When does the Riesz representation theorem hold?

The Riesz representation theorem

When Does the Subadditivity Theorem for Multiplier Ideals Hold?

Demailly, Ein and Lazarsfeld [DEL] proved the subadditivity theorem for multiplier ideals, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals, on non-singular varieties. We prove that, in two-dimensional case, the subadditivity theorem holds on log-terminal singularities. However, in higher dimensional case, we have several...

When Does Schwartz Conjecture Hold?

In 1990, Thomas Schwartz proposed the conjecture that every nonempty tournament has a unique minimal τ -retentive set (τ stands for tournament equilibrium set). A weak variant of Schwartz’s Conjecture was recently proposed by Felix Brandt. However, both conjectures were disproved very recently by two counterexamples. In this paper, we prove sufficient conditions for infinite classes of tourname...

When the extension property does not hold

A complete extension theorem for linear codes over a module alphabet and the symmetrized weight composition is proved. It is shown that an extension property with respect to arbitrary weight function does not hold for module alphabets with a noncyclic socle.

When Does the Zero - One Law Hold ?

In 1960 Paul Erdos and Alfred Renyi [ER] began the subject of random graphs. The 1985 book of Bela Bollobas [B) provides the standard reference for this field. In modern terminology the random graph G(n, p) is a graph on vertex set [n] = {I, ... , n} where each pair i, j of vertices are adjacent with independent probability p. More accurately, G(n, p) is a probability space over the space of gr...