Double (manifold)

In the subject of manifold theory in mathematics, if is a manifold with boundary, its double is obtained by gluing two copies of together along their common boundary. Precisely, the double is where for all .

Although the concept makes sense for any manifold, and even for some non-manifold sets such as the Alexander horned sphere, the notion of double tends to be used primarily in the context that is non-empty and is compact.

Doubles bound

Given a manifold , the double of is the boundary of . This gives doubles a special role in cobordism.

Examples

The n-sphere is the double of the n-ball. In this context, the two balls would be the upper and lower hemi-sphere respectively. More generally, if is closed, the double of is . Even more generally, the double of a disc bundle over a manifold is a sphere bundle over the same manifold. More concretely, the double of the Möbius strip is the Klein bottle.

If is a closed, oriented manifold and if is obtained from by removing an open ball, then the connected sum is the double of .

C data types

In the C programming language, data types are declarations for memory locations or variables that determine the characteristics of the data that may be stored and the methods (operations) of processing that are permitted involving them.

The C language provides basic arithmetic types, such as integer and real number types, and syntax to build array and compound types. Several header files in the C standard library contain definitions of support types, that have additional properties, such as providing storage with an exact size, independent of the implementation.

Basic types

The C language provides the four basic arithmetic type specifiers char, int, float and double, and the modifiers signed, unsigned, short and long. The following table lists the permissible combinations to specify a large set of storage size-specific declarations.

The actual size of integer types varies by implementation. The standard only requires size relations between the data types and minimum sizes for each data type:

Double-precision floating-point format

Double-precision floating-point format is a computer number format that occupies 8 bytes (64 bits) in computer memory and represents a wide, dynamic range of values by using a floating point.

Double-precision floating-point format usually refers to binary64, as specified by the IEEE 754 standard, not to the 64-bit decimal format decimal64.

IEEE 754 double-precision binary floating-point format: binary64

Double-precision binary floating-point is a commonly used format on PCs, due to its wider range over single-precision floating point, in spite of its performance and bandwidth cost. As with single-precision floating-point format, it lacks precision on integer numbers when compared with an integer format of the same size. It is commonly known simply as double. The IEEE 754 standard specifies a binary64 as having:

This gives 15–17 significant decimal digits precision. If a decimal string with at most 15 significant digits is converted to IEEE 754 double precision representation and then converted back to a string with the same number of significant digits, then the final string should match the original. If an IEEE 754 double precision is converted to a decimal string with at least 17 significant digits and then converted back to double, then the final number must match the original.

The Double (Gaelic games)

The Double is a term in Gaelic games that refers to a county winning the All-Ireland Senior Football Championship and the All-Ireland Senior Hurling Championship in the same year. Other major trophies won in combination in the same year at minor or under-21 levels are also often referred to as doubles. Similarly, the winning of the All-Ireland and the National League titles in the same year may also be referred to as the Double, albeit an inferior one.

The Double

Only two counties have achieved this rare distinction at Senior level, both on two separate occasions:

National League-All-Ireland double

This is a much more common double.

Near Doubles

A great many sides over the years have come close to winning the coveted senior double but narrowly failed to do so by losing one or both of the championships at the end of the season.

The full list of these teams: