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129.132.147.26: /* Version for Quine's New Foundations */

{{Refimprove|date=April 2012}}
[[File:Surreal number tree.svg|thumb|The surreal number tree visualization.]]
In [[mathematics]], the '''surreal number''' system is an [[Linear continuum|arithmetic continuum]] containing the [[real number]]s as well as [[Infinity|infinite]] and [[infinitesimal|infinitesimal numbers]], respectively larger or smaller in [[absolute value]] than any positive real number. The surreals share many properties with the reals, including a [[total order]] ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an [[ordered field]]. (Strictly speaking, the surreals are not a [[Set (mathematics)|set]], but a proper [[class (set theory)|class]].<ref>In the original formulation using [[von Neumann–Bernays–Gödel set theory]], the surreals form a proper class, rather than a set, so the term [[field (mathematics)|field]] is not precisely correct; where this distinction is important, some authors use Field or FIELD to refer to a proper class that has the arithmetic properties of a field. One can obtain a true field by limiting the construction to a [[Grothendieck universe]], yielding a set with the cardinality of some [[strongly inaccessible cardinal]], or by using a form of set theory in which constructions by [[transfinite recursion]] stop at some countable ordinal such as [[Epsilon numbers (mathematics)|epsilon nought]].</ref>) If formulated in [[Von Neumann–Bernays–Gödel set theory]], the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the [[rational function]]s, the [[Levi-Civita field]], the [[superreal number]]s, and the [[hyperreal number]]s, can be realized as subfields of the surreals.<ref name=bajnok>{{cite book|last=Bajnok|first=Béla|title=An Invitation to Abstract Mathematics|year=2013|quote=Theorem 24.29. The surreal number system is the largest ordered field}}</ref> It has also been shown (in [[Von Neumann–Bernays–Gödel set theory]]) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the [[axiom of global choice]], this need not be the case, and in such theories it is not necessarily true that the surreals are the largest ordered field. The surreals also contain all [[transfinite number|transfinite]] [[ordinal number]]s; the arithmetic on them is given by the [[Ordinal arithmetic#Natural operations|natural operations]].

The definition and construction of the surreals is due to [[John Horton Conway]]. They were introduced in [[Donald Knuth]]'s 1974 book ''Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness''. This book is a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea was first presented in a work of fiction. In his book, which takes the form of a dialogue, Knuth coined the term ''surreal numbers'' for what Conway had simply called ''numbers'' originally. Conway liked the new name, and later adopted it himself. Conway then described the surreal numbers and used them for analyzing games in his 1976 book ''[[On Numbers and Games]]''.

==Overview==

The surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers ''a'' and ''b'' either ''a'' ≤ ''b'' or ''b'' ≤ ''a''. (Both may hold, in which case ''a'' and ''b'' are equivalent and denote the same number.) Numbers are formed by pairing subsets of numbers already constructed: given subsets ''L'' and ''R'' of numbers such that all the members of ''L'' are strictly less than all the members of ''R'', then the pair {{nowrap begin}}{ ''L'' | ''R'' }{{nowrap end}} represents a number intermediate in value between all the members of ''L'' and all the members of ''R''.

Different subsets may end up defining the same number: {{nowrap begin}}{ ''L'' | ''R'' }{{nowrap end}} and {{nowrap begin}}{ ''L′'' | ''R′'' }{{nowrap end}} may define the same number even if ''L'' ≠ ''L′'' and ''R'' ≠ ''R′''. (A similar phenomenon occurs when [[rational numbers]] are defined as quotients of integers: 1/2 and 2/4 are different representations of the same rational number.) So strictly speaking, the surreal numbers are [[equivalence class]]es of representations of form {{nowrap begin}}{ ''L'' | ''R'' }{{nowrap end}} that designate the same number.

In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: {{nowrap begin}}{ | }{{nowrap end}}. This representation, where ''L'' and ''R'' are both empty, is called 0. Subsequent stages yield forms like:

:{{nowrap begin}}{ 0 | }{{nowrap end}} = 1

:{{nowrap begin}}{ 1 | }{{nowrap end}} = 2

:{{nowrap begin}}{ 2 | }{{nowrap end}} = 3

and

:{{nowrap begin}}{ | 0 }{{nowrap end}} = −1

:{{nowrap begin}}{ | −1 }{{nowrap end}} = −2

:{{nowrap begin}}{ | −2 }{{nowrap end}} = −3

The integers are thus contained within the surreal numbers. Similarly, representations arise like:

:{{nowrap begin}}{ 0 | 1 }{{nowrap end}} = 1/2

:{{nowrap begin}}{ 0 | 1/2 }{{nowrap end}} = 1/4

:{{nowrap begin}}{ 1/2 | 1 }{{nowrap end}} = 3/4

so that the [[dyadic rational]]s (rational numbers whose denominators are powers of 2) are contained within the surreal numbers.

After an infinite number of stages, infinite subsets become available, so that any [[real numbers|real number]] ''a'' can be represented by {{nowrap begin}}{ ''La'' | ''Ra'' },{{nowrap end}}
where ''La'' is the set of all dyadic rationals less than ''a'' and
''Ra'' is the set of all dyadic rationals greater than ''a'' (reminiscent of a [[Dedekind cut]]). Thus the real numbers are also embedded within the surreals.

But there are also representations like

:{{nowrap begin}}{ 0, 1, 2, 3, … | }{{nowrap end}} = ω

:{{nowrap begin}}{ 0 | 1, 1/2, 1/4, 1/8, … }{{nowrap end}} = ε

where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about 2ω or ω − 1 and so forth.

==Construction==

Surreal numbers are [[inductive definition|constructed inductively]] as [[equivalence class]]es of [[ordered pair|pair]]s of sets of surreal numbers, restricted by the condition that each element of the first set is smaller than each element of the second set. The construction consists of three interdependent parts: the construction rule, the comparison rule and the equivalence rule.

===Forms===
A ''form'' is a pair of sets of surreal numbers, called its ''left set'' and its ''right set''. A form with left set ''L'' and right set ''R'' is written {{nowrap begin}}{ ''L'' | ''R'' }{{nowrap end}}. When ''L'' and ''R'' are given as lists of elements, the braces around them are omitted.

Either or both of the left and right set of a form may be the empty set. The form {{nowrap begin}}{ { } | { } }{{nowrap end}} with both left and right set empty is also written {{nowrap begin}}{ | }{{nowrap end}}.

===Numeric forms===

'''Construction Rule'''
:A form { ''L'' | ''R'' } is ''numeric'' if the intersection of ''L'' and ''R'' is the empty set and each element of ''R'' is greater than every element of ''L'', according to the [[order theory|order relation]] ≤ given by the comparison rule below.

===Equivalence classes of numeric forms===

The numeric forms are placed in equivalence classes; each such equivalence class is a ''surreal number''. The elements of the left and right set of a form are drawn from the universe of the surreal numbers (not of ''forms'', but of their ''equivalence classes'').

'''Equivalence Rule'''
: Two numeric forms ''x'' and ''y'' are forms of the same number (lie in the same equivalence class) if and only if both ''x'' ≤ ''y'' and ''y'' ≤ ''x''.

An [[Order theory|ordering relationship]] must be [[antisymmetric relation|antisymmetric]], i.e., it must have the property that ''x'' = ''y'' (i. e., ''x'' ≤ ''y'' and ''y'' ≤ ''x'' are both true) only when ''x'' and ''y'' are the same object. This is not the case for surreal number ''forms'', but is true by construction for surreal ''numbers'' (equivalence classes).

The equivalence class containing {{nowrap begin}}{ | }{{nowrap end}} is labeled 0; in other words, {{nowrap begin}}{ | }{{nowrap end}} is a form of the surreal number 0.

===Order===

The recursive definition of surreal numbers is completed by defining comparison:

Given numeric forms ''x'' = { ''XL'' | ''XR'' } and ''y'' = { ''YL'' | ''YR'' }, ''x'' ≤ ''y'' if and only if:
*there is no <math>\scriptstyle x_L \in X_L</math> such that ''y'' ≤ <math>\scriptstyle x_L</math> (every element in the left part of ''x'' is smaller than ''y''), and
*there is no <math>\scriptstyle y_R \in Y_R</math> such that <math>\scriptstyle y_R</math> ≤ ''x'' (every element in the right part of ''y'' is bigger than ''x'').
A comparison ''y'' ≤ ''c'' between a form ''y'' and a surreal number ''c'' is performed by choosing a form ''z'' from the equivalence class ''c'' and evaluating ''y'' ≤ ''z''; and likewise for ''c'' ≤ ''x'' and for comparison ''b'' ≤ ''c'' between two surreal numbers.

===Induction===

This group of definitions is [[recursion|recursive]], and requires some form of [[mathematical induction]] to define the universe of objects (forms and numbers) that occur in them. The only surreal numbers reachable via ''finite induction'' are the [[Dyadic rational|dyadic fractions]]; a wider universe is reachable given some form of [[transfinite induction]].

'''Induction Rule'''
* There is a generation ''S''0 = { 0 }, in which 0 consists of the single form { | }.
* Given any [[ordinal number]] ''n'', the generation ''S''''n'' is the set of all surreal numbers that are generated by the construction rule from subsets of <math>\cup_{i < n} \scriptstyle S_i</math>.

The base case is actually a special case of the induction rule, with 0 taken as a label for the "least ordinal". Since there exists no ''Si'' with ''i'' < 0, the expression <math>\cup_{i < 0} \scriptstyle S_i</math> is the empty set; the only subset of the empty set is the empty set, and therefore ''S''0 consists of a single surreal form { | } lying in a single equivalence class 0.

For every finite ordinal number ''n'', ''Sn'' is [[well-order]]ed by the ordering induced by the comparison rule on the surreal numbers.

The first iteration of the induction rule produces the three numeric forms { | 0 } < { | } < { 0 | } (the form { 0 | 0 } is non-numeric because 0≤0). The equivalence class containing { 0 | } is labeled 1 and the equivalence class containing { | 0 } is labeled −1. These three labels have a special significance in the axioms that define a [[ring (mathematics)|ring]]; they are the additive identity (0), the multiplicative identity (1), and the additive inverse of 1 (−1). The arithmetic operations defined below are consistent with these labels.

For every ''i'' < ''n'', since every valid form in ''S''''i'' is also a valid form in ''S''''n'', all of the numbers in ''Si'' also appear in ''Sn'' (as supersets of their representation in ''Si''). (The set union expression appears in our construction rule, rather than the simpler form ''Sn-1'', so that the definition also makes sense when ''n'' is a [[limit ordinal]].) Numbers in ''Sn'' that are a superset of some number in ''Si'' are said to have been ''inherited'' from generation ''i''. The smallest value of α for which a given surreal number appears in ''S''α is called its ''birthday''. For example, the birthday of 0 is 0, and the birthday of −1 is 1.

A second iteration of the construction rule yields the following ordering of equivalence classes:

: { | −1 } = { | −1, 0 } = { | −1, 1 } = { | −1, 0, 1 }
: < { | 0 } = { | 0, 1 }
: < { −1 | 0 } = { −1 | 0, 1 }
: < { | } = { −1 | } = { | 1 } = { −1 | 1 }
: < { 0 | 1 } = { −1, 0 | 1 }
: < { 0 | } = { −1, 0 | }
: < { 1 | } = { 0, 1 | } = { −1, 1 | } = { −1, 0, 1 | }

Comparison of these equivalence classes is consistent irrespective of the choice of form. Three observations follow:
# ''S''2 contains four new surreal numbers. Two contain extremal forms: { | −1, 0, 1 } contains all numbers from previous generations in its right set, and { −1, 0, 1 | } contains all numbers from previous generations in its left set. The others have a form that partitions all numbers from previous generations into two non-empty sets.
# Every surreal number ''x'' that existed in the previous "generation" exists also in this generation, and includes at least one new form: a partition of all numbers ''other than'' ''x'' from previous generations into a left set (all numbers less than ''x'') and a right set (all numbers greater than ''x'').
# The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set.

The informal interpretations of { 1 | } and { | −1 } are "the number just after 1" and "the number just before −1" respectively; their equivalence classes are labeled 2 and −2. The informal interpretations of { 0 | 1 } and { −1 | 0 } are "the number halfway between 0 and 1" and "the number halfway between −1 and 0" respectively; their equivalence classes are labeled 1/2 and −1/2. These labels will also be justified by the rules for surreal addition and multiplication below.

The equivalence classes at each stage ''n'' of induction may be characterized by their ''n''-''complete forms'' (each containing as many elements as possible of previous generations in its left and right sets). Either this complete form contains ''every'' number from previous generations in its left or right set, in which case this is the first generation in which this number occurs; or it contains all numbers from previous generations but one, in which case it is a new form of this one number. We retain the labels from the previous generation for these "old" numbers, and write the ordering above using the old and new labels:
: −2 < −1 < −1/2 < 0 < 1/2 < 1 < 2.

The third observation extends to all surreal numbers with finite left and right sets. (For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element.) The number { 1, 2 | 5, 8 } is therefore equivalent to { 2 | 5 }; one can establish that these are forms of 3 by using the ''birthday property'', which is a consequence of the rules above.

'''Birthday property'''
: A form ''x'' = { ''L'' | ''R'' } occurring in generation ''n'' represents a number inherited from an earlier generation ''i'' < ''n'' if and only if there is some number in ''Si'' that is greater than all elements of ''L'' and less than all elements of the ''R''. (In other words, if ''L'' and ''R'' are already separated by a number created at an earlier stage, then ''x'' does not represent a new number but one already constructed.) If ''x'' represents a number from any generation earlier than ''n'', there is a least such generation ''i'', and exactly one number ''c'' with this least ''i'' as its birthday lies between ''L'' and ''R''. ''x'' is a form of this ''c'', i. e., it lies in the equivalence class in ''Sn'' that is a superset of the representation of ''c'' in generation ''i''.

==Arithmetic==

The addition, negation (additive inverse), and multiplication of surreal number ''forms'' ''x'' = { ''XL'' | ''XR'' } and ''y'' = { ''YL'' | ''YR'' } are defined by three recursive formulas.

===Negation===

Negation of a given number {{nowrap begin}}''x'' = { ''XL'' | ''XR'' }{{nowrap end}} is defined by
:<math>-x = - \{ X_L | X_R \} = \{ -X_R | -X_L \}</math>,
where the negation of a set ''S'' of numbers is given by the set of the negated elements of ''S'':
:<math>-S = \{ -s: s \in S \}</math>.

This formula involves the negation of the surreal ''numbers'' appearing in the left and right sets of ''x'', which is to be understood as the result of choosing a form of the number, evaluating the negation of this form, and taking the equivalence class of the resulting form. This only makes sense if the result is the same irrespective of the choice of form of the operand. This can be proven inductively using the fact that the numbers occurring in ''XL'' and ''XR'' are drawn from generations earlier than that in which the form ''x'' first occurs, and observing the special case:
: -0 = - { | } = { | } = 0.

===Addition===
The definition of addition is also a recursive formula:
: <math>x + y = \{ X_L | X_R \} + \{ Y_L | Y_R \} = \{ X_L + y, x + Y_L | X_R + y, x + Y_R \}</math>,
where

:<math>X + y = \{ x + y: x \in X \} , x + Y = \{ x + y: y \in Y \}</math>.

This formula involves sums of one of the original operands and a surreal ''number'' drawn from the left or right set of the other. These are to be understood as the result of choosing a form of the numeric operand, performing the sum of the two forms, and taking the equivalence class of the resulting form. This only makes sense if the result is the same irrespective of the choice of form of the numeric operand. This can also be proven inductively with the special cases:
: 0 + 0 = { | } + { | } = { | } = 0
: ''x'' + 0 = x + { | } = { ''XL'' + 0 | ''XR'' + 0 } = { ''XL'' | ''XR'' } = ''x''
: 0 + ''y'' = { | } + ''y'' = { 0 + ''YL'' | 0 + ''YR'' } = { ''YL'' | ''YR'' } = ''y''
(The latter two cases are of course themselves proven inductively.)

===Multiplication===

The recursive formula for multiplication contains arithmetic expressions involving the operands and their left and right sets, such as the expression <math>X_R y + x Y_R - X_R Y_R</math> that appears in the left set of the product of ''x'' and ''y''. This is to be understood as the set of surreal numbers resulting from choosing one number from each set that appears in the expression and evaluating the expression on these numbers. (In each individual evaluation of the expression, only one number is chosen from each set, and is substituted in each place where that set appears in the expression.)

This depends, in turn, on the ability to (a) multiply pairs of surreal ''numbers'' drawn from the left and right sets of ''x'' and ''y'' to get a surreal number, and negate the result; (b) multiply the surreal number ''form'' ''x'' or ''y'' and a surreal ''number'' drawn from the left or right set of the other operand to get a surreal number; and (c) add the resulting surreal numbers. This again involves special cases, this time containing 0 = { | }, the multiplicative identity 1 = { 0 | }, and its additive inverse -1 = { | 0 }.

: <math>\begin{align}
xy & = \{ X_L | X_R \} \{ Y_L | Y_R \} \\
& = \left\{ X_L y + x Y_L - X_L Y_L, X_R y + x Y_R - X_R Y_R | X_L y + x Y_R - X_L Y_R, x Y_L + X_R y - X_R Y_L \right\} \\
\end{align}</math>

===Consistency===

It can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that:
* addition and negation are defined recursively in terms of "simpler" addition and negation steps, so that operations on numbers with birthday ''n'' will eventually be expressed entirely in terms of operations on numbers with birthdays less than ''n'';
* multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday ''n'' will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than ''n'';
* as long as the operands are well-defined surreal number forms (each element of the left set is less than each element of the right set), the results are again well-defined surreal number forms;
* when forms are gathered into equivalence classes using the "birthday rule", the result of negating ''x'' or adding or multiplying ''x'' and ''y'' does not depend on the choice of form of ''x'' and ''y''; and
* these operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a [[field (mathematics)|field]], with additive identity 0 = { | } and multiplicative identity 1 = { 0 | }.

With these rules one can now verify that the numbers found in the first few generations were properly labeled. The construction rule is repeated to obtain more generations of surreals:

: ''S0'' = { 0 }
: ''S1'' = { −1 < 0 < 1 }
: ''S2'' = { −2 < −1 < −1/2 < 0 < 1/2 < 1 < 2}
: ''S3'' = { −3 < −2 < −3/2 < −1 < −3/4 < −1/2 < −1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 3/2 < 2 < 3 }
: ''S4'' = { -4 < −3 < ... < -1/8 < 0 < 1/8 < 1/4 < 3/8 < 1/2 < 5/8 < 3/4 < 7/8 < 1 < 5/4 < 3/2 < 7/4 < 2 < 5/2 < 3 < 4 }

===Arithmetic closure===

For each [[natural number]] (finite ordinal) ''n'', all numbers generated in ''Sn'' are [[dyadic fraction]]s, i.e., can be written as an [[irreducible fraction]] <math>\scriptstyle \,\frac{a}{2^b}</math>
where ''a'' and ''b'' are [[integer]]s and 0 ≤ ''b'' < ''n''.

The class of all surreal numbers that are generated in some ''Sn'' for finite ''n'' may be denoted as ''S*'' = <math>\cup_{n \in N} \scriptstyle S_n</math>. One may form the three classes ''S0'' = { 0 }, ''S+'' = <math>\scriptstyle { x \in S_*: x > 0 }</math>, and ''S-'' = <math>\scriptstyle { x \in S_*: x < 0 }</math>, and state that ''S*'' is the union of these three classes. No individual ''Sn'' is closed under addition and multiplication (except ''S0''), but ''S*'' is; it is the subring of the rationals consisting of all dyadic fractions.

At an appropriate stage of transfinite induction, the surreal numbers may be expected to form a category on which the addition and multiplication operations (as well as the surreal construction step) are closed, and in which the multiplicative inverse of every nonzero number can be found. Assuming that one can find such a class, the surreal numbers, with their ordering and these algebraic operations, constitute an [[ordered field]], with the caveat that they do not form a [[Set (mathematics)|set]] but a proper [[mathematical class|class]]. In fact, it is a very special ordered field: the biggest one. Every other ordered field can be embedded in the surreals. (See also the definition of [[rational number]]s and [[real number]]s.)

== "To Infinity ..." ==

Let there be an ordinal ω greater than the natural numbers, and define ''S''ω as the set of all surreal numbers generated by the construction rule from subsets of ''S*''. (This is the same inductive step as before, since the ordinal number ω is the smallest ordinal that is larger than all natural numbers; however, the set union appearing in the inductive step is now an infinite union of finite sets, and so this step can only be performed in a set theory that allows such a union.) A unique infinitely large positive number occurs in ''S''ω:
: <math>\scriptstyle \omega = \{ S_* | \} = \{ 1, 2, 3, 4, ... | \}. </math>
''S''ω also contains objects that can be identified as the [[rational number]]s. For example, the ω-complete form of the fraction 1/3 is given by:
: <math>\scriptstyle \tfrac{1} {3} = \{ y \in S_*: 3 y < 1 | y \in S_*: 3 y > 1 \}</math>.
The product of this form of 1/3 with any form of 3 is a form whose left set contains only numbers less than 1 and whose right set contains only numbers greater than 1; the birthday property implies that this product is a form of 1.

Not only do all the rest of the [[rational number]]s appear in ''S''ω; the remaining finite [[real number]]s do too. For example
: <math>\scriptstyle \pi = \{ 3, \frac{25}{8},\frac{201}{64}, ... | 4, \frac{7}{2}, \frac{13}{4}, \frac{51}{16},... \}</math>.

The only infinities in ''S''ω are ω and -ω; but there are other non-real numbers in ''S''ω among the reals. Consider the smallest positive number in ''S''ω:
: <math>\scriptstyle \epsilon = \{ S_- \cup S_0 | S_+ \} = \{ 0 | 1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, ... \} = \{ 0 | y \in S_* : y > 0 \}</math>.
This number is larger than zero but less than all positive dyadic fractions. It is therefore an [[infinitesimal]] number, often labeled ε. The ω-complete form of ε (resp. -ε) is the same as the ω-complete form of 0, except that 0 is included in the left (resp. right) set. The only "pure" infinitesimals in ''S''ω are ε and its additive inverse -ε; adding them to any dyadic fraction ''y'' produces the numbers ''y''±ε, which also lie in ''S''ω.

One can determine the relationship between ω and ε by multiplying particular forms of them to obtain:
: ω · ε = { ε · ''S''+ | ω · ''S''+ + ''S''* + ε · ''S''* }.
This expression is only well-defined in a set theory which permits transfinite induction up to <math>S_{\omega^2}</math>. In such a system, one can demonstrate that all the elements of the left set of ω · ε are positive infinitesimals and all the elements of the right set are positive infinities, and therefore ω · ε is the oldest positive finite number, i. e., 1. Consequently,
: 1/ε = ω.
Some authors systematically use ω−1 in place of the symbol ε.

===Contents of ''S''ω===

Given any ''x'' = { ''L'' | ''R'' } in ''S''ω, exactly one of the following is true:
* ''L'' and ''R'' are both empty, in which case ''x'' = 0;
* ''R'' is empty and some integer ''n''≥0 is greater than every element of ''L'', in which case ''x'' equals the smallest such integer ''n'';
* ''R'' is empty and no integer ''n'' is greater than every element of ''L'', in which case ''x'' equals +ω;
* ''L'' is empty and some integer ''n''≤0 is less than every element of ''R'', in which case ''x'' equals the largest such integer ''n'';
* ''L'' is empty and no integer ''n'' is less than every element of ''R'', in which case ''x'' equals -ω;
* ''L'' and ''R'' are both non-empty, and:
** some dyadic fraction ''y'' is "strictly between" ''L'' and ''R'' (greater than all elements of ''L'' and less than all elements of ''R''), in which case ''x'' equals the oldest such dyadic fraction ''y'';
** no dyadic fraction ''y'' lies strictly between ''L'' and ''R'', but some dyadic fraction <math>\scriptstyle y \in L</math> is greater than or equal to all elements of ''L'' and less than all elements of ''R'', in which case ''x'' equals ''y''+ε;
** no dyadic fraction ''y'' lies strictly between ''L'' and ''R'', but some dyadic fraction <math>\scriptstyle y \in R</math> is greater than all elements of ''L'' and less than or equal to all elements of ''R'', in which case ''x'' equals ''y''-ε;
** every dyadic fraction is either greater than some element of ''R'' or less than some element of ''L'', in which case ''x'' is some real number that has no representation as a dyadic fraction.

''S''ω is not an algebraic field, because it is not closed under arithmetic operations; consider ω+1, whose form <math>\scriptstyle \{ 1, 2, 3, 4, ... | \} + \{ 0 | \} = \{ 1, 2, 3, 4, ..., \omega | \}</math> does not lie in any number in ''S''ω. The maximal subset of ''S''ω that is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities ±ω, the infinitesimals ±ε, and the infinitesimal neighbors ''y''±ε of each nonzero dyadic fraction ''y''.

This construction of the real numbers differs from the [[Dedekind cut]]s of [[standard analysis]] in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in ''S''ω with its forms in previous generations. (The ω-complete forms of real elements of ''S''ω are in one-to-one correspondence with the reals obtained by Dedekind cuts, under the proviso that Dedekind reals corresponding to rational numbers are represented by the form in which the cut point is omitted from both left and right sets.) The rationals are not an identifiable stage in the surreal construction; they are merely the subset ''Q'' of ''S''ω containing all elements ''x'' such that ''x'' ''b'' = ''a'' for some ''a'' and some nonzero ''b'', both drawn from ''S''*. By demonstrating that ''Q'' is closed under individual repetitions of the surreal arithmetic operations, one can show that it is a field; and by showing that every element of ''Q'' is reachable from ''S''* by a finite series (no longer than two, actually) of arithmetic operations ''including multiplicative inversion'', one can show that ''Q'' is strictly smaller than the subset of ''S''ω identified with the reals.

== "... And Beyond." ==

Continuing to perform transfinite induction beyond ''S''ω produces more ordinal numbers α, each represented as the largest surreal number having birthday α. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is ω+1 = { ω | }. There is another positive infinite number in generation ω+1:
: ω−1 = { 1, 2, 3, 4, ... | ω }.
It is important to observe that the surreal number ω−1 is not an ordinal; the ordinal ω is not the successor of any ordinal. This is a surreal number with birthday ω+1, which is labeled ω−1 on the basis that it coincides with the sum of ω = { 1, 2, 3, 4, ... | } and −1 = { | 0 }. Similarly, there are two new infinitesimal numbers in generation ω+1:
: 2ε = ε + ε = { ε | 1+ε, 1/2+ε, 1/4+ε, 1/8+ε, ... } and
: ε/2 = ε · 1/2 = { 0 | ε }.

At a later stage of transfinite induction, there is a number larger than ω+''k'' for all natural numbers ''k'':
: 2ω = ω + ω = { ω+1, ω+2, ω+3, ω+4, ... | }
This number may be labeled ω + ω both because its birthday is ω + ω (the first ordinal number not reachable from ω by the successor operation) and because it coincides with the surreal sum of ω and ω; it may also be labeled 2ω because it coincides with the product of ω = { 1, 2, 3, 4, ... | } and 2 = { 1 | }. It is the second limit ordinal; reaching it from ω via the construction step requires a transfinite induction on <math>\bigcup_{k < \omega} S_{\omega + k}</math>. This involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required.

Note that the ''conventional'' addition and multiplication of ordinals does not always coincide with these operations on their surreal representations. The sum of ordinals 1 + ω equals ω, but the surreal sum is commutative and produces 1 + ω = ω + 1 > ω. The addition and multiplication of the surreal numbers associated with ordinals coincides with the [[ordinal arithmetic#Natural operations|natural sum and natural product]] of ordinals.

Just as 2ω is bigger than ω+''n'' for any natural number ''n'', there is a surreal number ω/2 that is infinite but smaller than ω−''n'' for any natural number ''n''.
: ω/2 = { ''S''* | ω − ''S''* }
where ''x'' − ''Y'' = { ''x'' − ''y'' | ''y'' in ''Y'' }. It can be identified as the product of ω and the form { 0 | 1 } of 1/2. The birthday of ω/2 is the limit ordinal 2ω.

== Powers of ω ==

To classify the "orders" of infinite and infinitesimal surreal numbers, also known as [[archimedean property|archimedean]] classes, Conway associated to each surreal number ''x'' the surreal number
* ω''x'' = { 0, ''r'' ω''x''L | ''s'' ω''x''R },
where ''r'' and ''s'' range over the positive real numbers. If 0 ≤ ''x'' < ''y'' then ω''y'' is "infinitely greater" than ω''x'', in that it is greater than ''r'' ω''x'' for all real numbers ''r''. Powers of ω also satisfy the conditions
* ω''x'' ω''y'' = ω''x+y'',
* ω−''x'' = 1/ω''x'',
so they behave the way one would expect powers to behave.

Each power of ω also has the redeeming feature of being the ''simplest'' surreal number in its archimedean class; conversely, every archimedean class within the surreal numbers contains a unique simplest member. Thus, for every positive surreal number ''x'' there will always exist some positive real number ''r'' and some surreal number ''y'' so that ''x'' − ''r'' ω''y'' is "infinitely smaller" than ''x''. The exponent ''y'' is the "base ω logarithm" of ''x'', defined on the positive surreals; it can be demonstrated that logω maps the positive surreals onto the surreals and that logω(''xy'') = logω(''x'') + logω(''y'').

This gets extended by transfinite induction so that every surreal number ''x'' has a "normal form" analogous to the [[Ordinal arithmetic#Cantor normal form|Cantor normal form]] for ordinal numbers. Every surreal number may be uniquely written as
* ''x'' = ''r''0 ω''y''0 + ''r''1 ω''y''1 + …,
where every ''r''α is a nonzero real number and the ''y''αs form a strictly decreasing sequence of surreal numbers. This "sum", however, may have infinitely many terms, and in general has the length of an arbitrary ordinal number. (Zero corresponds of course to the case of an empty sequence, and is the only surreal number with no leading exponent.)

Looked at in this manner, the surreal numbers resemble a [[Formal power series|power series field]], except that the decreasing sequences of exponents must be bounded in length by an ordinal and are not allowed to be as long as the class of ordinals.

== Surcomplex numbers ==

A '''surcomplex number''' is a number of the form <math>a+bi </math>, where ''a'' and ''b'' are surreal numbers.<ref>[http://jamespropp.org/surreal/text.ps.gz Surreal vectors and the game of Cutblock], James Propp, August 22, 1994.</ref><ref>N. L. Alling, ''Foundations of analysis over surreal number fields'', N. L. Alling, Amsterdam: North-Holland, 1987. ISBN 0-444-70226-1.</ref> The surcomplex numbers form an [[algebraically closed]] [[field (mathematics)|field]] (except for being a proper class), [[isomorphic (mathematics)|isomorphic]] to the [[algebraic closure]] of the field generated by extending the [[rational numbers]] by a [[proper class]] of [[algebraically independent]] [[transcendental (mathematics)|transcendental]] elements. Up to field [[isomorphism]], this fact characterizes the field of surcomplex numbers within any fixed set theory.<ref>Theorem 27, ''On Numbers and Games'', John H. Conway, 2nd ed., Natick, Massachusetts: [[A K Peters, Ltd.]], 2000. ISBN 1-56881-127-6.</ref>

==Games==
{{main|combinatorial game theory}}

The definition of surreal numbers contained one restriction: each element of L must be strictly less than each element of R. If this restriction is dropped we can generate a more general class known as ''games''. All games are constructed according to this rule:

:'''Construction Rule''' If ''L'' and ''R'' are two sets of games then { ''L'' | ''R'' } is a game.

Addition, negation, and comparison are all defined the same way for both surreal numbers and games.

Every surreal number is a game, but not all games are surreal numbers, e.g. the game [[star (game theory)|{ '''0''' | '''0''' }]] is not a surreal number. The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers. The class of surreal numbers forms a [[field (mathematics)|field]], but the class of games does not. The surreals have a [[total order]]: given any two surreals, they are either equal, or one is greater than the other. The games have only a [[partial order]]: there exist pairs of games that are neither equal, greater than, nor less than each other. Each surreal number is either positive, negative, or zero. Each game is either positive, negative, ''[[zero game|zero]]'', or ''[[fuzzy game|fuzzy]]'' (incomparable with zero, such as {1|−1}).

A move in a game involves the player whose move it is choosing a game from those available in L (for the left player) or R (for the right player) and then passing this chosen game to the other player. A player who cannot move because the choice is from the empty set has lost. A positive game represents a win for the left player, a negative game for the right player, a zero game for the second player to move, and a [[fuzzy game]] for the first player to move.

If ''x'', ''y'', and ''z'' are surreals, and ''x''=''y'', then ''x'' ''z''=''y'' ''z''. However, if ''x'', ''y'', and ''z'' are games, and ''x''=''y'', then it is not always true that ''x'' ''z''=''y'' ''z''. Note that "=" here means equality, not identity.

==Application to combinatorial game theory==

The surreal numbers were originally motivated by studies of the game [[Go (board game)|Go]],<ref>{{citation | url = http://www-history.mcs.st-andrews.ac.uk/Biographies/Conway.html | title = Conway Biography | last = O'Connor | first = J.J. | last2 = Robertson | first2 = E.F. | accessdate = 2008-01-24 }}</ref> and there are numerous connections between popular games and the surreals. In this section, we will use a capitalized ''Game'' for the mathematical object {L|R}, and the lowercase ''game'' for recreational games like [[Chess]] or [[Go (board game)|Go]].

We consider games with these properties:
* Two players (named ''Left'' and ''Right'')
* [[Deterministic]] (the game at each step will completely depend on the choices the players make, rather than a random factor)
* No hidden information (such as cards or tiles that a player hides)
* Players alternate taking turns (the game may or may not allow multiple moves in a turn)
* Every game must end in a finite number of moves
* As soon as there are no legal moves left for a player, the game ends, and that player loses

For most games, the initial board position gives no great advantage to either player. As the game progresses and one player starts to win, board positions will occur where that player has a clear advantage. For analyzing games, it is useful to associate a Game with every board position. The value of a given position will be the Game {L|R}, where L is the set of values of all the positions that can be reached in a single move by Left. Similarly, R is the set of values of all the positions that can be reached in a single move by Right.

The zero Game (called 0) is the Game where L and R are both empty, so the player to move next (L or R) immediately loses. The sum of two Games G = { L1 | R1 } and H = { L2 | R2 } is defined as the Game G + H = { L1 + H, G + L2 | R1 + H, G + R2 } where the player to move chooses which of the Games to play in at each stage, and the loser is still the player who ends up with no legal move. One can imagine two chess boards between two players, with players making moves alternatively, but with complete freedom as to which board to play on. If G is the Game {L | R}, -G is the game {-R | -L}, i.e. with the role of the two players reversed. It is easy to show G - G = 0 for all Games G (where G - H is defined as G + (-H)).

This simple way to associate Games with games yields a very interesting result. Suppose two perfect players play a game starting with a given position whose associated Game is ''x''. We can classify all Games into four classes as follows:

* If x > 0 then Left will win, regardless of who plays first.
* If x < 0 then Right will win, regardless of who plays first.
* If x = 0 then the player who goes second will win.
* If x || 0 then the player who goes first will win.

More generally, we can define G > H as G - H > 0, and similarly for <, = and ||.

The notation G || H means that G and H are incomparable. G || H is equivalent to G−H || 0, i.e. that G > H, G < H and G = H are all false. Incomparable games are sometimes said to be ''confused'' with each other, because one or the other may be preferred by a player depending on what is added to it. A game confused with zero is said to be [[fuzzy game|fuzzy]], as opposed to [[sign (mathematics)|positive, negative, or zero]]. An example of a fuzzy game is [[star (game theory)|star (*)]].

Sometimes when a game nears the end, it will decompose into several smaller games that do not interact, except in that each player's turn allows moving in only one of them. For example, in Go, the board will slowly fill up with pieces until there are just a few small islands of empty space where a player can move. Each island is like a separate game of Go, played on a very small board. It would be useful if each subgame could be analyzed separately, and then the results combined to give an analysis of the entire game. This doesn't appear to be easy to do. For example, you might have two subgames where whoever moves first wins, but when they are combined into one big game, it's no longer the first player who wins. Fortunately, there is a way to do this analysis. Just use the following remarkable theorem:

:If a big game decomposes into two smaller games, and the small games have associated Games of ''x'' and ''y'', then the big game will have an associated Game of ''x''+''y''.

A game composed of smaller games is called the [[disjunctive sum]] of those smaller games, and the theorem states that the method of addition we defined is equivalent to taking the disjunctive sum of the addends.

Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing [[Go terms#Yose|Go endgames]], and realized that it would be useful to have some way to combine the analyses of non-interacting subgames into an analysis of their [[disjunctive sum]]. From this he invented the concept of a Game and the addition operator for it. From there he moved on to developing a definition of negation and comparison. Then he noticed that a certain class of Games had interesting properties; this class became the surreal numbers. Finally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals.

==Alternative realizations==

Since Conway first introduced surreal numbers, several alternative constructions
have been developed.

===Sign expansion===

====Definitions====

In one alternative realization, called the ''sign-expansion'' or ''sign-sequence'' of a surreal number, a surreal number is a [[function (mathematics)|function]] whose [[domain of a function|domain]] is an [[ordinal number|ordinal]] and whose [[range (mathematics)|range]] is { − 1, + 1 }.

Define the binary predicate "simpler than" on numbers by ''x'' is simpler than ''y'' if ''x'' is a [[subset|proper subset]] of ''y'', ''i.e.'' if dom(''x'') < dom(''y'') and ''x''(α) = ''y''(α) for all α < dom(''x'').

For surreal numbers define the binary relation < to be lexicographic order (with the convention that "undefined values" are greater than −1 and less than 1). So ''x'' < ''y'' if one of the following holds:
* ''x'' is simpler than ''y'' and ''y''(dom(''x'')) = + 1;
* ''y'' is simpler than ''x'' and ''x''(dom(''y'')) = − 1;
* there exists a number ''z'' such that ''z'' is simpler than ''x'', ''z'' is simpler than ''y'', ''x''(dom(''z'')) = − 1 and ''y''(dom(''z'')) = + 1.

Equivalently, let δ(''x'',''y'') = min({ dom(''x''), dom(''y'')} ∪ { α :
α < dom(''x'') ∧ α < dom(''y'') ∧ ''x''(α) ≠ ''y''(α) }),
so that ''x'' = ''y'' if and only if δ(''x'',''y'') = dom(''x'') = dom(''y''). Then, for numbers ''x'' and ''y'', ''x'' < ''y'' if and only if one of the following holds:
* δ(''x'',''y'') = dom(''x'') ∧ δ(''x'',''y'') < dom(''y'') ∧ ''y''(δ(''x'',''y'')) = + 1;
* δ(''x'',''y'') < dom(''x'') ∧ δ(''x'',''y'') = dom(''y'') ∧ ''x''(δ(''x'',''y'')) = − 1;
* δ(''x'',''y'') < dom(''x'') ∧ δ(''x'',''y'') < dom(''y'') ∧ ''x''(δ(''x'',''y'')) = − 1 ∧ ''y''(δ(''x'',''y'')) = + 1.

For numbers ''x'' and ''y'', ''x'' ≤ ''y'' if and only if ''x'' < ''y'' ∨ ''x'' = ''y'', and ''x'' > ''y'' if and only if ''y'' < ''x''. Also ''x'' ≥ ''y'' if and only if ''y'' ≤ ''x''.

The relation < is [[transitive relation|transitive]], and for all numbers ''x'' and ''y'', exactly one of ''x'' < ''y'', ''x'' = ''y'', ''x'' > ''y'', holds (law of [[trichotomy (mathematics)|trichotomy]]). This means that < is a [[linear order]] (except that < is a proper class).

For sets of numbers, ''L'' and ''R'' such that ∀''x'' ∈ ''L'' ∀''y'' ∈ ''R'' (''x'' < ''y''), there exists a unique number ''z'' such that
* ∀''x'' ∈ ''L'' (''x'' < ''z'') ∧ ∀''y'' ∈ ''R'' (''z'' < ''y''),
* For any number ''w'' such that ∀''x'' ∈ ''L'' (''x'' < ''w'') ∧ ∀''y'' ∈ ''R'' (''w'' < ''y''), ''w'' = ''z'' or ''z'' is simpler than ''w''.

Furthermore, ''z'' is constructible from ''L'' and ''R'' by transfinite induction. ''z'' is the simplest number between ''L'' and ''R''. Let the unique number ''z'' be denoted by σ(''L'',''R'').

For a number ''x'', define its left set ''L''(''x'') and right set ''R''(''x'') by
* ''L''(''x'') = { ''x''|α : α < dom(''x'') ∧ ''x''(α) = + 1 };
* ''R''(''x'') = { ''x''|α : α < dom(''x'') ∧ ''x''(α) = − 1 },

then σ(''L''(''x''),''R''(''x'')) = ''x''.

One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike Conway's realization of the surreal numbers, however, the sign-expansion requires a prior construction of the ordinals, while in Conway's realization, the ordinals are constructed as particular cases of surreals.

However, similar definitions can be made that obviate the need for prior construction of the ordinals. For instance, we could let the surreals be the (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule ∀''g'' ∈ dom ''f'' (∀''h'' ∈ dom ''g'' (''h'' ∈ dom ''f'' )) and whose range is { −, + }. "Simpler than" is very simply defined now—''x'' is simpler than ''y'' if ''x'' ∈ dom ''y''. The total ordering is defined by considering ''x'' and ''y'' as sets of ordered pairs (as a function is normally defined): Either ''x'' = ''y'', or else the surreal number ''z'' = ''x'' ∩ ''y'' is in the domain of ''x'' or the domain of ''y'' (or both, but in this case the signs must disagree). We then have ''x'' < ''y'' if ''x''(''z'') = − or ''y''(''z'') = + (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements of dom ''f'' in order of simplicity (i.e., inclusion), and then write down the signs that ''f'' assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is { + }.

====Addition and multiplication====

The sum ''x'' + ''y'' of two numbers, ''x'' and ''y'', is defined by induction on dom(''x'') and dom(''y'') by ''x'' + ''y'' = σ(''L'',''R''), where
* ''L'' = { ''u'' + ''y'' : ''u'' ∈ ''L''(''x'') } ∪{ ''x'' + ''v'' : ''v'' ∈ ''L''(''y'') },
* ''R'' = { ''u'' + ''y'' : ''u'' ∈ ''R''(''x'') } ∪{ ''x'' + ''v'' : ''v'' ∈ ''R''(''y'') }.

The additive identity is given by the number 0 = { }, ''i.e.'' the number 0 is the unique function whose domain is the ordinal 0, and the additive inverse of the number ''x'' is the number − ''x'', given by dom(− ''x'') = dom(''x''), and, for α < dom(''x''), (− ''x'')(α) = − 1 if ''x''(α) = + 1, and (− ''x'')(α) = + 1 if ''x''(α) = − 1.

It follows that a number ''x'' is [[Positive number|positive]] if and only if 0 < dom(''x'') and ''x''(0) = + 1, and ''x'' is [[negative number|negative]] if and only if 0 < dom(''x'') and ''x''(0) = − 1.

The product ''xy'' of two numbers, ''x'' and ''y'', is defined by induction on dom(''x'') and dom(''y'') by ''xy'' = σ(''L'',''R''), where
* ''L'' = { ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''L''(''x''), ''v'' ∈ ''L''(''y'') } ∪ { ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''R''(''x''), ''v'' ∈ ''R''(''y'') },
* ''R'' = { ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''L''(''x''), ''v'' ∈ ''R''(''y'') } ∪ { ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''R''(''x''), ''v'' ∈ ''L''(''y'') }.

The multiplicative identity is given by the number 1 = { (0,+ 1) }, ''i.e.'' the number 1 has domain equal to the ordinal 1, and 1(0) = + 1.

====Correspondence with Conway====

The map from Conway's realization to sign expansions is given by ''f''({ ''L'' | ''R'' }) = σ(''M'',''S''), where ''M'' = { ''f''(''x'') : ''x'' ∈ ''L'' } and ''S'' = { ''f''(''x'') : ''x'' ∈ ''R'' }.

The inverse map from the alternative realization to Conway's realization is given by ''g''(''x'') = { ''L'' | ''R'' }, where ''L'' = { ''g''(''y'') : ''y'' ∈ ''L''(''x'') } and ''R'' = { ''g''(''y'') : ''y'' ∈ ''R''(''x'') }.

===Axiomatic approach===

In another approach to the surreals, given by Alling,<ref name="Alling">{{cite book | last = Alling | first = Norman L. | title = Foundations of Analysis over Surreal Number Fields | publisher = North-Holland | series = Mathematics Studies 141 | year = 1987 | isbn = 0-444-70226-1}}</ref> explicit construction is bypassed altogether. Instead, a set of axioms is given that any particular approach to the surreals must satisfy. Much like the [[Real numbers#Axiomatic approach|axiomatic approach]] to the reals, these axioms guarantee uniqueness up to isomorphism.

A triple <math>\langle \mathbf{No}, \mathrm{<}, b \rangle</math> is a surreal number system if and only if the following hold:

* < is a [[total order]] over '''No'''
* ''b'' is a function from '''No''' onto the class of all ordinals (''b'' is called the "birthday function" on '''No''').
* Let ''A'' and ''B'' be subclasses of '''No''' such that for all ''x'' ∈ ''A'' and ''y'' ∈ ''B'', ''x'' < ''y'' (using Alling's terminology, 〈 ''A'',''B'' 〉 is a "Conway cut" of '''No'''). Then there exists a unique ''z'' ∈ '''No''' such that ''b(z)'' is minimal and for all ''x'' ∈ ''A'' and all ''y'' ∈ ''B'', ''x'' < ''z'' < ''y''. (This axiom is often referred to as "Conway's Simplicity Theorem".)
* Furthermore, if an ordinal ''α'' is greater than ''b(x)'' for all ''x'' ∈ ''A'', ''B'', then ''b(z)'' ≤ ''α''. (Alling calls a system that satisfies this axiom a "full surreal number system".)

Both Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.

Given these axioms, Alling<ref name="Alling"/> derives Conway's original definition of ≤ and develops surreal arithmetic.

===Hahn series===

Alling<ref>Alling, ''op. cit.'', theorem of §6.55 (p. 246)</ref> also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of [[Hahn series]] with real coefficients on the value group of surreal numbers themselves (the series representation corresponding to the normal form of a surreal number, as defined above). This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.

==Relation to hyperreals==
[[Philip Ehrlich]] has constructed an isomorphism between Conway's maximal surreal number field and the maximal [[hyperreal field|hyperreals]] in [[von Neumann–Bernays–Gödel set theory]].<ref>{{cite journal
|author=Philip Ehrlich
|authorlink=Philip Ehrlich
|year=2012
|title=The absolute arithmetic continuum and the unification of all numbers great and small
|journal=The Bulletin of Symbolic Logic
|volume=18
|issue=1
|pages=1–45
|url=http://www.math.ucla.edu/~asl/bsl/1801-toc.htm
|accessdate=2012-01-29 }}</ref>

==See also==
{{Infinitesimal navbox}}
{{Portal|Mathematics}}
* [[Hyperreal number]]
* [[Non-standard analysis]]
* [[Dehn planes]]

==References==
{{Reflist|2}}

==Further reading==
* [[Donald Knuth]]'s original exposition: ''Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness''. 1974, ISBN 0-201-03812-9. More information can be found at [http://www-cs-faculty.stanford.edu/~knuth/sn.html the book's official homepage]
* An update of the classic 1976 book defining the surreal numbers, and exploring their connections to games: ''On Numbers And Games, 2nd ed.'', John Conway, 2001, ISBN 1-56881-127-6.
* An update of the first part of the 1981 book that presented surreal numbers and the analysis of games to a broader audience: ''Winning Ways for Your Mathematical Plays, vol. 1, 2nd ed.'', Berlekamp, Conway, and Guy, 2001, ISBN 1-56881-130-6.
* [[Martin Gardner]], ''Penrose Tiles to Trapdoor Ciphers,'' W. H. Freeman & Co., 1989. ISBN 0-7167-1987-8. Chapter 4 — not especially technical overview; reprints the 1976 <cite>Scientific American</cite> article.
* Polly Shulman, "[http://discovermagazine.com/1995/dec/infinityplusonea599 Infinity Plus One, and Other Surreal Numbers]". ''[[Discover (magazine)|Discover]],'' December 1995.
* A detailed, though somewhat technical, treatment of surreal numbers: ''Foundations of Analysis over Surreal Number Fields'', Alling, Norman L., 1987, ISBN 0-444-70226-1
* A treatment of surreals based on the sign-expansion realization: ''An Introduction to the Theory of Surreal Numbers'', Gonshor, Harry, 1986, ISBN 0-521-31205-1
* A detailed philosophical development of the concept of surreal numbers as a most general concept of number is [[Alain Badiou]], ''[[Number and Numbers]]'' (New York: Polity Press, 2008): ISBN 0-7456-3879-1 (paperback); ISBN 0-7456-3878-3 (hardcover)

==External links==
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ, by A. N. Walker]{{dead link|date=September 2012}}
* [http://www.tondering.dk/claus/surreal.html A gentle yet thorough introduction by Claus Tøndering]
* {{planetmath reference|id=3352|title=Surreal number}}
* [http://scientopia.org/blogs/goodmath/category/surreal-numbers/ Good Math, Bad Math: Surreal Numbers], a series of articles about surreal numbers, their variations, and their applications

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{{Number systems}}
{{Infinitesimals}}

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Eisenstein series

2014-01-21T15:34:11Z

129.132.147.99: /* Fourier series */

{{Intelligent Design}}

'''Specified complexity''' is an argument proposed by [[William Dembski]] and used by him and others to promote [[intelligent design]]. According to Dembski, the concept is intended to formalize a property that singles out patterns that are both ''specified'' and ''complex'', in specific senses defined by Dembski. Dembski states that specified complexity is a reliable marker of design by an intelligent agent, a central tenet to intelligent design which Dembski argues for in opposition to [[evolution|modern evolutionary theory]]. The concept of specified complexity is widely regarded as mathematically unsound and has not been the basis for further independent work in [[information theory]], the theory of [[complex systems]], or [[biology]].<ref>{{cite web |url=http://www.talkorigins.org/faqs/information/dembski.html |title=Information Theory and Creationism: William Dembski |author=Rich Baldwin |year=2005 |publisher=[[TalkOrigins Archive]] |accessdate=2010-05-10}}</ref><ref>Mark Perakh, (2005). ''[http://www.talkreason.org/articles/newmath.cfm Dembski "displaces Darwinism" mathematically -- or does he?]''</ref><ref>Jason Rosenhouse, (2001). ''[http://www.math.jmu.edu/~rosenhjd/sewell.pdf How Anti-Evolutionists Abuse Mathematics]'' The Mathematical Intelligencer, Vol. 23, No. 4, Fall 2001, pp. 3-8.</ref> Specified complexity is one of the two main arguments used by intelligent design proponents, the other being [[irreducible complexity]].

In Dembski's terminology, a ''specified'' pattern is one that admits short descriptions, whereas a ''complex'' pattern is one that is unlikely to occur by chance. Dembski argues that it is impossible for specified complexity to exist in patterns displayed by configurations formed by unguided processes. Therefore, Dembski argues, the fact that specified complex patterns can be found in living things indicates some kind of guidance in their formation, which is indicative of intelligence. Dembski further argues that one can rigorously show by applying [[No free lunch in search and optimization|no free lunch theorems]] the inability of evolutionary algorithms to select or generate configurations of high specified complexity.

In intelligent design literature, an [[intelligent designer|intelligent agent]] is one that chooses between different possibilities and has, by supernatural means and methods, caused life to arise.<ref>"no intelligent agent who is strictly physical could have presided over the origin of the universe or the origin of life." - Dembski [http://www.leaderu.com/offices/dembski/docs/bd-the_ac.html The Act of Creation: Bridging Transcendence and Immanence] Dembski has also said that "intelligent design should be understood as the evidence that God has placed in nature to show that the physical world is the product of intelligence and not simply the result of mindless material forces" [http://www.designinference.com/documents/2005.08.Commending_President_Bush.pdf Why President Bush Got It Right about Intelligent Design] [[William A. Dembski]] DesignInference.com, August 4, 2005. Dembski has also said that "Intelligent design is just the Logos theology of John's Gospel restated in the idiom of information theory." [http://touchstonemag.com/archives/issue.php?id=49 Signs of Intelligence] A Primer on the Discernment of Intelligent Design. William A. Dembski. Touchstone Journal, Volume 12, Issue 4, July/August 1999.</ref> Specified complexity is what Dembski terms an "explanatory filter" which can recognize design by detecting complex specified information (CSI). The filter is based on the premise that the categories of regularity, chance, and design are, according to Dembski, "mutually exclusive and exhaustive." Complex specified information detects design because it detects what characterizes intelligent agency; it detects the actualization of one among many competing possibilities.

A study by Wesley Elsberry and Jeffrey Shallit states that "Dembski's work is riddled with inconsistencies, equivocation, flawed use of mathematics, poor scholarship, and misrepresentation of others' results."<ref>Wesley Elsberry and Jeffrey Shallit, (2003). ''[http://www.talkreason.org/articles/eandsdembski.pdf Information Theory, Evolutionary Computation, and Dembski’s “Complex Specified Information”]''</ref> Another objection concerns Dembski's calculation of probabilities. According to [[Martin Nowak]], a Harvard professor of mathematics and evolutionary biology "We cannot calculate the probability that an eye came about. We don't have the information to make the calculation."<ref name="time.com">Martin Nowak (2005). [http://www.time.com/time/archive/preview/0,10987,1090909,00.html Time Magazine, 15 August 2005, page 32]</ref> Critics also reject applying specified complexity to infer design as an [[argument from ignorance]].

==Definition==
===Orgel's original use===
The term "specified complexity" was originally coined by [[origin of life]] researcher [[Leslie Orgel]] to denote what distinguishes living things from non-living things:

<blockquote>In brief, living organisms are distinguished by their ''specified'' complexity. Crystals are usually taken as the prototypes of simple well-specified structures, because they consist of a very large number of identical molecules packed together in a uniform way. Lumps of granite or random mixtures of polymers are examples of structures that are complex but not specified. The crystals fail to qualify as living because they lack complexity; the mixtures of polymers fail to qualify because they lack specificity.<ref>Leslie Orgel (1973). ''The Origins of Life'', p. 189.</ref></blockquote>

The term was later employed by physicist [[Paul Davies]] in a similar manner:

<blockquote>Living organisms are mysterious not for their complexity per se, but for their tightly specified complexity<ref>Paul Davies (1999). ''The Fifth Miracle'' p. 112</ref></blockquote>

===Dembski's definition===

For Dembski, specified complexity is a property which can be observed in living things. However, whereas Orgel used the term for biological features which are considered in science to have arisen through a process of evolution, Dembski says that it describes features which cannot form through "undirected" evolution—and concludes that it allows one to infer intelligent design. While Orgel employed the concept in a qualitative way, Dembski's use is intended to be quantitative. Dembski's use of the concept dates to his 1998 monograph ''[[The Design Inference]]''. Specified complexity is fundamental to his approach to intelligent design, and each of his subsequent books has also dealt significantly with the concept. He has stated that, in his opinion, "if there is a way to detect design, specified complexity is it."<ref>William A. Dembski (2002). ''[[No Free Lunch (book)|No Free Lunch]]'', p. 19.</ref>

Dembski asserts that specified complexity is present in a configuration when it can be described by a pattern that displays a large amount of independently specified information and is also complex, which he defines as having a low probability of occurrence. He provides the following examples to demonstrate the concept: "A single letter of the alphabet is specified without being complex. A long sentence of random letters is complex without being specified. A Shakespearean sonnet is both complex and specified." <ref>William A. Dembski (1999). ''Intelligent Design'', p. 47.</ref>

In his earlier papers Dembski defined ''complex specified information'' (CSI) as being present in a specified event whose probability did not exceed 1 in 10150, which he calls the [[universal probability bound]]. In that context, "specified" meant what in later work he called "pre-specified", that is specified before any information about the outcome is known. The value of the universal probability bound corresponds to the inverse of the upper limit of "the total number of [possible] specified events throughout cosmic history," as calculated by Dembski.<ref>William A. Dembski (2004). ''[[The Design Revolution|The Design Revolution: Answering the Toughest Questions About Intelligent Design]]'', p. 85.</ref> Anything below this bound has CSI. The terms "specified complexity" and "complex specified information" are used interchangeably. In more recent papers Dembski has redefined the universal probability bound, with reference to another number, corresponding to the total number of bit operations that could possibly have been performed in the entire history of the universe.

Dembski asserts that CSI exists in numerous features of living things, such as DNA and other functional biological molecules, and argues that it cannot be generated by the only known natural mechanisms of [[physical law]] and [[Randomness|chance]], or by their combination. He argues that this is so because laws can only shift around or lose information, but do not produce it, and chance can produce complex unspecified information, or simple specified information, but not CSI; he provides a mathematical analysis that he claims demonstrates that law and chance working together cannot generate CSI, either. Moreover, he claims that CSI is [[holism|holistic]], with the whole being greater than the sum of the parts, and that this decisively eliminates Darwinian evolution as a possible means of its creation. Dembski maintains that by process of elimination, CSI is best explained as being due to [[intelligence (trait)|intelligence]], and is therefore a reliable indicator of [[design]].

==Law of conservation of information==
Dembski formulates and proposes a [[law of conservation]] of [[information]] as follows:
<blockquote>
This strong proscriptive claim, that natural causes can only transmit CSI but never originate it, I call the Law of Conservation of Information.

Immediate corollaries of the proposed law are the following:
# The specified complexity in a closed system of natural causes remains constant or decreases.
# The specified complexity cannot be generated spontaneously, originate endogenously or [[self-organization|organize itself]] (as these terms are used in [[origins-of-life research]]).
# The specified complexity in a closed system of natural causes either has been in the system eternally or was at some point added exogenously (implying that the system, though now closed, was not always closed).
# In particular any closed system of natural causes that is also of finite duration received whatever specified complexity it contains before it became a closed system.<ref name="arn.org">William A. Dembski (1998) [http://www.arn.org/docs/dembski/wd_idtheory.htm Intelligent Design as a Theory of Information].</ref>
</blockquote>

Dembski notes that the term "Law of Conservation of Information" was previously used by [[Peter Medawar]] in his book <cite>The Limits of Science</cite> (1984) "to describe the weaker claim that deterministic laws cannot produce novel information."<ref>[http://www.designinference.com/documents/2005.03.Searching_Large_Spaces.pdf "Searching Large Spaces: Displacement and the No Free Lunch Regress (356k PDF)]", pp. 15-16, describing an argument made by [[Michael Shermer]] in ''How We Believe: Science, Skepticism, and the Search for God'', 2nd ed. (2003).</ref> The actual validity and utility of Dembski's proposed law are uncertain; it is neither widely used by the scientific community nor cited in mainstream scientific literature. A 2002 essay by Erik Tellgren provided a mathematical rebuttal of Dembski's law and concludes that it is "mathematically unsubstantiated." <ref>[http://www.talkreason.org/articles/dembski_LCI.pdf On Dembski's law of conservation of information] Erik Tellgren. talkreason.org, 2002. (PDF file)</ref>

==Specificity==

In a more recent paper,<ref>William A. Dembski (2005). [http://www.designinference.com/documents/2005.06.Specification.pdf ''Specification: The Pattern that Signifies intelligence'']</ref> Dembski provides an account which he claims is simpler and adheres more closely to the theory of [[statistical hypothesis testing]] as formulated by [[Ronald Fisher]]. In general terms, Dembski proposes to view design inference as a statistical test to reject a chance hypothesis P on a space of outcomes Ω.

Dembski's proposed test is based on the [[Kolmogorov complexity]] of a pattern ''T'' that is exhibited by an event ''E'' that has occurred. Mathematically, ''E'' is a subset of Ω, the pattern ''T'' specifies a set of outcomes in Ω and ''E'' is a subset of ''T''. Quoting Dembski<ref>(loc. cit. p 16)</ref>
<blockquote>Thus, the event ''E'' might be a die toss that lands six and ''T'' might be the composite event consisting of all die tosses that land on an even face.</blockquote>
Kolmogorov complexity provides a measure of the computational resources needed to specify a pattern (such as a DNA sequence or a sequence of alphabetic characters).<ref>Michael Sipser (1997). ''Introduction to the Theory of Computation'', PWS Publishing Company.</ref> Given a pattern ''T'', the number of other patterns may have Kolmogorov complexity no larger than that of ''T'' is denoted by φ(''T''). The number φ(''T'') thus provides a ranking of patterns from the simplest to the most complex. For example, for a pattern ''T'' which describes the bacterial [[flagellum]], Dembski claims to obtain the upper bound φ(''T'') ≤ 1020.

Dembski defines '''specified complexity''' of the pattern ''T'' under the chance hypothesis P as

:<math> \sigma= - \log_2 [R \times \varphi(T) \times \operatorname{P}(T)], </math>

where P(''T'') is the probability of observing the pattern ''T'', ''R'' is the number of "replicational resources" available "to witnessing agents". ''R'' corresponds roughly to repeated attempts to create and discern a pattern. Dembski then asserts that ''R'' can be bounded by 10120. This number is supposedly justified by a result of Seth Lloyd<ref>[[Seth Lloyd]] (2002), ''Computational capacity of the universe'', Phys. Rev. Lett. 88(23):790 1- 4. See also [http://arxiv.org/abs/quant-ph/0110141 arXiv:quant/ph0110141].</ref> in which he determines that the number of elementary logic operations that can have been performed in the universe over its entire history cannot exceed 10120 operations on 1090 bits.

Dembski's main claim is that the following test can be used to infer design for a configuration: There is a target pattern ''T'' that applies to the configuration and whose specified complexity exceeds 1. This condition can be restated as the inequality
:<math> 10^{120} \times \varphi(T) \times \operatorname{P}(T) < \frac{1}{2}. </math>

===Dembski's explanation of specified complexity===

Dembski's expression σ is unrelated to any known concept in information theory, though he claims he can justify its relevance as follows: An intelligent agent ''S'' witnesses an event ''E'' and assigns it to some reference class of events Ω and within this reference class considers it as satisfying a specification ''T''. Now consider the quantity φ(''T'') × P(''T'') (where P is the "chance" hypothesis):
[[Image:TargetReplicationalRsources.png|thumb|350px|Possible targets with complexity ranking and probability not exceeding those of attained target ''T''. Probability of [[union (set theory)|set-theoretic union]] does not exceed φ(''T'') × P(''T'')]]

<blockquote>
Think of S as trying to determine whether an archer, who has just shot an arrow at a large wall, happened to hit a tiny target on that wall by chance. The arrow, let us say, is indeed sticking squarely in this tiny target. The problem, however, is that there are lots of other tiny targets on the wall. Once all those other targets are factored in, is it still unlikely that the archer could have hit any of them by chance?
In addition, we need to factor in what I call the replicational resources associated with ''T'', that is, all the opportunities to bring about an event of ''T'''s descriptive complexity and improbability by multiple agents witnessing multiple events.
</blockquote>

According to Dembski, the number of such "replicational resources" can be bounded by "the maximal number of bit operations that the known, observable universe could have performed throughout its entire multi-billion year history", which according to Lloyd is 10120.

However, according to Elsberry and Shallit, "[specified complexity] has not been defined formally in any reputable peer-reviewed mathematical journal, nor (to the best of our knowledge) adopted by any researcher in information theory."<ref>Elsberry and Shallit, (2003)p 14</ref>

===Calculation of specified complexity===

Thus far, Dembski's only attempt at calculating the specified complexity of a naturally occurring biological structure is in his book ''No Free Lunch'', for the [[bacterial flagellum]] of [[E. coli]]. This structure can be described by the pattern "bidirectional rotary motor-driven propeller". Dembski estimates that there are at most 1020 patterns described by four basic concepts or fewer, and so his test for design will apply if

:<math> \operatorname{P}(T) < \frac{1}{2} \times 10^{-140}. </math>

However, Dembski says that the precise calculation of the relevant probability "has yet to be done", although he also claims that some methods for calculating these probabilities "are now in place".

These methods assume that all of the constituent parts of the flagellum must have been generated completely at random, a scenario that biologists do not seriously consider. He justifies this approach by appealing to [[Michael Behe]]'s concept of "[[irreducible complexity]]" (IC), which leads him to assume that the flagellum could not come about by any gradual or step-wise process. The validity of Dembski's particular calculation is thus wholly dependent on Behe's IC concept, and therefore susceptible to its criticisms, of which there are many.

To arrive at the ranking upper bound of 1020 patterns, Dembski considers a specification pattern for the flagellum defined by the (natural language) predicate "bidirectional rotary motor-driven propeller", which he regards as being determined by four independently chosen basic concepts. He furthermore assumes that English has the capability to express at most 105 basic concepts (an upper bound on the size of a dictionary). Dembski then claims that we can obtain the rough upper bound of

:<math> 10^{20}= 10^5 \times 10^5 \times 10^5 \times 10^5 </math>
for the set of patterns described by four basic concepts or fewer.

From the standpoint of Kolmogorov complexity theory, this calculation is problematic. Quoting Ellsberry and Shallit<ref>Elsberry and Shallit, (2003)p 12</ref> "Natural language specification without restriction, as Dembski tacitly permits, seems problematic. For one thing, it results in the [[Berry paradox]]". These authors add: "We have no objection to natural language specifications per se, provided there is some evident way to translate them to Dembski's formal framework. But what, precisely, is the space of events Ω here?"

==Criticisms==

{{Synthesis|section|date=May 2012}}

The soundness of Dembski's concept of specified complexity and the validity of arguments based on this concept are widely disputed. A frequent criticism (see Elsberry and Shallit) is that Dembski has used the terms "complexity", "information" and "improbability" interchangeably. These numbers measure properties of things of different types: Complexity measures how hard it is to describe an object (such as a bitstring), information measures how close to [[Uniform distribution (discrete)|uniform]] a random probability distribution is and improbability measures how unlikely an event is given a probability distribution.

When Dembski's mathematical claims on specific complexity are interpreted to make them meaningful and conform to minimal standards of mathematical usage, they usually turn out to be false.{{Citation needed|date=May 2012}} Dembski often sidesteps these criticisms by responding that he is not "in the business of offering a strict [[mathematical proof]] for the inability of material mechanisms to generate specified complexity".<ref>William A. Dembski, (2002). [http://www.designinference.com/documents/2002.08.Erik_Response.htm ''If Only Darwinists Scrutinized Their Own Work as Closely: A Response to "Erik"''].</ref> On page 150 of ''No Free Lunch'' he claims he can demonstrate his thesis mathematically: ''"In this section I will present an in-principle mathematical argument for why natural causes are incapable of generating complex specified information."'' Others have pointed out that a crucial calculation on page 297 of ''No Free Lunch'' is off by a factor of approximately 1065.<ref name=shallit>Jeffrey Shallit (2002) [http://www.cs.uwaterloo.ca/~shallit/nflr3.txt A review of Dembski's ''No Free Lunch'']</ref>

Dembski's calculations show how a simple [[smooth function]] cannot gain information. He therefore concludes that there must be a designer to obtain CSI. However, natural selection has a branching mapping from one to many (replication) followed by pruning mapping of the many back down to a few (selection). When information is replicated, some copies can be differently modified while others remain the same, allowing information to increase. These increasing and reductional mappings were not modeled by Dembski. In other words, Dembski's calculations do not model birth and death. This basic flaw in his modeling renders all of Dembski's subsequent calculations and reasoning in ''No Free Lunch'' irrelevant because his basic model does not reflect reality. Since the basis of ''No Free Lunch'' relies on this flawed argument, the entire thesis of the book collapses.<ref>Thomas D. Schneider. (2002) [http://www.lecb.ncifcrf.gov/~toms/paper/ev/dembski/specified.complexity.html Dissecting Dembski's "Complex Specified Information"]</ref>

According to Martin Nowak, a Harvard professor of mathematics and evolutionary biology "We cannot calculate the probability that an eye came about. We don't have the information to make the calculation".<ref name="time.com"/>

Dembski's critics note that specified complexity, as originally defined by Leslie Orgel, is precisely what Darwinian evolution is supposed to create. Critics maintain that Dembski uses "complex" as most people would use "absurdly improbable". They also claim that his argument is [[circular reasoning|circular]]: CSI cannot occur naturally because Dembski has defined it thus. They argue that to successfully demonstrate the existence of CSI, it would be necessary to show that some biological feature undoubtedly has an extremely low probability of occurring by any natural means whatsoever, something which Dembski and others have almost never attempted to do. Such calculations depend on the accurate assessment of numerous contributing probabilities, the determination of which is often necessarily subjective. Hence, CSI can at most provide a "very high probability", but not absolute certainty.

Another criticism refers to the problem of "arbitrary but specific outcomes". For example, if a coin is tossed randomly 1000 times, the probability of any particular outcome occurring is roughly one in 10300. For any particular specific outcome of the coin-tossing process, the ''a priori'' probability that this pattern occurred is thus one in 10300, which is astronomically smaller than Dembski's universal probability bound of one in 10150. Yet we know that the ''post hoc'' probability of its happening is exactly one, since we observed it happening. This is similar to the observation that it is unlikely that any given person will win a lottery, but, eventually, a lottery will have a winner; to argue that it is very unlikely that any one player would win is not the same as proving that there is the same chance that no one will win. Similarly, it has been argued that "a space of possibilities is merely being explored, and we, as pattern-seeking animals, are merely imposing patterns, and therefore targets, after the fact."<ref name="arn.org"/>

Apart from such theoretical considerations, critics cite reports of evidence of the kind of evolutionary "spontanteous generation" that Dembski claims is too improbable to occur naturally. For example, in 1982, B.G. Hall published research demonstrating that after removing a gene that allows sugar digestion in certain bacteria, those bacteria, when grown in media rich in sugar, rapidly evolve new sugar-digesting enzymes to replace those removed.<ref>B.G. Hall (1982). "Evolution of a regulated operon in the laboratory", ''[[Genetics (journal)|Genetics]]'', 101(3-4):335-44. [http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=6816666&query_hl=1 In PubMed.]</ref> Another widely cited example is the discovery of [[nylon eating bacteria]] that produce enzymes only useful for digesting synthetic materials that did not exist prior to the invention of [[nylon]] in 1935.

Other commentators have noted that evolution through selection is frequently used to design certain electronic, aeronautic and automotive systems which are considered problems too complex for human "intelligent designers".<ref>[http://www.newscientisttech.com/channel/tech/mg19526146.000-evolutionary-algorithms-now-surpass-human-designers.html Evolutionary algorithms now surpass human designers] New Scientist, 28 July 2007</ref> This contradicts the argument that an intelligent designer is required for the most complex systems. Such evolutionary techniques can lead to designs that are difficult to understand or evaluate since no human understands which trade-offs were made in the evolutionary process, something which mimics our poor understanding of biological systems.

Dembski's book ''No Free Lunch'' was criticised for not addressing the work of researchers who use computer simulations to investigate [[artificial life]]. According to [[Jeffrey Shallit]]:
<blockquote>
The field of artificial life evidently poses a significant challenge to Dembski's claims about the failure of evolutionary algorithms to generate complexity. Indeed, artificial life researchers regularly find their simulations of evolution producing the sorts of novelties and increased complexity that Dembski claims are impossible.<ref name="shallit"/>
</blockquote>

==See also==
*[[List of topics characterized as pseudoscience]]
*[[Teleological argument]]
*[[Texas sharpshooter fallacy]]

==Notes and references==
{{reflist}}

==References==
* [[Wesley Elsberry]] and Jeffrey Shallit, (2003). ''[http://www.talkreason.org/articles/eandsdembski.pdf Information Theory, Evolutionary Computation, and Dembski’s “Complex Specified Information”]''

==External links==
* [http://www.talkorigins.org/design/faqs/nfl/ Not a Free Lunch But a Box of Chocolates - A critique of William Dembski's book No Free Lunch] by Richard Wein, from TalkOrigins
* [http://www.talkorigins.org/faqs/information/dembski.html Information Theory and Creationism William Dembski] by Rich Baldwin, from Information Theory and Creationism, compiled by Ian Musgrave and Rich Baldwin
* [http://www.bostonreview.net/BR27.3/orr.html Critique of No Free Lunch by H. Allen Orr] from the Boston Review
* [http://www.lecb.ncifcrf.gov/~toms/paper/ev/dembski/specified.complexity.html Dissecting Dembski's "Complex Specified Information"] by Dr. Thomas D. Schneider.
* [http://www.talkreason.org/articles/jello.cfm William Dembski's treatment of the No Free Lunch theorems is written in jello] by No Free Lunch theorems co-founder, David Wolpert
* [http://evolutionlist.blogspot.com/2006/05/genetic-id-and-explanatory-filter.html The Evolution List - Genetic ID and the Explanatory Filter] by Allen MacNeill.
* [http://www.designinference.com/ Design Inference Website] - The writing of William A. Dembski
* [http://www.csicop.org/sb/2000-12/reality-check.html Committee for Skeptical Inquiry - Reality Check, The Emperor's New Designer Clothes] - Victor J. Stenger
* [http://www.darwinathome.org/ Darwin@Home Web site] - open-source software that demonstrates evolution in artificial life, written by Gerald de Jong

{{DEFAULTSORT:Specified Complexity}}
[[Category:Intelligent design]]
[[Category:Creationist objections to evolution]]

Doubling space

2014-01-21T14:03:39Z

129.132.147.26: /* Examples */

'''Work sampling''' is the statistical technique for determining the proportion of time spent by workers in various defined categories of activity (e.g. setting up a machine, assembling two parts, idle…etc.).<ref name="Groover" /> It is as important as all other statistical techniques because it permits quick analysis, recognition, and enhancement of job responsibilities, tasks, performance competencies, and organizational work flows. Other names used for it are 'activity sampling', 'occurrence sampling', and 'ratio delay study'.<ref name="Sheth" />

In a work sampling study, a large number of [[observation]]s are made of the workers over an extended period of time. For statistical accuracy, the observations must be taken at random times during the period of study, and the period must be representative of the types of activities performed by the subjects.

One important usage of the work sampling technique is the determination of the standard time for a manual [[manufacturing]] task. Similar techniques for calculating the standard time are [[Time and motion study|time study]], standard data, and [[predetermined motion time system]]s.

== Characteristics of work sampling study ==
The study of work sampling has some general characteristics related to the work condition. One of them is the sufficient time available to perform the study. A work sampling study usually requires a substantial period of time to complete. There must be enough time available (several weeks or more) to conduct the study. Another characteristic is multiple workers. Work sampling is commonly used to study the activities of multiple workers rather than one worker. The third characteristic is long cycle time. The job covered in the study has relatively a long cycle time. The last condition is the non-repetitive work cycles. The work is not highly repetitive. The jobs consist of various tasks rather than a single repetitive task. However, it must be possible to classify the work activities into a distinct number of categories.

== Steps in conducting a work sampling study ==
There are several recommended steps when starting to prepare a work sampling study:<ref name="Groover" />
# Define the manufacturing tasks for which the standard time is to be determined.
# Define the task elements. These are the defined broken-down steps of the task that will be observed during the study. Since a worker is going to be observed, additional categories will likely be included as well, such as "idle", "waiting for work", and "absent".
# Design the study. This includes designing the forms that will be used to record the observations, determining how many observations will be required, deciding on the number of days or shifts to be included in the study, scheduling the observations, and finally determining the number of observers needed.
# Identify the observers who will do the [[Sampling (statistics)|sampling]].
# Start the study. All those who are affected by the study should be informed about it.
# Make random visits to the plant and collect the observations.
# After completing the study, analyze and present the results. This is done by preparing a report that summarizes and analyzes all data and making recommendations when required.

==Determining the number of observations needed in work sampling==

After the work elements are defined, the number of observations for the desired accuracy at the desired confidence level must be determined.
The formula used in this method is:

<math>\sigma_P=\sqrt{\frac{pq}{n}}</math>

<math>n=\frac{pq}{{\sigma_P}^2}</math>

<math>\sigma_P=</math> standard error of proportion

<math>p=</math> percentage of idle time

<math>q=</math> percentage of working time

<math>n=</math> number of observations

==Additional applications of work sampling==
Work sampling was initially developed for determining time allocation among workers' tasks in manufacturing environments.<ref name="Tsai" /> However, the technique has also been applied more broadly to examine work in a number of different environments, such as healthcare<ref name="Ampt" /> and construction.<ref name="Buchholz" /> More recently, in the academic fields of [[organizational psychology]] and [[organizational behaviour]], the basic technique has been developed into a detailed [[job analysis]] method for examining a range of different research questions.<ref name="Robinson" />

==See also==
*[[Sampling (statistics)]]
*[[Profiling (computer programming)]] can be done by work sampling a computer program.

==References==
{{Reflist|refs=
<ref name="Groover">Groover, M. P. ''Work Systems and Methods, measurement, and Management of Work''. Pearson Education International, 2007 ISBN 978-0-13-140650-6</ref>
<ref name="Sheth">Sheth, V. ''Industrial Engineering Methods and Practices''. Penram International Publishing, 2000, ISBN 81-87972-18-1</ref>
<ref name="Tsai">Tsai, W-. H. (1996). A technical note on using work sampling to estimate the effort on activities under activity-based costing. International Journal of Production Economics, 43(1), 11-16.
http://dx.doi.org/10.1016/0925-5273(95)00189-1</ref>
<ref name="Ampt">Ampt, A., Westbrook, J., Creswick, N., & Mallock, N. (2007). A comparison of self-reported and observational work sampling techniques for measuring time in nursing tasks. Journal of Health Services Research & Policy, 12, 18–24. http://dx.doi.org/10.1258/135581907779497576</ref>
<ref name="Buchholz">Buchholz, B., Paquet, V., Punnett, L., Lee, D., & Moir, S. (1996). PATH: A work sampling-based approach to ergonomic job analysis for construction and other non-repetitive work. Applied Ergonomics, 27(3), 177-187. http://dx.doi.org/10.1016/0003-6870(95)00078-X</ref>
<ref name="Robinson">Robinson, M. A. (2010). Work sampling: Methodological advances and new applications. Human Factors and Ergonomics in Manufacturing & Service Industries, 20(1), 42–60. http://dx.doi.org/10.1002/hfm.20186</ref>
}}

==External links==
* [http://dx.doi.org/10.1002/hfm.20186 Work sampling method]

[[Category:Industrial engineering]]

Belief propagation

2014-01-19T11:39:37Z

129.132.45.163: /* Exact algorithm for trees */

{{technical|date=January 2014}}

[[Image:Conditional independence.svg|thumb|right|450px|These are two examples illustrating '''conditional independence'''. Each cell represents a possible outcome. The events ''R'', ''B'' and ''Y'' are represented by the areas shaded red, blue and yellow respectively. And the probabilities of these events are shaded areas with respect to the total area. In both examples ''R'' and ''B'' are conditionally independent given ''Y'' because:

<math>\Pr(R \cap B \mid Y) = \Pr(R \mid Y)\Pr(B \mid Y)\,</math><ref>To see that this is the case, one needs to realise that Pr(''R'' ∩ ''B'' | ''Y'') is the probability of an overlap of ''R'' and ''B'' (the purple shaded area) in the ''Y'' area. Since, in the picture on the left, there are two squares where ''R'' and ''B'' overlap within the ''Y'' area, and the ''Y'' area has twelve squares, Pr(''R'' ∩ ''B'' | ''Y'') = {{sfrac|2|12}} = {{sfrac|1|6}}. Similarly, Pr(''R'' | ''Y'') = {{sfrac|4|12}} = {{sfrac|1|3}} and Pr(''B'' | ''Y'') = {{sfrac|6|12}} = {{sfrac|1|2}}.</ref>

 but not conditionally independent given not ''Y'' because:

<math>\Pr(R \cap B \mid \text{not } Y) \not= \Pr(R \mid \mbox{not } Y)\Pr(B \mid \text{not } Y).\,</math>]]

{{Probability fundamentals}}
In [[probability theory]], two events ''R'' and ''B'' are '''conditionally independent''' given a third event ''Y'' precisely if the occurrence or non-occurrence of ''R'' ''and'' the occurrence or non-occurrence of ''B'' are [[statistical independence|independent]] events in their conditional [[probability distribution]] given ''Y''. In other words, ''R'' and ''B'' are conditionally independent given ''Y'' if and only if, given knowledge that ''Y'' occurs, knowledge of whether ''R'' occurs provides no information on the likelihood of ''B'' occurring, and knowledge of whether ''B'' occurs provides no information on the likelihood of ''R'' occurring.

In the standard notation of probability theory, ''R'' and ''B'' are conditionally independent given ''Y'' if and only if

:<math>\Pr(R \cap B \mid Y) = \Pr(R \mid Y)\Pr(B \mid Y),\,</math>

or equivalently,

:<math>\Pr(R \mid B \cap Y) = \Pr(R \mid Y).\,</math>

Two [[random variable]]s ''X'' and ''Y'' are '''conditionally independent''' given a third random variable ''Z'' if and only if they are independent in their conditional probability distribution given ''Z''. That is, ''X'' and ''Y'' are conditionally independent given ''Z'' if and only if, given any value of ''Z'', the probability distribution of ''X'' is the same for all values of ''Y'' and the probability distribution of ''Y'' is the same for all values of ''X''.

Two events ''R'' and ''B'' are '''conditionally independent''' given a [[sigma-algebra|σ-algebra]] Σ if

:<math>\Pr(R \cap B \mid \Sigma) = \Pr(R \mid \Sigma)\Pr(B \mid \Sigma)\ a.s.</math>

where <math>\Pr(A \mid \Sigma) </math> denotes the [[conditional expectation]] of the [[indicator function]] of the event <math>A</math>, <math>\chi_A</math>, given the sigma algebra <math>\Sigma</math>. That is,

:<math>\Pr(A \mid \Sigma) := \operatorname{E}[\chi_A\mid\Sigma].</math>

Two random variables ''X'' and ''Y'' are conditionally independent given a σ-algebra ''Σ'' if the above equation holds for all ''R'' in σ(''X'') and B in σ(''Y'').

Two random variables ''X'' and ''Y'' are conditionally independent given a random variable ''W'' if they are independent given σ(''W''): the σ-algebra generated by ''W''. This is commonly written:

:<math>X \perp\!\!\!\perp Y \,|\, W </math> or
:<math>X \perp Y \,|\, W</math>

This is read "X is independent of Y, given W"; the conditioning applies to the whole statement: "(X is independent of Y) given W".

:<math>(X \perp\!\!\!\perp Y) \,|\, W</math>

If ''W'' assumes a countable set of values, this is equivalent to the conditional independence of ''X'' and ''Y'' for the events of the form [''W'' = ''w''].
Conditional independence of more than two events, or of more than two random variables, is defined analogously.

The following two examples show that ''X'' {{Unicode|&perp;}} ''Y''
''neither implies nor is implied by'' ''X'' {{Unicode|&perp;}} ''Y | W''.
First, suppose ''W'' is 0 with probability 0.5 and is the value 1 otherwise. When
''W'' = 0 take ''X'' and ''Y'' to be independent, each having the value 0 with probability 0.99 and the value 1 otherwise. When ''W'' = 1, ''X'' and ''Y'' are again independent, but this time they take the value 1
with probability 0.99. Then ''X'' {{Unicode|&perp;}} ''Y'' | ''W''. But ''X'' and ''Y'' are dependent, because Pr(''X'' = 0) < Pr(''X'' = 0|''Y'' = 0). This is because Pr(''X'' = 0) = 0.5, but if ''Y'' = 0 then it's very likely that ''W'' = 0 and thus that ''X'' = 0 as well, so Pr(''X'' = 0|''Y'' = 0) > 0.5. For the second example, suppose ''X'' {{Unicode|&perp;}} ''Y'', each taking the values 0 and 1 with probability 0.5. Let ''W'' be the product ''X''{{Unicode|×}}''Y''. Then when ''W'' = 0, Pr(''X'' = 0) = 2/3, but Pr(''X'' = 0|''Y'' = 0) = 1/2, so ''X'' {{Unicode|&perp;}} ''Y'' | ''W'' is false.
This is also an example of Explaining Away. See Kevin Murphy's tutorial
<ref>http://people.cs.ubc.ca/~murphyk/Bayes/bnintro.html</ref>
where ''X'' and ''Y'' take the values "brainy" and "sporty".

==Uses in Bayesian inference==
Let ''p'' be the proportion of voters who will vote "yes" in an upcoming [[referendum]]. In taking an [[opinion poll]], one chooses ''n'' voters randomly from the population. For ''i'' = 1, ..., ''n'', let ''X''''i'' = 1 or 0 according as the ''i''th chosen voter will or will not vote "yes".

In a [[frequency probability|frequentist]] approach to [[statistical inference]] one would not attribute any probability distribution to ''p'' (unless the probabilities could be somehow interpreted as relative frequencies of occurrence of some event or as proportions of some population) and one would say that ''X''1, ..., ''X''''n'' are [[statistical independence|independent]] random variables.

By contrast, in a [[Bayesian inference|Bayesian]] approach to statistical inference, one would assign a [[probability distribution]] to ''p'' regardless of the non-existence of any such "frequency" interpretation, and one would construe the probabilities as degrees of belief that ''p'' is in any interval to which a probability is assigned. In that model, the random variables ''X''1, ..., ''X''''n'' are ''not'' independent, but they are '''conditionally independent''' given the value of ''p''. In particular, if a large number of the ''X''s are observed to be equal to 1, that would imply a high conditional probability, given that observation, that ''p'' is near 1, and thus a high conditional probability, given that observation, that the ''next'' ''X'' to be observed will be equal to 1.

==Rules of conditional independence==

A set of rules governing statements of conditional independence have been derived from the basic definition.<ref>{{cite journal
|first=A. P. |last=Dawid |authorlink=Philip Dawid
|title=Conditional Independence in Statistical Theory
|journal=[[Journal of the Royal Statistical Society, Series B]]
|year=1979
|volume=41 |issue=1 |pages=1–31
|mr=0535541
|jstor=2984718
}}</ref><ref>J Pearl, Causality: Models, Reasoning, and Inference, 2000, Cambridge University Press</ref>

Note: since these implications hold for any probability space, they will still hold if one considers a sub-universe by conditioning everything on another variable, say ''K''. For example, <math>X \perp\!\!\!\perp Y \Rightarrow Y \perp\!\!\!\perp X</math> would also mean that <math>X \perp\!\!\!\perp Y \mid K \Rightarrow Y \perp\!\!\!\perp X \mid K</math>.

Note: below, the comma can be read as an "AND".

===Symmetry===
: <math>
X \perp\!\!\!\perp Y
\quad \Rightarrow \quad
Y \perp\!\!\!\perp X
</math>

===Decomposition===
: <math>
X \perp\!\!\!\perp A,B
\quad \Rightarrow \quad
\text{ and }
\begin{cases}
X \perp\!\!\!\perp A \\
X \perp\!\!\!\perp B
\end{cases}
</math>

Proof:
* <math>
p_{X,A,B}(x,a,b) = p_X(x) p_{A,B}(a,b)
</math>      (meaning of <math>X \perp A,B</math>)
* <math>
\int_{B} \! p_{X,A,B}(x,a,b) = \int_{B} \! p_X(x) p_{A,B}(a,b)
</math>      (ignore variable ''B'' by integrating it out)
* <math>
p_{X,A}(x,a) = p_X(x) p_A(a)
</math>     

A similar proof shows the independence of ''X'' and ''B''.

===Weak union===

: <math>
X \perp\!\!\!\perp A,B
\quad \Rightarrow \quad
X \perp\!\!\!\perp A \mid B
</math>

===Contraction===

: <math>
\left.\begin{align}
X \perp\!\!\!\perp A \mid B \\
X \perp\!\!\!\perp B
\end{align}\right\}\text{ and }
\quad \Rightarrow \quad
X \perp\!\!\!\perp A,B
</math>

===Contraction-weak-union-decomposition===

Putting the above three together, we have:

: <math>
\left.\begin{align}
X \perp\!\!\!\perp A \mid B \\
X \perp\!\!\!\perp B
\end{align}\right\}\text{ and }
\quad \iff \quad
X \perp\!\!\!\perp A,B
\quad \Rightarrow \quad
\text{ and }
\begin{cases}
X \perp\!\!\!\perp A \mid B \\
X \perp\!\!\!\perp B \\
X \perp\!\!\!\perp B \mid A \\
X \perp\!\!\!\perp A \\
\end{cases}
</math>

===Intersection===

If the probabilities of ''X'', ''A'', ''B'' are all positive{{Citation needed|date=October 2010}}, then the following also holds:

: <math>
\left.\begin{align}
X \perp\!\!\!\perp A \mid B \\
X \perp\!\!\!\perp B \mid A
\end{align}\right\}\text{ and }
\quad \Rightarrow \quad
X \perp\!\!\!\perp A, B
</math>

==See also==
*[[Conditional dependence]]
*[[Cointelation]]
*[[de Finetti's theorem]]
*[[Conditional expectation]]

==References==
{{Reflist}}

{{DEFAULTSORT:Conditional Independence}}
[[Category:Probability theory]]
[[Category:Statistical dependence]]

Guruswami–Sudan list decoding algorithm

2013-11-07T15:01:23Z

129.132.57.154: /* Analysis */ Fixed a typesetting typo

{{context|date=May 2012}}

{{Other uses|Kansa (disambiguation){{!}}Kansa}}
E. J. Kansa in very early 1990s made the first attempt to extend [[radial basis function]] (RBF), which was then quite popular in scattered data processing and function approximation, to the solution of [[partial differential equations]] in the strong-form collocation formulation. His RBF collocation approach is inherently meshless, easy-to-program, and mathematically very simple to learn. Before long, this method is known as the Kansa method in academic community.

Due to the fact that the RBF uses the one-dimensional Euclidean distance variable irrespective of dimensionality, the '''Kansa method''' is independent of dimensionality and geometric complexity of problems of interest. The method is a domain-type numerical technique in the sense that the problem is discretized not only on the boundary to satisfy boundary conditions but also inside domain to satisfy governing equation.

In contrast, there is another type of RBF numerical methods, called boundary-type RBF collocation method, such as the [[method of fundamental solution]], [[boundary knot method]], [[singular boundary method]], [[boundary particle method]], and regularized meshless method, in which the basis functions, also known as kernel function, satisfy the governing equation and are often fundamental solution or general solution of governing equation. Consequently, only boundary discretization is required.

Since the RBF in the '''Kansa method''' does not necessarily satisfy the governing equation, one has more freedom to choose a RBF. The most popular RBF in the Kansa method is the multiquadric (MQ), which usually shows spectral accuracy if an appropriate shape parameter is chosen.

== Description ==

The '''Kansa method''', also called modified MQ scheme or MQ collocation method, originated from the well-known MQ interpolation. The efficiency and applicability of this method have been verified in a wide range of problems. Compared with the boundary-type RBF collocation methods, the '''Kansa method''' has wider applicability to problems whose fundamental and general solutions are not available, e.g., varying coefficient and nonlinear problems.

== Formulation ==

Let ''d''-dimensional physical domain <math>\Omega \subseteq^d</math> and consider the following [[boundary value problem]] (BVP)
:<math>Lu(X) = f(X),\quad X\in \Omega,\qquad (1)</math>
:<math>u(X) = g(X),\quad X\in\partial\Omega_D,\qquad(2)</math>
:<math>\frac{\partial u(X)}{\partial n} = h(X),\quad X\in\partial\Omega_N,\qquad (3)</math>
where ''L'' represents a differential operator and ''d'' is the dimensionality of the problem,<math>\partial \Omega_D,\quad \partial \Omega_N</math> denote the Dirichlet and Neumann boundaries, respectively, and <math>\partial \Omega_D \cup \partial \Omega_N =\partial \Omega </math>.
The '''Kansa method''' approximates the desired function by a linear combination of the RBF in the form:

:<math>{{u(X)}^{*}} = \sum\limits_{i=1}^N \alpha_i\phi \left( r_i \right),\qquad(4)</math>

where <math>{{\alpha }_{i}}</math> is the coefficients to be determined, <math>\phi \left( r_i \right)</math> denotes the RBF such as the MQ.

In order to guarantee the uniqueness of the solution, a polynomial term may be added as follows:

:<math>{{u(X)}^{*}} = \sum\limits_{i=1}^N \alpha_i\phi \left( r_i \right) + \sum\limits_{k=1}^M \alpha_{k+N}\gamma_k\left(X\right),\qquad (5)</math>

where <math>{{\gamma }_{k}}(X)</math> is the polynomial. The RBF interpolation (4) and (5) are both often used in practice. Mathematicians prefer the latter for its rigorous and solid theoretical foundation, while engineering users often employ the former since it is easier and simpler and produces the sound results in the most majority cases.
Substituting Eq. (4) or (5) into Eqs. (1–3) yields the resulting algebraic equation system:

: <math> \mathbf{A}\alpha = b,\qquad (6) </math>

where
: <math>\mathbf{A}=\left(\begin{matrix} L(\phi ) & L(\gamma)\\\phi & \gamma\\ \frac{\partial {\phi }}{\partial n}&\frac{\partial {\gamma}}{\partial n}\\ {\gamma}&0\\ \end{matrix} \right),\quad \mathbf{b}=\left(\begin{matrix} f\\g\\h\\0\\ \end{matrix} \right),\quad \phi = \phi\left( x_i,x_j\right),\quad \gamma = \gamma_k\left(X_i\right).\qquad (7)</math>
After expansion coefficients <math>\alpha_i</math> are evaluated, the desired function can be calculated from Eq. (4) or (5).

== History and recent developments ==

Numerical solutions of PDEs are usually obtained through the finite difference method (FDM), the [[finite element method]] (FEM) or [[boundary element method]] (BEM). It is known that the FDM is difficult to model an irregular domain for the reason that it usually requires a rectangular grid system. Although the FEM can accommodate a more flexible framework, the meshing and remeshing are not trivial. The BEM is an alternative method in some engineering problems, such as inverse, unbounded domain, and thin-walled structure problems. However, its applications are largely limited by the availability of the fundamental solution of the governing equation.

In the recent several decades, “meshless” or “element-free” methods attract great attention. The driving force behind the scene is that the mesh-based methods such as the standard FEM and BEM require prohibitively computational effort in handling high-dimensional, moving, and complex-shaped boundary problems. The Kansa method <ref name="Chena">E. J. Kansa, "Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations," Computers & Mathematics with Applications, vol. 19, pp. 147–161, 1990.</ref><ref>J. Li, et al., "A comparison of efficiency and error convergence of multiquadric collocation method and finite element method," Engineering Analysis with Boundary Elements, vol. 27, pp. 251–257, 2003.</ref> directly collocates the RBFs, especially the MQ, at the nodes without the need of mesh or elements and therefore is an inherently truly meshless method.

Despite great effort, the rigorous mathematical proof of the solvability of the Kansa method is still missing.<ref>Y. C. Hon and R. Schaback, "On unsymmetric collocation by radial basis functions," Applied Mathematics and Computation, vol. 119, pp. 177–186, 2001.</ref> In addition, the mixed boundary conditions also destroy the symmetry of its interpolation matrix. Refs.<ref>C. Franke and R. Schaback, "Solving partial differential equations by collocation using radial basis functions," Applied Mathematics and Computation, vol. 93, pp. 73–82, 1998.</ref><ref>G. E. Fasshauer, "Solving Partial Differential Equations by Collocation," 1996, p. 1.</ref> propose the symmetric Hermite RBF collocation scheme with sound mathematical analysis of solvability. One common issue in the Kansa method and symmetric Hermite method, however, is that the numerical solutions at nodes adjacent to boundary deteriorate by one to two orders of magnitude compared with those in central region. The PDE collocation on the boundary (PDECB) <ref>A. I. Fedoseyev, et al., "Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary," Computers & Mathematics with Applications, vol. 43, pp. 439–455, 2002.</ref> effectively remove this shortcoming. However, this strategy requires an additional set of nodes inside or outside of the domain adjacent to the boundary. The arbitrary placing of these additional nodes gives rise to troublesome issues in the simulation of complex and multiply-connected domain problems. The PDECB also lacks explicit theoretical endorsement. In fact, a similar strategy has also been proposed,<ref>X. Zhang, et al., "Meshless methods based on collocation with radial basis functions," Computational Mechanics, vol. 26, pp. 333–343, 2000.</ref> which collocates both governing and boundary equations on the same boundary nodes. However, the method is unsymmetrical and still lacks explicit theoretical foundation. By using the Green second identity, the modified Kansa method <ref>W. Chen, "New RBF Collocation Methods and Kernel RBF with Applications: Meshfree Methods for Partial Differential Equations." vol. 26, M. Griebel and M. A. Schweitzer, Eds., ed: Springer Berlin Heidelberg, 2002, pp. 75–86.</ref><ref>W. Chen and M. Tanaka, "New insights in boundary-only and domain-type RBF methods," Arxiv preprint cs/0207017, 2002.</ref> is devised to remedy all weaknesses aforementioned. For the MQ, its shape parameter largely determines its interpolation error. There exist a number of mathematical theories concerning the family of multiquadric radial basis functions and providing some suggestions on the choice of the shape parameter.<ref>R. L. Hardy, "Multiquadric Equations of Topography and Other Irregular Surfaces," J. Geophys. Res., vol. 76, pp. 1905–1915, 1971.</ref><ref>R. Franke, "Scattered Data Interpolation: Tests of Some Method," Mathematics of Computation, vol. 38, pp. 181-200, 1982.</ref><ref>E. J. Kansa and R. E. Carlson, "Improved accuracy of multiquadric interpolation using variable shape parameters," Computers & Mathematics with Applications, vol. 24, pp. 99–120, 1992.</ref><ref>C. A.H.-D, "Multiquadric and its shape parameter-A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation," Engineering Analysis with Boundary Elements, vol. 36, pp. 220–239, 2012.</ref>

The Kansa method are widely been applied in computational sciences. In,<ref name="Chena" /> the Kansa method is employed to address the parabolic, hyperbolic and elliptic partial differential equations. Kansa method has recently been extended to various ordinary and PDEs including the bi-phasic and triphasic mixture models of tissue engineering problems,<ref>Y. C. Hon, et al., "Multiquadric method for the numerical solution of a biphasic mixture model," Applied Mathematics and Computation, vol. 88, pp. 153–175, 1997.</ref><ref>Y. C. Hon, et al., "A new formulation and computation of the triphasic model for mechano-electrochemical mixtures," Computational Mechanics, vol. 24, pp. 155–165, 1999.</ref> 1D nonlinear Burger’s equation<ref>Y. C. Hon and X. Z. Mao, "An efficient numerical scheme for Burgers' equation," Applied Mathematics and Computation, vol. 95, pp. 37–50, 1998.</ref> with shock wave, shallow water equations <ref>Y.-C. Hon, et al., "Multiquadric Solution for Shallow Water Equations," Journal of Hydraulic Engineering, vol. 125, pp. 524–533, 1999.</ref> for tide and current simulation, heat transfer problems,<ref>M. Zerroukat, et al., "A numerical method for heat transfer problems using collocation and radial basis functions," International journal for numerical methods in engineering, vol. 42, pp. 1263–1278, 1998.</ref> free boundary problems,<ref>J. Perko, et al., "A polygon-free numerical solution of steady natural convection in solid-liquid systems," Computational Modelling of Moving and Free Boundary Problems, pp. 111–122, 2001.</ref> and fractional diffusion equations.<ref>W. Chen, et al., "Fractional diffusion equations by the Kansa method," Computers & Mathematics with Applications, vol. 59, pp. 1614–1620, 2010.</ref>

== See also ==
* [[Radial basis function]]
* [[Method of fundamental solutions]]
* [[Boundary knot method]]
* [[Boundary particle method]]
* [[Singular boundary method]]

== External links ==

* [http://em.hhu.edu.cn/chenwen/html/MKM.htm Modified Kansa method]

== References ==
{{Reflist}}

[[Category:Partial differential equations]]

Bicomplex number

2013-10-25T12:44:24Z

129.132.146.210: /* Direct sum C + C */

{{wiktionary|claw}}
A '''[[claw]]''' is a sharp growth at the end of a toe or finger.

'''Claw''' or '''Claws''' may also refer to:

*[[Chela (organ)]], a pincerlike organ terminating certain limbs of some arthropods like crabs
*[[Claw hammer]], a common carpentry tool
*[[Baltimore Claws]], a short-lived American basketball team
*[[Claw beaker]], a type of drinking vessel used in the Dark Ages in Europe
*[[Clothing, Laundry and Allied Workers Union of Aotearoa]], a trade union in New Zealand
*[[Claw (juggling)]] a ball juggling trick
*[[CLAW hypothesis]], a feedback loop between the climate and oceanic ecosystems
*Claw chisel or tool, used for [[stone carving]]

In '''entertainment''':
*[[Claws (play)|''Claws'' (play)]], a 1916 play by Sophie Treadwell
*[[The Claw (film)|''The Claw'' (film)]], a 1927 movie directed by [[Sidney Olcott]]
* "Claws", a 1976 episode of the television show ''[[The Bionic Woman]]''
*[[Claw (computer game)|''Claw'' (computer game)]], a 1997 computer game from Monolith Productions
*[[Claw vending machine]], a type of arcade game in the form of a vending machine
*[[Vega (Street Fighter)|''Claw'']], a name sometimes given to the character [[Vega (Street Fighter)|Vega]] from the [[Street Fighter]] games (so as to avoid confusion with his alternative name, [[Balrog (Street Fighter)|Balrog]])

In '''music'''
* [[C-L-A-W-S]], Gospel Claws album
* [[Claw (song)|"Claw" (song)]], by British metal band Motörhead from their 1986 album ''Orgasmatron''

In '''comics''' and '''animation''':
*[[C.L.A.W.]], a fictional weapon in the G.I. Joe universe
*[[Claw (comics)]], a supporting character in ''Daredevil''
*[[Doctor Claw]], the main villain of the animated television show ''Inspector Gadget''
*[[Claw the Unconquered]], a sword and sorcery character from DC Comics
*The Claw, the main villain in the animated series ''[[Karate Kommandos]]''
*[[Claw (Lev Gleason Publications)]], a villain in comics from Lev Gleason Publications

'''Other uses''':
*[[Claw (graph theory)]], in mathematics a complete bipartite graph <math>K_{1,3}</math>
* [[Claws Mail]], a GTK+-based e-mail client and news client for Linux
*"Claws", the nickname given to the [[Baryonyx]] find made in a clay pit in [[Surrey]]

==See also==
*[[Claw of Archimedes]], a weapon designed by Archimedes to destroy attacking ships
*[[Klaw (Marvel Comics)|Klaw]], a fictional character that appears in comic books published by Marvel Comics.
*[[The Claw (disambiguation)]]

{{disambiguation}}

Gauss–Laguerre quadrature

2013-09-30T14:57:08Z

129.132.146.210: /* For more general functions */

{{merge from|Thinning (morphology)|date=April 2013}}
In [[digital image processing]], '''morphological skeleton''' is a [[Topological skeleton|skeleton]] (or [[medial axis]]) representation of a [[shape]] or [[binary image]], computed by means of [[mathematical morphology|morphological operators]].

Morphological skeletons are of two kinds:
* Those defined and by means of [[opening (morphology)|morphological opening]]s, from which the original shape can be reconstructed,
* Those computed by means of the [[hit-or-miss transform]], which preserve the shape's [[topology]].

==Skeleton by openings==

===Lantuéjoul's formula===

====Continuous images====
In ([[#lantuejoul77|Lantuéjoul 1977]]),<ref>See also ([[#serra82|Serra's 1982 book]])</ref> Lantuéjoul derived the following morphological formula for the skeleton of a continuous binary image <math>X\subset \mathbb{R}^2</math>:

:<math>S(X)=\bigcup_{\rho >0}\bigcap_{\mu >0}\left[(X\ominus \rho B)-(X\ominus \rho B)\circ \mu \overline B\right]</math>,

where <math>\ominus</math> and <math>\circ</math> are the morphological [[erosion (morphology)|erosion]] and [[opening (morphology)|opening]], respectively, <math>\rho B</math> is an [[open ball]] of [[radius]] <math>\rho</math>, and <math> \overline B</math> is the closure of <math>B</math>.

====Discrete images====
Let <math>\{nB\}</math>, <math>n=0,1,\ldots</math>, be a family of shapes, where ''B'' is a [[structuring element]],
:<math>nB=\underbrace{B\oplus\ldots\oplus B}_{n\mbox{ times}}</math>, and
:<math>0B=\{o\}</math>, where ''o'' denotes the origin.

The variable ''n'' is called the ''size'' of the structuring element.

Lantuéjoul's formula has been discretized as follows. For a discrete binary image <math>X\subset \mathbb{Z}^2</math>, the skeleton ''S(X)'' is the [[set union|union]] of the '''skeleton subsets''' <math>\{S_n(X)\}</math>, <math>n=0,1,\ldots,N</math>, where:

:<math>S_n(X)=(X\ominus nB)-(X\ominus nB)\circ B</math>.

====Reconstruction from the skeleton====
The original shape ''X'' can be reconstructed from the set of skeleton subsets <math>\{S_n(X)\}</math> as follows:

:<math>X=\bigcup_n (S_n(X)\oplus nB)</math>.

Partial reconstructions can also be performed, leading to opened versions of the original shape:

:<math>\bigcup_{n\geq m} (S_n(X)\oplus nB)=X\circ mB</math>.

====The skeleton as the centers of the maximal disks====
Let <math>nB_z</math> be the translated version of <math>nB</math> to the point ''z'', that is, <math>nB_z=\{x\in E| x-z\in nB\}</math>.

A shape <math>nB_z</math> centered at ''z'' is called a '''maximal disk''' in a set ''A'' when:
* <math>nB_z\in A</math>, and
* if, for some integer ''m'' and some point ''y'', <math>nB_z\subseteq mB_y</math>, then <math>mB_y\not\subseteq A</math>.

Each skeleton subset <math>S_n(X)</math> consists of the centers of all maximal disks of size ''n''.

==Notes==

{{reflist}}

==References==
* <cite id=serra82>''Image Analysis and Mathematical Morphology'' by Jean Serra, ISBN 0-12-637240-3 (1982)</cite>
* ''Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances'' by Jean Serra, ISBN 0-12-637241-1 (1988)
* ''An Introduction to Morphological Image Processing'' by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)
* <cite id=lantuejoul77>Ch. Lantuéjoul, "Sur le modèle de Johnson-Mehl généralisé", ''Internal report of the Centre de Morph. Math.'', Fontainebleau, France, 1977.</cite>

[[Category:Mathematical morphology]]
[[Category:Digital geometry]]

Holonomic

2013-08-22T11:42:26Z

129.132.209.87: /* Mathematics */

{{refimprove|date=September 2010}}
In [[software engineering]], '''coupling''' or '''dependency''' is the degree to which each [[Module (programming)|program module]] relies on each one of the other modules.

Coupling is usually contrasted with [[cohesion (computer science)|cohesion]]. Low coupling often correlates with high cohesion, and vice versa. The [[software metric|software quality metrics]] of coupling and cohesion were invented by [[Larry Constantine]], an original developer of Structured Design<ref>W. Stevens, G. Myers, L. Constantine, "Structured Design", IBM Systems Journal, 13 (2), 115-139, 1974.</ref> who was also an early proponent of these concepts (see also [[Structured Systems Analysis and Design Methodology|SSADM]]). Low coupling is often a sign of a well-structured [[Computer|computer system]] and a good design, and when combined with high cohesion, supports the general goals of high readability and maintainability.{{Citation needed|date=October 2013}}

== Types of coupling ==
[[Image:Coupling sketches cropped 1.svg|thumb|300px|right|upright=2.0|Conceptual model of coupling]]
Coupling can be "low" (also "loose" and "weak") or "high" (also "tight" and "strong"). Some types of coupling, in order of highest to lowest coupling, are as follows:

=== Procedural programming ===
A module here refers to a subroutine of any kind, i.e. a set of one or more statements having a name and preferably its own set of variable names.

;Content coupling (high): Content coupling (also known as '''Pathological coupling''') occurs when one module modifies or relies on the internal workings of another module (e.g., accessing local data of another module).
:Therefore changing the way the second module produces data (location, type, timing) will lead to changing the dependent module.
;Common coupling: Common coupling (also known as '''Global coupling''') occurs when two modules share the same global data (e.g., a global variable).
:Changing the shared resource implies changing all the modules using it.
;External coupling: External coupling occurs when two modules share an externally imposed data format, communication protocol, or device interface. This is basically related to the communication to external tools and devices.
;Control coupling: Control coupling is one module controlling the flow of another, by passing it information on what to do (e.g., passing a what-to-do flag).
;Stamp coupling (Data-structured coupling): Stamp coupling occurs when modules share a composite data structure and use only a part of it, possibly a different part (e.g., passing a whole record to a function that only needs one field of it).
:This may lead to changing the way a module reads a record because a field that the module does not need has been modified.
;Data coupling: Data coupling occurs when modules share data through, for example, parameters. Each datum is an elementary piece, and these are the only data shared (e.g., passing an integer to a function that computes a square root).
;Message coupling (low): This is the loosest type of coupling. It can be achieved by state decentralization (as in objects) and component communication is done via parameters or message passing (see [[Message passing]]).
;No coupling: Modules do not communicate at all with one another.

=== Object-oriented programming ===
;Subclass Coupling: Describes the relationship between a child and its parent. The child is connected to its parent, but the parent is not connected to the child.

;Temporal coupling: When two actions are bundled together into one module just because they happen to occur at the same time.

In recent work various other coupling concepts have been investigated and used as indicators for different modularization principles used in practice.<ref>F. Beck, S. Diehl. On the Congruence of Modularity and Code Coupling. In Proceedings of the 19th ACM SIGSOFT Symposium and the 13th European Conference on Foundations of Software Engineering (SIGSOFT/FSE '11), Szeged, Hungary, September 2011. {{doi|10.1145/2025113.2025162}}</ref>

== Disadvantages ==
Tightly coupled systems tend to exhibit the following developmental characteristics, which are often seen as disadvantages:

# A change in one module usually forces a [[ripple effect]] of changes in other modules.
# Assembly of modules might require more effort and/or time due to the increased inter-module dependency.
# A particular module might be harder to reuse and/or test because dependent modules must be included.

== Performance issues ==
Whether loosely or tightly coupled, a system's performance is often reduced by message and parameter creation, transmission, translation (e.g. marshaling) and message interpretation (which might be a reference to a string, array or data structure), which require less overhead than creating a complicated message such as a [[SOAP]] message. Longer messages require more CPU and memory to produce. To optimize runtime performance, message length must be minimized and message meaning must be maximized.
;Message Transmission Overhead and Performance: Since a message must be transmitted in full to retain its complete meaning, message transmission must be optimized. Longer messages require more CPU and memory to transmit and receive. Also, when necessary, receivers must reassemble a message into its original state to completely receive it. Hence, to optimize runtime performance, message length must be minimized and message meaning must be maximized.

;Message Translation Overhead and Performance: Message protocols and messages themselves often contain extra information (i.e., packet, structure, definition and language information). Hence, the receiver often needs to translate a message into a more refined form by removing extra characters and structure information and/or by converting values from one type to another. Any sort of translation increases CPU and/or memory overhead. To optimize runtime performance, message form and content must be reduced and refined to maximize its meaning and reduce translation.

;Message Interpretation Overhead and Performance: All messages must be interpreted by the receiver. Simple messages such as integers might not require additional processing to be interpreted. However, complex messages such as [[SOAP]] messages require a parser and a string transformer for them to exhibit intended meanings. To optimize runtime performance, messages must be refined and reduced to minimize interpretation overhead.

== Solutions ==
One approach to decreasing coupling is [[functional design]], which seeks to limit the responsibilities of modules along functionality, coupling increases between two classes '''''A''''' and '''''B''''' if:

*'''''A''''' has an attribute that refers to (is of type) '''''B'''''.
*'''''A''''' calls on services of an object '''''B'''''.
*'''''A''''' has a method that references '''''B''''' (via return type or parameter).
*'''''A''''' is a subclass of (or implements) class '''''B'''''.

Low coupling refers to a relationship in which one module interacts with another module through a simple and stable interface and does not need to be concerned with the other module's internal implementation (see [[Information Hiding]]).

Systems such as [[CORBA]] or [[Component Object Model|COM]] allow objects to communicate with each other without having to know anything about the other object's implementation. Both of these systems even allow for objects to communicate with objects written in other languages.

== Coupling versus Cohesion ==
Coupling and [[cohesion (computer science)|Cohesion]] are terms which occur together very frequently. Coupling refers to the interdependencies between modules, while cohesion describes how related are the functions within a single module. Low cohesion implies that a given module performs tasks which are not very related to each other and hence can create problems as the module becomes large.

== Module coupling ==
Coupling in Software Engineering<ref name=Pressman>Pressman, Roger S. Ph.D. (1982). Software Engineering - A Practitioner's Approach - Fourth Edition. ISBN 0-07-052182-4</ref> describes a version of metrics associated with this concept.

For data and control flow coupling:

* '''di''': number of input data parameters
* '''ci''': number of input control parameters
* '''do''': number of output data parameters
* '''co''': number of output control parameters

For global coupling:

* '''gd''': number of global variables used as data
* '''gc''': number of global variables used as control

For environmental coupling:

* '''w''': number of modules called (fan-out)
* '''r''': number of modules calling the module under consideration (fan-in)

<math>\mathrm{Coupling}(C) = 1 - \frac{1}{d_{i} + 2\times c_{i} + d_{o} + 2\times c_{o} + g_{d} + 2\times g_{c} + w + r}</math>

<code>Coupling(C)</code> makes the value larger the more coupled the module is. This number ranges from approximately 0.67 (low coupling) to 1.0 (highly coupled)

For example, if a module has only a single input and output data parameter

<math>C = 1 - \frac{1}{1+0+1+0+0+0+1+0} = 1 - \frac{1}{3} = 0.67</math>

If a module has 5 input and output data parameters, an equal number of control parameters, and accesses 10 items of global data, with a fan-in of 3 and a fan-out of 4,

<math>C = 1 - \frac{1}{5 + 2\times 5 + 5 + 2\times 5 + 10 + 0 + 3 + 4} = 0.98</math>

==See also==
* [[Cohesion (computer science)]]
* [[Connascence (computer science)]]
* [[Dependency hell]]
* [[Efferent coupling]]
* [[Inversion of control]]
* [[List of object-oriented programming terms]]
* [[Loose coupling]]
* [[Make (software)]]
* [[Static code analysis]]

== References ==
<references/>

{{DEFAULTSORT:Coupling (Computer Science)}}
[[Category:Object-oriented programming]]
[[Category:Software architecture]]
[[Category:Software metrics]]

Real projective line

2013-07-01T12:59:13Z

129.132.183.94: /* Arithmetic operations which are defined */

{{Infobox Weapon
|name= Type 91 torpedo
|image= [[Image:Type 91 torpedo.JPG|250px]]
|caption= Type 91 torpedoes aboard an aircraft carrier.
|origin= [[Japan]]
|type= [[Aerial torpedo]]

|is_ranged=
|is_bladed=
|is_explosive=yes
|is_artillery=
|is_vehicle=
|is_missile=yes
|is_UK=

|service= 1931–1945
|used_by= [[Imperial Japanese Navy]]
|wars= [[World War II]]

|designer= Rear Admiral Shoji Naruse and his team
|design_date=1930–1945
|manufacturer=
|unit_cost= 20,000 yen (in the year 1941)
|production_date=
|number=
|variants=

|spec_label=
|weight= {{convert|848|kg|lb|abbr=on}}
|length= {{convert|5.270|m|abbr=on}}
|part_length=
|width=
|height=
|diameter={{convert|45|cm|in|abbr=on}}
|crew=

|cartridge=
|caliber=
|barrels=
|action=
|rate=
|velocity=
|range=
|max_range= 2,000 m (2,187 yd)
|feed=
|sights=

|breech=
|recoil=
|carriage=
|elevation=
|traverse=

|blade_type=
|hilt_type=
|sheath_type=
|head_type=
|haft_type=

|filling=
|filling_weight= 323.6 kg with high explosive 235 kg, (713.4 lb with 518 lb) for Type 91 warhead rev.3
|detonation=
|yield=

|armour=
|primary_armament=
|secondary_armament=
|engine=wet-heater type, 8-cylinder radial engine
|engine_power=200hp
|pw_ratio=
|transmission=
|payload_capacity=
|suspension=
|clearance=
|wingspan= 69 cm (27¼ in. in the air), 66 cm (26 in. in the water)
|propellant=
|fuel_capacity= 4 litres
|vehicle_range=
|ceiling=
|altitude=
|boost=
|speed=42 knots (77.8 km/h, 48.3 mile/h)
|guidance=
|steering=gyrocompass guided vertical rudder control system, gyroscope guided anti-rolling controller system
|accuracy=
|launch_platform=single-engine carrier-based attack aircraft, twin-engine land-based attack aircraft
|transport=
}}

The '''Type 91''' was an [[aerial torpedo]] of the [[Imperial Japanese Navy]] which was designed to be launched from an aircraft. It was used in [[naval battles]] in [[World War II]].

The Type 91 aerial torpedo had two unique characteristics:
* Wooden attachments on the tail fins, that acted as aerodynamic stabilizers, which were shed upon water entry.
* An angular acceleration control system ([[PID controller]]) to control rolling movements, which was very advanced for its time.
This system made it possible to release the Type 91 not only at a cruising speed of 180 knots (or 333 km/h, 207 mile/h) at an altitude of 20 m (66 ft), but also in a power-glide torpedo-bombing run at the [[Nakajima B5N|Nakajima B5N2]]'s maximum speed of 204 knots (or 378 km/h, 234 mile/h).

The Type 91 [[torpedo]] was 450 mm (17¾ in) diameter. There were five models put into service, with high explosive warheads weighing 213.5 kg to 526.0 kg (or 470.7 lb to 1160 lb) and having effective ranges of 1,500 m to 2,000 m (or 1,640 yd to 2,187 yd) at 42 knots (or 77.8 km/h).

Since the Type 91 torpedo was the only practical aerial torpedo of the [[Imperial Japanese Navy]], it was simply known as the ''Koku Gyorai'', or '''aerial torpedo'''. Surface warships and submarines used other types of torpedo, namely the [[Type 93 torpedo|Type 93]] and [[Type 95 torpedo|Type 95]] respectively, while the [[Type 97 torpedo]] was designed for use by [[midget submarine]]s.

==Thunder fish==
The '''Thunder fish''', type 91 torpedo (modification 2), was a shallow water [[aerial torpedo]] that was designed for and used in the [[attack on Pearl Harbor]] in 1941. Conventional torpedoes when launched from aircraft would dive to 100 ft before turning up to hit their targets. The waters of the lagoon at Pearl Harbor were much shallower so wooden fins  were added to allow for shallow launching at low altitude.<ref name="hellions">{{cite book|last=Gannon|first=Robert|title=Hellions of the Deep|publisher=Pennsylvania State University Press|date=30 April 1996|pages=49–50|isbn=978-0-271-01508-8|url=http://books.google.co.uk/books?id=5pJxc8Je2vsC&pg=RA1-PA49&lpg=RA1-PA49&dq=thunder+fish+torpedo&source=bl&ots=nS3-rDjgWR&sig=Gk9uvlCemjwcDsTfWU6Kl2h59p0&hl=en&ei=Fbl5Sv7hFOLLjAefuuSnBg&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false|accessdate=2009-08-05}}</ref>

The name '''Thunder fish''' is a literal translation of the Japanese ''gyorai'', meaning "torpedo" (''gyo'' = fish, ''rai'' = thunder).

The tactic was practised at a bay on [[Kyūshū]] island which resembled Pearl Harbor before being used in the attack in December 1941, launched from "Kate" [[B5N]] torpedo bombers.<ref name="hellions"/> The plan for attack was designed by naval air strategist [[Minoru Genda]] for [[Admiral]] [[Isoroku Yamamoto]] in the face of opposition from the majority Japanese navy view that victory could be achieved by the "The Great All-Out Battle" tactic of luring [[United States of America|United States]] [[battleships]] into a major action at sea.<ref>
{{cite web
|url=http://www.ess.uwe.ac.uk/genocide/reviewsw28.htm
|title=Review of Donald M. Goldstein and Katherine V. Dillon, eds. The Pearl Harbor Papers: Inside the Japanese Plans
|publisher=National Endowment for the Humanities, archived at [[UWE]]
|accessdate=2009-08-05
|last=Kolb
|first=Charles C.
}}
</ref>

The torpedo measured {{convert|18|ft|0|in|m}} in length, with a diameter of {{convert|1|ft|5.7|in|m}}, and weighed {{convert|1841|lb|kg}}, with an explosive charge of {{convert|452|lb|kg}}. It had a range of {{convert|2200|yd|m|0}} and a speed of 42 [[Knot (unit)|knot]]s. A slight variant was used to sink [[HMS Prince of Wales (53)|HMS Prince of Wales]] and [[HMS Repulse (1916)|HMS Repulse]], launched from "Betty" [[G4M]] bombers in [[sinking of Prince of Wales and Repulse|an action]] in the [[South China Sea]] three days after Pearl Harbor on December 10, 1941.<ref>
{{cite web
|url=http://www.navweaps.com/Weapons/WTJAP_WWII.htm
|title=Japan Torpedoes of World War II
|publisher=NavWeaps.com
|accessdate=2009-08-05
}}
</ref>

== Specifications ==
Here is the list of the series of Type 91 aerial torpedoes, production models.<ref>p.24, Ichikawa, Hidehiko; Table 1-2 List of Aerial Torpedo ''Koku Gyorai Note''</ref>

{| class="wikitable"
|+Type 91 Aerial Torpedo, rev.2 Specifications
|-
! Item
! -
|-
| High explosive
| 204 kg (449.7 lb)
|-
| Speed
| 42 knots (48.33 mile/h)
|-
| Range
| 2,000 meter (2,187 yard)
|-
| Diameter
| 45 cm (17¾ inch)
|-
| Weight
| 838 kg (1,847 lb)
|-
| Length
| 5.427 meter (17.81 ft)
|-
| Engine
| 200 hp, wet-heater type, 8-cylinder radial engine
|}

=== Variations ===
{| class="wikitable"
|+Type 91 Aerial Torpedo and Type 91 Warhead, operational models

|-
! Main body
! Warhead
! High explosive (kg)
! Speed (knots)
! Range (m)
! Total Length (m)
! Diameter (m)
! Total Weight (kg)
! Head Length (m)
! Head Weight (kg)
! Comments

|-
| Type91
| Type91
| 149.5
| 42
| 2,000
| 5.270
| 0.45
| 784
| 0.958
| 213.5
| -

|-
| Rev.1
| Rev.1
| 149.5
| 42
| 2,000
| 5.270
| 0.45
| 784
| 0.958
| 213.5
| Supported shedding wooden tail plates in 1936

|-
| Rev.2
| Rev.2
| 204.0
| 42
| 2,000
| 5.470
| 0.45
| 838
| 1.158
| 276.5
| Body reinforced in 1938, anti-rolling controller in 1941

|-
| Rev.3
| Rev.3
| 235.0
| 42
| 2,000
| 5.270
| 0.45
| 848
| 1.460
| 323.6
| -

|-
| Rev.3
| Rev.3_rev.
| 235.0
| 42
| 2,000
| 5.270
| 0.45
| 848
| 1.460
| 323.6
| Reinforced warhead

|-
| Rev.5
| Rev.3_rev.
| 235.0
| 41
| 1,500
| 5.270
| 0.45
| 848
| 1.460
| 323.6
| Precision forging and stainless cast steel in body

|-
| Rev.5
| Rev.7
| 420.0
| 41
| 1,500
| 5.710
| 0.45
| 1080
| 1.900
| 526.0
| Warhead designed to break bilge
|}
The later, heavier models had a decreased range; this was not an operational problem as close launching was required for accuracy in any case. There were two versions in the Type 91 warhead rev.3, differing in design maximum launch speeds.

=== Other Japanese aerial torpedoes ===
The Type 92 electrical (battery powered) aerial torpedo never made it beyond trial stage.

The Type 94 aerial torpedo was based on the highly successful [[Type 93 torpedo]]. The Type 93, called the "[[Long Lance]]" by the Allied press, was a massive (2.8 tonnes fueled) weapon of superior performance, due largely to the use of compressed [[oxygen]] as a propellant instead of compressed air; pure oxygen has approximately five times the reactant capacity in regard to common fuels as the same mass of mixed gasses found in air. The Type 94 emerged from development somewhat smaller, similar to the [[Type 95 torpedo]] - a type also derived from the Type 93 and used successfully as a submarine weapon. It was nonetheless a heavy, unwieldy device and never deployed operationally.

[[Yokosuka]] air arsenal began development of a 2 tonne large aerial torpedo for 4-engine [[Nakajima G8N]] land-based [[attack aircraft]], in spring 1944. It was called '''''Shisei Gyorai M''''' (Trial model torpedo M), or simply '''''2 tonne torpedo'''''. This was an enlarged version of Type 91 aerial torpedo, its diameter was 533mm (or 21 in, the diameter of standard Imperial Navy submarine torpedo tubes), length 7.10 m (about 23 ft 4in), total weight 2,070 kg (4,564 lb), with a huge 750 kg (1,653 lb - about 50% larger than similar weapons of the era) warhead.<ref>p.383, Minoru, Akimoto; "Nihon Gunyoki Kokusen Zenshi, volume 4", Green Arrow sha, June, 1995</ref>

But the Type 91 aerial torpedo project members did not regard it as a series of Type 91 aerial torpedo. It would have been the largest aerial torpedo in the Imperial Japanese Navy Air Force, but the since the operating concept became outdated, that torpedo remained uncompleted work. This [[Nakajima G8N|G8N]] or the 18th trial model 4-engine bomber was called "Type 18 prototype land-based attack aircraft".

== Tactics ==
The original Type 91 aerial torpedo entered service in 1931, corresponding to the year 2,591 of the Imperial Japanese calendar, leading to its model designation. This was the beginning of a protracted developing process toward a ''true'' aerial torpedo culminating in 1941.

=== Classic tactic ===
The first revision of the Type 91 aerial torpedo needed to be launched carefully, with the airspeed not exceeding 130 knots (or 240 km/h, 150 mile/h) and at an altitude no higher than 30 m (98 ft), with slower airspeeds resulting in better precision. This 'classic' approach was actually easier to carry out in obsolete biplanes or planes with fixed landing gears, whereas the modern, fast Nakajima B5N as used by the IJN carrier strike force's 1st Air Flotilla was not an easy plane to handle at these slow airspeeds.

The development project team at [[Yokosuka Naval Arsenal]] of ''Ko-Hon'' or [[Imperial Japanese Navy Air Service]] reasoned that the maximum range of any aerial torpedo could be less than 2,000m (or 2,187 yd, 1.8 nautical mile). When aircraft launches a torpedo in running speed 40 knots, the targeted ship steaming in 30 knots would surely turn hard to maneuver around. It is mandatory for torpedo-bomber pilot ''attacking'', or running into the target as close as possible.

=== The second tactic ===
Another approach, called ''the second tactic'', was developed for torpedo-bombing runs in shallow water ports. Here, the aircraft was supposed to fly in at an even slower 100 knots (or 185 km/h, 115 mile/h) and at an altitude of 10 m (32.8 ft) in the midst of intense AA fire. The only way to do this in a B5N2 was to lower the landing gear and flaps to increase drag and lift. The aviators of torpedo-bombing units trained this in the shallow waters of [[Kagoshima Bay]] by late August 1941 but felt uncertain about their chances of success.

=== The first tactic ===
The [[Torpedo bomber|torpedo-bomber]] unit of the [[Japanese aircraft carrier Akagi|carrier Akagi]] was the first to receive ten samples of the new rev.2 torpedo in August 1941. It was a marked improvement, allowing an approach in excess of 160 knots (or approx. 300 km/h, 185 mile/h) and at 20 m (66 ft) altitude.

Immediately all the torpedo units changed to ''the first tactic'', the gears and flaps are retracted in the wings and flew in faster speed at higher altitude .

* A torpedo released at a range of 800 meters (875 yd) from the target at a speed of 300 km/h, at height of 60 meters (196 ft), would dive into the water entry point 290 meters ahead in 324 km/h, at an entry angle of 22 degrees, after 3.5 seconds. That torpedo runs under the water for 500 meter, and hits the target after 21 second.

* A torpedo released at a range of 620 meters (678 yd) from the target at a speed of 300 km/h (185 mile/h), at height of 10 meters (33 ft), would dive into the water entry point 120 meter (130 yd) ahead in 304 km/h (190 mile/h), in entry angle of 9.5 degrees, after 1.4 seconds. The torpedo will then run under the water for 500 meters (547 yd), and hitting the target after 21 seconds.

In the morning of the [[Battle of the Coral Sea|Coral Sea]], on May 8, 1942, B5N torpedo-bomber units of ''The 5th Air Flotilla'' penetrated the American defenses at 0910[JST],<ref>Carrier Zuikaku Action Report</ref> swooped to [[USS Lexington (CV-2)]] and [[USS Yorktown (CV-5)]]. USS Yorktown CV-5 was attacked by four B5N2s in one unit of ''[[Japanese aircraft carrier Zuikaku|Zuikaku]] (Lucky Crane)'', led by squadron leader Lt Cmdr Shimazaki, and averted all four torpedoes. Large USS Lexington CV-2 was attacked by 3 units of total fourteen B5N2s, and was struck by the last two torpedoes to port. The Lt Sato unit of ''Zuikaku'' attacked first, followed by the Lt Iwamura unit, and the last was the Lt Ichihara unit of ''[[Japanese aircraft carrier Shōkaku|Shokaku]] (Flying Crane)'', in the crane wings attack formation. Those B5Ns were approaching to the ship in the full speed over 204 knots (or 378 km/h, 235 mile/h) faster than known torpedo-bomber aircraft should. Captain Frederick Carl Sherman in the bridge of CV-2 watched a B5N shot down with the torpedo near the ship. He saw that the tail section of the torpedo was covered with box-like attachment. He reported that he had found the reason why the B5N2s could launch Type 91 aerial torpedo at such high speed.

=== Fast torpedo-bombing tactic ===
As for a high speed torpedo bombing run in 300 knots (or 556 km/h, 345 mile/h), the maximum altitude for release was limited at 300 – 350 m (984 - 1,148 ft). The strength of contra-rotating propeller limited launch height . A torpedo, tested at 100 m in very high speed from [[Yokosuka P1Y|P1Y land-based attacker]] ''Ginga'' at Yokosuka arsenal, veered upon water entry because of a crack in a screw blade. The minimum release height was also limited to 40 m (131 ft) in a high speed run. If it were released lower than 30 m (98 ft) in a high speed run, the torpedo might skip on the water's surface.

With authorization from the Imperial Japanese Navy Air Force, Army Air Force pilot, Major Hideo Sakamoto established the ''fast torpedo-bombing tactic'', with his [[Mitsubishi Ki-67|Ki-67]] twin-engine bomber aircraft having good maneuverability, at Yokosuka Naval airbase in January 1944. He found the releasing parameters of the tactic after 300 tests. A Ki-67 with 1 tons torpedo starts steep dive at an altitude of 1,500m (approx. 5,000 ft) above water level and launches the torpedo in two styles.<ref>pp.196-222, Kyuno Joe Ozawa, "Mitsubishi Type 4 Army Bomber Aircraft", ''Document of Historical Aircraft with Japan Making'' SPECIAL THANKS 600 ISSUE OF AIRREVIEW, last volume, Kanto-sha, 1994.</ref>
# Launch in speed range 370–460 km/h, at 30-120m high (or speed 200-248knots (230-285mile/h) at altitude 98–394 ft)
# Launch in speed range 460–560 km/h at 50-120m high (or speed 248-302knots (285-348mile/h) at altitude 164–394 ft)

=== Skidding right and left ===
Tactics were needed because the ''killed in action'' rate of aviators in torpedo bombing squadrons was high, 30 - 50% in the beginning of World War II. In the latter stages of the Pacific War, the rate was up to 90% and 100% during daytime operations.

Skilled aviators had their own tactics to survive, [[Slip (aerodynamic)|skidding]] right and left with varying speed (180 knots to 70 knots) at less than 10 meters high in the midst of water splashes of AA gunfire, to avoid a curtain of intense AA barrage controlled by the [[Fire-control system]] typical of US Navy warships.<ref>p.105 - p.116, Seko, Tsutomu; "Totsugeki Junbi Taikei Tsukure (or Attack Formation!)", ''Raigeki no Tsubasa (Wings of Torpedo attack)'', Kojin-sha</ref>

A [http://worldwar2database.com/gallery3/index.php/wwii1247 photo] shows the typical tactic of torpedo bombing, a [[B6N]] ''Tenzan'' in torpedo bombing attack of [[USS Yorktown (CV-10)]]. The original series of photos shows the sequential right and left side-skidding tactic of the crew. In this second photo, the B6N is skidding to its left, or toward the right in the photo, and the flak shell explodes to its right, or to the left in the picture. It was coming up to counterattack near the Truk islands, in the evening of February 17, 1944. It was one of four B6N ''Tenzan'' of a torpedo squadron in 2nd A.G or 582 A.G. Two out of four aircraft ditched on their way back and the crews were rescued. The remaining two aircraft returned safely to base. Original photo is PD, the Property of US Government.

== Type 91 history ==
: '''Chronological Table'''
: 1931 Type 91 aerial torpedo is into service, start production.
: 1936 Revision 1. Self-detachable wooden plates are introduced.
: 1937 Launch-tests at 500m and 1,000m with wooden damper.
: 1939 Revision 2 starts production. The sinking level after water entry becomes big problem.
: 1941 Revision 2 cleared the test of shallow water launching by introduction of anti-rolling controller. [[Attack on Pearl Harbor|Battle of Pearl Harbor]], [[Sinking of the Prince of Wales and the Repulse|Battle of offshore Malay]].
: 1941 Revision 3 starts production.
: 1942 [[Indian Ocean raid|Battle of Indian Ocean]], [[Battle of the Coral Sea]], [[Battle of Midway]], [[Battle of the Santa Cruz Islands]].
: 1943 Revision 5 starts production.
: 1944 [[Battle of the Philippine Sea|Battle of offshore the Marianas Islands]], [[Aerial Battle of Taiwan-Okinawa|Aerial Battle of offshore Formosa]].

=== Scientists and engineers developing aerial torpedo ===
''Ninety One Association'' includes Rear Admiral Naruse, Lt Cmdr Haruo Hirota, Lt Cmdr Makoto Kodaira (Matsunawa), Naval assistant manager Iyeta, Naval engineer Noma, Naval engineer Moritoshi Maeda, Lieutenant Hidehiko Ichikawa, and Teruyuki Kawada, university student as naval apprentice.

Captain Fumio Aiko was assigned in charge of promoting the development project of Type 91 aerial torpedo since 1931. Capt F. Aiko concentrated [[human resources]] to make an aerial torpedo, ordered to analyze the cause and to make anti-rolling controller. He is very proud of the Type 91 aerial torpedo as his great achievement.

===Delay of the development===
In the beginning of 1934, ''Kan-Pon'' or [[Imperial Japanese Navy Technical Department]], an operating division of the Ministry of the Navy of Imperial Japanese government, which had the primary responsibility for naval weapon systems, had their own plan and their own project for a Japanese aerial torpedo. In their plan, a big flying boat was to carry heavy Type 93 ''oxygen torpedoes'', to launch at long range, and then turn back towards safety. This soon proved to be an unrealistic desk plan. ''Kan-Pon'' confidentially developed their own Type 94 aerial torpedo, an aerial version of the Type 93 ''oxygen torpedo''. Their original challenge of the latest type of giant [[Kawanishi H6K|H6K Type 97 Large Flying Boat, Mavis]] was just accomplished the test flight successfully in 1934.

They even ordered to stop production of Type 91 torpedo, which significantly delayed the development schedule based on Type 91. This frustrated the project members of [[Yokosuka Naval Arsenal]].

=== Wooden tail stabilizers ===
The project revised Type 91 aerial torpedoes, as the '''revision 1''' supporting wooden tail plates, taken off on water entry, in 1936. The team demonstrated the launching tests of Type 91 aerial torpedoes wearing wooden ''shock-damper'' objects at altitude 500m (1,640 ft) and 1,000m (3,281 ft) in the following year, 1937. The project came up again and resumed the development of Type 91 aerial torpedo.
Type 91 aerial torpedo of earlier revision had a frail body.

It was revised as the '''revision 2''' by reinforced the frail body structure, in 1938.

=== Anti-rolling controller ===
Type 91 aerial torpedoes won the admiration by the anti-[[Yaw, pitch, and roll|rolling]] controller with [[acceleration]] control system of early days. Type 91 already had a shed-off type wooden tail plates as aerodynamic stabilizer. But the running out problem merged as aircraft speed going up from 130 to 180 knots.

Before the anti-rolling controller was introduced, the early revision of Type 91 aerial torpedo had a serious problem as other aerial torpedoes had in those days. Roughly released in high speed, it even made double-roll in the air. When it dove into the swelling waves of the heavy sea, it got spinning moment at the hard impact on water entry. It was veering the running direction, going down to stick to the bottom of the shallow basin of a port, crashing at the depth limits 100m by the water pressure, jumping out of the water, skipping water surface, or even running backwards. Only the real razor aviators could make the sure torpedo-bombing run in the calm sea.

A tumbled torpedo runs out of control. Though the gyrocompass and the depth meter works well, the torpedo cannot control the running direction by tail rudders unless they are in the neutral position at first. Once the torpedo rolls, the horizontal and vertical rudders lose their positions, or even upside down, result in runaway.

Engineers and scientists of the project, led by Lieutenant Hirota, drew one conclusion from their years of tests and numerical analyses, in 1939. Since the specification of launch speed of aircraft was increased from 130knots to 180knots and faster, any aerial torpedo needed an anti-rolling system with not only damping stabilizer function but also acceleration controlling function, otherwise the torpedo would be falling into unstable-state. The idea of acceleration-control, or ''countersteering'' function was thought of as impossible in those days. Two years passed.

The breakthrough on aerial torpedo design was made with the anti-rolling controller invented first by Iyeda, Assistant Manager of arsenal workmen, in spring 1941.

Ten days later, while the test of Iyeda system was in practice, Naval Engineer Noma invented another system, and it was put into final test in August 1941. It functions all the same with different mechanism. During the prototype tests, the Noma's system found out the better for having less time lags in response, so that the Noma system was adopted for production type of Type 91 aerial torpedo.

It looked merely a tiny mechanical air valve object controlling small roll rudders on both sides in the aft of the torpedo, was really the innovation of the torpedo technology world. It was the breakthrough for aerial torpedo. Type 91 rev.2 aerial torpedo first made it possible to use in the high seas.

The anti-rolling controller is actually a steering controller to stabilize the rolling motion of the torpedo by roll rudders on both sides. The roll rudders work within angler range +/-22.5 degrees, twisting in ailerons manner.
When a torpedo is rolling or rolls to some degree, the anti-rolling controller twists those rudders in the counter-rolling direction. When a torpedo is rolling back to its neutral position of 0 degree, the controller sense it and switches roll rudders in the opposite direction to break angular velocity of torpedo rolling back, or ''[[countersteering]]''. They naval engineers called this operation as ''countersteer'' as they modeled it to steer a ship.

It enabled to keep Type 91 rev.2 aerial torpedo running under the water no deeper than 20 meter (65.6 ft). Actually the cutting-edge pilots of torpedo-bomber squadrons in ''Dai Ichi Koku Sentai'' or ''The 1st Air Flotilla'' of the carrier strike force were able to launch their torpedo so as to sink in the water depth no more than 10 meter (32.8 ft) after water entry.
Anti-rolling controller made aircraft possible to torpedo-bomb not only warships anchorage in shallow military port but also warships steam in chopped waves of heavy sea in full speed.

=== Increment of the explosive weight ===
The anti-rolling controller also made Type 91 aerial torpedo possible carrying heavier warhead section. Type 91 warhead and Type 91 rev.1 warhead, each weighs 213.5 kg (470.7 lb) with high explosive 149.5 kg (329.6 lb) only, but warhead rev.2 weighs 276 kg (595.2 lb) with high explosive 204 kg (449.7 lb). Warhead rev.7, which is for twin-engine bomber, weighs 526 kg (1160 lb) with high explosive 420 kg (925.9 lb). It was so designed to pierce the reinforced tough armor plates the latest USS warships developed during World War II.

=== Aerial torpedo technology of the Allies ===
Engineers and scientists of the Imperial Japanese Navy got a chance to inspect the torpedo technology of [[Allies of World War II|the Allies]], in aerial torpedoes captured at South West Pacific bases in early 1942. The [[United States Navy|US Navy]] aerial torpedo, [[Mark 13 torpedo]] was found at ''Sangley Point'', the military port in the [[Philippines]]. [[Fleet Air Arm]] aerial torpedoes of the [[Royal Navy]] were found at the base ''[[Kota Bharu]]'', northeastern [[Malaysia]], close to the [[Thailand|Thai]] border.

There were none like Type 91 aerial torpedo rev.2. They were disappointed with the work of their rivals, because those were as if they had few intentions to deal seriously with the development of aerial torpedo technology. The US Navy's aerial torpedo had little differences aside from being capable of being loaded in aircraft and looks almost the same as the bigger Mark 13 ship torpedo. The Royal Navy had their traditional type aerial torpedo, which had been originally designed by White Head Company in 1925.

== The structure of Type 91 aerial torpedo rev.2 ==
Type 91 aerial torpedo is the first aerial torpedo, which is able to use practically in ocean. The scientific approach consistent with experimental evidence was adopted to lead development.
[[Image:Type91 AerialTorpedo Rev3 StructuralDrawing.svg|360px|right|Type91_aerial torpedo rev3, structural drawing]]

=== Warhead ===
Length = 1,460 mm (57⅝in)

When a torpedo hits a ship, the inertia forces the aft mass in the initiator to thrust forward and ignite the high explosive in the initiator. The high explosive in the warhead will not detonate unless initiated as designed. A 20 mm explosive cannon shell can go through the warhead without igniting the stable high explosive in the warhead.

The warhead has '''T''' shaped strip parts to reinforce the internal-lower portion of the front shell against the heavy impact on water entry. For the production model, the warhead section needs five reinforced strip bands on the front-bottom of the inner shell, lap welded in a shape of cut lower half star, or the superpose of the letter '''T''' and the letter '''Λ''', instead. The warhead also has two tiny stitch lines aligned on the front-top of the shell to enhance the explosion. The latest version had two hooks on the nose.

Type 91 aerial torpedo is to be launched by aircraft power gliding from high in the sky. The aerial torpedo, released at altitude 100m, is falling in speed nearly [[Mach number|Mach]] 0.5 on water entries, and receives over 100[[gravitational acceleration|G]] at the hard impact on the water surface.

=== Air chamber ===
L = 1,068 mm (42⅛in)

The air chamber is a cylinder of thin shell made by alloy of nickel chromium-molybdenum steel. This tough steel alloy was originally developed for steel armor plate of battleship. The chamber is charged with highly compressed normal air at 175 - 215 [[atmosphere (unit)|atm]] (2,500 - 3,000 [[pounds per square inch|psi]]), which burns fuel oil to produce driving power. It loses the pressure down to around 50 atm (710 psi) while running 2,000 m (6,600 ft) under the water.

=== Front float ===
L = 733 mm (28⅞in)

Front float section has a pure water tank, a fuel oil tank and a depth meter.

The depth meter is placed at the inner bottom of the section to detect the water depth level. It detects the displacement level of the water depth and controls the tail horizontal rudders (or [[Elevator (aircraft)|elevators]]) proportionally, so that the torpedo keeps level running under the water.

=== Engine room ===
L = 427 mm (16⅞in)

This section is constructed free to coming in the water to help a cooling system of the engine in the torpedo.
It has a starter, a ''Chowaki'' or [[pressure regulator]], a wet-heat chamber, a main engine, and a horizontal rudder controller.

The ''starter'' starts controllers, one for vertical tail rudders, and another for roll rudders for anti-rolling in both side wing rudders, with horizontal tail rudders being locked at uppermost position, while the torpedo falls down to the water surface. It starts the main engine to propel when the torpedo hits the water. A thick bolt is stuck through the starter during loaded in aircraft as the lock. The bolt is pulled out from the torpedo when it is released. The bolt remains underneath the fuselage of aircraft.

The ''pressure regulator'' is called as ''Chowaki'' or ''harmonizing system'' for the engine, it actually is a two-stage pressure regulator with twin pressure-tunable regulation valves. It steps down the pressure of compressed air at 215 - 50 atm (3,000 - 711 psi) in the air chamber to the constant high-pressure air at 10 atm (142 psi). While the air pressure is declining as the torpedo is running under the water, the pressure regulator feeds the constant high-pressure air to the engine intake aspirator and keeps the constant running speed in 43 knots (or 80 km/h, 50 mile/h).

The ''wet-heat chamber'' is made by heat resistant steel. Type 91 aerial torpedoes use ''[[Torpedo#Wet-heater|wet-heater engine]]'' like almost all other torpedoes in World War II. The general wet heater burning method drastically improved the combustion efficiency of torpedo engines. It burns the mixed gas of fuel oil and the high-pressure air with spraying pure water in the wet-heat block to produce burning steam gas fed to the engine. The high-pressure fuel oil gas is burning at a temperature 800 degrees C (1,500 degF). The sprayed pure water mists into the combustion gas, which produces vapor explosion, results in completely gasified fuel oil combustion.

The ''main engine'' is an 8-cylinder single-row radial [[reciprocating engine|piston engine]].

The ''horizontal tail rudder controller'' is operated by the rod connection mechanism from the depth meter in the front float section.

=== Rear float ===
L = 1,002 mm (39½in)

A single [[drive shaft]] is going through the section to the tail. This rear float section has a machine oil tank, a rudder controller, an anti-rolling controller, and roll rudders on both sides.

The ''machine oil tank'' is center-mounted in the rear float section.

The ''rudder controller'' is a general [[gyrocompass]] controlled system, which is steering vertical rudders to keep the longitudinal axis of the torpedo in the sensed direction straight. Both the vertical rudder controller and the anti-rolling controller had their own [[gyroscope]], which is to start rotating when the torpedo is released from aircraft. Each gyro has dual ring support mechanisms to move freely.

==== Anti-rolling controller ====
[[Image:USS Lexington under attack at Coral Sea.jpg|thumb|right|Carrier Zuikaku B5N2s in torpedo attacking at Coral Sea on May 8, 1942. Water splashes in front center and left are made by water entries of Type 91 aerial torpedoes.]]

[[Image:Guadalcanal-Tulagi Operation, Mitsubishi G4M.jpg|thumb|right|G4M1s in torpedo attacking at Guadalcanal on August 8, 1942.]]

[[Image:BB-57 Kates Santa Cruz NAN4-2-43.jpg|thumb|right|B5N2s in torpedo attacking at Santa Cruz on October 26, 1942]]

[[Image:B6N2 in formation.jpg|thumb|right|B6N2s in formation flight with torpedoes wearing box type tail stabilizers]]

[[Image:B7A-Ryusei torpedo.jpg|thumb|right|Aichi B7A Ryusei carrying torpedo with cross type tail stabilizer plates, 1945]]

The ''stabilizer'' or anti-rolling controller is a mechanical control system, whose control design needs [[numerical analysis]] of physical mathematics theory to fit stability. A spinning gyroscope of control system senses the tilt degree of roll, and then the controller is centering the roll of the torpedo.

The anti-rolling controller with the gyroscope can steer roll-rudders on both sides in angle range of '''''+/- 22.5''''' ° . When the torpedo tilted, the anti-rolling controller steers roll-rudders (or [[aileron]]s) in twisting ailerons manner to produce counter-rolling moment.

When the torpedo tilted over 10 degree and is rolling back the angle toward the neutral center position, the tiny mechanical system in the control air valve works. When it rolls back within the tilt angle 10 degree, the controller now countersteers roll rudders switching back to the reverse angles to break the counter-rolling moment, and to prevent overshoot. The torpedo rolls over the neutral center position and keeps rolling to some degree. The torpedo then stops in opposite tilt at certain degree, and starts rolling back the angle toward the neutral center position. Then the controller senses the angle tilted and countersteers roll-rudders to break the moment. It is rolling over the neutral center position then stops at certain degree, vice versa, like an air cushion is bouncing and settled to the floor. The movement continues but damps to neutral the rolling angle within 2.0 to 3.6 second in the air.

In the test, the system working was observed and proved by the developed high-speed movie film, the top-view shot of the falling torpedo tested, taken from the bomb bay.<ref>Fig.4-5, p.68, Ichikawa, Hidehiko; Chap.4-5 The mechanism of roll controller, ''Koku Gyorai Note''</ref> The anti-rolling system also proved the controller functions effectively under the water surface by the running result after the water entry.

==== Roll rudders ====
''Stabilizing rudder'', or roll rudders (or [[aileron]]s) are put on both sides of the torpedo, being steered to produce counter-rolling moment. Each rudder is a small 8 cm2 size square metal wing.
Each roll rudder has been covered with a temporal wooden extended wing of 12 × 20 cm2 (4¾in × 7⅞in), tighten with six aluminum sharing pins on both side-edges (three pins each) to get enough aerodynamic force in the air, that is to be shed and broken off when the torpedo gets the hard impact from the water surface on water entry. The remained original metal roll rudders are steering in the water running to converge the rolling movement arisen on water entry.

=== Tail section and twin screws ===
L = 530 mm (to the tip end of propelling screw hub) (20⅞in)

There are [[bevel gears]] driving coaxial contra-rotating double 4 blades screws to propel torpedo running straight under the water. Tail section has vertical and horizontal stabilizer fins in cross. Each fin has a controlling rudder in aft. Horizontal fins and rudders or elevators have wide span in longitudinal direction and work proportionally, while vertical fins are small, and rudders have very short span.

==== Shed off aerodynamic wooden stabilizer plates ====
[[Image:Type91 AerialTorpedo TailSection and AeroDynamicTailStabilizer.PNG|thumb|right|Type91 aerial torpedo, tail section and aerodynamic tail stabilizer plates]]
Tail fins are covered with ''Kyoban'' or aerodynamic wooden stabilizer plates. They were introduced in 1936. It is shed off by the impact on water entry. This aerodynamic wooden stabilizer plates in the tail are in the shape of box cover for single-engine carrier-based torpedo-bombers [[Nakajima B5N]] and [[Nakajima B6N]]. In the case for twin-engine land-based torpedo-bombers [[Mitsubishi G3M|G3M]], [[Mitsubishi G4M|G4M]], [[Yokosuka P1Y|P1Y]], and [[Mitsubishi Ki-67|Ki-67]], the torpedo wears another type of aerodynamic wooden stabilizer plates covering tail fins in cross to extend their tail length in the air, which is less in drag resistance loss but needs more clearance heights in the bomb bay underneath the fuselage. In the case of land-based torpedo-bomber aircraft, a plate is needed to be set inside the bomb bay to groom the airflow, because the vortex coming in the bomb bay gives the turbulence behavior to the torpedo released.

A torpedo is released at a speed greater than 160 knots (or 300 km/h, 184 mph) in the air and then follows a parabolic path free-falling to the water. The aerodynamically designed wooden plates stabilize the up-and-down motion of the torpedo in the air keeping it aligned to the diving course. The wooden plates are shed as the torpedo hits the water, and the elevators or horizontal rudders set farthest forward lift the nose of the torpedo after water entry to start level running. The structure is simple and works well. The wooden head cap had been used before the anti-rolling controller system was introduced.

==== Screws ====
[[Propeller|Screws]] are coaxial contra-rotating double screw, with 4 propeller blades each.

Each screw is wrought from a cube steel alloy mass into bold cross shape and punched through the center. Hammering punches of 1 tons and 3 tons shape 4 blades.

Propel section is compactly designed so that the front screw and the rear screw are put in 5 mm close to each other. A trouble happened in 1943, when a P1Y tested the torpedo released at altitude 100 m in high-speed power gliding. That torpedo veered the running direction. The hard impact on water entry made a front blade get cracked by the hit of a rear blade. The project members shared the recognition on the importance of annealing, quench hardening, and normalizing process of the screw blades, and so was done.

In the first [[Annealing (metallurgy)|annealing]] process, the metal is left in the oven at 700 °C (or 1,300 °F) for 2 hours then slowly cooled in lime powder. The metal is hammered out and machined in the shape of screw. Then the screw is put in the [[Quench|quench hardening]] process, kept at 850 °C (or 1,560 °F) for 1.5 hours, and is cooled in oil. In the last thermal normalizing process, the screw is put in 180 °C (or 356 °F) hot oil for 2 hours, then is left cooling in the air.

: Material: SK chromium-molybdenum alloy steel <ref>The alloy is used as the material for [[Turbine|turbine blade]] in 21st century.</ref>
: Process: Hammer out.

=== Rudder steering means ===
: 1. Full steering system
:: Vertical rudder system steers rudders tri-stated to full-right / neutral / full-left as the gyroscope senses. Type 91 aerial torpedo has long time constant period with respect to the longitudinal axis turn moment in the water.

: 2. Proportional steering system
::Horizontal rudder system elevates the angle of rudders proportional to the angle of displacement the depth meter detect. Type 91 aerial torpedo has medium time constant period in longitudinal axis lift moment under the water.

: 3. Angular velocity steering system
:: Anti-rolling controller system steers both roll rudders tri-stated to full-up / neutral / full-down in twisting aileron manner. When the controller detect the rolling is coming back to the center position, the system countersteers roll rudders in opposite direction. This system uses countersteering function so as to dump unstable rolling oscillation movement. Type 91 aerial torpedoes has fast period in approx. 0.5 s time constant in rolling moment.

== Structure of Anti-Rolling Controller ==
: Haruo Hirota, Naval Lieutenant Commander
: Makoto Kodaira, Naval Lieutenant <ref>''Kodaira'' is his adopted family name, his birth name is ''Matsunawa''</ref>
[[Image:Type91 AerialTorpedo RollRudder.svg|thumb|right|Type91 aerial torpedo, roll rudder]]

[[Image:Type91 AerialTorpedo MainRollController.svg|thumb|right|Type91 aerial torpedo, main roll controller]]

[[Image:Type91 AerialTorpedo RollControllerMovement.GIF|thumb|right|Type91 aerial torpedo, roll controller movement]]

The structure of the anti-rolling controller is a set of gyro-controlled air valve system to steer the roll rudders on both sides of a torpedo.

The anti-rolling controller is composed with a gyroscope, a main controller, and an output booster. The most significant part is the main controller.

=== Gyroscope ===
The [[gyroscope]] senses the rolling tilt degree in real-time. It inputs push-pull control operation force to a pilot valve, sliding it inside the main controller to switch two output valves exclusively to the rudders.

=== Main Controller ===
The main controller controls two output air valves exclusively to steer roll rudders.
It steers and countersteers those roll rudders. It steers the rolling-rudders with detecting the tilted degree of the torpedo rolling by the control of pilot valve. It countersteers those roll rudders when the torpedo is rolling back to neutral position, which results in ''detecting'' the acceleration of the rolling angular velocity derivative with respect to time.


=== Output Booster ===
The output booster or auxiliary valve has two inlets and two outlets ports. The output booster is working as a pair of air shutoff valves. It is connected in cascade to two output ports of the main anti-rolling controller, switches on and off directly the two powerful high pressure controlling air flows one for clockwise twist and the other for counter clockwise twist of roll rudders, exclusively to each other. It is so as to save the main controller system and to ensure proper operation under the heavy impact condition.

== Sequential operation steps in aerial torpedo bombing ==
# A release button is switched on in the cockpit.
# Cartridge is ignited to cut a loading wire band. Torpedo is released and wire is falling freely.
# The torpedo is falling and the safety bolt is pulled out. It starts gyroscopes of both the vertical rudder controller and the anti-rolling controller.
#: Vertical rudders are kept in straight direction.
#: Horizontal rudders (or elevators) are locked in uppermost position to prepare the water entry.
#: Roll rudders start steering by the anti-rolling controller.
#: ---Water Entry---
# Hard impact to the water surface breaks off wooden air wing covers of side roll rudders and tail aerodynamic stabilizer plates or box.
# Doubly rotate screws are unlocked at propel block.
# Propel engine starts ''cool idling'' while running. (Engine starts rotating with high-pressure air only.)
# Brakes on horizontal rudders (or elevators) are released. Depth meter starts working.
#: Water pressure after the entry downs a plate to start firing combustion chamber of the engine.
# Propel wet-heater engine starts ''hot'' running by burning fuel air gas, mixed with sprayed water.
# Safety lock in the warhead is released while running.
# High explosive explodes on the hit of the target.

----

== Theory: Aerial torpedo motion equation ==
[[Image:Type91 AerialTorpedo Vectors of MotionEqs.svg|thumb|right|Vectors of motion equations for aerial torpedo in the air]]
Rear Admiral Shoji Naruse explained in his class as follows.

The torpedo motion equation is the set of simultaneous ordinary differential equations, which is to model pitch motion of airborne aerial torpedo as follows.
<ref>Equations in the private book ''Koku Gyorai Note'' has errors in typescript symbol letters in privately printed edition service.</ref>

: ''Eq.''1: Falling velocity of the torpedo equation

: ''Eq.''2: Horizontal vector velocity of the torpedo mass equation

: ''Eq.''3: Vertical vector velocity of the torpedo equation

: ''Eq.''4: Vertical vector acceleration of the torpedo mass equation

: ''Eq.''5: Angular velocity equation with respect to time

: ''Eq.''6: Angular velocity differential equation with respect to time


: <math>
\begin{array}{lcll}
dx/dt &= &V_{X} &\cdots(Eq.1)\\
W/g \times \left(dV_{X}/dt \right) &= &- D \cos \varphi - L \sin \varphi &\cdots(Eq.2)\\
dz/dt &= &V_{Z} &\ldots (Eq.3)\\
W/g \times \left(dV_{Z}/dt\right) &= &D \sin \varphi - L \cos \varphi + W &\cdots(Eq.4)\\
d\theta/dt &= &\omega &\cdots(Eq.5)\\
I \times \left(d\omega/dt\right) &= &57.3M - bV\omega &\cdots(Eq.6)
\end{array}
</math>

: where constant '''57.3''' of ''Eq.''6 is the coefficient from '''1''' (radian) = '''57.2958°'''

'''''b V ω''''' of ''Eq.''6 is the damping moment, where '''''b''''' is defined as;

: <math> b = - (\delta C_{mgH} / \delta \alpha_{H}) \times ( \rho S \mathit{l}/2) \times \mathit{l}_{H}\,</math>

Since the angular moment of a torpedo here is the lifting movement as follows;

: <math> \omega\mathit{l}_{H} /V = ( \delta C_{mgH} /\delta\alpha_{H}) \times ( \rho S \mathit{l} /2 ) \times \mathit{l}_{H} \times V \times \omega \,</math>

:: '''''V''''' : The velocity of the torpedo
:: '''''VH''''' : The horizontal axis velocity of the torpedo
:: '''''VZ''''' : The vertical axis velocity of the torpedo
:: '''''φ''''' : The moving vector angle of the torpedo in reference to horizontal axis
:: '''''θ''''' : The posture angle of the torpedo in reference to the horizontal axis
:: '''''α''''' : The internal angle between '''''φ''''' and '''''θ''''', which is equal to the lift angle of tail fins of the torpedo
:: '''''W''''' : Weight of the torpedo
:: '''''g''''' : Acceleration of gravity, '''9.8 m/s2'''
:: '''''I''''' : Inertia coefficient with respect to lift moment at the gravity center of the torpedo
:: '''''ω''''' : Lifting angular velocity (radian)
:: '''''D''''' : Drag moment of force
:: '''''L''''' : Lift moment of force
:: '''''M''''' : Roll moment of force around the longitudial axis of the torpedo
:: '''''ρ''''' : Air density
:: '''''S''''' : Cross-section area of the torpedo
:: '''''l''''' : Total length of the torpedo
:: '''''lH''''' : The length between the gravity center and the center of lift moment of tail fins of the torpedo

The value '''''lH''''' is measured in wind-tunnel test,
as drag moment of force coefficient differences between the torpedoes with and without box type attachment of wooden tail stabilizer plates;

: <math> \mathit{l}_{H} = (-C_{mgH}\,\mathit{l}+C_{tHd})/C_{nH}\,</math>

where each coefficient is defined as;

: '''''CnH'' = ''CXH'' sin ''α'' + ''CZH'' cos ''α '''''
: '''''CtH'' = ''CXH'' cos ''α'' + ''CZH'' sin ''α '''''

:: '''''CX''''' : Drag moment of force coefficient '''''D'' / (1/2 ''ρ V''2 ''S'')'''
:: '''''CZ''''' : Lift moment of force coefficient '''''L'' / (1/2 ''ρ V''2 ''S'')'''
:: '''''Cmg''''' : Roll moment of force coefficient around the gravity center of the torpedo '''''M'' / (1/2 ''ρ V''2 ''l'')'''
:: '''''CXH''''' : Drag moment of force coefficient of tail stabilizer box of torpedo
:: '''''CZH''''' : Lift moment of force coefficient of tail stabilizer box of torpedo
:: '''''CmgH''''' : Roll moment of force coefficient around the gravity center of tail stabilizer box of the torpedo

Solving ''Eq.''1 through ''Eq.''4 with respect to movement under certain [[Boundary value problem|boundary conditions]], we could derive the set of equations '''''t''''', '''''X''''', '''''Z''''' in definite integral equation form.

: <math> X = \int_{\lambda_0}^{\lambda}\,\frac{d \lambda}{gC}\,+\,k\,\varphi(x), \,\, Z = \int_{\lambda_0}^{\lambda}\,\lambda\,\frac{d \lambda}{gC}\,+\,k\,\varphi(x) </math>

: where, '''''λ'' = - tan ''φ''''', '''''λ''0 = - tan ''φ''0''', at time '''''t'' = 0 '''

[[Integral|Definite integrals]] can be numerically solved by ''[[Simpson's rule|Composite Simpson's rule]]'' in the ordinary differential equations field.

Lifting motion ''Eq.''5 can be numerically solved by [[Runge-Kutta methods|The common fourth-order Runge-Kutta method]] to get values of '''''ω'''''.

Lifting stability of Torpedo ''Eq.''6 can be numerically solved by [[Moving average|Exponential moving average method]] for exponential equations.

=== Practical equation for lift moment ===
Lieutenant Commander Hirota proved the practical torpedo motions by his equations.

In the pitching movement, tail stabilizer box has the function not only aligning the longitudinal axis of torpedo in the moving direction or the vector of the center gravity, but also damping the lifting movement or pitching oscillation. The latter effect is the change of lift moment with respect to the angular velocity of the center gravity as follows.

:<math> -(1/2) \times 57.3 C_1 \, \rho \, V^2 \, b \, S_1 \, (b/V) \, (d\theta / dt) \,</math>

:: '''''ρ''''' : Air density
:: '''''V''''' : Speed of the torpedo (constant)
:: '''''C''1''' : Lift moment coefficient by one degree with respect to the airflow vector to the plates of the aerodynamic tail stabilizer
:: '''''b''''' : The length between gravity center of torpedo and the lift moment center of the plates of the tail stabilizer
:: '''''S''1''' : Area sum of the horizontal plates of the tail stabilizer
:: '''''θ''''' : Angle of the longitudinal axis of the torpedo to the vector of the center gravity, where the angle is in radian.

The 1st [[derivative]] of the lift angle with respect to time, '''57.3 (''b/V'') (''dθ/dt'')''' is the key factor to damp the pitch movement. The characteristics to damp the pitching factor improved the airborne stability and the course of the torpedo.

=== Practical equation for roll moment ===
First, the roll moment equation is set, then it is simplified in 2nd derivative of '''''θ''''' with respect to '''''t''''', as follows:
:<math>
\begin{array}{lcll}
I_2(d^2\theta/dt^2)&= &-K-M(d\theta/dt) &\cdots(Eq.7)\\
(d^2\theta/dt^2)+M/I_2(d\theta/dt)&= &0 &\cdots(Eq.8)
\end{array}
</math>

Here we set the initial conditions at '''''t'' = 0''' as '''''dθ/dt'' = ''ω''0''', '''''θ'' = 0''' to analyze the angular movement on the top neutral point,
then we can simplify ''Eq.''7 and ''Eq.''8 as follows:

:<math>
\begin{array}{lcll}
d\theta / dt&= &-(I_2\,/\,M)\,(K\,/\,I_2)\,+\,(\omega_0\,+\,K\,/\,M)\,e^{-(M/I_2)t} &\cdots(Eq.9)\\
\theta &= &-(K/M)t\,+\,(I_2/M)\,(\omega_0\,+\,K/M)\,(1\,-\,e^{-(M/I_2)t}) &\cdots(Eq.10)
\end{array}
</math>

:where '''''K''''' and '''''M''''' are defined as follows:
::'''''K'' = - (1/2) ''C''2 ''ρ V''2 ''S''2 a '''
::'''''M'' = (1/2) × 57.3 ''C''1 ''S''3 (''b''22 / 12 )''ρ V'''''

:and the symbols above are the constants and variables as follows:
:: '''''ρ''''' : Air density
:: '''''V''''' : Speed of the torpedo (constant)
:: '''''C''2''' : Lift coefficient of '''22.5°''' with respect to the airflow vector to the side roll rudder plates
:: '''''a''''' : The length between the aerodynamic centers of the lift moment force in two roll rudders
:: '''''S''2''' : The area of one roll rudder
:: '''''b''2''' : The width of the tail stabilizer
:: '''''C''1''' : Lift coefficient by one degree with respect to the airflow vector to the plates of the aerodynamic tail stabilizer
:: '''''S''3''' : Area sum of the plates of the tail stabilizer, where '''''S''3 / 4''' is used in the case of box type aerodynamic tail stabilizer.
:: '''''I''2''' : Inertia coefficient with respect to lifting moment at the gravity center of the torpedo
:: '''''θ''''' : Roll angle of the torpedo at a right angle with the longitudinal axis

In the practical iterative process in analyses, set certain angle value '''''θ''°''' to ''Eq.''10 and gets value '''''t'''''. Put the value '''''t''''' to ''Eq.''9 then we can get the angular velocity passing through the top neutral point.

Since numerical calculation analyses are plain and difficult to gain public understanding, Hirota made a very simple alternative qualitative explanation of ''Eq.''7 for the countersteering theory to the outsiders of the project, then. It made sense to all by the characteristics of the equation with respect to roll angle, which was obtained through the results of his numerical analyses. ''Eq.''7 represents the change of the rolling angular velocity of the torpedo, that is the 2nd order derivative of angle with respect to time.

The torpedo with anti-rolling controller can converge a big roll angle moment of the torpedo to the small ''to and fro'' rolling motion by the factor '''''K''''' and '''''M''''' of ''Eq.''7. Anti-rolling controller can converge a big roll angle moment of the torpedo to the small to and fro rolling motion by changing '''''K''''' factor in the right hand side in ''Eq.''7, which represents the roll moment produced by twisting the side roll rudders, as follows:

When the roll angle is over the range of '''+/-10''' degree, the sign of the first term '''''K''''' factor in ''Eq.''7 works always in positive to steer roll rudders. When the roll angle is coming back within the range of '''+/-10''' degree, the first term '''''K''''' factor in ''Eq.''7 changes the sign of value in negative to countersteer.

While the 2nd term '''''M''''' factor of right hand side in ''Eq.''7, which represents the drag force of wooden tail stabilizer plates, always damps the rolling moment.

That is how Type 91 aerial torpedo can converge the rolling moment oscillation while falling in the air and running under the water.

== Miscellaneous stories ==

=== Provision of the aerial torpedo technology to Germany ===
Germany approached Japan to transfer Japanese aerial torpedo technology and [[List of World War II torpedoes of Germany#List of WWII Luftwaffe torpedoes|Type 91 aerial torpedoes]]. The Imperial Japanese Navy accepted that approach, and brought not only the technology but also a number of Type 91 aerial torpedoes to Germany in response, designated the ''Luftorpedo'' '''LT 850''' in German service.

Germany needed to know the '''aerial torpedo''' technology because the [[Italian battleship Littorio]] was heavily damaged in the [[Battle of Taranto]] on November 11, 1940, and the [[German battleship Bismarck]] was hit by a single torpedo, which jammed its [[rudder]] and steering gear for hours on May 26, 1941. Germany also needed aerial torpedoes to attack the Allied transport ships steaming in the [[Mediterranean Sea]].<ref>p.13, Fumio Aikō; ''Koku Gyorai Note''</ref> It had previously imported Italian-made aerial torpedoes.

=== Kawatana Naval Arsenal: Firm of Type 91 aerial torpedo ===
Kawatana Naval Arsenal was the production firm of Type 91 aerial torpedo.
Type 91 aerial torpedo was first produced at Nagasaki Weapon Factory of [[Mitsubishi Heavy Industries]] in the beginning. Aerial torpedo was researched and developed at [[Yokosuka Naval Arsenal]] in [[Kanagawa Prefecture]].

Later, Imperial Japanese Navy established two branch arsenals. One was Suzuka Naval Arsenal in [[Mie Prefecture]]. Another was Kawatana Naval Arsenal, the branch of [[Sasebo Naval Arsenal]] in [[Nagasaki Prefecture]]. [[Kawatana, Nagasaki|Kawatana]] was specialized to torpedo production.
In 1945, a strange map of Japan, carried by aviators of [[United States Army Air Forces]], noted Kawatana Naval Arsenal as a beer factory. Kawatana Arsenal got no air raid. After the war was over, Kawatana could not help but refused the order of rendering beer by the occupied military force.

[[Nagasaki]] City and Nagasaki Weapon Factory were perished from the face of the earth by ''[[Fat Man]]''. Casualties and burned citizens were carried over to a small town, Kawatana by rail. Kawatana Naval Arsenal accommodated them all. All high school and junior-high girl students working at the arsenal nursed them. They have fresh memories of Kawatana with killed or burned casualties by ''Fat Man''.

=== Postwar ===
The Imperial Japanese Navy was perished in 1945. Japan decleared the no-war commitment. The torpedo technology was prohibited by the war-renouncing [[Constitution of Japan|Constitution]] Article 9. A few scientists got job in some universities, most were working in nonpublic corporations in the chaos of postwar Japan.

About 15 years later, Type 91 aerial torpedo project members knew the industrial machineries imported from United States to Japan powered by [[PID controller|proportional-integral-derivative control devices]] at last. They saw those machineries, which proclaimed "high-sounding" title, with mingled feelings.<ref>p.78, Ichikawa, Hidehiko; Chap.4-5 The mechanism of roll controller, ''Koku Gyorai Note''</ref>

Some dacades later, they called together at a historic inn named '''Yōshin-tei''', Izu, Kanagawa-prefecture, Japan to establish a small association on January 16, 1978. They decided to raise money to make a tiny book ''Koku Gyorai Note'', or Aerial Torpedo Notebook by private book service.
They selected Yuta Tanaka as the 1st chairman. He soon died in 1979, and the major member Toshimori Maeda, died too. Satoshi Suzuki was selected as the 2nd chairman to publish the private book.<ref>p.278, Ichikawa, Hidehiko; Postface, ''Koku Gyorai Note''</ref>
<ref>''Hotel Yoshin-tei'' was a venerable inn since 1989, famous for its amenity, sanitary modern building, having beautiful scenery by the resort beach of Hayama, Zushi-city, in Kanagawa prefecture, Japan. It located close to the Imperial family resort villa, ''Hayama Goyo Tei''. When [[Emperor Taishō]] had been going to die at Hayama Imperial villa on December 1926, ministors of Imperial Japanese government had stayed at Yoshin-tei to serve Emperor Taishō. Traditional ''Hotel Yoshin-tei'' was closed in 1984 and an Italian restaurant ''Cantina'' is opened at the lot since February, 2004. Only a beach cabana, ''Branch of Yoshin-tei'' is opening on nearby sand beach every summer.</ref>

Type 91 torpedoes were now displayed at [[Naval Academy Edashima|Etajima school of Japan Maritime Self-Defense]] (The Maritime Self Defense Force 1st Technical School) and Shimofusa Base. They had lost the roll rudders.

Type 91 torpedo retaining original form was once displayed at personal exhibition stand by a small wayside drive-inn cafe, named ''Yoroi-ya'' (armor house), in the pile of other dusty military junk parts the owner had, the engine of [[Nakajima B6N|B6N2]] ''Tenzan'', broken radio communication systems, used cups, and rusted mountain artilleries from the Imperial Japanese army and navy, in Hyogo prefecture, Japan, till 2005. The stand was closed in 2005 and the exhibited junk articles were sold to neighboring farmers or military collectors.

An excavated Type 91 aerial torpedo was preserved at Resource Museum in [[Naha Airport|JGSDF Camp Naha]], 1st Combined Brigade of The Western Army, [[Japan Ground Self-Defense Force|JGSDF]], located in [[Naha, Okinawa|Naha]] city, Okinawa, Japan. It is retaining the original form. It was picked up as an [[unexploded ordnance]] by a bomb-disposal Squadron of JGSDF.

A captured Type 91 aerial torpedo is displayed at the U. S. Naval Academy, Annapolis, Maryland. It rests on two supports flanking a pathway in a small park in front of the Academy's Dahlgren Hall. Displayed on the other side of the pathway is a Type 93 Japanese Long Lance ship-launched torpedo.

==References==

===Bibliography===
*{{cite book
| last = Ichikawa
| first = Hidehiko
| coauthors= Kodaira, Makoto; Kawada, Teruyuki
| date = July 25, 1985
| chapter =
| title = "Kyu Ichi Kai - Koku Gyorai Note" or 91 Association - Aerial Torpedo Notebook
| publisher = ''Iyeno Hikari'' Private Publishing Service
| location = Tokyo, Japan
| id = ISBN
}} - text in Japanese, privately printed book.
*{{cite book
| last =
| first =
| authorlink =
|date=July 2002
| chapter = Warship Carrier Zuikaku Action Report No.7, Battle of the Coral Sea
| title = "Kaigun Koku Bokan Sento Kiroku" or Naval Aircraft Carrier Action Reports
| publisher = Athen-shobo
| location = Tokyo, Japan
| id =
}} - photographic print copies of Imperial Japanese Navy Action Reports, text in Japanese.
*{{cite book
| last =
| first =
| coauthors=
| month = | year = 1994
| chapter = p196 - p222, Ozawa, Kyuno Joe; "Mitsubishi Type 4 Army Bomber Aircraft"
| title = Document of Historical Aircraft with Japan Making, SPECIAL THANKS 600 ISSUE OF AIRREVIEW, last volume
| publisher = Kanto-sha
| location = Tokyo, Japan
| id = ISBN
}} - text in Japanese, Prof. Ozawa is the designer of Ki-69.
*{{cite book
| last = Seko
| first = Tsutomu
| coauthors=
|date=December 1986
| chapter =
| title = "Raigeki no Tsubasa" or Wings of Torpedo Bombers
| publisher = Kojin-sha
| location = Tokyo, Japan
| id = ISBN
}} - text in Japanese, Seko was one of the last torpedo bombardiers of B6Ns.
*{{cite book
| last = Akimoto
| first = Minoru
| coauthors=
|date=June 1995
| chapter =
| title = "Nihon Gunyoki Kokusen Zenshi, Dai 4 kan" or Japanese military aircraft air combat complete history, volume 4
| publisher = Green arrow sha
| location = Tokyo, Japan
| isbn = 4-7663-3174-5
}} - text in Japanese.
* (August, 1945), Resources from Torpedo bombing section, Kawatana branch, Naval aerial technology arsenal, Imperial Japanese Navy.
* (August, 1945), Resources from the 1st torpedo section, Kawatana naval arsenal production firm, Imperial Japanese Navy.

=== Notes ===
{{reflist}}

== External links ==
*[http://www.navweaps.com/Weapons/WTJAP_WWII.htm A page with many statistics on Japanese WWII torpedoes.]



{{IJN}}

{{DEFAULTSORT:Type 91 Torpedo}}
[[Category:Torpedoes of Japan]]
[[Category:World War II weapons of Japan]]
[[Category:World War II naval weapons]]
[[Category:Aerial torpedoes]]

Tertiary ideal

2013-06-03T10:21:24Z

129.132.146.160: /* References */

{{Context|date=December 2009}}

Photon transport theories, such as the [[Monte Carlo methods|Monte Carlo method]], are commonly used to model [[Monte Carlo method for photon transport|light propagation in tissue]]. The responses to a [[pencil beam]] incident on a scattering medium are referred to as [[Green's function|Green’s functions]] or [[impulse response]]s. Photon transport methods can be directly used to compute broad-beam responses by distributing photons over the cross section of the beam. However, [[convolution]] can be used in certain cases to improve computational efficiency.

==General convolution formulas==
In order for convolution to be used to calculate a broad-beam response, a system must be [[time-invariant system|time invariant]], [[linear]], and [[Translational symmetry|translation invariant]]. Time invariance implies that a photon beam delayed by a given time produces a response shifted by the same delay. Linearity indicates that a given response will increase by the same amount if the input is scaled and obeys the property of [[Superposition principle|superposition]]. Translational invariance means that if a beam is shifted to a new location on the tissue surface, its response is also shifted in the same direction by the same distance. Here, only spatial convolution is considered.

Responses from photon transport methods can be physical quantities such as [[Absorption (electromagnetic radiation)|absorption]], [[fluence]], [[reflectance]], or [[transmittance]]. Given a specific physical quantity, ''G(x,y,z)'', from a pencil beam in Cartesian space and a collimated light source with beam profile ''S(x,y)'', a broad-beam response can be calculated using the following 2-D convolution formula:

:<math>C(x,y,z)=\int_{-\infty}^\infty \int_{-\infty}^\infty \ G( x-x',y-y',z)S(x',y')\, dx'\,dy'. \qquad(1) </math>

Similar to 1-D convolution, 2-D convolution is commutative between ''G'' and ''S'' with a change of variables <math>x''=x-x'\,</math> and <math>y''=y-y'\,</math>:

:<math> C(x,y,z)=\int_{-\infty}^{\infty}\!\int_{-\infty}^{\infty}\ G( x'',y'',z)S(x-x'',y-y'')\, dx''\,dy''. \qquad(2)</math>

Because the broad-beam response <math>C(x,y,z)\,</math> has cylindrical symmetry, its convolution integrals can be rewritten as:

:<math> C(r,z) = \int_0^\infty \ S(r')r' \left [ \int_{0}^{2\pi} \ G\left (\sqrt{r^2+r'\,^2-2rr'cos\phi'},z \right )\, d\phi' \right ]dr' \qquad(3)</math>

:<math>C(r,z)=\int_0^\infty G(r'',z)r'' \left [ \int_{0}^{2\pi} S \left ( \sqrt{r^2+r''\,^2-2rr''cos\phi''} \right ) \, d\phi'' \right ]dr''\qquad(4)</math>

where <math> r'=\sqrt{x'^2+y'^2}</math>. Because the inner integration of Equation 4 is independent of ''z'', it only needs to be calculated once for all depths. Thus this form of the broad-beam response is more computationally advantageous.

==Common beam profiles==

===Gaussian beam===

For a [[Gaussian beam]], the intensity profile is given by
:<math> S(r') = S_0 \exp \left [ -2 \left( \frac{r'}{R} \right )^2 \right]. \qquad(5)</math>

Here, ''R'' denotes the <math>\tfrac{1}{e^2} \,</math> radius of the beam, and ''S''0 denotes the intensity at the center of the beam. ''S''0 is related to the total power ''P''0 by

:<math> S_0 = \frac{2P_0}{\pi R^2}.\qquad(6)</math>
Substituting Eq. 5 into Eq. 4, we obtain
:<math> C(r,z) = 2\pi S(r)\int_{0}^{\infty} G(r'',z)\exp\left [-2\left (\frac{r''}{R} \right )^2 \right ]I_0\left (\frac{4rr''}{R^2} \right ) r'' \, dr'', \qquad(7)</math>

where ''I''0 is the zeroth-order [[modified Bessel function]].

===Top-hat beam===
For a [[tophat beam|top-hat beam]] of radius ''R'', the source function becomes

:<math> S(r') =
\begin{cases}
S_0, & \text{if }r'\leq R \\
\,0, & \text{if }r'> R
\end{cases}\qquad(8)</math>

where ''S''0 denotes the intensity inside the beam. ''S''0 is related to the total beam power ''P''0 by

:<math> S_0 = \frac{P_0}{\pi R^2}.\qquad(9)</math>

Substituting Eq. 8 into Eq. 4, we obtain
:<math>C(r,z) = 2\pi S_0\int_{0}^{\infty}G(r'',z)I_\phi (r,r'')r''\,dr'',\qquad(10)</math>
where
:<math> I_\phi (r,r'') =
\begin{cases}
1, & \mbox{if }R\geq r+r'', \\
\tfrac{1}{\pi}\cos^{-1} \left(\tfrac{r^2+r''^2-R^2}{2rr''} \right ), & \mbox{if } \left |r-r'' \right |\leq R < r+r'', \\
0, & \mbox{if }R < \left | r+r'' \right |.
\end{cases}\qquad(11)</math>

==Errors in numerical evaluation==

===First interactions===
First photon-tissue interactions always occur on the z axis and hence contribute to the specific absorption or related physical quantities as a [[Dirac delta function]]. Errors will result if absorption due to the first interactions is not recorded separately from absorption due to subsequent interactions. The total impulse response can be expressed in two parts:

:<math> C(r,z)=G_1(0,z)\frac{\delta (r)}{2\pi r}+ G_2(r,z),\qquad(12)</math>

where the first term results from the first interactions and the second, from subsequent interactions.
For a Gaussian beam, we have

:<math> C(r,z)=G_1(0,z)S(r)+2\pi S_0\int_{0}^{\infty} G_2(r'',z)\,exp\left [-2\left (\frac{r''-r}{R} \right )^2 \right ]I_{0e}\left(\frac{4rr''}{R^2} \right )r''\,dr''.\qquad(13)</math>

For a top-hat beam, we have

:<math> C(r,z)=G_1(0,z)S(r)+2\pi S_0\int_{0}^{\infty} G_2(r'',z)I_\phi (r,r'')r''\,dr''.\qquad(14)</math>

===Truncation error===
For a top-hat beam, the upper integration limits may be bounded by ''r''max, such that ''r'' ≤ ''r''max − ''R''. Thus, the limited grid coverage in the ''r'' direction does not affect the convolution. To convolve reliably for physical quantities at ''r'' in response to a top-hat beam, we must ensure that ''r''max in photon transport methods is large enough that ''r'' ≤ ''r''max − ''R'' holds.
For a Gaussian beam, no simple upper integration limits exist because it theoretically extends to infinity. At ''r'' >> ''R'', a Gaussian beam and a top-hat beam of the same ''R'' and ''S''0 have comparable convolution results. Therefore, ''r'' ≤ ''r''max − ''R'' can be used approximately for Gaussian beams as well.

==Implementation of convolution==
There are two common methods used to implement discrete convolution: the definition of convolution and [[fast Fourier transform]]ation (FFT and IFFT) according to the [[convolution theorem]]. To calculate the optical broad-beam response, the impulse response of a pencil beam is convolved with the beam function. As shown by Equation 4, this is a 2-D convolution. To calculate the response of a light beam on a plane perpendicular to the z axis, the beam function (represented by a ''b × b'' matrix) is convolved with the impulse response on that plane (represented by an ''a'' × ''a'' matrix). Normally ''a'' is greater than ''b''. The calculation efficiency of these two methods depends largely on ''b'', the size of the light beam.

In direct convolution, the solution matrix is of the size (''a'' + ''b'' − 1) × (''a'' + ''b'' − 1). The calculation of each of these elements (except those near boundaries) includes ''b'' × ''b'' multiplications and ''b'' × ''b'' − 1 additions, so the [[analysis of algorithms|time complexity]] is [[big O notation|O]][(''a'' + ''b'')2''b''2]. Using the FFT method, the major steps are the FFT and IFFT of (''a'' + ''b'' − 1) × (''a'' + ''b'' − 1) matrices, so the time complexity is O[(''a'' + ''b'')2 log(''a'' + ''b'')]. Comparing O[(''a'' + ''b'')2''b''2] and O[(''a'' + ''b'')2 log(''a'' + ''b'')], it is apparent that direct convolution will be faster if ''b'' is much smaller than ''a'', but the FFT method will be faster if ''b'' is relatively large.

==Computational examples==

The fate of photons can be modeled using a Matlab implementation of the Monte Carlo method (''n''rel = 1, ''μ''a = 0.1, ''μ''s=100, ''g'' = 0.9, 100,000 photons). Using this Matlab model, the fluence of a 3 × 3 × 3 cm3 region is recorded and the fluence distribution of a broad-beam response is plotted. Figure 1 and Figure 2 show the responses to a pencil beam and a 1-cm top-hat broad-beam, respectively. Direct convolution was used to calculate the broad-beam response in Figure 2. Figure 3 shows the broad-beam response calculated using the FFT method. When the diameter of the light beam is 0.2 cm, direct convolution costs 1.93 seconds, and the FFT method costs 7.35 seconds. When the diameter of the light beam is 2 cm, direct convolution costs 90.1 seconds, and FFT method costs 16.8 seconds. Of course, the absolute computation time depends on the processing speed of the computer being used. These two comparisons were made on the same computer. Although the computation times differ, the plots in Figures 2 and 3 are indistinguishable.
{|
|
| 
|
|}

==See also==
*[[Radiative transfer equation and diffusion theory for photon transport in biological tissue]]
*[[Monte Carlo method]]
*[[Monte Carlo method for photon transport]]

==Links to other Monte Carlo resources==
*[http://labs.seas.wustl.edu/bme/Wang/mc.html Optical Imaging Laboratory at Washington University in St. Louis (MCML)]
*[http://omlc.ogi.edu/software/mc/ Oregon Medical Laser Center]

==References==
*L.-H. Wang and H.-I. Wu. Biomedical Optics: Principles and Imaging. Wiley 2007.
*L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, "Monte Carlo modeling of photon transport in multi-layered tissues," Computer Methods and Programs in Biomedicine 47, 131–146 (1995).
*L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, "Convolution for responses to a finite diameter photon beam incident on multi-layered tissues," Computer Methods and Programs in Biomedicine 54, 141–150 (1997). [http://labs.seas.wustl.edu/bme/Wang/epub/1997LWCMPBConv.pdf Download article].

[[Category:Scattering theory]]

Ligand cone angle

2013-05-24T14:12:06Z

129.132.74.93: citations for cone angle values

[[File:GaussianBeamWaist.svg|thumb|350px|right|Gaussian beam width <math>w(z)</math> as a function of the axial distance <math>z</math>. <math>w_0</math>: beam waist; <math>b</math>: confocal parameter; <math>z_\mathrm{R}</math>: Rayleigh length; <math>\Theta</math>: total angular spread]]

In [[optics]] and especially [[laser science]], the '''Rayleigh length''' or '''Rayleigh range''' is the distance along the propagation direction of a [[light beam|beam]] from the [[beam waist|waist]] to the place where the area of the [[Cross_section_(geometry)|cross section]] is doubled.<ref name="Siegman1986">{{cite book | last = Siegman | first = A. E. | title = Lasers | publisher = University Science Books | date = 1986 | pages = 664–669 | isbn = 0-935702-11-3 }}</ref> A related parameter is the '''confocal parameter''', ''b'', which is twice the Rayleigh length.<ref name="Damask" /> The Rayleigh length is particularly important when beams are modeled as [[Gaussian beam]]s.

==Explanation==
{{more|Gaussian beam}}
For a Gaussian beam propagating in free space along the <math>\hat{z}</math> axis, the Rayleigh length is given by <ref name="Damask">{{cite book | last = Damask | first = Jay N. | title = Polarization Optics in Telecommunications | publisher = [[Springer Science+Business Media|Springer]] | date = 2004 | pages = 221–223 | isbn = 0-387-22493-9 }}</ref>

:<math>z_\mathrm{R} = \frac{\pi w_0^2}{\lambda} ,</math>
where <math>\lambda</math> is the [[wavelength]] and <math>w_0</math> is the [[beam waist]], the radial size of the beam at its narrowest point. This equation and those that follow assume that the waist is not extraordinarily small; <math>w_0 \ge 2\lambda/\pi</math>.<ref>Siegman (1986) p. 630.</ref>

The radius of the beam at a distance <math>z</math> from the waist is <ref>{{cite book | last = Meschede | first = Dieter | title = Optics, Light and Lasers: The Practical Approach to Modern Aspects of Photonics and Laser Physics | publisher = Wiley-VCH | date = 2007 | pages = 46–48 | isbn = 3-527-40628-X }}</ref>

:<math>w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 } . </math>

The minimum value of <math>w(z)</math> occurs at <math>w(0) = w_0</math>, by definition. At distance <math>z_\mathrm{R}</math> from the beam waist, the beam radius is increased by a factor <math>\sqrt{2}</math> and the cross sectional area by 2.

==Related quantities==
The total angular spread of a Gaussian beam in [[radian]]s is related to the Rayleigh length by<ref name="Siegman1986"/>

:<math>\Theta_{\mathrm{div}} \simeq 2\frac{w_0}{z_R}.</math>

The [[Beam diameter|diameter]] of the beam at its waist (focus spot size) is given by

:<math>D = 2\,w_0 \simeq \frac{4\lambda}{\pi\, \Theta_{\mathrm{div}}}</math>.

These equations are valid within the limits of the [[paraxial approximation]]. For beams with much larger divergence the Gaussian beam model is no longer accurate and a [[physical optics]] analysis is required.

==See also==
* [[Beam divergence]]
* [[Beam parameter product]]
* [[Gaussian function]]
* [[Electromagnetic wave equation]]
* [[John Strutt, 3rd Baron Rayleigh]]
* [[Robert Strutt, 4th Baron Rayleigh]]
* [[Depth of field]]

==References==
{{reflist}}

*[http://www.rp-photonics.com/rayleigh_length.html Rayleigh length] ''RP Photonics Encyclopedia of Optics''

[[Category:Optics]]

Potential of mean force

2013-04-04T10:46:57Z

129.132.218.135:

{{Unreferenced|date=December 2008}}
The '''Pareto priority index (PPI)''', so named because of its connection with the [[Pareto principle]], which is in turn named after the economist [[Vilfredo Pareto]], can be used to prioritize several (quality improvement) projects. It is especially used in the surroundings of [[six sigma]] projects. It has first been established by [[AT&T]].{{Fact|date=December 2008}}

The PPI is calculated as follows:

: <math> \text{PPI} = \frac{\text{savings} \times \text{probability of success}}{\text{cost} \times \text{time of completion}}</math>

A high PPI suggests a high project priority.

[[Category:Industry]]
[[Category:Business terms]]

Jack function

2013-03-01T11:36:17Z

129.132.146.66: /* External links */

[[File:NineZonesPlus.png|thumb|400px|right|Map & traveler views of one-gee proper-acceleration from rest for one year.]]
[[File:TwentyFiveZones.png|thumb|300px|right|Traveler spacetime for a constant-acceleration roundtrip.]]

In relativity theory, '''proper acceleration'''<ref>Edwin F. Taylor & John Archibald Wheeler (1966 1st ed. only) ''Spacetime Physics'' (W.H. Freeman, San Francisco) ISBN 0-7167-0336-X, Chapter 1 Exercise 51 page 97-98: "Clock paradox III" ([http://www.eftaylor.com/pub/spacetime/STP1stEdExercP81to100.pdf pdf]).</ref> is the physical acceleration (i.e., measurable acceleration as by an [[accelerometer]]) experienced by an object. It is thus acceleration relative to a [[free-fall]], or inertial, observer who is momentarily at rest relative to the object being measured. Gravitation therefore does not cause proper acceleration, since gravity acts upon the inertial observer that any proper acceleration must depart from (accelerate from). A corollary is that all inertial observers always have a proper acceleration of zero.

Proper acceleration contrasts with [[acceleration|coordinate acceleration]], which is dependent on choice of [[coordinate system]]s and thus upon choice of observers.

In the standard inertial coordinates of special relativity, for unidirectional motion, proper acceleration is the rate of change of [[proper velocity]] with respect to coordinate time.

In an inertial frame in which the object is momentarily at rest, the proper acceleration 3-vector, combined with a zero time-component, yields the object's [[four-acceleration]], which makes proper-acceleration's magnitude [[Lorentz invariant|Lorentz-invariant]]. Thus the concept is useful: (i) with accelerated coordinate systems, (ii) at relativistic speeds, and (iii) in curved spacetime.

In an accelerating rocket after launch, or even in a rocket standing at the gantry, the proper acceleration is the acceleration felt by the occupants, and which is described as [[g-force]] (which is ''not'' a force but rather an acceleration; see that article for more discussion of proper acceleration).<ref>Relativity By Wolfgang Rindler pg 71</ref> The "acceleration of gravity" ("force of gravity") never contributes to proper acceleration in any circumstances, and thus the proper acceleration felt by observers standing on the ground is due to the mechanical force ''from the ground'', not due to the "force" or "acceleration" of gravity. If the ground is removed and the observer allowed to free-fall, the observer will experience coordinate acceleration, but no proper acceleration, and thus no g-force. Generally, objects in such a fall or generally any such ballistic path (also called inertial motion), including objects in orbit, experience no proper acceleration (neglecting small tidal accelerations for inertial paths in gravitational fields). This state is also known as "zero gravity," ("zero-g") or "free-fall," and it always produces a sensation of [[weightlessness]].

Proper acceleration reduces to coordinate [[acceleration]] in an inertial coordinate system in flat spacetime (i.e. in the absence of gravity), provided the magnitude of the object's proper-velocity<ref>Francis W. Sears & Robert W. Brehme (1968) ''Introduction to the theory of relativity'' (Addison-Wesley, NY) [http://catalog.loc.gov/webvoy.htm LCCN 680019344], section 7-3</ref> (momentum per unit mass) is much less than the speed of light ''c''. Only in such situations is coordinate acceleration ''entirely'' felt as a "g-force" (i.e., a proper acceleration, also defined as one that produces measurable weight).

In situations in which gravitation is absent but the chosen coordinate system is not inertial, but is accelerated with the observer (such as the accelerated reference frame of an accelerating rocket, or a frame fixed upon objects in a centrifuge), then g-forces and corresponding proper accelerations felt by observers in these coordinate systems are caused by the mechanical forces which resist their [[weight]] in such systems. This weight, in turn, is produced by [[fictitious force]]s or "inertial forces" which appear in all such accelerated coordinate systems, in a manner somewhat like the weight produced by the "force of gravity" in systems where objects are fixed in space with regard to the gravitating body (as on the surface of the Earth).

The total (mechanical) force which is calculated to induce the proper acceleration on a mass at rest in a coordinate system that has a proper acceleration, via Newton's law '''F''' = ''m'' '''a''', is called the '''proper force'''. As seen above, the proper force is equal to the opposing reaction force that is measured as an object's "operational weight" (i.e., its weight as measured by a device like a spring scale, in vacuum, in the object's coordinate system). Thus, the proper force on an object is always equal and opposite to its measured weight.

==Examples==

For instance, when holding onto a carousel that turns at constant [[angular velocity]] you experience a radially inward ([[centripetal acceleration|centripetal]]) proper-acceleration due to the interaction between the hand-hold and your hand. This cancels the radially outward ''geometric acceleration'' associated with your [[Rotating reference frame|spinning coordinate frame]]. This outward acceleration (from the spinning frame's perspective) will become the coordinate acceleration when you let go, causing you to fly off along a zero proper-acceleration ([[geodesic]]) path. Unaccelerated observers, of course, in their frame simply see your equal proper and coordinate accelerations vanish when you let go.

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!''Animation:'' lost grip on a carousel
|-
|[[File:Spinframe.gif|thumb|360px|center|Map and spin frame perspectives of proper (red) and geometric (blue) accelerations for an object released from a carousel.]]
From the map frame perspective, what's dangerous is your tangential velocity. From the spin frame perspective, the danger instead may lie with that geometric acceleration.''Note:'' With some browsers, you can hit [Esc] to freeze the motion for more detailed analysis. However you may have to reload the page to restart.
|}

Similarly, standing on a non-rotating planet (and on earth for practical purposes) we experience an upward proper-acceleration due to the normal-force exerted by the earth on the bottom of our shoes. This cancels the downward ''geometric acceleration'' due to our choice of coordinate system (a so-called shell-frame<ref name="TaylorWheeler2003">Edwin F. Taylor and John Archibald Wheeler (2000) ''Exploring black holes'' (Addison Wesley Longman, NY) ISBN 0-201-38423-X</ref>). That downward acceleration becomes coordinate if we inadvertently step off a cliff into a zero proper-acceleration (geodesic or rain-frame) trajectory.

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!''Animation:'' ball that rolls off a cliff
|-
|[[File:Shellframe.gif|thumb|360px|center|Rain and shell frame perspectives of proper (red) and geometric (blue) accelerations for an object that rolls off a cliff.]] ''Note:'' The rain frame perspective, rather than being that of a raindrop, is more like that of a trampoline jumper whose trajectory tops out just as the ball reaches the edge of the cliff. The shell frame perspective may be familiar to planet dwellers who rely minute by minute on upward physical accelerations from their environment, to protect them from that geometric acceleration due to curved spacetime. No wonder micro-gravity environments may seem scary to them at first.
|}

Note that ''geometric accelerations'' (due to the [[affine connection|connection]] term in the coordinate system's [[covariant derivative]] below) act on ''every ounce of our being'', while proper-accelerations are usually caused by an external force. Introductory physics courses often treat gravity's downward (geometric) acceleration as one that's due to a [[Gravitational force|mass-proportional force]]. This, along with diligent avoidance of unaccelerated frames, allows them to treat proper and coordinate acceleration as the same thing.

Even then if an object maintains a ''constant proper-acceleration'' from rest over an extended period in flat spacetime, observers in the rest frame will see the object's coordinate acceleration decrease as its coordinate velocity approaches lightspeed. The rate at which the object's proper-velocity goes up, nevertheless, remains constant.

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!''Animation:'' high speed trip up then down
|-
|[[File:Tripframe.gif|thumb|201px|center|Map-frame perspective of proper (red) and ''coordinate'' (green) accelerations/decelerations in the vertical direction.]]
Here our object first accelerates upward for a time period of 2*c/α on traveler clocks,
where c is lightspeed and α is the (red) proper acceleration's magnitude. This first leg takes about 2 years if the acceleration's magnitude is about 1-gee. It then accelerates downward (first slowing and then speeding up) over twice that period, followed by a 2*c/α upward deceleration to return to the original height. Note that the coordinate acceleration (green) is significant only during the low-speed segments of this voyage.
|}

Thus the distinction between proper-acceleration and coordinate acceleration<ref name="MisnerThorneWheeler">cf. C. W. Misner, K. S. Thorne and J. A. Wheeler (1973) ''Gravitation'' (W. H. Freeman, NY) ISBN 0-7167-0334-0 {{Please check ISBN|reason=Check digit (0) does not correspond to calculated figure.}}, section 1.6</ref> allows one to track the experience of accelerated travelers from various non-Newtonian perspectives. These perspectives include those of accelerated coordinate systems (like a carousel), of high speeds (where proper time differs from coordinate time), and of curved spacetime (like that associated with gravity on earth).

==Classical applications==

At low speeds in the [[inertial reference frame|inertial coordinate systems]] of [[Newton's Laws|Newtonian physics]], proper acceleration simply equals the coordinate acceleration '''a'''=d2'''x'''/dt2. As reviewed above, however, it differs from coordinate acceleration if one chooses (against Newton's advice) to describe the world from the perspective of an accelerated coordinate system like a motor vehicle accelerating from rest, or a stone being spun around in a slingshot. If one chooses to recognize that gravity is caused by the curvature of spacetime (see below), proper acceleration also differs from coordinate acceleration in a gravitational field.

For example, an object subjected to physical or proper acceleration '''a'''o will be seen by observers in a coordinate system undergoing constant acceleration '''a'''frame to have coordinate acceleration:
:<math>\vec{a}_{acc} = \vec{a}_{o} - \vec{a}_{frame}</math>.
Thus if the object is accelerating with the frame, observers fixed to the frame will see no acceleration at all.

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!''Animation:'' driving from block to block
|-
|[[File:Carframe.gif|thumb|360px|center|Map and car frame perspectives of physical (red) and geometric (blue) accelerations for a car driving from one stop sign to the next.]]
In this illustration the car accelerates after a stop sign until midway up the block, at which point the driver is immediately off the accelerator and onto the brake so as to make the next stop.
|}

Similarly, an object undergoing physical or proper acceleration '''a'''o will be seen by observers in a frame rotating with angular velocity '''ω''' to have coordinate acceleration:
:<math>\vec{a}_{rot} =
\vec{a}_{o} - \vec\omega \times (\vec\omega \times \vec{r} ) - 2 \vec\omega \times \vec{v}_{rot} - \frac{d \vec\omega}{dt} \times \vec{r}
</math>.
In the equation above, there are three geometric acceleration terms on the right hand side. The first "centrifugal acceleration" term depends only on the radial position '''r''' and not the velocity of our object, the second "Coriolis acceleration" term depends only on the object's velocity in the rotating frame '''v'''rot but not its position, and the third "Euler acceleration" term depends only on position and the rate of change of the frame's angular velocity.

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!''Newtonian example:'' constant speed slingshot
|-
|[[File:Cforces.gif|thumb|360px|center|Map and spin frame perspectives of accelerations and forces associated with a stone released after being spun around on a massless rope.]]
Forces ''on the stone'' include the inward centripetal (red) force seen in both frames, as well as the geometric (blue) force seen in the spin frame. Before the stone is released, the blue geometric force is purely centrifugal (pointing radially outward), while after release the geometric force is a sum of centrifugal and Coriolis components.

Note that after release in the spin frame that the centrifugal component (light blue) is always radial, while the Coriolis component (green) is always perpendicular to spin frame velocity. Also seen in both frames is the force ''on the rope's anchor point'' (magenta) caused by [[Newton's law|Newton's 3rd Law]] action-reaction to the centripetal force on the stone.

===Before projectile launch===
The following alternate analyses of motion ''before'' the stone is released consider only forces acting in the radial direction. Both analyses predict that string tension ''T''=''mv''2/''r''. For example if the radius of the sling is ''r''=1 metre, the velocity of the stone in the map frame is ''v''=25 metres per second, and the stone's mass ''m''=0.2 kilogram, then the tension in the string will be 125 newtons.

*'''Map frame story before launch'''
:<math>-T_{centripetal}=\Sigma F_{radial} = m a_{radial} = - m \frac{v^2}{r}.</math>
Here the stone is seen to be continually accelerated inward so as to follow a circular path of radius r. The inward radial acceleration of aradial=v2/r is caused by a single ''unbalanced'' centripetal force T. The fact that the tension force is unbalanced means that, in this frame, the centrifugal (radially-outward) force on the stone is zero.
*'''Spin frame story before launch'''
:<math>m \frac{v^2}{r} - T_{centripetal}=\Sigma F_{rot} = m a_{rot} = 0.</math>
From the spin frame perspective the stone may be said to experience balanced inward centripetal (''T'') and outward centrifugal (''mv''2/''r'') forces, which result in no acceleration at all ''from the perspective of that frame''. Unlike the centripetal force, the frame-dependent centrifugal force acts on every bit of the circling stone much as gravity acts on every ounce of you. Moreover the centrifugal force magnitude is proportional to the stone's mass so that, if allowed to cause acceleration, the acceleration would be mass-independent.

===After projectile launch===
After the stone is released, in the spin frame both centripetal and Coriolis forces act in a delocalized way on all parts of the stone with accelerations that are independent of the stone's mass. By comparison in the map frame, after release no forces are acting on the projectile at all.
|}

In each of these cases, physical or proper acceleration differs from coordinate acceleration because the latter can be affected by your choice of coordinate system as well as by physical forces acting on the object. Those components of coordinate acceleration ''not'' caused by physical forces (like direct contact or electrostatic attraction) are often attributed (as in the Newtonian example above) to forces that: (i) act on every ounce of the object, (ii) cause mass-independent accelerations, and (iii) don't exist from all points of view. Such geometric (or improper) forces include [[Coriolis force|Coriolis]] forces, [[Euler force|Euler]] forces, [[g force|g-forces]], [[centrifugal force]]s and (as we see below) [[gravity]] forces as well.

==Viewed from a flat spacetime slice==

Proper-acceleration's relationships to coordinate acceleration in a specified slice of flat spacetime follow<ref>P. Fraundorf (1996) "A one-map two-clock approach to teaching relativity in introductory physics" ([http://xxx.lanl.gov/abs/physics/9611011 arXiv:physics/9611011])</ref> from [[Hermann Minkowski|Minkowski]]'s flat-space metric equation (''c''d''τ'')2 = (''c''d''t'')2 - (d'''x''')2. Here a single reference frame of yardsticks and synchronized clocks define map position '''x''' and map time ''t'' respectively, the traveling object's clocks define [[proper time]] ''τ'', and the "d" preceding a coordinate means infinitesimal change. These relationships allow one to tackle various problems of "anyspeed engineering", albeit only from the vantage point of an observer whose extended map frame defines simultaneity.

===Acceleration in (1+1)D===

[[File:Roundtriptimes.png|thumb|300px|right|This plot shows how a spaceship capable of 1-gee (10 m/s2 or about 1.0 lightyear per year squared) acceleration for 100 years might power a round trip to most anywhere in the visible universe and back in a lifetime.]]

In the unidirectional case i.e. when the object's acceleration is parallel or anti-parallel to its velocity in the spacetime slice of the observer, proper acceleration '''α''' and coordinate acceleration '''a''' are related<ref>A. John Mallinckrodt (1999) [http://www.csupomona.edu/~ajm/professional/talks.html What happens when a*t>c?] (AAPT Summer Meeting, San Antonio TX)</ref> through the [[Lorentz factor]] γ by '''α'''=γ3'''a'''. Hence the change in proper-velocity w=dx/dτ is the integral of proper acceleration over map-time t i.e. Δ''w''=''α''Δ''t'' for constant ''α''. At low speeds this reduces to the [[Equation of motion|well-known relation]] between coordinate [[velocity]] and coordinate acceleration times map-time, i.e. Δ''v''=''a''Δ''t''.

For constant unidirectional proper-acceleration, similar relationships exist between [[rapidity]] ''η'' and elapsed proper time Δ''τ'', as well as between Lorentz factor ''γ'' and distance traveled Δ''x''. To be specific:

:<math>\alpha=\frac{\Delta w}{\Delta t}=c \frac{\Delta \eta}{\Delta \tau}=c^2 \frac{\Delta \gamma}{\Delta x}</math>,
where the various velocity parameters are related by
:<math>\eta = \sinh^{-1}\left(\frac{w}{c}\right) = \tanh^{-1}\left(\frac{v}{c}\right) = \pm \cosh^{-1}\left(\gamma\right) </math>.

These equations describe some consequences of accelerated travel at high speed. For example, imagine a spaceship that can accelerate its passengers at "1 gee" (10 m/s2 or about 1.0 lightyear per year squared) halfway to their destination, and then decelerate them at "1 gee" for the remaining half so as to provide earth-like artificial gravity from point A to point B over the shortest possible time.<ref>E. Eriksen and Ø. Grøn (1990) Relativistic dynamics in uniformly accelerated reference frames with application to the clock paradox, ''Eur. J. Phys.'' '''39''':39-44</ref><ref>C. Lagoute and E. Davoust (1995) The interstellar traveler, ''Am. J. Phys.'' '''63''':221-227</ref> For a map-distance of Δ''x''AB, the first equation above predicts a midpoint Lorentz factor (up from its unit rest value) of ''γ''mid=1+''α''(Δ''x''AB/2)/c2. Hence the round-trip time on traveler clocks will be Δ''τ'' = 4(''c''/''α'') cosh−1(''γ''mid), during which the time elapsed on map clocks will be Δ''t'' = 4(''c''/α'') sinh[cosh−1(''γ''mid)].

This imagined spaceship could offer round trips to [[Proxima Centauri]] lasting about 7.1 traveler years (~12 years on Earth clocks), round trips to the [[Milky Way]]'s central [[black hole]] of about 40 years (~54,000 years elapsed on earth clocks), and round trips to [[Andromeda Galaxy]] lasting around 57 years (over 5 million years on Earth clocks). Unfortunately, sustaining 1-gee acceleration for years is easier said than done, as illustrated by the maximum payload to launch mass ratios shown in the figure at right.

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!''Animation:'' roundtrip to a star 6.9 ly away
|-
|[[File:Tripframe2.gif|thumb|396px|center|Map and traveler perspectives of a roundtrip at constant 1 gee proper-acceleration (red arrow in the traveler frame) between the sun (yellow) and a hypothetical star (cyan) 6.9 lightyears away. Proxima Centauri (orange) 4 lightyears from the sun is shown in orange toward the upper left.]]

From each perspective a year should elapse about every two seconds or every 100/17.4 frames. After each round trip ship-pilots on this shuttle-run will have aged only half as much as colleagues stationed on earth. This is [[time dilation]] in action.

Other differences include the distance changes between co-moving stars, seen in the traveler frame. This is [[length contraction]] in action. Coordinate acceleration (green) seen in the map frame is only significant in the year before and after each launch, while the proper-acceleration (red) felt by the traveler is significant throughout the voyage.

Note also the trace of a light signal initiated from each launch point, but 0.886 map years after launch. This pulse reaches the traveler at the voyage midpoint to remind them to begin deceleration. In the map frame Proxima Centauri sees the turnaround pulse before the destination star does, but the converse is true in the traveler frame. This is [[Relativity of simultaneity|relative simultaneity]] in action. Nonetheless both observers agree on the sequence of events along any time-like world line.
|}

==In curved spacetime==

In the language of [[general relativity]], the components of an object's acceleration four-vector ''A'' (whose magnitude is proper acceleration) are related to elements of the [[four-velocity]] via a [[covariant derivative]] ''D'' with respect to proper time τ:

:<math>A^\lambda := \frac{DU^\lambda }{d\tau} = \frac{dU^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu U^\nu </math>

Here ''U'' is the object's [[four-velocity]], and ''Γ'' represents the coordinate system's 64 connection coefficients or [[Christoffel symbols]]. Note that the Greek subscripts take on four possible values, namely 0 for the time-axis and 1-3 for spatial coordinate axes, and that repeated indices are used to indicate [[summation convention|summation]] over all values of that index. Trajectories with zero proper acceleration are referred to as [[geodesic]]s.

The left hand side of this set of four equations (one each for the time-like and three spacelike values of index λ) is the object's proper-acceleration 3-vector combined with a null time component as seen from the vantage point of a reference or book-keeper coordinate system in which the object is at rest. The first term on the right hand side lists the rate at which the time-like (energy/''mc'') and space-like (momentum/''m'') components of the object's four-velocity ''U'' change, per unit time ''τ'' on traveler clocks.

Let's solve for that first term on the right since at low speeds its spacelike components represent the coordinate acceleration. More generally, when that first term goes to zero the object's coordinate acceleration goes to zero. This yields...

:<math>\frac{dU^\lambda }{d\tau } =A^\lambda - \Gamma^\lambda {}_{\mu \nu}U^\mu U^\nu</math>.

Thus, as exemplified with the first two animations above, coordinate acceleration goes to zero whenever proper-acceleration is exactly canceled by the connection (or ''geometric acceleration'') term on the far right.<ref>cf. R. J. Cook (2004) Physical time and physical space in general relativity, ''Am. J. Phys.'' '''72''':214-219</ref> ''Caution:'' This term may be a sum of as many as sixteen separate velocity and position dependent terms, since the repeated indices ''μ'' and ''ν'' are by convention summed over all pairs of their four allowed values.

===Force and equivalence===

The above equation also offers some perspective on forces and the [[equivalence principle]]. Consider ''local'' book-keeper coordinates<ref name="TaylorWheeler2003" /> for the metric (e.g. a local Lorentz tetrad<ref name="MisnerThorneWheeler" /> like that which [[global positioning system]]s provide information on) to describe time in seconds, and space in distance units along perpendicular axes. If we multiply the above equation by the traveling object's rest mass m, and divide by Lorentz factor ''γ'' = d''t''/d''τ'', the spacelike components express the rate of momentum change for that object from the perspective of the coordinates used to describe the metric.

This in turn can be broken down into parts due to proper and geometric components of acceleration and force. If we further multiply the time-like component by lightspeed ''c'', and define coordinate velocity as '''v''' = d'''x'''/d''t'', we get an expression for rate of energy change as well:

:<math>\frac{dE}{dt}=\vec{v}\cdot\frac{d\vec{p}}{dt}</math> (timelike) and <math>\frac{d\vec{p}}{dt}=\Sigma\vec{f_o}+\Sigma\vec{f_g}=m(\vec{a_o}+\vec{a_g}) </math> (spacelike).

Here ''a''''o'' is an acceleration due to proper forces and ''a''''g'' is, by default, a geometric acceleration that we see applied to the object because of our coordinate system choice. At low speeds these accelerations combine to generate a coordinate acceleration like '''a'''=d2'''x'''/d''t''2, while for unidirectional motion ''at any speed'' ''a''''o'''s magnitude is that of proper acceleration ''α'' as in the section above where ''α'' = ''γ''3''a'' when ''a''''g'' is zero. In general expressing these accelerations and forces can be complicated.

Nonetheless if we use this breakdown to describe the connection coefficient (Γ) term above in terms of geometric forces, then the motion of objects from the point of view of ''any coordinate system'' (at least at low speeds) can be seen as locally Newtonian. This is already common practice e.g. with centrifugal force and gravity. Thus the equivalence principle extends the local usefulness of Newton's laws to accelerated coordinate systems and beyond.

===Surface dwellers on a planet===

For low speed observers being held at fixed radius from the center of a spherical planet or star, coordinate acceleration '''a'''shell is approximately related to proper acceleration '''a'''o by:

:<math>\vec{a}_{shell} = \vec{a}_o - \sqrt{\frac{r}{r-r_s}} \frac{G M}{r^2} \hat{r} </math>

where the planet or star's [[Schwarzschild radius]] rs=2GM/c2. As our shell observer's radius approaches the Schwarzschild radius, the proper acceleration ao needed to keep it from falling in becomes intolerable.

On the other hand for r>>rs, an upward proper force of only GMm/r2 is needed to prevent one from accelerating downward. At the Earth's surface this becomes:

:<math>\vec{a}_{shell} = \vec{a}_o - g \hat{r}</math>

where g is the downward 9.8 m/s2 acceleration due to gravity, and <math>\hat{r}</math> is a unit vector in the radially outward direction from the center of the gravitating body. Thus here an outward proper force of mg is needed to keep one from accelerating downward.

===Four-vector derivations===

The spacetime equations of this section allow one to address ''all deviations'' between proper and coordinate acceleration in a single calculation. For example, let's calculate the [[Christoffel symbols]]:<ref>Hartle, James B. (2003). Gravity: an Introduction to Einstein's General Relativity. San Francisco: Addison-Wesley. ISBN 0-8053-8662-9.</ref>

:<math>\left(
\begin{array}{llll}
\left\{\Gamma _{tt}^t,\Gamma _{tr}^t,\Gamma _{t\theta }^t,\Gamma _{t\phi }^t\right\} & \left\{\Gamma _{rt}^t,\Gamma _{rr}^t,\Gamma
_{r\theta }^t,\Gamma _{r\phi }^t\right\} & \left\{\Gamma _{\theta t}^t,\Gamma _{\theta r}^t,\Gamma _{\theta \theta }^t,\Gamma _{\theta
\phi }^t\right\} & \left\{\Gamma _{\phi t}^t,\Gamma _{\phi r}^t,\Gamma _{\phi \theta }^t,\Gamma _{\phi \phi }^t\right\} \\
\left\{\Gamma _{tt}^r,\Gamma _{tr}^r,\Gamma _{t\theta }^r,\Gamma _{t\phi }^r\right\} & \left\{\Gamma _{rt}^r,\Gamma _{rr}^r,\Gamma
_{r\theta }^r,\Gamma _{r\phi }^r\right\} & \left\{\Gamma _{\theta t}^r,\Gamma _{\theta r}^r,\Gamma _{\theta \theta }^r,\Gamma _{\theta
\phi }^r\right\} & \left\{\Gamma _{\phi t}^r,\Gamma _{\phi r}^r,\Gamma _{\phi \theta }^r,\Gamma _{\phi \phi }^r\right\} \\
\left\{\Gamma _{tt}^{\theta },\Gamma _{tr}^{\theta },\Gamma _{t\theta }^{\theta },\Gamma _{t\phi }^{\theta }\right\} & \left\{\Gamma
_{rt}^{\theta },\Gamma _{rr}^{\theta },\Gamma _{r\theta }^{\theta },\Gamma _{r\phi }^{\theta }\right\} & \left\{\Gamma _{\theta t}^{\theta
},\Gamma _{\theta r}^{\theta },\Gamma _{\theta \theta }^{\theta },\Gamma _{\theta \phi }^{\theta }\right\} & \left\{\Gamma _{\phi
t}^{\theta },\Gamma _{\phi r}^{\theta },\Gamma _{\phi \theta }^{\theta },\Gamma _{\phi \phi }^{\theta }\right\} \\
\left\{\Gamma _{tt}^{\phi },\Gamma _{tr}^{\phi },\Gamma _{t\theta }^{\phi },\Gamma _{t\phi }^{\phi }\right\} & \left\{\Gamma _{rt}^{\phi
},\Gamma _{rr}^{\phi },\Gamma _{r\theta }^{\phi },\Gamma _{r\phi }^{\phi }\right\} & \left\{\Gamma _{\theta t}^{\phi },\Gamma _{\theta
r}^{\phi },\Gamma _{\theta \theta }^{\phi },\Gamma _{\theta \phi }^{\phi }\right\} & \left\{\Gamma _{\phi t}^{\phi },\Gamma _{\phi
r}^{\phi },\Gamma _{\phi \theta }^{\phi },\Gamma _{\phi \phi }^{\phi }\right\}
\end{array}
\right)</math>

for the far-coordinate [[Schwarzschild metric]] {{nowrap|1=(''c'' d''τ'')2 = (1−''r''s/''r'')(''c'' d''t'')2 − (1/(1−''r''s/''r''))d''r''2 − ''r''2 d''θ''2 − (''r'' sin''θ'')2 d''φ''2}}, where ''r''s is the [[Schwarzschild radius]] 2''GM''/''c''2. The resulting array of coefficients becomes:
:<math>\left(
\begin{array}{llll}
\left\{0,\frac{r_s}{2 r (r - r_s)},0,0\right\} & \left\{\frac{r_s}{2 r (r - r_s)},0,0,0\right\} & \{0,0,0,0\} & \{0,0,0,0\} \\
\left\{\frac{r_s c^2 (r-r_s)}{2 r^3},0,0,0\right\} & \left\{0,\frac{r_s}{2 r (r_s-r)},0,0\right\} & \{0,0,r_s-r,0\} & \left\{0,0,0,(r_s-r) \sin ^2\theta
\right\} \\
\{0,0,0,0\} & \left\{0,0,\frac{1}{r},0\right\} & \left\{0,\frac{1}{r},0,0\right\} & \{0,0,0,-\cos \theta \sin \theta \} \\
\{0,0,0,0\} & \left\{0,0,0,\frac{1}{r}\right\} & \{0,0,0,\cot (\theta )\} & \left\{0,\frac{1}{r},\cot \theta ,0\right\}
\end{array}
\right)</math>.

From this you can obtain the shell-frame proper acceleration by setting coordinate acceleration to zero and thus requiring that proper acceleration cancel the geometric acceleration of a stationary object i.e. <math>A^\lambda = \Gamma^\lambda {}_{\mu \nu}U^\mu U^\nu = \{0,GM/r^2,0,0\}</math>. This does not solve the problem yet, since [[Schwarzschild coordinates]] in curved spacetime are book-keeper coordinates<ref name="TaylorWheeler2003" /> but not those of a local observer. The magnitude of the above proper acceleration 4-vector, namely <math>\alpha=\sqrt{1/(1-r_s/r)}GM/r^2</math>, is however precisely what we want i.e. the upward frame-invariant proper acceleration needed to counteract the downward geometric acceleration felt by dwellers on the surface of a planet.

A special case of the above Christoffel symbol set is the flat-space [[spherical coordinate]] set obtained by setting ''r''s or ''M'' above to zero:

:<math>\left(
\begin{array}{llll}
\left\{0,0,0,0\right\} & \left\{0,0,0,0\right\} & \{0,0,0,0\} & \{0,0,0,0\} \\
\left\{0,0,0,0\right\} & \left\{0,0,0,0\right\} & \{0,0,-r,0\} & \left\{0,0,0,-r \sin ^2\theta
\right\} \\
\{0,0,0,0\} & \left\{0,0,\frac{1}{r},0\right\} & \left\{0,\frac{1}{r},0,0\right\} & \{0,0,0,-\cos \theta \sin \theta \} \\
\{0,0,0,0\} & \left\{0,0,0,\frac{1}{r}\right\} & \{0,0,0,\cot \theta \} & \left\{0,\frac{1}{r},\cot \theta ,0\right\}
\end{array}
\right)</math>.

From this we can obtain, for example, the centri''petal'' proper acceleration needed to cancel the centri''fugal'' geometric acceleration of an object moving at constant angular velocity ''ω''=d''φ''/d''τ'' at the equator where ''θ''=''π''/2. Forming the same 4-vector sum as above for the case of d''θ''/d''τ'' and d''r''/d''τ'' zero yields nothing more than the classical acceleration for rotational motion given above, i.e. <math>A^\lambda = \Gamma^\lambda {}_{\mu \nu}U^\mu U^\nu = \{0,-r(d\phi/d\tau)^2,0,0\}</math> so that ''a''o=''ω''2''r''. Coriolis effects also reside in these [[Covariant derivative#Coordinate description|connection coefficients]], and similarly arise from coordinate-frame geometry alone.

==See also==
* [[Acceleration]]
*[[Fictitious force]]: one name for mass times ''geometric acceleration''
*[[Four-vector]]: making the connection between space and time explicit
*[[Kinematics]]: for studying ways that position changes with time
*[[Uniform acceleration]]: holding coordinate acceleration fixed

==Footnotes==
{{reflist}}

==External links==
* Excerpts from the first edition of ''Spacetime Physics'', and other [http://www.eftaylor.com/download.html#special_relativity resources posted by Edwin F. Taylor]
* [http://www.physics.ucsb.edu/~gravitybook/ James Hartle's gravity book page] including Mathematica programs to calculate Christoffel symbols.
* Andrew Hamilton's [http://casa.colorado.edu/~ajsh/phys5770_08/notes.html notes and programs] for working with local tetrads at U. Colorado, Boulder.

{{DEFAULTSORT:Proper Acceleration}}
[[Category:Minkowski spacetime]]
[[Category:Acceleration]]

Ovoid (projective geometry)

2013-02-20T10:00:10Z

129.132.146.160:

{{refimprove|date=April 2011}}
'''Background Radiation Equivalent Time''', or '''BRET''', is a unit of measurement of [[ionizing radiation]] dosage. One BRET is the equivalent of one day worth of average human exposure to [[background radiation]]. The unit is also referred to as BERT (Background Equivalent Radiation Time).

BRET units are used as a measure of low level radiation exposure. The health hazards of low doses of ionizing radiation are unknown and controversial, because the effects, mainly [[cancer]] and [[genetic damage]], take many years to appear, and the incidence due to radiation exposure can't be statistically separated from the many other causes of these diseases. The purpose of the BRET measure is to allow a low level dose to be easily compared with a universal yardstick: the average dose of [[background radiation]], mostly from natural sources, that every human unavoidably receives during daily life. Background radiation level is widely used in radiological health fields as a standard for setting exposure limits.<ref>http://radiographics.rsna.info/content/28/5/1439.full</ref> Presumably, a dose of radiation which is equivalent to what a person would receive in a few days of ordinary life will not increase his rate of disease measurably.

==Definition==
The BRET is the creation of Professor J R Cameron.<ref>http://www.chiropractic-uk.co.uk/gfx/uploads/textbox/Members%20area/Xray%20guidelines/BERT%20table%20reference.pdf</ref> It corresponding to a dose of radiation is the number of days of average background dose it is equivalent to. It is calculated from the [[equivalent dose]] in [[sievert]]s by dividing by the average annual [[background radiation]] dose in Sv, and multiplying by 365:

:<math>BRET = \frac{{SV}_{dose}}{{SV}_{background}} \cdot 365 \,</math>

The definition of the BRET unit is apparently unstandardized, and depends on what value is used for the average annual [[background radiation]] dose, which varies greatly across time and location. The 2000 [[UNSCEAR]] estimate for worldwide average natural background radiation dose is 2.4 mSv (240 mrem), with a range from 1 to 13 mSv. A small area in India as high as 30 mSv (3 rem).<ref>[http://www.doh.wa.gov/ehp/rp/factsheets/factsheets-htm/fs10bkvsman.htm "Background Radiation: Natural versus Man-Made"] Washington Stet Department of Health</ref><ref>[http://www.hindu.com/seta/2009/01/01/stories/2009010150131600.htm "Monazite sand does not cause excess cancer incidence "], ''The Hindu''</ref> Using the 2.4 mSv value each BRET unit equals 6.6 μSv.

BRET values range from 2 BRET for a [[Dentistry|dental]] [[x-ray]] to around 400 for a [[barium enema]] study.

==See also==
* [[Radiology]]

==References==

{{reflist}}
* http://www.radiationcontrol.utah.gov/XRAY/BRET.HTM

[[Category:Radioactivity]]
[[Category:Background radiation]]

{{med-sign-stub}}

Texture mapping

2012-08-20T15:57:38Z

129.132.62.17:

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Exner function

2012-08-16T11:53:58Z

129.132.221.152:

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Saffman–Delbrück model

2012-07-19T16:26:19Z

129.132.228.179: /* Beyond the Saffman–Delbrück length */

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2012-07-10T11:18:05Z

129.132.210.31: /* Equivalency of agreement problems */

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2012-06-13T11:57:38Z

129.132.146.194: /* Definitions */ "genuine surgery" is not mentioned in the linked article, and after some search, the term looks to me like a Wikipedia invention

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2012-06-11T08:27:21Z

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2012-06-07T13:21:37Z

129.132.146.160: /* The sheaves of forms with values in a holomorphic vector bundles */

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2012-06-07T12:57:44Z

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2012-05-31T09:43:17Z

129.132.146.160: /* Examples */

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Oh yes, and I don�t wear trainers, let alone state-of-the-art techno-trainers with motion-control, forefoot cushioning, stability cradles and ex-Nasa shock absorption. A pair of those flimsy, five-toed Vibram foot-gloves - barely distinguishable from running barefoot - gives me all the comfort I need, and lets me enjoy the feel of the ground benea 

Talking of which, wherever possible I avoid smooth, hard surfaces (tarmac, pavement, man-made paths), preferring instead to take my chances with mud, puddles, roots, nettles, thistles, fences and livestock in my local fiel .
Strangest of all, I enjoy it. In fact, I can�t remember the last time a month went by when I didn�t think: this is my favourite time of ye .
I hardly need to spell out the many ways in which my habits offend against modern norms. They are, obviously, an affront to health and safety. They�re also an outrage against decent standards of personal hygiene (all that mud!) and comfort. What normal approach to sport leaves you with such cold feet at the end of each winter session that the only sensible way to clean up afterwards is under a cold show 

Above all, they are an offence against the running industry, a retail sector currently in breathtakingly good health. You�ve probably noticed. Up and down Britain, in allegedly moribund high streets, specialist running stores keep sprouting, new and vigorous: Sweatshop, Runners Need, Up & Running - it�s hard to miss the brandi 

'There is no more need for a running industry than there is for a tree-climbing industry or a hide-and-seek industry' (Alam 

They�re just the visible showcases. The products they and others sell are everywhere, in shops, online, on billboards, in glossy magazine ads: running shoes, running clothes, running books, running DVDs, specialist running kit, medications, drinks and foo .
If you thought running was just a question of going outside and putting one foot in front of another, you are, like me, out of step. Big Running preaches another gospel, as simple as it is profitable: as a runner, you are what you spe 

Estimates and definitions vary, but a low-end calculation puts the 2012 value of the UK running market at �425m at retail alone. That�s just the physical stuff you buy. There�s also a vast invisible running economy. Think of the bright young things who work for bodies such as UK Athlet s.
Think of the gym workers and personal trainers; the go-getters who organise for-profit races; the people who sell race T-shirts and race goodie bags; the people who put on running festivals and running holidays; the therapists who treat running-related injuries; the sponsorship brokers and superstar agents; the corporate ladder-climbers at the running-shoe multination 

It�s hard to put precise figures on it. How many of the 26,000 people who work in the British gym industry, for instance, should be counted as part of the [http://www.Bing.com/search?q=running+economy&form=MSNNWS&mkt=en-us&pq=running+economy running economy]? But it�s clear that, even without its increasingly well-paid televised elite, running adds up to a big, big busin s.
Globally, according to a 2013 report by Bloomberg, the running industry is worth nearly twice as much as the football industry; the consultants NPD Inc value the retail part [http://www.tullochphotography.co.nz/Images/nike-shoes-for-kids.asp nike free womens online] of it at �16.5bn a year - and say that running is �definitely� the world�s fastest growing sports bus s.

It�s a long-term boom, too. In the past 20 years, the running market has grown by 500 per ent.
It�s easy to see why. There are more runners: membership of running clubs in England alone has increased by 25 per cent in the past five years. More importantly, there are more products. Running shops and specialist running kit barely existed before the 1970s, when Bill Bowerman, founder of Nike, simultaneously launched a concept - jogging - and a product - thick-heeled running-shoes - without which the jogger�s heel-striking action was prohibitively uncomfor e.

Since then, barely a year has passed without a new, allegedly indispensable hi-tech product being launched (including innumerable products whose raison-d��tre is to cure the injuries that the previous generation of hi-tech products mysteriously failed to pre nt).
None comes cheaply, and with each passing year the threshold of what it seems reasonable to ask an ordinary recreational runner to pay shifts subtly upwards. Of course the industry is bo g.

Yet the boom is also perplexing - because how, when you think about it, can running be an industry at all? Can you think of any other human activity, apart from eating, drinking, sex and defecating, that is so utterly natural as running? It�s as simple, spontaneous and life-enhancing as si ing.
We�ve been doing it for tens of thousands of years, for necessity (hunting, fleeing, etc.) and, over countless generations, for pleasure. A child doesn�t need special training or equipment to discover the joy of running for fun, any more than a dog or a horse s.

There�s no more need for a running industry than there is for a tree-climbing industry or a hide-and-seek ind try.
The fact that we none the less have one tells us much about the forces that shape modern life. First, persuade people of a hitherto unnoticed need; then sell them a product that meets that need; and then, inexorably, keep cranking up the level of need and the price they must pay to have i met.
Apple does it with its endlessly evolving iPhones and operating systems. Big Running does it by plundering the technologies of the Olympic elite and then persuading ordinary runners that they�d be fools to run without them. The fact that these hi-tech, high-price-tag products would have been unimaginable and unaffordable for most runners in history is irrelevant; as is the fact that, for all but the tiniest handful of us, the �marginal gains� they confer are irrelevant (since it makes not the blindest difference if our personal best for 10k is or isn�t a hundredth of a second f r).

To quote from the blurb of one leading manufacturer: �As they put on their technologically advanced garments, athletes also put on a desire to perform beyond expectation and be greater than ever before�� In other words, buying this stuff makes us feel er.

Meanwhile, the kind of running I do - back to basics and close to nature - has almost died out. It hasn�t become any harder. Given the state of most people�s finances, it should be a more attractive approach than ever. Yet when I try to convey to my fellow runners the appeal of �natural� running, first reactions usually range from amused scepticism [http://www.tullochphotography.co.nz/Images/nike-shoes-for-kids.asp nike free womens online] to mild alarm - as if I were some kind of crazed fitness surv ist.

I tell them about the texture of my daily excursions in rural Northamptonshire, which on recent mornings have included wading through flooded meadows, leaping fallen trees, splashing through so much mud that I might as well have been playing rugby, squeezing through barbed wire and spiky hawthorn to avoid frightening some newborn lambs - and wondering at enough natural beauty to fill several volumes of try.

If my listeners haven�t edged away, I go on to enthuse about my preferred kinds of organised running challenge: not mass-participation road-races with �50 entry fees but obscure, uncommercial landscape-based events such as cheese-rolling, man-hunting (with or without hounds), ploughed-field-racing, and the kind of rough cross-country races, over moor or mountain, whose distances are given to the nearest mile - since no two people will take the sa oute.

Expensive running technology is irrelevant in such activities - it�s what you�re made of that counts, not what you�ve bought. Yet for many runners this very lack of expenditure seems to create anxiety. Never mind if those ever-more-sophisticated products give us anything wort ving.

Never mind if we can afford them or not. Buying products is what normal runners do - it says so in the advert ements.
This isn�t just a running issue; it�s true of many of life�s basic pleasures (food, holidays, music, clothing). It�s still theoretically possible to enjoy them without spending a fortune on equipment, ingredients, packaging and extras, but the 21st century�s agencies of persuasion encourage us to consider it weird o so.

Hence all those modern children who believe that a packaged sandwich is more desirable than a home-made one. It is, I suppose, a rational response to a lifetime of being carpet-bombed by advertising. The bought things in life are better - more desirable, more trustworthy, perhaps even more real - than those that you improvise for self.

Indeed, that may, sadly, turn out to have been the most enduring legacy of London 2012. Long after the athletes and volunteers and others who created the magic had gone home, the sponsors were still booming out the same deafening message from the billboards: if you want to amount to anything in your chosen sport - or, by implication, in life generally - then, obviously, you�ll need to buy the same state-of-the-art clothing, equipment, nutritional aid, shampoo or even junk food that your particular Olympic hero or heroine has been paid to orse.

The strange thing is, the more we spend on running, the less fit and more fat we collectively become. In the past 20 years (when, remember, expenditure on running has increased fivefold), the incidence of obesity in the UK has oubled.
Or perhaps it isn�t so strange. If even the simplest kind of outdoor exercise has been converted into a form of consumption, is it really surprising if those without money to burn opt to remain on the sofa? On the contrary: being a couch potato is the logical onse.

�They took all the trees,� sang Joni Mitchell, �and put �em in a tree museum,/ And charged the people a dollar-and-a-half just to see �em.� It�s much the same with running, except that it will cost you a lot more than a dollar-and-a-half to get yourself kitted out as a proper modern runner; and more still to start running in the way that politicians and health professionals generally recommend - that is, on a treadmill in a member ly gym.

The joy of my outdoor, lo-tech alternative is not just that it costs nothing. It�s also free in another sense. The expensively packaged version of our sport that Big Running sells us herds us into a few easily managed channels: mass-participation road-running; marathons; gyms; the kind of running in which times and results are all t t matter.
Nothing wrong with that, if that�s what you really want; but there are other kinds of ru g, too.

�Running Free�, as I call it, is the kind of running that often involves getting wet, muddy or lost; the kind of running where you�ll be more concerned with the [http://www.Dailymail.co.uk/home/search.html?sel=site&searchPhrase=rhythms rhythms] of the agricultural year and the ways of wild and domesticated animals than you will be with split times and recovery rates; the kind where you look outwards rather than in. You won�t need a stopwatch for it, or a heart-rate monitor, or any other special equipment (or, for that matter, skills), just a sense of wonder and a resolution to live in the moment, as natu ntended.

I think of it as Running Outside The Box (the box in my mind�s eye being one of those big shiny ones that expensive new traine come in).
I can�t vouch for its likely effect on your fitness goals. But if it brings you joy it�s unlikely to reduce your enthusiasm for running; and that, in turn, is unlikely to make ou slower.
The real point, though, is liberation. Runners are born free, and everywhere they run in chains - or, if you prefer, in chain stores. As spring approaches, what do we have to lose by breaking out and going bac nature?

Adapted from �Running Free: A Runner�s Journey Back to Nature�, by Richard Askwith (published on 6 March by Yellow Jersey, �16.99). To buy the book for �12.99 free P&P, call 08430 600030 or visit independentbooksdirect.co.uk

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Schrödinger equation

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129.132.71.139: /* Time independent equation */

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