A dynamical Borel–Cantelli lemma via improvements to Dirichlet’s theorem

The Hairy Ball Theorem via Sperner's Lemma

It is impossible to comb all the hairs of a fuzzy ball so that: i) each hair lies tangent to the surface of the ball, and ii) the angles of the hairs vary continuously over the surface of the ball. (By this we mean that the angle between two hairs at positions p and q say can be made arbitrarily small by choosing q sufficiently close to p.) Any attempt to accomplish this feat must produce a cow...

Algorithmic Improvements of the Lovász Local Lemma via Cluster Expansion

The Lovász Local Lemma (LLL) is a powerful tool that can be used to prove that an object having none of a set of bad properties exists, using the probabilistic method. In many applications of the LLL it is also desirable to explicitly construct the combinatorial object. Recently it was shown that this is possible using a randomized algorithm in the full asymmetric LLL setting [R. Moser and G. T...

A TRANSLATION THEOREM 1 Lemma

A Universal Coefficient Theorem for Gauss’s Lemma

We shall prove a version of Gauß’s Lemma that recursively constructs polynomials {ck}k=0,...,m+n in Z[ai, Ai, bj , Bj ]i=0,...,m, j=0,...,n, of degree at most ( m+n n ) , such that whenever ∑ k CkX k = ∑ i AiX i · ∑ j BjX j and 1 = ∑

Some improvements of numerical radius inequalities via Specht’s ratio

We obtain some inequalities related to the powers of numerical radius inequalities of Hilbert space operators. Some results that employ the Hermite-Hadamard inequality for vectors in normed linear spaces are also obtained. We improve and generalize some inequalities with respect to Specht's ratio. Among them, we show that, if $A, Bin mathcal{B(mathcal{H})}$ satisfy in some conditions, it follow...