Arc (projective geometry)

A (simple) arc in finite projective geometry is a set of points which satisfies, in an intuitive way, a feature of curved figures in continuous geometries. Loosely speaking, they are sets of points that are far from "line-like" in a plane or far from "plane-like" in a three-dimensional space. In this finite setting it is typical to include the number of points in the set in the name, so these simple arcs are called k-arcs. An important generalization of the k-arc concept, also referred to as arcs in the literature, are the (k, d)-arcs.

k-arcs in a projective plane

In a finite projective plane π (not necessarily Desarguesian) a set A of k (k ≥ 3) points such that no three points of A are collinear (on a line) is called a k - arc. If the plane π has order q then k ≤ q + 2, however the maximum value of k can only be achieved if q is even. In a plane of order q, a (q + 1)-arc is called an oval and, if q is even, a (q + 2)-arc is called a hyperoval.

Every conic in the Desarguesian projective plane PG(2,q), i.e., the set of zeros of an irreducible homogeneous quadratic equation, is an oval. A celebrated result of Beniamino Segre states that when q is odd, every (q + 1)-arc in PG(2,q) is a conic. This is one of the pioneering results in finite geometry.

Arc (geometry)

In Euclidean geometry, an arc (symbol: ⌒) is a closed segment of a differentiable curve. A common example in the plane (a two-dimensional manifold), is a segment of a circle called a circular arc. In space, if the arc is part of a great circle (or great ellipse), it is called a great arc.

Every pair of distinct points on a circle determines two arcs. If the two points are not directly opposite each other, one of these arcs, the minor arc, will subtend an angle at the centre of the circle that is less than π radians (180 degrees), and the other arc, the major arc, will subtend an angle greater than π radians.

Circular arcs

Length of an arc of a circle

The length (more precisely, arc length), L, of an arc of a circle with radius and subtending an angle (measured in radians) with the circle center — i.e., the central angle — equals . This is because

Substituting in the circumference

and, with being the same angle measured in degrees, since the arc length equals

A practical way to determine the length of an arc in a circle is to plot two lines from the arc's endpoints to the center of the circle, measure the angle where the two lines meet the center, then solve for L by cross-multiplying the statement:

ARC21

ARC21 is a local government body in Northern Ireland that is tasked with coordination of the waste management services in the North East of Northern Ireland. The ARC21 waste management region includes the city, borough and district councils of:

ARC21 is accountable to the Northern Ireland Department for the Environment.

See also

References

Music

Music is an art form and cultural activity whose medium is sound and silence. The common elements of music are pitch (which governs melody and harmony), rhythm (and its associated concepts tempo, meter, and articulation), dynamics (loudness and softness), and the sonic qualities of timbre and texture (which are sometimes termed the "color" of a musical sound). Different styles or types of music may emphasize, de-emphasize or omit some of these elements. Music is performed with a vast range of instruments and with vocal techniques ranging from singing to rapping, and there are solely instrumental pieces, solely vocal pieces and pieces that combine singing and instruments. The word derives from Greek μουσική (mousike; "art of the Muses"). In its most general form, the activities describing music as an art form include the production of works of music (songs, tunes, symphonies, and so on), the criticism of music, the study of the history of music, and the aesthetic examination of music. Ancient Greek and Indian philosophers defined music as tones ordered horizontally as melodies and vertically as harmonies. Common sayings such as "the harmony of the spheres" and "it is music to my ears" point to the notion that music is often ordered and pleasant to listen to. However, 20th-century composer John Cage thought that any sound can be music, saying, for example, "There is no noise, only sound."

Music (short story)

"Music" is a short story by Russian American author Vladimir Nabokov originally published in Russian in 1932.

Summary

The story uses third-person narration and tells the story of Victor, a self-conscious man for whom "music he did not know... could be likened to the patter of a conversation in a strange tongue." When Victor arrives at a party, he finds the other guests listening with varying degrees of engagement to a man named Wolfe play the piano. As Victor does not know the song being played, he loses interest. He catches a glimpse of his ex-wife at the party, but cannot look at her. He laments the fact that now he must "start all over" the long task of forgetting her (in a flashback, it's revealed that she left him for another, who may or may not be at the party). Throughout the entire story, Victor views the music as a structure that has him encaged in an awkward situation with his ex-wife; it had seemed to him "a narrow dungeon" until it ends, thus giving his ex-wife the opportunity to leave, which she does. Victor then realizes that the music was not a dungeon, but actually "incredible bliss, a magic glass dome that had embraced and imprisoned him and her," and which allowed him to "breathe the same air as she." After she leaves, another party-goer comments to Victor that he looked immune to the music and that he didn't think such a thing possible. His own inanity is revealed when Victor asks him what was played and he cannot tell whether it was Beethoven's Kreutzer Sonata or Tekla Bądarzewska-Baranowska's rather easy piece, Maiden's Prayer.

Music (Erick Sermon and Marvin Gaye song)

"Music" is a 2001 hit single by Erick Sermon featuring archived vocals from Marvin Gaye.

The song was thought of by Sermon after buying a copy of Gaye's Midnight Love and the Sexual Healing Sessions album, which overlook some of the original album's earlier mixes. After listening to an outtake of Gaye's 1982 album track, "Turn On Some Music" (titled "I've Got My Music" in its initial version), Sermon decided to mix the vocals (done in a cappella) and add it into his own song. The result was similar to Natalie Cole's interpolation of her father, jazz great Nat "King" Cole's hit, "Unforgettable" revisioned as a duet. The hip hop and soul duet featuring the two veteran performers was released as the leading song of the soundtrack to the Martin Lawrence & Danny DeVito comedy, "What's the Worst That Could Happen?" The song became a runaway success rising to #2 on Billboard's R&B chart and was #1 on the rap charts. It also registered at #21 pop giving Sermon his highest-charted single on the pop charts as a solo artist and giving Gaye his first posthumous hit in 10 years following 1991's R&B-charted single, "My Last Chance" also bringing Gaye his 41st top 40 pop hit. There is also a version that's played on Adult R&B stations that removes Erick Sermon's rap verses. The song was featured in the 2011 Matthew McConaughey film The Lincoln Lawyer.