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Lift coefficient

2014-03-04T21:28:21Z

131.215.118.133: /* Section lift coefficient */

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2014-03-04T21:16:36Z

131.215.118.133: added Cd of sphere at 10^5

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2014-02-22T21:25:58Z

131.215.248.46: /* Example 2 */

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2014-02-10T17:43:07Z

131.215.195.89: remove extraneous bullet point

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Moment magnitude scale

2014-02-05T00:28:05Z

131.215.74.78: Undid revision 591640811 by LulaBeast (talk) The constant is 6, not 10.7, because we are using SI units (N.m, not dyn.cm)

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2014-01-29T09:46:54Z

131.215.166.126: /* Orthogonality */ typo fix

{{See introduction|Ramsey theory}}

In [[combinatorics]], '''Ramsey's theorem''' states that in any colouring of the [[edge (graph theory)|edges]] of a sufficiently large [[complete graph]], one will find monochromatic complete subgraphs. For two colours, Ramsey's theorem states that for any pair of positive integers (''r'',''s''), there exists a least positive integer ''R''(''r'',''s'') such that for any [[complete graph]] on ''R''(''r'',''s'') vertices, whose edges are coloured ''red'' or ''blue'', there exists either a complete subgraph on ''r'' vertices which is entirely blue, or a complete subgraph on ''s'' vertices which is entirely red. Here ''R''(''r'',''s'') signifies an integer that depends on both ''r'' and ''s''. It is understood to represent the smallest integer for which the theorem holds.

Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by [[F. P. Ramsey]]. This initiated the combinatorial theory, now called [[Ramsey theory]], that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of '''''monochromatic subsets''''', that is, subsets of connected edges of just one colour.

An extension of this theorem applies to any finite number of colours, rather than just two. More precisely, the theorem states that for any given number of colours ''c'', and any given integers ''n''1,...,''nc'', there is a number, ''R''(''n''1, ..., ''nc''), such that if the edges of a complete graph of
order ''R''(''n''1, ..., ''n''''c'') are coloured with ''c'' different colours, then for some ''i'' between 1 and ''c'', it must contain a complete subgraph of order ''n''''i'' whose edges are all colour ''i''. The special case above has ''c'' = 2 (and ''n''1 = ''r'' and ''n''2 = ''s'').

==Example: ''R''(3,3) = 6==
[[Image:RamseyTheory K5 no mono K3.svg|frame|A 2-colouring of ''K''5 with no monochromatic ''K''3]]

In the following example, the formula ''R''(3,3) provides a solution to the question which asks the minimum number of vertices a graph must contain in order to ensure that either (1) at least 3 vertices in the graph are connected '''or''' (2) at least 3 vertices in the graph are unconnected. Note that owing to the symmetrical nature of the problem space, ''R''(''r'',''s'') is equal to ''R''(''s'',''r'').

Suppose the edges of a complete graph on 6 vertices are coloured red and blue. Pick a vertex ''v''. There are 5 edges incident to ''v'' and so (by the [[pigeonhole principle]]) at least 3 of them must be the same colour. [[Without loss of generality]] we can assume at least 3 of these edges, connecting to vertices ''r'', ''s'' and ''t'', are blue. (If not, exchange red and blue in what follows.) If any of the edges (''r'', ''s''), (''r'', ''t''), (''s'', ''t'') are also blue then we have an entirely blue triangle. If not, then those three edges are all red and we have an entirely red triangle. Since this argument works for any colouring, ''any'' ''K''6 contains a monochromatic ''K''3, and therefore ''R''(3,3) ≤ 6. The popular version of this is called the [[theorem on friends and strangers]].

An alternative proof works by [[double counting (proof technique)|double counting]]. It goes as follows: Count the number of ordered triples of vertices ''x'', ''y'', ''z'' such that the edge (''xy'') is red and the edge (''yz'') is blue. Firstly, any given vertex will be the middle of either 0 × 5 = 0 (all edges from the vertex are the same colour), 1 × 4 = 4 (four are the same colour, one is the other colour), or 2 × 3 = 6 (three are the same colour, two are the other colour) such triples. Therefore there are at most 6 × 6 = 36 such triples. Secondly, for any non-monochromatic triangle ('''xyz'''), there exist precisely two such triples. Therefore there are at most 18 non-monochromatic triangles. Therefore at least 2 of the 20 triangles in the ''K''6 are monochromatic.

Conversely, it is possible to 2-colour a ''K''5 without creating any monochromatic ''K''3, showing that ''R''(3,3) > 5. The unique colouring is shown to the right. Thus ''R''(3,3) = 6.

The task of proving that ''R''(3,3) ≤ 6 was one of the problems of [[William Lowell Putnam Mathematical Competition]] in 1953.

==Proof of the theorem==
===2-colour case===
First we prove the theorem for the 2-colour case, by [[Mathematical induction|induction]] on ''r'' + ''s''. It is clear from the definition that for all ''n'', ''R''(''n'', 1) = ''R''(1, ''n'') = 1. This starts the induction. We prove that ''R''(''r'', ''s'') exists by finding an explicit bound for it. By the inductive hypothesis ''R''(''r'' − 1, ''s'') and ''R''(''r'', ''s'' − 1) exist.

<blockquote>'''Lemma 1.''' ''R''(''r'', ''s'') ≤ ''R''(''r'' − 1, ''s'') + ''R''(''r'', ''s'' − 1):</blockquote>

'''Proof.''' Consider a complete graph on ''R''(''r'' − 1, ''s'') + ''R''(''r'', ''s'' − 1) vertices whose edges are coloured with two colours. Pick a vertex ''v'' from the graph, and partition the remaining vertices into two sets ''M'' and ''N'', such that for every vertex ''w'', ''w'' is in ''M'' if (''v'', ''w'') is blue, and ''w'' is in ''N'' if (''v'', ''w'') is red. Because the graph has ''R''(''r'' − 1, ''s'') + ''R''(''r'', ''s'' − 1) = |''M''| + |''N''| + 1 vertices, it follows that either |''M''| ≥ ''R''(''r'' − 1, ''s'') or |''N''| ≥ ''R''(''r'', ''s'' − 1). In the former case, if ''M'' has a red ''Ks'' then so does the original graph and we are finished. Otherwise ''M'' has a blue ''K''''r''−1 and so ''M'' ∪ {''v''} has blue ''Kr'' by definition of ''M''. The latter case is analogous.Thus the claim is true and we have completed the proof for 2 colours.

'''Note.''' In the 2-colour case, if ''R''(''r'' − 1, ''s'') and ''R''(''r'', ''s'' − 1) are both even, the induction inequality can be strengthened to:<ref>{{cite web|url=http://www.cut-the-knot.org/Curriculum/Combinatorics/ThreeOrThree.shtml#inequality2 |title=Party Acquaintances}}</ref>
:''R''(''r'', ''s'') ≤ ''R''(''r'' − 1, ''s'') + ''R''(''r'', ''s'' − 1) − 1.

===General case===
We now prove the result for the general case of ''c'' colours. The proof is again by induction, this time on the number of colours ''c''. We have the result for ''c'' = 1 (trivially) and for ''c'' = 2 (above). Now let ''c'' > 2.

<blockquote>'''Lemma 2.''' ''R''(''n''1, ..., ''n''''c'') ≤ ''R''(''n''1, ..., ''n''''c''−2, ''R''(''n''''c''−1, ''n''''c'')).</blockquote>

'''Proof.''' Consider a graph on ''t'' vertices and colour its edges with ''c'' colours. Now 'go colour-blind' and pretend that ''c'' − 1 and ''c'' are the same colour. Thus the graph is now (''c'' − 1)-coloured. By the inductive hypothesis, it contains either a ''Kni'' monochromatically coloured with colour ''i'' for some 1 ≤ ''i'' ≤ ''c'' − 2 or a ''K''''R''(''n''''c''−1,''n''''c'')-coloured in the 'blurred colour'. In the former case we are finished. In the latter case, we recover our sight again and see from the definition of ''R''(''nc−1'', ''nc'') we must have either a (''c'' − 1)-monochrome ''K''''n''''c''−1 or a ''c''-monochrome ''K''''n''''c''. In either case the proof is complete.

The right hand side of the inequality in Lemma 2 only contains Ramsey numbers for ''c'' − 1 colours and 2 colours, and therefore exists and is a finite number ''t'', by the inductive hypothesis. Thus, proving the claim will prove the theorem.

==Ramsey numbers==
The numbers ''R''(''r'',''s'') in Ramsey's theorem (and their extensions to more than two colours) are known as '''Ramsey numbers'''. An upper bound for ''R''(''r'',''s'') can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first lower bound was obtained by [[Paul Erdős]] using the [[probabilistic method]].) However, there is a vast gap between the tightest lower bounds and the tightest upper bounds. Consequently, there are very few numbers ''r'' and ''s'' for which we know the exact value of ''R''(''r'',''s''). Computing a lower bound ''L'' for ''R''(''r'',''s'') usually requires exhibiting a blue/red colouring of the graph ''K''''L''−1 with no blue ''Kr'' subgraph and no red ''K''''s'' subgraph. Upper bounds are often considerably more difficult to establish: one either has to check all possible colourings to confirm the absence of a counterexample, or to present a mathematical argument for its absence. A sophisticated computer program does not need to look at all colourings individually in order to eliminate all of them; nevertheless it is a very difficult computational task that existing software can only manage on small sizes. The complexity for searching all possible graphs (via [[brute-force search|brute force]]) is [[Big O notation|O]](2(''n''−1)(''n''−2)/2) for an upper bound of ''n'' nodes.<ref>[http://www.learner.org/channel/courses/mathilluminated/units/2/textbook/06.php 2.6 Ramsey Theory from Mathematics Illuminated]</ref>

As described above, ''R''(3,3) = 6. It is easy to prove that ''R''(4,2) = 4, and, more generally, that ''R''(''s'',2) = ''s'' for all ''s'': a graph on ''s'' − 1 nodes with all edges coloured red serves as a counterexample and proves that ''R''(''s'',2) ≥ ''s'' ; among colourings of a graph on ''s'' nodes, the colouring with all edges coloured red contains a ''s''-node red subgraph, and all other colourings contain a 2-node blue subgraph (that is, a pair of nodes connected with a blue edge.) Using induction inequalities, it can be concluded that ''R''(4,3) ≤ ''R''(4,2) + ''R''(3,3) − 1 = 9, and therefore ''R''(4,4) ≤ ''R''(4,3) + ''R''(3,4) ≤ 18. There are only two (4,4,16) graphs (that is, 2-colourings of a complete graph on 16 nodes without 4-node red or blue complete subgraphs) among 6.4×1022 different 2-colourings of 16-node graphs, and only one (4,4,17) graph (the [[Paley graph]] of order 17) among 2.46×1026 colourings.<ref>{{cite web|url=http://cs.anu.edu.au/~bdm/data/ramsey.html|title=Ramsey Graphs}}</ref> (This was proven by Evans, Pulham and Sheehan in 1979.) It follows that ''R''(4,4) = 18.

The fact that ''R''(4,5)=25 was first established by [[Brendan McKay]] and [[Stanisław Radziszowski]] in 1995.<ref>{{cite journal|title=R(4,5) = 25|author=Brendan D. McKay, Stanislaw P. Radziszowski|journal=Journal of Graph Theory|date=May 1995}}</ref>

The exact value of ''R''(5,5) is unknown, although it is known to lie between 43 (Geoffrey Exoo) and 49 (McKay and Radziszowski) (inclusive).

{{cquote|[[Paul Erdős|Erdős]] asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of ''R''(5,5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for ''R''(6,6). In that case, he believes, we should attempt to destroy the aliens.|||[[Joel Spencer]]<ref>{{citation|title=Ten Lectures on the Probabilistic Method|page=4|author=[[Joel H. Spencer]]|year=1994|publisher=[[Society for Industrial and Applied Mathematics|SIAM]]|isbn=978-0-89871-325-1}}</ref>}}

McKay, Radziszowski and Exoo employed computer-assisted graph generation methods to conjecture in 1997 that ''R''(5,5) is exactly 43. They were able to construct exactly 656 (5,5,42) graphs, arriving at the same set of graphs through different routes. None of the 656 graphs can be extended to a (5,5,43) graph.<ref>{{cite journal| url=http://cs.anu.edu.au/~bdm/papers/r55.pdf| title=Subgraph Counting Identities and Ramsey Numbers|author=Brendan D. McKay, Stanisław P. Radziszowski|journal=Journal of Combinatorial Theory}}</ref>

For ''R''(''r'',''s'') with ''r'', ''s'' > 5, only weak bounds are available. Lower bounds for ''R''(6,6) and ''R''(8,8) have not been improved since 1965 and 1972, respectively.<ref name="Survey">{{cite web|url=http://www.combinatorics.org/ojs/index.php/eljc/issue/view/Surveys|title=Dynamic Surveys}}</ref>

''R''(''r'',''s'') for values of ''r'' and ''s'' up to 10 are shown in the table below. Where the exact value is unknown, the table lists the best known bounds. ''R''(''r'',''s'') for values of ''r'' and ''s'' less than 3 are given by ''R''(1,''s'') = 1 and ''R''(2,''s'') = ''s'' for all values of ''s''. The standard survey on the development of Ramsey number research has been written by Stanisław Radziszowski.

{| class="wikitable"
! ''r'',''s''
! width="80" | 1
! width="80" | 2
! width="80" | 3
! width="80" | 4
! width="80" | 5
! width="80" | 6
! width="80" | 7
! width="80" | 8
! width="80" | 9
! width="80" | 10
|-
! 1
| '''1'''
|
|
|
|
|
|
|
|
|
|-
! 2
| 1
| '''2'''
|
|
|
|
|
|
|
|
|-
! 3
| 1
| 3
| '''6'''
|
|
|
|
|
|
|
|-
! 4
| 1
| 4
| 9
| '''18'''
|
|
|
|
|
|
|-
! 5
| 1
| 5
| 14
| 25
| '''43–49'''
|
|
|
|
|
|-
! 6
| 1
| 6
| 18
| 36–41
| 58–87
| '''102–165'''
|
|
|
|
|-
! 7
| 1
| 7
| 23
| 49–61
| 80–143
| 113–298
| '''205–540'''
|
|
|
|-
! 8
| 1
| 8
| 28
| 58–84
| 101–216
| 132–495
| 217–1031
| '''282–1870'''
|
|
|-
! 9
| 1
| 9
| 36
| 73–115
| 126–316
| 169–780
| 241–1713
| 317–3583
| '''565–6588'''
|
|-
! 10
| 1
| 10
| 40–42
| 92–149
| 144–442
| 179–1171
| 289–2826
| 331–6090
| 581–12677
| '''798–23556'''
|}

Since ''R(r, s) = R(s, r)'', there is a trivial symmetry across the diagonal.

This table is extracted from a larger table compiled by Stanisław Radziszowski.<ref name="Survey" />

==Asymptotics==
The inequality ''R''(''r'', ''s'') ≤ ''R''(''r'' − 1, ''s'') + ''R''(''r'', ''s'' − 1) may be applied inductively to prove that
:<math>R(r,s) \leq \binom{r+s-2}{r-1}. </math>
In particular, this result, due to [[Paul Erdős|Erdős]] and [[George Szekeres|Szekeres]], implies that when ''r'' = ''s'',
:<math>R(s,s) \leq (1 + o(1))\frac{4^{s-1}}{\sqrt{\pi s}}.</math>
An exponential lower bound,
:<math>R(s,s) \geq (1 + o(1)) \frac{s}{\sqrt{2} e} 2^{\frac{s}{2}},</math>
was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. There is obviously a huge gap between these two bounds: for example, for ''s'' = 10, this gives 101 ≤ ''R''(10,10) ≤ 48620. Nevertheless, exponential growth factors of either bound have not been improved to date and still stand at 4 and <math>\sqrt{2}</math> respectively. There is no known explicit construction producing an exponential lower bound. The best known lower and upper bounds for diagonal Ramsey numbers currently stand at
:<math>(1 + o(1)) \frac{\sqrt{2} s}{e} 2^{\frac{s}{2}} \leq R(s,s) \leq s^{-\frac{c \log s}{\log \log s}} 4^{s},</math>
due to [[Joel Spencer|Spencer]] and Conlon respectively.

For the off-diagonal Ramsey numbers ''R''(3,''t''), it is known that they are of order <math>\tfrac{t^2}{\log t}</math>; this may be stated equivalently as saying that the smallest possible [[Independent set (graph theory)|independence number]] in an ''n''-vertex [[triangle-free graph]] is <math>\Theta(\sqrt{n\log n})</math>. The upper bound for ''R''(3,''t'') is given by [[Miklós Ajtai|Ajtai]], [[János Komlós (mathematician)|Komlós]], and [[Endre Szemerédi|Szemerédi]], the lower bound by [[Jeong Han Kim|Kim]]. More generally, for off-diagonal Ramsey numbers ''R''(''s'', ''t'') with ''s'' fixed and ''t'' growing, the best known bounds are
:<math> c'_s \frac{t^{\frac{s+1}{2}}}{(\log t)^{\frac{s+1}{2} - \frac{1}{s-2}}} \leq R(s,t) \leq c_s \frac{t^{s-1}}{(\log t)^{s-2}},</math>
due to Bohman and Keevash and [[Miklós Ajtai|Ajtai]], [[János Komlós (mathematician)|Komlós]] and [[Endre Szemerédi|Szemerédi]] respectively.

==A multicolour example: ''R''(3,3,3) = 17==
[[Image:Sixteens.gif|right|thumb|The only two 3-colourings of K16 with no monochromatic K3. The untwisted colouring (top) and the twisted colouring (bottom).]]

A multicolour Ramsey number is a Ramsey number using 3 or more colours. There is only one nontrivial multicolour Ramsey number for which the exact value is known, namely ''R''(3,3,3) = 17.

Suppose that you have an edge colouring of a complete graph using 3 colours, red, yellow and green. Suppose further that the edge colouring has no monochromatic triangles. Select a vertex ''v''. Consider the set of vertices that have a green edge to the vertex ''v''. This is called the green neighborhood of ''v''. The green neighborhood of ''v'' cannot contain any green edges, since otherwise there would be a green triangle consisting of the two endpoints of that green edge and the vertex ''v''. Thus, the induced edge colouring on the green neighborhood of ''v'' has edges coloured with only two colours, namely yellow and red. Since ''R''(3,3) = 6, the green neighborhood of ''v'' can contain at most 5 vertices. Similarly, the red and yellow neighborhoods of ''v'' can contain at most 5 vertices each. Since every vertex, except for ''v'' itself, is in one of the green, red or yellow neighborhoods of ''v'', the entire complete graph can have at most 1 + 5 + 5 + 5 = 16 vertices. Thus, we have ''R''(3,3,3) ≤ 17.

To see that ''R''(3,3,3) ≥ 17, it suffices to draw an edge colouring on the complete graph on 16 vertices with 3 colours that avoids monochromatic triangles. It turns out that there are exactly two such colourings on ''K''16, the so-called untwisted and twisted colourings. Both colourings are shown in the figure to the right, with the untwisted colouring on the top, and the twisted colouring on the bottom. In both colourings in the figure, note that the vertices are labeled, and that the vertices ''v''11 through ''v''15 are drawn twice, on both the left and the right, in order to simplify the drawings.

Thus, ''R''(3,3,3) = 17.

[[Image:Clebsch graph.svg|thumb|right|[[Clebsch graph]]]]If you select any colour of either the untwisted or twisted colouring on ''K''16, and consider the graph whose edges are precisely those edges that have the specified colour, you will get the [[Clebsch graph]].

It is known that there are exactly two edge colourings with 3 colours on ''K''15 that avoid monochromatic triangles, which can be constructed by deleting any vertex from the untwisted and twisted colourings on ''K''16, respectively.

It is also known that there are exactly 115 edge colourings with 3 colours on ''K''14 that avoid monochromatic triangles, provided that we consider edge colourings that differ by a permutation of the colours as being the same.

==Extensions of the theorem==
The theorem can also be extended to [[hypergraph]]s. An ''m''-hypergraph is a graph whose "edges" are sets of ''m'' vertices – in a normal graph an edge is a set of 2 vertices. The full statement of Ramsey's theorem for hypergraphs is that for any integers ''m'' and ''c'', and any integers ''n''1,...,''n''''c'', there is an integer ''R''(''n''1, ..., ''n''''c'';''c'',''m'') such that if the hyperedges of a complete ''m''-hypergraph of order ''R''(''n''1,...,''n''''c'';''c'',''m'') are coloured with ''c'' different colours, then for some ''i'' between 1 and ''c'', the hypergraph must contain a complete sub-''m''-hypergraph of order ''n''''i'' whose hyperedges are all colour ''i''. This theorem is usually proved by induction on ''m'', the 'hyper-ness' of the graph. The base case for the proof is ''m'' = 2, which is exactly the theorem above.

==Infinite Ramsey theorem==
A further result, also commonly called ''Ramsey's theorem'', applies to infinite graphs. In a context where finite graphs are also being discussed it is often called the "Infinite Ramsey theorem". As intuition provided by the pictorial representation of a graph is diminished when moving from finite to infinite graphs, theorems in this area are usually phrased in [[set theory|set-theoretic]] terminology.

<blockquote>'''Theorem.''' Let ''X'' be some [[countable set|countably]] infinite set and colour the elements of ''X''(''n'') (the subsets of ''X'' of size ''n'') in ''c'' different colours. Then there exists some infinite subset ''M'' of ''X'' such that the size ''n'' subsets of ''M'' all have the same colour.</blockquote>

'''Proof''': The proof is given for ''c'' = 2. It is easy to prove the theorem for an arbitrary number of colours using a 'colour-blindness' argument as above. The proof is by (complete) induction on ''n'', the size of the subsets. For ''n'' = 1,the statement is equivalent to saying that if you split an infinite set into two sets, one of them is infinite. This is evident. Assuming the theorem is true for ''n'' ≤ ''r'', we prove it for ''n'' = ''r'' + 1. Given a 2-colouring of the (''r'' + 1)-element subsets of ''X'', let ''a''0 be an element of ''X'' and let ''Y'' = ''X''\{''a''0}. We then induce a 2-colouring of the ''r''-element subsets of ''Y'', by just adding ''a''0 to each ''r''-element subset (to get an (''r'' + 1)-element subset of ''X''). By the induction hypothesis, there exists an infinite subset ''Y''1 within ''Y'' such that every ''r''-element subset of ''Y''1 is coloured the same colour in the induced colouring. Thus there is an element ''a''0 and an infinite subset ''Y''1 such that all the (''r'' + 1)-element subsets of ''X'' consisting of ''a''0 and ''r'' elements of ''Y''1 have the same colour. By the same argument, there is an element ''a''1 in ''Y''1 and an infinite subset ''Y''2 of ''Y''1 with the same properties. Inductively, we obtain a sequence {''a''0,''a''1,''a''2,...} such that the colour of each (''r'' + 1)-element subset (''a''''i''(1),''a''''i''(2),...,''a''''i''(''r'' + 1)) with ''i''(1) < ''i''(2) < ... < ''i''(''r'' + 1) depends only on the value of ''i''(1). Further, there are infinitely many values of ''i''(''n'') such that this colour will be the same. Take these ''a''''i''(''n'')'s to get the desired monochromatic set.

==Infinite version implies the finite==
It is possible to deduce the finite Ramsey theorem from the infinite version by a [[proof by contradiction]]. Suppose the finite Ramsey theorem is false. Then there exist integers ''c'', ''n'', ''T'' such that for every integer ''k'', there exists a ''c''-colouring of <math>[k]^{(n)}</math> without a monochromatic set of size ''T''. Let ''Ck'' denote the ''c''-colourings of <math>[k]^{(n)}</math> without a monochromatic set of size ''T''.

For any ''k'', the restriction of a colouring in ''Ck+1'' to <math>[k]^{(n)}</math> (by ignoring the colour of all sets containing ''k''+1) is a colouring in ''Ck''. Define <math>C^{1}_k</math> to be the colourings in ''Ck'' which are restrictions of colourings in ''Ck+1''. Since ''Ck+1'' is not empty, neither is <math>C^{1}_k</math>.

Similarly, the restriction of any colouring in <math>C^{1}_{k+1}</math> is in <math>C^{1}_k</math>, allowing one to define <math>C^2_k</math> as the set of all such restrictions, a non-empty set. Continuing so, define <math>C^{m}_k</math> for all integers ''m'', ''k''.

Now, for any integer ''k'', <math>C_k\supseteq C^1_k\supseteq C^2_k\supseteq \dots</math>, and each set is non-empty. Furthermore, ''Ck'' is finite as <math>|C_k|\le c^{\frac{k!}{n!(k-n)!}}</math>. It follows that the intersection of all of these sets is non-empty, and let <math>D_k=C_k\cap C^1_k\cap C^2_k\cap \dots</math>. Then every colouring in ''Dk'' is the restriction of a colouring in ''Dk+1''. Therefore, by unrestricting a colouring in ''Dk'' to a colouring in ''Dk+1'', and continuing doing so, one constructs a colouring of <math>\mathbb N^{(n)}</math> without any monochromatic set of size ''T''. This contradicts the infinite Ramsey theorem.

If a suitable topological viewpoint is taken, this argument becomes a standard [[compactness theorem|compactness argument]] showing that the infinite version of the theorem implies the finite version.

==Directed graph Ramsey numbers==
It is also possible to define Ramsey numbers for ''directed'' graphs. (These were introduced by [[Paul Erdős|P. Erdős]] & L. Moser.) Let ''R''(''n'') be the smallest number ''Q'' such that any complete graph with singly directed arcs (also called a "tournament") and with ≥ ''Q'' nodes contains an acyclic (also called "transitive") ''n''-node subtournament.

This is the directed-graph analogue of what (above) has been called ''R''(''n'',''n'';2), the smallest number ''Z'' such that any 2-colouring of the edges of a complete ''un''directed graph with ≥ ''Z'' nodes, contains a monochromatic complete graph on n nodes. (The directed analogue of the two possible arc ''colours'' is the two ''directions'' of the arcs, the analogue of "monochromatic" is "all arc-arrows point the same way," i.e. "acyclic.")

Indeed many {{Who|date=January 2010}} find the directed graph problem to actually be ''more'' elegant than the undirected one. We have ''R''(0)=0, ''R''(1)=1, ''R''(2)=2, ''R''(3)=4, ''R''(4)=8, ''R''(5)=14, ''R''(6)=28, 32≤''R''(7)≤55, and ''R''(8) is again a problem you do not want powerful aliens to pose.

==See also==
*[[Paris–Harrington theorem]]
*[[Sim (pencil game)]]
*[[Infinite Ramsey theory]]
*[[Van der Waerden number]]

==Notes==
{{reflist}}

==References==
*{{citation
| last1 = Ajtai | first1 = Miklós | author1-link = Miklós Ajtai
| last2 = Komlós | first2 = János | author2-link = János Komlós (mathematician)
| last3 = Szemerédi | first3 = Endre | author3-link = Endre Szemerédi
| journal = J. Combin. Theory Ser. A
| pages = 354–360
| title = A note on Ramsey numbers
| volume = 29
| year = 1980
| doi = 10.1016/0097-3165(80)90030-8
| issue = 3}}.
*{{citation
| last1 = Bohman | first1 = Tom
| last2 = Keevash | first2 = Peter
| journal = Invent. Math.
| pages = 291–336
| title = The early evolution of the H-free process
| volume = 181
| year = 2010
| doi = 10.1007/s00222-010-0247-x
| issue = 2}}.
*{{citation
| last = Conlon | first = D.
| journal = Annals of Mathematics
| pages = 941–960
| title = A new upper bound for diagonal Ramsey numbers
| volume = 170
| year = 2009
| doi = 10.4007/annals.2009.170.941
| issue = 2}}.
* {{cite journal| first1=Norman | last1=Do | title= Party problems and Ramsey theory
| journal= Austr. Math. Soc. Gazette | year=2006 | volume=33 | number=5 | pages=306-312
| url=http://www.austms.org.au/Publ/Gazette/2006/Nov06/Nov06.pdf#page=48
}}
*{{citation
| last = Erdős | first = Paul | author-link = Paul Erdős
| journal = Bull. Amer. Math. Soc.
| pages = 292–294
| title = Some remarks on the theory of graphs
| volume = 53
| year = 1947
| doi = 10.1090/S0002-9904-1947-08785-1
| issue = 4}}.
*{{citation
| last1 = Erdős | first1 = Paul | author1-link = Paul Erdős
| last2 = Szekeres | first2 = George | author2-link = George Szekeres
| journal = [[Compositio Mathematica]]
| pages = 463–470
| title = A combinatorial problem in geometry
| volume = 2
| year = 1935
| doi = 10.1007/978-0-8176-4842-8_3
| isbn = 978-0-8176-4841-1}}.
*{{citation
| last = Exoo | first = G.
| journal = Journal of Graph Theory
| pages = 97–98
| title = A lower bound for R(5,5)
| volume = 13
| year = 1989
| doi = 10.1002/jgt.3190130113}}.
*{{citation
| last1 = Graham | first1 = R. | author1-link = Ronald Graham
| last2 = Rothschild | first2 = B.
| last3 = Spencer | first3 = J. H. | author3-link = Joel Spencer
| location = New York
| publisher = John Wiley and Sons
| title = Ramsey Theory
| year = 1990}}.
*{{citation
| doi =10.1112/plms/s2-30.1.264
| last = Ramsey | first = F. P. | author-link = Frank P. Ramsey
| journal = Proceedings of the London Mathematical Society
| pages = 264–286
| title = On a problem of formal logic
| volume = 30
| year = 1930}}.
*{{citation
| last = Spencer | first = J. |author-link = Joel H. Spencer
| journal = J. Combin. Theory Ser. A
| pages = 108–115
| title = Ramsey's theorem – a new lower bound
| volume = 18
| year = 1975
| doi = 10.1016/0097-3165(75)90071-0}}.
*{{citation
| last1 = Bian | first1 = Zhengbing
| last2 = Chudak | first2 = Fabian
| last3 = Macready | first3 = William G.
| last4 = Clark | first4 = Lane
| last5 = Gaitan | first5 = Frank
| title = Experimental determination of Ramsey numbers
| journal = Physical Review Letters
| volume = 111
| pages = 130505-1 - 130505-6
| year = 2013
| doi = 10.1103/PhysRevLett.111.130505}}.

==External links==
{{Wikibooks|Combinatorics|Bounds for Ramsey numbers|Ramsey numbers}}
* {{springer|title=Ramsey theorem|id=p/r077240}}
* [http://www.ramseyathome.com/ramsey/ Ramsey@Home] is a [[distributed computing]] project designed to find new lower bounds for various Ramsey numbers using a host of different techniques.
* [http://www.combinatorics.org/Surveys/ds1/sur.pdf Radziszowski's survey of small Ramsey numbers]
* this [http://www.cs.bham.ac.uk/~mmk/demos/ramsey-upper-limit.lisp lisp code] computes upper bounds for particular Ramsey numbers given the general rules in Radziszowski
* [http://RangeVoting.org/PuzzRamsey.html Survey of directed-graph Ramsey numbers]
* [http://mathworld.wolfram.com/RamseyNumber.html Ramsey Number – from MathWorld] (contains lower and upper bounds up to R(19,19))
* [http://ginger.indstate.edu/ge/RAMSEY/index.html Ramsey Number – Geoffrey Exoo] (Contains R(5,5) > 42 counter-proof)
* [http://www.math.sinica.edu.tw/post-doctor/cariolaro/r36.pdf A proof that R(3,6) = 18]

[[Category:Ramsey theory]]
[[Category:Theorems in graph theory]]
[[Category:Articles containing proofs]]

Kakutani fixed-point theorem

2014-01-23T17:35:52Z

131.215.220.166: /* Infinite dimensional generalizations */ (m) add hyphens

In [[mathematics]], the Fourier '''sine and cosine transforms''' are forms of the [[Fourier transform|Fourier integral transform]] that do not use [[complex number]]s. They are the forms originally used by [[Joseph Fourier]] and are still preferred in some applications, such as [[signal processing]] or [[statistics]].

==Definition==

The '''Fourier sine transform''' of <math> f (t) </math>, sometimes denoted by either <math> {\hat f}^s </math> or <math> {\mathcal F}_s (f) </math>, is

:<math> 2 \int\limits_{-\infty}^\infty f(t)\sin\,{2\pi \nu t} \,dt.</math>

If <math> t </math> means time, then <math> \nu</math> is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other.

This transform is necessarily an [[odd function]] of frequency, i.e.,
:<math> {\hat f}^s(\nu) = - {\hat f}^s(-\nu) </math> for all <math>\nu</math>.

The numerical factors in the [[Fourier transform]]s are defined uniquely only by their product. Here, in order that the Fourier inversion formula not have any numerical factor, the factor of 2 appears because the sine function has <math> L^2 </math> norm of <math> \frac 1 {\sqrt2} </math>.

The '''Fourier cosine transform''' of <math> f (t) </math>, sometimes denoted by either <math> {\hat f}^c </math> or <math> {\mathcal F}_c (f) </math>, is

:<math> 2 \int\limits_{-\infty}^\infty f(t)\cos\,{2\pi \nu t} \,dt.</math>

It is necessarily an [[even function]] of <math>\nu</math>, i.e.,
<math> {\hat f}^c(\nu) = {\hat f}^c(-\nu) </math> for all <math>\nu</math>.

Some authors<ref>[[Mary L. Boas]], ''[[Mathematical Methods in the Physical Sciences]]'', 2nd Ed, John Wiley & Sons Inc, 1983. ISBN 0-471-04409-1</ref> only define the cosine transform for [[even function]]s of <math>t </math>, in which case its sine transform is zero. Since cosine is also even, a simpler formula can be used, <math> 4 \int\limits_0^\infty f(t)\cos\,{2\pi \nu t} \,dt.</math> Similarly, if <math>f</math> is an [[odd function]], then the cosine transform is zero and the sine transform can be simplified to <math> 4 \int\limits_0^\infty f(t)\sin\,{2\pi \nu t} \,dt.</math>

==Fourier inversion==
The original function <math> f(t) </math> can be recovered from its transforms under the usual hypotheses, that <math> f </math> and both of its transforms should be absolutely integrable. For more details on the different hypotheses, see [[Fourier inversion theorem]].

The inversion formula is<ref>{{cite book|last=[[Poincaré]]|first=Henri|title=Theorie analytique de la propagation de chaleur|year=1895|publisher=G. Carré|location=Paris|pages=pp. 108ff.|url=http://gallica.bnf.fr/ark:/12148/bpt6k5500702f/f115.image}}</ref>

:<math> f(t) = \int _0^\infty {\hat f}^c \cos (2\pi \nu t) d\nu + \int _0^\infty {\hat f}^s \sin (2\pi \nu t) d\nu,</math>

which has the advantage that all frequencies are positive and all quantities are real. If the numerical factor 2 is left out of the definitions of the transforms, then the inversion formula is usually written as an integral over both negative and positive frequencies.

Using the addition formula for [[cosine]], this is sometimes rewritten as

:<math> \frac\pi2 (f(x+0)+f(x-0)) = \int _0^\infty \int_{-\infty}^\infty \cos \omega (t-x) f(t) dt d\omega, </math>

where <math> f(x+0) </math> denotes the one-sided [[limit]] of <math>f</math> as <math> x </math> approaches zero from above, and
<math> f(x-0) </math> denotes the one-sided limit of <math>f</math> as <math> x </math> approaches zero from below.

If the original function <math> f</math> is an [[even function]], then the sine transform is zero; if <math> f</math> is an [[odd function]], then the cosine transform is zero. In either case, the inversion formula simplifies.

==Relation with complex exponentials==

The form of the [[Fourier transform]] used more often today is

:<math>
\hat f(\nu)
= \int\limits_{-\infty}^\infty f(t) e^{-2\pi i\nu t}\,dt.
</math>

Expanding the [[integrand]] by means of [[Euler's formula]] results in

:<math> = \int\limits_{-\infty}^\infty f(t)(\cos\,{2\pi\nu t} - i\,\sin{2\pi\nu t})\,dt,</math>

which may be written as the [[sum]] of two [[integral]]s

:<math> = \int\limits_{-\infty}^\infty f(t)\cos\,{2\pi \nu t} \,dt - i \int\limits_{-\infty}^\infty f(t)\sin\,{2\pi \nu t}\,dt,</math>

:<math> = \frac 12 {\hat f}^c (\nu) - \frac i2 {\hat f}^s (\nu). </math>

==See also==
*[[Discrete cosine transform]]
*[[Discrete sine transform]]

==References==

* Whittaker, Edmund, and James Watson, ''A Course in Modern Analysis'', Fourth Edition, Cambridge Univ. Press, 1927, pp. 189, 211

<references />

[[Category:Integral transforms]]
[[Category:Fourier analysis]]

Gentzen's consistency proof

2014-01-19T20:27:27Z

131.215.220.166: typo

{{redirect3|Exponential type|For exponential types in type theory and programming languages, see [[Function type]]}}

In [[mathematics]], in the area of [[complex analysis]], '''Nachbin's theorem''' (named after [[Leopoldo Nachbin]]) is commonly used to establish a bound on the growth rates for an [[analytic function]]. This article will provide a brief review of growth rates, including the idea of a '''function of exponential type'''. Classification of growth rates based on type help provide a finer tool than [[big O notation|big O]] or [[Landau notation]], since a number of theorems about the analytic structure of the bounded function and its [[integral transform]]s can be stated. In particular, Nachbin's theorem may be used to give the domain of convergence of the '''generalized Borel transform''', given below.

==Exponential type==
{{Main|Exponential type}}
A function ''f''(''z'') defined on the [[complex plane]] is said to be of exponential type if there exist constants ''M'' and τ such that

:<math>|f(re^{i\theta})|\le Me^{\tau r}</math>

in the limit of <math>r\to\infty</math>. Here, the [[complex variable]] ''z'' was written as <math>z=re^{i\theta}</math> to emphasize that the limit must hold in all directions θ. Letting τ stand for the [[infimum]] of all such τ, one then says that the function ''f'' is of ''exponential type τ''.

For example, let <math>f(z)=\sin(\pi z)</math>. Then one says that <math>\sin(\pi z)</math> is of exponential type π, since π is the smallest number that bounds the growth of <math>\sin(\pi z)</math> along the imaginary axis. So, for this example, [[Carlson's theorem]] cannot apply, as it requires functions of exponential type less than π.

==Ψ type==
Bounding may be defined for other functions besides the exponential function. In general, a function <math>\Psi(t)</math> is a '''comparison function''' if it has a series

:<math>\Psi(t)=\sum_{n=0}^\infty \Psi_n t^n</math>

with <math>\Psi_n>0</math> for all ''n'', and

:<math>\lim_{n\to\infty} \frac{\Psi_{n+1}}{\Psi_n} = 0.</math>

Note that comparison functions are necessarily [[entire function|entire]], which follows from the [[ratio test]]. If <math>\Psi(t)</math> is such a comparison function, one then says that ''f'' is of Ψ-type if there exist constants ''M'' and ''τ'' such that

:<math>\left|f\left(re^{i\theta}\right)\right| \le M\Psi(\tau r)</math>

as <math>r\to \infty</math>. If τ is the infimum of all such ''τ'' one says that ''f'' is of Ψ-type ''τ''.

==Nachbin's theorem==
Nachbin's theorem states that a function ''f''(''z'') with the series

:<math>f(z)=\sum_{n=0}^\infty f_n z^n</math>

is of Ψ-type τ if and only if

:<math>\limsup_{n\to\infty} \left| \frac{f_n}{\Psi_n} \right|^{1/n} = \tau.</math>

==Borel transform==
Nachbin's theorem has immediate applications in [[Cauchy's integral formula|Cauchy theorem]]-like situations, and for [[integral transforms]]. For example, the '''generalized Borel transform''' is given by

:<math>F(w)=\sum_{n=0}^\infty \frac{f_n}{\Psi_n w^{n+1}}.</math>

If ''f'' is of Ψ-type ''τ'', then the exterior of the domain of convergence of <math>F(w)</math>, and all of its singular points, are contained within the disk

:<math>|w| \le \tau.</math>

Furthermore, one has

:<math>f(z)=\frac{1}{2\pi i} \oint_\gamma \Psi (zw) F(w)\, dw</math>

where the [[contour of integration]] γ encircles the disk <math>|w| \le \tau</math>. This generalizes the usual '''Borel transform''' for exponential type, where <math>\Psi(t)=e^t</math>. The integral form for the generalized Borel transform follows as well. Let <math>\alpha(t)</math> be a function whose first derivative is bounded on the interval <math>[0,\infty)</math>, so that

:<math>\frac{1}{\Psi_n} = \int_0^\infty t^n\, d\alpha(t)</math>

where <math>d\alpha(t)=\alpha^{\prime}(t)\,dt</math>. Then the integral form of the generalized Borel transform is

:<math>F(w)=\frac{1}{w} \int_0^\infty f \left(\frac{t}{w}\right) \, d\alpha(t).</math>

The ordinary Borel transform is regained by setting <math>\alpha(t)=e^{-t}</math>. Note that the integral form of the Borel transform is just the [[Laplace transform]].

==Nachbin resummation==
Nachbin resummation (generalized Borel transform) can be used to sum divergent series that escape to the usual [[Borel resummation]] or even to solve (asymptotically) integral equations of the form:

:<math> g(s)=s\int_0^\infty K(st) f(t)\,dt </math>

where ''f''(''t'') may or may not be of exponential growth and the kernel ''K''(''u'') has a [[Mellin transform]]. The solution, pointed out by L. Nachbin himself, can be obtained as <math> f(x)= \sum_{n=0}^\infty \frac{a_n}{M(n+1)}x^n </math> with <math> g(s)= \sum_{n=0}^\infty a_n s^{-n} </math> and ''M''(''n'') is the Mellin transform of ''K''(''u''). an example of this is the Gram series <math> \pi (x) \approx \sum_{n=1}^{\infty} \frac{\log^{n}(x)}{n\cdot n!\zeta (n+1)} </math>

==Fréchet space==
Collections of functions of exponential type <math>\tau</math> can form a [[complete space|complete]] [[uniform space]], namely a [[Fréchet space]], by the [[topological space|topology]] induced by the countable family of [[norm (mathematics)|norm]]s

:<math> \|f\|_{n} = \sup_{z \in \mathbb{C}} \exp \left[-\left(\tau + \frac{1}{n}\right)|z|\right]|f(z)| </math>

==See also==
* [[Divergent series]]
* [[Borel summation]]
* [[Euler summation]]
* [[Cesàro summation]]
* [[Lambert summation]]
* [[Nachbin resummation]]
* [[Phragmén–Lindelöf principle]]
* [[Abelian and tauberian theorems]]
* [[Van Wijngaarden transformation]]

==References==
* L. Nachbin, "An extension of the notion of integral functions of the finite exponential type", ''Anais Acad. Brasil. Ciencias.'' '''16''' (1944) 143–147.
* Ralph P. Boas, Jr. and R. Creighton Buck, ''Polynomial Expansions of Analytic Functions (Second Printing Corrected)'', (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. ''(Provides a statement and proof of Nachbin's theorem, as well as a general review of this topic.)''
* {{springer|author=A.F. Leont'ev|id=F/f041990|title=Function of exponential type}}
* {{springer|author=A.F. Leont'ev|id=B/b017190|title= Borel transform}}
* Garcia J. Borel Resummation & the Solution of Integral Equations '' Prespacetime Journal '' nº 4 Vol 4. 2013 http://prespacetime.com/index.php/pst/issue/view/42/showToc

[[Category:Integral transforms]]
[[Category:Theorems in complex analysis]]
[[Category:Summability methods]]

System size expansion

2014-01-16T01:27:24Z

131.215.142.224:

== Summary ==
{{Information
|Description = An illustration of a uniformly tended division of a curve. The term <math>C_\beta ^{(i)} </math> denotes the <math>i^{th}</math> sub-curve, whose tendency is <math>\beta </math>. The tendency indicator vectors of the uniformly tended curves are:
:<math> \vec{s}(C_{+1}^{(1)}) = [+1,-1,-1,+1], </math>
:<math> \vec{s}(C_{-1}^{(2)}) = [-1,+1,-1,+1], </math>
:<math> \vec{s}(C_{-1}^{(3)}) = [-1,+1,-1,+1], </math>
:<math> \vec{s}(C_{0}^{(4)}) = [0,0,+1,-1], </math>
:<math> \vec{s}(C_{0}^{(5)}) = [+1,-1,0,0], </math>
:<math> \vec{s}(C_{-1}^{(6)}) = [+1,-1,+1,-1]. </math>
</math>
|Source = I ([[User:Amiruchka|Amiruchka]] ([[User talk:Amiruchka|talk]])) created this work entirely by myself.
|Date = 14:36, 1 December 2010 (UTC)
|Author = [[User:Amiruchka|Amiruchka]] ([[User talk:Amiruchka|talk]])
|other_versions =
}}

== Licensing ==
{{self|cc-by-sa-3.0|GFDL|migration=redundant}}

{{Orphan image}}

{{Copy to Wikimedia Commons|bot=Fbot}}

Supersymmetric gauge theory

2014-01-02T05:51:55Z

131.215.220.166: /* Theories with 8 or more SUSY generators */

In [[theoretical physics]], one often analyzes theories with [[supersymmetry]] in which '''superfields''' play a very important role. A superfield is a function defined in [[superspace]] which properly packages the various fields of a [[supermultiplet]], namely, the array of [[fermion]] and [[boson]] fields related among themselves by [[supersymmetry]]. Superfields were introduced by [[Abdus Salam]] and John Strathdee in 1979.<ref>{{cite doi|10.1016/0550-3213(74)90537-9|noedit}}</ref>

==Overview==
In four dimensions, the simplest example—namely, the minimal ''N = 1'' supersymmetry—may be written as a vector in a [[superspace]] with four extra fermionic coordinates <math>\theta^1,\theta^2,\bar\theta^1,\bar\theta^2</math>, transforming as a two-component [[spinor]] and its conjugate. More generally, there are 4N extra fermionic ([[Grassmann number]]) coordinates .

A more coordinate-free description of the superspace is that it's the [[quotient space]] of the [[super-Poincaré group]] divided by the [[Lorentz group]].

Every superfield, i.e. a field that depends on all coordinates of the superspace (or in other words, an element of a [[Module (mathematics)|module]] of the algebra of functions over superspace), may be expanded with respect to the new fermionic coordinates. There exists a special kind of superfields, the so-called ''chiral superfields'', that, in the chiral representation of supersymmetry, depend only on the variables ''θ'' but not their conjugates. The last term in the corresponding expansion, namely <math>F \theta^1\theta^2</math>, is called the ''F-term''. Other superfields include [[vector superfield]]s.

There also exist superfields in theories with larger supersymmetry.

Manifestly supersymmetric actions may also be written as [[integral]]s over the whole superspace. Some special terms, such as the [[superpotential]], may be written as integrals over ''θ''s only. They are also referred to as ''F''-terms, much like the terms in the ordinary potential that arise from these terms of the supersymmetric Lagrangian.

==References==
{{Reflist|30em}}

{{quantum-stub}}

[[Category:Supersymmetry]]
[[Category:Quantum field theory]]
[[Category:Theoretical physics]]

Optical ring resonators

2013-10-25T01:41:23Z

131.215.220.186: /* References */

{{Multiple issues|unreferenced = December 2009|technical = March 2012}}

A '''sensor array''' is a group of sensors deployed in a certain geometry pattern. The advantage of using a sensor array over using a single sensor lies in the factor that an array can increase the [[antenna gain]] in the direction of the signal while decreasing the gain in the directions of noise and interferences. In other words, sensor arrays can increase [[signal-to-noise ratio]] ('''SNR'''): magnify the signal while suppressing the noise. Another ability of sensor array is that is can detect the direction and distance of impinging signal sources. The technology to achieve this is called '''Array Signal Processing'''. Application examples of array signal processing include [[RADAR]]/[[SONAR]], wireless communications, [[seismology]], machine condition monitoring and [[fault diagnosis]], etc.

Using array signal processing, the temporal and spatial properties (or parameters) of the impinging signals contaminated with noise and hidden in the data collected by the sensor array can be estimated and revealed. This is known as [[parameter estimation]].

[[File:Illustration linear sensor array and incident angle.jpg|frame|Figure 1: Linear array and incident angle]]

Figure 1 illustrates a six-element uniform linear array. In this example, the impinging signal is assumed far-field so that it can be treated as planar wave.

Parameter estimation takes advantage of the fact that the distance from the source to each microphone in the array is different, which means that the signal recorded by each microphone will be phase-shifted replicas of each other. Eq. (1) shows the calculation for the extra time it takes to reach each microphone in the array relative to the array center, where c is the sound speed.

<math>\Delta t_i = \frac{(i-1)d \cos \theta}{c}, i = 1, 2, ..., M......(1) </math>

Each sensor is associated with a different delay. Although the delays are small but not trivial. In frequency domain, the delays display as phase shift among the signals received by the sensors. The delays are closely related to the incident angle and the geometry of the sensor array. Given the geometry of the array, the delays or phase differences can be used to estimate the incident angle. This {{clarify|date=August 2012}} the mathematical basis behind the array signal processing. Simply summing the signals received by the sensors and calculating the mean value give the following result:

<math>y = \frac{1}{M}\sum_{i=1}^{M}[x(t-\Delta t_i)]......(2) </math>

Because the received signals are out of phase, this mean value does not give an enhanced signal compared with the original source. Heuristically, if we can find weights multiplying to the received signals to make them in phase before summing them together, the mean value will give an enhanced signal:

<math>y = \frac{1}{M}\sum_{i=1}^{M}[w_i*x(t-\Delta t_i)]......(3) </math>

The process of multiplying a well selected set of weights to the signals received by the sensor array so that the signal is added constructively while suppressing the noise is called '''[[beamforming]]'''.
There are a variety of beamforming algorithms for sensor arrays, such as The delay-and-sum approach, spectral based (non-parametric) approaches and parametric approaches. These beamforming algorithms are briefly described as follows.

==Delay-and-Sum Beamforming==

If a time delay is added to the recorded signal from each microphone that is equal and opposite of the delay caused by the extra travel time, it will result in signals that are perfectly in-phase with each other. Summing these in-phase signals will result in constructive interference that will amplify the result by the number of microphones in the array. This is known as delay-and-sum beamforming. For DOA (direction of arrival) estimation, one can iteratively test time delays for all possible directions. If the guess is wrong, the signal will destructively interfere, resulting in a diminished output signal, but the correct guess will result in the signal amplification described above.

The problem is, before the incident angle is estimated, how could it be possible to a time delay that is 'equal' and opposite of the delay caused by the extra travel time? It is impossible. The solution is to try a series of angles <math>\hat{\theta} \in [0, \pi]</math> at sufficiently high resolution, and calculate the resulted mean output signal of the array using Eq. (3). The trial angle that maximizes the mean output is an estimation of DOA given by the delay-and-sum beamformer.
Adding an opposite delays to the input signals is equivalent to physically turning the sensor array. Therefore, it is also known as '''beam steering'''.

==Spectrum-Based Beamforming==

Delay and sum beamforming is a time domain approach. It is simple to implement, but it may poorly estimate direction of arrival (DOA): if the signal is contaminated with strong noise, there may be practical difficulty in implementing the algorithm. The solution to this is a frequency domain approach. The [[Fourier transform]] transforms the time domain signal to the frequency domain. This converts the time delays between adjacent sensors into phase shifts. Thus, the array output vector at any time ''t'' can be denoted as <math>x(t) = x_1(t)\begin{bmatrix} 1 & e^{-j\omega\Delta t} & \cdots & e^{-j\omega(M-1)\Delta t} \end{bmatrix}^T </math>, where <math>x_1(t)</math> stands for the signal received by the first sensor. Frequency domain beamforming algorithms use the spatial covariance matrix, represented by <math>R=E{x(t)x^T(t)}</math>. This ''M'' by ''M'' matrix carries the spatial and spectral information of the incoming signals. Assuming zero-mean Gaussian [[white noise]], the frequency signal-noise snapshot model of the spatial covariance matrix is given by

<math> R = VSV^H + \sigma^2I ......(4) </math>

where <math>\sigma^2 </math> is the variance of the white noise, I is the identity matrix and V is the array manifold vector: <math> V = \begin{bmatrix} 1 & e^{-j\omega\Delta t} & \cdots & e^{-j\omega(M-1)\Delta t} \end{bmatrix}^T </math>. This model is of central use in frequency domain beamforming algorithms.

Some spectrum-based beamforming approaches are listed below.

=== Conventional (Barlett) Beamformer ===

The Barlett beamformer is a natural extension of conventional spectral analysis ([[spectrogram]]) to the sensor array. Its spectral power is represented by

<math> \hat{P}_{Barlett}(\theta)=V^HRV ......(5) </math>

The angle that maximize this power is an estimation of the angle of arrival.

=== MVDR (Capon) Beamformer ===

The Minimum Variance Distortionless Response beamformer, also known as the Capon beamforming algorithm, has a power is given by

<math> \hat{P}_{Capon}(\theta)=\frac{1}{V^HR^{-1}V} ......(6) </math>

Though the MVDR/Capon beamformer can achieve better resolution than the conventional/Bartlett approach, its algorithm is much more computationally intensive due to the full-rank matrix inversion. This said, advancements in [[General-purpose computing on graphics processing units|GPU computing]] have begun to narrow this gap and make real-time Capon beamforming possible.<ref>http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6689777&sortType%3Dasc_p_Sequence%26filter%3DAND%28p_IS_Number%3A6689765%29</ref>

=== MUSIC Beamformer ===

MUSIC ([[Multiple signal classification|MUltiple SIgnal Classification]]) beamforming algorithm is derived from the Capon algorithm by decomposing the covariance matrix as given by Eq. (4) for both the signal part and the noise part. The eigen-decomposition of is represented by

<math> R = U_s\Lambda_s{U_s}{^H} + U_n\Lambda_n{U_n}^H ......(7) </math>

The MUSIC uses the noise sub-space of the spatial covariance matrix in the denominator of the Capon algorithm:

<math> \hat{P}_{MUSIC}(\theta)=\frac{1}{V^H(U_n{U_n}^H)V} ......(8) </math>

Therefore MUSIC beamformer is also known as subspace beamformer. Compared Capon beamformer, it gives much better DOA estimation while avoiding matrix inversion. Computational intensity is reduced significantly if the number of sensors (''M'') is large.

==Parametric Beamformers==

One of the major pros of the spectral based beamformers is their relatively lighter computational complexity, but they may not give accurate DOA estimation if the signals are correlated or coherent. An alternative approach is the parametric beamformers, also known as '''[[Maximum likelihood|Maximum Likelihood (ML)]]''' beamformers. One example of maximum likelihood method commonly used in engineering is [[least squares]] method. In the least square approach, a quadratic penalty function is used. To get the minimum value (or least square) of the quadratic penalty function (or [[objective function]]), take its derivative (which is linear) and let it equal zero, and solve linear equations.

In ML Beamformers, quadratic penalty functions are used to the spatial covariance matrix and the signal/noise model. One example of ML beamformer penalty function is:

<math>L_{ML}(\theta)=\sum_{i=1}^{N} \|\hat{R}-(VSV^H)\|^2 ......(9) </math>

where ''N'' is the number of snapshots and <math>\| \|^2 </math> is the Euclidean norm. It can be seen from Eq. (4) that minimizing the penalty function of Eq. (9) is to make the noise term as much as possible, or the signal model as close to the sample covariance matrix as much as possible. In other words, the Maximum Likelihood beamformer is to find the DOA <math>\theta</math>, the independent variable of vector '''V''', so that the penalty function such as Eq. (9) is minimized. In practice, the penalty functions used look different, depending on the signal/noise model employed, but they are the same in essence. For this reason, there are two major categories of maximum likelihood beamformers: Deterministic ML beamformers and Stochastic ML beamformers, corresponding to the deterministic noise model and the [[stochastic]] noise model, respectively.

Another factor to change the former of the penalty equation is the consideration of simplifying the minimization of the penalty function. Although the derivative of a quadratic function is linear, matrix operations and trigonometric operations make it highly non-linear. In order to simplify the [[optimization]] algorithm, logarithmic operation and [[Probability density function|probability density function (PDF)]] of the observations may be used in some ML beamformers.

In ML beamformer, the optimizing problem becomes finding the roots of the equation of zeroing the derivative of the penalty function. Because the equation is highly non-linear, usually a numerical searching approach, such as [[Newton-Raphson method]], is used. The Newton-Raphson method is an iterative root search method as given by

<math> x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}......(10)</math>

The search starts from an initial guess <math>x_0</math>. If the Newton-Raphson search method is employed in minimizing the penalty function in beamforming, the resulting beamformer is called Newton ML beamformer. Several best-known ML beamformers are described below without giving their formulas due to the complexity of expression.

* Deterministic Maximum Likelihood Beamformer
In Deterministic Maximum Likelihood Beamformer ('''DML'''), the noise is modeled as a stationary Gaussian white random processes while the signal waveform as deterministic (but arbitrary) and unknown.

* Stochastic Maximum Likelihood Beamformer
In Stochastic Maximum Likelihood Beamformer ('''SML'''), the noise are modeled as a stationary Gaussian white random processes (the same as in DML) whereas the signal waveform as Gaussian random processes.

* Method of Direction Estimation
Method of Direction Estimation ('''MODE''') is subspace maximum likelihood beamformer, just as '''MUSIC''' is the subspace spectral based beamformer. Subspace ML beamforming is obtained by [[Eigendecomposition of a matrix|eigen-decomposition]] of the sample covariance matrix.

==References==
* H. L. Van Trees, “Optimum array processing – Part IV of detection, estimation, and modulation theory”, John Wiley, 2002
* H. Krim and M. Viberg, “Two decades of array signal processing research”, IEEE Transactions on Signal Processing Magazine, July 1996
* S. Haykin, Ed., “Array Signal Processing”, Eaglewood Cliffs, NJ: Prentice-Hall, 1985
* S. U. Pillai, “Array Signal Processing”, New York: Springer-Verlag, 1989
* P. Stoica and R. Moses, “Introduction to Spectral Analysis", Prentice-Hall, Englewood Cliffs, USA, 1997. [http://user.it.uu.se/~ps/SAS-new.pdf available for download.]
* J. Li and P. Stoica, “Robust Adaptive Beamforming", John Wiley, 2006.
* J. Cadzow, “Multiple Source Location—The Signal Subspace Approach”, IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 38, No. 7, July 1990
* G. Bienvenu and L. Kopp, “Optimality of high resolution array processing using the eigensystem approach”, IEEE Transactions on Acoustics, Speech and Signal Process, Vol. ASSP-31, pp. 1234–1248, October 1983
* I. Ziskind and M. Wax, “Maximum likelihood localization of multiple sources by alternating projection”, IEEE Transactions on Acoustics, Speech and Signal Process, Vol. ASSP-36, pp. 1553–1560, October 1988
* B. Ottersten, M. Verberg, P. Stoica, and A. Nehorai, “Exact and large sample maximum likelihood techniques for parameter estimation and detection in array processing”, Radar Array Processing, Springer-Verlag, Berlin, pp. 99–151, 1993
* M. Viberg, B. Ottersten, and T. Kailath, “Detection and estimation in sensor arrays using weighted subspace fitting”, IEEE Transactions on Signal Processing, vol. SP-39, pp 2346–2449, November 1991
* M. Feder and E. Weinstein, “Parameter estimation of superimposed signals using the EM algorithm”, IEEE Transactions on Acoustic, Speech and Signal Proceeding, vol ASSP-36, pp. 447–489, April 1988
* Y. Bresler and Macovski, “Exact maximum likelihood parameter estimation of superimposed exponential signals in noise”, IEEE Transactions on Acoustic, Speech and Signal Proceeding, vol ASSP-34, pp. 1081–1089, October 1986
* R. O. Schmidt, “New mathematical tools in direction finding and spectral analysis”, Proceedings of SPIE 27th Annual Symposium, San Diego, California, August 1983
{{reflist}}

{{DEFAULTSORT:Sensor Array}}
[[Category:Sensors]]

Summation by parts

2013-10-11T19:06:34Z

131.215.220.166: /* Newton series */

In [[linear algebra]], a [[square matrix]] ''A'' is called '''diagonalizable''' if it is [[similar (linear algebra)|similar]] to a [[diagonal matrix]], i.e., if there exists an [[invertible matrix]] ''P'' such that ''P''−1''AP'' is a diagonal matrix. If ''V'' is a finite-[[dimension (linear algebra)|dimension]]al [[vector space]], then a [[linear operator|linear map]] ''T'' : ''V'' → ''V'' is called '''diagonalizable''' if there exists an [[Basis (linear algebra)#Ordered bases and coordinates|ordered basis]] of ''V'' with respect to which ''T'' is represented by a diagonal matrix. '''Diagonalization''' is the process of finding a corresponding [[diagonal matrix]] for a diagonalizable matrix or linear map.<ref>Horn & Johnson 1985</ref> A square matrix which is not diagonalizable is called ''[[Defective matrix|defective]].''

Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle: [[Transformation_matrix#Eigenbasis_and_diagonal_matrix|their eigenvalues and eigenvectors are known]] and one can raise a diagonal matrix to a power by simply raising the diagonal entries to that same power. Geometrically, a diagonalizable matrix is an '''[[inhomogeneous dilation]]''' (or ''anisotropic scaling'') — it [[Scaling (geometry)|scales]] the space, as does a ''[[homogeneous dilation]]'', but by a different factor in each direction, determined by the scale factors on each axis (diagonal entries).

== Characterization ==
The fundamental fact about diagonalizable maps and matrices is expressed by the following:

* An ''n''×''n'' matrix ''A'' over the [[field (mathematics)|field]] ''F'' is diagonalizable [[if and only if]] the sum of the [[dimension (linear algebra)|dimension]]s of its eigenspaces is equal to ''n'', which is the case if and only if there exists a [[basis (linear algebra)|basis]] of ''F''''n'' consisting of eigenvectors of ''A''. If such a basis has been found, one can form the matrix ''P'' having these basis vectors as columns, and ''P''−1''AP'' will be a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of ''A''.
* A linear map ''T'' : ''V'' → ''V'' is diagonalizable if and only if the sum of the [[dimension (linear algebra)|dimension]]s of its eigenspaces is equal to dim(''V''), which is the case if and only if there exists a basis of ''V'' consisting of eigenvectors of ''T''. With respect to such a basis, ''T'' will be represented by a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of ''T''.

Another characterization: A matrix or linear map is diagonalizable over the field ''F'' if and only if its [[minimal polynomial (linear algebra)|minimal polynomial]] is a product of distinct linear factors over ''F''. (Put in another way, a matrix is diagonalizable if and only if all of its [[elementary divisor]]s are linear.)

The following sufficient (but not necessary) condition is often useful.

* An ''n''×''n'' matrix ''A'' is diagonalizable over the field ''F'' if it has ''n'' distinct eigenvalues in ''F'', i.e. if its [[characteristic polynomial]] has ''n'' distinct roots in ''F''; however, the converse may be false. Let us consider
:: <math>\begin{bmatrix} -1 & 3 & -1 \\ -3 & 5 & -1 \\ -3 & 3 & 1 \end{bmatrix},</math>
: which has eigenvalues 1, 2, 2 (not all distinct) and is diagonalizable with diagonal form ( [[similar (linear algebra)|similar]] to ''A'')
:: <math>\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}</math>
: and [[change of basis|change of basis matrix]] ''P''
:: <math>\begin{bmatrix} 1 & 1 & -1 \\ 1 & 1 & 0 \\ 1 & 0 & 3 \end{bmatrix}.</math>
: The converse fails when ''A'' has an eigenspace of dimension higher than 1. In this example, the eigenspace of ''A'' associated with the eigenvalue 2 has dimension 2.

* A linear map ''T'' : ''V'' → ''V'' with ''n'' = dim(''V'') is diagonalizable if it has ''n'' distinct eigenvalues, i.e. if its characteristic polynomial has ''n'' distinct roots in ''F''.

Let ''A'' be a matrix over ''F''. If ''A'' is diagonalizable, then so is any power of it. Conversely, if ''A'' is invertible, ''F'' is algebraically closed, and ''An'' is diagonalizable for some ''n'' that is not an integer multiple of the characteristic of ''F'', then ''A'' is diagonalizable. Proof: If ''An'' is diagonalizable, then ''A'' is annihilated by some polynomial <math>(x^n - \lambda_1) \cdots (x^n - \lambda_k)</math>, which has no multiple root (since <math>\lambda_j \ne 0</math>) and is divided by the minimal polynomial of ''A''.

As a rule of thumb, over '''C''' almost every matrix is diagonalizable. More precisely: the set of complex ''n''×''n'' matrices that are ''not'' diagonalizable over '''C''', considered as a [[subset]] of '''C'''''n''×''n'', has [[Lebesgue measure]] zero. One can also say that the diagonalizable matrices form a dense subset with respect to the [[Zariski topology]]: the complement lies inside the set where the [[discriminant]] of the characteristic polynomial vanishes, which is a [[hypersurface]]. From that follows also density in the usual (''strong'') topology given by a [[norm (mathematics)|norm]]. The same is not true over '''R'''.

The [[Jordan–Chevalley decomposition]] expresses an operator as the sum of its semisimple (i.e., diagonalizable) part and its [[nilpotent]] part. Hence, a matrix is diagonalizable if and only if its nilpotent part is zero. Put in another way, a matrix is diagonalizable if each block in its [[Jordan form]] has no nilpotent part; i.e., each "block" is a one-by-one matrix.

== Diagonalization ==
If a matrix ''A'' can be diagonalized, that is,

:<math>P^{-1}AP=\begin{pmatrix}\lambda_{1}\\
& \lambda_{2}\\
& & \ddots\\
& & & \lambda_{n}\end{pmatrix},</math>

then:

:<math>AP=P\begin{pmatrix}\lambda_{1}\\
& \lambda_{2}\\
& & \ddots\\
& & & \lambda_{n}\end{pmatrix} .</math>

Writing ''P'' as a [[block matrix]] of its column vectors <math>\vec{\alpha}_{i}</math>

:<math>P=\begin{pmatrix}\vec{\alpha}_{1} & \vec{\alpha}_{2} & \cdots & \vec{\alpha}_{n}\end{pmatrix},</math>

the above equation can be rewritten as

:<math>A\vec{\alpha}_{i}=\lambda_{i}\vec{\alpha}_{i}\qquad(i=1,2,\cdots,n).</math>

So the column vectors of ''P'' are [[right eigenvector]]s of ''A'', and the corresponding diagonal entry is the corresponding [[eigenvalue]]. The invertibility of ''P'' also suggests that the eigenvectors are [[linearly independent]] and form a basis of ''F''''n''. This is the necessary and sufficient condition for diagonalizability and the canonical approach of diagonalization. The row vectors of ''P''−1 are the [[left eigenvector]]s of ''A''.

When the matrix ''A'' is a [[Hermitian matrix]] (resp. [[symmetric matrix]]), eigenvectors of ''A'' can be chosen to form an [[orthonormal basis]] of '''C'''''n'' (resp. '''R'''''n''). Under such circumstance ''P'' will be a [[unitary matrix]] (resp. [[orthogonal matrix]]) and ''P''−1 equals the [[conjugate transpose]] (resp. [[transpose]]) of ''P''.

== Simultaneous diagonalization ==
{{See also|Triangular matrix#Simultaneous triangularisability|l1=Simultaneous triangularisability|Weight (representation theory)|Positive definite matrix#Simultaneous_diagonalization|l3=Positive definite matrix}}

A set of matrices are said to be ''simultaneously diagonalisable'' if there exists a single invertible matrix ''P'' such that ''P''−1''AP'' is a diagonal matrix for every ''A'' in the set. The following theorem characterises simultaneously diagonalisable matrices: A set of diagonalizable [[Commuting matrices|matrices commutes]] if and only if the set is simultaneously diagonalisable.<ref>Horn & Johnson 1985, pp. 51–53</ref>

The set of all ''n''×''n'' diagonalisable matrices (over '''C''') with ''n'' > 1 is not simultaneously diagonalisable. For instance, the matrices

:<math> \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \quad\text{and}\quad \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} </math>

are diagonalizable but not simultaneously diagonalizable because they do not commute.

A set consists of commuting [[normal matrix|normal matrices]] if and only if it is simultaneously diagonalisable by a [[unitary matrix]]; that is, there exists a unitary matrix ''U'' such that ''U*AU'' is diagonal for every ''A'' in the set.

In the language of [[Lie theory]], a set of simultaneously diagonalisable matrices generate a [[toral Lie algebra]].

== Examples ==
=== Diagonalizable matrices ===
* [[Involution (mathematics)|Involution]]s are diagonalisable over the reals (and indeed any field of characteristic not 2), with ±1 on the diagonal
* Finite order [[endomorphism]]s are diagonalisable over '''C''' (or any algebraically closed field where the characteristic of the field does not divide the order of the endomorphism) with [[roots of unity]] on the diagonal. This follows since the minimal polynomial is [[separable polynomial|separable]], because the roots of unity are distinct.
* [[Projection (linear algebra)|Projections]] are diagonalizable, with 0's and 1's on the diagonal.
* Real [[symmetric matrices]] are diagonalizable by [[orthogonal matrix|orthogonal matrices]]; i.e., given a real symmetric matrix ''A'', ''QTAQ'' is diagonal for some orthogonal matrix ''Q''. More generally, matrices are diagonalizable by [[unitary matrix|unitary matrices]] if and only if they are [[normal matrix|normal]]. In the case of the real symmetric matrix, we see that <math>A=A^T</math>, so clearly <math>AA^T=A^TA</math> holds. Examples of normal matrices are real symmetric (or skew-symmetric) matrices (e.g. covariance matrices) and [[Hermitian matrix|Hermitian matrices]] (or skew-Hermitian matrices). See [[spectral theorem]]s for generalizations to infinite-dimensional vector spaces.

=== Matrices that are not diagonalizable ===
In general, a [[rotation matrix]] is not diagonalizable over the reals, but all [[rotation matrix#Independent planes|rotation matrices]] are diagonalizable over the complex field. Even if a matrix is not diagonalizable, it is always possible to "do the best one can", and find a matrix with the same properties consisting of eigenvalues on the leading diagonal, and either ones or zeroes on the superdiagonal - known as [[Jordan Normal Form|Jordan normal form]].

Some matrices are not diagonalizable over any field, most notably nonzero [[nilpotent matrix|nilpotent matrices]]. This happens more generally if the [[Eigenvalue,_eigenvector_and_eigenspace#Algebraic and geometric multiplicities|algebraic and geometric multiplicities]] of an eigenvalue do not coincide. For instance, consider

:<math> C = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. </math>

This matrix is not diagonalizable: there is no matrix ''U'' such that ''U''−1''CU'' is a diagonal matrix. Indeed, ''C'' has one eigenvalue (namely zero) and this eigenvalue has algebraic multiplicity 2 and geometric multiplicity 1.

Some real matrices are not diagonalizable over the reals. Consider for instance the matrix

:<math> B = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}. </math>

The matrix ''B'' does not have any real eigenvalues, so there is no '''real''' matrix ''Q'' such that ''Q''−1''BQ'' is a diagonal matrix. However, we can diagonalize ''B'' if we allow complex numbers. Indeed, if we take

:<math> Q = \begin{bmatrix} 1 & \textrm{i} \\ \textrm{i} & 1 \end{bmatrix}, </math>

then ''Q''−1''BQ'' is diagonal.

Note that the above examples show that the sum of diagonalizable matrices need not be diagonalizable.

=== How to diagonalize a matrix ===
Consider a matrix

:<math>A=\begin{bmatrix}
1& 2 & 0 \\
0 & 3 & 0 \\
2 & -4 & 2 \end{bmatrix}.</math>

This matrix has [[eigenvalues]]

: <math> \lambda_1 = 3, \quad \lambda_2 = 2, \quad \lambda_3= 1. </math>

''A'' is a 3×3 matrix with 3 different eigenvalues; therefore, it is diagonalizable. Note that if there are exactly ''n'' distinct eigenvalues in an ''n''×''n'' matrix then this matrix is diagonalizable.

These eigenvalues are the values that will appear in the diagonalized form of matrix ''A'', so by finding the eigenvalues of ''A'' we have diagonalized it. We could stop here, but it is a good check to use the [[eigenvectors]] to diagonalize ''A''.

The [[eigenvectors]] of ''A'' are

:<math>v_1 = \begin{bmatrix} -1 \\ -1 \\ 2 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \quad v_3 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix}.</math>

One can easily check that <math>A v_k = \lambda_k v_k.</math>

Now, let ''P'' be the matrix with these eigenvectors as its columns:

:<math>P= \begin{bmatrix}
-1 & 0 & -1 \\
-1 & 0 & 0 \\
2 & 1 & 2 \end{bmatrix}.</math>

Note there is no preferred order of the eigenvectors in ''P''; changing the order of the [[eigenvectors]] in ''P'' just changes the order of the [[eigenvalues]] in the diagonalized form of ''A''.<ref>{{cite book| last1=Anton |first1=H. |last2= Rorres|first2= C. |title=Elementary Linear Algebra (Applications Version) |publisher=John Wiley & Sons|edition=8th|date=22 Feb 2000| ISBN= 978-0-471-17052-5}}</ref>

Then ''P'' diagonalizes ''A'', as a simple computation confirms, having calculated ''P'' −1 using any [[Invertible_matrix#Methods_of_matrix_inversion|suitable method]]:

:<math>P^{-1}AP =
\begin{bmatrix}
0 & -1 & 0 \\
2 & 0 & 1 \\
-1 & 1 & 0 \end{bmatrix}
\begin{bmatrix}
1 & 2 & 0 \\
0 & 3 & 0 \\
2 & -4 & 2 \end{bmatrix}
\begin{bmatrix}
-1 & 0 & -1 \\
-1 & 0 & 0 \\
2 & 1 & 2 \end{bmatrix} =
\begin{bmatrix}
3 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 1\end{bmatrix}.</math>

Note that the eigenvalues <math>\lambda_k</math> appear in the diagonal matrix.

== An application ==
Diagonalization can be used to compute the powers of a matrix ''A'' efficiently, provided the matrix is diagonalizable. Suppose we have found that

:<math>P^{-1}AP = D</math>

is a diagonal matrix. Then, as the matrix product is associative,

:<math>\begin{align}
A^k &= (PDP^{-1})^k = (PDP^{-1}) \cdot (PDP^{-1}) \cdots (PDP^{-1}) \\
&= PD(P^{-1}P) D (P^{-1}P) \cdots (P^{-1}P) D P^{-1} \\
&= PD^kP^{-1} \end{align} </math>

and the latter is easy to calculate since it only involves the powers of a diagonal matrix. This approach can be generalized to [[matrix exponential]] and other [[matrix function]]s since they can be defined as power series.

This is particularly useful in finding closed form expressions for terms of [[linear recursive sequences]], such as the [[Fibonacci number]]s.

=== Particular application ===
For example, consider the following matrix:

:<math>M =\begin{bmatrix}a & b-a \\ 0 &b \end{bmatrix}.</math>

Calculating the various powers of ''M'' reveals a surprising pattern:

:<math>M^2 = \begin{bmatrix}a^2 & b^2-a^2 \\ 0 &b^2 \end{bmatrix},\quad
M^3 = \begin{bmatrix}a^3 & b^3-a^3 \\ 0 &b^3 \end{bmatrix},\quad
M^4 = \begin{bmatrix}a^4 & b^4-a^4 \\ 0 &b^4 \end{bmatrix},\quad \ldots</math>

The above phenomenon can be explained by diagonalizing ''M''. To accomplish this, we need a basis of '''R'''2 consisting of eigenvectors of ''M''. One such eigenvector basis is given by

:<math>\mathbf{u}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}=\mathbf{e}_1,\quad \mathbf{v}=\begin{bmatrix} 1 \\ 1 \end{bmatrix}=\mathbf{e}_1+\mathbf{e}_2,</math>

where '''e'''i denotes the standard basis of '''R'''n. The reverse change of basis is given by

:<math> \mathbf{e}_1 = \mathbf{u},\qquad \mathbf{e}_2 = \mathbf{v}-\mathbf{u}.</math>

Straightforward calculations show that

:<math>M\mathbf{u} = a\mathbf{u},\qquad M\mathbf{v}=b\mathbf{v}.</math>

Thus, ''a'' and ''b'' are the eigenvalues corresponding to '''u''' and '''v''', respectively. By linearity of matrix multiplication, we have that

:<math> M^n \mathbf{u} = a^n\, \mathbf{u},\qquad M^n \mathbf{v}=b^n\,\mathbf{v}.</math>

Switching back to the standard basis, we have

:<math> M^n \mathbf{e}_1 = M^n \mathbf{u} = a^n \mathbf{e}_1,</math>
:<math> M^n \mathbf{e}_2 = M^n (\mathbf{v}-\mathbf{u}) = b^n \mathbf{v} - a^n\mathbf{u} = (b^n-a^n) \mathbf{e}_1+b^n\mathbf{e}_2.</math>

The preceding relations, expressed in matrix form, are

:<math> M^n = \begin{bmatrix}a^n & b^n-a^n \\ 0 &b^n \end{bmatrix}, </math>

thereby explaining the above phenomenon.

== Quantum mechanical application ==
In [[quantum mechanics|quantum mechanical]] and [[quantum chemistry|quantum chemical]] computations matrix diagonalization is one of the most frequently applied numerical processes. The basic reason is that the time-independent [[Schrödinger equation]] is an eigenvalue equation, albeit in most of the physical situations on an infinite dimensional space (a [[Hilbert space]]). A very common approximation is to truncate Hilbert space to finite dimension, after which the Schrödinger equation can be formulated as an eigenvalue problem of a real symmetric, or complex Hermitian, matrix. Formally this approximation is founded on the [[variational principle]], valid for Hamiltonians that are bounded from below.
But also first-order perturbation theory for degenerate states leads to a matrix eigenvalue problem.

== See also ==
* [[Defective matrix]]
* [[Scaling (geometry)]]
* [[Triangular matrix]]
* [[Semisimple operator]]
* [[Diagonalizable group]]
* [[Jordan normal form]]
* [[Weight module]] – associative algebra generalization

== Notes ==
{{Reflist}}

== References ==
* {{cite book | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher= Cambridge University Press | isbn=978-0-521-38632-6 | year=1985}}

== External links ==
* {{planetmath reference|id=1960|title=Diagonalization}}

[[Category:Matrices]]

Second-order fluid

2013-07-22T17:40:49Z

131.215.116.144: Link to upper-convected derivative.

{{multiple issues|
{{More footnotes|date=April 2009}}
{{Cleanup|date=April 2009}}
}}
{{merge from|Factor of adhesion|date=January 2014}}

[[File:Steam locomotive driving wheel.jpg|thumb|right|Driving wheel of steam locomotive]]

The term '''adhesion railway''' or '''adhesion traction''' describes the most common type of [[railway]], where power is applied by driving some or all of the [[wheel]]s of the locomotive. Thus, it relies on the [[friction]] between a steel wheel and a steel rail. The term is particularly used when discussing conventional railways to distinguish from other forms of traction such as [[funicular]] or [[Rack railway|cog railway]] For example, the [[Berner Oberland Bahn|Bernese Oberland Railway]] "is a mixed rack and adhesion railway".<ref>http://www.rail-info.ch/BOB/index.en.html</ref>

[[Traction (engineering)|Traction]] or [[friction]] can be reduced when the rails are greasy, because of [[rain]], [[oil]] or decomposing [[leaf|leaves]] which compact into a hard slippery [[lignin]] coating. Measures against reduced adhesion due to leaves include application of '[[Sandite]]' (a gel-sand mix) by special sanding trains, scrubbers and water jets, and long-term management of railside vegetation. On an adhesion railway, most locomotives will have a sand containment vessel. Properly dried sand can be dropped onto the rail to improve traction under slippery conditions. The sand is most often applied using compressed air. When an engine slips, particularly when starting a heavy train, sand applied at the front of the driving wheels greatly aids in tractive effort causing the train to 'lift', or to commence the motion intended by the engine driver.

== Effect of adhesion limits ==
{{see also|Traction (engineering)}}
Adhesion is caused by [[friction]], with maximum tangential force produced by a driving wheel before slipping given by:

:Fmax= coefficient of friction x Weight on wheel

Usually the force needed to start sliding is greater than that needed to continue sliding. The former is concerned with static friction, referred colloquially to as '[[stiction]]', or 'limiting friction', whilst the latter is called 'sliding friction'.

For steel on steel, the coefficient of friction can be as high as 0.78, under laboratory conditions, but typically on railways it is between 0.35 - 0.5,<ref>http://documents.epfl.ch/users/a/al/allenbac/www/documents/ResuTE31.pdf</ref> whilst under extreme conditions it can fall to as low as 0.05. Thus a 100 tonne locomotive could have a tractive effort of 350 kilonewton, under the ideal conditions (assuming sufficient force can be produced by the engine), falling to a 50 kilonewton under the worst conditions.

== All-weather adhesion ==
The term ''all weather adhesion'' is usually used in North America, and refers to the adhesion available during traction mode with 99% reliability in all weather conditions.

== Toppling conditions ==
The maximum speed a train can proceed around a turn is limited by the radius of turn, the position of the centre of mass of the units, the [[Axle track#Rail|wheel gauge]] and whether the track is ''superelevated'' or ''canted'' (see [[cant (road/rail)]]).

[[File:TopplingLimit.png|thumb|right|Toppling limit on tight turn radius]]

Toppling will occur when the overturning moment due to the side force ([[centrifugal force|centrifugal]] acceleration) is sufficient to cause the inner wheel to begin to lift off the rail. This may result in loss of adhesion - causing the train to slow, preventing toppling. Alternatively, the inertia may be sufficient to cause the train to continue to move at speed causing the vehicle to topple completely.

For a wheel gauge of 1.5 m, no canting, a centre of gravity height of 3 m and speed of 30 m/s (108 km/h), the radius of turn is 360 m. For a modern high speed train at 80 m/s, the toppling limit would be about 2.5 km. In practice, the minimum radius of turn is much greater than this, as contact between the wheel flanges and rail at high speed could cause significant damage to both. For very high speed, the minimum adhesion limit again appears appropriate, implying a radius of turn of about 13 km. In practice, curved lines used for high speed travel are ''superelevated'' or ''canted'' so that the turn limit is closer to 7 km.

During the 19th century, it was widely believed that coupling the drive wheels would compromise performance and was avoided on engines intended for express passenger service. With a single drive wheelset, the Herzian contact [[stress (physics)|stress]] between the wheel and rail necessitated the largest diameter wheels that could be accommodated. The weight of locomotive was restricted by the stress on the rail and sandboxes were required, even under reasonable adhesion conditions.

==Directional stability and hunting instability==
{{multiple image
| align = right
| direction = vertical
| width = 300

| image1 = Wheelset.svg
| alt1 = Diagram of a railway wheelset in the central position
| caption1 = Wheelset in the Central position

| image2 = Wheelset2.png
| alt2 = Diagram of a railway wheelset showing the effects of lateral displacement
| caption2 = The effect of lateral displacement
}}
While common belief dictates that the wheels are kept on the tracks by the flanges, the flanges in reality make few contacts with the track and, when they do, most of the contact is sliding. The rubbing of a flange on the track dissipates large amounts of energy, mainly as heat but also including noise and, if sustained, would lead to excessive wheel wear.

Close examination of a typical railway wheel reveals that the tread is burnished but the flange is not; the flanges rarely make contact with the rail. The tread of the wheel is slightly tapered. When the train is in the centre of the track, the region of the wheels in contact with the rail traces out a circle which has the same diameter for both wheels. The velocities of the two wheels are equal, so the train moves in a straight line.

If, however, the wheelset is displaced to one side, the diameters of the regions of contact and hence the (linear) tangential velocities of the wheels (at the running surfaces) are different and the wheelset tends to steer back towards the centre. Also, when the train encounters an [[banked turn|unbanked turn]], the wheelset displaces laterally slightly, so that the outer wheel tread speeds up (linearly) and the inner wheel tread slows down, causing the train to turn the corner. It should be noted that some railway systems employ a flat wheel and track profile, relying on [[Cant (Road/rail)|cant]] alone to reduce/eliminate flange contact.

Understanding how the train stays on the track, it becomes evident why Victorian locomotive engineers were averse to coupling wheelsets. This simple coning action is possible only with wheelsets where each can have some free motion about its vertical axis. If wheelsets are rigidly coupled together, this motion is restricted, so that coupling the wheels would be expected to introduce sliding, resulting in increased rolling losses. This problem was alleviated to a great extent by ensuring the diameter of all coupled wheels was very closely matched.

With perfect rolling contact between the wheel and rail, this coning behaviour manifests itself as a swaying of the train from side to side. See also [[Hunting oscillation]] for more details. In practice, the swaying is [[Damping|damped]] out below a critical speed, but is amplified by the forward motion of the train above the critical speed. This lateral swaying is known as '[[Hunting oscillation|hunting]]'. The phenomenon of hunting was known by the end of the 19th century, although the cause was not fully understood until the 1920s and measures to eliminate it were not taken until the late 1960s. As is often the case, the limitation on maximum speed was imposed not by raw power but by encountering an instability in the motion.

The kinematic description of the motion of tapered treads on the two rails is insufficient to describe hunting well enough to predict the critical speed. It is necessary to deal with the forces involved. There are two phenomena which must be taken into account. The first is the inertia of the wheelsets and vehicle bodies, giving rise to forces proportional to acceleration; the second is the distortion of the wheel and track at the point of contact, giving rise to elastic forces. The kinematic approximation corresponds to the case which is dominated by contact forces.

A fairly straightforward analysis of the kinematics of the coning action yields an estimate of the wavelength of the lateral oscillation:

::<math>\lambda={2\pi}\sqrt{\frac{rd}{2k}}</math>

where ''d'' is the wheel gauge, ''r'' is the nominal wheel radius and ''k'' is the taper of the treads. For a given speed, the longer the wavelength and the lower the inertial forces will be, so the more likely it is that the oscillation will be damped out. Since the wavelength increases with reducing taper, increasing the critical speed requires the taper to be reduced, which implies a large minimum radius of turn.

A more complete analysis, taking account of the actual forces acting, yields the following result for the critical speed of a wheelset:

::<math>V^2=\frac{Wrad^2}{k \left(4C+md^2 \right)}</math>

where W is the axle load for the wheelset, a is a shape factor related to the amount of wear on the wheel and rail, C is the moment of inertia of the wheelset perpendicular to the axle, m is the wheelset mass.

The result is consistent with the kinematic result in that the critical speed depends inversely on the taper. It also implies that the weight of the rotating mass should be minimised compared with the weight of the vehicle. The wheel gauge implicitly appears in both the numerator and denominator, implying that it has only a second-order effect on the critical speed.

The true situation is much more complicated, as the response of the vehicle suspension must be taken into account. Restraining springs, opposing the yaw motion of the wheelset, and similar restraints on bogies, may be used to raise the critical speed further. However, in order to achieve the highest speeds without encountering instability, a significant reduction in wheel taper is necessary, so there is little prospect of reducing the turn radius of high speed trains much below the current value of 7 km.

=={{anchor|Creep}}Forces on wheels, creep==


The behaviour of adhesion railways is determined by the [[force]]s arising between two surfaces in contact. This may appear trivially simple from a superficial glance but it becomes extremely complex when studied to the depth necessary to predict useful results.

The first error to address is the assumption that wheels are round. A glance at the tyres of a parked car will immediately show that this is not true: the region in contact with the road is noticeably flattened, so that the wheel and road conform to each other over a region of contact. If this were not the case, the contact stress of a load being transferred through a point contact would be infinite. Rails and railway wheels are much stiffer than pneumatic tyres and tarmac but the same distortion takes place at the region of contact. Typically, the area of contact is elliptical, of the order of 15 mm across.

[[File:Illustration of creepage for a railway wheel.png|thumb|A torque <math>M</math> applied on the axle causes creepage: difference between forward velocity <math>V</math> and circumferential velocity <math>\omega R</math>, with resulting creep force <math>F_w</math>.]]

The distortion is small and localised but the forces which arise from it are large. In addition to the distortion due to the weight, both wheel and rail distort when braking and accelerating forces are applied and when the vehicle is subjected to side forces. These tangential forces cause distortion in the region where they first come into contact, followed by a region of slippage. The net result is that, during traction, the wheel does not advance as far as would be expected from rolling contact but, during braking, it advances further. This mix of elastic distortion and local slipping is known as "creep" (not to be confused with the [[Creep (deformation)|creep]] of materials under constant load). The definition of creep<ref>Wikins (2003) p.6 1.3 Creep (see Fig. 1.5a)</ref> in this context is:

::<math>\mbox{creep}=\frac{(\mbox{actual displacement} - \mbox{rolling displacement})}{(\mbox{rolling displacement})} \,</math>

In analysing the dynamics of wheelsets and complete rail vehicles, the contact forces can be treated as linearly dependent on the creep <ref>See *Деев В.В., Ильин Г.А., Афонин Г.С. {{ru icon}} "Тяга поездов" (Traction of trains) Учебное пособие. - М.: Транспорт, 1987. - Fig. 2.3 p.30 for a curve (which is linear at first) relating creep to tangential force </ref>(Kalker's linear theory, valid for small creepage) or more advanced theories can be used from [[frictional contact mechanics]].

The forces which result in directional stability, propulsion and braking may all be traced to creep. It is present in a single wheelset and will accommodate the slight kinematic incompatibility introduced by coupling wheelsets together, without causing gross slippage, as was once feared.

Provided the radius of turn is sufficiently great (as should be expected for express passenger services), two or three linked wheelsets should not present a problem. However, 10 drive wheels (5 main wheelsets) are usually associated with heavy freight locomotives.

== Booster ==
Some steam locomotives were fitted with [[booster engine]]s on the rear trailing wheels. These were turned on as required at starting to give additional adhesive effort.

== See also ==
* [[Curve resistance]]
* [[Friction]]
* [[Frictional contact mechanics]]
* [[Hunting oscillation]]
* [[Rolling resistance]]
* [[Slippery rail]]
* [[Traction (engineering)|Traction]]
* [[Wheelset (railroad)|Wheelset]]
* [[Wheelskate]]

== References ==
{{reflist}}
{{refbegin}}
* {{cite book|first=Sir Charles|last=Inglis|title=Applied Mathematics for Engineers|pages=194–195|publisher=Cambridge University Press|year=1951}}
* {{cite book|last=Carter|first=F W|title=On the Stability of Running of Locomotives|series=Proc. Royal Society|date=July 25, 1928}}
* {{cite journal|last=Wickens|first=A H|title=The Dynamics of Railway Vehicles on Straight Track: Fundamental Considerations of Lateral Stability|journal=[[Proceedings of the Institution of Mechanical Engineers|Proc. Inst. Mech. Eng.]]|year=1965–1966|page=29}}
* {{cite book|last1=Wickens|first1=A H|last2=Gilchrist|first2=A O|first3=A E W|last3=Hobbs|work=Suspension Design for High-Performance Two-Axle Freight Vehicles|series=Proc. Inst. Mech. Eng.|year=1969–1970|page=22}}
by A H Wickens
* {{cite book|last=Wikins|first=A H|title=Fundamentals of rail vehicle dynamics : guidance and stability|publisher=Swets & Zeitlinger|date=Jan 1, 2003}}
{{refend}}

{{DEFAULTSORT:Rail Adhesion}}
[[Category:Rail technologies]]
[[Category:Railways by type]]

{{Link GA|ja}}

Ihara zeta function

2013-07-03T01:54:12Z

131.215.220.166: Fixed errors in formula

{{Unreferenced stub|auto=yes|date=December 2009}}
A '''transition radiation detector''' (TRD) is a [[particle detector]] using the <math>\scriptstyle \gamma</math>-dependent threshold of transition radiation in a stratified material. It contains many layers of materials with different [[Index of refraction|indices of refraction]]. At each interface between materials, the probability of [[transition radiation]] increases with the relativistic [[Lorentz factor|gamma factor]]. Thus particles with large <math>\scriptstyle \gamma</math> give off many [[photons]], and small <math>\scriptstyle \gamma</math> give off few. For a given energy, this allows a discrimination between a lighter particle (which has a high <math>\scriptstyle \gamma</math> and therefore radiates) and a heavier particle (which has a low <math>\scriptstyle \gamma</math> and radiates much less).

{{DEFAULTSORT:Transition Radiation Detector}}
[[Category:Particle detectors]]

{{Particle-stub}}

Wiedemann–Franz law

2013-07-02T18:20:03Z

131.215.51.210: Lorentz -> Lorenz. Similar names, different people.

In [[category theory]], a branch of [[mathematics]], a '''connected category''' is a [[category (category theory)|category]] in which, for every two objects ''X'' and ''Y'' there is a [[finite sequence]] of objects
:<math>X = X_0, X_1, \ldots, X_{n-1}, X_n = Y</math>
with morphisms
:<math>f_i : X_i \to X_{i+1}</math>
or
:<math>f_i : X_{i+1} \to X_i</math>
for each 0 ≤ ''i'' < ''n'' (both directions are allowed in the same sequence). Equivalently, a category ''J'' is connected if each [[functor]] from ''J'' to a [[discrete category]] is constant. In some cases it is convenient to not consider the empty category to be connected.

A stronger notion of connectivity would be to require at least one morphism ''f'' between any pair of objects ''X'' and ''Y''. Clearly, any category which this property is connected in the above sense.

A [[small category]] is connected [[if and only if]] its underlying graph is [[connected graph|weakly connected]].

Each category ''J'' can be written as a disjoint union (or [[coproduct]]) of a connected categories, which are called the '''connected components''' of ''J''. Each connected component is a [[full subcategory]] of ''J''.

==References==
*{{cite book | first = Saunders | last = Mac Lane | authorlink = Saunders Mac Lane | year = 1998 | title = [[Categories for the Working Mathematician]] | series = Graduate Texts in Mathematics '''5''' | edition = (2nd ed.) | publisher = Springer-Verlag | isbn = 0-387-98403-8}}

[[Category:Category theory]]

{{categorytheory-stub}}

Standard probability space

2013-05-19T04:12:57Z

131.215.108.224:

{{Unreferenced|date=August 2010}}

A '''well kill''' is the operation of placing a column of heavy fluid into a [[Oil well|well bore]] in order to prevent the flow of [[Oil reservoir|reservoir]] fluids without the need for pressure control equipment at the surface. It works on the principle that the weight of the "kill fluid" or "kill mud" will be enough to suppress the [[pressure]] of the formation fluids. Well kills may be planned in the case of advanced [[Well intervention|interventions]] such as [[workover]]s, or be contingency operations. The situation calling for a well kill will dictate the method taken.

Not all well kills are deliberate. Sometimes, the unintended build up of fluids, either from injection of chemicals like [[methanol]] from surface, or from liquids produced from the reservoir, can be enough to kill the well, particularly [[Natural gas|gas]] wells, which are notoriously easy to kill.

==Principles==
The principle of a well kill revolves around the weight of a column of fluid and hence the pressure exerted at the bottom.

<math>P=hg\rho</math>

Where P is the pressure at depth h in the column, g is the [[acceleration of gravity]] and ρ is the [[density]] of the fluid. It is common in the [[oil industry]] to use [[Specific weight|weight density]], which is the product of mass density and the acceleration of gravity. This reduces the equation to:

<math>P=h\gamma</math>

Where γ is the weight density. Weight density may also be described as the pressure gradient because it directly determines how much extra pressure will be added by increasing depth of the column of fluid.

The objective in a well kill, is to make the pressure at the bottom of the kill fluid equal (or slightly greater) than the pressure of the reservoir fluids.

===Example===
The pressure of the reservoir fluids at the bottom of the hole is 38[[Pascal (unit)|MPa]]. We have a kill fluid with a weight density of 16[[Newton (unit)|kN]].m-3. What will need to be the height of the hydrostatic head in order to kill the well?

From the equation:
<math>h=\frac{P}{\gamma}</math>

<math>h=\frac{38\,MPa}{16\,kNm^{-3}}</math>

<math>h=2375\,m</math>

Therefore, a column of 2375m of this fluid is needed. This refers to the true vertical depth of the column, not the measured depth, which is always larger than true vertical depth due to deviations from vertical.

===Maths in the oil field===
In the [[oil industry]], a pure [[International System of Units|SI system]] is far from being used. Weight densities are commonly either given as [[specific gravity]] or in [[Pound (mass)|pounds]] per [[gallon]]. Simple conversion factors (0.433 for specific gravity and 0.052 for ppg) convert these values to a pressure gradient in [[Pounds per square inch|psi]] per [[foot (length)|foot]]. Multiplying by the depth in feet gives the pressure at the bottom of the column.

==Methods of well kill==
===Reverse circulation===
This is often the tidiest way of making a pre-planned well kill. It involves pumping kill fluid down the 'A' [[Annulus (oil well)|annulus]] of the well, through a point of communication between it and the [[production tubing]] just above the [[production packer]] and up the tubing, displacing the lighter well bore fluids, which are allowed to flow to production.

The point of communication was traditionally a device called a sliding sleeve, or sliding side door, which is a hydraulically operated device, built into the [[production tubing]]. During normal operation, it would remain closed sealing off the tubing and the annulus, but for events such as this, it would be opened to allow the free flow of fluids between the two regions. These components have fallen out of favour as they were prone to leaking. Instead, it is now more common to punch a hole in the tubing for circulation kills. Although this permanently damages the tubing, given that most pre-planned well kills are for [[workover]]s, this is not an issue, since the tubing is being pulled for replacement anyway.

===Bullheading===
This is the most common method of a contingency well kill. If there is a sudden need to kill a well quickly, without the time for rigging up for circulation, the more blunt instrument of bullheading may be used. This involves simply pumping the kill fluid directly down the well bore, forcing the well bore fluids back into the [[Oil reservoir|reservoir]]. This can be effective at achieving the central aim of a well kill; building up a sufficient hydrostatic head in the well bore. However, it can risk damaging the reservoir, by forcing undesired materials into it. The principal advantage is that it can be done with little advanced planning.

===Forward circulation===
This is similar to reverse circulation, except the kill fluid is pumped into the [[production tubing]] and circulated out through the [[Annulus (oil well)|annulus]]. Though effective, it is not as desirable since it is preferred that the well bore fluids be displaced out to production, rather than the annulus.

===Lubricate and bleed===
This is the most time consuming form of well kill. It involves repeatedly pumping in small quantities of kill mud into the well bore and then bleeding off excess pressure. It works on the principle that the heavier kill mud will sink below the lighter well bore fluids and so bleeding off the pressure will remove the latter leaving an increasing quantity of kill mud in the well bore with successive steps.

===Well kills during drilling operations===
During drilling, pressure control is maintained through the use of precisely concocted [[drilling fluid]], which balances out the pressure at the bottom of the hole. In the event of suddenly encountering a high pressure pocket of, say [[Natural gas|gas]] (called a "kick"), it can become necessary to kill the well. This is done by pumping kill mud down the [[Drill string|drill pipe]], where it circulates out the bottom and into the well bore.

==Reversing a well kill==
The intention of a well kill (or the reality of an unintentional well kill) is to stop reservoir fluids flowing to surface. This of course creates problems when it is desirable to get the well flowing again. In order to reverse the well kill, the kill fluid must be displaced from the well bore. This involves injecting a gas at high pressure, usually [[nitrogen]] since it is inert and cheap. A gas can be put under sufficient pressure to allow it to push heavy kill fluid, but will then expand and become light once pressure is removed. This means that having displaced the kill fluid, it will not itself kill the well. The reservoir fluids should be able to flow to surface, displacing the gas.

The cheapest way to do it is similar to bullheading, where the nitrogen is pumped in under high pressure to force the kill fluid into the reservoir. This, of course, runs a high risk of causing well damage. The most effective way is to use [[coiled tubing]], pumping the gas down the coil and circulating out the bottom into the well bore, where it will displace the kill mud to production.

==See also==
* [[Blowout preventer]]
* [[Coiled tubing]]
* [[Oil well]]
* [[Top kill]]
* [[Well drilling]]
* [[Well intervention]]

==External links==
*[http://www.iwcf.org/IWCF/PDF/QA-WIS%20WI%20Syllabus,%20Rev%203.2,%201-08-2004,%20B&W.pdf IWCF Well Intervention Pressure Control syllabus (pdf)]

[[Category:Petroleum production]]
[[Category:Oil wells]]

Geodesics as Hamiltonian flows

2013-02-25T03:11:18Z

131.215.143.239:

A '''self-organizing list''' is a [[list (computing)|list]] that reorders its elements based on some [[self-organizing heuristic]] to improve [[average]] [[access time]].
The aim of a self-organizing list is to improve efficiency of linear search by moving more frequently accessed items towards the head of the list. A self-organizing list achieves near constant time for element access in the best case. A self-organizing list uses a reorganizing algorithm to adapt to various query distributions at runtime.

==History==
The concept of self-organizing lists was introduced by McCabe in 1965.<ref>{{Citation
| last = Knuth
| first = Donald
| author-link = Donald Knuth
| series = [[The Art of Computer Programming]]
| title = Sorting and Searching
| place =
| publisher = Addison-Wesley
| year = 1998
| volume = Volume 3
| edition = Second
| page = 402
| pages =
| url =
| doi =
| isbn = 0-201-89685-0 }}</ref> In a pioneering work, he introduced two heuristics- the MTF rule and the transposition rule. Further improvements were made, and algorithms suggested by Ronald Rivest, Tenenbaum and Nemes, D. Knuth and so on.

==Introduction==
The simplest implementation of a self-organizing list is as a [[linked list]] and thus while being efficient in random node inserting and memory allocation, suffers from inefficient accesses to random nodes. A self-organizing list reduces the inefficiency by dynamically rearranging the nodes in the list based on access frequency.

===Inefficiency of linked list traversals===

If a particular node is to be searched for in the list, each node in the list must be sequentially compared till the desired node is reached. In a linked list, retrieving the nth element is an O(n) operation. This is highly inefficient when compared to an array for example, where accessing the nth element is an O(1) operation.

===Efficiency of self-organizing lists===

A self organizing list rearranges the nodes keeping the most frequently accessed ones at the head of the list. Generally, in a particular query, the chances of accessing a node which has been accessed many times before are higher than the chances of accessing a node which historically has not been so frequently accessed. As a result, keeping the commonly accessed nodes at the head of the list results in reducing the number of comparisons required in an average case to reach the desired node. This leads to better efficiency and generally reduced query times.

==Implementation of a self-organizing list==
The implementation and methods of a self-organizing list are identical to the those for a standard [[linked list]]. The linked list and the self-organizing list differ only in terms of the organization of the nodes; the interface remains the same.

==Analysis of Running Times for Access/ Search in a List==

===Average Case===

It can be shown that in the average case, the time required to a search on a self-organizing list of size n is
:<math>Tavg = 1 * p(1) + 2 * p(2) + 3 * p(3) + . . . + n * p(n).</math>
where p(i) is the probability of accessing the ith element in the list, thus also called the access probability.
If the access probability of each element is the same (i.e. p(1) = p(2) = p(3) = ... = p(n) = 1/n) then the ordering of the elements is irrelevant and the average time complexity is given by
:<math>T(n) = 1/n + 2/n + 3/n + ... + n/n = (1 + 2 + 3 + ... + n)/n = (n+1)/2</math>
and T(n) does not depend on the individual access probabilities of the elements in the list in this case.
However in the case of searches on lists with non uniform record access probabilities (i.e. those lists in which the probability of accessing one element is different from another), the average time complexity can be reduced drastically by proper positioning of the elements contained in the list. 
This is done by pairing smaller i with larger access probabilities so as to reduce the overall average time complexity.
 
This may be demonstrated as follows: 
Given List: A(0.1), B(0.1), C(0.3), D(0.1), E(0.4) 
Without rearranging, average search time required is:
:<math>T(n) = 1*0.1 + 2*0.1 + 3*0.3 + 4*0.1 + 5*0.4 = 3.6</math>
Now suppose the nodes are rearranged so that those nodes with highest probability of access are placed closest to the front so that the rearranged list is now: 
E(0.4), C(0.3), D(0.1), A(0.1), B(0.1) 
Here, average search time is: 
:<math>T(n) = 1*0.4 + 2*0.3 + 3*0.1 + 4*0.1 + 5*0.1 = 2.2</math>
Thus the average time required for searching in an organized list is (in this case) around 40% less than the time required to search a randomly arranged list.
 
This is the concept of the self-organized list in that the average speed of data retrieval is increased by rearranging the nodes according to access frequency.

===Worst Case===

In the worst case, the element to be located is at the very end of the list be it a normal list or a self-organized one and thus n comparisons must be made to reach it. Therefore the worst case running time of a linear search on the list is O(n) independent of the type of list used.
Note that the expression for the average search time in the previous section is a probabilistic one. Keeping the commonly accessed elements at the head of the list simply reduces the probability of the worst case occurring but does not eliminate it completely. Even in a self-organizing list, if a lowest access probability element (obviously located at the end of the list) is to be accessed, the entire list must be traversed completely to retrieve it. This is the worst case search.

===Best Case===

In the best case, the node to be searched is one which has been commonly accessed and has thus been identified by the list and kept at the head. This will result in a near constant time operation. In big-oh notation, in the best case, accessing an element is an O(1) operation.

==Techniques for Rearranging Nodes==
While ordering the elements in the list, the access probabilities of the elements are not generally known in advance. This has led to the development of various heuristics to approximate optimal behavior. The basic heuristics used to reorder the elements in the list are:

===Move to Front Method (MTF)===

This technique moves the element which is accessed to the head of the list. This has the advantage of being easily implemented and requiring no extra memory. This heuristic also adapts quickly to rapid changes in the query distribution. On the other hand, this method may prioritize infrequently accessed nodes-for example, if an uncommon node is accessed even once, it is moved to the head of the list and given maximum priority even if it is not going to be accessed frequently in the future. These 'over rewarded' nodes destroy the optimal ordering of the list and lead to slower access times for commonly accessed elements. Another disadvantage is that this method may become too flexible leading to access patterns that change too rapidly. This means that due to the very short memories of access patterns even an optimal arrangement of the list can be disturbed immediately by accessing an infrequent node in the list.

<div class="center">[[File:MTF Algorithm.png|330px|Move To Front Algorithm]] If the 5th node is selected, it is moved to the front</div>

At the t-th item selection:
'''if''' item i is selected:
move item i to head of the list

===Count Method===

In this technique, the number of times each node was searched for is counted i.e. every node keeps a separate counter variable which is incremented every time it is called. The nodes are then rearranged according to decreasing count. Thus, the nodes of highest count i.e. most frequently accessed are kept at the head of the list. The primary advantage of this technique is that it generally is more realistic in representing the actual access pattern. However, there is an added memory requirement, that of maintaining a counter variable for each node in the list. Also, this technique does not adapt quickly to rapid changes in the access patterns. For example: if the count of the head element say A is 100 and for any node after it say B is 40, then even if B becomes the new most commonly accessed element, it must still be accessed at least (100 - 40 = 60) times before it can become the head element and thus make the list ordering optimal.
 
<div class="center">[[File:CountAlgorithm.png|330px|Count Algorithm]]
If the 5th node in the list is searched for twice, it will be swapped with the 4th
</div>

'''init:''' count(i) = 0 for each item i
At t-th item selection:
'''if''' item i is searched:
count(i) = count(i) + 1
rearrange items based on count

===Transpose Method===

This technique involves swapping an accessed node with its predecessor. Therefore, if any node is accessed, it is swapped with the node in front unless it is the head node, thereby increasing its priority. This algorithm is again easy to implement and space efficient and is more likely to keep frequently accessed nodes at the front of the list. However, the transpose method is more cautious. i.e. it will take many accesses to move the element to the head of the list. This method also does not allow for rapid response to changes in the query distributions on the nodes in the list. 
 
<div class="center">[[File:Transpose Algorithm.png|330px|Transpose Algorithm]]
If the 5th node in the list is selected, it will be swapped with the 4th
</div>
<div>

At the t-th item selection:
'''if''' item i is selected:
'''if''' i is not the head of list:
swap item i with item (i - 1)

</div>

===Other Methods===

Research has been focused on fusing the above algorithms to achieve better efficiency.<ref>http://www.springerlink.com/content/978-3-540-34597-8/#section=508698&page=1&locus=3 Lists on Lists: A Framework for Self Organizing-Lists in Environments with Locality of Reference</ref> Bitner's Algorithm uses MTF initially and then uses transpose method for finer rearrangements. Some algorithms are randomized and try to prevent the over-rewarding of infrequently accessed nodes that may occur in the MTF algorithm. Other techniques involve reorganizing the nodes based on the above algorithms after every n accesses on the list as a whole or after n accesses in a row on a particular node and so on. Some algorithms rearrange the nodes which are accessed based on their proximity to the head node, for example: Swap-With-Parent or Move-To-Parent algorithms.
Another class of algorithms are used when the search pattern exhibits a property called locality of reference whereby in a given interval of time, only a smaller subset of the list is probabilistically most likely to be accessed. This is also referred to as dependent access where the probability of the access of a particular element depends on the probability of access of its neighboring elements. Such models are common in real world applications such as database or file systems and memory management and caching. A common framework for algorithms dealing with such dependent environments is to rearrange the list not only based on the record accessed but also on the records near it. This effectively involves reorganizing a sublist of the list to which the record belongs.

==Applications of self-organizing lists==
Language translators like compilers and interpreters use self-organizing lists to maintain [[symbol table]]s during compilation or interpretation of program source code. Currently research is underway to incorporate the self-organizing list data structure in [[embedded system]]s to reduce bus transition activity which leads to power dissipation in those circuits. These lists are also used in [[artificial intelligence]] and [[neural networks]] as well as self-adjusting programs. The algorithms used in self-organizing lists are also used as [[caching algorithm]]s as in the case of LFU algorithm.

== References ==
{{reflist}}
*{{Citation|last=Vlajic|first=N|title=Self-Organizing Lists|url=http://www.cse.yorku.ca/course_archive/2003-04/F/2011/2011A/DatStr_071_SOLists.pdf|format=pdf|year=2003}}
*{{Citation|title=Self Organization|url=http://courses.cs.vt.edu/~cs2604/spring04/Notes/C16.SelfOrganizingLists.pdf|format=pdf|year=2004}}
* NIST [http://www.nist.gov/dads/HTML/selfOrganizingList.html DADS entry]
* A Drozdek, Data Structures and Algorithms in Java Third edition
* {{Citation|last=Amer|first=Abdelrehman|coauthors=B. John Oommen|title=Lists on Lists: A Framework for Self-organizing Lists in Environments with Locality of Reference|url=http://www.springerlink.com/content/978-3-540-34597-8/#section=508698&page=1&locus=3 Lists on Lists: A Framework for Self Organizing-Lists in Environments with Locality of Reference|format=pdf|year=2006}}

{{Data structures}}

{{DEFAULTSORT:Linked List}}
[[Category:Linked lists|S]]
[[Category:Articles with example C code]]

Friedel oscillations

2013-01-20T01:50:32Z

131.215.42.233: /* Scattering Description */

{{good article}}
{{Planetbox begin
| name= Kepler-4b
}}
{{Planetbox image
| image = Exoplanet Comparison Kepler-4 b.png
| caption = Size comparison of Kepler-4b (right) with [[Neptune]] (left).
}}
{{Planetbox star
| star = [[Kepler-4]]<ref name=datatable />
| constell= [[Draco (constellation)|Draco]]
| RA = {{RA|19|2|27.7}}
| DEC= {{DEC|+50|8|8.7}}
| app_mag = 12.6<ref name=datatable />
| dist_pc = 550<ref name=datatable />
| dist_ly = 1631
| mass = 1.223<ref name=datatable />
| radius = 1.487<ref name=datatable />
| temperature = 5857<ref name=datatable />
| luminosity = 0.354 ± 0.1
| metallicity = 0.17<ref name=datatable />
| age = 4.5<ref name="EPE"/>
}}
{{Planetbox character
| radius_earth = 3.878
| radius = 0.357<ref name=datatable />
| mass = .077<ref name=datatable />
| temperature = 1650<ref name=datatable />
| stellar_flux = ~168
}}
{{Planetbox orbit
| period = 3.2135<ref name=datatable />
| semimajor = .04558<ref name=datatable />
| eccentricity = 0.25 ± 0.12<ref name="Kipping2010"/>
| inclination = 89.76<ref name=datatable />
}}
{{Planetbox discovery
| discovery_date = 2010-01-04<ref name=Talcott />
| discoverers =
| discovery_method = [[Methods of detecting extrasolar planets#Transit method|Transit]] ([[Kepler Mission]])<ref name=QuickGuide />
| detection_methods = Radial velocity
| discovery_status = Conference announcement<ref name=Talcott />
}}
{{Planetbox end}}

'''Kepler-4b''', initially known as '''KOI 7.01''', is an [[exoplanet|extrasolar planet]] first detected as a [[astronomical transit|transit]] by the [[Kepler (spacecraft)|Kepler spacecraft]]. Its [[radius]] and [[mass]] are similar to that of [[Neptune]]; however, due to its proximity to its host star, it is substantially hotter than any planet in the Solar System.<ref name="datatable"/><ref name="sciencenews"/> The planet's discovery was announced on January 4, 2010 in [[Washington, D.C.]] along with four other planets that were initially detected by the Kepler spacecraft and subsequently confirmed by telescopes at the [[W.M. Keck Observatory]].

==Nomenclature and history==

Kepler-4b was named because it was the first planet discovered in the orbit of its star, [[Kepler-4]]. The star was, in turn, named for the Kepler Mission, a [[NASA]] satellite whose purpose is to discover Earth-like planets in a section of the sky between constellations [[Cygnus (constellation)|Cygnus]] and [[Lyra]] using the transit method. Using this method, Kepler notes small and steady decreases in a star's brightness that are measured as a planet crosses in front of it.<ref name=QuickGuide>{{cite web |url=http://kepler.nasa.gov/Mission/QuickGuide/ |title=Kepler: About the Mission |author= |year=2011 |work=Kepler Mission |publisher=NASA |accessdate=11 March 2011}}</ref> Initially, Kepler-4b was detected as a transit event by the [[Kepler telescope]] and considered a [[Kepler Object of Interest]] with the designation KOI 7.01.<ref name="Borucki2011"/>

Subsequent [[radial velocity]] measurements by the [[High Resolution Echelle Spectrometer]] on the telescopes of W.M. Keck Observatory confirmed the planetary nature of the transit event and established a mass estimate for the planet.<ref name="Borucki2010"/> The planet's existence was announced on January 4, 2010 along with four other planets detected by Kepler: [[Kepler-5b]], [[Kepler-6b|6b]], [[Kepler-7b|7b]] and [[Kepler-8b|8b]]<ref name="sciencenews"/> at the [[American Astronomical Society 215th meeting|215th]] meeting of the [[American Astronomical Society]] in Washington, D.C.<ref name=Talcott>{{cite web |url=http://cs.astronomy.com/asycs/blogs/astronomy/archive/2010/01/05/215th-aas-meeting-update-kepler-discoveries-the-talk-of-the-town.aspx |title=215th AAS meeting update: Kepler discoveries the talk of the town |author=Rich Talcott |date=5 January 2010 |work=Astronomy.com |publisher=''[[Astronomy (magazine)|Astronomy]]'' magazine |accessdate=11 March 2011}}</ref>

==Host star==
{{main|Kepler-4}}
Kepler-4 is located within the [[Draco (constellation)|Draco constellation]] on the sky, and is approximately 550 [[parsecs]] from our solar system. It has an effective temperature almost identical to the [[Sun]] at 5857 [[Kelvin]], but a mass and radius that are somewhat larger than the sun: respectively, 1.223 [[solar mass|<math>\begin{smallmatrix}M_\odot\end{smallmatrix}</math>]] and 1.487 [[solar radius|<math>\begin{smallmatrix}R_\odot\end{smallmatrix}</math>]]. The star is thought to be around 4.5 billion years old, and at or very near the end of its [[main-sequence]] [[hydrogen burning]] phase. In several tens of millions of years it will likely become a [[subgiant]] star.<ref name="Borucki2010"/>

==Characteristics==
Kepler-4b orbits its host star in 3.213 days at a distance of 0.046 [[astronomical unit|AU]].<ref name="Borucki2010"/> This places it almost 10 times closer to its star than Mercury is to the Sun. Consequently, Kepler-4b is thought to be extremely hot, with an [[equilibrium temperature]] greater than 1700 [[Kelvin]] (2600 Fahrenheit).<ref name="Kipping2010"/> The planet is estimated to be 25 times more massive than the Earth with a radius that is 4 times greater than the Earth.<ref name="Borucki2010"/> This makes it similar to Neptune in terms of size and mass, but with a temperature that is not comparable to any planet in our solar system (Venus, the hottest planet is only 735 Kelvin). Kepler-4b's [[orbital eccentricity|eccentricity]] was assumed to be 0, however an independent reanalysis of the discovery data found a value of 0.25 ± 0.12.<ref name="Kipping2010"/>

[[File:Kepler first five exoplanet size.jpg|thumb|300px|A picture showing the relative sizes of the first five planets discovered by Kepler. Kepler-4b is the smallest of the five, highlighted in purple.]]

== See also ==
* [[Kepler-5b]]
* [[Kepler-6b]]
* [[Kepler-7b]]
* [[Kepler-8b]]
* [[Kepler-9b]]
* [[Kepler-10b]]

== References ==
{{reflist|refs=

<ref name="Borucki2010">{{cite journal | title=Kepler-4b: A Hot Neptune-like Planet of a G0 Star Near Main-sequence Turnoff | url=http://iopscience.iop.org/2041-8205/713/2/L126/fulltext/ | last1=Borucki | first1=William J. | last2=Koch | first2=David G. | last3=Brown | first3=Timothy M. | last4=Basri | first4=Gibor | last5=Batalha | first5=Natalie M. | last6=Caldwell | first6=Douglas A. | last7=Cochran | first7=William D. | last8=Dunham | first8=Edward W. | last9=Gautier III | first9=Thomas N. | display-authors=1 | journal=The Astrophysical Journal Letters | year=2010 | volume=713 | issue=2 | page=L126 | arxiv=1001.0604 | bibcode=2010ApJ...713L.126B | doi=10.1088/2041-8205/713/2/L126 }}</ref>

.<ref name ="Borucki2011">{{cite web| author=Borucki, William J; et al. | title=Characteristics of planetary candidates observed by Kepler, II: Analysis of the first four months of data | url=http://kepler.nasa.gov/files/mws/FebDataRelease_revised_020211.pdf |publisher=http://kepler.nasa.gov | date=2011-02-01 |accessdate=2011-03-07}}</ref>

<ref name="EPE">{{cite web | url=http://exoplanet.eu/catalog/kepler-4_b/ | title=Planet Kepler-4 b
| author=Jean Schneider | year=2010 | work=[[Extrasolar Planets Encyclopaedia]] | publisher=Jean Schneider | accessdate=5 August 2012}}</ref>

<ref name="Kipping2010">{{cite journal | title=An Independent Analysis of Kepler-4b through Kepler-8b | url=http://iopscience.iop.org/0004-637X/730/1/50/fulltext/ | last1=Kipping | first1=David | last2=Bakos | first2=Gáspár | journal=The Astrophysical Journal | year=2011 | volume=730 | issue=1 | at=50 | arxiv=1004.3538 | bibcode=2011ApJ...730...50K | doi=10.1088/0004-637X/730/1/50 }}</ref>

<ref name="sciencenews">
{{cite web |url=http://www.usnews.com/science/articles/2010/01/04/space-telescope-finds-its-first-extrasolar-planets |title=Kepler space telescope finds its first extrasolar planets |author=Ron Cowen |date=2010-01-04 |work= |publisher=Science News|accessdate=2010-01-04}}</ref>

<ref name="datatable">
{{cite web |url=http://kepler.nasa.gov/Mission/discoveries/ |title=Summary Table of Kepler Discoveries |date=2010-03-15 |work= |publisher=NASA|accessdate=2010-03-18}}</ref>

}}

== External links ==
{{commonscat-inline|Kepler-4 b}}
*http://kepler.nasa.gov/Mission/discoveries/kepler4b/

{{Sky|19|2|27.7|+|50|8|8.7| 550}}

[[Category:Exoplanets discovered in 2010]]
[[Category:Hot Neptunes]]
[[Category:Gas giants]]
[[Category:Transiting exoplanets]]
[[Category:Draco (constellation)]]
[[Category:Exoplanets discovered by Kepler (spacecraft)|4b]]

Slater determinant

2012-08-09T20:53:12Z

131.215.220.166: /* Two-particle case */

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{{Infobox economist
| name = Michał Kalecki
| school_tradition = [[Neo-Marxian economics]]
| color = red
| image = Michal Kalecki.jpg
| image_size = 250px
| birth_date = {{birth date|1899|6|22|df=yes}}
| birth_place =[[Łódź]], [[Congress Poland]]
| death_date = {{Death date and age|1970|4|18|1899|6|22|df=yes}}
| death_place = [[Warsaw]], [[People's Republic of Poland]]
| nationality = [[Polish people|Polish]]
| field = [[Macroeconomics]]
| influences = [[Karl Marx]] (1885), [[Rosa Luxemburg]] (1913), [[Mikhail Tugan-Baranovsky]] (1905)
| influenced = [[Nicholas Kaldor]], [[Joan Robinson]], [[Richard Kahn, Baron Kahn|Richard Kahn]], [[Luigi Pasinetti]], [[Alfred Eichner]], [[John Eatwell]], [[Josef Steindl]], [[Piero Sraffa]], [[Hyman Minsky]], [[Marc Lavoie]], [[Amit Bhaduri]], [[Malcom Sawyer]], [[Lance Taylor]], [[Lawrence Klein]],<ref name="google"/> [[Jan Kregel]], [[George Feiwel]], [[Peter Kriesler]], [[John Maynard Keynes]],<ref name="economicoutlook"/> [[Athanasios Asimakopulos]], [[Paul Sweezy]], [[Paul A. Baran]], [[Richard M. Goodwin]], [[Robin Hahnel]], [[Bill Mitchell (economist)|Bill Mitchell]], [[Jonathan Nitzan]], [[Shimshon Bichler]], [[Maria da Conceição Tavares]]
| contributions = The [[#The profit equation|profit equation]], Theories of [[Markup (business)|mark-up]] and [[effective demand]]
| awards =
| signature = 
}}
'''Michał Kalecki''' ({{IPAc-pl|'|m|i|h|a|U|-|k|a|'|l|e|c|k|i}}; 22 June 1899 – 18 April 1970) was a [[Poland|Polish]] [[Economics|economist]] who specialized in [[macroeconomics]] of a broadly-defined [[Keynesian]] sort. Over the course of his life, Kalecki worked at the [[London School of Economics]], [[University of Cambridge]], [[University of Oxford]] and [[Warsaw School of Economics]] as well as an economic advisor to governments of [[Cuba]], [[Israel]], [[Mexico]] and [[India]].

Kalecki has been called "one of the most distinguished economists of the 20th century." It is often claimed that he developed many of the same ideas as Keynes, before Keynes; however, since he published in [[Polish language|Polish]], he remains much less known to the English-speaking world. He offered a synthesis that integrated [[Marxism|Marxist]] [[class analysis]] and the then-new literature on [[oligopoly theory]], and his work had a significant influence on both [[Neo-Marxian economics|Neo-Marxian]] ([[Monopoly Capital]])<ref name="google1"/> and [[Post Keynesian economics|Post Keynesian]] schools of economic thought. He was also one of the first macroeconomists to apply [[mathematical models]] and [[statistical data]] to economic questions.

==Biography==
===Early years: 1899–1932===
Michal Kalecki was born in [[Łódź]], [[Tsardom of Poland|Poland]], on 22 June 1899. Information about his early years of life is very sparse, part of it being lost during the [[Nazi]] occupation. In 1917 Kalecki finished a Bachelor's degree, in order to join later the [[University of Warsaw]], where he began [[civil engineering]]. He was a very clever student, and in this period he formalized a generalization of [[Pascal's theorem]], concerning a [[hexagon]] drawn within a second degree curve. Kalecki generalized this for a [[polygon]] of 2n sides.

However, after his father had lost a small [[textile]] workshop, and though he obtained a job as an [[accountant]], the young Kalecki had to search for another job in order to earn some money. During his first year in [[Warsaw]] he continued working sporadic jobs. After finishing his first year of engineering, he had to abandon his studies and from 1918 to 1921 he was in military service. Upon leaving the military he joined [[Gdańsk University of Technology|Gdańsk Polytechnic]], staying there until 1924. Kalecki was then 25 years old.

During these years he first approached [[economics]], although informally. He read mostly "unorthodox" works, particularly those of [[Mikhail Tugan-Baranovsky]] and [[Rosa Luxemburg]]. Years later, this early influence of these two economists would be felt in some of his own writings related to the potential growth of a [[capitalist]] [[system]].

[[File:Myhaylo Tuhan-Baranovsky.jpg|left|200px|thumb|Statue of [[Mikhail Tugan-Baranovsky]], near Donetsk Commercial University. Tugan-Baranovsky was one of the few economists read by the young Kalecki.]]

In 1924 [[Kalecki]] was about to finish his studies when his father lost his job again. This forced him to leave the university again and this time permanently, because he needed to find a better paid job. His first job (which was also economic in nature) was to collect data on companies which asked for credit. In this same period he tried unsuccessfully to start a newspaper and then was forced to write articles for two economic newspapers, the ''Polska gospodarcza'' and ''Przeglad gospodarczy''. It was probably in writing these articles that he began to acquire skills in obtaining and analyzing [[empirical]] information which was later included in his writings.

After five years and many articles he applied in 1929 for work at the Research Institute of Business Cycle and Prices (RIBCP). The experience he had acquired in the use of statistics got him the job. On 18 June 1930 he married Ada Szternfeld. In RIBCP he met Ludwig Landau, whose knowledge of [[statistics]] influenced the way that years later Kalecki presented the statistical part of his works. As a result his first works had a practical character particularly in establishing relationships between macro-magnitudes.
In fact the first article that anticipated many subsequent contributions was published in 1932 in a magazine (which disappeared the same year) called ''Przeglad socjalistyczny'' (Socialist Review), under the pseudonym of Henryk Braun. The article dealt with the subject of the impact of wage cuts during an [[economic downturn]]. It was the first step towards the contributions he would make the following year.

===The revolution of Kalecki and Keynes: 1933–1939===
[[File:Cam colls from johns.jpg|right|thumb|150px|[[Trinity College, Cambridge]].]]
In 1933 [[Kalecki]] wrote an essay that brought together many of the issues which would dominate his thoughts for the rest of his life. This essay was ''Proba teorii koniunktury'' (An Essay on the Theory of the Business Cycle), which was published by the RIBCP and in which for the first time Kalecki was able to develop a comprehensive theory of [[business cycles]]. In October of the same year Kalecki read his essay to the ''International Econometrics Association'' and in 1935 published it in two major journals: ''Revue d’Economie Politique'' and ''[[Econometrica]]''. Although the readers of both journals were not particularly impressed, Kalecki's article received favourable comments from such leading economists as [[Ragnar Frisch]] and [[Jan Tinbergen]].

The following year he was granted a [[scholarship]], which enabled him to travel to [[Sweden]] with his wife, where the followers of [[Knut Wicksell]] were trying to formalize a theory similar to Kalecki's. About that time he learnt of the publication of [[John Maynard Keynes]]'s ''[[The General Theory of Employment, Interest and Money|General Theory]]''. This was most likely his motive for traveling to [[England]]. He first visited the ''[[London School of Economics]]'' and afterward went to [[Cambridge]]. Thus began his friendship with [[Richard Kahn, Baron Kahn|Richard Kahn]], [[Joan Robinson]] and [[Piero Sraffa]], leaving an indelible mark on all of them.
In 1937 he met Keynes. The meeting was cooler than expected, Keynes keeping aloof. Although the conclusions they had arrived at in their respective works were very similar, their characters could not have been more different. Kalecki very graciously neglected to mention that he enjoyed [[Priority right|priority of publication]]. As [[Joan Robinson]] reminds us:
{{quote|“Michal Kalecki’s claim to priority of publication is indisputable. With proper scholarly dignity (which, however, is unfortunately rather rare among scholars) he never mentioned this fact. And, indeed, except for the authors concerned, it is not particularly interesting to know who first got into print. The interesting thing is that two thinkers, from completely different political and intellectual starting points, should come to the same conclusion. For us in Cambridge it was a great comfort.”<ref name="Problems of Economic Dynamics and Planning: Essays in Honour of Michal Kalecki"/>}}

In 1939 Kalecki wrote one of his most important works, ''Essays in the Theory of Economic Fluctuations''. Although his conception changed through the years, all the essential elements of Kaleckian economics were already present in this work: in a sense his subsequent work would consist of mere elaborations on ideas he had already propounded.

While Kalecki was generally enthusiastic about the [[Keynesian revolution]], he predicted that it would not endure, in his article "Political Aspects of Full Employment", which [[Anatole Kaletsky]] has called one of the "most prescient economic paper ever published."<ref name="cap4"/> In the article Kalecki predicted that the full employment delivered by Keynesian policy would eventually lead to a more assertive working class and weakening of the social position of business leaders, causing the elite to use their political power to force the [[Post-war displacement of Keynesianism|displacement of the Keynesian policy]] even though profits would be higher than under a laissez faire system: The erosion of social prestige and political power would be unacceptable to the elites despite higher profits.<ref name = "cap4"/><ref name="Political Aspects of Full Employment"/>

===The war years: 1940–1945===
[[File:John Maynard Keynes.jpg|left|200px|thumb|During his stay in England, Kalecki met [[John Maynard Keynes]] (pictured), and was able to discuss with him some of the ideas they shared.]]

In 1937 his friend Ludwig Landau was ejected from the RIBCP for political motives, which moved [[Kalecki]] to resign in protest and extend his stay abroad. But for this fortuitous fact, the war would have caught Kalecki in Poland and given his [[Jewish]] origins he would probably not have survived.

In any case, the Oxford Institute of Statistics (OIS) hired Kalecki in 1940. His job there consisted mainly of writing reports for the [[British Government]] concerning the management of the [[war economy]]. This did not prevent him from giving the occasional lecture at [[Oxford University]]. However, despite the elaborate reports prepared by Kalecki for the Government (most of them were concerned with the operation of the rationing of goods) the economists working for the Government often disregarded these reports. They did so, among other reasons, because many economists at the OIS were refugees. In the words of G. Feiwel: “As a consequence of this void in historical reporting, Kalecki's work of the war period is far less known than it deserves to be”.<ref name="george"/>

However, some of Kalecki’s major works were written during this period: in 1943 he wrote two articles, one of which dealt with new additions made to traditional [[Business cycle|business cycle theory]]. The second article presented Kalecki’s completely original theory of business cycles caused by political events. The latter was published in 1944 and was based on the premise of full [[employment]]. Actually that article was a compilation of studies by Kalecki and his colleagues at the OIS, who, however, were strongly influenced by Kalecki’s thinking.<ref name="eumed"/>

In 1945 Kalecki left the OIS, upset because he felt that his talents were not sufficiently appreciated. Kalecki displayed great modesty about his work and did not expect to be paid splendid tribute for his accomplishments. He was offended at being discriminated against on account of his immigrant status. In fact, one reason why he left the OIS and was not appointed to a more senior position was that he had not applied to become a British subject. (“Subject” was the term then used in Britain instead of “citizen”.)

===The postwar years: 1945–1968===
Leaving OIS, [[Kalecki]] went to [[París]], where he stayed not long, moving to [[Montreal]] later, where he stayed fifteen months. In July 1946 he accepted the Polish government invitation to head the Central Planning Office of the Ministry of Economics, but he left some months later. At the end of 1946 he decided to move to [[New York]]. He liked the position offered him in the Economic Department of the [[United Nations Secretariat]]. He remained there until 1954, allowing him to develop his work as a political advisor. As in 1945, Kalecki resigned the position as a protest signal. It was argued instead that he was punished on political grounds (a non merited economic planner stance was attributed to him). In any case, the [[Senator Joseph McCarthy]]’s [[witch hunt]] depressed him as many of his closest friends were directly affected. In 1955 he returned to [[Poland]] and basically never went back to work abroad for any extended period.

[[File:Wroclaw AE Lange 2005.jpg|right|300px|thumb|Statue of [[Oskar Lange]], at the Wrocław University of Economics. In Poland, Michał Kalecki and Lange, the other great Polish economist of the time, collaborated in economic seminars.]]

When he arrived in Poland, Kalecki was quite hopeful with the ability to make reforms that were socially advantageous. In 1957 he was appointed chairman of the Committee for the Perspective Plan. The plan had a horizon covering 1961 to 1975, and was basically a practical level embodiment of Kalecki's theories of growth in [[socialist economies]]. However, final plan developed by Kalecki was dismissed by board members as too gloomy and tinged as defeatist. Then things got worse, as related by G. Feiwel:
{{quote|“By 1959 the policy makers had forsaken rationality altogether and had reverted to "hurrah planning". Constraints on the growth rate were disregarded under the spell of optimism engendered by the good performance in 1956-57. Although Kalecki remained with the Commission of the Perspective Plan for another year beyond 1959, all concerned knew that it was a pro forma function. The end of 1958 had marked the beginning of the erosion of his influence.”<ref name="george2"/>| G.Feiwel (1975)}}

Thereafter, Kalecki spent much time of the rest of his life in teaching and research. There were however issues that he had neglected during his time at the Commission. In 1959 he began directing a seminar on the socio-economic problems of the [[Third World]] along with [[Oskar Lange]] and [[Czesław Bobrowski]]. For him the problem was not new, having already written outstanding articles dealing with these issues thoroughly, especially those related to development.

He also devoted this period to the study of [[mathematics]]. In fact, this was in part a continuation of the interest he had when he was young and generalized Pascal’s theorem. The investigations were in [[number theory]] and [[probability]]. For him these periods of time dedicated to mathematics were a release of extreme disappointment caused by the lack of power to help his country in [[economic policy]].

===Later years 1968–1970===
[[File:B-30 ship in shipyard.jpg|left|150px|thumb|[[Gdansk Shipyard]], ca 1972. Kalecki devoted the last years of his life to advising the Polish Government.]]

[[John Maynard Keynes]] had said years before that knowledge of the laws governing the [[capitalist economy]] would make people more prosperous, happy and more responsible respect the economic decisions taken. However, [[Kalecki]] contested this view, arguing that in fact the idea of [[Business cycle|political business cycle]] (governments can force situations to their advantages) seems to point in the opposite direction.
As he grew older, Kalecki was ever more convinced of this, showing an increasingly pessimistic view of humanity.

Michal Kalecki died on 4 April 1970 at the age of 70 years, and although he was bitterly disappointed with all political developments, he lived long enough to see recognition of the value of his many original contributions to economics. In his last visit to [[Cambridge]] he gave a conference where he was greatly applauded for the clearness of his explanation as well as by the trajectory of his life.
Probably one of the most moving paragraphs concerning the end of his life we owe Feiwel, an excellent summary of the life of this remarkable man:
{{quote|“With Michal Kalecki's death, the world lost a unique individual of extremely high principles, powerful energy, and brilliant mind, and economics lost a model and inspiration. His legacy, however, cannot be erased. […] He demanded perfection, or at least an unalloyed commitment to that goal, he could not tolerate slovenly thought or superficial minds, and, most significant, he simply would not compromise his principles. Looking back over his troubled years, Kalecki once made the sad but true observation that the story of his life could be compressed into a series of resignations in protest-against [[tyranny]], [[prejudice]], and [[oppression]].”<ref name="george3"/>}}

==Theoretical contributions==
===The profit equation===
The volume of economic literature written by [[Kalecki]] during his life was very large. Although in most of his articles he returned to the same subjects ([[business cycles]], determinants of investment or [[Planned economy|socialist planning]]), he often did it from a slightly unusual perspective and with original contributions.

The literature written about him has tried to distil certain aspects of his work, which already can be considered almost a reference in studies on his theoretical work.

Kalecki's most famous contribution is his profit equation. Kalecki, whose early influences always came from [[Marxian economics|Marxian economists]],<ref name="google4"/> thought that the volume and profit sharing in a capitalist society were vital points to be treated. This followed from [[Marx]]'s work on certain relationships such as the rate of [[surplus value]] or the [[organic composition of capital]] (and even a forecast about the overall trend of [[Profit (economics)|profits]]). However, Marx was never able to make any meaningful statement about the total volume of profits in a given period.

Kalecki derived this relationship in an extremely concise, elegant and intuitive way. He starts by making simplifications which he later progressively eliminates. These assumptions are:

* Divide the whole economy into two groups: [[workers]], who earn only [[wages]] and [[capitalists]], who earns only profits.
* [[Workers]] do not save.
* The economy is closed (there is no [[international trade]]) and there is no [[public sector]].

With these assumptions Kalecki derives the following [[accounting identity]]:

<math> \textstyle P+W=C_W+C_P+I \,\ </math>

where <math>P \,\ </math> is the volume of gross profits (profits plus depreciation), <math>W \,\ </math> is the volume of total wages, <math>C_P \,\ </math> is capitalists consumption, <math>C_W \,\ </math> is workers consumption and <math>I \,\ </math> is the gross investment that have been made in the economy. Since we have supposed workers who do not save (that is, to say <math>W=C_W \,\ </math> in the preceding equation), we can simplify the two terms and arrive at:

<math> \textstyle P=C_P+I</math>

This is the famous profits equation, which says that profits are equal to the sum of [[investment]] and capitalist’s consumption.

At this point, Kalecki goes on to determine the causal link between the two sides of the equation: Does capitalist’s consumption and investment determine profits or profits instead determine capitalist consumption and investment? In answer to this, Kalecki says

{{quote|“The answer to this question depends on which of these items is directly subject to the decisions of capitalists. Now, it is clear that capitalists may decide to consume and to invest more in a given period than in the preceding one, but they cannot decide to earn more. It is, therefore, their investment and consumption decisions which determine profits, and not vice versa”.<ref name="kalecki"/>|}}

For someone who has not seen before the preceding relationship, it might after a rigorous examination seem somewhat paradoxical. If the capitalists consume more, obviously the amount of funds which they have at the end of the year should be less. However, this reasoning, obvious to the individual entrepreneur, is not true for the business class as a whole, as the consumption of one capitalist becomes part of the profits of another. In a way, ''they are masters of their fate''.<ref name="maurice"/>

On the other hand, we should point out that if in the preceding equation we move capitalist consumption to the left, the equation becomes:

<math> \textstyle S=I\,\ </math>

Since profits minus capitalist’s consumption are the total saving in the economy because the workers do not save. The previous causal relationship still applies, and goes from investment to saving. That is to say, total savings are determined once investment has been determined. So, in some way, investment generates sufficient resources. ''Investment finances itself'',<ref name="kalecki5"/> so that equality between savings and investment is not caused by any interest rate mechanism as earlier economists thought.
Finally, we can eliminate the assumptions of the original equation: the economy can be open, there may be a [[Public sector|government sector]] and we can let [[workers]] save something. The resulting equation is:

<math> \textstyle P_N=C_P+I+D_g+E_e-S_w\,\ </math>

In this model total profits (net taxes this time) are the sum of capitalist consumption, investment, public deficit, net external surplus (exports minus imports) minus workers savings.
Before trying to explain income distribution, Kalecki introduces some behavioural assumptions in his simplified equation of profits. For him, investment is determined by a combination of many factors difficult to explain, which are considered given, [[exogenous]]. Regarding capitalist consumption, he considers that a simplified form is the following equation:

<math> \textstyle C_P=A+q \cdot P\,\ </math>

That is, capitalist consumption depends on a fixed part (independent part), the term <math> A \,\ </math>, and a proportional share of profits, the term <math>q \,\ </math>, which is called the [[marginal propensity to consume]] of the capitalists. If this consumption function is substituted into the profit equation, we have:

<math> \textstyle P=A+q \cdot P+I\,\ </math>

And, if we finally express in terms of <math> P \,\ </math>, this give us:

<math> \textstyle P= \frac{A+I}{1-q} \,\ </math> 

The advantage of this manipulation is that we have reduced the two earnings determinants (capitalist’s consumption and investment) to only one (investment).

===Income distribution and the constancy of the share of wages===
Income distribution is the other pillar of Kalecki’s efforts to build a business cycle theory. To do this, [[Kalecki]] assumes that the industries compete in imperfectly competitive markets, more particularly in [[oligopolistic]] markets where the firms set a [[markup (business)|mark-up]] on its variable average costs (raw materials, wages of employees on the shop floor that are supposed to be variable) in order to cover their [[overhead (business)|overhead]] costs (salaries to senior management and administration) to obtain a certain amount of profit. The mark-up fixed by firms is higher or lower depending on the degree of monopoly, or the ease with which firms raise the price without seeing reduced the quantity demanded.
This can be summarized in the next equation:

<math> \textstyle P+B=k \cdot (W+M) \,\ </math>

where <math>P\,\ </math> and <math>W\,\ </math> are respectively again profits and wages, <math>k\,\ </math> is the average mark-up for the whole economy, <math>M\,\ </math> is the cost of raw materials and <math>B\,\ </math> is the total amount of salaries (which must be distinguished from wages, since these are variables and salaries are considered fixed).
The preceding equation allows us to derive the wage share in the national income. If we add to both members <math>W\,\ </math>, and pass one to the other side, we have:

<math> \textstyle Y=k \cdot (W+M)+W \,\ </math>

If we multiply each side by <math>W\,\ </math>, and we pass to the other term, we have:

<math> \alpha= \frac{W}{W+k \cdot (W+M)} \,\ </math>

or, which is the same:

<math> \alpha= \frac{1}{1+k \cdot (1+j)} \,\ </math>

where <math>\alpha \,\ </math> is the wage share in the national income and <math>j \,\ </math> is the relation between the cost of raw materials and wages. It follows that the wage share in the national income depends negatively on mark-up and on the relationship of raw material costs to wages.
At this point Kalecki’s interest is in finding out what’s happen to the wage share during the [[business cycle]]. During recessions, firms collaborate among themselves to cope with the fall of profits, so the degree of [[monopoly]] increases and this increases the mark-up. The <math>k \,\ </math> parameter goes up. Nonetheless, the lack of demand during recessions causes a fall in the price of raw materials, so the <math>j \,\ </math> parameter goes down. The argument is symmetrical during the boom: prices of raw materials rise (<math>j \,\ </math> parameter increases) meanwhile the strength of unions due to increased occupation of labour causes the degree of [[monopoly]] to fall and thereby the mark-up level. We conclude therefore that α parameter is roughly constant over the [[business cycle]].

Finally, we need an equation that determines the total product of an economy. We can state that:

<math> 1-\alpha= \frac{P+B}{Y} \,\ </math>

that is simply to say that the share of profits and salaries are the complement of the share of wages. Solving for<math>Y \,\ </math> we have:

<math> Y= \frac{P+B}{1-\alpha} \,\ </math>

Now we have the three components necessary to determine total product: an equation of profits, a theory of income distribution and an equation that links the product with profits and income distribution. Now it only remains to substitute the equation of <math>P \,\ </math> which we obtained before:

<math> Y= \frac{\frac{I+A}{1-q}+B}{1-\alpha} \,\ </math>

The preceding equation finally shows the determination of income in a closed system without [[public sector]]. It shows that output is completely determined by [[investment]]. How will output change from one period to the next? Insofar we have assumed that <math>A \,\ </math> and <math>B \,\ </math> are constants, the above formula comes down to the [[multiplier (economics)|multiplier]]:

<math> \Delta Y= \frac{\Delta I}{(1-q) \cdot (1-\alpha)} \,\ </math>

That is to say, the problem of the change in output and hence the [[business cycle]] it is due to changes in the volume of [[investment]]. Therefore it is in [[investment]] where we must find the reasons for the fluctuations of a [[capitalist economy]].

===Determinants of investment===
All this allow us to see the crucial role played by [[investment]] in a [[Capitalism|capitalist system]]. If we can find an [[investment function]] which somehow was well specified, then surely we could resolve many problems of the [[capitalist economy]]. This subject was treated for a long time by [[Kalecki]], and he was never completely satisfied with his solutions. This is because the factors that determine the [[investment]] decisions are multiple and not always are clear.
A thorough analysis of this subject would be intolerably long, so the best way to continue is to introduce the solution that Kalecki gave in one of his books.

Kalecki’s [[investment function]] in the study of [[business cycles]] is the following:

<math> D=a \cdot S+b \cdot \frac {\Delta P}{\Delta t}-c \cdot\frac {\Delta K}{\Delta t}+d \,\ </math>

where <math>D \,\ </math> is the amount of investment decisions in fixed capital, <math>a \,\ </math>, <math>b \,\ </math> and <math>c \,\ </math> are parameters that specify a linear relation, <math>d \,\ </math> is a constant which can vary in the long-run, <math>P \,\ </math> are [[Profit (economics)|profits]], <math>S \,\ </math> is the gross saving generated by the [[firm]], and <math>K \,\ </math> is the stock of [[fixed capital]].
The previous equation shows that investment decisions depend positively on the savings generated by the firm, on the rate of change of profits and a constant subject to long-term changes, and negatively on the increase of fixed capital.

We can see that the above equation is able to generate cycles by itself. During booms, firms are able to generate more [[cash-flow]] and enjoy increases in profits. However, the increase in orders for capital investment increases the stock of capital, until it proves unprofitable to make more investments. Ultimately, the variations in the level of investment generate the business cycles. As [[Kalecki]] would say later:
{{quote|“The tragedy of investment is that it causes crisis because it is useful. Doubtless many people will consider this paradoxical. But it is not the theory which is paradoxical, but its subject –the capitalist economy.”<ref name="dobb"/>|}}

==Influence==
In the first half of the 1990s, [[Oxford University Press]] published 7 volumes of the ''Collected Works of Michal Kalecki'', referring to him as "one of the most distinguished economists of the 20th century." Many of his works were translated into English for the first time in this collection.

Kalecki's work would inspire the Cambridge (UK) Post-Keynesians, especially [[Joan Robinson]], [[Nicholas Kaldor|Kaldor]] and [[Richard M. Goodwin|Goodwin]], as well as modern American Post Keynesian economics.<ref name="hetw"/>

==Publications==
=== In Polish ===
* ''Próba teorii koniunktury'' (1933)
* ''Szacunek dochodu społecznego w roku 1929'' (1934, z Ludwikiem Landauem)
* ''Dochód społeczny w roku 1933 i podstawy badań periodycznych nad zmianami dochodu'' (1935, z Landauem)
* ''Teoría cyklu koniunkturalnego'' (1935)
* ''Płace nominalne i realne'' (1939)
* ''Teoría dynamiki gospodarczej'' (1958)
* ''Zagadnienia finansowania rozwoju ekonomicznego'' (1959, w: Problemy wzrostu ekonomicznych krajów słabo rozwiniętych pod redakcją Ignacego Sachsa i Jerzego Zdanowicza)
* ''Uogólnienie wzoru efektywności inwestycji'' (1959, z Mieczysławem Rakowskim)
* ''Polityczne aspekty pełnego zatrudnienia'' (1961)
* ''O podstawowych zasadach planowania wieloletniego'' (1963)
* ''Zarys teorii wzrostu gospodarki socjalistycznej'' (1963)
* ''Model ekonomiczny a materialistyczne pojmowanie dziejów'' (1964)
* ''Dzieła'' (1979–1980, 2 tomy)

=== In English ===
* "Mr Keynes's Predictions", 1932, ''Przegląd Socjalistyczny''.
* An Essay on the Theory of the Business Cycle (''Próba teorii koniunktury''), 1933.
* "On foreign trade and domestic exports", 1933, ''Ekonomista''.
* "Essai d'une theorie du mouvement cyclique des affaires", 1935, ''Revue d'economie politique''.
* "A Macrodynamic Theory of Business Cycles", 1935, ''Econometrica''.
* "The Mechanism of Business Upswing" (El mecanismo del auge económico), 1935, ''Polska Gospodarcza''.
* "Business upswing and the balance of payments" (El auge económico y la balanza de pagos), 1935, ''Polska Gospodarcza''.
* "Some Remarks on Keynes's Theory", 1936, ''Ekonomista''.
* "A Theory of the Business Cycle", 1937, ''Review of Economic Studies''.
* "A Theory of Commodity, Income and Capital Taxation", 1937, ''Economic Journal''.
* "The Principle of Increasing Risk", 1937, ''Económica''.
* "The Determinants of Distribution of the National Income", 1938, ''Econometrica''.
* ''Essays in the Theory of Economic Fluctuations'', 1939.
* "A Theory of Profits", 1942, ''Economic Journal''.
* ''Studies in Economic Dynamics'', 1943.
* [http://www.cfeps.org/ss2006/readings/Courvisanos_c.pdf "Political Aspects of Full Employment", 1943, ''Political Quarterly''].
* ''Economic Implications of the Beveridge Plan'' (1943)
* "Professor Pigou on the Classical Stationary State", 1944, ''Economic Journal''.
* "Three Ways to Full Employment", 1944 in ''Economics of Full Employment''.
* "On the Gibrat Distribution", 1945, ''Econometrica''.
* "A Note on Long Run Unemployment", 1950, ''Review of Economic Studies''.
* ''Theory of Economic Dynamics: An essay on cyclical and long-run changes in capitalist economy'', 1954. 1965 reprint.
* "Observations on the Theory of Growth", 1962, ''Economic Journal''.
* ''Studies in the Theory of Business Cycles, 1933–1939'', 1966.
* "The Problem of Effective Demand with Tugan-Baranovski and Rosa Luxemburg", 1967, ''Ekonomista''.
* "The Marxian Equations of Reproduction and Modern Economics", 1968, ''Social Science Information''.
* "Trend and the Business Cycle", 1968, ''Economic Journal''.
* "Class Struggle and the Distribution of National Income" (Lucha de {{Not a typo|clases}} y distribución del ingreso), 1971, ''Kyklos''.
* ''Selected Essays on the Dynamics of the Capitalist Economy'', 1933–1970, 1971.
* ''Selected Essays on the Economic Growth of the Socialist and the Mixed Economy'', 1972.
* ''The Last Phase in the Transformation of Capitalism'', 1972.
* ''Essays on Developing Economies'', 1976.
* ''Collected Works of Michał Kalecki'' (seven volumes..), Oxford University Press, 1990–1997.

=== In Spanish ===
* ''Teoría de la dinámica económica: ensayo sobre los movimientos cíclicos y a largo plazo de la economía capitalista'', Fondo de Cultura Económica, 1956
* ''El Desarrollo de la Economía Socialista'', Fondo de Cultura Económica, 1968
* ''Estudios sobre la Teoría de los Ciclos Económicos'', Ariel, 1970
* ''Economía socialista y mixta: selección de ensayos sobre crecimiento económico'', Fondo de Cultura Económica, 1976
* ''Ensayos escogidos sobre dinámica de la economía capitalista 1933–1970'', Fondo de Cultura Económica, 1977
* ''Ensayos sobre las economías en vías de desarrollo'', Crítica, 1980

==See also==
* [[Neo-Ricardianism]]
* [[Monopoly Capital]]
* [[Cost-of-production theory of value]]

==References==
{{reflist|refs=

<ref name = "cap4">{{cite book |author = [[Anatole Kaletsky]] |title=Capitalism 4.0: The Birth of a New Economy |year=2011 |pages = 52–53, 173, 262 |isbn=978-1-4088-0973-0 |publisher= Bloomsbury }}</ref>

<ref name="dobb">Dobb (1973), p. 222</ref>

<ref name="economicoutlook">[[Bill Mitchell (economist)|Bill Mitchell]] (August 13, 2010), [http://bilbo.economicoutlook.net/blog/?p=11127 "Michal Kalecki – The Political Aspects of Full Employment"]: "... those scholars who do not see Keynes as being the central figure in the development of the theory of effective demand [...] lean to the view that the transition from the ''[[A Treatise on Money|Treatise]]'' (1930) to the ''[[The General Theory of Employment, Interest and Money|General Theory]]'' (1936) was so great that it is likely that Keynes knew what Kalecki had written and published and was influenced by it."</ref>

<ref name="eumed">[http://www.eumed.net/ce/2011a/jfbr3.htm Confianza, Reformas y Crisis Económica]</ref>

<ref name="george">George R. Feiwel (1975). p. 239.</ref>

<ref name="george2">George R. Feiwel (1975).: p. 297.</ref>

<ref name="george3">[[George Feiwel]]: 1975, p.455.</ref>

<ref name="google">[[Don Patinkin]], [http://books.google.gr/books?id=dlyLxR56kLIC&dq ''Anticipations of the General Theory?: And Other Essays on Keynes''], University of Chicago Press, 1984, p. 61.</ref>

<ref name="google1">[[Jonathan Nitzan]] and [[Shimshon Bichler]]. ''[http://books.google.gr/books?id=-qRKIpvTO6IC&dq= Capital as power: a study of order and creorder]''. Taylor & Francis, 2009, p. 50–51.</ref>

<ref name="google4">Alessandro Roncaglia, [http://books.google.gr/books?id=o_-hlxiJPbkC&dq= ''The wealth of ideas: a history of economic thought''], Cambridge University Press, 2005, p. 411.</ref>

<ref name="kalecki">Michal Kalecki (1971), p. 78-79.</ref>

<ref name="kalecki5">Michal Kalecki (1971), p.84.</ref>

<ref name="maurice">Maurice Dobb (1973), p.223.</ref>

<ref name="hetw">"Michal Kalecki, 1899–1970". ''Profiles''. ''[http://www.hetwebsite.org/het/profiles/kalecki.htm The History of Economic Thought]''. ''[[The New School]]''. Retrieved 12 March 2009. http://www.hetwebsite.org/het/profiles/kalecki.htm</ref>

<ref name="Political Aspects of Full Employment">{{cite web |url= http://mrzine.monthlyreview.org/2010/kalecki220510.html |title= Political Aspects of Full Employment |work= [[Monthly Review]] |publisher = [[The Political Quarterly]] |author= Kalecki |year = 1943 |accessdate=2012-05-02 }}</ref>

<ref name="Problems of Economic Dynamics and Planning: Essays in Honour of Michal Kalecki">{{cite book |title=Problems of Economic Dynamics and Planning: Essays in Honour of Michal Kalecki |chapter=Kalecki and Keynes |last=Robinson |first=Joan |authorlink=Joan Robinson |year=1966 |page=337 |publisher=Polish Scientific Publishers}}</ref>
}}

==Sources==
* [[Maurice Dobb|Dobb, Maurice]] (1973). ''Theories of value and distribution since Adam Smith''. Cambridge University Press.
* Feiwel, George R. (1975). ''The Intellectual Capital of Michal Kalecki: A Study in Economic Theory and Policy''. Fondo de Cultura Económica, México.
* Kalecki, M. (2009) ''Theory of Economic Dynamics: An Essay on Cyclical and Long-Run Changes in Capitalist Economy'', Monthly Review Press
* Kalecki, M. ''Collected Works of Michal Kalecki'', Oxford University Press, USA.
* Kalecki, M. (1971) ''Selected essays on the dynamics of the capitalist economy'', Cambridge University Press.

==Further reading==
* {{cite book |title=Kalecki's Economics Today |last=Sadowski |first=Zdzislaw L. |authorlink= |coauthors=Szeworski, Adam |year=2004 |publisher=Routledge |location=London |isbn=0-415-29993-4 |pages= |url= }}
* {{cite book |chapter=An economist from Poland |title=A History of Post Keynesian Economics Since 1936 |last=King |first=J. E. |authorlink= |coauthors= |year=2003 |publisher=Edward Elgar |location=Cheltenham |isbn=1-84376-650-7 |pages=35–55 |url= }}
* {{cite book |chapter=Keynes, Kalecki and The General Theory |title=A "Second Edition" of the General Theory |last=Kriesler |first=Peter |authorlink= |editors=Harcourt, Geoffrey Colin; Riach R. A.; Riach, P. A. (ed.) |year=1997 |publisher=Routledge |location=London |isbn=0-415-14943-6 |pages= |url= }}
* [[Fernando Vianello|Vianello]], F. [1989], “Effective Demand and the Rate of Profits: Some Thoughts on [[Karl Marx|Marx]], Kalecki and [[Piero Sraffa|Sraffa]]”, in: Sebastiani, M. (ed.), ''Kalecki's Relevance Today'', London, Macmillan, ISBN 978-03-12-02411-6.

==External links==
{{wikiquote|Michal Kalecki}}
* [http://www.hetwebsite.org/het/profiles/kalecki.htm Biography]
* [http://www.hetwebsite.org/het/essays/multacc/kalecyc.htm Kalecki Distribution Cycle]
* [http://csf.colorado.edu/pkt/pktauthors/Kriesler.Peter/Keynes,Kalecki-and-General-Theory Peter Kriesler's Keynes, Kalecki and the General Theory"]
* [http://csf.colorado.edu/pkt/pktauthors/Kriesler.Peter/microfoundations-a-kaleckianperspective Peter Kriesler's "Microfoundations: A Kaleckian perspective"]
* [http://wueconb.wustl.edu/eprints/mac/papers/9805/9805001.abs Malcolm Sawyer's "The Kaleckian Analysis and the New Mellium"]
* [http://www.u-paris10.fr/ActuelMarx/economarx/ab120601.htm Review (by Gary Dymski) of Sebastiani's book, Kalecki and Unemployment Equilibrium in JEL]
* [http://ideas.repec.org/p/wpa/wuwpmh/0305001.html Alberto Chilosi "Kalecki's Theory of Income Determination: A Reconstruction and an Assessment"]
* [http://digamo.free.fr/kalecki2010.pdf Julio López G. and Michaël Assous"Michal Kalecki"]
* [http://monthlyreview.org/press/books/pb0818/ Theory of Economic Dynamics] – ''[[Monthly Review]]''
* [http://www.featuredplus.ca/Degree%20of%20monopoly%20and%20class%20struggle%20political%20aspects%20of%20Kalecki%E2%80%99s%20pricing%20and%20distribution%20theory.pdf Kalecki’s pricing and distribution theory F.M. Rugitsky]
{{Authority control|VIAF=76331472}}
{{Persondata 
| NAME = Kalecki, Michal
| ALTERNATIVE NAMES = Kalecki, Michał
| SHORT DESCRIPTION = Polish economist
| DATE OF BIRTH = June 22, 1899
| PLACE OF BIRTH = Łódź, Poland
| DATE OF DEATH = April 18, 1970
| PLACE OF DEATH = Warsaw, Poland
}}
{{DEFAULTSORT:Kalecki, Michal}}
[[Category:1899 births]]
[[Category:1970 deaths]]
[[Category:People from Łódź]]
[[Category:People from Congress Poland]]
[[Category:Polish Jews]]
[[Category:Polish economists]]
[[Category:Marxian economists]]
[[Category:Post-Keynesian economists]]
[[Category:Gdańsk University of Technology alumni]]

Schrödinger equation

2009-02-12T02:19:16Z

131.215.30.140: /* Variational principle */

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Schrödinger equation

2008-03-16T21:33:23Z

131.215.220.132: /* Historical background and development */

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