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Circular polarization

In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude but its direction rotates at a constant rate in a plane perpendicular to the direction of the wave. In electrodynamics the strength and direction of an electric field is defined by its electric field vector. In the case of a circularly polarized wave, as seen in the accompanying animation, the tip of the electric field vector, at a given point in space, describes a circle as time progresses. At any instant of time, the electric field vector of the wave indicates a point on a helix oriented along the direction of propagation. A circularly polarized wave can rotate in one of two possible senses: right circular polarization in which the electric field vector rotates in a right-hand sense with respect to the direction of propagation, left circular polarization in which the vector rotates in a left-hand sense. Circular polarization is a limiting case of the more general condition of elliptical polarization.

The other special case is the easier-to-understand linear polarization. The phenomenon of polarization arises as a consequence of the fact that light behaves as a two-dimensional transverse wave. On the right is an illustration of the electric field vectors of a circularly polarized electromagnetic wave; the electric field vectors have a constant magnitude but their direction changes in a rotary manner. Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane, perpendicular to the axis. Given that this is a circularly polarized plane wave, these vectors indicate that the electric field, from plane to plane, has a constant strength while its direction rotates. Refer to these two images in the plane wave article to better appreciate this; this light is considered to be right-hand, clockwise circularly polarized. Since this is an electromagnetic wave each electric field vector has a corresponding, but not illustrated, magnetic field vector, at a right angle to the electric field vector and proportional in magnitude to it.

As a result, the magnetic field vectors would trace out a second helix. Circular polarization is encountered in the field of optics and in this section, the electromagnetic wave will be referred to as light; the nature of circular polarization and its relationship to other polarizations is understood by thinking of the electric field as being divided into two components which are at right angles to each other. Refer to the second illustration on the right; the vertical component and its corresponding plane are illustrated in blue while the horizontal component and its corresponding plane are illustrated in green. Notice that the rightward horizontal component leads the vertical component by one quarter of a wavelength, it is this quadrature phase relationship which creates the helix and causes the points of maximum magnitude of the vertical component to correspond with the points of zero magnitude of the horizontal component, vice versa. The result of this alignment is that there are select vectors, corresponding to the helix, which match the maxima of the vertical and horizontal components.

To appreciate how this quadrature phase shift corresponds to an electric field that rotates while maintaining a constant magnitude, imagine a dot traveling clockwise in a circle. Consider how the vertical and horizontal displacements of the dot, relative to the center of the circle, vary sinusoidally in time and are out of phase by one quarter of a cycle; the displacements are said to be out of phase by one quarter of a cycle because the horizontal maximum displacement is reached one quarter of a cycle before the vertical maximum displacement is reached. Now referring again to the illustration, imagine the center of the circle just described, traveling along the axis from the front to the back; the circling dot will trace out a helix with the displacement toward our viewing left, leading the vertical displacement. Just as the horizontal and vertical displacements of the rotating dot are out of phase by one quarter of a cycle in time, the magnitude of the horizontal and vertical components of the electric field are out of phase by one quarter of a wavelength.

The next pair of illustrations is that of left-handed, counter-clockwise circularly polarized light when viewed by the receiver. Because it is left-handed, the rightward horizontal component is now lagging the vertical component by one quarter of a wavelength rather than leading it. To convert a given handedness of polarized light to the other handedness one can use a half-waveplate. A half-waveplate shifts a given linear component of light one half of a wavelength relative to its orthogonal linear component; the handedness of polarized light is reversed when it is reflected off a surface at normal incidence. Upon such reflection, the rotation of the plane of polarization of the reflected light is identical to that of the incident field. However, with propagation now in the opposite direction, the same rotation direction that would be described as "right handed" for the incident beam, is "left-handed" for propagation in the reverse direction, vice versa. Aside from the reversal of handedness, the ellipticity of polarization is preserved.

Note that this principle only holds for light reflected at normal incidence. For instance, right circularly polarized light reflected from a dielectric surface at grazing incidence (an angle beyond the

2013 Southern Conference football season

The 2013 Southern Conference football season, part of the 2013 NCAA Division I FCS football season competition of college football, began on Thursday, August 29, 2013 with Chattanooga hosting Tennessee–Martin. The regular season concluded on November 23, while Samford and Furman qualified for the NCAA Division I Football Championship. Samford was eliminated in the first round by Jacksonville State. Furman defeated South Carolina State in the opening round, but fell to eventual champion North Dakota State 38–7 in the second round. Appalachian State and Georgia Southern played their final seasons as members of the Southern Conference but were ineligible to win the conference championship or participate in the playoffs as they transitioned to the Football Bowl Subdivision. Appalachian State and Georgia Southern were ineligible for the Coaches' Poll due the additional scholarship players on the rosters as part of their transition to FBS. All times Eastern time. Rankings reflect that of the Sports Network poll for that week.

Players of the Week Players of the Week Players of the Week Players of the Week Players of the Week

Wetumpka Municipal Airport

Wetumpka Municipal Airport is a city-owned, public-use airport located six nautical miles west of the central business district of Wetumpka, a city in Elmore County, United States. It is included in the FAA's National Plan of Integrated Airport Systems for 2011–2015, which categorized it as a general aviation facility. During World War II the airport, known as Elmore Auxiliary Field, served as an auxiliary field for Gunter Army Airfield training operations. In 2013 the airport was listed in the Alabama Register of Heritage. Wetumpka Municipal Airport covers an area of 312 acres at an elevation of 197 feet above mean sea level, it has two runways: 9/27 is 3,011 by 80 feet with an asphalt surface. For the 12-month period ending December 7, 2010, the airport had 39,400 general aviation aircraft operations, an average of 107 per day. At that time there were 80 aircraft based at this airport: 89% single-engine, 9% multi-engine, 1% helicopter and 1% glider. FAA Terminal Procedures for 08A, effective February 27, 2020 Resources for this airport: FAA airport information for 08A AirNav airport information for 08A FlightAware airport information and live flight tracker SkyVector aeronautical chart for 08A

Grandselve Abbey

Grandselve Abbey was a Cistercian monastery in south-west France, at Bouillac, Tarn-et-Garonne. It was one of the most important Cistercian abbeys in the south of France. Grandselve was founded as a hermitage under the Benedictine rule in 1114 by Gerald of Sales, who placed it under the supervision of Cadouin Abbey. In 1117 Bishop Amelius Raymond du Puy of Toulouse recognized it as a monastery, he authorized the monks to build a church, gave them the lands, required them to follow the rule as practiced at Cîteaux Abbey. Over time, the monks began to detach themselves from their connection to Cadouin, in 1135 Bishop Amelius, at the request of Pope Innocent II, reminded them of their required obedience. Grandselve joined the Cistercian movement as a daughter house of Clairvaux Abbey in 1145; the church was dedicated in 1253. The land was cultivated, mills were established on the rivers and vineyards were planted. By the fourteenth century, the abbey owned two wine cellars in Bordeaux, it became one of the most famous abbeys of the south.

Grandselve founded the College of St. Bernard in Toulouse to teach theology. William VI and William VII, counts of Montpellier, were buried at Grandselve, where William VII's son, Raimond de Montpellier was a monk. In 1231 Bishop of Toulouse Folquet de Marselha was buried, beside the tomb of William VII of Montpellier, at the abbey of Grandselves, near Toulouse, where his sons and Petrus had been abbots; the abbey properties suffered during the Hundred Years' War such that John II of France temporarily exempted the abbey from taxes. By the late fifteenth century, commendatory abbots further depleted the abbey's resources while neglecting maintenance and repair. By 1790 there were only fourteen religious left; the abbey was suppressed during the French Revolution. It was sold in 1791 to private owners. List of abbots

The X Factor (Greek TV series)

The X Factor is the Greek version of The X Factor, a show originating in the United Kingdom. It is a television music talent show contested by aspiring pop singers drawn from public auditions. Auditions are held in Greece in the cities of Athens and Thessaloniki, as well as in Cyprus in the city of Nicosia. Auditions for the third X Factor were held in New York City. Applicants from the Greek diaspora are accepted; the show was broadcast live in Greece and Cyprus, as well as abroad via ANT1's international stations, for the first three seasons. For the fourth and fifth season was broadcast live in Greece via Skai TV and in Cyprus via Sigma TV. From sixth season is broadcast live in Cyprus via Omega; the winners of the show are: in the 1st season was Loukas Yorkas, in the 2nd was Stavros Michalakakos, in the 3rd was Harry Antoniou, in the 4th was Andreas Leontas, in the 5th was Panagiotis Koufogiannis and in the 6th was Giannis Grosis. There are four stages to the competition: Stage 1: Judges' auditions – either in an audition room or an arena Stage 2: Bootcamp Stage 3: Judges Houses Stage 3: Four-Chair Challenge Stage 4: Live Shows To date, five series have been broadcast, as summarised below.

Contestant in "Boys" category Contestant in "Girls" category Contestant in "Groups" category Contestant in "Over 25s" category Key: – Winning judge/category. Winners are in other contestants in small font; the first series of the Greek X Factor started airing in October 2008 on ANT1 and was hosted by singer Sakis Rouvas. The judges were Giorgos Levendis, songwriter Giorgos Theofanous, marketing executive of ANT1 TV, Katerina Gagaki and music critic Nikos Mouratidis; the winner of the 1st X-Factor was Loukas Yorkas, who released his debut EP album on September 2009. A second series of X Factor was broadcated by ANT1 TV; the live shows debuted on October 30. The host and the judges remain the same as in the first series; the winner of the 2nd X-Factor was Stavros Michalakakos. A third season of X Factor was broadcast from October 2010 to February 2011; the third series of X Factor was won by Greek-Cypriot Haris Antoniou. The program was broadcast by ANT1 TV and the host and the judges remained the same as the previous year.

A fourth season of X Factor was broadcast from 4 April to 8 July 2016 on Skai TV. The host Sakis Rouvas remained with Evaggelia Aravani in backstage. Giorgos Theofanous remained from the judges of the past seasons with Peggy Zina and Thodoris Marantinis from Onirama being the new judges; the winner of this series was Andreas Leontas from Cyprus. A fifth season of X Factor was broadcast from April 27 to July 20, 2017 on Skai TV; the host Sakis Rouvas remained with Evaggelia Aravani in backstage. Tamta remained from the judges of the past seasons with Giorgos Mazonakis, Giorgos Papadopoulos and Babis Stokas being the new judges; the winner of this series was Panagiotis Koufogiannis from Cyprus. A sixth season of X Factor began broadcast from 11 September 2019 on Open TV; the judges are Giorgos Theofanous, who returned for his fifth series as judge, former The Voice of Greece coach Melina Aslanidou, former Rising Star judge Christos Mastoras and Athens DeeJay Michael Tsaousopoulos. Despina Vandi will be the host.

Aris Makris is presenting the Backstage of Live Shows. Greek Idol The Voice of Greece Official Website

K├Âthe conjecture

In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2010. It is formulated in various ways. Suppose that R is a ring. One way to state the conjecture is that if R has no nil ideal, other than it has no nil one-sided ideal, other than; this question was posed in 1930 by Gottfried Köthe. The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings and right Noetherian rings, but a general solution remains elusive; the conjecture has several different formulations: In any ring, the sum of two nil left ideals is nil. In any ring, the sum of two one-sided nil ideals is nil. In any ring, every nil left or right ideal of the ring is contained in the upper nil radical of the ring. For any ring R and for any nil ideal J of R the matrix ideal Mn is a nil ideal of Mn for every n. For any ring R and for any nil ideal J of R the matrix ideal M2 is a nil ideal of M2. For any ring R, the upper nilradical of Mn is the set of matrices with entries from the upper nilradical of R for every positive integer n.

For any ring R and for any nil ideal J of R, the polynomials with indeterminate x and coefficients from J lie in the Jacobson radical of the polynomial ring R. For any ring R, the Jacobson radical of R consists of the polynomials with coefficients from the upper nilradical of R. A conjecture by Amitsur read: "If J is a nil ideal in R J is a nil ideal of the polynomial ring R." This conjecture, if true, would have proven the Köthe conjecture through the equivalent statements above, however a counterexample was produced by Agata Smoktunowicz. While not a disproof of the Köthe conjecture, this fueled suspicions that the Köthe conjecture may be false in general. In, it was proven that a ring, the direct sum of two nilpotent subrings is itself nilpotent; the question arose whether or not "nilpotent" could be replaced with "locally nilpotent" or "nil". Partial progress was made when Kelarev produced an example of a ring which isn't nil, but is the direct sum of two locally nilpotent rings; this demonstrates that Kegel's question with "locally nilpotent" replacing "nilpotent" is answered in the negative.

The sum of a nilpotent subring and a nil subring is always nil. Köthe, Gottfried, "Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständig reduzibel ist", Mathematische Zeitschrift, 32: 161–186, doi:10.1007/BF01194626 PlanetMath page Survey paper