A signed graph is a pair that consists of a graph and a sign mapping called signature from E to the sign group . In this paper, we discuss the t-path product signed graph where vertex set of is the same as that of and two vertices are adjacent if there is a path of length t, between them in the signed graph . The sign of an edge in the t-path product signed graph is determined by the product of marks of the vertices in the signed graph , where the mark of a vertex is the product of signs of all edges incident to it. In this paper, we provide a characterization of which are switching equivalent to t-path product signed graphs for which are switching equivalent to and also the negation of the signed graph ŋ that are switching equivalent to for . We also characterize signed graphs that are switching equivalent to -distance signed graph for where 2-distance signed graph defined as follows: the vertex set is same as the original signed graph and two vertices , are adjacent if and only if there exists a distance of length two in . The edge is negative if and only if all the edges, in all the distances of length two in are negative otherwise the edge is positive. The t-path network along with these characterizations can be used to develop model for the study of various real life problems communication networks.•t-path product signed graph.•t-distance signed graph.

Our research identified residues in AtPSKR1 and OsPSKR15 critical for kinase activity and interaction with ABA receptors, revealing PSKRs as core ABA signaling members that phosphorylate AtPYL4 and AtPYL9 receptors. Abscisic acid (ABA) serves as a vital signaling molecule that help plants respond to various environmental stresses, ensuring their survival and adaptability. The ABA signaling pathway begins when ABA is recognized by receptors known as PYR/PYL/RCAR. Upon ABA binding, these receptors undergo structural changes, but the precise modifications occurring during post-translational stages and their impact on ABA signaling are not fully understood. In this study, we have identified and characterized the ABA receptor family as target of PSKRs in both Arabidopsis and rice. In addition, we pinpointed the critical active sites in AtPSKR1 (N865) and OsPSKR15 (N892) that are responsible for kinase activity of the respective receptors and also important for direct interaction with ABA receptors. In vitro kinase experiments demonstrated phosphorylation of ABA receptors at S99 in AtPYL4, and S79 in AtPYL9. In addition, our genetic analysis demonstrated that PSKR plays a positive role in regulating ABA-mediated physiological responses, and promotes ABA-dependent leaf senescence in Arabidopsis. Phenotypic studies and expression analysis of ABA-related genes in complementation lines (AtPSKR1:pyl9 and OsPSKR15:pyl9) suggested that the overexpression of PSKR can partially restore the insensitivity of pyl9 mutant plants to ABA. These findings underscore the critical role of PSKR in enhancing ABA signaling via phosphorylation of PYL4/PYL9 in Arabidopsis.

ABSTRACTThis manuscript provides the theoretical explanation of the nonlinear scattering of an EMIC into an IAW a scattered ion‐cyclotron wave (SICW) in a dusty plasma using fluid theory. The low‐frequency nonlinearity arises through the parallel ponderomotive force on ions and the high‐frequency nonlinearity arises through nonlinear current density of ions. For the linked modes, a nonlinear dispersion relation is obtained. A formula for the SICW growth rate has been derived. The estimation of the turbulence growth rate takes into account for the typical parameters of dusty plasma. It is detected that an increased growth rate can be seen with a rise in the pump wave amplitude and the number density of dust grains. However, a decline in the growth rate has been reported with growing relative density dust grains, magnetic field, and the dust grain's size. Therefore, these wave modes have many applications in space and astrophysics as well as in laboratory experiments.

The electrostatic low energy beam transport (ELEBT) is used to handle low energy and intense beams from ion sources. It is preferred over magnetostatic low energy beam transport (MLEBT) for many reasons such as compactness, no space charge compensation and no transverse phase space coupling, more beam current handling capacity and less emittance growth as well as beam losses. The MLEBT has served in Linear accelerator 4 (LINAC4) at CERN and has been in operation at the test stand for about a decade but with demands of increasing luminosity and source upgradation to provide more beam current, there is a need of compact LEBT design which may overcome the limitations of MLEBT. The paper describes the novel design of an ELEBT by means of numerical simulation with many advantages over previous configurations and is designed to deliver the 70 mA H- beams at 45 keV energy within targeted rms normalized emittances of 0.5 π mm-mrad. We adopt the most reliable method for multiparticle beam dynamics simulation by first extracting 3D electric field data of optimized ELEBT generated by FEM computational tool and then running multiparticles with the appropriate grid. The numerical studies show the 100 % beam transmission in the proposed ELEBT. It has lower output beam emittances compared to measured beam emittances from existing MLEBT. With more numerical optimization, the energy spread and transverse twiss parameters at the end of the ELEBT are maintained within the existing RFQ acceptance. The beam optical, electrical and hardware specifications of the ELEBT section are presented in detail along with thermal analysis.