Maximum usable frequency: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
No edit summary
en>Sonĝanto
mNo edit summary
 
Line 1: Line 1:
{{Cleanup|date=June 2007}}
In [[combinatorial game theory]], the '''Sprague–Grundy theorem''' states that every [[impartial game]] under the [[normal play convention]] is equivalent to a [[nimber]].  The '''Grundy value''' or '''nim-value''' of an impartial game is then defined as the unique nimber that the game is equivalent to.  In the case of a game whose positions (or summands of positions) are indexed by the natural numbers (for example the possible heap sizes in nim-like games), the sequence of nimbers for successive heap sizes is called the '''nim-sequence''' of the game.


The theorem was discovered independently by [[Roland Sprague|R. P. Sprague]] (1935) and [[Patrick Grundy|P. M. Grundy]] (1939).


==Definitions==
We think we're so unique yet, we all want to be healthy, all of us want to be financially comfortable and complete thing . These individuals are based in south Delhi and all they have to provide you with a solution for all your problems.<br><br>The big corporate offices, the us government offices, [http://Wikivapia.com/Turn_Your_Dvr_Cctv_Software_For_Mac_Into_A_High_Performing_Machine railway] stations, and other alike other places use this kind of cameras hugely. So suppose I had to vacuum the household room three times to have out all the crumbs.<br><br>If the crime has already occurred, the recording recorded by cameras CCTC will be  eneo dtr useful for investigators  dvr camera software free download who have committed the crime. The security DVR is a bit more sophisticated and reliable than those for your television.<br><br>Simple, the existing components, like cameras, DVRs, cables, embark on sale as the new equipment begins to replace that old. But just how do museums start letting people know they're still the very best place to see and observe history.<br><br>The forms of CCTV Doncaster security companies install are for surveillance purposes. Secretly installed cameras around ATM machines can track the pin quantity of an unsuspecting user and transmit the info wirelessly on the criminal. Employing vehicle CCTV systems can bring down unnecessary fleet mileage and restrict fleet drivers wasting fleet resources. [http://drmarket.ru/redirect.php?url=http://cctvdvrreviews.com cctv dvr search and playback with client software] Wireless CCTVs operate on the principle of sensors and have remote cameras that might be operated offline. These two-in-one cameras, as earlier mentioned, not merely cut the quantity of cameras needed, but additionally divide the quantity of domes needed for the device in half. Cameras have been around for a long time of course, before this development.<br><br>Merriam Webster defines rape as &ldquo;[http://Doc.froza.ru/index.php/Uncommon_Article_Gives_You_The_Facts_on_Bosch_Dvr4c_That_Only_A_Few_People_Know_Exist sexual intercourse] forced over a person without his / her [http://www.thomsonsav.co.uk/wiki/The_Birth_of_16_Channel_Dvr_Card_Suppliers permission]. This trend would be a while back predicted with the Sun Corporation then CEO, Scott Mc - Nealy stating "the network is the computer. To consider whether you require it you aren't, decide what your difficulty actually is. Nowadays, shops, commercial buildings, casinos and military bases are choosing CCTV access control systems for properly guarding their premises. In the pop-up login box, type in the default name and password.<br><br>has earth's highest rape rate of all countries that publish statistics on rape. Battery daily life is extended in a Dictaphone compared to a Electronic Words Recorder. To protect your house you need to have a CCTV system available but choosing one could be difficult if you're not very technologically minded. Growing rate of crime and theft are forcing the people to search for better security options.<br><br>You'll ought to contend with walls, crevices, and in many cases weather when wanting to get clear signals all of enough time for security. Capturing satellite or cable TV and ultizing your computer being a DVR unit might be everything you're going for. When with such methods, their effectiveness is frequently defined through the responses they generate.<br><br>Here are the services offered and billing as an example, AT & T Residential bundled service plan. The latest DVRs are effective at recording as much as 75 hours of HD programming, and it might be as much as 2,000 hours for Standard programming. However, while using technology becoming more and more affordable, and easier to use, everybody is actually installing CCTV cameras inside their homes and businesses for really security.<br><br>You can evaluate the DVR with your web browser from anywhere inside the world. Developments in technology make security solutions more efficient and efficient and can be a real step towards deterring and catching criminals. The most effective way of doing that is through an automated procedure for Disaster Recovery, offsite data backup and hard disk drive recovery to be sure rapid and price effective execution. But is there any real advantage to using these sorts of cameras over other types of visual security. Digital Video clip Recorders are accessible as Standalone units or PC based units. You've most likely noticed that safety cameras are starting to get a many more and additional popular, some cities keep these things on each street corner.<br><br>Security cameras or CCTV video cameras play an important role in providing security with a business place or residence. This gadget could be operated with all the aid of your special remote controller. It can be possible that save video recording at a remote site. Cat5 cable are capable of transmitting signals down each of its twisted cores.<br><br>This is important inside the world of today's data centers which require constant uptime for operations. Information such as time stamping was usually not available, as well as the fidelity of the video was poor. These are few pros and cons that you simply might face on installing the CCTV cameras in India.
For the purposes of the Sprague&ndash;Grundy theorem, a ''game'' is a two-player game of [[perfect information]] satisfying the ''ending condition'' (all games come to an end: there are no infinite lines of play) and the ''normal play condition'' (a player who cannot move loses).
 
An ''[[impartial game]]'' is one such as [[nim]], in which each player has exactly the same available moves as the other player in any position. Note that games such as [[tic-tac-toe]], [[checkers]], and [[chess]] are ''not'' impartial games.  In the case of checkers and chess, for example, players can only move their own pieces, not their opponent's pieces.  And in tic-tac-toe, one player puts down X's, while the other puts down O's.  Impartial games fall into two ''outcome classes'': either the next player wins (an ''N-position'') or the previous player wins (a ''P-position'').
 
An impartial game can be identified with the set of positions that can be reached in one move (these are called the ''options'' of the game). Thus the game with options ''A'', ''B'', or ''C'' is the set {''A'', ''B'', ''C''}.
 
The normal play convention is where the last player to move wins. Alternatively, the player who first does not have any valid move loses. The opposite - the [[misère]] convention is where the last person to have a valid move or makes the last move loses.
 
A ''[[nimber]]'' is a special game denoted *''n'' for some ordinal ''n''. We define *0 = {} (the empty set), then *1 = {*0}, *2 = {*0, *1}, and *(''n''+1) = *''n'' ∪ {*''n''}. When ''n'' is an integer, the nimber *''n'' = {*0, *1, ..., *(''n''&minus;1)}. This corresponds to a heap of ''n'' counters in the game of [[nim]], hence the name.
 
Two games ''G'' and ''H'' can be ''added'' to make a new game ''G''+''H'' in which a player can choose either to move in ''G'' or in ''H''. In set notation, ''G''+''H'' means {''G''+''h'' for ''h'' in ''H''} ∪ {''g''+''H'' for ''g'' in ''G''}, and thus game addition is commutative and associative.
 
Two games ''G'' and ''G&#39;'' are ''equivalent'' if for every game ''H'', the game ''G''+''H'' is in the same outcome class as ''G&#39;''+''H''. We write ''G'' ≈ ''G&#39;''.
 
A game can refer to two things. It can define a set of possible positions and their moves through its rules, for example, chess, or nim. It can also refer to a certain position, for example, the game *5. Generally, the meaning to be taken is clear from the context.
 
==Lemma==
For impartial games, ''G'' ≈ ''G&#39;'' if and only if ''G''+''G&#39;'' is a ''P-position''.
 
First, we note that ≈ is an [[equivalence relation]] since equality of outcome classes is an equivalence relation.  
 
We now show that for every game G, and ''P-position'' game ''A'', ''A''+''G'' ≈ ''G''. By the definition of ≈, we need to show that ''G''+''H'' is in the same outcome-class as ''A''+''G''+''H'' for all games ''H''.
If ''G''+''H'' is ''P-position'', then the previous player has a winning strategy in ''A''+''G''+''H'': to every move in ''G''+''H'' he responds according to his winning strategy in ''G''+''H'', and to every move in ''A'' he responds with his winning strategy there. If ''G''+''H'' is ''N-position'', then the next player in ''A''+''G''+''H'' makes a winning move in ''G''+''H'', and then reverts to responding to his opponent in the manner described above.
 
Also, ''G''+''G'' is ''P-position'' for any game ''G''. For every move made in one copy of ''G'', the previous player can respond with the same move in the other copy, which means he always makes the last move.
 
Now, we can prove the lemma.  
 
If ''G'' ≈ ''G&#39;'', then ''G''+''G&#39;'' is of the same outcome-class as ''G''+''G'', which is ''P-position''.
 
On the other hand, if ''G''+''G&#39;'' is ''P-position'', then since ''G''+''G'' is also ''P-position'', ''G'' ≈ ''G''+(''G''+''G&#39;'') ≈ (''G''+''G'')+''G&#39;'' ≈ ''G&#39;'', thus ''G'' ≈ ''G&#39;''.
 
==Proof==
We prove the theorem by [[structural induction]] on the set representing the game.
 
Consider a game <math>G = \{G_1, G_2, \ldots, G_k\}</math>. By the induction hypothesis, all of the options are equivalent to nimbers, say <math>G_i \approx *n_i</math>. We will show that <math>G \approx *m</math>, where <math>m</math> is the [[mex (mathematics)|mex]] of the numbers <math>n_1, n_2, \ldots, n_k</math>, that is the smallest non-negative integer not equal to some <math>n_i</math>.
 
Let <math>G'=\{*n_1, *n_2, \ldots, *n_k\}</math>. The first thing we need to note is that <math>G \approx G'</math>. Consider <math>G+G'</math>. If the first player makes a move in <math>G</math>, then the second player can move to the equivalent <math>*n_i</math> in <math>G'</math>, and conversely if the first player makes a move in <math>G'</math>. After this the game is a P-position (by the lemma), since it's the sum of some option of <math>G</math> and a nim pile equivalent to that option. Therefore, <math>G+G'</math> is a P-position, and by another application of our lemma, <math>G \approx G'</math>.
 
So now, by our lemma, we need to show that <math>G+*m</math> is a P-position. We do so by giving an explicit strategy for the second player in the equivalent <math>G'+*m</math>.
 
Suppose that the first player moves in the component <math>*m</math> to the option <math>*m'</math> where <math>m'<m</math>. But since <math>m</math> was the ''minimal'' excluded number, the second player can move in <math>G'</math> to <math>*m'</math>.
 
Suppose instead that the first player moves in the component <math>G'</math> to the option <math>*n_i</math>. If <math>n_i < m</math> then the second player moves in <math>*m</math> to <math>*n_i</math>. If <math>n_i > m</math> then the second player, moves in <math>*n_i</math> to <math>*m</math>. It's not possible that <math>n_i = m</math> because <math>m</math> was defined to be different from all the <math>n_i</math>.
 
Therefore, <math>G'+*m</math> is a P-position, and hence so is <math>G+*m</math>. By our lemma, <math>G \approx *m</math> as desired.
 
==Development==
The Sprague&ndash;Grundy theorem has been developed into the field of [[combinatorial game theory]], notably by [[E. R. Berlekamp]], [[John Horton Conway]] and others. The field is presented in the books ''[[Winning Ways for your Mathematical Plays]]'' and ''[[On Numbers and Games]]''.
 
==See also==
*[[Genus theory]]
*[[Indistinguishability quotient]]
 
==References==
*{{cite journal
| author = Sprague, R. P.
| title = Über mathematische Kampfspiele
| url = http://www.jstage.jst.go.jp/article/tmj1911/41/0/41_0_438/_article
| journal = [[Tohoku Mathematical Journal]]
| year = 1935–36
| volume = 41
| pages = 438–444}}
*{{cite journal
| author = Grundy, P. M.
| title = Mathematics and games
| journal = [[Eureka (University of Cambridge magazine)|Eureka]]
| url = http://www.archim.org.uk/eureka/27/games.html
  | year = 1939
| volume = 2
| pages = 6–8 |archiveurl = http://web.archive.org/web/20070927192024/http://www.archim.org.uk/eureka/27/games.html |archivedate = 2007-09-27}} Reprinted, 1964, '''27''': 9–11.
*{{cite journal
| author = Schleicher, Dierk; Stoll, Michael
| title = An introduction to Conway's games and numbers
| year = 2004
| arxiv = math.CO/0410026}}
*{{cite journal
| author = Milvang-Jensen, Brit C. A.
| year = 2000
| url = http://www.itu.dk/people/brit/Brits%20thesis.pdf
| title = Combinatorial Games, Theory and Applications}}
 
==External links==
* [http://www.cut-the-knot.org/Curriculum/Games/Grundy.shtml Grundy's game] at [[cut-the-knot]]
* [http://www.math.ucla.edu/~tom/Game_Theory/comb.pdf Easily readable, introductory account from the UCLA Math Department]
* [http://sputsoft.com/blog/2009/04/the-game-of-nim.html The Game of Nim] at [http://sputsoft.com sputsoft.com]
 
{{DEFAULTSORT:Sprague-Grundy theorem}}
[[Category:Combinatorial game theory]]
[[Category:Theorems in discrete mathematics]]

Latest revision as of 16:41, 30 July 2014


We think we're so unique yet, we all want to be healthy, all of us want to be financially comfortable and complete thing . These individuals are based in south Delhi and all they have to provide you with a solution for all your problems.

The big corporate offices, the us government offices, railway stations, and other alike other places use this kind of cameras hugely. So suppose I had to vacuum the household room three times to have out all the crumbs.

If the crime has already occurred, the recording recorded by cameras CCTC will be eneo dtr useful for investigators dvr camera software free download who have committed the crime. The security DVR is a bit more sophisticated and reliable than those for your television.

Simple, the existing components, like cameras, DVRs, cables, embark on sale as the new equipment begins to replace that old. But just how do museums start letting people know they're still the very best place to see and observe history.

The forms of CCTV Doncaster security companies install are for surveillance purposes. Secretly installed cameras around ATM machines can track the pin quantity of an unsuspecting user and transmit the info wirelessly on the criminal. Employing vehicle CCTV systems can bring down unnecessary fleet mileage and restrict fleet drivers wasting fleet resources. cctv dvr search and playback with client software Wireless CCTVs operate on the principle of sensors and have remote cameras that might be operated offline. These two-in-one cameras, as earlier mentioned, not merely cut the quantity of cameras needed, but additionally divide the quantity of domes needed for the device in half. Cameras have been around for a long time of course, before this development.

Merriam Webster defines rape as “sexual intercourse forced over a person without his / her permission. This trend would be a while back predicted with the Sun Corporation then CEO, Scott Mc - Nealy stating "the network is the computer. To consider whether you require it you aren't, decide what your difficulty actually is. Nowadays, shops, commercial buildings, casinos and military bases are choosing CCTV access control systems for properly guarding their premises. In the pop-up login box, type in the default name and password.

has earth's highest rape rate of all countries that publish statistics on rape. Battery daily life is extended in a Dictaphone compared to a Electronic Words Recorder. To protect your house you need to have a CCTV system available but choosing one could be difficult if you're not very technologically minded. Growing rate of crime and theft are forcing the people to search for better security options.

You'll ought to contend with walls, crevices, and in many cases weather when wanting to get clear signals all of enough time for security. Capturing satellite or cable TV and ultizing your computer being a DVR unit might be everything you're going for. When with such methods, their effectiveness is frequently defined through the responses they generate.

Here are the services offered and billing as an example, AT & T Residential bundled service plan. The latest DVRs are effective at recording as much as 75 hours of HD programming, and it might be as much as 2,000 hours for Standard programming. However, while using technology becoming more and more affordable, and easier to use, everybody is actually installing CCTV cameras inside their homes and businesses for really security.

You can evaluate the DVR with your web browser from anywhere inside the world. Developments in technology make security solutions more efficient and efficient and can be a real step towards deterring and catching criminals. The most effective way of doing that is through an automated procedure for Disaster Recovery, offsite data backup and hard disk drive recovery to be sure rapid and price effective execution. But is there any real advantage to using these sorts of cameras over other types of visual security. Digital Video clip Recorders are accessible as Standalone units or PC based units. You've most likely noticed that safety cameras are starting to get a many more and additional popular, some cities keep these things on each street corner.

Security cameras or CCTV video cameras play an important role in providing security with a business place or residence. This gadget could be operated with all the aid of your special remote controller. It can be possible that save video recording at a remote site. Cat5 cable are capable of transmitting signals down each of its twisted cores.

This is important inside the world of today's data centers which require constant uptime for operations. Information such as time stamping was usually not available, as well as the fidelity of the video was poor. These are few pros and cons that you simply might face on installing the CCTV cameras in India.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Maximum_usable_frequency&oldid=221886"