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It was originally made from the years 1956 to 1958 but was re-issued in the late 90s, in 2006 in a slightly modified form as the '56 Pro, and again in 2010 as the '56 Single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

CN U-1-f

Canadian National Railways U-1-f class locomotives, were a class of twenty 4-8-2 or Mountain type locomotives built by Montreal Locomotive Works in 1944. They were numbered 6060–6079 by CN and nicknamed “Bullet Nose Bettys” due to their distinctive cone-shape smokebox door cover.

Construction history

The order for these engines came during World War II when steel was of extreme value. The mountain type locomotive was a step down in size from the much more prevalent Northern Type (4-8-4). As a result of the step down in size the mountain type had less power but more speed and served well as a general purpose workhorse.

Modifications

Half the class had been converted to oil-firing by October 1944. This resulted in the 18-ton coal/11,700-gallon tender being exchanged for a 5,000-gallon oil/11,000-gallon water tender. In later years several locomotives lost the distinctive cone-shaped smokebox door cover.

Preservation

Of the twenty locomotives that were built, only three remain in existence: 6060 owned by the Rocky Mountain Rail Society at the Alberta Prairie Railway, Stettler, Alberta; 6069 at Sarnia, Ontario; and 6077 at the Northern Ontario Railroad Museum, at Capreol, Ontario.