Mathematical Foundations for Computational Engineering: A Handbook
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books.google.com/books?isbn=3540679952
Peter Jan Pahl, Rudolf Damrath - 2001 - Computers
Canonical decomposition of a permutation into cycles : Every permutation 4> of a symmetric group Sn may be represented as a product of disjoint cycles ify .
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mathworld.wolfram.com › ... › Combinatorics › Permutations
sorted in lowest canonical order (first by cycle length, and then by lowest ... The cyclic decomposition of a permutation can be computed in Mathematica with ... symbols can be uniquely expressed as a product of disjoint cycles (Skiena 1990, p.
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en.wikipedia.org/wiki/Cycle_notation
For the cyclic decomposition of graphs, see Cycle decomposition (graph theory) ... convention for writing down a permutation in terms of its constituent cycles. ... 1 Definition; 2 Permutation as product of cycles; 3 Example; 4 See also; 5 Notes; 6 References ... Since disjoint cycles commute with each other, the meaning of this
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www.math.jhu.edu/mathcourses/401/Homework/signature.pdf
Sep 24, 2008 - product of transpositions is not unique! However, the parity ... considering the canonical decomposition of σ into disjoint cycles. Let assume that ...
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uk.answers.yahoo.com › ... › Science & Mathematics › Mathematics
I understand how to put a permutation into a product of disjoint cycles however I ... I have a few examples but I just don't understand how these are decomposed!
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www.proofwiki.org/wiki/Definition:Permutation_on.../Cycle_Notation
Jun 9, 2013 - 1.1 Canonical Representation ... As Disjoint Permutations Commute, the order in which they are performed does not matter. ... of all the disjoint cycles into which $\
Mathematical Foundations for Computational Engineering: A Handbook
books.google.com/books?isbn=3540679952
Peter Jan Pahl, Rudolf Damrath - 2001 - Computers
Canonical decomposition of a permutation into cycles : Every permutation 4> of a symmetric group Sn may be represented as a product of disjoint cycles ify .- mathworld.wolfram.com › ... › Combinatorics › Permutations
- en.wikipedia.org/wiki/Cycle_notation
- www.math.jhu.edu/mathcourses/401/Homework/signature.pdf
- uk.answers.yahoo.com › ... › Science & Mathematics › Mathematics
- www.proofwiki.org/wiki/Definition:Permutation_on.../Cycle_Notation
- arxiv.org/pdf/1306.5708
- by R Moreno - 2013
- Jun 24, 2013 - blocks in cycles in the decomposition of β as a product of disjoint cycles. ... 3 Permutations that k-commute with a cycle of a permutation ...... Let π be a permutation written in canonical cycle notation, the transition function of π.
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- chronicle.com/.../cycles-and-the-cycle-decomposition-of-a-permutation/
Nov 28, 2011 - Let's go back to a simpler permutation to get the basic concept. ... a product of disjoint cycles like this, it's tantamount to making a table of values ...
- www.math.ucsd.edu/~revans/Even.pdf
- by JL BRENNER - 1987 - Cited by 3 - Related articles
- n> 15, every permutation in A,, is the product of two elements of order 5 in A,. .... the canonical factorization of P into disjoint cycles Ci in S,,; each factor Q,.
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- groupprops.subwiki.org/.../Cycle_decomposition_theorem_for_permutat...
Aug 18, 2008 - Any permutation on a finite set admits a cycle decomposition: it can be expressed as a product of a finite number of pairwise disjoint cycles. ... with the existence of a rational canonical form and a Jordan canonical form. A common framework that unifies results in both contexts is the framework of APS theory.
The decomposition of a permutation into a product of transpositions?
I understand how to put a permutation into a product of disjoint cycles however I don't understand how how then put this into a product of transpositions.
I have a few examples but I just don't understand how these are decomposed!
1) (1 4 5 2 6)(3) = (1 2)(1 5)(2 6)(1 4)
Why is this the correct answer?
and..
2) (1 3 4 2 5) = (1 5)(1 2)(1 4)(1 3)
Again, I just don't understand why this is the answer!
Please help, my final exam is tomorrow and there is a question like this in every past paper!
Thanks in advance!
I have a few examples but I just don't understand how these are decomposed!
1) (1 4 5 2 6)(3) = (1 2)(1 5)(2 6)(1 4)
Why is this the correct answer?
and..
2) (1 3 4 2 5) = (1 5)(1 2)(1 4)(1 3)
Again, I just don't understand why this is the answer!
Please help, my final exam is tomorrow and there is a question like this in every past paper!
Thanks in advance!
Best Answer - Chosen by Voters
It seems that what you are asking is how to go from a long cycle into a product of transpositions (which is a 2-cycle). I'll show this by example, let's do (1 3 4 2 5).
First, write down the digits in order:
1 2 3 4 5. You know that "1 goes to 3" (at least that's how I say it in my head). That means that in the next line, you should have the three go below the 1, by swapping the 1 and the 3. It looks like:
1 2 3 4 5 and now do (1 3) to get the 3 under the 1:
3 2 1 4 5. So far so good, we went from 1 to 3. Now we want to go from 3 to 4 (the next element in the original cycle). That means that you want to get a 4 (in the 3rd row - to be created) underneath the 3 of the first row. To do that swap the 1 and 4 in the 2nd row (the row above the one being created).
1 2 3 4 5 and now do (1 3) to get the 3 under the 1:
3 2 1 4 5 and now do (1 4) to get the 4 under the 3
3 2 4 1 5. So far so good. Now look at the destination cycle (1 3 4 2 5) to see that you now want to go from 4 (in the first row) to 2 in the next row to be created. Therefore, swap the 2 and 1 in the 3rd row (the row above the one being created):
1 2 3 4 5 and now do (1 3) to get the 3 under the 1:
3 2 1 4 5 and now do (1 4) to get the 4 under the 3:
3 2 4 1 5 and now do (1 2) to get the 2 under the 4:
3 1 4 2 5. Finally, the 2 wants to go to 5. In other words, in the next row to be created, the 5 should appear underneath the 2 of the first row, which you get by swapping the 1 and the 5 in the 4ht row, the row above the row to be created:
1 2 3 4 5. (1 3) =>
3 2 1 4 5. (1 4) =>
3 2 4 1 5. (1 2) =>
3 1 4 2 5. (1 5) =>
3 5 4 2 1
So what we saw was going from a 'long cycle' to transpositions. Start with the elements of the 'long cycle' in order, and get the first element to go to whatever follows it in the long cycle by swapping those two elements. Remember to write down the transpositions in backwards order. And also, remember that the way to "read" these cycles is to look at the top row and see what winds up on the bottommost row below a given element. Then find that bottommost element in the top row and see what lies below it in the bottommost row. Keep repeating till you cycle back to the starting element.
Now, what about if you have the transpositions and want to get to the cycle (ie. you want to know what cycle it corresponds to). The answer is that you do the same thing. Write down
1 2 3 4 5 and now apply the transpositions, in turn, going from right to left (ie. starting with (1 3) and finishing with (1 5) in the 2nd example). You get the same sequence as before. That's not surprising because that's how the two are equivalent.
By the way, regarding the test. Once you understand it this process is very mechanical, but until you get the mechanics, it can be really confusing. That's why I didn't do the first example. You should be able to get it using the step by step process I showed you. If you go into the test without this process down pat, it is really easy to crash and burn (ie. Just reading my explanation is not sufficient) so my suggestion is that you work a few examples.
Here's one: What 'long cycle' is represented in going from the first row to the 2nd one:
G H I R T
H T G I R
Nobody said they must be numbers. Hopefully you'll realize when you've got it right. :)
Well, it would be nice to hear how your test went.
First, write down the digits in order:
1 2 3 4 5. You know that "1 goes to 3" (at least that's how I say it in my head). That means that in the next line, you should have the three go below the 1, by swapping the 1 and the 3. It looks like:
1 2 3 4 5 and now do (1 3) to get the 3 under the 1:
3 2 1 4 5. So far so good, we went from 1 to 3. Now we want to go from 3 to 4 (the next element in the original cycle). That means that you want to get a 4 (in the 3rd row - to be created) underneath the 3 of the first row. To do that swap the 1 and 4 in the 2nd row (the row above the one being created).
1 2 3 4 5 and now do (1 3) to get the 3 under the 1:
3 2 1 4 5 and now do (1 4) to get the 4 under the 3
3 2 4 1 5. So far so good. Now look at the destination cycle (1 3 4 2 5) to see that you now want to go from 4 (in the first row) to 2 in the next row to be created. Therefore, swap the 2 and 1 in the 3rd row (the row above the one being created):
1 2 3 4 5 and now do (1 3) to get the 3 under the 1:
3 2 1 4 5 and now do (1 4) to get the 4 under the 3:
3 2 4 1 5 and now do (1 2) to get the 2 under the 4:
3 1 4 2 5. Finally, the 2 wants to go to 5. In other words, in the next row to be created, the 5 should appear underneath the 2 of the first row, which you get by swapping the 1 and the 5 in the 4ht row, the row above the row to be created:
1 2 3 4 5. (1 3) =>
3 2 1 4 5. (1 4) =>
3 2 4 1 5. (1 2) =>
3 1 4 2 5. (1 5) =>
3 5 4 2 1
So what we saw was going from a 'long cycle' to transpositions. Start with the elements of the 'long cycle' in order, and get the first element to go to whatever follows it in the long cycle by swapping those two elements. Remember to write down the transpositions in backwards order. And also, remember that the way to "read" these cycles is to look at the top row and see what winds up on the bottommost row below a given element. Then find that bottommost element in the top row and see what lies below it in the bottommost row. Keep repeating till you cycle back to the starting element.
Now, what about if you have the transpositions and want to get to the cycle (ie. you want to know what cycle it corresponds to). The answer is that you do the same thing. Write down
1 2 3 4 5 and now apply the transpositions, in turn, going from right to left (ie. starting with (1 3) and finishing with (1 5) in the 2nd example). You get the same sequence as before. That's not surprising because that's how the two are equivalent.
By the way, regarding the test. Once you understand it this process is very mechanical, but until you get the mechanics, it can be really confusing. That's why I didn't do the first example. You should be able to get it using the step by step process I showed you. If you go into the test without this process down pat, it is really easy to crash and burn (ie. Just reading my explanation is not sufficient) so my suggestion is that you work a few examples.
Here's one: What 'long cycle' is represented in going from the first row to the 2nd one:
G H I R T
H T G I R
Nobody said they must be numbers. Hopefully you'll realize when you've got it right. :)
Well, it would be nice to hear how your test went.
如何理解“正则/canonical”？
作者: cenwanglai (站内联系TA) 发布: 2011-09-12
中文正则，从canonical翻译过来的？不是音译呢。有没有词源？如何理解？
linux中有正则表达式
物理化学中有正则系综。
正则是什么意思？为什么把一些类型命名为“正则”？
正则似乎来源于理论力学。请路过的人给予提示。
linux中有正则表达式
物理化学中有正则系综。
正则是什么意思？为什么把一些类型命名为“正则”？
正则似乎来源于理论力学。请路过的人给予提示。
有可能是从canonical 翻译来的，意思为“ 标准的，权威的, 典型的”，比如正则表达式就是“在计算机科学中，是指一个用来描述或者匹配一系列符合某个句法规则的字符串的单个字符串”的意思，（呵呵，俺瞎猜的，纯粹凑热闹，希望知道的虫子指教~）
Mathematics
Mathematicians have for perhaps a century or more used the word canonical to refer to concepts that have a kind of uniqueness or naturalness. Examples include the canonical prime factorization of positive integers, the Jordan canonical form of matrices (that is built out of the irreducible factors of the characteristic polynomial of the matrix), and the canonical decomposition of a permutation into a product of disjoint cycles. Various functions in mathematics are also canonical, like the canonical homomorphism of a group onto any of its quotient groups, or the canonical isomorphism between a finite-dimensional vector space and its double dual. Although a finite-dimensional vector space and its dual space are isomorphic, there is no canonical isomorphism. This lack of a canonical isomorphism can be made precise in terms of category theory; see natural transformation. But at a simpler level one could say that "any isomorphism you can think of here depends on choosing a basis." As stated by Goguen, "To any canonical construction from one species of structure to another corresponds an adjunction between the corresponding categories."
Being canonical in mathematics is stronger than being a conventional choice. For instance, the vector space Rn has a standard basis, which is canonical in the sense that it is not just a choice that makes certain calculations easy; in fact most linear operators on Euclidean space take on a simpler form when written as a matrix relative to some basis other than the standard one (see Jordan form). In contrast, an abstract n-dimensional real vector space V would not have a canonical basis; it is isomorphic to Rn of course, but the choice of isomorphism is not canonical.
The word canonical is also used for a preferred way of writing something, see the main article canonical form.
In set theory, the term "canonical" identifies an element as representative of a set. If a set is partitioned into equivalence classes, then one member can be chosen from each equivalence class to represent that class. That representative member is the canonical member. If you have a canonicalizing function, f(x), that maps x to the canonical member of the equivalence class which contains it, then testing whether two items, a and b, are equivalent is the same as testing whether f(a) is identical to f(b).
Physics
In theoretical physics, the concept of canonical (or conjugate, or canonically conjugate) variables is of major importance. They always occur in complementary pairs, such as spatial location x and linear momentum p, angle φ and angular momentum L, and energy E and time t. They can be defined as any coordinates whose Poisson brackets give a Kronecker delta (or a Dirac delta in the case of continuous variables). The existence of such coordinates is guaranteed under broad circumstances as a consequence of Darboux's theorem. Canonical variables are essential in the Hamiltonian formulation of physics, which is particularly important in quantum mechanics. For instance, the Schrödinger equation and the Heisenberg uncertainty relation always incorporate canonical variables. Canonical variables in physics are based on the aforementioned mathematical structure and therefore bear a deeper meaning than being just convenient variables. One facet of this underlying structure is expressed by Noether's theorem, which states that a (continuous) symmetry in a variable implies an invariance of the conjugate variable, and vice versa; for instance symmetry under spatial displacement leads to conservation of momentum, and time-independence implies energy conservation.
In statistical mechanics, the grand canonical ensemble, canonical ensemble, and the microcanonical ensemble are archetypal probability distributions for the (unknown) microscopic state of a thermal system, applying respectively in the physical cases of:
An open system at fixed temperature (able to exchange both energy and particles with the environment);
A closed system at fixed temperature (able to exchange energy with its environment);
A closed thermally isolated system (able to exchange neither). These probability distributions can be applied directly to practical problems in thermodynamics.
Occasionally, certain derivations of important ideas from more basic facts are known as "canonical derivations", and can be contrasted with "alternative derivations". One example from statistical mechanics is the canonical derivation of the Boltzmann factors appearing in the partition function.
Canonical Theory
The Canonical Theory (also named the "canonical form") was developed by Joel E. Keizer and coworkers. They showed that his molecular theory explains many physical, chemical, and biological processes in an unified and canonical way, unlike the other theories cited in previous sections. Ronald F. Fox and Keizer showed the application of the canonical theory to chaos.
Prof. Keizer used the canonical form for the first formulation of statistical thermodynamics valid in far from equilibrium regimes, where the Onsager reciprocal relations and the Einstein formula for the fluctuations do not work. Keizer also provided fluctuating generalizations of the Boltzmann equation and of hidrodynamics (fluctuating hydrodynamics). The applications of his work to biology are the reason that he was considered as one of the pioneers in the field of computational biology. Cosma Shalizi wrote: "Chapter five applies the canonical theory to various chemical and electrochemical processes. There is a detailed comparison of a model based on the formalism to actual experimental data for a calcium-regulated potassium channel in muscle cells, yielding remarkably close agreement (especially since the channel is really just a single molecule!)... Keizer was, until his premature death in May, 1999, an active and talented scientist who played a significant role not merely in the development of the formal structure of far from equilibrium thermodynamics, but also in its application to experiment, especially in biology. Unlike a number of others who have attempted such cross-overs, he made it work."
Besides the unification of disparate topics as chemical reactions, hydrodynamics, or heat transport in solids, the canonical theory has been applied to solving the problems of traditional disciplines as statistical mechanics.
Mathematicians have for perhaps a century or more used the word canonical to refer to concepts that have a kind of uniqueness or naturalness. Examples include the canonical prime factorization of positive integers, the Jordan canonical form of matrices (that is built out of the irreducible factors of the characteristic polynomial of the matrix), and the canonical decomposition of a permutation into a product of disjoint cycles. Various functions in mathematics are also canonical, like the canonical homomorphism of a group onto any of its quotient groups, or the canonical isomorphism between a finite-dimensional vector space and its double dual. Although a finite-dimensional vector space and its dual space are isomorphic, there is no canonical isomorphism. This lack of a canonical isomorphism can be made precise in terms of category theory; see natural transformation. But at a simpler level one could say that "any isomorphism you can think of here depends on choosing a basis." As stated by Goguen, "To any canonical construction from one species of structure to another corresponds an adjunction between the corresponding categories."
Being canonical in mathematics is stronger than being a conventional choice. For instance, the vector space Rn has a standard basis, which is canonical in the sense that it is not just a choice that makes certain calculations easy; in fact most linear operators on Euclidean space take on a simpler form when written as a matrix relative to some basis other than the standard one (see Jordan form). In contrast, an abstract n-dimensional real vector space V would not have a canonical basis; it is isomorphic to Rn of course, but the choice of isomorphism is not canonical.
The word canonical is also used for a preferred way of writing something, see the main article canonical form.
In set theory, the term "canonical" identifies an element as representative of a set. If a set is partitioned into equivalence classes, then one member can be chosen from each equivalence class to represent that class. That representative member is the canonical member. If you have a canonicalizing function, f(x), that maps x to the canonical member of the equivalence class which contains it, then testing whether two items, a and b, are equivalent is the same as testing whether f(a) is identical to f(b).
Physics
In theoretical physics, the concept of canonical (or conjugate, or canonically conjugate) variables is of major importance. They always occur in complementary pairs, such as spatial location x and linear momentum p, angle φ and angular momentum L, and energy E and time t. They can be defined as any coordinates whose Poisson brackets give a Kronecker delta (or a Dirac delta in the case of continuous variables). The existence of such coordinates is guaranteed under broad circumstances as a consequence of Darboux's theorem. Canonical variables are essential in the Hamiltonian formulation of physics, which is particularly important in quantum mechanics. For instance, the Schrödinger equation and the Heisenberg uncertainty relation always incorporate canonical variables. Canonical variables in physics are based on the aforementioned mathematical structure and therefore bear a deeper meaning than being just convenient variables. One facet of this underlying structure is expressed by Noether's theorem, which states that a (continuous) symmetry in a variable implies an invariance of the conjugate variable, and vice versa; for instance symmetry under spatial displacement leads to conservation of momentum, and time-independence implies energy conservation.
In statistical mechanics, the grand canonical ensemble, canonical ensemble, and the microcanonical ensemble are archetypal probability distributions for the (unknown) microscopic state of a thermal system, applying respectively in the physical cases of:
An open system at fixed temperature (able to exchange both energy and particles with the environment);
A closed system at fixed temperature (able to exchange energy with its environment);
A closed thermally isolated system (able to exchange neither). These probability distributions can be applied directly to practical problems in thermodynamics.
Occasionally, certain derivations of important ideas from more basic facts are known as "canonical derivations", and can be contrasted with "alternative derivations". One example from statistical mechanics is the canonical derivation of the Boltzmann factors appearing in the partition function.
Canonical Theory
The Canonical Theory (also named the "canonical form") was developed by Joel E. Keizer and coworkers. They showed that his molecular theory explains many physical, chemical, and biological processes in an unified and canonical way, unlike the other theories cited in previous sections. Ronald F. Fox and Keizer showed the application of the canonical theory to chaos.
Prof. Keizer used the canonical form for the first formulation of statistical thermodynamics valid in far from equilibrium regimes, where the Onsager reciprocal relations and the Einstein formula for the fluctuations do not work. Keizer also provided fluctuating generalizations of the Boltzmann equation and of hidrodynamics (fluctuating hydrodynamics). The applications of his work to biology are the reason that he was considered as one of the pioneers in the field of computational biology. Cosma Shalizi wrote: "Chapter five applies the canonical theory to various chemical and electrochemical processes. There is a detailed comparison of a model based on the formalism to actual experimental data for a calcium-regulated potassium channel in muscle cells, yielding remarkably close agreement (especially since the channel is really just a single molecule!)... Keizer was, until his premature death in May, 1999, an active and talented scientist who played a significant role not merely in the development of the formal structure of far from equilibrium thermodynamics, but also in its application to experiment, especially in biology. Unlike a number of others who have attempted such cross-overs, he made it work."
Besides the unification of disparate topics as chemical reactions, hydrodynamics, or heat transport in solids, the canonical theory has been applied to solving the problems of traditional disciplines as statistical mechanics.
理论力学里面有个哈密顿正则方程。描述的哈密顿量和广义坐标以及广义速度的关系。也算是第一性原理的一部分，在统计力学，量子力学里面有着广泛的应用。基本上可以认为是描述物体运动的一种手段。
linux中的正则表达式叫做regular expression，不叫做canonical
5楼: Originally posted by iamikaruk at 2011-09-13 09:53:20:
linux中的正则表达式叫做regular expression，不叫做canonical
呵呵，为什么翻译成正则表达式呢？linux中的正则表达式叫做regular expression，不叫做canonical
与物理中的正则是否有相同之处？
从正交那里来取得理解吧，
我是这样理解的：所用的正则基本都是把一些参数对体系的影响独立出来，一般各参量之间是有相互关系的，然后来研究这个参数对体系的影响。比如说正则坐标系，三个分量正交后都是独立的，还有两个弹簧振子串在一起的处理等。所以这样理解程序的就是写入一个表达式，这个表达式只考了式子中的参量的变化，而这个参量引起其他参量的变化的影响就不考虑了。
不知道这样理解对不对……
我是这样理解的：所用的正则基本都是把一些参数对体系的影响独立出来，一般各参量之间是有相互关系的，然后来研究这个参数对体系的影响。比如说正则坐标系，三个分量正交后都是独立的，还有两个弹簧振子串在一起的处理等。所以这样理解程序的就是写入一个表达式，这个表达式只考了式子中的参量的变化，而这个参量引起其他参量的变化的影响就不考虑了。
不知道这样理解对不对……
我觉得没有相同之处
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