Search

Let be a simple graph of order . A set is a perfect dominating set of a graph if every vertex is adjacent to exactly one vertex in . That is, every vertex outside has exactly one neighbor in . Every graph has at least the trivial perfect dominating sets consisting of all vertices in . The perfect domination number is the minimal cardinality of dominating sets in . Let be the family of perfect dominating sets for a graph with cardinality and . The perfect domination polynomial of a graph of order is where is the number of perfect dominating sets of of size . In this paper, we studied the perfect domination polynomial of homogeneous caterpillar graphs and of full binary trees. [ABSTRACT FROM AUTHOR]

Regular multilinear operators and regular homogeneous polynomials acting between Banach lattices are automatically continuous, but the converse, in general, is not true. The problem arises of characterizing Banach lattices for which the classes of continuous and regular multilinear operators (or homogeneous polynomials) coincide. The aim of this note is to extend two results in this direction, earlier obtained for linear operators, to the above-mentioned classes of operators and polynomials. The main method is linearization with the use of the Fremlin tensor product of Banach lattices. [ABSTRACT FROM AUTHOR]

We introduce the class of operators of almost Hadamard type, that is, linear continuous operators that act on a locally convex space containing all polynomials and have the property that the homogeneous polynomials of any given degree form an invariant subspace. The Hadamard-type (diagonal) operators, for which each monomial is an eigenvector, are a special case of operators of almost Hadamard type. The operators of almost Hadamard type are studied in the space of all entire functions of several complex variables. The results are used to describe all linear operators continuous in this space and commuting in it with the multidimensional analog of the Hardy–Littlewood operator. [ABSTRACT FROM AUTHOR]

Unitary transformations and canonical representatives of a family of real-valued harmonic fourth-degree polynomials in three complex variables are studied. The subject relates to the study of Moser normal equations for real hypersurfaces of four-dimensional complex spaces and isotropy groups (holomorphic stabilizers) of such surfaces. The dimension of the stabilizer for a particular strictly pseudo-convex hypersurface is estimated from above by the dimension of a unitary subgroup preserving the fourth-degree polynomial from its normal equation. [ABSTRACT FROM AUTHOR]

The exact value of the convergence exponent of the special integral in the two-dimensional Tarry problem with a homogeneous polynomial of second degree in the exponent of the imaginary exponential is obtained. The result is based on a representation of the trigonometric integral as a Fourier transform. [ABSTRACT FROM AUTHOR]

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δ u + q( x) u = 0, where either q( x) ≠ const or q( x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established. [ABSTRACT FROM AUTHOR]

In the paper, a classification of aspherical compact homogeneous manifolds of dimension less than or equal to 7 is discussed. [ABSTRACT FROM AUTHOR]

The author's results concerning the null subspaces of arbitrary odd polynomials in several variables are generalized to the case of common null subspaces for several odd polynomials as well as to the complex case. [ABSTRACT FROM AUTHOR]

We obtain a generalization of the Aron-Hájek theorem about the null subspaces of homogeneous odd polynomials (in several variables) to the case of arbitrary odd polynomials. [ABSTRACT FROM AUTHOR]