in a different linguistic environment may have a different contextual meaning from another. The lower expressions of uncertainty during the last State of the Nation, which ended the tenure of office of the ex-president, show that he was sure he had performed well and was bold enough to make categorical statements. The study exhibited some occurrences of expression of tentativeness with the use of modal verbs alone. All the categories recorded any instance of tentativeness. It is also recommended that politicians, spokespersons, and speech writers be cautious when indicating their information sources and how they obtain them. This helps maintain political credibility, which is the dominant index of the electorate's quest for people who know the conditions and problems prevailing in the country and their solutions.

With respect to the exciton moving in the two-dimensional quantum well, the paper presents a scheme which can rigorously calculate out the binding energy of the exciton in the two-dimensional semiconductor quantum well by simply using the relation $\left|z_e-z_h\right|=\rho \tan \alpha$, which is much simpler than the complex calculation of Ref.[1-2]. Concerning the calculation result eq.(13), the paper discusses the results for two significant cases of $\left|z_e-z_h\right|$<<$\rho$and $\left|z_e-z_h\right|\to \infty$.