OTHER WORK

My transcriptions of E.S.’ solos that I think are crucial for understanding the artist. Some are complete (Decade – Leucocyte, I Mean You Introduction – Jazz Baltica ‘99), others (Say Hello To Mr. D., Somewhere Else Before, …) only focusing on certain things I was looking for at the time.

In Euclidean data clustering, we seek to assign input data points into clusters, such that the maximum distance between points within the same cluster is minimized (or alternatively, such that the maximum distance between points from their cluster’s center is minimized). Through various reductions of graph problems (\(k\)-coloring, vertex cover), it is possible to show that not only is this task in \(\ge 3\) dimensions NP-complete, but also its approximations are NP-complete. Our aim in this project will be to improve known approximation factors for NP-completeness or polynomial complexity.

Consider a finite set of vertices \(V\) and suppose that we only know distances between some of them. In other words, consider a random graph \( G \) over \(V\) and a function \(d: E(G) \to \mathbb{R}^+ \cup {0}\). What properties of \(G\) make it possible to uniquely reconstruct the configuration of \(V\) (up to isometry)? We will show that the minimum degree \(\ge 2\) is alone sufficient with high probability.

A (commutative, unital) ring is the 3-tuple \( (R, +, \cdot) \) such that \( (R, +) \) is an Abelian group, \( (R, \cdot) \) is a commutative monoid, and the two satisfy mutually distributive laws. An \( R \)-module is an Abelian group tied to a scalar multiplication ring \( R \), satisfying mutual distributivite laws and multiplicative compatibility.

A (commutative, unital) ring is the 3-tuple \( (R, +, \cdot) \) such that \( (R, +) \) is an Abelian group, \( (R, \cdot) \) is a commutative monoid, and the two satisfy mutually distributive laws. \( S \subset R \) is a subring of \( R \) if there exists a ring morphism \(R \to S \). An ideal \(I \subset R \) is a additive subgroup of \( R \) that absorbs all multiplication: \( \forall i \in I, r \in R: ir \in R \).