apply

1 put into service; make work or employ (something) for a particular purpose or for its inherent or natural purpose; "use your head!"; "we only use Spanish at home"; "I can't make use of this tool"; "Apply a magnetic field here"; "This thinking was applied to many projects"; "How do you utilize this tool?"; "I apply this rule to get good results"; "use the plastic bags to store the food"; "He doesn't know how to use a computer" [syn: use, utilize, utilise, employ]

2 be pertinent or relevant or applicable; "The same laws apply to you!"; "This theory holds for all irrational numbers"; "The same rules go for everyone" [syn: hold, go for]

3 ask (for something); "He applied for a leave of absence"; "She applied for college"; "apply for a job"

4 apply to a surface; "She applied paint to the back of the house"; "Put on make-up!" [syn: put on]

5 be applicable to; as to an analysis; "This theory lends itself well to our new data" [syn: lend oneself] [ant: defy]

6 give or convey physically; "She gave him First Aid"; "I gave him a punch in the nose" [syn: give]

7 avail oneself to; "apply a principle"; "practice a religion"; "use care when going down the stairs"; "use your common sense"; "practice non-violent resistance" [syn: practice, use]

8 ensure observance of laws and rules; "Apply the rules to everyone"; [syn: enforce, implement] [ant: exempt]

9 refer (a word or name) to a person or thing; "He applied this racial slur to me!"

10 apply oneself to; "Please apply yourself to your homework" [also: applied]

English

Etymology

aplier, appliquer, from applico; ad + plico See applicant, ply.

Pronunciation

• Rhymes: -aɪ

Verb

1. To lay or place; to put or adjust (one thing to another);—with to; as, to apply the hand to the breast; to apply medicaments to a diseased part of the body.
   • Dryden,

     He said, and the sword his throat applied.

2. To put to use; to use or employ for a particular purpose, or in a particular case; to appropriate; to devote; as, to apply money to the payment of a debt.
3. To make use of, declare, or pronounce, as suitable, fitting, or relative; as, to apply the testimony to the case; to apply an epithet to a person.
   • Milton,

     Yet God at last To Satan, first in sin, his doom applied.

4. To fix closely; to engage and employ diligently, or with attention; to attach; to incline.
   • 1611, Authorized King James Version, Proverbs 23:12,

     Apply thine heart unto instruction, and thine ears to the words of knowledge.

5. To betake; to address; to refer;—used reflexively.
   • Johnson,

     I applied myself to him for help.

6. To submit oneself as a candidate for.

   I recently applied for a job as a bartender at the tavern.

   Most of the colleges she applied to were ones she thought she had a good chance of getting into.

   Many of them don't know it, but almost a third of the inmates are eligible to apply for parole or work-release programs.

7. To pertain or be relevant to a specified individual or group.

   That rule only applies to foreigners.

Derived terms

• appliance
• applicable
• application

References

In mathematics and computer science, Apply is a function that applies functions to arguments. It is a central concept in programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. In particular, it has a role in the study of the denotational semantics of computer programs, by virtue of the fact that it is a continuous function on complete partial orders.

In category theory, Apply is important in Cartesian closed categories, (and thus, also in Topos theory), where it is a universal morphism, right adjoint to currying.

Programming

In computer programing, Apply is simply the notion of applying a function to arguments. Thus, given a procedure g and some arguments x, the result of Apply(g,x) is the same as directly invoking g(x).

Universal property

Consider a function g:X\times Y\to Z, that is, g\isin [X\times Y\to Z] where the bracket notation [A\to B] denotes the space of functions from A to B. By means of currying, there is a unique function \mbox(g) :X\to [Y\to Z]. Then Apply provides the universal morphism

\mbox:[Y\to Z]\times Y \to Z,

so that

\mbox (f,y)=f(y)

or, equivalently one has the commuting diagram

\mbox \circ \left( \mbox(g) \times \mbox_Y \right) = g

The notation [A\to B] for the space of functions from A to B occurs more commonly in computer science. In category theory, however, [A\to B] is known as the exponential object, and is written as B^A. There are other common notational differences as well; for example Apply is often called Eval, even though in computer science, these are not the same thing, with eval distinguished from Apply, as being the evaluation of the quoted string form of a function with its arguments, rather than the application of a function to some arguments.

Also, in category theory, curry is commonly denoted by \lambda, so that \lambda g is written for curry(g). This notation is in conflict with the use of \lambda in lambda calculus, where lambda is used to denote free variables. With all of these notational changes accounted for, the adjointness of Apply and curry is then expressed in the commuting diagram

The articles on exponential object and Cartesian closed category provide a more precise discussion of the category-theoretic formulation of this idea. Thus use of lambda here is not accidental; Cartesian close categories provide the general, natural setting for lambda calculus.

Topological properties

In order theory, in the category of complete partial orders endowed with the Scott topology, both curry and apply are continuous functions (that is, they are Scott continuous). This property helps establish the foundational validity of the study of the denotational semantics of computer programs.

References

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