An application of Fourier series expansion of a function in a non-polar spherical coordinate system

Cubed sphere is one of the main tools used to avoid pole problems those arise in the selection of spherical polar coordinates. In this respect, earlier we considered a recently developed cubed sphere based on coordinate mapping over the cubed surface. The function on the sphere was treated as an ordered set of six-tuples. In that work, we established weakly orthogonal and completely orthogonal spherical harmonics of the system and developed corresponding symmetric and linear relations. Also, we found the norm of the orthogonal spherical harmonics. In this work, we explore the Fourier representation of a spherical function on this coordinate system in terms of weakly orthogonal spherical harmonics. The advantages of the linear relation between the two sets of spherical harmonics and diagonal property of the norm of the fully orthogonal spherical harmonics were in cooperated for this work. We also strength our work by giving an example to demonstrate how Fourier coefficients can be computed to represent a given spherical function in terms of the spherical harmonics of the coordinate system

Relational properties of weakly orthogonal and orthogonal spherical harmonics in cubed sphere

Numerical computations on the sphere in solving problems defined on the sphere suffer from many difficulties near the poles when using spherical polar coordinate system for the spherical surface. For example, in the computations of global weather prediction models, concentrated grid points near the poles increase the amount of computations in the pole region where quantities of interest are of less important than in other parts of the globe. Such problems are collectively called as the 'pole problems'. Avoiding pole problems have attracted some researches in the recent past. One of the recent development in this direction is to define grid meshes on the sphere which do not contain polar concentrated points. Among these the 'cubed sphere' defined from the surface of a unit cube has been used by some authors for approximating weather prediction models by finite difference and finite element methods. In a recent paper, one of the present authors has constructed weakly orthogonal spherical harmonics in a non-polar spherical co-ordinate system based on the cube sphere concept. This can be used for approximating functions on the sphere by spectral methods without the pole problems. In this work, we establish some Lin ear and recurrence relations between these two sets of spherical harmonics. We also exploit linear relations between harmonic components defined in the six faces of the cubed sphere

Synthesis, X-ray structure, Hirshfeld surface analysis and antimicrobial assessment of tetranuclear s-triazine hydrazine Schiff base ligand

Funding: The Deputyship for Research and Innovation, “Ministry of Education”, King Saud University (IFKSUOR3-188-3), Saudi Arabia.The unexpected tetranuclear [Cu4(DPPT)2Cl6] complex was obtained by self-assembly of CuCl2.2H2O and (E)-2,4-di(piperidin-1-yl)-6-(2-(1-(pyridin-2-yl)ethylidene)hydrazinyl)-1,3,5-triazine, ( HDPPT ) in ethanol. In this tetranuclear [Cu4(DPPT)2Cl6] complex, the organic ligand acts as mononegative chelate bridging two crystallographically independent Cu(II) sites. The DPPT− anion acts as a bidentate ligand with respect to Cu(1), while it is a tridentate for Cu(2). The Cu(1)N2Cl3 and Cu(2)N3Cl spheres have square pyramidal and square planar coordination geometries with some distortion, respectively. Two of the chloride ions coordinating the Cu(1) are bridging between two crystallographically related Cu(1) sites connecting two [Cu2(DPPT)Cl3] units together, leading to the tetranuclear formula [Cu4(DPPT)2Cl6]. The packing of the [Cu4(DPPT)2Cl6] complex is dominated by C-H…Cl contacts, leading to one-dimensional hydrogen-bond polymeric structure. According to Hirshfeld surface analysis of molecular packing, the non-covalent interactions H…H, Cl…H, Cl…C, C…H, and N…H are the most significant. Their percentages are 52.8, 19.0, 3.2, 7.7, and 9.7%, respectively. Antimicrobial assessment showed good antifungal activity of the Cu(II) complex against A. fumigatus and C. albicans compared to Ketoconazole as positive control. Moreover, the [Cu4(DPPT)2Cl6] complex has higher activity against Gram-positive bacteria than Gentamycin as positive control. The opposite was observed when testing the tetranuclear [Cu4(DPPT)2Cl6] complex against the Gram-negative bacteria.Publisher PDFPeer reviewe

New Triazoloquinoxaline Ligand and its Polymeric 1D Silver(I) complex Synthesis, Structure, and Antimicrobial activity

The organic ligand 4-Benzyl-1-(N,N-dimethylamino)-[1,2,4]triazolo[4,3a]quinoxaline 1 (L) and its polymeric silver(I) complex, [Ag2L(NO3)2]n (2), have been synthesized and characterized. The organic ligand 1 crystallizes in the triclinic space group P¯1. The unit cell contains two parallel-stacked molecules. The complex [Ag2L(NO3)2]n (2) crystallizes in the monoclinic space group P21/n. The structure contains two different silver(I) ions. Ag(2) is coordinated by three oxygens (involving two nitrate groups) and to a nitrogen of the triazole ring of 1. These ligands form a strongly distorted tetrahedral, nearly planar coordination sphere. Ag(1) has an approximately tetrahedral geometry. It is bonded to one oxygen of a nitrate anion and a nitrogen of two different L; this aspect giving rise to an infinite chain structure. A final bond to Ag(1) involves the carbon of a phenyl group. It is more weakly bonded to the phenyl carbons on either side of this, so that the Ag(1)-phenyl bonding has aspects of an Ag-allyl bond. Ag(1) and Ag(2) participate in bonding to a common nitrate anion and alternate, the two distinct modes of bridging between them lead to a zig-zag chain structure. In addition to spectroscopic studies, the biological activities of the ligand and of the complex were scanned over a wide range of Gram positive and Gram negative flesh- and bone-eating bacteria. The results are discussed in comparison with well-known antibiotics

Synthesis, structure and antimicrobial activity of new Co(II) complex with bis-morpholino/benzoimidazole -s-triazine Ligand

Funding: The authors would like to extend their sincere appreciation to the Researchers Supporting Project (RSP2023R64), King Saud University, Riyadh, Saudi Arabia.A new Co(II) perchlorate complex of the bis-morpholino/benzoimidazole-s-triazine ligand, 4,4′-(6-(1H-benzo[d]imidazol-1-yl)-1,3,5-triazine-2,4-diyl)dimorpholine ( BMBIT ), was synthesized and characterized. The structure of the new Co(II) complex was approved to be [Co(BMBIT)2(H2O)4](ClO4)2*H2O using single-crystal X-ray diffraction. The Co(II) complex was found crystallized in the monoclinic crystal system and P21/c space group. The unit cell parameters are a = 22.21971(11) Å, b = 8.86743(4) Å, c = 24.38673(12) Å and β = 113.4401(6)°. This heteroleptic complex has distorted octahedral coordination geometry with two monodenatate BMBIT ligand units via the benzoimidazole N-atom and four water molecules as monodentate ligands. The hydration water and perchlorate ions participated significantly in the supramolecular structure of the [Co( BMBIT )2(H2O)4](ClO4)2*H2O complex. Analysis of dnorm map and a fingerprint plot indicated the importance of O···H, N···H, C···H, C···O, C···N and H···H contacts. Their percentages are 27.5, 7.9, 14.0, 0.9, 2.8 and 43.5%, respectively. The sensitivity of some harmful microbes towards the studied compounds was investigated. The Co(II) complex has good antifungal activity compared to the free BMBIT which has no antifungal activity. The Co(II) complex has good activity against B. subtilis, S. aureus, P. vulgaris and E. coli while the free BMBIT ligand has limited activity only towards B. subtilis and P. vulgaris. Hence, the [Co( BMBIT )2(H2O)4](ClO4)2*H2O complex has broad spectrum antimicrobial action compared to the free BMBIT ligand.Publisher PDFPeer reviewe

Elucidating glycosaminoglycan–protein–protein interactions using carbohydrate microarray and computational approaches

Glycosaminoglycan polysaccharides play critical roles in many cellular processes, ranging from viral invasion and angiogenesis to spinal cord injury. Their diverse biological activities are derived from an ability to regulate a remarkable number of proteins. However, few methods exist for the rapid identification of glycosaminoglycan–protein interactions and for studying the potential of glycosaminoglycans to assemble multimeric protein complexes. Here, we report a multidisciplinary approach that combines new carbohydrate microarray and computational modeling methodologies to elucidate glycosaminoglycan–protein interactions. The approach was validated through the study of known protein partners for heparan and chondroitin sulfate, including fibroblast growth factor 2 (FGF2) and its receptor FGFR1, the malarial protein VAR2CSA, and tumor necrosis factor-α (TNF-α). We also applied the approach to identify previously undescribed interactions between a specific sulfated epitope on chondroitin sulfate, CS-E, and the neurotrophins, a critical family of growth factors involved in the development, maintenance, and survival of the vertebrate nervous system. Our studies show for the first time that CS is capable of assembling multimeric signaling complexes and modulating neurotrophin signaling pathways. In addition, we identify a contiguous CS-E-binding site by computational modeling that suggests a potential mechanism to explain how CS may promote neurotrophin-tyrosine receptor kinase (Trk) complex formation and neurotrophin signaling. Together, our combined microarray and computational modeling methodologies provide a general, facile means to identify new glycosaminoglycan–protein–protein interactions, as well as a molecular-level understanding of those complexes