On the relation between torsion submodule and Fitting ideals

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a submodule of [Formula: see text] which consists of columns of a matrix [Formula: see text] with [Formula: see text] for all [Formula: see text], [Formula: see text], where [Formula: see text] is an index set. For every [Formula: see text], let I[Formula: see text] be the ideal generated by subdeterminants of size [Formula: see text] of the matrix [Formula: see text]. Let [Formula: see text]. In this paper, we obtain a constructive description of [Formula: see text] and we show that when [Formula: see text] is a local ring, [Formula: see text] is free of rank [Formula: see text] if and only if I[Formula: see text] is a principal regular ideal, for some [Formula: see text]. This improves a lemma of Lipman which asserts that, if [Formula: see text] is the [Formula: see text]th Fitting ideal of [Formula: see text] then [Formula: see text] is a regular principal ideal if and only if [Formula: see text] is finitely generated free and [Formula: see text] is free of rank [Formula: see text]

Constructive description of Hölder classes on some multidimensional compact sets

We give a constructive description of Hölder classes of functions on certain compacts in Rm (m > 3) in terms of a rate of approximation by harmonic functions in shrinking neighborhoods of these compacts. The considered compacts are a generalization to the higher dimensions of compacts that are subsets of a chord-arc curve in R3. The size of the neighborhood is directly related to the rate of approximation it shrinks when the approximation becomes more accurate. In addition to being harmonic in the neighborhood of the compact the approximation functions have a property that looks similar to Hölder condition. It consists in the fact that the difference in values at two points is estimated in terms of the size of the neighborhood, if the distance between these points is commensurate with the size of the neighborhood (and therefore it is estimated in terms of the distance between the points).

MAXIMAL INTERACTION OF BINARY FACTORS

In the formal model of the binary sufficient cause theory based on the theory of finite Boolean algebras a dependence of integer invariant from a number of joined acting factors is studied. A constructive description of such a minimal k for which attains a maximal value is given.

On the Construction of Substitutes

Gross substitutability is a central concept in economics and is connected to important notions in discrete convex analysis, number theory, and the analysis of greedy algorithms in computer science. Many different characterizations are known for this class, but providing a constructive description remains a major open problem. The construction problem asks how to construct all gross substitutes from a class of simpler functions using a set of operations. Because gross substitutes are a natural generalization of matroids to real-valued functions, matroid rank functions form a desirable such class of simpler functions. Shioura proved that a rich class of gross substitutes can be expressed as sums of matroid rank functions, but it is open whether all gross substitutes can be constructed this way. Our main result is a negative answer showing that some gross substitutes cannot be expressed as positive linear combinations of matroid rank functions. En route, we provide necessary and sufficient conditions for the sum to preserve substitutability, uncover a new operation preserving substitutability, and fully describe all substitutes with at most four items.