Wedge sum

In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces (i.e. topological spaces with distinguished basepoints x₀ and y₀) the wedge sum of X and Y is the quotient space of the disjoint union of X and Y by the identification x₀ ∼ y₀:

where ∼ is the equivalence closure of the relation {(x₀,y₀)}. More generally, suppose (X_i )_i ∈ I is a family of pointed spaces with basepoints {p_i }. The wedge sum of the family is given by:

where ∼ is the equivalence relation {(p_i , p_j ) | i,j ∈ I }. In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints {p_i}, unless the spaces {X_i } are homogeneous.

The wedge sum is again a pointed space, and the binary operation is associative and commutative (up to isomorphism).

Sometimes the wedge sum is called the wedge product, but this is not the same concept as the exterior product, which is also often called the wedge product.

Submarine sandwich

A submarine sandwich, also known as a sub, wedge, hoagie, hero, grinder, or one of the many regional naming variations, is a type of sandwich that consists of a long roll of bread split widthwise into two pieces, and filled with a variety of meats, cheeses, vegetables, seasonings, and sauces. The sandwich has no standardized name, and many U.S. regions have their own names for it; one study found 13 different names for the sandwich in the United States. The usage of the several terms varies regionally but not in any pattern, as they have been used variously by the people and enterprises who make and sell them. The terms submarine and sub are widespread and not assignable to any certain region, though many of the localized terms are clustered in the northeastern United States.

History and etymology

The sandwich originated in several different Italian American communities in the Northeastern United States from the late 19th to mid-20th centuries. Portland, Maine claims to be the birthplace of the "Italian sandwich" and it is considered Maine's signature sandwich. The popularity of this Italian-American cuisine has grown from its origins in Connecticut, Pennsylvania, Delaware, New York, New Jersey, and Massachusetts to most parts of the United States, Canada, and with the advent of chain restaurants, is now available in many parts of the world.

Wedge prism

The wedge prism is a prism with a shallow angle between its input and output surfaces. This angle is usually 3 degrees or less. Refraction at the surfaces causes the prism to deflect light by a fixed angle. When viewing a scene through such a prism, objects will appear to be offset by an amount that varies with their distance from the prism.

Deflection angle

For a wedge prism in air, rays of light passing through the prism are deflected by the angle δ, which is approximately given by

where n is the index of refraction of the prism material, and α is the angle between the prism's surfaces.

Applications

The term "optical wedge" refers to any shallow angle between two plane surfaces of a window. This wedge may range from a few millionths of a degree of perfect parallelism to as much as three degrees of angle. Even though high-precision optics, such as optical flats, may be lapped and polished to extremely high levels of parallelism, nearly all optics with parallel faces have some slight wedge. This margin of error is usually listed in minutes or seconds of arc. Windows manufactured with an intentional wedge are often referred to as wedge prisms, and typically come with wedge angles of one, two, or three degrees. Many applications exist for wedge prisms, including laser-beam steering, rangefinding and variable focusing.

Wedge (symbol)

Wedge (∧) is a symbol used to represent:

In Unicode the symbol is encoded U+2227 ∧ LOGICAL AND (HTML ∧ · ∧) and by \wedge and \land in TeX. The opposite symbol (∨) is called a vel, or sometimes a (descending) wedge. Some authors who call the descending wedge 'vel' often call the ascending wedge ac (the corresponding Latin word for "and", also spelled "atque") keeping their usage parallel.

The wedge looks similar to the caret (^).

See also

Sum

Sum may refer to:

As an acronym, SUM may refer to:

Sumé

Sumé is a municipality in the state of Paraíba in the Northeast Region of Brazil.

See also

References

Coordinates: 7°39′S 36°55′W / 7.650°S 36.917°W / -7.650; -36.917

Åsum

Åsum is a village and eastern suburb of Odense, Funen, Denmark. It contains a church.

References