Embedding Of Non-Simple Lie-Groups, Coupling-Constant Relations And Non-Uniqueness Of Models Of Unification

A general derivation of the coupling constant relations which result on embedding a non-simple group like SU L (2) @ U(1) in a larger simple group (or graded Lie group) is given. It is shown that such relations depend only on the requirement (i) that the multiplet of vector fields form an irreducible representation of the unifying algebra and (ii) the transformation properties of the fermions under SU L (2). This point is illustrated in two ways, one by constructing two different unification groups containing the same fermions and therefore have same Weinberg angle; the other by putting different SU L (2) structures on the same fermions and consequently have different Weinberg angles. In particular the value sin~0=3/8 is characteristic of the sequential doublet models or models which invoke a large number of additional leptons like E 6, while addition of extra charged fermion singlets can reduce the value of sin ~ 0 to 1/4. We point out that at the present time the models of grand unification are far from unique.

The algebra and geometry of SU(3) matrices

We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular group SU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real Linear vector space, are developed in an SU(3) covariant manner. The f and d symbols of SU(3) lead to two ways of 'multiplying' two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization of SU(3) is developed as a generalization of that for SU(2), and the specifically new features are brought out. Application to the dynamics of three-level systems is outlined.

Beta-Turn Conformations In Crystal-Structures Of Model Peptides Containing Alpha,Alpha-Di-N-Propylglycine And Alpha,Alpha-Di-N-Butylglycine

The crystal state conformations of three peptides containing the alpha, alpha-dialkylated residues, alpha,alpha-di-n-propylglycine (Dpg) and alpha,alpha-di-n-butylglycine (Dbg), have been established by x-ray diffraction. Boc-Ala-Dpg-Ala-OMe (I) and Boc-Ala-Dbg-Ala-OMe (III) adopt distorted type II beta-turn conformations with Ala (1) and Dpg/Dbg (2) as the corner residues. In both peptides the conformational angles at the Dxg residue (I: phi = 66.2 degrees, psi = 19.3 degrees; III: phi = 66.5 degrees, psi = 21.1 degrees) deviate appreciably from ideal values for the i + 2 residue in a type II beta-turn. In both peptides the observed (N...O) distances between the Boc CO and Ala(3) NH groups are far too long (I: 3.44 Angstrom; III: 3.63 Angstrom) for an intramolecular 4 --> 1 hydrogen bond. Boc-Ala-Dpg-Ala-NHMe (II) crystallizes with two independent molecules in the asymmetric unit. Both molecules IIA and IIB adopt consecutive beta-turn (type III-III in IIA and type III-I in IIB) or incipient 3(10)-helical structures, stabilized by two intramolecular 4 --> 1 hydrogen bonds. In all four molecules the bond angle N-C-alpha-C' (tau) at the Dxg residues are greater than or equal to 110 degrees. The observation of conformational angles in the helical region of phi,psi space at these residues is consistent with theoretical predictions.

NMR Investigations on Di- and Tri-Azanaphthalenes Oriented in Liquid Crystals

Proton NMR spectra of 1,3-diazanaphthalene and 1,2,4-triazanaphthalene have been investigated in the nematic phase of three liquid crystals. The spectral analysis provided direct dipole-dipole couplings which have been used to derive the molecular structure. Geometry of the phenyl ring in both the molecules deviates from the regular hexagonal structure. Signs of the order parameter of the largest magnitude are opposite in liquid crystals with positive diamagetic anisotropies.

Synthesis and Reactions of Binuclear Diolefin Rhodium(I) Compounds bridged by Nitrogen Heterocycles, Crystal Structure of Dichloro-Di-1,5-Cyclooctadiene(Mu-Pyrazine)-Dirhodium(I) [[Rh(1,5-C8h12)Cl]2(Mu-C4h4n2)]

Binuclear complexes of rhodium(I) of the type [(dien)(X)Rh(μ-N-N)Rh(X)(dien)] (dien = 1,5-cyclooctadiene or norbornadiene; N-N = pyrazine, 4,4′-bipyridine or Phenazine and X = Cl or Br) with bridging heterocycles have been isolated and their reactions with carbon monoxide, 2,2′-bipyridine and 1,10-phenanthroline investigated. The crystal structure of [(COD)(Cl)Rh(μ-pyrazine)Rh(Cl)(COD)] has been determined.

Structure-reactivity correlation in inclusion complexes: deoxycholic acid di-tert-butyl thioketone

Ultraviolet irradiation of crystalline molecular inclusion complexes of deoxycholic acid with di-tert-butyl thioketone results in no reaction. The structure of the above complex has been determined via X-ray diffraction. The absence of expected photoreactions. namely, photoreduction and photooxidation, is rationalized on the basis of the X-ray structure analysis of the complex.

Representations and properties of para-Bose oscillator operators. I. Energy position and momentum eigenstates

Para-Bose commutation relations are related to the SL(2,R) Lie algebra. The irreducible representation [script D]alpha of the para-Bose system is obtained as the direct sum Dbeta[direct-sum]Dbeta+1/2 of the representations of the SL(2,R) Lie algebra. The position and momentum eigenstates are then obtained in this representation [script D]alpha, using the matrix mechanical method. The orthogonality, completeness, and the overlap of these eigenstates are derived. The momentum eigenstates are also derived using the wave mechanical method by specifying the domain of the definition of the momentum operator in addition to giving it a formal differential expression. By a careful consideration in this manner we find that the two apparently different solutions obtained by Ohnuki and Kamefuchi in this context are actually unitarily equivalent. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Hydrogen bonding with mono- and di-carbonyls

The electronic absorption and i.r. spectroscopic studies are reported for the hydrogen bonding systems involving alcohol and various ketones. It is shown that the hydrogen bonding abilities of ketones are determined by the extent of delocalization of the lone pair electrons in their non-bonding molecular orbitals. Evidence for the formation of very weak intermolecular hydrogen bonds between alcohol and the π-electron part of the dicarbonyls has also been presented from the i.r. studies in the 3400–3700 cm−1 region.

Geometrical interpretation of Inönü-Wigner contractions

The Inönü-Wigner contractions which interrelate the Lie algebras of the isometry groups of metric spaces are discussed with reference to deformations of the absolutes of the spaces. A general formula is derived for the Lie algebra commutation relations of the isometry group for anyN-dimensional metric space. These ideas are illustrated by a discussion of important particular cases, which interrelate the four-dimensional de Sitter, Poincaré, and Galilean groups.

Iron(II, III) complexes with N, N'-DI(2-)Pyridyl Thiourea. Moumlssbauer and other spectroscopic studies

Several iron(II, III) complexes of N, N'-di(2-)pyridyl thiourea have been synthesized. The preparation of the complexes from iron(III) salts proceeds through a reduction of iron(III) to iron(II) followed by a subsequent reoxidation. The Moumlssbauer, electronic and infrared spectra of these complexes have been measured. The results are concordant with the coordination of pyridine nitrogens and thiocarbonyl sulfur yielding polymeric complexes. A variable temperature NMR study of the free ligand shows that two conformation are accessible for it in solution at subambient temperatures.

Complexes of rare earth perchlorates with di-2-pyridyl ketone

Abstract is not availabe.

Multipliers of Sobolev spaces

Let Wm,p denote the Sobolev space of functions on Image n whose distributional derivatives of order up to m lie in Lp(Image n) for 1 less-than-or-equals, slant p less-than-or-equals, slant ∞. When 1 < p < ∞, it is known that the multipliers on Wm,p are the same as those on Lp. This result is true for p = 1 only if n = 1. For, we prove that the integrable distributions of order less-than-or-equals, slant1 whose first order derivatives are also integrable of order less-than-or-equals, slant1, belong to the class of multipliers on Wm,1 and there are such distributions which are not bounded measures. These distributions are also multipliers on Lp, for 1 < p < ∞. Moreover, they form exactly the multiplier space of a certain Segal algebra. We have also proved that the multipliers on Wm,l are necessarily integrable distributions of order less-than-or-equals, slant1 or less-than-or-equals, slant2 accordingly as m is odd or even. We have obtained the multipliers from L1(Image n) into Wm,p, 1 less-than-or-equals, slant p less-than-or-equals, slant ∞, and the multiplier space of Wm,1 is realised as a dual space of certain continuous functions on Image n which vanish at infinity.

Formal description, compression and transformation of digital pictures II. Picture algebra and geometry

In an earlier paper (Part I) we described the construction of Hermite code for multiple grey-level pictures using the concepts of vector spaces over Galois Fields. In this paper a new algebra is worked out for Hermite codes to devise algorithms for various transformations such as translation, reflection, rotation, expansion and replication of the original picture. Also other operations such as concatenation, complementation, superposition, Jordan-sum and selective segmentation are considered. It is shown that the Hermite code of a picture is very powerful and serves as a mathematical signature of the picture. The Hermite code will have extensive applications in picture processing, pattern recognition and artificial intelligence.

Complexes of lanthanide perchlorates with N,N,N′,N′-tetramethyl-α-carboxamido-O-anisamide and N,N′-di-t-butyl-α-carboxamido-O-anisamide

A survey of the literature on lanthanide coordination compounds reveals that ligands involving ether oxygens as donor atoms have received very little attention [ 11. Only recently have the complexes of lanthanides with cyclic polyethers been characterized [l-3]. We report in this communication that interaction of rareearth perchlorates with two new ligands namely N,N,N’,N’-tetramethyl-u-carboxamido-Oanisamide (TMCA) and N,N’-di-t-butyl-crcarboxamido- 0-anisamide (DTBCA). The two ligands are potentially tridentate possessing two amide moieties and an ether linkage in between. The isolated complexes have been characterized by analysis, electrolytic conductance, infrared and electronic spectra. The ‘H and “C NMR spectra for the diamagnetic La3+ and Y3+ complexes are also discussed.