Fixed Points of Mappings Satisfying a Weakly Contractive Type Condition

In this paper, we discuss a fixed point theorem for mappings derived by a pair of mappings satisfying weak(k, k/) contractive type condition on the tensor product spaces. Let X and Y be Banach spaces and T_1 : X γ Y → X and T_2: X γ Y → Y be two operators which satisfy weak(k, k/) contractive type condition. Using T_1 and T_2, we construct an operator T on X γ Y and show that T has a unique fixed point in a closed and bounded subset of X γ Y.We derive an iteration scheme converging to this unique fixed point of T. Conversely, using a weakly contractive mapping T, we construct a pair of mappings(T_1, T_2) satisfying weak(k, k/)contractive type condition on X γ Y and from this pair, we also obtain two self mappings S_1 and S_2 on X and Y respectively with unique fixed points.     

Distance signless Laplacian eigenvalues of graphs

Suppose that the vertex set of a graph G is V(G)=\{v\mathrm{1},v\mathrm{2},\mathrm{...},vn\}. The transmission Tr(vi) (or D_i) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let Tr(G) be the n\times n diagonal matrix with its (i, i)-entry equal to {{\displaystyle Tr}}_G(vi). The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as L(G)=Tr(G)+D(G), where D(G) is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.     

Nuclear multifragmentation: Basic concepts

We present a brief overview of nuclear multifragmentation reaction. Basic formalism of canonical thermodynamical model based on equilibrium statistical mechanics is described. This model is used to calculate basic observables of nuclear multifragmentation like mass distribution, fragment multiplicity, isotopic distribution and isoscaling. Extension of canonical thermodynamical model to a projectile fragmentation model is outlined. Application of the projectile fragmentation model for calculating average number of intermediate mass fragments and the average size of the largest cluster at different Z _bound, differential charge distribution and cross-section of neutron-rich nuclei of different projectile fragmentation reactions at different energies are described. Application of nuclear multifragmentation reaction in basic research as well as in other domains is outlined.     

On Berezin Number Inequalities for Operator Matrices

In this article, we refine certain earlier existing bounds for Berezin number of operator matrices. We also prove some new Berezin number inequalities for general n × n operator matrices.Further, we establish several upper bounds for Berezin number and generalized Euclidean Berezin number for off-diagonal operator matrices. Finally, some interesting examples are discussed.     

A Priori Error Estimates in H1 and H2 Norms for Galerkin Approximations to a Single-Phase Nonlinear Stefan Problem in One Space Dimension

Optimal error estimates in H₁ and H₂ norms for continuous-timeGalerkin approximations of a single-phase nonlinear Stefan problemin one space dimension are derived.     

Structure elucidation of two new bisbenzylisoquinoline alkaloids and NMR assignments of the alkaloids from the fruits of Tiliacora racemosa

Besides three known biphenyldibenzodioxinbisbenzylisoquinoline alkaloids, two novel alkaloids of the same group have been isolated from the fruits of Tiliacora racemosa. The structures of these new compounds designated tiliaimine and nordinklacorine were established on the basis of 1D and 2D NMR spectroscopy, including COSY, TOCSY, ROESY, HMBC and HSQC experiments. Preliminary antibacterial activity studies on the known compounds (tiliarine, 2′‐nortiliacorinine, 2′‐nortiliacorine) from the fruits of the plant showed strong activity against three bacterial strains viz. Escherichia coli strain VT3 (Verotoxigenic E.coli), Staphylococcus aureus ATCC 25923 and Bacillus subtilis ATCC 6623 but were inactive against the strains Vibrio cholerae O1 (NB2) and Pseudomonas aeruginosa ATCC 15442. Copyright © 2010 John Wiley & Sons, Ltd.     

Dynamics of OH radical generation in laser-induced photodissociation of tetrahydropyran at 193 nm

Tetrahydropyran (THP) undergoes photodissociation on excitation with ArF laser at 193 nm, generating OH radical as one of the transient photoproducts. Laser-induced fluorescence technique is used to detect the nascent OH radical and measure its energy state distribution. The OH radical is formed mostly in the ground vibrational level (v"=0), with low rotational excitation. The rotational distribution of OH (v"=0,J) is characterized by a temperature of 433+/-31 K, corresponding to a rotational energy of 0.86+/-0.06 kcalmol. Two Lambda-doublet levels, 2Pi+(A') and 2Pi-(A"), and the two spin-orbit states, the 2Pi(3/2) and 2Pi(1/2), of OH are populated statistically for all rotational levels. The relative translational energy associated with the photoproducts in the OH channel is calculated to be 21.9+/-3.2 kcal mol(-1), from the Doppler-broadened linewidth, giving an ft value of approximately 43%, and most of the remaining 57% of the available energy is distributed in the internal modes of the other photofragment, C5H9. The observed distribution of the available energy is explained well, using a hybrid model of energy partitioning, with an exit barrier of 40 kcal mol(-1). The potential-energy surface of the reaction channel was mapped by ab initio molecular-orbital calculations. Based on experimental and theoretical results, a mechanism for OH formation is proposed. Electronically excited THP relaxes to the ground electronic state, and from there, a sequence of reactions takes place, generating OH. The proposed mechanism first involves C-O bond scission, followed by a 1,3 H atom migration to O atom, and finally, the C-OH bond cleavage giving OH.     

Synthesis and crystal structure of bis(benzimidazolium) tetrahalocobaltate(II)

The crystal structures of benzimidazolium tetrahalocobaltates (HBz)₂[CoX₄] (X = Cl and Br) have been determined. The chloride salt is triclinic, P-1, with a = 7.670(3) Å, b = 8.307(3) Å, c = 15.730(2) Å, and = 87.37(3)°, = 84.99(3)°, = 67.72(2)°. The bromide salt is monoclinic, C2/c, with a = 15.568(2) Å, b = 8.063(3) Å, c = 5.762(2) Å and = 91.36(3)°. The structures of the two salts are closely related. Both the compounds contain isolated tetrahedral CoX₄ ^2– anions and benzimidazolium cations. In the chloride salt, three chloride ions are involved in strong hydrogen bonding while only two bromide ions participate in the bromide salt. The greater deviation from the ideal tetrahedral geometry in CoBr₄ ^2– can be related to the less extensive hydrogen-bonding network compared to the chloro complex.     

Kinetic studies on the periodate oxidation of an iron(II) oxime‐imine complex

The iron(II) complex of H₂L (H₂L=3, 14‐dimethyl‐4, 7, 10, 13‐tetraazahexadeca‐3,13‐diene‐2,15‐dione dioxime, Coord. Chem. Rev., 33, 87 (1980)) is oxidized by periodate very rapidly in the range pH 2.0–7.0, and the kinetics of the reaction has been followed by stopped‐flow spectrophotometry at 30°C and ionic strength I=0.20 mol L⁻¹ (NaClO₄). The reaction is found to follow a simple second‐order kinetics as −d/dt [Fe^II(H₂L)²⁺]=k [Fe^II(H₂L)²⁺] [I(VII)], giving [Fe^III(L)]⁺ and IO₃⁻ as the final products. The reaction has been proposed to occur through a H‐bonded transition state formed probably between the protonated oxime group of the ligand and the oxygen atom on the periodate species, followed by an electron transfer from Fe^II centre to I^VII in a rate‐determining step. The I^VI species thus generated reacts in a fast step with another Fe^II complex. © 1999 John Wiley & Sons, Inc. Int J Chem Kinet 31: 23–28, 1999     

Large deformation extensional rheology of bread dough

The stress–strain curves of bread dough were derived under uniaxial compression, uniaxial tension and equi-biaxial tension loading conditions. In uniaxial compression, a lubricant was used to eliminate frictional effects between the loading platens and the sample. In uniaxial tension, cylindrical samples with thin flat discs at both ends (‘I’ samples) were tested. The discs at both ends were allowed to air-dry and were subsequently glued onto the loading platens. In equi-biaxial tension, a thin disc of dough was inflated into a bubble using pressurised air. The thickness at the top of the bubble was measured by shining a light through the walls of the bubble and recording the change in light intensity as the wall becomes thinner. All methods ensured that uniform deformation was obtained. Stress and strain were accurately evaluated using image analysis techniques. The tests were performed at various strain rates and speeds that defined the time dependence of the material. A non-linear viscoelastic model based on the Prony series and Van der Waals hyperelasticity was used to predict all test data. The model had a total of five material parameters and two time constants, which were set to represent the actual time scales of the experiments. A reasonable agreement between the experimental data and the chosen material model was observed.