A dominating set S in a graph G is said to be perfect if every vertex of G not in S is adjacent to just one vertex of S. Given a vertex subset S of a side Pm of an m×n grid graph G, the perfect dominating sets S in G with S = S ∩ V (Pm) can be determined via an exhaustive algorithm Θ of running time O(2m+n). Extending Θ to infinite grid graphs of width m − 1, periodicity makes the binary decisi...

We consider the setting of linear regression in high dimension. We focus on the problem of constructing adaptive and honest confidence sets for the sparse parameter θ, i.e. we want to construct a confidence set for θ that contains θ with high probability, and that is as small as possible. The l2 diameter of a such confidence set should depend on the sparsity S of θ the larger S, the wider the c...

This paper considers the problem of inference for partially identified econometric models. The class of models studied are defined by a population objective function Q(θ, P ) for θ ∈ Θ. The second argument indicates the dependence of the objective function on P , the distribution of the observed data. Unlike the classical extremum estimation framework, it is not assumed that Q(θ, P ) has a uniq...

We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F = F (x1, ..., xn) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X = {x1, ..., xn}, EndF be the semigroup of endomorphisms of F , and Aut EndF be the group of automorphisms of the semigroup EndF ...

The macroscopic Hausdorff dimension DimH(E) of a set E⊂Rd was introduced by Barlow and Taylor to quantify “fractal at large scales” behavior unbounded, possibly discrete, sets E. We develop method based on potential theory in order estimate this Rd. Then, we apply obtain Marstrand-like projection theorems: given E⊂R2, for almost every θ∈[0,2π], the E straight line passing through 0 with angle θ...

Let mIO(X) be the family of ⋆-open (resp. α-I-open, pre-I-open, semi-I-open, β-I-open, etc.) sets in an ideal topological space (X, τ, I). By using mIO(X), we introduce and investigate notions m-I-continuous multifunction F : I) → (Y, σ) mi⋆ -continuous σ, J). The notion -continuity is a generalization m-I-continuity i⋆-continuity [9].