Average Pace and Horizontal Chords

Average salary of OR directors tops $90,000; raises pace inflation.

Chords not required: Incorporating horizontal and vertical aspects independently in a computer improvisation algorithm

Most previous computer improvisation algorithms for tonal jazz create an output based primarily on the underlying chord progression. This approach may partly ignore melodic continuity often seen in transcriptions of traditional jazz artists. Here we suggest a different approach implemented as a computer algorithm that creates material solely based on probabilities related to past note choices. ...

$\chi$-bounds, operations and chords

A long unichord in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is long-unichord-free if it does not contain any long-unichord. We prove a structure theorem for long-unichord-free graph. We give an O(nm)-time algorithm to recognize them. We show that any long-unichord-free graph G can be colored with at most O(ω) colors, where ω is the maximum number o...

Requiring chords in cycles

R. E. Jamison proved that every cycle of length greater than three in a graph has a chord—in other words, the graph is chordal—if and only if every k-cycle is the sum of k − 2 triangles. This result generalizes to having or not having crossing chords and to having strong chords, with similar characterizations of a variety of graph classes that includes chordal bipartite, distance-hereditary, an...

Intersecting Chords Theorem

This entry provides a geometric proof of the intersecting chords theorem. The theorem states that when two chords intersect each other inside a circle, the products of their segments are equal. After a short review of existing proofs in the literature [1, 3–5], I decided to use a proof approach that employs reasoning about lengths of line segments, the orthogonality of two lines and Pythagoras ...