Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane or simply the unit complex numbers

The circle group forms a subgroup of C^×, the multiplicative group of all nonzero complex numbers. Since C^× is abelian, it follows that T is as well. The circle group is also the group U(1) of 1×1 unitary matrices; these act on the complex plane by rotation about the origin. The circle group can be parametrized by the angle θ of rotation by

This is the exponential map for the circle group.

The circle group plays a central role in Pontryagin duality, and in the theory of Lie groups.

The notation T for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally Tⁿ (the direct product of T with itself n times) is geometrically an n-torus.

Elementary introduction

One way to think about the circle group is that it describes how to add angles, where only angles between 0° and 360° are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer should be 150° + 270° = 420°, but when thinking in terms of the circle group, we need to "forget" the fact that we have wrapped once around the circle. Therefore we adjust our answer by 360° which gives 420° = 60° (mod 360°).

U1

U1, U.I or U-1 may refer to:

Mathematics

Military

Entertainment

People

Railways

U-Bahn lines

Danelectro U2

The Danelectro U2 is a dual-pickup hollow bodied guitar made of Masonite and shaped similar to a Les Paul model guitar.

It was originally made from the years 1956 to 1958 but was re-issued in the late 90s, in 2006 in a slightly modified form as the '56 Pro, and again in 2010 as the '56 Single Cutaway.

The sound of a U2 is distinctive of the Danelectro guitar, coming from its lipstick pickups which when both selected are wired in series rather than the more standard parallel used today by most big brands.

Construction materials used by Danelectro in this period are quite unusual for guitar making. The U2's body is made of a poplar wood frame with Masonite used for both the top and back, with the side of the body being bound with creme coloured vinyl.

A single-pickup version, the U1, was manufactured and sold alongside the U2. The U1 and U2 were reissued from 1998 to 2001, along with a new version, the U3 with three pickups and a six-position switch which would allow the player to select bridge, middle, neck, bridge and neck, bridge and middle, middle and neck, or with the "blow" switch, all three pickups at once.

German submarine U-1 (1935)

German submarine U-1 was the first U-boat (or submarine) built for Nazi Germany's Kriegsmarine following Adolf Hitler's abrogation of the terms of the Treaty of Versailles in 1935, which banned Germany possessing a submarine force.

A Type IIA U-boat, she was built at the Deutsche Werke shipyards in Kiel, yard number 236. Her keel being laid on 11 February 1935 amid celebration. She was completed on 29 June 1935 after a very rapid construction, and was manned by crews trained in the Netherlands.

Design

German Type II submarines were based on the Finnish submarine Vesikko. U-1 had a displacement of 254 tonnes (250 long tons) when at the surface and 303 tonnes (298 long tons) while submerged. Officially, the standard tonnage was 250 long tons (250 t), however. The U-boat had a total length of 40.90 m (134 ft 2 in), a pressure hull length of 27.80 m (91 ft 2 in), a beam of 4.08 m (13 ft 5 in), a height of 8.60 m (28 ft 3 in), and a draught of 3.83 m (12 ft 7 in). The submarine was powered by two MWM RS 127 S four-stroke, six-cylinder diesel engines of 700 metric horsepower (510 kW; 690 shp) for cruising, two Siemens-Schuckert PG VV 322/36 double-acting electric motors producing a total of 360 metric horsepower (260 kW; 360 shp) for use while submerged. She had two shafts and two 0.85 m (3 ft) propellers. The boat was capable of operating at depths of up to 80–150 metres (260–490 ft).

SM U-1 (Germany)

SM U-1, also known in English as the German Type U 1 submarine, was the first U-boat class of the U-boat series of submarines produced for the German Empire's Imperial German Navy. Only one was built. The U-1 was constructed by Germaniawerft in Kiel and was commissioned on 14 December 1906. When World War I began in 1914, the U-1 was deemed obsolete and was used only for training until 19 February 1919, when it was struck by another vessel while on an exercise.

Design

The U-Boat was a redesigned Karp class submarine by Spanish engineer Raimondo Lorenzo d'Equevilley Montjustin working for the German armaments company Krupp. The main improvements over the export Karp class included trim tanks instead of a moveable weight, a redesigned forecastle to improve seagoing ability, a 10 cm (3.9 in) larger diameter and strengthened pressure hull which prevented oil leakage from the external tanks, a rearrangement of the internal equipment and a heavier ballast keel.

The Imperial German Navy avoided the use of gasoline due to the perceived risk of fires and explosions that had caused many accidents in early submarines, and instead of the gasoline engines that had powered the Karp boats, U-1 was given much safer Körting kerosene engines. While normally kerosene engines were started using gasoline, the U-1's engines avoided even this and instead used electrically-heated air.

U-1 (semi-trailer)

The U-1 was a liquid hydrogen trailer designed to carry cryogenic liquid hydrogen (LH₂) on roads being pulled by a powered vehicle. It was constructed by the Cambridge Corporation and had a capacity of 26 500 liters with a hydrogen loss rate of approximately 2 percent per day. The U-1 was a single axle semi-trailer, The specifications for its successor the U-2 a double axle semi-trailer were issued on 15 March 1957.

See also

References