Finding the unknown boundary in the initial boundary-value problem for the heat equation

The article considers the determination of the boundary of a two-dimensional region in which an initial boundary-value problem for the heat equation is defined, given the solution of the problem for all time instants at some points of the region. The direct problem is reduced to an integral equation, and numerical solutions of the inverse problem are obtained for the case when the boundary is an ellipse. We investigate the sensitivity of the observed variables to the location (relative to the boundary) of the point where the right-hand side of the equation is specified. Translated from Prikladnaya Matematika i Informatika, No. 30, 2008, pp. 18–24.     

Synthesis of lactide from lactic acid and its esters in the presence of rare-earth compounds

A procedure is described for the synthesis of lactide by dehydration of L-lactic acid and subsequent depolymerization of its oligomer mixture in the presence of yttrium(III) and praseodymium(III) oxides, as well as of cerium(III) chloride heptahydrate. The catalytic activity of yttrium and praseodymium sesquioxides was determined at different temperatures at the oligomerization and deoligomerization stages. Ethyl lactate was prepared in the presence of Purolite C100 MB cation exchange resin and subjected to oligomerization followed by thermal decomposition of oligoester and oligolactic acid mixture in the presence of yttrium(III) and praseodymium(III) oxides and aqueous cerium(III) chloride.     

On the Caccetta–Häggkvist Conjecture with Forbidden Subgraphs

The Caccetta–Häggkvist conjecture developed in 1978 asserts that every oriented graph on n vertices without oriented cycles of length must contain a vertex of outdegree at most . It has a rather elaborate set of (conjectured) extremal configurations. In this paper, we consider the case that received quite a significant attention in the literature. We identify three oriented graphs on four vertices each that are missing as an induced subgraph in all known extremal examples and prove the Caccetta–Häggkvist conjecture for oriented graphs missing as induced subgraphs any of these oriented graphs, along with . Using a standard method, we can also lift the restriction of being induced, though this makes graphs in our list slightly more complicated.     

On Turán’s (3,4)-problem with forbidden subgraphs

We identify three 3-graphs on five vertices that are missing in all known extremal configurations for Turán’s (3,4)-problem and prove Turán’s conjecture for 3-graphs that are additionally known not to contain any induced copies of these 3-graphs. Our argument is based on an (apparently) new technique of “indirect interpretation” that allows us to retrieve additional structure from hypothetical counterexamples to Turán’s conjecture, but in rather loose and limited sense. We also include two miscellaneous calculations in flag algebras that prove similar results about some other additional forbidden subgraphs.     

Almost Euclidean subspaces of ℓ 1 N VIA expander codes

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X⊆ℝ ^N of dimension (1−o(1))N such that for every x∈X,

Determining the boundary of a two-dimensional region from the solution of the external initial boundary-value problem for the heat equation

We determine the boundary of a two-dimensional region using the solution of the external initial boundary-value problem for the nonhomogeneous heat equation. The initial values for the boundary determination include the right-hand side of the equation and the solution of the initial boundary-value problem given for finitely many points outside the region. The inverse problem is reduced to solving a system of two integral equations nonlinear in the function defining the sought boundary. An iterative procedure is proposed for numerical solution of the problem involving linearization of integral equations. The efficiency of the proposed procedure is investigated by a computer experiment.     

One-Electron Reduction of 2-Mono(2,6-diisopropylphenylimino)acenaphthene-1-one (dpp-mian)

The electrochemical characteristics of 2-mono(2,6-diisopropylphenylimino)acenaphthene-1-one (dpp-mian) have been investigated. One-electron reduction of dpp-mian involves the iminoketone fragment, which is revealed by the EPR spectrum obtained after the electrolysis of the dpp-mian solution in tetrahydrofuran (THF). The reduction of dpp-mian with one equivalent of metallic potassium leads to a similar EPR spectrum. The sodium complex [(dpp-mian)Na(dme)]₂ ( 1 ) produces an EPR signal with hyperfine coupling on the nitrogen atom of the iminoketone fragment of the dpp-mian ligand. Dpp-mian can also be reduced in a one-electron process by SnCl₂×(dioxane). In this case, complex (dpp-mian)₂SnCl₂ ( 2 ) is formed, with the tin atom displaying an oxidation state of +4. Tin(II) chloride dihydrate, SnCl₂×2(H₂O), also reduces dpp-mian, but the two ligands bound to tin in the product form a new carbon–carbon bond between the ketone moieties of the dpp-mian monoanions to form complex (bis-dpp-mian)HSnCl₃ ( 3 ). Metallic tin reduces dpp-mian to form the (bis-dpp-mian)₂Sn ( 4 ) species. Compounds 1 – 4 were characterized by X-ray diffraction.