Capillary wave: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
Newer edit →
en>Crowsnest
Undid revision 509912627 by 188.26.22.131 (talk) he did his PhD thesis on capillary waves, but it is only one of many contributions to the theory of capillary waves
 
No edit summary
Line 1: Line 1:
If you have to accelerate the PC then you have come to the appropriate area. I will show you, right now, five quick methods to dramatically enhance a computer's performance.<br><br>Firstly, you need to utilize a Antivirus or safety tool and run a scan on your computer. It is possible which a computer is afflicted with virus or malware that slows down the computer. If there is nothing found inside the scanning report, it can be the RAM which cause the problem.<br><br>H/w connected error handling - whenever hardware causes BSOD installing latest fixes for the hardware and/ or motherboard may enable. You could equally add new hardware that is compatible with the system.<br><br>Review the files plus clean it up regularly. Destroy all unnecessary plus unused files because they just jam your computer program. It might surely improve the speed of the computer and be careful which a computer never afflicted by a virus. Remember constantly to update a antivirus software each time. If you never utilize the computer surprisingly usually, we can take a free antivirus.<br><br>To fix the problem that is caused by registry error, you have to utilize a [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities]. That is the safest and simplest technique for average PC consumers. But there are thousands of registry products available out there. You should discover a advantageous 1 that can definitely solve a issue. If you use a terrible one, we may expect more issues.<br><br>Why this problem happens frequently? What are the causes of it? In fact, there are 3 main causes which may lead to the PC freezing problem. To resolve the problem, we should take 3 procedures in the following paragraphs.<br><br>You need a choice to automatically delete unwanted registry keys. This might help save you hours of laborious checking from the registry keys. Automatic deletion is a key element whenever we compare registry cleaners.<br><br>A system and registry cleaner is downloaded from the web. It's convenient to use and the task does not take long. All it does is scan plus then whenever it finds mistakes, it usually fix and clean those errors. An error free registry can protect the computer from errors and give you a slow PC fix.
In [[mathematics]] and [[computer science]], the '''syntactic monoid''' ''M''(''L'') of a [[formal language]] ''L'' is the smallest [[monoid]] that [[recognizable set|recognizes]] the language ''L''.
 
==Syntactic quotient==
Given <math>S\subset M</math> of a monoid ''M'' of every string over some alphabet, one may define sets that consist of formal left or right [[Inverse element#In an unital magma|inverses of elements]] in ''S''.  These are called [[string operations|quotients]], and one may define right or left quotients, depending on which side one is concatenating. Thus, the '''right quotient''' of ''S'' by an element <math>m\in M</math> is the set
:<math>S \ / \ m=\{u\in M \;\vert\; um\in S \}.</math>
 
Similarly, the '''left quotient''' is
 
:<math>m \setminus S=\{u\in M \;\vert\; mu\in S \}.</math>
 
==Syntactic equivalence==
The syntactic quotient induces an [[equivalence relation]] on ''M'', called the '''syntactic relation''', or '''syntactic equivalence'''  (induced by ''S''). The right syntactic equivalence is the equivalence relation
 
:<math>\sim_S \;= \{(s,t)\in M\times M \,\vert\; S \ / \ s = S \ / \ t \}.</math>
 
Similarly, the left syntactic relation is
 
:<math>\,_S\sim \;= \{(s,t)\in M\times M \,\vert\; s\setminus S = t \setminus S \}.</math>
 
The '''syntactic congruence'''{{clarify|reason='Syntactic congruence' and 'right syntactic equivalence' share the same symbolic notation. Please indicate clearly if the are meant to be identical notions (which is not obvious from their definitions). Otherwise, use different symbolic notations, and comment on the relation between both concepts, if appropriate.|date=August 2013}} may be defined as<ref name=Law210>Lawson (2004) p.210</ref>
 
:<math>u \sim_S v \Leftrightarrow \forall x, y\in M (xuy \in S \Leftrightarrow xvy \in S).</math>
 
The definition extends to a congruence defined by a subset ''S'' of a general monoid ''M''. A '''disjunctive set''' is a subset ''S'' such that the syntactic congruence defined by ''S'' is the equality relation.<ref name=Law232>Lawson (2004) p.232</ref>
 
==Syntactic monoid==
The syntactic quotient is [[Quotient algebra|compatible]] with concatenation in the monoid, in that one has
 
:<math>(M \ / \ s) \ / \ t = M \ / \ (ts)</math>
 
for all <math>s,t\in M</math> (and similarly for the left quotient). Thus, the syntactic quotient is a [[monoid morphism]], and induces a [[quotient monoid]]
 
:<math>M(S)= M \ / \ \sim_S.</math>
 
This monoid <math>M(S)</math> is called the '''syntactic monoid''' of ''S''.
It can be shown that it is the smallest [[monoid]] that [[recognizable set|recognizes]] ''S'' ; that is, ''M''(''S'') recognizes ''S'', and for every monoid ''N'' recognizing ''S'', ''M''(''S'') is a quotient of a [[submonoid]] of ''N''. The syntactic monoid of ''S'' is also the [[transition monoid]] of the [[minimal automaton]] of ''S''.<ref name=Law210/><ref name=S55>Straubing (1994) p.55</ref>
 
Similarly, a language ''L'' is regular if and only if the family of quotients
 
:<math>\{m \setminus  L \,\vert\; m\in M\}</math>
 
is finite. The proof showing equivalence is quite easy. Assume that a string ''x'' is read by a [[deterministic finite automaton]], with the machine proceeding into state ''p''. If ''y'' is another string read by the machine, also terminating in the same state ''p'', then clearly one has <math>x \setminus  L\,= y \setminus  L</math>. Thus, the number of elements in <math>\{m \setminus L \,\vert\; m\in M\}</math> is just exactly equal to the number of states of the automaton and <math>\{m \setminus L \,\vert\; m\in L\}</math> is equal to number of final states.  Assume the converse: that the number of elements in <math>\{m \setminus L \,\vert\; m\in M\}</math> is finite. One can then construct an automaton where <math>Q=\{m \setminus L \,\vert\; m\in M\}</math> is the set of states, <math>F=\{m \setminus L \,\vert\; m\in L\}</math> is the set of final states, the language ''L'' is the initial state, and the transition function is given by <math>y\setminus(x \setminus L)  =(xy) \setminus L</math>. Clearly, this automaton recognizes ''L''. Thus, a language ''L'' is recognizable if and only if the set <math>\{m \setminus L \,\vert\; m\in M\}</math> is finite.
 
Given a [[regular expression]] ''E'' representing ''S'', it is easy to compute the syntactic monoid of ''S''.
 
A '''group language''' is one for which the syntactic monoid is a [[Group (mathematics)|group]].<ref name=Sak342>Sakarovitch (2009) p.342</ref>
 
==Examples==
* Let ''L'' be the language over ''A'' = {''a'',''b''} of words of even length.  The syntactic congruence has two classes, ''L'' itself and ''L''<sub>1</sub>, the words of odd length.  The syntactic monoid is the group of order 2 on {''L'',''L''<sub>1</sub>}.<ref name=S54>Straubing (1994) p.54</ref> 
* The [[bicyclic monoid]] is the syntactic monoid of the [[Dyck language]] (the language of balanced sets of parentheses).
* The [[free monoid]] on ''A'' is the syntactic monoid of the language { ''ww''<sup>R</sup> | ''w'' in ''A''* }, where ''w''<sup>R</sup> denotes the reversal of word ''w''.
* Every finite monoid is homomorphic to the syntactic monoid of some non-trivial language,<ref name=MP48>{{cite book | last1=McNaughton | first1=Robert | last2 = Papert | first2=Seymour | author2-link=Seymour Papert | others=With an appendix by William Henneman | series=Research Monograph | volume=65 | year=1971 | title=Counter-free Automata | publisher=MIT Press | isbn=0-262-13076-9 | zbl=0232.94024 | page=48 }}</ref> but not every finite monoid is isomorphic to a syntactic monoid.<ref name=Law233>Lawson (2004) p.233</ref>
* Every finite group is isomorphic to the syntactic monoid of some non-trivial language.<ref name=MP48/>
* The language over {''a'',''b''} in which the number of occurrences of ''a'' and ''b'' are congruent modulo 2<sup>''n''</sup> is a group language with syntactic monoid '''Z'''/2<sup>''n''</sup>.<ref name=Sak342/>
* [[Trace monoid]]s are examples of syntactic monoids.
* [[Marcel-Paul Schützenberger]]<ref>{{cite journal | author=[[Marcel-Paul Schützenberger]] | title=On finite monoids having only trivial subgroups | journal=Information and Computation| year=1965| volume=8 | issue=2 | pages=190–194|url=http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1965-4TrivialSubgroupsIC.pdf}}</ref> characterized [[star-free language]]s as those with finite [[Aperiodic_monoid|aperiodic]] syntactic monoids.<ref name=S60>Straubing (1994) p.60</ref>
 
==References==
{{reflist}}
* {{cite book | last=Anderson | first=James A. | title=Automata theory with modern applications | others=With contributions by Tom Head | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-61324-8 | zbl=1127.68049 }}
* {{cite book | last=Lawson | first=Mark V. | title=Finite automata | publisher=Chapman and Hall/CRC | year=2004 | isbn=1-58488-255-7 | zbl=1086.68074 }}
* {{cite book | editor1-last=Rozenberg | editor1-first=G. | editor2-last=Salomaa | editor2-first=A. | first=Jean-Eric | last=Pin | url=http://www.liafa.jussieu.fr/~jep/PDF/HandBook.pdf | chapter=10. Syntactic semigroups | title=Handbook of Formal Language Theory | volume=1 | publisher=[[Springer-Verlag]] | year=1997 | pages=679-746 | zbl=0866.68057 }}
* {{cite book | last=Sakarovitch | first=Jacques | title=Elements of automata theory | others=Translated from the French by Reuben Thomas | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-84425-3 | zbl=1188.68177 }}
* {{cite book | last=Straubing | first=Howard | title=Finite automata, formal logic, and circuit complexity | series=Progress in Theoretical Computer Science | location=Basel | publisher=Birkhäuser | year=1994 | isbn=3-7643-3719-2 | zbl=0816.68086 }}
 
[[Category:Formal languages]]
[[Category:Semigroup theory]]

Revision as of 18:03, 18 January 2014

In mathematics and computer science, the syntactic monoid M(L) of a formal language L is the smallest monoid that recognizes the language L.

Syntactic quotient

Given S⊂M of a monoid M of every string over some alphabet, one may define sets that consist of formal left or right inverses of elements in S. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating. Thus, the right quotient of S by an element m∈M is the set

S/m={u∈M|um∈S}.

Similarly, the left quotient is

m∖S={u∈M|mu∈S}.

Syntactic equivalence

The syntactic quotient induces an equivalence relation on M, called the syntactic relation, or syntactic equivalence (induced by S). The right syntactic equivalence is the equivalence relation

∼S={(s,t)∈M×M|S/s=S/t}.

Similarly, the left syntactic relation is

S∼={(s,t)∈M×M|s∖S=t∖S}.

The syntactic congruenceTemplate:Clarify may be defined as[1]

u∼Sv⇔∀x,y∈M(xuy∈S⇔xvy∈S).

The definition extends to a congruence defined by a subset S of a general monoid M. A disjunctive set is a subset S such that the syntactic congruence defined by S is the equality relation.[2]

Syntactic monoid

The syntactic quotient is compatible with concatenation in the monoid, in that one has

(M/s)/t=M/(ts)

for all s,t∈M (and similarly for the left quotient). Thus, the syntactic quotient is a monoid morphism, and induces a quotient monoid

M(S)=M/∼S.

This monoid M(S) is called the syntactic monoid of S. It can be shown that it is the smallest monoid that recognizes S ; that is, M(S) recognizes S, and for every monoid N recognizing S, M(S) is a quotient of a submonoid of N. The syntactic monoid of S is also the transition monoid of the minimal automaton of S.[1][3]

Similarly, a language L is regular if and only if the family of quotients

{m∖L|m∈M}

is finite. The proof showing equivalence is quite easy. Assume that a string x is read by a deterministic finite automaton, with the machine proceeding into state p. If y is another string read by the machine, also terminating in the same state p, then clearly one has x∖L=y∖L. Thus, the number of elements in {m∖L|m∈M} is just exactly equal to the number of states of the automaton and {m∖L|m∈L} is equal to number of final states. Assume the converse: that the number of elements in {m∖L|m∈M} is finite. One can then construct an automaton where Q={m∖L|m∈M} is the set of states, F={m∖L|m∈L} is the set of final states, the language L is the initial state, and the transition function is given by y∖(x∖L)=(xy)∖L. Clearly, this automaton recognizes L. Thus, a language L is recognizable if and only if the set {m∖L|m∈M} is finite.

Given a regular expression E representing S, it is easy to compute the syntactic monoid of S.

A group language is one for which the syntactic monoid is a group.[4]

Examples

  • Let L be the language over A = {a,b} of words of even length. The syntactic congruence has two classes, L itself and L1, the words of odd length. The syntactic monoid is the group of order 2 on {L,L1}.[5]
  • The bicyclic monoid is the syntactic monoid of the Dyck language (the language of balanced sets of parentheses).
  • The free monoid on A is the syntactic monoid of the language { wwR | w in A* }, where wR denotes the reversal of word w.
  • Every finite monoid is homomorphic to the syntactic monoid of some non-trivial language,[6] but not every finite monoid is isomorphic to a syntactic monoid.[7]
  • Every finite group is isomorphic to the syntactic monoid of some non-trivial language.[6]
  • The language over {a,b} in which the number of occurrences of a and b are congruent modulo 2n is a group language with syntactic monoid Z/2n.[4]
  • Trace monoids are examples of syntactic monoids.
  • Marcel-Paul Schützenberger[8] characterized star-free languages as those with finite aperiodic syntactic monoids.[9]

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  1. ↑ 1.0 1.1 Lawson (2004) p.210
  2. ↑ Lawson (2004) p.232
  3. ↑ Straubing (1994) p.55
  4. ↑ 4.0 4.1 Sakarovitch (2009) p.342
  5. ↑ Straubing (1994) p.54
  6. ↑ 6.0 6.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  7. ↑ Lawson (2004) p.233
  8. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  9. ↑ Straubing (1994) p.60
Retrieved from "https://en.formulasearchengine.com/index.php?title=Capillary_wave&oldid=6105"