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In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz. Lp spaces form an important class of Banach spaces in functional analysis, of topological vector spaces; because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used in the theoretical discussion of problems in physics, finance and other disciplines. In statistics, measures of central tendency and statistical dispersion, such as the mean and standard deviation, are defined in terms of Lp metrics, measures of central tendency can be characterized as solutions to variational problems. In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the L1 norm of a solution's vector of parameter values, or its L2 norm. Techniques which use an L1 penalty, like LASSO, encourage solutions.

Techniques which use an L2 penalty, like ridge regression, encourage solutions where most parameter values are small. Elastic net regularization uses a penalty term, a combination of the L1 norm and the L2 norm of the parameter vector; the Fourier transform for the real line, maps Lp to Lq where 1 ≤ p ≤ 2 and 1/p + 1/q = 1. This is a consequence of the Riesz–Thorin interpolation theorem, is made precise with the Hausdorff–Young inequality. By contrast, if p > 2, the Fourier transform does not map into Lq. Hilbert spaces are central from quantum mechanics to stochastic calculus; the spaces L2 and ℓ2 are both Hilbert spaces. In fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ℓ2, where E is a set with an appropriate cardinality; the length of a vector x = in the n-dimensional real vector space Rn is given by the Euclidean norm: ‖ x ‖ 2 = 1 / 2. The Euclidean distance between two points x and y is the length ||x − y||2 of the straight line between the two points.

In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space. An analogy to this is suggested by taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distance, which takes into account that streets are either orthogonal or parallel to each other; the class of p-norms generalizes these two examples and has an abundance of applications in many parts of mathematics and computer science. For a real number p ≥ 1, the p-norm or Lp-norm of x is defined by ‖ x ‖ p = 1 / p; the absolute value bars are unnecessary when p is a rational number and, in reduced form, has an numerator. The Euclidean norm from above falls into this class and is the 2-norm, the 1-norm is the norm that corresponds to the rectilinear distance; the L∞-norm or maximum norm is the limit of the Lp-norms for p → ∞. It turns out that this limit is equivalent to the following definition: ‖ x ‖ ∞ = max See L-infinity.

For all p ≥ 1, the p-norms and maximum norm as defined above indeed satisfy the properties of a "length function", which are that: only the zero vector has zero length, the length of the vector is positive homogeneous with respect to multiplication by a scalar, the length of the sum of two vectors is no larger than the sum of lengths of the vectors. Abstractly speaking, this means that Rn together with the p-norm is a Banach space; this Banach space is the Lp-space over Rn. The grid distance or rectilinear distance between two points is never shorter than the length of the line segment between them. Formally, this means that the Euclidean norm of any vector is boun

German Bohemians known as the Sudeten Germans, were ethnic Germans living in the Czech lands of the Bohemian Crown, which became an integral part of the state of Czechoslovakia. Before 1945, Czechoslovakia was inhabited by over three million such German Bohemians, comprising about 23 percent of the population of the whole republic and about 29.5 percent of the population of Bohemia and Moravia. Ethnic Germans migrated into the Kingdom of Bohemia, an electoral territory of the Holy Roman Empire, from the 11th century in the border regions of what would be called the "Sudetenland", named after the Sudeten Mountains; this process of German expansion was known as Ostsiedlung. The name "Sudeten Germans" was adopted amidst rising nationalism in the aftermath of the fall of the Austro-Hungarian monarchy, a consequence of the First World War. All Czechoslovak Jews, most of whom were German-speaking, were deported and murdered by the Nazi authorities following the German invasion in 1938. At the conclusion of the Second World War, the remainder of the German-speaking population was expelled from Czechoslovakia to Germany and Austria.

The area that became known as the Sudetenland possessed chemical works and lignite mines, as well as textile and glass factories. The Bohemian border with Bavaria was inhabited by Germans; the Upper Palatine Forest, which extends along the Bavarian frontier and into the agricultural areas of southern Bohemia, was an area of German settlement. Moravia contained patches of "locked" German territory to the south. More characteristic were the German language islands: towns inhabited by German minorities and surrounded by Czechs. Sudeten Germans were Roman Catholics, a legacy of centuries of Austrian Habsburg rule. Not all ethnic Germans lived in well-defined areas. Since the second half of the 19th century and Germans created separate cultural, educational and economic institutions which kept both groups semi-isolated from each other; this form of separation continued until the end of the Second World War, when all Germans were expelled. In the English language, ethnic Germans who originated in the Kingdom of Bohemia were traditionally referred to as "German Bohemians".

This appellation utilizes the broad definition of Bohemia, which includes all of the three Bohemian crown lands: Bohemia and Silesia. In the German language, it is more common to distinguish among the three lands, hence the prominent terms Deutschböhmen, Deutschmährer and Deutschschlesier. In German, the broader use of "Bohemian" is found; the term "Sudeten Germans" came about during rising ethnic nationalism in the early 20th century, after the fall of the Austro-Hungarian Empire in the First World War. It coincided with the rise of another new term, "the Sudetenland", which referred only to the parts of the former Kingdom of Bohemia that were inhabited predominantly by ethnic Germans; these names were derived from the Sudeten Mountains, which form the northern border of the Bohemian lands. As these terms were used by the Nazi German regime to push forward the creation of a Greater Germanic Reich, many contemporary Germans avoid them in favour of the traditional names. There have been ethnic Germans living in the Bohemian crown lands since the Middle Ages.

In the late 12th and in the 13th century the Přemyslid rulers promoted the colonization of certain areas of their lands by German settlers from the adjacent lands of Bavaria, Upper Saxony and Austria during the Ostsiedlung migration. In 1348, the Luxembourg king Charles I King of the Romans and Holy Roman Emperor from 1355, founded the Charles University in Prague, the first in Central Europe, attended by large German student nations, while the language of education was Latin. Czechs made up about 20 percent of the student body at the time of its founding, while the rest was German. A culturally significant example of German Bohemian prose from the Middle Ages is the story Der Ackermann aus Böhmen, written in Early New High German by Johannes von Tepl in Žatec, who had studied the liberal arts in Prague. For centuries, German Bohemians played important roles in the economy and politics of the Bohemian lands. For example, forest glass production was a common industry among German Bohemians. Though they were living beyond the medieval Kingdom of Germany, an independent German Bohemian awareness, was not widespread and for a long time it played no decisive role in everyday life.

Individuals were seen as Bohemians, Silesians. Defining events in German Bohemian history were the Hussite Wars, the occupation of Bohemia by the Czech Brethren, the Thirty Years’ War, during which the Lands of the Bohemian Crown were affected, forwarding the immigration of further German settlers. After the death of King Louis II of Hungary and Bohemia in the 1526 Battle of Mohács, the Habsburg archduke Ferdinand of Austria had become King of Bohemia and the country became a constituent state of the Habsburg Monarchy. With the rise of the Habsburgs in Bohemia after the 1620 Battle of White Mountain, the old Bohemian nobility became meaningless; the Bohemian crown lands were ruled from the Austrian capital Vienna, which favored the dominance of both the German language and German culture. On the other hand, the 18th century Silesian Wars of King Frederick II of Prussia against Austria, resul

WEMP is an FM radio station licensed to Two Rivers, Wisconsin carrying a mixed easy listening and soft oldies format. The station transmits from the WLKN tower in Newton and covers the Manitowoc/Two Rivers market, along with eastern Sheboygan County, including Sheboygan; the station is owned by Mark Seehafer through Seehafer Broadcasting Corporation, the station's licensee. The station's allocation has been proposed by the FCC since 1996, going through three owners who failed to build the facilities before previous owner Mark Heller's purchase of the license in 2013; the station began broadcasting via program test authority on the evening of December 7, 2013 at 10pm, broadcasting until December 9 at midnight, using a limited hour loop of easy listening music with some Christmas music mixed in due to music licensing concerns, along with weather and sports updates and small talk from Heller, pre-recorded station identifications done by WGN's Orion Samuelson and Max Armstrong. Part of the PTA were several jingles from the original WEMP in Milwaukee during their prime in the 1960s.

The PTA happened nine days ahead of the expiration of the station's construction permit after the station's original plan to build a new tower in Newton was rejected by the town board. The test was performed using the studio facilities of Cleveland's WLKN, along with their antenna and transmitter just west of Newton. WLKN went off the air to allow the PTA to go forward, streamed the PTA in full using their website; the test ended with WLKN's programming resuming as scheduled. WEMP began permanent service a year on December 7, 2014, broadcasting from the WLKN tower with its current format, remained commercial free during a testing period; the station was simulcast on their sister station New Holstein-licensed WLAK, which operates as a daytime-only signal. WEMP utilized the basement studio of WLKN, along with sharing their post office box for correspondence their first few months on the air. In February 2015, limited commercial advertising began, along with the addition of top-of-the-hour newscasts from ABC News Radio and half-hour weather updates.

The station was sold by Heller to Seehafer Broadcasting in June 2015 and moved their operations to the WOMT facility in Manitowoc, with no major changes to the format or commercial scheduling. The WLAK simulcast was dropped shortly thereafter with the end of common ownership. WLKN would itself move to Manitowoc two years with Seehafer's purchase of the Cub Radio stations and consolidation of Seehafer's five area stations into one facility. Official website WEMP on Facebook Query the FCC's FM station database for WEMP Radio-Locator information on WEMP Query Nielsen Audio's FM station database for WEMP

The communicating branch to the anterior cutaneous and saphenous branches of the femoral is continued down, as a cutaneous branch, to the thigh and leg, as the cutaneous branch of the obturator nerve. When this is so, it emerges from beneath the lower border of the Adductor longus, descends along the posterior margin of the sartorius to the medial side of the knee, where it pierces the deep fascia, communicates with the saphenous nerve, is distributed to the skin of the tibial side of the leg as low down as its middle. Cutaneous innervation of the lower limbs#Thigh This article incorporates text in the public domain from page 954 of the 20th edition of Gray's Anatomy Anatomy photo:11:05-0205 at the SUNY Downstate Medical Center - "Superficial Anatomy of the Lower Extremity: Cutaneous Nerves of the Anterior Thigh and Leg"

Rieneck Castle is a hill castle located in the town of Rieneck, in Bavaria, southern Germany. It was built by Ludwig I, Count of Loon and Rieneck, around the year 1150, it is today used by the German Christian Scout Association. Rieneck Castle sits on a hill in the town of Rieneck, it is located in the Spessart hills, in the Main-Spessart district of Bavaria, about 80 km east of Frankfurt. In 1150 Ludwig I, Count of Loon and Rieneck, ordered the building of the castrum Rinecke on the northeastern boundary of his territory, with the aim of safeguarding the lands of this aristocratic family against the neighbouring lordships of Mainz, Würzburg and Fulda; the little hill in the Sinn valley offered excellent conditions: there was only one direction where the castle required additional protection by a defensive ditch, offered as narrow a front as possible to attack. The latter can be seen in the ground plan of the keep, the 19-metre high "Thick Tower", outwardly an irregular, seven-sided polygon, whose tip points towards the nearby hill.

The castle complex consisted of a courtyard surrounded by defensive walls, the keep, with its 4 to 8-metre thick walls. Inside the walls half-timbered buildings were constructed as living quarters, store rooms, stables. Of these only what we now know; the castle today serves as a scouting facility and is owned by the German Christian Guide and Scout Association, VCP. The castle's programme includes scouting activities such as arts and crafts and Medieval trades and offers accommodation and a campsite. Scouting in Germany Official homepage of Rieneck Castle Rieneck Castle on the website of the Haus der Bayerischen Geschichte