Equivariant Spectral Triples on the Quantum SU(2) Group

We characterize all equivariant odd spectral triples for the quantum SU(2) group acting on its L ₂-space and having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and given any element of the K-homology group of SU_q(2), there is an equivariant odd spectral triple of dimension 3 inducing that element. The method employed to get equivariant spectral triples in the quantum case is then used for classical SU(2), and we prove that for p < 4, there does not exist any equivariant spectral triple with nontrivial K-homology class and dimension p acting on the L ₂-space.The first author would like to acknowledge support from the National Board of Higher Mathematics, India.     

Self-assembled metallasupramolecular cages towards light harvesting systems for oxidative cyclization

Designing artificial light harvesting systems with the ability to utilize the output energy for fruitful application in aqueous medium is an intriguing topic for the development of clean and sustainable energy. We report here facile synthesis of three prismatic molecular cages as imminent supramolecular optoelectronic materials via two-component coordination-driven self-assembly of a new tetra-imidazole donor (L) in combination with 180°/120° di-platinum(ii) acceptors. Self-assembly of 180° trans-Pt(ii) acceptors A1 and A2 with L leads to the formation of cages Pt₄L₂(1a) and Pt₈L₂(2a) respectively, while 120°-Pt(ii) acceptor A3 with L gives the Pt₈L₂(3a) metallacage. PF₆⁻ analogues (1b, 2b and 3b) of the metallacages possess a high molar extinction coefficient and large Stokes shift. 1b–3b are weakly emissive in dilute solution but showed aggregation induced emission (AIE) in a water/MeCN mixture as well as in the solid state. AIE active 2b and 3b in aqueous (90% water/MeCN mixture) medium act as donors for fabricating artificial light harvesting systems via Förster resonance energy transfer (FRET) with organic dye rhodamine-B (RhB) with high energy efficiency and good antenna effect. The metallacages 2b and 3b represent an interesting platform to fabricate new generation supramolecular aqueous light harvesting systems with high antenna effect. Finally, the harvested energy of the LHSs (2b + RhB) and (3b + RhB) was utilized successfully for efficient visible light induced photo-oxidative cross coupling cyclization of N,N-dimethylaniline (4) with a series of N-alkyl/aryl maleimides (5) in aqueous acetonitrile with dramatic enhancement in yields compared to the reactions with RhB or cages alone.

Synthesis of Pt(ii) based metallacages as aggregation induced emissive supramolecular architectures for fabricating artificial light harvesting systems for cross coupling cyclization under visible light is achieved.     

Photopolymerization of methyl methacrylate with the use of bromine as photoinitiator

Low concentrations of bromine (0.008–0.06M) were used to initiate photopolymerization of MMA in bulk and in diluted (near bulk) systems, the diluents or solvents used being benzene, toluene, dioxane, tetrahydrofuran, carbon tetrachloride, chloroform, methylene chloride, and methanol. Polymerization in bulk follows usual free-radical kinetics. Inert solvents (benzene, toluene) as well as the other solvents used enhance the rate of polymerization MMA even when used in the range of catalytic concentrations (0.04–0.4M). An initiation mechanism involving solvent molecules appears to be predominant in diluted systems.     

Structural analyses and magnetic properties of 3D coordination polymeric networks of nickel(II) maleate and manganese(II) adipate with the flexible 1,2-bis(4-pyridyl)ethane ligand

Two novel inorganic-organic hybrid 3D extended networks of Ni(II) and Mn(II) having molecular formulas [(maleate)(2)Ni(3)(bpe)(4)(H(2)O)(4)](NO(3))(2).H(2)O (1) and [(adipate)Mn(bpe)] (2) (bpe = 1, 2-bis(4-pyridyl)ethane), respectively, have been synthesized and characterized by single-crystal X-ray diffraction studies and low-temperature (300-2 K) magnetic measurements. Compound 1 crystallizes in the monoclinic system, space group C2/c (No. 15), with chemical formula C(56)H(62)N(10)Ni(3)O(19), a = 30.955(4) A, b = 12.705(3) A, c = 17.058(5) A, beta = 117.26(2) degrees, and Z = 4. Compound 2 crystallizes in the triclinic system, space group Ponemacr; (No. 2), with chemical formula C(18)H(20)MnN(2)O(4), a = 8.492(2) A, b = 9.444(2) A, c = 11.533(3) A, alpha = 97.19(1) degrees, beta = 94.64(1) degrees, gamma = 105.02(1) degrees, and Z = 2. The structure determination reveals for both a 3D network. Compound 1 contains two crystallographically independent Ni(II) ions in different octahedral environments. Ni(1) lies on an inversion center, and its coordination environment comprises two chelating maleate anions and two bpe nitrogen donors, while the Ni(2) ion is surrounded by meridionally disposed three bpe N atoms, two water molecules, and one oxygen donor from the dicarboxylate anion. Of the three crystallographic independent bpe ligand, one presents an anti and the others a gauche conformation. The corresponding N-to-N distances are 9.344, 6.543, and 6.187 A. Variable-temperature magnetic susceptibility measurement of the complex reveals the existence of a dominant ferromagnetic interaction within the molecule. Compound 2 is composed of Mn(2) dimer units linked by adipate anions to form corrugated 2D sheets which, on interconnection through bpe (anti conformation, N-to-N distance of 9.391 A), produces an interpenetrated 3D alpha-polonium-related type net. Complex 2 reveals to be antiferromagnetic fitting data using a dimeric Mn(II) model that considers negligible magnetic transmission through the carbon skeleton of adipate and the bpe pathway.     

Fluorination of polymers by sulfur hexafluoride gas under electric discharge

Fluorination of low-density polyethylene, polyacetylene, and poly(vinyl alcohol) was carried out using SF₆ gas under electric discharge. The polymers were partially fluorinated and the extent of fluorination was more in the case of poly (vinyl alcohol) than the other two polymers. The fluorinated polymers were characterized by elemental analysis (Fluorine), IR, and x-ray diffraction. Optical transparency of the films was also measured. The fluorinated polymers show better solvent resistance and decreased transparency than the virgin polymer. © 1994 John Wiley & Sons, Inc.     

A note on Hardy’s theorem and rotations

Semigroup Forum - We give a very short proof, using the Hermite semigroup, to a generalized version of Hardy’s theorem due to Hogan and Lakey. We characterize $$f\in L^2({\mathbb {R}}^n)$$...     

Three‐Component Self‐Assembly of a Series of Triply Interlocked Pd12 Coordination Prisms and Their Non‐Interlocked Pd6 Analogues

Template‐assisted formation of multicomponent Pd₆ coordination prisms and formation of their self‐templated triply interlocked Pd₁₂ analogues in the absence of an external template have been established in a single step through Pd? N/Pd? O coordination. Treatment of cis‐[Pd(en)(NO₃)₂] with K₃tma and linear pillar 4,4′‐bpy (en=ethylenediamine, H₃tma=benzene‐1,3,5‐tricarboxylic acid, 4,4′‐bpy=4,4′‐bipyridine) gave intercalated coordination cage [{Pd(en)}₆(bpy)₃(tma)₂]₂[NO₃]₁₂ ( 1 ) exclusively, whereas the same reaction in the presence of H₃tma as an aromatic guest gave a H₃tma‐encapsulating non‐interlocked discrete Pd₆ molecular prism [{Pd(en)}₆(bpy)₃(tma)₂(H₃tma)₂][NO₃]₆ ( 2 ). Though the same reaction using cis‐[Pd(NO₃)₂(pn)] (pn=propane‐1,2‐diamine) instead of cis‐[Pd(en)(NO₃)₂] gave triply interlocked coordination cage [{Pd(pn)}₆(bpy)₃(tma)₂]₂[NO₃]₁₂ ( 3 ) along with non‐interlocked Pd₆ analogue [{Pd(pn)}₆(bpy)₃(tma)₂](NO₃)₆ ( 3′ ), and the presence of H₃tma as a guest gave H₃tma‐encapsulating molecular prism [{Pd(pn)}₆(bpy)₃(tma)₂(H₃tma)₂][NO₃]₆ ( 4 ) exclusively. In solution, the amount of 3′ decreases as the temperature is decreased, and in the solid state 3 is the sole product. Notably, an analogous reaction using the relatively short pillar pz (pz=pyrazine) instead of 4,4′‐bpy gave triply interlocked coordination cage [{Pd(pn)}₆(pz)₃(tma)₂]₂[NO₃]₁₂ ( 5 ) as the single product. Interestingly, the same reaction using slightly more bulky cis‐[Pd(NO₃)₂(tmen)] (tmen=N,N,N′,N′‐tetramethylethylene diamine) instead of cis‐[Pd(NO₃)₂(pn)] gave non‐interlocked [{Pd(tmen)}₆(pz)₃(tma)₂][NO₃]₆ ( 6 ) exclusively. Complexes 1 , 3 , and 5 represent the first examples of template‐free triply interlocked molecular prisms obtained through multicomponent self‐assembly. Formation of the complexes was supported by IR and multinuclear NMR (¹H and ¹³C) spectroscopy. Formation of guest‐encapsulating complexes ( 2 and 4 ) was confirmed by 2D DOSY and ROESY NMR spectroscopic analyses, whereas for complexes 1 , 3 , 5 , and 6 single‐crystal X‐ray diffraction techniques unambiguously confirmed their formation. The gross geometries of H₃tma‐encapsulating complexes 2 and 4 were obtained by universal force field (UFF) simulations.     

Anomalous behaviour in the SmA*–SmCA* pre‐transitional regime of a chiral swallow‐tailed antiferroelectric liquid crystal

The temperature‐ and electric field‐dependent dielectric relaxation and polarisation of a new chiral swallow tailed antiferroelectric liquid crystal, i.e. 1‐ethylpropyl (S)‐2‐{6‐[4‐(4′‐decyloxyphenyl)benzoyloxy]‐2‐naphthyl}propionate (abbreviated as EP10PBNP), were investigated. The electric field‐induced dielectric loss spectra of EP10PBNP revealed electroclinic and anomalous dielectric behaviour in the chiral smectic A (SmA*)–chiral antiferroelectric smectic C (SmC_A*) pre‐transitional regime. From an analysis of thermal hysteresis of the dielectric constant, electric field‐induced polarisation and dielectric loss spectra, the appearance of a ferrielectric‐like mesophase is observed followed by an unstable SmC_A* phase in the SmA*–SmC_A* pre‐transitional regime.