Implicit functionals and eigenvalue problems

An approach to nonlinear, positively homogeneous eigenvalue problems based on the use of the spectral parameter as a functional of Euler type is suggested. It allows one to present the spectral-parameter domain as a bifurcation diagram of the problem. Fučik spectrum problems (classical and for p-Laplacians) and the problem with a nonlinear dependence of the weight function on the spectral parameter are considered. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 169–186, 2005.     

Eigenvalue Problem for the Laplace Operator with Nonlocal Boundary Conditions

The suggested approach to maximizing the difference between the first and second eigenvalues of the Laplace operator is based on the introduction of nonlocal boundary conditions of a special form. It is shown that the difference can be arbitrarily large.     

Raman Spectroscopic Study of Aluminum Silicate Complexes at 20°C in Basic Solutions

Raman spectroscopic measurements were performed at ambient temperature onaqueous silica-bearing solutions (0.005 < m _Si < 0.02; 0 < pH < 14). The spectraare consistent with the formation of monomeric Si(OH)^o ₄, SiO(OH)^– ₃ andSiO₂(OH)^2– ₂ species at acid to neutral, basic, and strongly basic pH, respectively.Raman spectra of aqueous Al-bearing solutions at basic pH confirm thepredominance of the Al(OH)^– ₄ species in a wide concentration range (0.01 < m _Al < 0.1).Raman spectra of basic solutions (12.4 < pH < 14.3), containing both Al andSi, exhibit a strong decrease in intensities of SiO(OH)^– ₃, SiO₂(OH)^2– ₂, andAl(OH)^– ₄ bands in comparison with Al-free Si-bearing and Si-free Al-bearingsolutions of the same metal concentration and pH, suggesting the formation ofsoluble Al—Si complexes. The amounts of complexed Al and Si derived fromthe measurements of the Al and Si band intensities in strongly basic solutions(pH 14) are consistent with the formation, between Al(OH)^– ₄ andSiO₂(OH)^2– ₂, of the single Al—Si dimer SiAlO₃(OH)^3– ₄ according to the reactionSiO₂(OH)^2– ₂ + Al(OH)^– ₄ SiAlO₃(OH)^3– ₄ + H₂OAt lower pH ( 12.5) the changes in band intensities are consistent with theformation of several, likely more polymerized, Al—Si complexes.     

Gallium(III) adsorption on carbonates and oxides: X-ray absorption fine structure spectroscopy study and surface complexation modeling

Adsorption of Ga on calcite, magnesite, amorphous silica, and manganese oxide as a function of pH and gallium concentration in solution was studied using a batch adsorption technique. Adsorbed complexes of Ga on calcite, magnesite, and delta-MnO2 were further characterized using XAFS spectroscopy. At high surface loadings from supersaturated solutions, Ga is likely to form a polymeric network at the surface (edge- and corner-sharing octahedra). At low surface loadings, Ga presents as isolated octahedra, probably attached to the Me-O sites on the surface, and coordinated by water molecules and hydroxide groups at 1.90-1.94 A. At pH>6, Ga therefore changes its coordination from 4 to 6 when adsorbing from solution (Ga(OH)(-)4(aq)) onto metal surface sites (MeOGa(OH)n(H2O)2-n(5-n), Me = Ca, Mg, or Mn, and n=1 and 2 for carbonate minerals and MnO2, respectively). Because the EXAFS is not capable of seeing hydrogen atoms, the protonation of surface complexes was determined by fitting the experimental pH-dependent Ga adsorption edge. A surface complexation model which assumes the constant capacitance of the electric double layer (CCM) and postulates the formation of positively charged, neutral and negatively charged surface complexes for carbonates, manganese oxide and silica, respectively, was used to describe the dependence of adsorption equilibria on aqueous solution composition in a wide range of pH and Ga concentration.     

Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator

Let be the spectrum of in L ²(ℝ), where q is an even almost-periodic complex-valued function with bounded primitive and derivative. It is known that , where is the spectrum of the unperturbed operator. Suppose that the asymptotic approximation to the first asymptotic correction is given. We prove the formula that recovers the frequencies and the Fourier coefficients of q in terms of Δμ _n .