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Function (mathematics)

Function (mathematics)

In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). The concept of a function is fundamental to virtually every branch of mathematics and every quantitative science. The terms function, mapping, map and transformation are usually used synonymously. The term operation is frequently used for binary functions; functions whose domain is a set of functions, or a vector space, are often called operators (see also operator (programming)).

Intuitive introduction

Essentially, a function is a "rule" or procedure that assigns an "output" value to each given "input" value. The following are examples of functions:
- In a group of people, each person has a favorite colour—from the set of red, orange, yellow, green, cyan, blue, indigo, or violet. Here, the input is the person, and the output is one of the 8 colours. The favorite colour is a function of the person. For example, John has favorite colour red, while Kim has favorite colour violet. Note that more than one person may be associated with a given colour (e.g., John and Kim may both like red), but one person cannot have more or less than one favorite color.
- A stone is dropped from different stories of a tall building. The dropped stone may take 2 seconds to fall from the second story, and 4 seconds to fall from the 8th story. Here, the input is the story, and the output is the number of seconds. The relevant function describes the relationship between the time it takes the stone to reach the ground and the story. (See acceleration) The "rule" defining a function can be specified by a formula, a relationship, or simply a table listing the outputs against inputs. The most important feature of a function is that it is consistent, or deterministic, always producing the same output from a given input. In this way, a function may be thought of as a mechanism or "machine" (a "black box") consistently converting a given valid input into its unique associated output. In certain technical contexts, the input is often called the argument of the function, and the output the value of the function. A very common type of function occurs when the argument (input) and the value (output) are both numbers, the functional relationship is expressed by a formula, and the value (output) of the function is obtained by direct substitution of the argument into the formula. Consider for example :

f(x)=x^

which for any number x, assigns to x the associated value the square of x. A straightforward generalization is to allow functions depending on several arguments. For instance, :

g(x,y) = xy

is a function which takes the input, two expressions x and y, and assigns to it its product (output), xy. It might seem that this is not really a function as we described above, because this "rule" depends on two inputs. However, if we think of the two inputs together as a single pair (x, y), then we can interpret g as a function -- the argument (unified single input) is the ordered pair (x, y), and the function value (output) is xy. Such functions whose input consists of ordered pairs are called "binary" or "2-ary". In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on earth over time: this is a function which takes location and time as arguments and gives as output value the temperature at the indicated location at the indicated moment in time. We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that every element of the domain is associated to precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, the function concept is fundamental to virtually every branch of mathematics and the quantitative sciences.

History

As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, and these measurements are the basis of calculus. The word function was later used by Euler during the mid-18th century to describe an expression or formula involving various arguments, e.g. f(x) = sin(x) + x³. During the 19th century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis). By broadening the definition of functions, mathematicians were then able to study "strange" mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only theoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from functional analysis have shown that these functions are in some sense "more common" than differentiable functions. Such functions have since been applied to the modeling of physical phenomena such as Brownian motion. Towards the end of the 19th century, mathematicians started trying to formalize all of mathematics using set theory, and they sought to define every mathematical object as a set. Dirichlet and Lobachevsky independently and almost simultaneously gave the modern "formal" definition of function (see formal definition below). In this definition, a function is a special case of a relation. In most cases of practical interest, however, the differences between the modern definition and Euler's definition are negligible. The notion of function as a rule for computing, rather than a special kind of relation, has been formalized in mathematical logic and theoretical computer science by means of several systems, including the lambda calculus, the theory of recursive functions and the Turing machine.

Formal definition

Formally a function f from a set X to a set Y, written f : X → Y, is an ordered triple (X, Y, G(f)), where G(f) is a subset of the cartesian product X × Y, such that for each x in X, there is a unique y in Y such that the ordered pair (x, y) is in G(f). X is called the domain of f, Y is called the codomain of F, and G(f) is called the graph of f. For each "input value" x in the domain, the corresponding unique "output value" y in the codomain is denoted by f(x). Equivalently a function f can be defined as a relation between X and Y which satisfies: # f is total, or entire: for all x in X, there exists a y in Y such that x f y (x is f-related to y), i.e. for each input value, there is at least one output value in Y. # f is many-to-one, or functional: if x f y and x f z, then y = z. i.e., many input values can be related to one output value, but one input value cannot be related to many output values. A relation between X and Y that satisfies condition (1) is a multivalued function. Every function is a multivalued function, but not every multivalued function is a function. A relation between X and Y that satisfies condition (2) is a partial function. Every function is a partial function, but not every partial function is a function. In this encyclopedia, the term "function" will mean a relation satisfying both conditions (1) and (2), unless otherwise stated. Consider the following three examples:

image:notMap1.png This relation is total but not many-to-one; the element 3 in X is related to two elements b and c in Y. Therefore, this is a multivalued function, but not a function.

image:notMap2.png This relation is many-to-one but not total; the element 1 in X is not related to any element of Y. Therefore, this is a partial function, but not a function.

image:mathmap2.png

This relation is both total and many-to-one, and so it is a function from X to Y. Note that the emphasis is on "-to-one" as "many" may actually mean "one". The function can be given explicitly by specifying its graph G(f) = or as :

f(x)=\left\

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on: - misunderstanding of the implications of mathematical rigour; - attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias; - lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999). - Courant, R. and H. Robbins, What Is Mathematics? (1941); - Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics. - Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics. - Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language. - Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM. - Kline, M., Mathematical Thought from Ancient to Modern Times (1973). - Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot - Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics. - Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics. - Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics. - Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations. - A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics] - [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup. - [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education. - [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc... - [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations. - [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics - [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians. - [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate. - [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae. - [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians. - [http://www.physicsmathforums.com/ Physics Math Forums] - Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Relation (mathematics)

In mathematics, an n-ary relation (or n-place relation or often simply relation) is a generalization of binary relations such as "=" and "<" which occur in statements such as "5 < 6" or "2 + 2 = 4". It is also the fundamental notion in the relational model for databases.

Definition

A relation over the sets X₁, ..., X_n is an (n + 1)-tuple R=(X₁, ..., X_n, G(R)) where G(R) is a subset of X₁ × ... × X_n (the Cartesian product of these sets). If X=X₁=X₂=...=X_n, R is simply called a relation over X. G(R) is called the graph of R and, similar to the case of binary relations, R is often identified with its graph. An n-ary predicate is a truth-valued function of n variables.

Remarks

Because a relation as above defines uniquely an n-ary predicate that holds for x₁, ..., x_n if (x₁, ..., x_n) is in G(R), and vice versa, the relation and the predicate are often denoted with the same symbol. So, for example, the following two statements are considered to be equivalent: :

(x_1,x_2,\dotsb)\in G(R)

R(x_1,x_2,\dotsb)

Relations are classified according to the number of sets in the Cartesian product; in other words the number of terms in the expression:
- unary relation or property: R(x)
- binary relation: R(x, y) or x R y
- ternary relation: R(x, y, z)
- quaternary relation: R(x, y, z, w) Relations with more than 4 terms are usually called n-ary; for example "a 5-ary relation".

Domain (mathematics)

In mathematics, the domain of a function is the set of all input values to the function. X, the set of input values, is called the domain of f, and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs . Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values. Given a function f : A → B, the set A is called the domain, or domain of definition of f. The set of all values in the codomain that f maps to is called the range of f, written f(A). A well-defined function must map every element of the domain to an element of its codomain. For example, the function f defined by : f(x) = 1/x has no value for f(0). Thus, the set R of real numbers cannot be its domain. In cases like this, the function is usually either defined on R\, or the "gap" is plugged by specifically defining f(0). If we extend the definition of f to : f(x) = 1/x, for x ≠ 0 : f(0) = 0, then f is defined for all real numbers and we can choose its domain to be R. Any function can be restricted to a subset of its domain. The restriction of g : A → B to S, where S ⊆ A, is written g |_S : S → B. Some well-known domains are as follows (note that each successive domain includes those above it):

Category theory

In category theory, instead of functions, one deals with morphisms, which are simply arrows from one object to another. The domain of any morphism is then simply the object where the arrow starts. In this context, many set theoretic ideas about domains have to be abandoned, or at least formulated more abstractly. For example, the notion of restricting a morphism to a subset of its domain must be modified. See subobject for more.

Complex analysis

In complex analysis, a domain is an open connected subset of the complex numbers.

Range (mathematics)

In mathematics, the range of a function is the set of all "output" values produced by that function. Sometimes called the image, or more precisely, the image of the domain of the function.

Formal definition

Given a function

f\colon A\rightarrow B

, the range of

f

is defined to be the set :

\.

The range should not be confused with the codomain B. The range is a subset of the codomain, but is not necessarily equal to the codomain, since there may be elements of the codomain which are not elements of the range. Another way to think about this is to consider the codomain to be the set of all possible output values, while the range is the set of all actual outputs. Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values. A function whose range equals its codomain is called onto or surjective.

Examples

Let the function f be a function on the real numbers: :

f\colon \mathbb\rightarrow\mathbb

defined by :

f(x) = x^2

The codomain of f is R, and f takes all nonnegative values but never takes negative values, and thus the range is in fact the set R⁺—non-negative reals, i.e. the interval [0,∞): :

0\leq f(x)<\infty.

Now let g be a function on the real numbers: :

g\colon \mathbb\rightarrow\mathbb

defined by :

g(x) = 2x

In this case the image of g equals R, its codomain, since, for any real number y, :

g(y/2) = y.

In other words, g is onto R.

Map (mathematics)

In mathematics and related technical fields, the term map or mapping is often a synonym for function. Along these lines, a partial map is a partial function, and a total map is a total function. In many specific branches of mathematics, the term is used for a function with a specific property relevant to that branch, such as a continuous function in topology, a linear transformation in linear algebra, etc. In formal logic, the term is sometimes used for a functional predicate, whereas a function is a model of such a predicate in set theory. A mapping m which has domain A and codomain B can be denoted symbolically as :

m : A \rightarrow B \

. If element a belongs to the domain A and if element b belongs to the codomain B and if mapping m maps element a to element b, this can be denoted symbolically as :

m : a \mapsto b

, "m maps a to b", which means the same as m(a) = b, "function m of a is b", in function notation. In such case, element b can be referred to as the image of element a. For any element a in a mapping's domain, a has one and only one image. That is, there exists a b such that

m : a \mapsto b

and if

m : a \mapsto c

then b = c. A mapping m is an injective mapping m : A ↪ B if each element of A is mapped to a different element of B (note the use of arrow with hook). Injective mappings have the property that the domain's cardinality is less than or equal to the codomain's cardinality: card A ≤ card B. A mapping m is a surjective mapping m : A ↠ B if every element of B is an image of some element of A (note the use of two headed arrow): that is, if the mapping's range is equal to the mapping's codomain. Surjective mappings have the property that the domain's cardinality is greater than or equal to the codomain's cardinality: card A ≥ card B. A bijective mapping is both injective and surjective, and has the property that the domain's cardinality equals the codomain's cardinality. A mapping has an inverse mapping if and only if the mapping is bijective. Why? If the mapping is bijective, then it is evident that it has an inverse. Now, suppose that a mapping

m : A \rightarrow B

has an inverse

m^ : B \rightarrow A

. This inverse can only be a mapping if every element of its domain B has an image, which can only be the case if mapping m is surjective. Also, the inverse can be a mapping only if every element of its domain has no more than one image, which can be the case only if mapping m is injective. So if mapping m has an inverse mapping, then m must be bijective. A mapping's domain and codomain are uniquely defined, so that if m : A → B is some mapping, and if n : B ↪ C is an inclusion mapping, C being a proper superset of B, then the mapping n o m which is their composite is a different mapping from m. See also:
- Homomorphism
- Morphism Category:Set theory ja:写像

Transformation (mathematics)

In mathematics, a transformation in elementary terms is any of a variety of different operations from geometry, such as rotations, reflections and translations. These can be carried out in Euclidean space, particularly in dimensions 2 and 3. They are also operations that can be performed using linear algebra, and explicitly using matrix theory. For example, in 3D computer graphics, the operations of moving, scaling, or rotating a 3D model are commonly called transformations. More generally, a transformation in mathematics is one facet of the mathematical function; the term mapping is also used in ways that are quite close synonyms. A transformation is, most often, an invertible function from a set X to itself; but this is not always assumed. In a sense the term transformation only flags that a function's more geometric aspects are being considered (for example, with attention paid to invariants).

Binary function

In mathematics, a binary function, or function of two variables, is a function which takes two inputs. To be specific, suppose X, Y, and Z are sets. Suppose that, given any elements x of X and y of Y, f(x,y) is a unique element of Z. Then f is a binary function from X and Y to Z. For example, if Z is the set of integers, N⁺ is the set of natural numbers (except for zero), and Q is the set of rational numbers, then division is a binary function from Z and N⁺ to Q. Set-theoretically, one may represent a binary function as a subset of the Cartesian product X × Y × Z, where (x,y,z) belongs to the subset if and only if f(x,y) = z. Conversely, a subset R defines a binary function if and only if, for any x in X and y in Y, there exists a unique z in Z such that (x,y,z) belongs to R. We then define f(x,y) to be this z. Alternatively, a binary function may be interpreted as simply a function from X × Y to Z. Even when thought of this way, however, one generally writes f(x,y) instead of f((x,y)). (That is, the same pair of parentheses is used to indicate both function application and the formation of an ordered pair.) In turn, one can also derive ordinary functions of one variable from a binary function. Given any element x of X, there is a function f^x, or f(x,·), from Y to Z, given by f^x(y) := f(x,y). Similarly, given any element y of Y, there is a function f_y, or f(·,y), from X to Z, given by f_y(x) := f(x,y). (In computer science, this identification between a function from X × Y to Z and a function from X to Z^Y is called Currying.) The various concepts relating to functions can also be generalised to binary functions. For example, the division example above is surjective (or onto) because every rational number may be expressed as a quotient of an integer and a natural number. This example is injective in each input separately, because the functions f^x and f_y are always injective. However, it's not injective in both variables simultaneously, because (for example) f(2,4) = f(1,2). One can also consider partial binary functions, which may be defined only for certain values of the inputs. For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero. But this function is undefined when the second input is zero. A binary operation is a binary function where the sets X, Y, and Z are all equal; binary operations are often used to define algebraic structures. In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f^x and f_y are all linear transformations. A bilinear transformation, like any binary function, can be interpreted as a function from X × Y to Z, but this function in general won't be linear. However, the bilinear transformation can also be interpreted as a single linear transformation from the tensor product X (×) Y to Z. The concept of binary function generalises to ternary (or 3-ary) function, quaternary (or 4-ary) function, or more generally to n-ary function for any natural number n. A 0-ary function to Z is simply given by an element of Z. One can also define an A-ary function where A is any set; there is one input for each element of A. In category theory, n-ary functions generalise to n-ary morphisms in a multicategory. The interpretation of an n-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original n-ary morphism will work in a monoidal category. The construction of the derived morphisms of one variable will work in a closed monoidal category. The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above. Category:Abstract algebra

Operator

:This article is about operators in mathematics, for other kinds of operators see operator (disambiguation). In mathematics, an operator is some kind of function; if it comes with a specified type of operand as function domain, it is no more than another way of talking of functions of a given type. The most frequently met usage is a mapping between vector spaces; this kind of operator is distinguished by taking one vector and returning another. For example, consider an enlargement, say by a factor of √2; such as is required to take one size of DIN paper to another. It can also be applied geometrically to vectors as operands. In many important cases, operators transform functions into other functions. We also say an operator maps a function to another. The operator itself is a function, but has an attached type indicating the correct operand, and the kind of function returned. This extra data can be defined formally, using type theory; but in everyday usage saying operator flags its significance. Functions can therefore conversely be considered operators, for which we forget some of the type baggage, leaving just labels for the domain and codomain.

Operators and levels of abstraction

To begin with, the usage of operator in mathematics is subsumed in the usage of function: an operator can be taken to be some special kind of function. The word is generally used to call attention to some aspect of its nature as a function. Since there are several such aspects that are of interest, there is no completely consistent terminology. Common are these:
- To draw attention to the function domain, which may itself consist of vectors or functions, rather than just numbers. The expectation operator in probability theory, for example, has random variables as domain (and is also a functional).
- To draw attention to the fact that the domain consists of pairs or tuples of some sort, in which case operator is synonymous with the usual mathematical sense of operation.
- To draw attention to the function codomain; for example a vector-valued function might be called an operator. A single operator might conceivably qualify under all three of these. Other important ideas are:
- Overloading, in which for example addition, say, is thought of as a single operator able to act on numbers, vectors, matrices ... .
- Operators are often in practice just partial functions, a common phenomenon in the theory of differential equations since there is no guarantee that the derivative of a function exists.
- Use of higher operations on operators, meaning that operators are themselves combined. These are abstract ideas from mathematics, and computer science. They may however also be encountered in quantum mechanics. There Dirac drew a clear distinction between q-number or operator quantities, and c-numbers which are conventional complex numbers. The manipulation of q-numbers from that point on became basic to theoretical physics.

Describing operators

Operators are described usually by the number of operands:
- monadic or unary operators take one argument.
- dyadic or binary operators take two arguments.
- triadic or ternary/trinary/tertiary operators take three arguments. The number of operands is also called the arity of the operator. If an operator has an arity given as n-ary (or n-adic), then it takes n arguments. In programming, outside than functional programming, the -ary terms are more often used than the other variants. See arity for an extensive list of the -ary endings.

Notations

There are five major systematic ways of writing operators and their arguments. These are
- prefix: where the operator name comes first and the arguments follow, for example: ::Q(x₁, x₂,...,x_n). : In prefix notation, the brackets are sometimes omitted if it is known that Q is an n-ary operator.
- postfix: where the operator name comes last and the arguments precede, for example: ::(x₁, x₂,...,x_n) Q : In postfix notation, the brackets are sometimes omitted if it is known that Q is an n-ary operator.
- infix: where the operator name comes between the arguments. This is awkward and not commonly used for operators other than binary operators. Infix style is written, for example, as :: x₁ Q x₂.
- juxtaposition, for example for multiplication of numbers, scalar multiplication, matrix multiplication, and function composition, and also for "multiplication" of a numerical value and a physical unit, and of two physical units, e.g. 3Nm.
- a superscript or subscript on the right or on the left; the main uses are selection (an index), such as a coordinate of a vector, and, in the case of a superscript on the right, for exponentiation of numbers and square matrices, and multiple function composition. For operators on a single argument, prefix notation such as −7 is most common, but postfix such as 5! (factorial) or x
- is also usual. There are other notations commonly met. Writing exponents such as 2⁸ is really a law unto itself, since it is postfix only as a unary operator applied to 2, but on a slant as binary operator. In some literature, a circumflex is written over the operator name. In certain circumstances, they are written unlike functions, when an operator has a single argument or operand. For example, if the operator name is Q and the operand a function f, we write Qf and not usually Q(f); this latter notation may however be used for clarity if there is a product — for instance, Q(fg). Later on we will use Q to denote a general operator, and x_i to denote the i-th argument. Notations for operators include the following. If f(x) is a function of x and Q is the general operator we can write Q acting on f as (Qf)(x) also. Operators are often written in calligraphy to differentiate them from standard functions. For instance, the Fourier transform (an operator on functions) of f(t) (a function of t), which produces another function F(ω) (a function of ω), would be represented as

\mathcal(f(t)) = F(\omega).

Examples of mathematical operators

This section concentrates on illustrating the expressive power of the operator concept in mathematics. Please refer to individual topics pages for further details.

Linear operators

Main article: Linear transformation The most common kind of operator encountered are linear operators. In talking about linear operators, the operator is signified generally by the letters T or L. Linear operators are those which satisfy the following conditions; take the general operator T, the function acted on under the operator T, written as f(x), and the constant a: :

T\ (f(x)+g(x)) = T\ (f(x))+T\ (g(x))

T\ (af(x)) = a\,T\ (f(x))

Many operators are linear. For example, the differential operator and Laplacian operator, which we will see later. Linear operators are also known as linear transformations or linear mappings. Many other operators one encounters in mathematics are linear, and linear operators are the most easily studied (Compare with nonlinearity). Such an example of a linear transformation between vectors in R² is reflection, given a vector x=(x₁, x₂) :Q(x₁, x₂)=(-x₁, x₂) We can also make sense of linear operators between generalisations of finite-dimensional vector spaces. For example, there is a large body of work dealing with linear operators on Hilbert spaces and on Banach spaces. See also operator algebra.

Operators in probability theory

Main article: Probability theory Operators are also involved in probability theory. Such operators as expectation, variance, covariance, factorials, et al.

Operators in calculus

Calculus is, essentially, the study of one particular operator, and its behavior embodies and exemplifies the idea of the operator very clearly. The key operator studied is the differential operator. It is linear, as are many of the operators constructed from it.

The differential operator

Main article: Differential operator The differential operator is an operator which is fundamentally used in Calculus to denote the action of taking a derivative. Common notations are such d/dx, y'(x) to denote the derivative of y(x). However here we will use the notation that is closest to the operator notation we have been using, that is, using D f to represent the action of taking the derivative of f.

Integral operators

Given that integration is an operator as well (inverse of differentiation), we have some important operators we can write in terms of integration.

Convolution

Main article: Convolution The convolution of two functions is a mapping from two functions to one other, defined by an integral as follows: If x₁=f(t) and x₂=g(t), define the operator Q such that; :

Q\ x_1\ x_2\ = \int f(t) g(\tau - t) dt

which we write as

(f - g)(\tau)

Fourier transform

Main article: Fourier transform The Fourier transform is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few. It is another integral operator; it is useful mainly because it converts a function on one (spatial) domain to a function on another (frequency) domain, in a way that is effectively invertible. Nothing significant is lost, because there is an inverse transform operator. In the simple case of periodic functions, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine waves and cosine waves: :

f(t) =  + \sum_^

When dealing with general function R → C, the transform takes on an integral form: :

f(t) =  \int_^

Laplacian transform

Main article: Laplace transform The Laplace transform is another integral operator and is involved in simplifying the process of solving differential equations. Given f=f(s), it is defined by: :

F(s) = (\mathcalf)(s) =\int_0^\infty e^ f(t)\,dt.

Fundamental operators on scalar and vector fields

Main articles: vector calculus, scalar field, gradient, divergence, and curl Three main operators are key to vector calculus, the operator ∇, known as gradient, where at a certain point in a scalar field forms a vector which points in the direction of greatest change of that scalar field. In a vector field, the divergence is an operator that measures a vector field's tendency to originate from or converge upon a given point. Curl, in a vector field, is a vector operator that shows a vector field's tendency to rotate about a point.

Operators in physics

Main article: Operator (physics) In physics, an operator often takes on a more specialized meaning than in mathematics. Operators as observables are a key part of the theory of quantum mechanics. In that context operator often means a linear transformation from a Hilbert space to another, or (more abstractly) an element of a C
- -algebra.

Operator (programming)

Programming languages generally have a set of operators that are similar to operators in mathematics: they are somehow special functions. In addition to arithmetic operations they often perform boolean operations on truth values and string operations on strings of text. Unlike functions, operators often provide the primitive operations of the language, their name consists of punctuation rather than alphanumeric characters, and they have special infix syntax and irregular parameter passing conventions. The exact terminology, however, varies from language to language. Conventionally, the computing usage of "operator" goes beyond the set of common arithmetic operators. The C programming language for example also supports operators like &, ++ and sizeof. Operators like sizeof, which are alphanumeric rather than a mathematical symbol or a punctuation character, are sometimes called named operators. See Operators in C and C++. Operators in C are primitive operations of the language that the compiler can fairly directly map into the machine instructions of microprocessors. In Haskell, any combination of symbols and punctuation can be used as a binary operator. The operation is defined like a function, and precedence and associativity can be set. Operators are a purely syntactic concept. An operator can be used as a function and vice versa by putting the name into parentheses or backticks respectively. In certain programming languages, such as PostScript, the use of the word "operator" has more specific meaning, in that an operator is an executable element in the stack. Because operators here are always written postfix, the need for parentheses is redundant as the way objects are taken from the stack ensures correct evaluation. This is an example of Reverse Polish notation.

Operator syntax

Computers are mathematical devices, but compilers and interpreters require a full syntactic theory of all operations in order to parse formulae involving any combinations correctly. In particular they depend on operator precedence rules, on order of operations, that are tacitly assumed in mathematical writing. Operators are binary or unary. Binary operators ("bi" as in "two") have two operands. In "A
- B" the
- operator has two operands: A and B. In "!B" the "!" operator (meaning boolean NOT) has only one operand, and is therefore a unary operator. The "-" and "+" operators can be both binary and unary, in "-4" or "+4" it denotes a negative or a positive number, in "0-4" it acts as the subtraction operator. Operators have associativity. This defines the correct evaluation order if a sequence of the same operator is used one after another: whether the evaluator will evaluate the left operations first or the right. For example, in 8 - 4 - 2, since subtraction in most languages is left associative, the so that expression evaluates left to right. 8 - 4 is evaluated first making restult 2, and not 6. The ^ operator (sometime written
- ) is often right assosiative. So 4^3^2 equals 4^9, not 64^2. : See Infix notation for more information about infix operator syntax.

Operator overloading

Main article: Operator overloading In some programming languages an operator may work with more than one kind of data, (such as in Java where the + operator is used both for the addition of numbers and for the concatenation of strings). Such an operator is said to be overloaded. In languages that support operator overloading by the programmer, such as C++, one can define customized uses for operators; in Prolog, one can also define new operators.

Operand coercion

Some languages also allow for the operands of an operator to be implicitly converted or coerced to suitable data types for the operation to occur. For example, in Perl coercion rules lead into 12 + "3.14" producing the result of 15.14. The text "3.14" is converted to the number 3.14 before addition can take place. Further, 12 is an integer and 3.14 is either a floating or fixed point number (a number that has a decimal place in it) so the integer is then converted to a floating point or fixed point number respectively. In the presence of coercions in a language, the programmer must be aware of the specific rules regarding operand types and the operation result type to avoid subtle programming mistakes. : See Type conversion for more information about coercion. Category:Programming constructs ja:演算子

Procedure

A procedure is a series of activities, tasks, steps, decisions, calculations and other processes, that when undertaken in the sequence laid down produces the described result, product or outcome. Following a procedure should produce repeatable results for the same input conditions. For this reason, formal written procedures are usually used in manufacturing and process industry operations to ensure safety and consistency. However, constant review or continuous improvement along with proper change management is usually advisable to ensure the continued applicability of written procedures in the face of change in supplies, product specifications, or surrounding work processes. Recorded procedures are usually called instructions or recipes. High-level procedures are sometimes known as methods. For example, the procedure for baking a cake from a packet mix may consist of the following steps: # Pre-heating the oven. # Opening the mix box. # Mixing the batter. # Putting the cake in. # Removing it when it's done. In mathematics and science, a procedure, or algorithm, is a sequence of tasks or calculations that accomplish some goal. In computer science, procedure is a subprogram, generally one which returns or is intended to return no direct result to its caller, and a stored procedure is a series of instructions that are stored within a database. In law, procedure is the body of law and rules used in the administration of justice in the court system, see: civil procedure, criminal procedure, administrative procedure, labour procedure. In government, parliamentary procedure is the process used for decision making by a legislative assembly.

Input

The term input has a variety of uses in different fields.

Information processing

In information processing, input refers to either information received or the process of receiving it:
- In human-computer interaction, input is the information produced by the user with the purpose of controlling the computer program. The user interface determines what kinds of input the programs accepts (for example, control strings or text typed with keyboard and mouse clicks).
- Input also comes from networks and storage devices such as disk drives.

Control theory

In control theory, the inputs of a system are the signals that can be observed or affected that feed into the system. Specifically, inputs are differentiated from states.

Equity theory

In equity theory, inputs are the skills, time, effort, expertise, experience or qualifications that an employee brings to his job.

Acceleration

In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. It is thus a vector quantity with dimension length/time². In SI units, this is meter/second².

Explanation

To accelerate an object is to change its velocity over a period of time. In this strict scientific sense, acceleration can have positive and negative values – respectively called acceleration and deceleration (or retardation) in common speech – as well as change of direction. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and is given by the equation :

\mathbf =

where :a is the acceleration vector :v is the velocity vector expressed in m/s :t is time expressed in seconds. This equation gives a the units of m/(s·s), or m/s² (read as "metres per second per second", or "metres per second squared"). An alternative equation is: :

\mathbf =

where :ā is the average acceleration (m/s²) :u is the initial velocity (m/s) :v is the final velocity (m/s) :t is the time interval (s) Transverse acceleration (perpendicular to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have :

\mathbf = - \frac \frac = - \omega^2 \mathbf

One common unit of acceleration is g, one g being the acceleration caused by the gravity of Earth at sea level at 45° latitude (Paris), or about 9.81 m/s². Jerk is the rate of change of an object's acceleration over time. In classical mechanics, acceleration

a \

is related to force

F \

and mass

m \

(assumed to be constant) by way of Newton's second law: :

F = m \cdot a

As a result of its invariance under the Galilean transformations, acceleration is an absolute quantity in classical mechanics.

Relation to relativity

After defining his theory of special relativity, Albert Einstein realized that forces felt by objects undergoing constant acceleration are indistinguishable from those in a gravitational field, and thus defined general relativity that also explained how gravity's effects could be limited by the speed of light. If you accelerate away from your friend, you could say (given your frame of reference) that it is your friend who is accelerating away from you, although only you feel any force. This is also the basis for the popular Twin paradox, which asks why only one twin ages when moving away from his sibling at near light-speed and then returning, since the aging twin can say that it is the other twin that was moving. General relativity solved the "why does only one object feel accelerated?" problem which had plagued philosophers and scientists since Newton's time (and caused Newton to endorse absolute space). In special relativity, only inertial frames of reference (non-accelerated frames) can be used and are equivalent; general relativity considers all frames, even accelerated ones, to be equivalent. With changing velocity, accelerated objects exist in warped space (as do those that reside in a gravitational field). Therefore, frames of reference must include a description of their local spacetime curvature to qualify as complete. Acceleration can be measured using an accelerometer.

References

-
-

External links and references

- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- [http://www.sandia.gov/LabNews/labs-accomplish/2005/pulp.html Experiments on Z produced a world-record peak velocity of 34 km/s] (that is about 76,000 mph)
- [http://www.sandia.gov/media/NewsRel/NR2001/flyer.htm Magnetic field shocklessly shoots pellets 20 times faster than rifle bullet] Category:Physical quantity Category:Classical mechanics ko:가속도 ja:加速度 simple:Acceleration th:ความเร่ง

Formula

: For help with typing formulas in Wikipedia markup, see help:formula. In mathematics and in the sciences, a formula (formulas or formulae in plural form) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. One of many famous formulae is Albert Einstein's E=mc² (see special relativity). When one attempts to write a chemical formula, it is important to know whether the substance in question actually exists. For example, one can easily write the formula of carbon nitrate, but no chemist has ever prepared this compound. Here are basic rules for writing chemical formulae: 1. Represent the symbols of the components, placing the positive part first, and then the negative part. 2. Indicate the respective oxidation numbers above and to the right of each symbol. (Enclose radicals in parentheses for the time being.) 3. For each symbol, write a subscript number equal to the oxidation number of the other element or radical. This is the same as the mechanical crisscross method. Since the positive oxidation number shows the number of electrons that may be lost or shared and the negative oxidation number shows the number of electrons that may be gained or shared, you must have just as many electrons lost (or partially lost in sharing) as are gained (or partially gained in sharing). 4. Now rewrite the formulas, omitting the subscript 1, the parentheses of the radicals that have the subscript 1, and the plus and minus numbers. 5. As a general rule, the subscript numbers in the final formula are reduced to their lowest terms. There are, however, certain exception, such as hydrogen peroxide(H₂O₂, and acetylene C₂H₂). For these exceptions, you must have more specific information about the compound. The only way to become proficient at writing formulae is to memorize the oxidations numbers of common elements (or learn to use the periodic chart group numbers) and practice writing formulae.

Other meanings

- Formula auto racing (for example, Formula One)
- Formula Language (Lotus Notes programming language)
- Infant formula (baby feeding substitute for breast milk)
- In a biological/chemical lab, formula can also mean something like a recipe, similar to the term for infant formula ja:公式

Determinism

:This article is about the general notion of determinism in philosophy. For other uses of the word "determinism" see: Deterministic (disambiguation). Determinism is the philosophical proposition that every event, including human cognition and action, is causally determined by an unbroken chain of prior occurrences. No mysterious miracles or wholly random events occur. If there has been even one indeterministic event since the beginning of time, then determinism is false.

Philosophy of determinism

The principal consequence of deterministic philosophy is that free will (except as defined in strict compatibilism) becomes an illusion. It is a popular misconception that determinism necessarily entails that all future events have already been determined (a position known as Fatalism); this is not obviously the case, and the subject is still debated among metaphysicians. Determinism is associated with, and relies upon, the ideas of Materialism and Causality. Some of the philosophers who have dealt with this issue are Omar Khayyám, David Hume, Thomas Hobbes, Immanuel Kant, and, more recently, John Searle. ::: With Earth's first Clay They did the Last Man's knead, ::: And then of the Last Harvest sow'd the Seed: ::: Yea, the first Morning of Creation wrote ::: What the Last Dawn of Reckoning shall read. :::::(Rubaiyat of Omar Khayyam, LIII, rendered into English verse by Edward FitzGerald)

The nature of determinism

The exact meaning of the term "determinism" has historically been subject to various interpretations. Some view determinism and free will as mutually exclusive, whereas others, labelled "Compatibilists", believe that the two ideas can be coherently reconciled. Most of this disagreement is due to the fact that the definition of "free will," like determinism, varies. Some feel it refers to the metaphysical truth of independent agency, whereas others simply define it as the feeling of agency that humans experience when they act. For example, David Hume argued that while it is possible that one does not freely arrive at one's set of desires and beliefs, the only meaningful interpretation of freedom relates to one's ability to translate those desires and beliefs into voluntary action.

Determinism in Western tradition

The idea that the entire universe is a deterministic system has been articulated in both Western and non-Western religion, philosophy, and literature. The Ancient Greek atomists Leucippus and Democritus were the first to anticipate determinism when they theorized that all processes in the world were due to the mechanical interplay of atoms, but this theory did not gain much support at the time. Determinism in the West is often associated with Newtonian physics, which depicts the physical matter of the universe as operating according to a set of fixed, knowable laws. The "billiard ball" hypothesis, a product of Newtonian physics, argues that once the initial conditions of the universe have been established the rest of the history of the universe follows inevitably. If it were actually possible to have complete knowledge of physical matter and all of the laws governing that matter at any one time, then it would be theoretically possible to compute the time and place of every event that will ever occur (Laplace's demon). In this sense, the basic particles of the universe operate in the same fashion as the rolling balls on a billiard table, moving and striking each other in predictable ways to produce predictable results. Whether or not it is all-encompassing in so doing, Newtonian mechanics deals only with caused events, e.g.: If the original position of an object is x, y, z, and if it is hit dead on by an object moving along some vector V, then it will be pushed straight toward another point x', y', z'. If it goes somewhere else, the Newtonians argue, one must question one's measurements of the original position of the

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Map (mathematics)

Transformation (mathematics)

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Binary function

Operator

Operators and levels of abstraction

Describing operators

Notations

Examples of mathematical operators

Linear operators

Operators in probability theory

Operators in calculus

The differential operator

Integral operators

Convolution

Fourier transform

Laplacian transform

Fundamental operators on scalar and vector fields

Operators in physics

See also

Operator (programming)

Operator syntax

Operator overloading

Operand coercion

Procedure

See also

Input

Information processing

Control theory

Equity theory

See also

Acceleration

Explanation

Relation to relativity

References

External links and references

Formula

Other meanings

Determinism

Philosophy of determinism

The nature of determinism

Determinism in Western tradition