End

End or Ending may refer to:

End

END

Ending

See also

End (graph theory)

In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit-evasion games on the graph, or (in the case of locally finite graphs) as topological ends of topological spaces associated with the graph.

Ends of graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition for groups with more than one end.

Definition and characterization

Ends of graphs were defined by Rudolf Halin (1964) in terms of equivalence classes of infinite paths. A ray in an infinite graph is a semi-infinite simple path; that is, it is an infinite sequence of vertices v₀, v₁, v₂, ... in which each vertex appears at most once in the sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph. According to Halin's definition, two rays r₀ and r₁ are equivalent if there is another ray r₂ (not necessarily different from either of the first two rays) that contains infinitely many of the vertices in each of r₀ and r₁. This is an equivalence relation: each ray is equivalent to itself, the definition is symmetric with regard to the ordering of the two rays, and it can be shown to be transitive. Therefore, it partitions the set of all rays into equivalence classes, and Halin defined an end as one of these equivalence classes.

End (category theory)

In category theory, an end of a functor is a universal extranatural transformation from an object e of X to S.

More explicitly, this is a pair , where e is an object of X and

is an extranatural transformation such that for every extranatural transformation

there exists a unique morphism

of X with

for every object a of C.

By abuse of language the object e is often called the end of the functor S (forgetting ) and is written

Characterization as limit: If X is complete, the end can be described as the equaliser in the diagram

where the first morphism is induced by and the second morphism is induced by .

Coend

The definition of the coend of a functor is the dual of the definition of an end.

Thus, a coend of S consists of a pair , where d is an object of X and

is an extranatural transformation, such that for every extranatural transformation

there exists a unique morphism

of X with

for every object a of C.

The coend d of the functor S is written

Characterization as colimit: Dually, if X is cocomplete, then the coend can be described as the coequalizer in the diagram

Flight dynamics

Flight dynamics is the study of the performance, stability, and control of vehicles flying through the air or in outer space. It is concerned with how forces acting on the vehicle influence its speed and attitude with respect to time.

In fixed-wing aircraft, the changing orientation of the vehicle with respect to the local air flow is represented by two critical parameters, angle of attack ("alpha") and angle of sideslip ("beta"). These angles describe the vector direction of airspeed, important because it is the principal source of modulations in the aerodynamic forces and moments applied to the aircraft.

Spacecraft flight dynamics involve three forces: propulsive (rocket engine), gravitational, and lift and drag (when traveling through the earths or any other celestial atmosphere). Because aerodynamic forces involved with spacecraft flight are very small, this leaves gravity as the dominant force.

Aircraft and spacecraft share a critical interest in their orientation with respect to the earth horizon and heading, and this is represented by another set of angles, "yaw," "pitch" and "roll" which angles match their colloquial meaning, but also have formal definition as an Euler sequence. These angles are the product of the rotational equations of motion, where orientation responds to torque, just as the velocity of a vehicle responds to forces. For all flight vehicles, these two sets of dynamics, rotational and translational, operate simultaneously and in a coupled fashion to evolve the vehicle's state (orientation and velocity) trajectory.

Mail on Sunday (album)

Mail on Sunday is the debut studio album by American hip hop artist Flo Rida, and was released on March 18, 2008 via Atlantic, and Poe Boy Entertainment. It spawned three singles; the first, "Low" was number-one on the U.S. Billboard Hot 100 for ten weeks. The second, and third singles, "Elevator", and "In the Ayer", were successful as well, being top twenty hits. "Roll" featuring Sean Kingston was not an official single, but it managed to peak at number fifty-nine on the US Billboard Hot 100, and number forty-three on the Canadian Hot 100 due to digital sales in both countries.

Music

T-Pain, who laced the first single, is only one of the many featured guests throughout the album. Timbaland, who produced the second single "Elevator", is also featured on the track. Rick Ross, and Trey Songz made appearances as well.Lil Wayne has also been added to the list with assistance by young Memphian, Jamil Smith- while Sean Kingston appears on the J. R. Rotem-produced "Roll" the concept was created, and co-written by Compton rapper Spitfiya, from The Bullets Production Team, Various other guests include Birdman, Brisco. The third single is "In the Ayer" featuring will.i.am. The fourth single was scheduled to be "Money Right" featuring Brisco, and Rick Ross, but was canceled due to the release of his upcoming second studio album R.O.O.T.S. His second collaboration with T-Pain, "I Bet", as well as his collaboration with Trina, named "Bout It", both didn't make the final tracklist, but were recorded. As for production, J. R. Rotem, DJ Montay, and Timbaland all provided tracks for Mail on Sunday, among others.

Banjo roll

In bluegrass music, a banjo roll or roll is an accompaniment pattern played by the banjo that uses a repeating eighth-note arpeggio – a broken chord – that by subdividing the beat 'keeps time'. "Each ["standard"] roll pattern is a right hand fingering pattern, consisting of eight (eighth) notes, which can be played while holding any chord position with the left hand."

"When used as back-up, the same pattern can be repeated over and over throughout an entire song, (...[with chord changes] as required), or the roll patterns can be combined with one another and with [back-up licks]... The roll patterns can also be used to embellish the vamping style of back-up, especially when the chords are played high... These roll patterns can be used as back-up for any song played at any tempo." The banjo is commonly played in open tunings, such as open G (as are all of the examples): G'DGBD', allowing rolls to be practiced on all open strings (without fretting). Rolls are a distinguishing characteristic of Scruggs style banjo playing, with older styles being more melodic. The older style of banjo playing has been described as "in 'phrases,'" and with, "really no connective tissue between four notes and next four notes...sorta like a gallop. Snuffy [Jenkins] was a little like that...old sounding, choppy...".

Standard (1911 automobile)

The Standard was a German automobile manufactured between 1911 and 1912. The car was produced at Berlin-Charlottenburg using a rotary valve engine built by Henriod, which was unreliable and had not been fully developed; consequently, it was very unpopular.

References