Prod

Prod or PROD may refer to:

Product integral, often shown using the symbol

Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity 1 (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.

The term "empty product" is most often used in the above sense when discussing arithmetic operations. However, the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products in computer programming; these are discussed below.

Nullary arithmetic product

Justification

Let a₁, a₂, a₃,... be a sequence of numbers, and let

be the product of the first m elements of the sequence. Then

for all m = 1,2,... provided that we use the following conventions: and . In other words, a "product" with only one factor evaluates to that factor, while a "product" with no factors at all evaluates to 1. Allowing a "product" with only one or zero factors reduces the number of cases to be considered in many mathematical formulas. Such "products" are natural starting points in induction proofs, as well as in algorithms. For these reasons, the "empty product is one convention" is common practice in mathematics and computer programming.

Rest

Rest may refer to:

Rest may also refer to:

REST written as an abbreviation may refer to:

Rest (album)

Rest, released April 1, 2008, is the second full-length album by Virginian post-rock band Gregor Samsa. The band posted the tracks online at Imeem before the album was released. The album was released in five formats, digital (April 1), unlimited (May 13), limited (April 24), collector's, and vinyl.

Track listing

Contributors

Rest (music)

A rest is an interval of silence in a piece of music, marked by a symbol indicating the length of the pause. Each rest symbol corresponds with a particular note value:

The quarter (crotchet) rest may also be found as a form in older music.

(The four-measure rest or longa rest is a symbol found in Western musical notation denoting a silence four times the duration of a whole rest. They are only used in long silent passages which are not divided into bars.

The two-measure rest or breve rest is another symbol found in Western musical notation denoting a silence twice the duration of a whole rest.

The combination of rests used to mark a pause follows the same rules as for notes. For more details see note value.

One-bar rests

When an entire bar is devoid of notes, a whole (semibreve) rest is used, regardless of the actual time signature. The only exceptions are for a 4/2 time signature (four half notes per bar), when a double whole rest is typically used for a bar's rest, and for time signatures shorter than 3/16, when a rest of the actual measure length would be used. For a 4/2 bar rest, it is also common to use the whole rest instead of the double whole rest, so that a whole-bar rest for all time signatures starting from 3/16 is notated using a whole note rest. Some published (usually earlier) music places the numeral "1" above the rest to confirm the extent of the rest.