Articles

A residual-set framework is introduced for analyzing additive prime conjectures, with particular emphasis on the Strong Goldbach Conjecture (SGC). For each even integer <math><semantics><mrow><msub><mi>E</mi><mi>n</mi></msub><mo>&#x2265;</mo><mn>4</mn></mrow></semantics></math>, the residual set <math><semantics><mrow><mi>&#x211B;</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow><mo>=</mo><mo>&#x007B;</mo><msub><mi>E</mi><mi>n</mi></msub><mo>&#x2212;</mo><mi>p</mi><mo>&#x2223;</mo></mrow></semantics></math> <math><semantics><mrow><mi>p</mi><mo>&#x003C;</mo><msub><mi>E</mi><mi>n</mi></msub><mo>,</mo><mi>p</mi><mo>&#x2208;</mo><mi>&#x2119;</mi><mo>&#x007D;</mo></mrow></semantics></math> is defined, and the universal residual set <math><semantics><mrow><msub><mi>&#x211B;</mi><mi mathvariant='double-struck'>E</mi></msub><mo>=</mo><msub><mrow><msup><mstyle mathsize='140%' displaystyle='true'><mo>&#x222A;</mo></mstyle><mtext>&#x200B;</mtext></msup></mrow><mrow><msub><mi>E</mi><mi>n</mi></msub></mrow></msub><mi>&#x211B;</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math> is constructed. It is shown that <math><semantics><mrow><msub><mi>&#x211B;</mi><mi mathvariant='double-struck'>E</mi></msub></mrow></semantics></math> contains infinitely many primes. A nontrivial constructive lower bound is derived, establishing that the number of Goldbach partitions satisfies <math><semantics><mrow><mi>G</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>&#x2265;</mo><mn>2</mn></mrow></semantics></math> for all <math><semantics><mrow><mi>E</mi><mo>&#x2265;</mo><mn>8</mn></mrow></semantics></math>, and that the cumulative partition count satisfies <math><semantics><mrow><munder><mstyle mathsize='140%' displaystyle='true'><mo>&#x2211;</mo></mstyle><mrow><mi>E</mi><mo>&#x2264;</mo><mi>N</mi></mrow></munder><mi>G</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>&#x226B;</mo><mfrac><mrow><msup><mi>N</mi><mn>2</mn></msup></mrow><mrow><msup><mrow><mtext>log</mtext></mrow><mn>4</mn></msup><mi>N</mi></mrow></mfrac></mrow></semantics></math>. An optimized deterministic algorithm is implemented to verify the SGC for even integers up to 16,000 digits. Each computed partition <math><semantics><mrow><msub><mi>E</mi><mi>n</mi></msub><mo>=</mo><mi>p</mi><mo>+</mo><mi>q</mi></mrow></semantics></math> 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 <math><semantics><mrow><mi>n</mi><mo>&#x2265;</mo><mn>7</mn></mrow></semantics></math> can be expressed as <math><semantics><mrow><mi>n</mi><mo>=</mo><mi>p</mi><mo>+</mo><mn>2</mn><mi>q</mi></mrow></semantics></math>, where <math><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>&#x2208;</mo><mi>&#x2119;</mi></mrow></semantics></math>. A corresponding residual formulation is developed, and it is proven that at least two valid partitions exist for all <math><semantics><mrow><mi>n</mi><mo>&#x2265;</mo><mn>9</mn></mrow></semantics></math>. 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 <math><semantics><mrow><mover accent="true"><mrow><mi mathvariant="normal">n</mi></mrow><mo>-</mo></mover></mrow></semantics></math>.

Starting with Planck scale it is developed the Relativistic Radial Density Theory (<i>RRDT</i>). In this theory, the Planck and gravitational parameters can be described as the functions of the radial mass (energy) density value. This density is maximal at the minimal radius and minimal at the maximal radius. This conclusion is based on the fact that the ratio of Planck mass and Planck length (radius) is constant. These radiuses can be described as the function of the energy conservation constant <i>κ</i>. Using <i>RRD</i><i>T</i>, it is possible to develop the connections between Planck’s and gravitational parameters as function of the maximal and minimal radial mass (energy) density values. In that sense, the gravitational length, time, energy and temperature can be presented as the function of the Planck length, time, energy and temperature, respectively. This opens possibility to merge of Quantum Field Theory (<i>QFT</i>) and the General Theory of Relativity (<i>GTR</i>) at the quantum scale in gravitational field. The existence of the maximal radial mass (energy) density value at the minimal radius in gravitational field means that no singularity in that field. Further, the existence of the minimal radial mass (energy) density value at the maximal radius in gravitational field means that no infinity in that field. It follows the postulation: the most minimal radius in a gravitational field belongs to the minimal mass (energy). Since the Planck mass is not the minimal mass in space-time, the Planck length/radius is not the minimal length/radius in the space-time. If the calculated minimal (or maximal) radius is the bigger than the related official radius it means that there exists a dark matter in this object. In that sense, the black holes are presenting the state of the matter at the minimal radius where we have the maximal radial mass (energy) density value. Further, the maximal possible radius of the matter is presenting the state with the minimal radial mass (energy) density value. Thus, the maximal and minimal radial mass (energy) density values are constants and conserved items. Now the question is: do motion of the Universe follows the <i>RRD</i><i>T</i>?

Carboxylates and their antimony(III) complexes experimentally scanned earlier for anti-leishmanial activity (IC₅₀) against five leishmanial strains viz., <i>L. major, L. major (Pak),</i><i> </i><i>L. tropica, L. mex mex, </i>and<i> L. donovani</i>. These activities have been theoretically predicted by DFT method along with quantitative structure-activity relationship (QSAR) study. Molecular modeling and geometry optimization of the all the eight compounds have been performed on workspace program of CAChe Pro software of Fujitsu by opting B88-PW91 (Becke '88; Perdew &#x26; Wang '91) GGA (generalized-gradient approximation) energy functional with DZVP (double-zeta valence polarized ) basis set in DFT (Density Functional Theory). For QSAR, multiple linear regression (MLR) analysis has been performed on Project Leader Program associated with CAChe. The reliability of correlation between experimental activities and predicted activities are r²= 0.826, r²_CV= 0.426 (<i>L. major</i>); r²= 0.905, r²_CV= 0.507 (<i>L. major (Pak)</i>); r²= 0.980, r²_CV= 0.932 (<i>L. tropica</i>); r²= 0.781, r²_CV= 0.580 (<i>L. mex mex</i>) and r²= 0.634, r²_CV= 0.376 (<i>L. donovani</i>), and a comparison of the experimental values and the values obtained by theoretical calculations has been presented pictorially that shows close resemblance.

<b>Background:</b> Coronavirus disease has caused global turmoil especially causing huge impact on human life all over the world. Current reports states more than 3 million people have lost life and more than 160 million people are known to be suspected with the SARS-CoV-2. Transmission and disease incidence rates are indicators to assess the seriousness of COVID-19 pandemic and studies to understand the factors that aid in this direction are very vital to curb the disease. <b>Methods:</b> The study intends to discover the relationship by performing statistical analysis using correlation and multiple linear regression analysis between the variable’s population density, temperature, relative humidity, and active time of virus and find out the parameters that predict the cases reported per million population in 83 countries. <b>Results:</b> Analysis indicates active time of virus in days is very positively associated with the COVID -19 cases in all the countries <i>r </i>= .604, <i>p</i> < .01. Active time of virus shows strong negative correlation with temperature <i>r</i> = -.930, <i>p</i> < .01 revealing that rise in temperature will reduce the virus activity in the population. Together, these variables will account for 36.2% variance in the cases per million population with no significant prediction estimated from any factor. <b>Conclusion:</b> The study outcomes clearly state that population density alone is insufficient to estimate the extent of influence on COVID -19 cases as the number of persons living per sq. km of land is a dynamic quantity tend to fluctuate over time and space due to migration of population. In conjunction to the previous studies reported on the environmental and climatic factors influencing the cases reported, population dynamics does not show much significance on the disease spread and incidence.<b> Contribution:</b> The rise in confirmed cases and the high incidence rate reported in countries can be attributed to the active time of virus life expectancy as there is a positive correlation observed between the COVID-19 cases reported and the virus active time in the examined countries. Also, environment and climatic factors play a role in modulating the infection and transmission rate with less significant influence of population density on the COVID-19.