Vol 51, No 2 (2017)

Under some additional restrictions we find dimensions and bases of moduli algebras of isolated singularities of polynomials in n variables that are sums of n monomials of equal weighted degrees and one monomial of lower degree.

In the present paper we consider bounded linear operators which have orbits dense relative to nontrivial subspaces. We give nontrivial examples of such operators and establish many of their basic properties. An example of an operator which has an orbit dense relative to a certain subspace but is not subspace-hypercyclic for this subspace is given. This, in turn, provides a new answer to a question posed in [18]. Other hypercyclic-like properties of such operators are also considered.

Let H₁, H₂, and H₃ be complex separable Hilbert spaces. Given A ∈ B(H₁), B ∈ B(H₂), and C ∈ B(H₃), write \({M_{D,E,F}} = \left( {\begin{array}{*{20}{c}} A&D&E \\ 0&B&F \\ 0&0&C \end{array}} \right)\), where D ∈ B(H₂,H₁), E ∈ B(H₃,H₁), and F ∈ B(H₃,H₂) are unknown operators. This paper gives a complete description of the intersection ∩_D,E,Fσ(M_D,E,F), where D, E, and F range over the respective sets of bounded linear operators. Further, we show that σ(A) ∪ σ(B) ∪ σ(C) = σ(M_D,E,F) ∪ W, where W is the union of certain gaps in σ(M_D,E,F), which are subsets of (σ(A) ∩ σ(B)) ∪ (σ(B) ∩ σ(C)) ∪ (σ(A) ∩ σ(C)). Finally, we obtain a necessary and sufficient condition for the relation σ(M_D,E,F) = σ(A)∪σ(B)∪σ(C) to hold for any D, E, and F.

We present an example of a C¹ Anosov diffeomorphism with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure.

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R^d of the form A^ε = −divA(x, x/ε)∇. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A^ε − μ)⁻¹, including one with a corrector, and for (−Δ)^s/2(A^ε − μ)⁻¹ in the operator norm on L₂(R^d)ⁿ. For s ≠ 0, we also give estimates of the rates of approximation.