Homology of Digraphs

Mathematical Notes - A theory of singular cubic homology of digraphs is developed; the obtained homology groups are proved to be functorial and homotopy invariant. Commutative diagrams of exact...     

The π-π-theorem for manifold pairs with boundaries

The surgery obstruction of a normal map to a simple Poincaré pair (X, Y) lies in the relative surgery obstruction group L _*(π ₁(Y) → π ₁(X)). A well-known result of Wall, the so-called π-π-theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincaré pair with π ₁(X) ? π ₁(Y) is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced the surgery obstruction groups LP _* for manifold pairs and splitting obstruction groups LS _*. In the present paper, we formulate and prove for manifold pairs with boundary results similar to the π-π-theorem. We give direct geometric proofs, which are based on the original statements of Wall’s results and apply obtained results to investigate surgery on filtered manifolds.     

Invariant stabilization of pulse systems with a nonlinear right-hand side

The Lyapunov function of the quadratic form is used for the analytical synthesis of a robust stabilizing control for two classes of nonlinear pulse modulated systems described by functional differential equations with a lagging argument.     

Fundamental groupoids of digraphs and graphs

We introduce the notion of fundamental groupoid of a digraph and prove its basic properties. In particular, we obtain a product theorem and an analogue of the Van Kampen theorem. Considering the category of (undirected) graphs as the full subcategory of digraphs, we transfer the results to the category of graphs. As a corollary we obtain the corresponding results for the fundamental groups of digraphs and graphs. We give an application to graph coloring.     

Investigation of properties of polymeric composition materials around a heterogeneous matrix

The results of microstructure analysis of polymeric material around an epoxy oligomer cured by diaminodiphenylsulfone, as well as of thermoplastic polymers like polysulfone and polyarylene ester ketone, are presented. Calculation of the operating life of items produced from this material during multicycle loading is carried out.     

On iterated Browder-Livesay invariants

Mathematical Notes - The Browder-Livesay invariants provide obstructions to the realization of elements of Wall groups by normal maps of closed manifolds. A generalization of the iterated...     

New monomer based on thienopyrazine with fluorocarbazole substituents as a promising building block for organic electronics

A new monomer, 2,3-bis[6-fluoro-9-(2-octyldodecyl)-9H-carbazol-3-yl]-3,3'-[5,7-di(5-bromothienyl-2)thieno[3,4-b]pyrazine, M1, based on thienopyrazine containing fluorocarbazole substituents in the pyrazine ring has been synthesized. The structure of the compound has been proved by ¹H and ¹³C NMR and elemental analysis. The HOMO and LUMO energies for monomer M1 and its precursor 3 determined by the electrochemical method are–5.03 and–3.31 eV, as well as–5.28 and–3.36 eV, respectively. Band gap widths E _g ^ec are 1.72 and 1.92 eV for compounds M1 and 3, respectively. The new structural fragment has rather deep energy levels of frontier molecular orbitals and a small band gap width; therefore, it is a promising building block for the synthesis of polymers for organic electronics.     

K-groups of quadratic extensions of rings

We construct the Levin braid connecting the Tate cohomology ofK-groups for quadratic extensions of rings with antistructures. Explicit formulas for isomorphisms of relative cohomology groups for the induced mapping and for the transfer mapping are obtained; these formulas are necessary in the construction of the Levin braid.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 272–280, August, 1995.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-1402.     

Alexandroff Homology and Path Homology

Mathematical Notes -     

Stability of pulse-modulated systems with a monotone equivalent nonlinearity in the modulator

A pulse-modulated system described by a functional differential equation is considered. Sufficient frequency conditions for global stability are obtained by using the averaging method and a frequency theorem.