Sequence

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an index set that may not be numbers to another set of elements.

For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...).

The position of an element in a sequence is its rank or index; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In mathematical analysis, a sequence is often denoted by letters in the form of ${\displaystyle a_{n}}$, ${\displaystyle b_{n}}$ and ${\displaystyle c_{n}}$, where the subscript n refers to the nth element of the sequence; for example, the nth element of the Fibonacci sequence ${\displaystyle F}$ is generally denoted as ${\displaystyle F_{n}}$.

In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

An infinite sequence of real numbers (in blue). This sequence is neither increasing, decreasing, convergent, nor Cauchy. It is, however, bounded.

Examples and notation

A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right, and can be studied as patterns or puzzles, such as in the study of prime numbers.

There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify a sequence is to list all its elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation is used for infinite sequences as well. For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Because notating sequences with ellipsis leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Other ways of denoting a sequence are discussed after the examples.

Examples

A tiling with squares whose sides are successive Fibonacci numbers in length.

The prime numbers are the natural numbers greater than 1 that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in mathematics, particularly in number theory where many results related to them exist.

The Fibonacci numbers comprise the integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...).

Other examples of sequences include those made up of rational numbers, real numbers and complex numbers. The sequence (.9, .99, .999, .9999, ...), for instance, approaches the number 1. In fact, every real number can be written as the limit of a sequence of rational numbers (e.g. via its decimal expansion). As another example, π is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. A related sequence is the sequence of decimal digits of π, that is, (3, 1, 4, 1, 5, 9, ...). Unlike the preceding sequence, this sequence does not have any pattern that is easily discernible by inspection.

The On-Line Encyclopedia of Integer Sequences comprises a large list of examples of integer sequences.

Indexing

Other notations can be useful for sequences whose pattern cannot be easily guessed or for sequences that do not have a pattern such as the digits of π. One such notation is to write down a general formula for computing the nth term as a function of n, enclose it in parentheses, and include a subscript indicating the set of values that n can take. For example, in this notation the sequence of even numbers could be written as ${\displaystyle (2n)_{n\in \mathbb {N} }}$. The sequence of squares could be written as ${\displaystyle (n^{2})_{n\in \mathbb {N} }}$. The variable n is called an index, and the set of values that it can take is called the index set.

It is often useful to combine this notation with the technique of treating the elements of a sequence as individual variables. This yields expressions like ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$, which denotes a sequence whose nth element is given by the variable ${\displaystyle a_{n}}$. For example:

${\displaystyle {\begin{aligned}a_{1}&=1{\text{st element of }}(a_{n})_{n\in \mathbb {N} }\\a_{2}&=2{\text{nd element }}\\a_{3}&=3{\text{rd element }}\\&\;\;\vdots \\a_{n-1}&=(n-1){\text{th element}}\\a_{n}&=n{\text{th element}}\\a_{n+1}&=(n+1){\text{th element}}\\&\;\;\vdots \end{aligned}}}$

One can consider multiple sequences at the same time by using different variables; e.g. ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$ could be a different sequence than ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$. One can even consider a sequence of sequences: ${\displaystyle ((a_{m,n})_{n\in \mathbb {N} })_{m\in \mathbb {N} }}$ denotes a sequence whose mth term is the sequence ${\displaystyle (a_{m,n})_{n\in \mathbb {N} }}$.

An alternative to writing the domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values. For example, the notation ${\displaystyle (k^{2})_{k=1}^{10}}$ denotes the ten-term sequence of squares ${\displaystyle (1,4,9,\ldots ,100)}$. The limits ${\displaystyle \infty }$ and ${\displaystyle -\infty }$ are allowed, but they do not represent valid values for the index, only the supremum or infimum of such values, respectively. For example, the sequence ${\displaystyle (a_{n})_{n=1}^{\infty }}$ is the same as the sequence ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$, and does not contain an additional term "at infinity". The sequence ${\displaystyle (a_{n})_{n=-\infty }^{\infty }}$ is a bi-infinite sequence, and can also be written as ${\displaystyle (\ldots ,a_{-1},a_{0},a_{1},a_{2},\ldots )}$.

In cases where the set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes ${\displaystyle (a_{k})}$ for an arbitrary sequence. Often, the index k is understood to run from 1 to ∞. However, sequences are frequently indexed starting from zero, as in

${\displaystyle (a_{k})_{k=0}^{\infty }=(a_{0},a_{1},a_{2},\ldots ).}$

In some cases, the elements of the sequence are related naturally to a sequence of integers whose pattern can be easily inferred. In these cases, the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of odd numbers could be denoted in any of the following ways.

• ${\displaystyle (1,9,25,\ldots )}$
• ${\displaystyle (a_{1},a_{3},a_{5},\ldots ),\qquad a_{k}=k^{2}}$
• ${\displaystyle (a_{2k-1})_{k=1}^{\infty },\qquad a_{k}=k^{2}}$
• ${\displaystyle (a_{k})_{k=1}^{\infty },\qquad a_{k}=(2k-1)^{2}}$
• ${\displaystyle \left((2k-1)^{2}\right)_{k=1}^{\infty }}$

Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations, if the indexing set was understood to be the natural numbers. In the second and third bullets, there is a well-defined sequence ${\displaystyle (a_{k})_{k=1}^{\infty }}$, but it is not the same as the sequence denoted by the expression.

Defining a sequence by recursion

Main article: Recurrence relation

Sequences whose elements are related to the previous elements in a straightforward way are often defined using recursion. This is in contrast to the definition of sequences of elements as functions of their positions.

To define a sequence by recursion, one needs a rule, called recurrence relation to construct each element in terms of the ones before it. In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by successive applications of the recurrence relation.

The Fibonacci sequence is a simple classical example, defined by the recurrence relation

${\displaystyle a_{n}=a_{n-1}+a_{n-2},}$

with initial terms ${\displaystyle a_{0}=0}$ and ${\displaystyle a_{1}=1}$. From this, a simple computation shows that the first ten terms of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34.

A complicated example of a sequence defined by a recurrence relation is Recamán's sequence, defined by the recurrence relation

${\displaystyle {\begin{cases}a_{n}=a_{n-1}-n,\quad {\text{if the result is positive and not already in the previous terms,}}\\a_{n}=a_{n-1}+n,\quad {\text{otherwise}},\end{cases}}}$

with initial term ${\displaystyle a_{0}=0.}$

A linear recurrence with constant coefficients is a recurrence relation of the form

${\displaystyle a_{n}=c_{0}+c_{1}a_{n-1}+\dots +c_{k}a_{n-k},}$

where ${\displaystyle c_{0},\dots ,c_{k}}$ are constants. There is a general method for expressing the general term ${\displaystyle a_{n}}$ of such a sequence as a function of n; see Linear recurrence. In the case of the Fibonacci sequence, one has ${\displaystyle c_{0}=0,c_{1}=c_{2}=1,}$ and the resulting function of n is given by Binet's formula.

A holonomic sequence is a sequence defined by a recurrence relation of the form

${\displaystyle a_{n}=c_{1}a_{n-1}+\dots +c_{k}a_{n-k},}$

where ${\displaystyle c_{1},\dots ,c_{k}}$ are polynomials in n. For most holonomic sequences, there is no explicit formula for expressing ${\displaystyle a_{n}}$ as a function of n. Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a Taylor series whose sequence of coefficients is holonomic. The use of the recurrence relation allows a fast computation of values of such special functions.

Not all sequences can be specified by a recurrence relation. An example is the sequence of prime numbers in their natural order (2, 3, 5, 7, 11, 13, 17, ...).

Formal definition and basic properties

There are many different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the definitions and notations introduced below.

Definition

In this article, a sequence is formally defined as a function whose domain is an interval of integers. This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions of these kinds of sequences). However, many authors use a narrower definition by requiring the domain of a sequence to be the set of natural numbers. This narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition. In some contexts, to shorten exposition, the codomain of the sequence is fixed by context, for example by requiring it to be the set R of real numbers, the set C of complex numbers, or a topological space.

Although sequences are a type of function, they are usually distinguished notationally from functions in that the input is written as a subscript rather than in parentheses, that is, a_n rather than a(n). There are terminological differences as well: the value of a sequence at the lowest input (often 1) is called the "first element" of the sequence, the value at the second smallest input (often 2) is called the "second element", etc. Also, while a function abstracted from its input is usually denoted by a single letter, e.g. f, a sequence abstracted from its input is usually written by a notation such as ${\displaystyle (a_{n})_{n\in A}}$, or just as ${\displaystyle (a_{n}).}$ Here A is the domain, or index set, of the sequence.

Sequences and their limits (see below) are important concepts for studying topological spaces. An important generalization of sequences is the concept of nets. A net is a function from a (possibly uncountable) directed set to a topological space. The notational conventions for sequences normally apply to nets as well.

Finite and infinite

See also: ω-language

The length of a sequence is defined as the number of terms in the sequence.

A sequence of a finite length n is also called an n-tuple. Finite sequences include the empty sequence ( ) that has no elements.

Normally, the term infinite sequence refers to a sequence that is infinite in one direction, and finite in the other—the sequence has a first element, but no final element. Such a sequence is called a singly infinite sequence or a one-sided infinite sequence when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a bi-infinite sequence, two-way infinite sequence, or doubly infinite sequence. A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( ..., −4, −2, 0, 2, 4, 6, 8, ... ), is bi-infinite. This sequence could be denoted ${\displaystyle (2n)_{n=-\infty }^{\infty }}$.

Increasing and decreasing

A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For example, the sequence ${\displaystyle (a_{n})_{n=1}^{\infty }}$ is monotonically increasing if and only if a_n+1 ${\displaystyle \geq }$ a_n for all n ∈ N. If each consecutive term is strictly greater than (>) the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonically decreasing if each consecutive term is less than or equal to the previous one, and is strictly monotonically decreasing if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a monotonic function.

The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively.

Bounded

If the sequence of real numbers (a_n) is such that all the terms are less than some real number M, then the sequence is said to be bounded from above. In other words, this means that there exists M such that for all n, a_n ≤ M. Any such M is called an upper bound. Likewise, if, for some real m, a_n ≥ m for all n greater than some N, then the sequence is bounded from below and any such m is called a lower bound. If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded.

Subsequences

A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of the positive integers (1, 2, 3, ...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved.

Formally, a subsequence of the sequence ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ is any sequence of the form ${\displaystyle (a_{n_{k}})_{k\in \mathbb {N} }}$, where ${\displaystyle (n_{k})_{k\in \mathbb {N} }}$ is a strictly increasing sequence of positive integers.

Other types of sequences

Some other types of sequences that are easy to define include:

• An integer sequence is a sequence whose terms are integers.
• A polynomial sequence is a sequence whose terms are polynomials.
• A positive integer sequence is sometimes called multiplicative, if a_nm = a_n a_m for all pairs n, m such that n and m are coprime. In other instances, sequences are often called multiplicative, if a_n = na₁ for all n. Moreover, a multiplicative Fibonacci sequence satisfies the recursion relation a_n = a_n−1 a_n−2.
• A binary sequence is a sequence whose terms have one of two discrete values, e.g. base 2 values (0,1,1,0, ...), a series of coin tosses (Heads/Tails) H,T,H,H,T, ..., the answers to a set of True or False questions (T, F, T, T, ...), and so on.

Limits and convergence

Main article: Limit of a sequence

 

The plot of a convergent sequence (a_n) is shown in blue. From the graph we can see that the sequence is converging to the limit zero as n increases.

An important property of a sequence is convergence. If a sequence converges, it converges to a particular value known as the limit. If a sequence converges to some limit, then it is convergent. A sequence that does not converge is divergent.

Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value ${\displaystyle L}$ (called the limit of the sequence), and they become and remain arbitrarily close to ${\displaystyle L}$, meaning that given a real number ${\displaystyle d}$ greater than zero, all but a finite number of the elements of the sequence have a distance from ${\displaystyle L}$ less than ${\displaystyle d}$.

For example, the sequence ${\textstyle a_{n}={\frac {n+1}{2n^{2}}}}$ shown to the right converges to the value 0. On the other hand, the sequences ${\textstyle b_{n}=n^{3}}$ (which begins 1, 8, 27, …) and ${\displaystyle c_{n}=(-1)^{n}}$ (which begins −1, 1, −1, 1, …) are both divergent.

If a sequence converges, then the value it converges to is unique. This value is called the limit of the sequence. The limit of a convergent sequence ${\displaystyle (a_{n})}$ is normally denoted ${\textstyle \lim _{n\to \infty }a_{n}}$. If ${\displaystyle (a_{n})}$ is a divergent sequence, then the expression ${\textstyle \lim _{n\to \infty }a_{n}}$ is meaningless.

Formal definition of convergence

A sequence of real numbers ${\displaystyle (a_{n})}$ converges to a real number ${\displaystyle L}$ if, for all ${\displaystyle \varepsilon >0}$, there exists a natural number ${\displaystyle N}$ such that for all ${\displaystyle n\geq N}$ we have

${\displaystyle |a_{n}-L|<\varepsilon .}$

If ${\displaystyle (a_{n})}$ is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define convergence, with the provision that ${\displaystyle |\cdot |}$ denotes the complex modulus, i.e. ${\displaystyle |z|={\sqrt {z^{*}z}}}$. If ${\displaystyle (a_{n})}$ is a sequence of points in a metric space, then the formula can be used to define convergence, if the expression ${\displaystyle |a_{n}-L|}$ is replaced by the expression ${\displaystyle \operatorname {dist} (a_{n},L)}$, which denotes the distance between ${\displaystyle a_{n}}$ and ${\displaystyle L}$.

Applications and important results

If ${\displaystyle (a_{n})}$ and ${\displaystyle (b_{n})}$ are convergent sequences, then the following limits exist, and can be computed as follows:

• ${\displaystyle \lim _{n\to \infty }(a_{n}\pm b_{n})=\lim _{n\to \infty }a_{n}\pm \lim _{n\to \infty }b_{n}}$
• ${\displaystyle \lim _{n\to \infty }ca_{n}=c\lim _{n\to \infty }a_{n}}$ for all real numbers ${\displaystyle c}$
• ${\displaystyle \lim _{n\to \infty }(a_{n}b_{n})=\left(\lim _{n\to \infty }a_{n}\right)\left(\lim _{n\to \infty }b_{n}\right)}$
• ${\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}={\frac {\lim \limits _{n\to \infty }a_{n}}{\lim \limits _{n\to \infty }b_{n}}}}$, provided that ${\displaystyle \lim _{n\to \infty }b_{n}\neq 0}$
• ${\displaystyle \lim _{n\to \infty }a_{n}^{p}=\left(\lim _{n\to \infty }a_{n}\right)^{p}}$ for all ${\displaystyle p>0}$ and ${\displaystyle a_{n}>0}$

Moreover:

• If ${\displaystyle a_{n}\leq b_{n}}$ for all ${\displaystyle n}$ greater than some ${\displaystyle N}$, then ${\displaystyle \lim _{n\to \infty }a_{n}\leq \lim _{n\to \infty }b_{n}}$.
• (Squeeze Theorem)
  If ${\displaystyle (c_{n})}$ is a sequence such that ${\displaystyle a_{n}\leq c_{n}\leq b_{n}}$ for all ${\displaystyle n>N}$ and ${\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}=L}$,
  then ${\displaystyle (c_{n})}$ is convergent, and ${\displaystyle \lim _{n\to \infty }c_{n}=L}$.
• If a sequence is bounded and monotonic then it is convergent.
• A sequence is convergent if and only if all of its subsequences are convergent.

Cauchy sequences

Main article: Cauchy sequence

 

The plot of a Cauchy sequence (X_n), shown in blue, as X_n versus n. In the graph the sequence appears to be converging to a limit as the distance between consecutive terms in the sequence gets smaller as n increases. In the real numbers every Cauchy sequence converges to some limit.

A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is Cauchy characterization of convergence for sequences:

A sequence of real numbers is convergent (in the reals) if and only if it is Cauchy.

In contrast, there are Cauchy sequences of rational numbers that are not convergent in the rationals, e.g. the sequence defined by x₁ = 1 and x_n+1 = x_n + 2/x_n/2 is Cauchy, but has no rational limit, cf. here. More generally, any sequence of rational numbers that converges to an irrational number is Cauchy, but not convergent when interpreted as a sequence in the set of rational numbers.

Metric spaces that satisfy the Cauchy characterization of convergence for sequences are called complete metric spaces and are particularly nice for analysis.

Infinite limits

In calculus, it is common to define notation for sequences which do not converge in the sense discussed above, but which instead become and remain arbitrarily large, or become and remain arbitrarily negative. If ${\displaystyle a_{n}}$ becomes arbitrarily large as ${\displaystyle n\to \infty }$, we write

${\displaystyle \lim _{n\to \infty }a_{n}=\infty .}$

In this case we say that the sequence diverges, or that it converges to infinity. An example of such a sequence is a_n = n.

If ${\displaystyle a_{n}}$ becomes arbitrarily negative (i.e. negative and large in magnitude) as ${\displaystyle n\to \infty }$, we write

${\displaystyle \lim _{n\to \infty }a_{n}=-\infty }$

and say that the sequence diverges or converges to negative infinity.

Series

Main article: Series (mathematics)

A series is, informally speaking, the sum of the terms of a sequence. That is, it is an expression of the form ${\textstyle \sum _{n=1}^{\infty }a_{n}}$ or ${\displaystyle a_{1}+a_{2}+\cdots }$, where ${\displaystyle (a_{n})}$ is a sequence of real or complex numbers. The partial sums of a series are the expressions resulting from replacing the infinity symbol with a finite number, i.e. the Nth partial sum of the series ${\textstyle \sum _{n=1}^{\infty }a_{n}}$ is the number

${\displaystyle S_{N}=\sum _{n=1}^{N}a_{n}=a_{1}+a_{2}+\cdots +a_{N}.}$

The partial sums themselves form a sequence ${\displaystyle (S_{N})_{N\in \mathbb {N} }}$, which is called the sequence of partial sums of the series ${\textstyle \sum _{n=1}^{\infty }a_{n}}$. If the sequence of partial sums converges, then we say that the series ${\textstyle \sum _{n=1}^{\infty }a_{n}}$ is convergent, and the limit ${\textstyle \lim _{N\to \infty }S_{N}}$ is called the value of the series. The same notation is used to denote a series and its value, i.e. we write ${\textstyle \sum _{n=1}^{\infty }a_{n}=\lim _{N\to \infty }S_{N}}$.

Use in other fields of mathematics

Topology

Sequences play an important role in topology, especially in the study of metric spaces. For instance:

• A metric space is compact exactly when it is sequentially compact.
• A function from a metric space to another metric space is continuous exactly when it takes convergent sequences to convergent sequences.
• A metric space is a connected space if and only if, whenever the space is partitioned into two sets, one of the two sets contains a sequence converging to a point in the other set.
• A topological space is separable exactly when there is a dense sequence of points.

Sequences can be generalized to nets or filters. These generalizations allow one to extend some of the above theorems to spaces without metrics.

Product topology

The topological product of a sequence of topological spaces is the cartesian product of those spaces, equipped with a natural topology called the product topology.

More formally, given a sequence of spaces ${\displaystyle (X_{i})_{i\in \mathbb {N} }}$, the product space

${\displaystyle X:=\prod _{i\in \mathbb {N} }X_{i},}$

is defined as the set of all sequences ${\displaystyle (x_{i})_{i\in \mathbb {N} }}$ such that for each i, ${\displaystyle x_{i}}$ is an element of ${\displaystyle X_{i}}$. The canonical projections are the maps p_i : X → X_i defined by the equation ${\displaystyle p_{i}((x_{j})_{j\in \mathbb {N} })=x_{i}}$. Then the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections p_i are continuous. The product topology is sometimes called the Tychonoff topology.

Analysis

In analysis, when talking about sequences, one will generally consider sequences of the form

${\displaystyle (x_{1},x_{2},x_{3},\dots ){\text{ or }}(x_{0},x_{1},x_{2},\dots )}$

which is to say, infinite sequences of elements indexed by natural numbers.

A sequence may start with an index different from 1 or 0. For example, the sequence defined by x_n = 1/log(n) would be defined only for n ≥ 2. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given N.

The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This type can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often function spaces. Even more generally, one can study sequences with elements in some topological space.

Sequence spaces

Main article: Sequence space

A sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K, where K is either the field of real numbers or the field of complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

The most important sequences spaces in analysis are the ℓ^p spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of L^p spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c₀, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called an FK-space.

Linear algebra

Sequences over a field may also be viewed as vectors in a vector space. Specifically, the set of F-valued sequences (where F is a field) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.

Abstract algebra

Abstract algebra employs several types of sequences, including sequences of mathematical objects such as groups or rings.

Free monoid

Main article: Free monoid

If A is a set, the free monoid over A (denoted A^*, also called Kleene star of A) is a monoid containing all the finite sequences (or strings) of zero or more elements of A, with the binary operation of concatenation. The free semigroup A⁺ is the subsemigroup of A^* containing all elements except the empty sequence.

Exact sequences

Main article: Exact sequence

In the context of group theory, a sequence

${\displaystyle G_{0}\;{\xrightarrow {f_{1}}}\;G_{1}\;{\xrightarrow {f_{2}}}\;G_{2}\;{\xrightarrow {f_{3}}}\;\cdots \;{\xrightarrow {f_{n}}}\;G_{n}}$

of groups and group homomorphisms is called exact, if the image (or range) of each homomorphism is equal to the kernel of the next:

${\displaystyle \mathrm {im} (f_{k})=\mathrm {ker} (f_{k+1})}$

The sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms.

Spectral sequences

Main article: Spectral sequence

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.

Set theory

An ordinal-indexed sequence is a generalization of a sequence. If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. In this terminology an ω-indexed sequence is an ordinary sequence.

Computing

In computer science, finite sequences are called lists. Potentially infinite sequences are called streams. Finite sequences of characters or digits are called strings.

Streams

Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science. They are often referred to simply as sequences or streams, as opposed to finite strings. Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0, 1}). The set C = {0, 1}^∞ of all infinite binary sequences is sometimes called the Cantor space.

An infinite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to 1 if and only if the n th string (in shortlex order) is in the language. This representation is useful in the diagonalization method for proofs.

Anthropometry

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Anthropometry 

The field of ergonomics employs anthropometry to optimize human interaction with equipment and workplaces.

Anthropometry (from Ancient Greek ἄνθρωπος (ánthrōpos) 'human', and μέτρον (métron) 'measure') refers to the measurement of the human individual. An early tool of physical anthropology, it has been used for identification, for the purposes of understanding human physical variation, in paleoanthropology and in various attempts to correlate physical with racial and psychological traits. Anthropometry involves the systematic measurement of the physical properties of the human body, primarily dimensional descriptors of body size and shape. Since commonly used methods and approaches in analysing living standards were not helpful enough, the anthropometric history became very useful for historians in answering questions that interested them.

Today, anthropometry plays an important role in industrial design, clothing design, ergonomics and architecture where statistical data about the distribution of body dimensions in the population are used to optimize products. Changes in lifestyles, nutrition, and ethnic composition of populations lead to changes in the distribution of body dimensions (e.g. the rise in obesity) and require regular updating of anthropometric data collections.

History

Main article: History of anthropometry

 

A Bertillon record for Francis Galton, from a visit to Bertillon's laboratory in 1893

The history of anthropometry includes and spans various concepts, both scientific and pseudoscientific, such as craniometry, paleoanthropology, biological anthropology, phrenology, physiognomy, forensics, criminology, phylogeography, human origins, and cranio-facial description, as well as correlations between various anthropometrics and personal identity, mental typology, personality, cranial vault and brain size, and other factors.

At various times in history, applications of anthropometry have ranged vastly—from accurate scientific description and epidemiological analysis to rationales for eugenics and overtly racist social movements—and its points of concern have been numerous, diverse, and sometimes highly unexpected.

Individual variation

Auxologic

Main article: Auxology

Auxologic is a broad term covering the study of all aspects of human physical growth.

Height

Main article: Human height

Human height varies greatly between individuals and across populations for a variety of complex biological, genetic, and environmental factors, among others. Due to methodological and practical problems, its measurement is also subject to considerable error in statistical sampling.

The average height in genetically and environmentally homogeneous populations is often proportional across a large number of individuals. Exceptional height variation (around 20% deviation from a population's average) within such a population is sometimes due to gigantism or dwarfism, which are caused by specific genes or endocrine abnormalities. It is important to note that a great degree of variation occurs between even the most 'common' bodies (66% of the population), and as such no person can be considered 'average'.

In the most extreme population comparisons, for example, the average female height in Bolivia is 142.2 cm (4 ft 8.0 in) while the average male height in the Dinaric Alps is 185.6 cm (6 ft 1.1 in), an average difference of 43.4 cm (1 ft 5.1 in). Similarly, the shortest and tallest of individuals, Chandra Bahadur Dangi and Robert Wadlow, have ranged from 1 ft 9 in (53 cm) to 8 ft 11.1 in (272 cm), respectively.

The age range where most females stop growing is 15–⁠18 years and the age range where most males stop growing is 18–⁠21 years.

Weight

Main article: Human weight

Human weight varies extensively both individually and across populations, with the most extreme documented examples of adults being Lucia Zarate who weighed 4.7 lb (2.1 kg), and Jon Brower Minnoch who weighed 1,400 lb (640 kg), and with population extremes ranging from 109.3 lb (49.6 kg) in Bangladesh to 192.7 lb (87.4 kg) in Micronesia.

Organs

Adult brain size varies from 974.9 cm³ (59.49 cu in) to 1,498.1 cm³ (91.42 cu in) in females and 1,052.9 cm³ (64.25 cu in) to 1,498.5 cm³ (91.44 cu in) in males, with the average being 1,130 cm³ (69 cu in) and 1,260 cm³ (77 cu in), respectively. The right cerebral hemisphere is typically larger than the left, whereas the cerebellar hemispheres are typically of more similar size.

Size of the human stomach varies significantly in adults, with one study showing volumes ranging from 520 cm³ (32 cu in) to 1,536 cm³ (93.7 cu in) and weights ranging from 77 grams (2.7 oz) to 453 grams (16.0 oz).

Male and female genitalia exhibit considerable individual variation, with penis size differing substantially and vaginal size differing significantly in healthy adults.

Aesthetic

Main articles: Human physical appearance and physical attractiveness

Human beauty and physical attractiveness have been preoccupations throughout history which often intersect with anthropometric standards. Cosmetology, facial symmetry, and waist–hip ratio are three such examples where measurements are commonly thought to be fundamental.

Evolutionary science

Anthropometric studies today are conducted to investigate the evolutionary significance of differences in body proportion between populations whose ancestors lived in different environments. Human populations exhibit climatic variation patterns similar to those of other large-bodied mammals, following Bergmann's rule, which states that individuals in cold climates will tend to be larger than ones in warm climates, and Allen's rule, which states that individuals in cold climates will tend to have shorter, stubbier limbs than those in warm climates.

On a microevolutionary level, anthropologists use anthropometric variation to reconstruct small-scale population history. For instance, John Relethford's studies of early 20th-century anthropometric data from Ireland show that the geographical patterning of body proportions still exhibits traces of the invasions by the English and Norse centuries ago.

Similarly, anthropometric indices, namely comparison of the human stature was used to illustrate anthropometric trends. This study was conducted by Jörg Baten and Sandew Hira and was based on the anthropological founds that human height is predetermined by the quality of the nutrition, which used to be higher in the more developed countries. The research was based on the datasets for Southern Chinese contract migrants who were sent to Suriname and Indonesia and included 13,000 individuals.

Measuring instruments

3D body scanners

Today anthropometry can be performed with three-dimensional scanners. A global collaborative study to examine the uses of three-dimensional scanners for health care was launched in March 2007. The Body Benchmark Study will investigate the use of three-dimensional scanners to calculate volumes and segmental volumes of an individual body scan. The aim is to establish whether the Body Volume Index has the potential to be used as a long-term computer-based anthropometric measurement for health care. In 2001 the UK conducted the largest sizing survey to date using scanners. Since then several national surveys have followed in the UK's pioneering steps, notably SizeUSA, SizeMexico, and SizeThailand, the latter still ongoing. SizeUK showed that the nation had become taller and heavier but not as much as expected. Since 1951, when the last women's survey had taken place, the average weight for women had gone up from 62 to 65 kg. However, recent research has shown that posture of the participant significantly influences the measurements taken, the precision of 3D body scanner may or may not be high enough for industry tolerances, and measurements taken may or may not be relevant to all applications (e.g. garment construction). Despite these current limitations, 3D Body Scanning has been suggested as a replacement for body measurement prediction technologies which (despite the great appeal) have yet to be as reliable as real human data.

Baropodographic

Main article: Baropodography

 

Example insole (in-shoe) foot pressure measurement device

Baropodographic devices fall into two main categories: (i) floor-based, and (ii) in-shoe. The underlying technology is diverse, ranging from piezoelectric sensor arrays to light refraction, but the ultimate form of the data generated by all modern technologies is either a 2D image or a 2D image time series of the pressures acting under the plantar surface of the foot. From these data other variables may be calculated (see data analysis).

The spatial and temporal resolutions of the images generated by commercial pedobarographic systems range from approximately 3 to 10 mm and 25 to 500 Hz, respectively. Sensor technology limits finer resolution. Such resolutions yield a contact area of approximately 500 sensors (for a typical adult human foot with surface area of approximately 100 cm²). For a stance phase duration of approximately 0.6 seconds during normal walking, approximately 150,000 pressure values, depending on the hardware specifications, are recorded for each step.

Neuroimaging

See also: Neuroimaging

Direct measurements involve examinations of brains from corpses, or more recently, imaging techniques such as MRI, which can be used on living persons. Such measurements are used in research on neuroscience and intelligence. Brain volume data and other craniometric data are used in mainstream science to compare modern-day animal species and to analyze the evolution of the human species in archeology.

Epidemiology and medical anthropology

Anthropometric measurements also have uses in epidemiology and medical anthropology, for example in helping to determine the relationship between various body measurements (height, weight, percentage body fat, etc.) and medical outcomes. Anthropometric measurements are frequently used to diagnose malnutrition in resource-poor clinical settings.

Forensics and criminology

An early set of finger- and handprints by Sir William Herschel, 2nd Baronet (1833–1917)

 

Further information: Forensic anthropology and anthropological criminology

Forensic anthropologists study the human skeleton in a legal setting. A forensic anthropologist can assist in the identification of a decedent through various skeletal analyses that produce a biological profile. Forensic anthropologists utilize the Fordisc program to help in the interpretation of craniofacial measurements in regards to ancestry determination.

One part of a biological profile is a person's ancestral affinity. People with significant European or Middle Eastern ancestry generally have little to no prognathism; a relatively long and narrow face; a prominent brow ridge that protrudes forward from the forehead; a narrow, tear-shaped nasal cavity; a "silled" nasal aperture; tower-shaped nasal bones; a triangular-shaped palate; and an angular and sloping eye orbit shape. People with considerable African ancestry typically have a broad and round nasal cavity; no dam or nasal sill; Quonset hut-shaped nasal bones; notable facial projection in the jaw and mouth area (prognathism); a rectangular-shaped palate; and a square or rectangular eye orbit shape. A relatively small prognathism often characterizes people with considerable East Asian ancestry; no nasal sill or dam; an oval-shaped nasal cavity; tent-shaped nasal bones; a horseshoe-shaped palate; and a rounded and non-sloping eye orbit shape. Many of these characteristics are only a matter of frequency among those of particular ancestries: their presence or absence of one or more does not automatically classify an individual into an ancestral group.

Ergonomics

Main article: Human factors and ergonomics

Today, ergonomics professionals apply an understanding of human factors to the design of equipment, systems and working methods to improve comfort, health, safety, and productivity. This includes physical ergonomics in relation to human anatomy, physiological and bio mechanical characteristics; cognitive ergonomics in relation to perception, memory, reasoning, motor response including human–computer interaction, mental workloads, decision making, skilled performance, human reliability, work stress, training, and user experiences; organizational ergonomics in relation to metrics of communication, crew resource management, work design, schedules, teamwork, participation, community, cooperative work, new work programs, virtual organizations, and telework; environmental ergonomics in relation to human metrics affected by climate, temperature, pressure, vibration, and light; visual ergonomics; and others.

Biometrics

2009 photo showing a man having a retinal scan taken by a U.S. Army soldier

 

Main article: Biometrics

Biometrics refers to the identification of humans by their characteristics or traits. Biometrics is used in computer science as a form of identification and access control. It is also used to identify individuals in groups that are under surveillance. Biometric identifiers are the distinctive, measurable characteristics used to label and describe individuals. Biometric identifiers are often categorized as physiological versus behavioral characteristics. Subclasses include dermatoglyphics and soft biometrics.

United States military research

The US Military has conducted over 40 anthropometric surveys of U.S. Military personnel between 1945 and 1988, including the 1988 Army Anthropometric Survey (ANSUR) of men and women with its 240 measures. Statistical data from these surveys encompasses over 75,000 individuals.

Civilian American and European Surface Anthropometry Resource Project — CAESAR

CAESAR began in 1997 as a partnership between government (represented by the US Air Force and NATO) and industry (represented by SAE International) to collect and organize the most extensive sampling of consumer body measurements for comparison.

The project collected and organized data on 2,400 U.S. & Canadian and 2,000 European civilians and a database was developed. This database records the anthropometric variability of men and women, aged 18–65, of various weights, ethnic groups, gender, geographic regions, and socio-economic status. The study was conducted from April 1998 to early 2000 and included three scans per person in a standing pose, full-coverage pose and relaxed seating pose.

Data collection methods were standardized and documented so that the database can be consistently expanded and updated. High-resolution measurements of body surfaces were made using 3D Surface Anthropometry. This technology can capture hundreds of thousands of points in three dimensions on the human body surface in a few seconds. It has many advantages over the old measurement system using tape measures, anthropometers, and other similar instruments. It provides detail about the surface shape as well as 3D locations of measurements relative to each other and enables easy transfer to Computer-Aided Design (CAD) or Manufacturing (CAM) tools. The resulting scan is independent of the measurer, making it easier to standardize. Automatic landmark recognition (ALR) technology was used to extract anatomical landmarks from the 3D body scans automatically. Eighty landmarks were placed on each subject. More than 100 univariate measures were provided, over 60 from the scan and approximately 40 using traditional measurements.

Demographic data such as age, ethnic group, gender, geographic region, education level, and present occupation, family income and more were also captured.

Fashion design

Scientists working for private companies and government agencies conduct anthropometric studies to determine a range of sizes for clothing and other items. For just one instance, measurements of the foot are used in the manufacture and sale of footwear: measurement devices may be used either to determine a retail shoe size directly (e.g. the Brannock Device) or to determine the detailed dimensions of the foot for custom manufacture (e.g. ALINEr).

In popular culture

In art Yves Klein termed his performance paintings anthropometries, where he covered nude women with paint and used their bodies as paintbrushes.