Recovering variable order differential operators on star-type graphs from spectra

We consider inverse spectral problems for ordinary differential operators on compact star-type graphs for the case in which the differential equations have different orders on different edges. We study inverse problems of recovering potentials from a system of spectra. We provide algorithms for constructing solutions of these inverse problems and prove their uniqueness.     

Inverse problem for differential pencils on a hedgehog graph

We study boundary value problems on a hedgehog graph for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of spectral characteristics and consider the inverse spectral problem of reconstructing the coefficients of a differential pencil on the basis of spectral data. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing its solution.     

Molecular structure and barriers to internal rotation of α-naphthalenesulfonyl chloride: a study by gas-phase electron diffraction and quantum chemical calculations

α-Naphthalenesulfonyl chloride, α-NaphSC, was studied by gas-phase electron diffraction (GED) and quantum chemical calculations (HF/6-311 + G**, HF/aug-cc-pVDZ, B3LYP/cc-pVDZ, B3LYP/cc-pVTZ, B3LYP/aug-cc-pVDZ, B3LYP/aug-cc-pVTZ, MP2/cc-pVDZ, and MP2/cc-pVTZ). The calculations predict the existence of two conformers with C ₁ (I) and C _s (II) symmetries. The most stable conformer I has an enantiomer. The experimental data of α-NaphSC obtained at 370(5) K could be best fitted by a C ₁ symmetry model indicating that only this form exists in the gas-phase. In this model the C_α–S–Cl plane deviates from the perpendicular orientation relative to the plane of the naphthalene skeleton. Under the applied experimental conditions, the mole fraction of a second less stable conformer II of α-NaphSC predicted by calculations is no more than 1 %. The following geometrical parameters of conformer I were obtained from the experiment (Å and °; uncertainties are in parentheses): r _h1(C–H) = 1.082(6), r _h1(C–C)_cp = 1.407(3), r _h1(C–S) = 1.764(5), r _h1(S–O)_av = 1.425(3), r _h1(S–Cl) = 2.051(5), ∠C–C_α–C = 122.5(1), ∠C_α–S–Cl = 101.5(10); C9–C1–S–Cl = 71.4(21). The calculated barriers to internal rotation of the sulfonyl chloride group exceed considerably the thermal energy values corresponding to the temperatures of the GED experiments. Natural bond orbitals analysis of the electron density distribution was carried out to explain the peculiarities of the molecular structure of the studied compound and the deviation from the structures of β-NaphSHal molecules and their benzene analogs.     

Recovering differential pencils on compact graphs

We study boundary value problems on compact graphs without circles (i.e. on trees) for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering the coefficients of the differential equation from the so-called Weyl vector which is a generalization of the Weyl function (m-function) for the classical Sturm-Liouville operator. For this inverse problem we prove the uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mappings.     

Inverse problems on star-type graphs: differential operators of different orders on different edges

We study inverse spectral problems for ordinary differential equations on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.     

Statistical Analysis of Morphology of Low Density Polyethylene/Polyamide 6 Blends with Addition of Organoclay and Maleic Anhydride-Grafted Polystyrene-b-Poly(Ethylene-co-Butene-1)-b-Polystyrene Copolymer As Compatibilizers

Statistical analysis of the size distribution of a polymer minor phase droplets was successfully applied for the characterization of the morphology in a LDPE/PA6 blend (75/25 wt/wt) obtained after mixing with added SEBS-g-MA(S) compatibilizer and/or organoclay 20A. It was shown that the developed approach provided detailed analysis of the morphology development in the polymer blends, including the primary droplets formation of the minor polymer phase and their break-up and coalescence. The introduction of organoclay increased the break-up of primary droplets and completely suppressed their coalescence. The addition of compatibilizer S, in contrast to nanoclay, did not suppress coalescence but the mean size of the primary droplets as well as the droplets formed at coalescence was strongly reduced. The combined addition of compatibilizer S and nanoclay did not change the morphology development of the LDPE/PA6 blend. Both processes of the droplet transformation were accelerated, similar to the system with addition of compatibilizer S only. However, an increase of nanoclay amount disturbed the break-up of the primary droplets, and the mean size of these droplets increases. Thereafter, the mean size of droplets formed at coalescence also increased. The results of statistical analysis of the phase morphology were found to correlate with the mechanical properties of the polymer blends. The fine dispersion of the minor polymer phase improved the stiffness of the polymer blends. For enhanced impact properties, the presence of relatively broad distribution of the minor polymer phase was necessary.     

The correlations in a star molecule fluid. Integral equation theory and Monte Carlo study

The structure of a starlike molecule (SLM) fluid with four arms of different length is studied by applying the associative Percus–Yevick integral equation (IE) theory and canonical Monte Carlo (MC) simulations. In the IE study the SLM fluid is modelled by a fluid of hard spheres with four associative sites on each sphere while the MC has been performed for a freely-joined tangent hard sphere fluid. The total radial distribution functions have been calculated in both approaches for different volume fraction regimes and different arm lengths. It is shown that the associative IE theory predicts the structure of SLM fluid best for relatively long arms and at high densities. Additionally, the dependence of the SLM centre–centre correlations on the functionality and fluid particle density has been analysed using the MC results.     

Inverse spectral problems for differential pencils on the half-line with turning points

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.