Rectangular function

The rectangular function (also known as the rectangle function, rect function, Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as:

Alternative definitions of the function define to be 0, 1, or undefined.

Relation to the boxcar function

The rectangular function is a special case of the more general boxcar function:

Where u is the Heaviside function; the function is centered at X and has duration Y, from X-Y/2 to X+Y/2.

Another example is this: rect((t - (T/2)) / T ) goes from 0 to T, so in terms of Heaviside function u(t) - u((t-T) / T )

Fourier transform of the rectangular function

The unitary Fourier transforms of the rectangular function are

using ordinary frequency f, and

using angular frequency ω, where is the unnormalized form of the sinc function.

Note that as long as the definition of the pulse function is only motivated by the time-domain experience of it, there is no reason to believe that the oscillatory interpretation (i.e. the Fourier transform function) should be intuitive, or directly understood by humans. However, some aspects of the theoretical result may be understood intuitively, finiteness in time domain corresponds to an infinite frequency response. (Vice versa, a finite fourier transform will correspond to infinite time domain response.)

Opening

Opening may refer to:

See also

Chess opening

A chess opening is the group of initial moves of a chess game. Recognized sequences of initial moves are referred to as openings by White, or defenses by Black, but opening is also used as the general term. There are many dozens of different openings, and hundreds of named variants. The Oxford Companion to Chess lists 1,327 named openings and variants. These vary widely in character from quiet positional play to wild tactical play. In addition to referring to specific move sequences, the opening is the first phase of a chess game, the other phases being the middlegame and the endgame.

A sequence of opening moves that is considered standard (often catalogued in a reference work such as the Encyclopaedia of Chess Openings) is referred to as "the book moves", or simply "book". These reference works often present these move sequences in simple algebraic notation, opening trees, or theory tables. When a game begins to deviate from known opening theory, the players are said to be "out of book". In some opening lines, the moves considered best for both sides have been worked out for twenty to twenty-five moves or more. Some analysis goes to thirty or thirty-five moves, as in the classical King's Indian Defense and in the Sveshnikov and Najdorf variations of the Sicilian Defense. Professional chess players spend years studying openings, and continue doing so throughout their careers, as opening theory continues to evolve. Players at the club level also study openings but the importance of the opening phase is smaller there since games are rarely decided in the opening. The study of openings can become unbalanced if it is to the exclusion of tactical training and middlegame and endgame strategy.

Balance of Power (album)

Balance of Power is the twelfth studio album by the Electric Light Orchestra (ELO) released in 1986.

Overview

Balance of Power was the last studio album by the Electric Light Orchestra before their disbanding (the band would later reform in 2000). By this time Kelly Groucutt had departed and the group was pared down to a trio of Jeff Lynne, Richard Tandy, and Bev Bevan. Recording for the album began in 1984, with a planned release in spring 1985. The addition of several synthesizer tracks to the album caused its release to be pushed back to early 1986. ELO played some live concerts in the UK and Europe (their last for fifteen years), and for one UK show George Harrison performed as guest guitarist.

The video for the single "Calling America" was shot in Paris, and contains shots of the band playing in front of Centre Georges Pompidou. A video was also made for the single "So Serious." The track "Endless Lies" was intended as an homage to Roy Orbison, and was originally recorded for the never-released double LP version of Secret Messages; it reappeared on this album in a slightly reworked and more compact form. The album was ELO's first to not feature any strings, the first to feature a saxophone solo, and the first released on CD. The original release has the shortest running time of any ELO album.