Fredholm Properties of Band-dominated Operators on Periodic Discrete Structures

Let (X, ~) be a combinatorial graph the vertex set X of which is a discrete metric space. We suppose that a discrete group G acts freely on (X, ~) and that the fundamental domain with respect to the action of G contains only a finite set of points. A graph with these properties is called periodic with respect to the group G. We examine the Fredholm property and the essential spectrum of band-dominated operators acting on the spaces l ^p (X) or c_0(X), where (X, ~) is a periodic graph. Our approach is based on the thorough use of band-dominated operators. It generalizes the necessary and sufficient results obtained in [39] in the special case and in [42] in case X = G is a general finitely generated discrete group. Submitted: May 21, 2007. Revised: September 25, 2007. Accepted: November 5, 2007.     

Shape of the vapor—liquid interface in narrow slit-shaped pores

The shape of the meniscus (vapor—liquid interface) and the transient region in narrow slit-shaped pores of different width were studied. The adsorbate was modeled by spherical particles, the interaction between them being described by the Lennard-Jones potential model. The calculation was carried out on the basis of the lattice-gas model in the quasichemical approximation for accounting the intermolecular interactions between the nearest neighbors. The greatest meniscus curvature was found for narrow pores. As the pore width increases, the meniscus curvature decreases. The effect of the surface potential on the meniscus shape in narrow pores was discussed. Dedicated to the 90th anniversary of the L. Ya. Karpov Institute of Physical Chemistry. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1118–1129, June, 2008.     

Synthesis of fragment 8–12 of the natural sequence of ACTH

Two new variants of the synthesis of the pentapeptide 8–12 of the natural sequence of ACTH are presented. In some stages, the trimethylsilyl group was used as temporary protection of the carboxy group of the amino component. The final and intermediate compounds were obtained with good yields, were distinguished by chromatographic homogeneity, and were characterized by their angles of optical rotation, melting points, and electrophoretic behavior. Their purity was checked by TLC and by high-pressure liquid chromatography. Some physicochemical characteristics (angles of optical rotation, melting points, chromatographic mobilities) of the compounds obtained are given.     

Rheology of the gelation of fluorine-doped silica sols

Dynamic oscillatory shear measurements are used to probe the gelation kinetics of aqueous sols composed of either particulate silica or silicon alkoxide solution. Unlike steady shear measurements, these dynamic tests do not alter the structure and kinetics of the sol-gel process. The dynamic storage moduli of both systems show sharp transitions at the onset of gelation. However, the gelation kinetics of the two systems are very different; the modulus of alkoxide system remains unchanged until the gel point is reached, whereas that of the particulate system increases with time even below the gel point as the colloid cluster grow. Unlike the alkoxide gel, the particulate gel reverses to a sol upon shearing. The gelation kinetics of a resheared sample is slower than that of a fresh sample. The structures associated with the sol-gel transition for this particulate system have been monitored using freeze fracture microscopy and correlate with rheological observations. The gelation times for both sols are varied by changing the fluoride ion content.     

Molecular structure of ThBr4(DME)2 (DME = 1,2-dimethoxyethane)

The molecular structure of ThBr₄(DME)₂ (DME = 1,2-dimethoxyethane), the first structurally characterized thorium complex containing a simple bidentate ether ligand, is described. The eight-coordinate complex presents a distorted dodecahedral geometry, with Th—Br and Th—O bond lengths in the ranges 2.8516(13)–2.8712(13) Å and 2.564(8)–2.620(8) Å, respectively. ThBr₄(DME)₂ is monoclinic, space group P2₁/n, a = 7.672(1), b = 14.581(1), c = 15.847(2) Å, = 102.24(1)°, V = 1732.4(3) ų, and Z = 4.     

Synthesis of Numerical Methods for Pareto Set Approximation Based on a Universal Procedure

Computational Mathematics and Mathematical Physics - A universal computational procedure is used to construct numerical methods for Pareto set approximation. The numerical methods are developed...     

Essential spectrum and exponential estimates of eigenfunctions of lattice Schrödinger and Dirac operators

The main aim of the paper is the investigation of a relation between the essential spectrum and the exponential decay at infinity of eigenfunctions of the lattice analogs of Schrödinger and Dirac operators.     

Pseudodifferential Operators on Periodic Graphs

The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph Γ which is periodic with respect to the action of the group \mathbb Zⁿ{{\mathbb {Z}}^n} . The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class OPS ⁰ on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of Γ. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.