Existence and approximations of solutions to some fractional order functional integral equations

In this paper we shall study a fractional order functional integral equation. In the first part of the paper, we proved the existence and uniqueness of mile and global solutions in a Banach space. In the second part of the paper, we used the analytic semigroups theory oflinear operators and the fixed point method to establish the existence, uniqueness and convergence of approximate solutions of the given problem in a separable Hilbert space. We also proved the existence and convergence of Faedo-Galerkin approximate solution to the given problem. Finally, we give an example.

Phase behavior of concentrated aqueous solutions of cetyltrimethylammonium bromide (CTAB) and sodium hydroxy naphthoate (SHN)

We have characterized the phase behavior of mixtures of the cationic surfactant cetyltrimethylammonium bromide (CTAB) and the organic salt 3-sodium-2-hydroxy naphthoate (SHN) over a wide range of surfactant concentrations using polarizing optical microscopy and X-ray diffraction. A variety of liquid crystalline phases, such as hexagonal, lamellar with and without curvature defects, and nematic, are observed in these mixtures. At high temperatures the curvature defects in the lamellar phase are annealed gradually on decreasing the water content. However, at lower temperatures these two lamellar structures are separated by an intermediate phase, where the bilayer defects appear to order into a lattice. The ternary phase diagram shows a high degree of symmetry about the line corresponding to equimolar CTAB/SHN composition, as in the case of mixtures of cationic and anionic surfactants.

Structural Ordering of Disordered Ligand-Binding Loops of Biotin Protein Ligase into Active Conformations as a Consequence of Dehydration

Mycobacterium tuberculosis (Mtb), a dreaded pathogen, has a unique cell envelope composed of high fatty acid content that plays a crucial role in its pathogenesis. Acetyl Coenzyme A Carboxylase (ACC), an important enzyme that catalyzes the first reaction of fatty acid biosynthesis, is biotinylated by biotin acetyl-CoA carboxylase ligase (BirA). The ligand-binding loops in all known apo BirAs to date are disordered and attain an ordered structure only after undergoing a conformational change upon ligand-binding. Here, we report that dehydration of Mtb-BirA crystals traps both the apo and active conformations in its asymmetric unit, and for the first time provides structural evidence of such transformation. Recombinant Mtb-BirA was crystallized at room temperature, and diffraction data was collected at 295 K as well as at 120 K. Transfer of crystals to paraffin and paratone-N oil (cryoprotectants) prior to flash-freezing induced lattice shrinkage and enhancement in the resolution of the X-ray diffraction data. Intriguingly, the crystal lattice rearrangement due to shrinkage in the dehydrated Mtb-BirA crystals ensued structural order of otherwise flexible ligand-binding loops L4 and L8 in apo BirA. In addition, crystal dehydration resulted in a shift of similar to 3.5 angstrom in the flexible loop L6, a proline-rich loop unique to Mtb complex as well as around the L11 region. The shift in loop L11 in the C-terminal domain on dehydration emulates the action responsible for the complex formation with its protein ligand biotin carboxyl carrier protein (BCCP) domain of ACCA3. This is contrary to the involvement of loop L14 observed in Pyrococcus horikoshii BirA-BCCP complex. Another interesting feature that emerges from this dehydrated structure is that the two subunits A and B, though related by a noncrystallographic twofold symmetry, assemble into an asymmetric dimer representing the ligand-bound and ligand-free states of the protein, respectively. In-depth analyses of the sequence and the structure also provide answers to the reported lower affinities of Mtb-BirA toward ATP and biotin substrates. This dehydrated crystal structure not only provides key leads to the understanding of the structure/function relationships in the protein in the absence of any ligand-bound structure, but also demonstrates the merit of dehydration of crystals as an inimitable technique to have a glance at proteins in action.

Study of molecular reorientation and quantum rotational tunneling in tetramethylammonium selenate by H-1 NMR

H-1 NMR spin-lattice relaxation time measurements have been carried out in [(CH3)(4)N](2)SeO4 in the temperature range 389-6.6K to understand the possible phase transitions, internal motions and quantum rotational tunneling. A broad T, minimum observed around 280K is attributed to the simultaneous motions of CH3 and (CH3)(4)N groups. Magnetization recovery is found to be stretched exponential below 72 K with varying stretched exponent. Low-temperature T-1 behavior is interpreted in terms of methyl groups undergoing quantum rotational tunneling. (c) 2007 Elsevier Inc. All rights reserved.

Sound Velocity Anomaly at the Mott Transition: Application to Organic Conductors and V2O3

Close to the Mott transition, lattice degrees of freedom react to the softening of electron degrees of freedom. This results in a change of lattice spacing, a diverging compressibility, and a critical anomaly of the sound velocity. These effects are investigated within a simple model, in the framework of dynamical mean-field theory. The results compare favorably to recent experiments on the layered organic-conductor kappa-(BEDT-TTF)(2)Cu[N(CN)(2)]Cl. We predict that effects of a similar magnitude are expected for V2O3, despite the much larger value of the elastic modulus of this material.

The vibration spectrum of rutile

The long-wave lattice dynamics of rutile has been studied using a rigid ion model. The vibration frequencies for the zero wavevector have been calculated using the expressions for the frequencies of the normal modes derived group theoretically. The observed Raman and infrared frequencies have been explained.

Localized modes due to impurities in caesium iodide

The nature of the localized modes due to single substitutional impurities in a caesium iodide lattice is investigated using the models of Elliott, Dawber and Maradudin. The infra-red absorption due to U centres and their relation to the lattice spacing is also discussed.

Molecular model for U centres in caesium iodide

A molecular model has been developed to study the vibrations of U centres in caesium iodide. Employing the rigid ion model with nearest-neighbour short-range forces, the dynamical matrix of order 27 × 27 was solved to obtain the frequencies of the localized modes and the perturbed lattice modes. The results are compared with those obtained from the Green function method.

Molecular model for localised modes in zincblende structures

A molecular model for substitutional defects in a zincblende lattice has been worked out. The infrared absorption due to A1 in InSb and Li in GaAs are interpreted on the basis of this model.

Raman and infra-red spectra of crystalline ammonium sulphamate

Raman spectrum of a single crystal of ammonium sulphamate has been recorded for the two different orientations using λ 2537 resonance radiation of the mercury as the exciter. Thirty-four Raman lines have been observed of which eight belong to the lattice oscillations. Weak hydrogen bonding of NH2 group in the crystal was predicted. The infra-red absorption spectrum of the substance was taken in the powder form in potassium bromide disc, using Carl Zeiss UR10 IR spectrometer. Thirty-five absorption maxima could be identified.

Raman and infra-red spectra of crystalline potassium sulphamate

Raman spectrum of a single crystal of potassium sulphamate has been recorded for the first time using λ 2536 radiation of mercury as the exciter. Thirty-three Raman lines have been observed of which nine belong to the lattice oscillations. The infra-red absorption spectrum of the substance was taken in the powder form in potassium bromide disc using Carl Zeiss UR 10 IR spectrometer. Thirty-six absorption maxima could be identified of which twenty-five have been recorded for the first time. The analysis clearly shows that the N-H bond in the crystalline potassium sulphamate is not hydrogen-bonded to any appreciable extent.

Raman spectra of alkali halides

In this article, we present a comparative study of the Raman spectra of alkali halides in relation to the lattice dynamics ofBorn andRaman. It is shown that the experimentally observed limit of the second-order spectra in almost all the cases can be explained well by the Lyddane-Sachs-Teller relation. It is also seen, while, an explanation of the second-order Raman spectrum of a crystal of diamond or zinc blende structure requires the frequencies from the critical points,W, Gamma, X andL inBorn's analysis, the frequencies fromGamma, X andL alone are sufficient and necessary for an interpretation of the same onRaman's model. Some similarities in the determination of the long wave properties of crystals like elastic constants and limiting frequencies of the lattice vibrations in the symmetry directions in both the models are pointed out.

Raman spectrum of crystalline aminosulphonic acid

The Raman spectrum of a single crystal of sulphamic acid has been recorded withλ 2537 excitation. Thirty-eight lines have been observed, of which twenty-nine have been recorded for the first time. Seven Raman lines with shifts in the region 50–155 cm.−1 have been assigned to the lattice oscillations, two at 177 and 240 cm.−1 have been attributed to the low-frequency hydrogen bond vibrations.. The splitting of the degenerate modes and the appearance of N-H....O bonded stretching vibrations are consistent with the structural data which expect the presence of the free molecule as a Zwitter ion with only slight distortion from C3v symmetry.

Raman spectrum of crystalline cadmium acetate dihydrate

The Raman spectrum of a single crystal of cadmium acetate dihydrate has been recorded for the first time using λ 2537 excitation. Twenty-three lines have been observed out of which ten have been attributed to the internal oscillations of the acetate ion, nine to the lattice modes, two to low-frequency hydrogen bond vibrations. A line at 308 cm.−1 and the continuum 3250–3560 cm.−1 have been assigned to the Cd-O6 and internal vibrations of the water molecules.

Raman spectra of ferro-electric crystals Part IV. Lithium hydrazinium sulphate (LiN2H5SO4)

The Raman spectrum of lithium hydrazinium sulphate has been recorded both in the single crystal form and in aqueous solutions. The crystal exhibits thirty-eight Raman lines having the frequency shifts 52, 70, 104, 146, 174, 220, 260, 302, 350, 454, 470, 610, 630, 715, 977, 1094, 1115, 1132, 1177, 1191, 1260, 1444, 1493, 1577, 1630, 1670, 2205, 2484, 2553, 2655, 2734, 2848, 2894, 2939, 3028, 3132, 3290 and 3330 cm.−1 The aqueous solution gave rise to six Raman lines at 452, 980, 1050–1200, 1260, 1425 and 1570 cm.−1 apart from a maximum at 180 cm.−1 in the ‘wing’ accompanying the Rayleigh line. The observed Raman lines have been assigned as arising from the vibrations of the SO4 ion, N2H5+ ion, Li-O4 group, hydrogen bond and the lattice. The influence of the hydrogen bond on the N-H stretching vibrations has been pointed out. The various features of the observed spectrum strongly support the hypothesis that the NH3 group in the crystal is rotating around the N-N axis at room temperature.