The investigation of determining solutions for the Diophantine equation  over the Gaussian integer ring for the specific case of  is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

This paper proposes a compact, plasmonic-based 4 × 4 nonblocking switch for optical networks. This device uses six 2 × 2 plasmonic Mach-Zehnder switch (MZS), whose arm waveguide is supported by a JRD1 polymer layer as a high electro-optic coefficient material. The 4 × 4 switch is designed in COMSOL environment for 1550 nm wavelength operation. The performance of the proposed switch outperforms those of conventional (nonplasmonic) counterparts. The designed switch yields a compact structure ( 500 × 70   µ m 2 ) having V π L = 12   V · µ m , 1.5 THz optical bandwidth, 7.7 dB insertion loss, and −26.5 dB crosstalk. The capability of the switch to route 8 × 40 Gbps WDM signal is demonstrated successfully.

Metal corrosion is a destructive process for many industrial operations, including oil well acidizing and acid pickling. Therefore, numerous efforts made by many researchers to control the steel corrosion. In the present work, A (E)-4-(((4-(5-mercapto-1,3,4-oxadiazol-2-yl) phenyl) amino) methyl)-2-methoxyphenol (MOPM) has been synthesized and characterized as a new corrosion inhibitor for mild steel in 0.1 M hydrochloric acid. FTIR and 1 HNMR were used in the diagnosis of MOPM, while electrochemical polarization technique was employed to test the performance of inhibitor at various temperatures and inhibitor concentrations. Electrochemical studies showed that MOPM acts as a mixed-type inhibitor with a maximum inhibition efficiency of

Topological indices provide important insights into the structural characteristics of molecular graphs. The present investigation proposes and explores a creative graph on a finite group G, which is known as the RIG. This graph is designated as ΓRS G2(4) indicating a simple undirected graph containing elements of G. Two distinct ertices are regarded as nearly the same if and only if their sum yields a non-trivial involution element in G. RIGs have been discovered in various finite groups. We examine several facets of the RIG by altering the graph through the conjugacy classes of G. Furthermore, we investigate the topological indices as applications in graph theory applying the distance matrix of the G2(4) group.

Two series of Schiff Bases [VI]n and thiazolidin-4-one derivatives[VII]n were synthesized by many steps starting from cyclization of 4- hydroxyacetophenon with thiourea in iodine to yield 1,3-thiazole compound which was reacted with pentoxy bromide in anhydrous potassium carbonate to converted compound[II] and this reacted with Phenol to yield azo compound[III]. The azo compound reacted with ethyl chloro acetate in basic medium to get a new easter compound[IV] which is converted to their acid hydrazid[V]. The later compound condensation with n-alkoxy benzaldehyde to give new Schiff bases[VI]n . Imine group undergoes addition cyclization with thioglycolic acid to get thiazolidinone compounds[VII]n .Also, two new series of Schiff Bases [XII]n