Lev Yu. Glebsky

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It has been long conjectured that the crossing number of Cm × Cn is (m−2)n, for allm,n such that n ≥ m ≥ 3. In this paper it is shown that if n ≥ m(m+1) andm ≥ 3, then this conjecture holds. That is, the crossing number of Cm × Cn is as conjectured for all but finitely many n, for each m. The proof is largely based on techniques from the theory of(More)
We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem(More)
We prove the existence of small localized stationary solutions for the generalized Swift-Hohenberg equation and find under some assumption a part of a boundary of their existence in the parameter plane. The related stationary equation creates a reversible Hamiltonian system with two degrees of freedom that undergoes the Hamiltonian-Hopf bifurcation with an(More)
We introduce and discuss a definition of approximation of a topological algebraic system A by finite algebraic systems of some class K. For the case of a discrete algebraic system this definition is equivalent to the well-known definition of a local embedding of an algebraic system A in a class K of algebraic systems. According to this definition A is(More)
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