A new algebraic approach to the eigenvalue problems of linear differential operators without integrations

In this work, a new approximation scheme based on the evaluation of the pointwise expectation of the Hamiltonian (H) via a conveniently chosen basis set is proposed. This scheme does not necessitate integration; however, physical and mathematical considerations in choosing the basis set are considerably important when very precise and rapidly convergent results are desired. In this method, the best linear combination of “well-selected” basis functions are sought in a way such that H ψ / ψ is flat in the neighborhood of a conveniently chosen point in the domain of H. This yields an algebraic eigenvalue problem. Some concrete applications that have already been realized confirm the efficiency of this approach.     

Complex Calculation of Solution Pulsed Nuclear Reactor VIR-2M

Physics of Atomic Nuclei - The results of complex calculation of the solution pulsed nuclear reactor (PNR) VIR-2M are presented. The simulation of the fuel solution dynamics is described, and...     

Intramolecular noncovalent interactions: Bis(toluene)chromium(0) conformers

The structure of trans- and cis-conformers of bis(toluene)chromium(0) and the intramolecular interactions in them are studied by means of MP2 and density functional theory along with topological and NBO analyses. It was concluded for the first time that the locations of two hydrogen atoms of each methyl group between the phenyl ring planes in the most stable cis-conformer of bis(toluene)chromium(0) are largely determined by the stabilizing interactions of methyl C-H bonds and their hydrogen atoms with the metal atom and chromium-carbon bonds. It was concluded from the obtained data that the C-H...Cr contacts in the studied conformers should be considered as hydrogen bonds rather than agostic interactions. Contrary to the existing conception of the quantum theory of atoms in molecules, repulsive interaction is shown to occur between the pairs of hydrogen atoms of the two methyl groups in bis(toluene)chromium(0) cis-conformers rather than the stabilizing hydrogen-hydrogen interactions.     

Numerical solution of ordinary differential equations by Fluctuationlessness theorem

This paper presents a numerical method based on Fluctuationlessness Theorem for the solution of Ordinary Differential Equations over appropriately defined Hilbert Spaces. We focus on the linear differential equations in this work. The approximated solution is written in the form of an nth degree polynomial of the independent variable. The unknown coefficients are obtained by setting up a system of linear equations which satisfy the initial or boundary conditions and the differential equation at the grid points, which are constructed as the independent variable’s matrix representation restricted to an n dimensional subspace of the Hilbert Space. An error comparison of the numerical solution and the MacLaurin series with the analytical solution is performed. The results show that the numerical solution obtained here converges to the analytical solution without using too many mesh points.     

Change in the crystal structure of germanium-containing lithiophosphate upon heating

The crystal structure of lithiophosphate with phosphorus atoms partially replaced by germanium, Li_3.17(P_0.69Ge_0.24Mo_0.07)O₄, at temperatures of 25, 150, 300, 450, and 600°C has been refined by the Rietveld method using powder data. New additional Bragg reflections are observed at T = 600°C, which indicate a change in the crystal structure of this compound.     

Cluster self-organization of Li- and TR-Containing silicate systems: Suprapolyhedral precursors and self-assembly of LiY<Superscript>[7]</Superscript>SiO<Subscript>4</Subscript> and LiTR<Superscript>[6]</Superscript>SiO<Subscript>4</Subscript> (TR = Y,Sc, Tm) dimorphic crystal structures

The combinatorial and topologic analysis (the TOPOS 4.0 program package, the coordination sequences method) is carried out for silicates LiY^[7]SiO₄ (a synthetic phase (SIN), space group P2₁/b) and LiTR^[6]SiO₄ (TR = Y, Sc, Tm; the olivine structure type (OLI), space group Pnma). The framework of these silicates is built of polyhedral structural units YO₇ + SiO₄ and YO₆(ScO₆, TmO₆) + SiO₄, respectively. The framework structures TRTO₄, built of Y,O- and Si,O-polyhedra in SIN and OLI, respectively, are represented as three-dimensional (3D) Y,Si-nets with distal oxygen atoms. The crystal-forming 2D Y,Si-net is the four-nodal net Y(1) (43333) + Y(2) (4334334) + Si(1) (434) + Si(2) (3343) for SIN and the two-nodal net Y 443333 + Si 4433 for OLI. In the 2D Y,Si, nets, equivalent chains of four-nodal precursor clusters Y-Si-Y-Si with three shared bonds are identified. In the dimorphic crystals structures of LiTRTO₄, polyhedral precursor clusters Li₂TR₂T₂ are linked into a ring with two Li atoms lying one above the other below the center of the ring, thus retaining the center of symmetry\(\bar 1\). Precursor clusters Li₂TR₂T₂ through the matrix self-assembly mechanism control the evolution of high-level crystal-forming clusters. Cluster coordination numbers in layers are equal to six.     

Adsorption of uranium ions by crosslinked polyester resin functionalized with acrylic acid from aqueous solutions

In this paper, the crosslinked polyester resin containing acrylic acid functional groups was used for the adsorption of uranium ions from aqueous solutions. For this purpose, the crosslinked polyester resin of unsaturated polyester in styrene monomer (Polipol 353, Poliya) and acrylic acid as weight percentage at 80 and 20%, respectively was synthesized by using methyl ethyl ketone peroxide (MEK_p, Butanox M60, Azo Nobel)-cobalt octoate initiator system. The adsorption of uranium ions on the sample (0.05 g copolymer and 5 mL of U(VI) solution were mixed) of the crosslinked polyester resin functionalized with acrylic acid was carried out in a batch reactor. The effects of adsorption parameters of the contact time, temperature, pH of solution and initial uranium(VI) concentration for U(VI) adsorption on the crosslinked polyester resin functionalized with acrylic acid were investigated. The adsorption data obtained from experimental results depending on the initial U(VI) concentration were analyzed by the Freundlich, Langmuir and Dubinin–Radushkevich (D–R) adsorption isotherms. The adsorption capacity and free energy change were determined by using D–R isotherm. The obtained experimental adsorption data depending on temperature were evaluated to calculate the thermodynamic parameters of enthalpy (ΔH°), entropy (ΔS°) and free energy change (ΔG°) for the U(VI) adsorption on the crosslinked polyester resin functionalized with acrylic acid from aqueous solutions. The obtained adsorption data depending on contact time were analyzed by using adsorption models such as the modified Freundlich, Elovich, pseudo-first order and pseudo-second-order kinetic models.     

Characteristic Regimes of Transitional Separation Bubbles in Unsteady Flow

This experimental investigation deals with transition phenomena of a separated boundary layer under unsteady inlet flow conditions. The main purpose of this investigation is to understand the influence of the rotor-stator interaction in turbomachinery on the subsequent, highly loaded boundary layer. The research project is divided into two phases. In the first phase, which has been completed recently, only the variation of mean velocity caused by upstream blades was simulated in the experiments while the free-stream turbulence intensity was retained at a constant low level. The experiments are carried out in an Eifel-type wind tunnel to investigate the laminar separated boundary layer of a flat plate under oscillating inlet conditions. The adverse pressure gradient, similar to that of turbomachines, is generated by the contoured upper wall. The unsteadiness is produced by a rotating flap located downstream of the test section. The reduced frequency, the amplitude and the mean Reynolds number are varied to simulate the conditions prevailing in turbomachines. In addition to the Kelvin–Helmholtz instability of the separated shear layer, a lower frequency instability was observed. This is frequently referred to as `free shear layer flapping' and results in two distinctly different ways of re-attachment, depending primarily on the Reynolds number. For low momentum thickness Reynolds numbers at the separation point, large-scale vortices locked to the frequency of the unsteady main flow are identified. They originate nearly at the top of the separation bubble and are ejected downstream. A fully turbulent boundary layer develops after these vortices mix out. For higher Reynolds numbers, transition is completed within a short length of the free shear layer and there-attachment region. The characteristic momentum thickness Reynolds number separating these two regimes in unsteady flow is about 125. The Strouhal number (reduced frequency) does not appear to have any significant effect. Based on the experimental results, this behaviour is discussed in some detail. This revised version was published online in July 2006 with corrections to the Cover Date.