Harmonic wavelet transform

In mathematics and computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute, but others are not. It generalizes the concept of a string, by not forcing the letters to always be in a fixed order, but allowing certain reshufflings to take place. Traces were introduced by Cartier and Foata in 1969 to give a combinatorial proof of MacMahon's Master theorem. Traces are used in theories of concurrent computation, where commuting letters stand for portions of a job that can execute independently of one another, while non-commuting letters stand for locks, synchronization points or thread joins.

The trace monoid or free partially commutative monoid is a monoid of traces. In a nutshell, it is constructed as follows: sets of commuting letters are given by an independency relation. These induce an equivalence relation of equivalent strings; the elements of the equivalence classes are the traces. The equivalence relation then partitions up the free monoid (the set of all strings of finite length) into a set of equivalence classes; the result is still a monoid; it is a quotient monoid and is called the trace monoid. The trace monoid is universal, in that all homomorphic monoids are in fact isomorphic.

Trace monoids are commonly used to model concurrent computation, forming the foundation for process calculi. They are the object of study in trace theory. The utility of trace monoids comes from the fact that they are isomorphic to the monoid of dependency graphs; thus allowing algebraic techniques to be applied to graphs, and vice-versa. They are also isomorphic to history monoids, which model the history of computation of individual processes in the context of all scheduled processes on one or more computers.

Trace

Let ${\mathrm{\Sigma }}^{*}$ denote the free monoid, that is, the set of all strings written in the alphabet $\mathrm{\Sigma }$. Here, the asterisk denotes, as usual, the Kleene star. An independency relation $I$ then induces a binary relation $\sim$ on ${\mathrm{\Sigma }}^{*}$, where $u\sim v$ if and only if there exist $x,y\in {\mathrm{\Sigma }}^{*}$, and a pair $(a,b)\in I$ such that $u=xaby$ and $v=xbay$. Here, $u,v,x$ and $y$ are understood to be strings (elements of ${\mathrm{\Sigma }}^{*}$), while $a$ and $b$ are letters (elements of $\mathrm{\Sigma }$).

The trace is defined as the symmetric, reflexive and transitive closure of $\sim$. The trace is thus an equivalence relation on ${\mathrm{\Sigma }}^{*}$, and is denoted by ${\equiv }_D$. The subscript D on the equivalence simply denotes that the equivalence is obtained from the independency I induced by the dependency D. Clearly, different dependencies will give different equivalence relations.

The transitive closure simply implies that $u\equiv v$ if and only if there exists a sequence of strings $(w_0,w_1,\cdots ,w_n)$ such that $u\sim w_0$ and $v\sim w_n$ and $w_i\sim w_{i+1}$ for all $0\le i<n$.

Trace monoid

The trace monoid, commonly denoted as $\mathbb{𝕄}(D)$, is defined as the quotient monoid

$\mathbb{𝕄}(D)={\mathrm{\Sigma }}^{*}/{\equiv }_D.$

The homomorphism

${\varphi }_D:{\mathrm{\Sigma }}^{*}\to \mathbb{𝕄}(D)$

is commonly referred to as the natural homomorphism or canonical homomorphism. That the terms natural or canonical are deserved follows from the fact that this morphism embodies a universal property, as discussed in a later section.

Examples

Consider the alphabet $\mathrm{\Sigma }=\{a,b,c\}$. A possible dependency relation is

$\begin{array}{ccc}D& =& \{a,b\}\times \{a,b\}\cup \{a,c\}\times \{a,c\}\\ & =& \{a,b{\}}^2\cup \{a,c{\}}^2\\ & =& \{(a,b),(b,a),(a,c),(c,a),(a,a),(b,b),(c,c)\}\end{array}$

The corresponding independency is

$I_D=\{(b,c),(c,b)\}$

Therefore, the letters $b,c$ commute. Thus, for example, a trace equivalence class for the string $abababbca$ would be

$[abababbca{]}_D=\{abababbca,abababcba,ababacbba\}$

The equivalence class $[abababbca{]}_D$ is an element of the trace monoid.

Properties

The cancellation property states that equivalence is maintained under right cancellation. That is, if $w\equiv v$, then $(w÷a)\equiv (v÷a)$. Here, the notation $w÷a$ denotes right cancellation, the removal of the first occurrence of the letter a from the string w, starting from the right-hand side. Equivalence is also maintained by left-cancellation. Several corollaries follow:

• Embedding: $w\equiv v$ if and only if $xwy\equiv xvy$ for strings x and y. Thus, the trace monoid is a syntactic monoid.

• Independence: if $ua\equiv vb$ and $a\ne b$, then a is independent of b. That is, $(a,b)\in I_D$. Furthermore, there exists a string w such that $u=wb$ and $v=wa$.

• Projection rule: equivalence is maintained under string projection, so that if $w\equiv v$, then ${\pi }_{\mathrm{\Sigma }}(w)\equiv {\pi }_{\mathrm{\Sigma }}(v)$.

A strong form of Levi's lemma holds for traces. Specifically, if $uv\equiv xy$ for strings u, v, x, y, then there exist strings $z_1,z_2,z_3$ and $z_4$ such that $(w_2,w_3)\in I_D$ for all letters $w_2\in \mathrm{\Sigma }$ and $w_3\in \mathrm{\Sigma }$ such that $w_2$ occurs in $z_2$ and $w_3$ occurs in $z_3$, and

$u\equiv z_1{z}_2,v\equiv z_3{z}_4,$

$x\equiv z_1{z}_3,y\equiv z_2{z}_4.$

Universal property

A dependency morphism (with respect to a dependency D) is a morphism

$\psi :{\mathrm{\Sigma }}^{*}\to M$

to some monoid M, such that the "usual" trace properties hold, namely:

1. $\psi (w)=\psi (\epsilon )$ implies that $w=\epsilon$

2. $(a,b)\in I_D$ implies that $\psi (ab)=\psi (ba)$

3. $\psi (ua)=\psi (v)$ implies that $\psi (u)=\psi (v÷a)$

4. $\psi (ua)=\psi (vb)$ and $a\ne b$ imply that $(a,b)\in I_D$

Dependency morphisms are universal, in the sense that for a given, fixed dependency D, if $\psi :{\mathrm{\Sigma }}^{*}\to M$ is a dependency morphism to a monoid M, then M is isomorphic to the trace monoid $\mathbb{𝕄}(D)$. In particular, the natural homomorphism is a dependency morphism.

Normal forms

There are two well-known normal formsTemplate:Disambiguation needed for words in trace monoids. One is the lexicographic normal form, due to Anatolij V. Anisimov and Donald Knuth, and the other is the Foata normal form due to Pierre Cartier and Dominique Foata who studied the trace monoid for its combinatorics in the 1960s.

Trace languages

Just as a formal language can be regarded as a subset of ${\mathrm{\Sigma }}^{*}$ the set of all possible strings, so then a trace language is defined as subset of $\mathbb{𝕄}(D)$ all possible traces.

A language $L\subset {\mathrm{\Sigma }}^{*}$ is a trace language, or is said to be consistent with dependency D if

$L=\bigcup [L{]}_D$

where

$[L{]}_D=\{[w{]}_D|w\in L\}$

is the trace closure of a set of strings, and

$\bigcup T=\{w|[w{]}_D\in T\}$

is the set of strings in a set of traces.

Notes

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References

General references

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Antoni Mazurkiewicz, "Introduction to Trace Theory", pp 3–41, in The Book of Traces, V. Diekert, G. Rozenberg, eds. (1995) World Scientific, Singapore ISBN 981-02-2058-8
• Volker Diekert, Combinatorics on traces, LNCS 454, Springer, 1990, ISBN 3-540-53031-2, pp. 9–29
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Seminal publications

• Pierre Cartier and Dominique Foata, Problèmes combinatoires de commutation et réarrangements, Lecture Notes in Mathematics 85, Springer-Verlag, Berlin, 1969, Free 2006 reprint with new appendixes
• Antoni Mazurkiewicz, Concurrent program schemes and their interpretations, DAIMI Report PB 78, Aarhus University, 1977