Germ (mathematics): Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
Newer edit →
en>Daniele.tampieri
m →Basic definition: Minor lexical changes
 
en>TakuyaMurata
Line 1: Line 1:
When you were younger it didnt appear to matter what you ate we usually had a lot of energy and for most people which meant the fat would be kept down moreover. But as you get elder our bodies changes plus the processes that you took for granted when we were young dont function too. Such is the case for metabolism. Unfortunately metabolism has become an convenient target over time because something to blame weight gain on. But that really isnt the case; we require metabolism particularly when you are elder. One method to increase it's through strength training.<br><br>Plan out your food [http://safedietplans.com/bmr-calculator bmr calculator] for the day. Having a program reduces a risk of eating quick foods plus unhealthy snacks. Plan to consume 4-5 times a day. A mid-morning plus mid-afternoon meal will help keep you from overeating at lunch plus dinner. Try to include a protein into each of the food throughout the day. Including a protein with meals and snacks may enable we feel happy and hold off the hunger. Try pre-cooking foods to lower preparation time. I have found that reducing the prep time for meals will lower the likelihood which we will opt for a quicker plus less healthy meal.<br><br>BMR stands for basal metabolic rate plus is the amount of calories the body burns at rest to maintain normal body functions, regardless of exercise. These calories are utilized for a body's general everyday functions like breathing, digestion, keeping all organs functioning, blood pumping, etc. Physical activity directly affects a BMR raising it throughout (plus keeping it raised even after) moderate exercise. Additionally, your BMR increases with all the increase of muscle tissue (another reason to lift those weights!).<br><br>11) Simply as you only must eat whilst you may be hungry, only sip on sweet-tasting liquids when you may be hungry. Consider everything you place inside the mouth with all the exception of water to be food, so focus found on the proper food, balanced food throughout the course of your day.<br><br>Body Surface Area: Your height plus fat lead a lot inside determining bmr. The greater is the body surface area, the higher is the BMR. Thin, tall people have a high BMR.<br><br>To calculate the activity level, see this free online calculator: http://exercise.about.com/cs/fitnesstools/l/blcalorieburn.htm which might employ a current fat, the kind of exercise, and the amount of time you are able to afford to invest doing that exercise to tell we how various calories you're going to burn. Take note of the results - if you utilize this calculator realistically, you can plug it right into the BMR results.<br><br>I will hear a gears grinding inside the background. Screeching even! Am I suggesting which we now have to begin counting calories and journaling what we eat every day? Well, that is 1 method to create sure the eating less than the daily calorie requirement. But pay a visit to the 3rd installment of this series plus see how to conveniently discover your BMR plus start using this useful info. Remember, this BMR-thing enabled me to beat childhood weight whilst the principles behind it have assisted me keep a extremely healthy, balanced fat for over fifteen years. I have never been a big calorie counter either. Once we get severe and start paying attention to what the body requirements, advantageous practices easily follow  I promise!
{{For|other uses of the term|Magic number (disambiguation)}}
In certain [[sports]], a '''magic number''' is a number used to indicate how close a front-running team is to clinching a division title and/or a playoff spot. It represents the total of additional wins by the front-running team or additional losses (or any combination thereof) by the rival team after which it is mathematically impossible for the rival team to capture the title in the remaining games.  This assumes that each game results in a win or a loss, but not a [[tie (draw)|tie]]. Teams other than the front-running team have what is called an '''elimination number''' (or '''"tragic number"''') (often abbreviated ''E#'').  This number represents the number of wins by the leading team or losses by the trailing team which will eliminate the trailing team. The elimination number for the second place team is exactly the magic number for the leading team.
 
The magic number is calculated as ''G'' + 1 − ''W''<sub>''A''</sub> − ''L''<sub>''B''</sub>, where
*''G'' is the total number of games in the [[season (sport)|season]]
*''W''<sub>''A''</sub> is the number of wins that Team A has in the season
*''L''<sub>''B''</sub> is the number of losses that Team B has in the season
 
For example, in [[Major League Baseball]] there are 162 games in a season.  Suppose the top of the [[division (sport)|division]] standings late in the season are as follows:
 
{| class="wikitable"
|-
|Team||Wins||Losses
|-
|A||96||58
|-
|B||93||62
|}
 
Then the magic number for Team A to win the division is 162 + 1 − 96 − 62 = 5.
 
Any combination of wins by Team A and losses by Team B totaling to 5 makes it impossible for Team B to win the division title.
 
The "+1" in the formula serves the purpose of eliminating ties; without it, if the magic number were to decrease to zero and stay there, the two teams in question would wind up with identical records. If circumstances dictate that the front-running team would win the tiebreaker regardless of any future results, then the additional constant 1 can be eliminated.  For example, the [[National Basketball Association|NBA]] uses complicated formulae for breaking ties, using several other statistics of merit besides overall win/loss record; however the ''first'' tiebreaker between two teams is their head-to-head record; if the frontrunning team has already clinched the better head-to-head record, then the +1 is unnecessary.
 
The magic number can also be calculated as ''W''<sub>''B''</sub> + ''GR''<sub>''B''</sub> - ''W''<sub>''A''</sub> + 1, where
*''W''<sub>''B''</sub> is the number of wins that Team B has in the season
*''GR''<sub>''B''</sub> is the number of games remaining for Team B in the season
*''W''<sub>''A''</sub> is the number of wins that Team A has in the season
 
This second formula basically says: ''Assume Team B wins every remaining game. Calculate how many games team A needs to win to surpass team B's maximum total by 1''.  Using the example above and with the same 162-game season, team B has 7 games remaining.
 
The magic number for Team A to win the division is still "5": 93 + 7 − 96 + 1 = 5.
 
Team B can win as many as 100 games. If Team A wins 101, Team B is eliminated. The magic number would decrease with a Team A win and would also decrease with a Team B loss, as its maximum win total would decrease by one.
 
A variation of the above looks at the relation between the losses of the two teams. The magic number can be calculated as ''L''<sub>''A''</sub> + ''GR''<sub>''A''</sub> - ''L''<sub>''B''</sub> + 1, where
*''L''<sub>''A''</sub> is the number of losses that Team A has in the season
*''GR''<sub>''A''</sub> is the number of games remaining for Team A in the season
*''L''<sub>''B''</sub> is the number of losses that Team B has in the season
 
This third formula basically says: ''Assume Team A loses every remaining game.  Calculate how many games team B needs to lose to surpass team A's maximum total by 1''.  Using the example above and with the same 162-game season, team A has 8 games remaining.
 
The magic number for Team A to win the division is still "5": 58 + 8 − 62 + 1 = 5.  As you can see, the magic number is the same whether calculating it based on potential wins of the leader or potential losses of the trailing team. Indeed, mathematical proofs will show that the three formulas presented here are mathematically equivalent.
 
Team A can lose as many as 66 games. If Team B loses 67, Team B is eliminated.  Once again, the magic number would decrease with a Team A win and would also decrease with a Team B loss.
 
In some sports, ties are broken by an additional [[one-game playoff]](s) between the teams involved. When a team gets to the point where its magic number is 1, it is said to have "clinched a tie" for the division or the wild card. However, if they end the season tied with another team, and only one is eligible for the playoffs, the extra playoff game will erase that "clinching" for the team that loses the playoff game.
 
By convention, the magic number typically is used to describe the first place team only, relative to the teams it leads. However, the same mathematical formulas could be applied to any team, teams that are tied for the lead, as well as teams that trail. In these cases, a team that is not in first place will depend on the leading team to lose some games so that it may catch up, so the magic number will be larger than the number of games remaining. Ultimately, for teams that are no longer in contention, their magic number would be larger than their remaining games + the remaining games for the first place team&nbsp;— which would be impossible to overcome.
 
==Derivation==
The formula for the magic number is derived straightforwardly as follows. As before, at some particular point in the season let Team A have ''W''<sub>''A''</sub> wins and ''L''<sub>''A''</sub> losses. Suppose that at some later time, Team A has ''w''<sub>''A''</sub> additional wins and ''l''<sub>''A''</sub> additional losses, and define similarly ''W''<sub>''B''</sub>, ''L''<sub>''B''</sub>, ''w''<sub>''B''</sub>, ''l''<sub>''B''</sub> for Team B. The total number of wins that Team B needs to make up is thus given by (''W''<sub>''A''</sub> + ''w''<sub>''A''</sub>) − (''W''<sub>''B''</sub> + ''w''<sub>''B''</sub>). Team A clinches when this number exceeds the number of games Team B has remaining, since at that point Team B cannot make up the deficit even if Team A fails to win any more games. If there are a total of ''G'' games in the season, then the number of games remaining for Team B is given by ''G'' − (''W''<sub>''B''</sub> + ''w''<sub>''B''</sub> + ''L''<sub>''B''</sub> + ''l''<sub>''B''</sub>). Thus the condition for Team A to clinch is that (''W''<sub>''A''</sub> + ''w''<sub>''A''</sub>) − (''W''<sub>''B''</sub> + ''w''<sub>''B''</sub>) = 1 + ''G'' − (''W''<sub>''B''</sub> + ''w''<sub>''B''</sub> + ''L''<sub>''B''</sub> + ''l''<sub>''B''</sub>). Canceling the common terms, we obtain ''w''<sub>''A''</sub> + ''l''<sub>''B''</sub> = ''G'' + 1 − ''W''<sub>''A''</sub> − ''L''<sub>''B''</sub>, which establishes the magic number formula.
 
==Subtlety==
Sometimes a team can ''appear'' to have a mathematical chance to win even though they have actually been eliminated already, due to scheduling. In this Major League Baseball scenario, there are three games remaining in the season. Teams A, B and C are assumed to be eligible only for the division championship; teams with better records in other divisions have already clinched the two available "wild card" spots:
 
{| class="wikitable"
|-
|Team||Wins||Losses
|-
|A||87||72
|-
|B||87||72
|-
|C||85||74
|}
 
If Team C were to win all three remaining games, it would finish at 88-74, and if both Teams A and B were to lose their three remaining games, they would finish at 87-75, which would make Team C the division winner. However if Teams A and B are playing against ''each other'' in the final weekend (in a 3 game series), it would be impossible for both teams to lose the three remaining games. One of them will win at least two games and thereby clinch the division title with a record of either 90-72 or 89-73. The more direct consequence of this situation is that it is also not possible for Teams A and B to finish in a tie with each other, and Team C can't win the division.
 
One can say definitely whether a team has been eliminated by use of the algorithm for the [[maximum flow problem]].<ref>{{cite book | last1 = Kleinberg | first1 = Jon | authorlink1 = Jon Kleinberg | last2 = Tardos | first2 = Éva | authorlink2 = Éva Tardos | title = Algorithm Design | year = 2005 | publisher = Addison-Wesley | isbn = 978-0321295354}}</ref>
 
==Alternative Method==
 
Another method can be used to determine the Elimination Number which uses only the Games Remaining (<math>GR_L, GR_T</math>) and Games Behind Leader (GBL) statistics, as follows:
<math>E= \frac{GR_L+GR_T}{2}-GBL+1</math>,
<br>where <math>GR_L</math> means Games Remaining for Leader (similarly, <math>GR_T</math> means Games Remaining for Trailer).
 
<br>Refer back to the example presented above.  The elimination number for Team B is once again "5": <math>E= \frac{8+7}{2}-3.5+1</math>.
 
<br>It is necessary to use this method if the teams play different numbers of games in the full season, for instance due to cancellations or ties that will not be replayed. Note that this algorithm also is limited by the aforementioned subtleties.
 
==See also==
*[[Games behind]]
*[[Playoff berth]]
*[[Elimination from possibility of reaching postseason]]
 
==References==
 
{{Reflist}}
 
==External links==
*[http://www.obsoletecomputermuseum.org/magic/magicexpo.shtml Comparison of several equivalent formulas]
*[http://lyle.smu.edu/~olinick/riot/baseball_main.html RIOT] an operations research approach applied to Major League Baseball
 
[[Category:Terminology used in multiple sports]]

Revision as of 00:21, 24 November 2013

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In certain sports, a magic number is a number used to indicate how close a front-running team is to clinching a division title and/or a playoff spot. It represents the total of additional wins by the front-running team or additional losses (or any combination thereof) by the rival team after which it is mathematically impossible for the rival team to capture the title in the remaining games. This assumes that each game results in a win or a loss, but not a tie. Teams other than the front-running team have what is called an elimination number (or "tragic number") (often abbreviated E#). This number represents the number of wins by the leading team or losses by the trailing team which will eliminate the trailing team. The elimination number for the second place team is exactly the magic number for the leading team.

The magic number is calculated as G + 1 − WA − LB, where

  • G is the total number of games in the season
  • WA is the number of wins that Team A has in the season
  • LB is the number of losses that Team B has in the season

For example, in Major League Baseball there are 162 games in a season. Suppose the top of the division standings late in the season are as follows:

Team Wins Losses
A 96 58
B 93 62

Then the magic number for Team A to win the division is 162 + 1 − 96 − 62 = 5.

Any combination of wins by Team A and losses by Team B totaling to 5 makes it impossible for Team B to win the division title.

The "+1" in the formula serves the purpose of eliminating ties; without it, if the magic number were to decrease to zero and stay there, the two teams in question would wind up with identical records. If circumstances dictate that the front-running team would win the tiebreaker regardless of any future results, then the additional constant 1 can be eliminated. For example, the NBA uses complicated formulae for breaking ties, using several other statistics of merit besides overall win/loss record; however the first tiebreaker between two teams is their head-to-head record; if the frontrunning team has already clinched the better head-to-head record, then the +1 is unnecessary.

The magic number can also be calculated as WB + GRB - WA + 1, where

  • WB is the number of wins that Team B has in the season
  • GRB is the number of games remaining for Team B in the season
  • WA is the number of wins that Team A has in the season

This second formula basically says: Assume Team B wins every remaining game. Calculate how many games team A needs to win to surpass team B's maximum total by 1. Using the example above and with the same 162-game season, team B has 7 games remaining.

The magic number for Team A to win the division is still "5": 93 + 7 − 96 + 1 = 5.

Team B can win as many as 100 games. If Team A wins 101, Team B is eliminated. The magic number would decrease with a Team A win and would also decrease with a Team B loss, as its maximum win total would decrease by one.

A variation of the above looks at the relation between the losses of the two teams. The magic number can be calculated as LA + GRA - LB + 1, where

  • LA is the number of losses that Team A has in the season
  • GRA is the number of games remaining for Team A in the season
  • LB is the number of losses that Team B has in the season

This third formula basically says: Assume Team A loses every remaining game. Calculate how many games team B needs to lose to surpass team A's maximum total by 1. Using the example above and with the same 162-game season, team A has 8 games remaining.

The magic number for Team A to win the division is still "5": 58 + 8 − 62 + 1 = 5. As you can see, the magic number is the same whether calculating it based on potential wins of the leader or potential losses of the trailing team. Indeed, mathematical proofs will show that the three formulas presented here are mathematically equivalent.

Team A can lose as many as 66 games. If Team B loses 67, Team B is eliminated. Once again, the magic number would decrease with a Team A win and would also decrease with a Team B loss.

In some sports, ties are broken by an additional one-game playoff(s) between the teams involved. When a team gets to the point where its magic number is 1, it is said to have "clinched a tie" for the division or the wild card. However, if they end the season tied with another team, and only one is eligible for the playoffs, the extra playoff game will erase that "clinching" for the team that loses the playoff game.

By convention, the magic number typically is used to describe the first place team only, relative to the teams it leads. However, the same mathematical formulas could be applied to any team, teams that are tied for the lead, as well as teams that trail. In these cases, a team that is not in first place will depend on the leading team to lose some games so that it may catch up, so the magic number will be larger than the number of games remaining. Ultimately, for teams that are no longer in contention, their magic number would be larger than their remaining games + the remaining games for the first place team — which would be impossible to overcome.

Derivation

The formula for the magic number is derived straightforwardly as follows. As before, at some particular point in the season let Team A have WA wins and LA losses. Suppose that at some later time, Team A has wA additional wins and lA additional losses, and define similarly WB, LB, wB, lB for Team B. The total number of wins that Team B needs to make up is thus given by (WA + wA) − (WB + wB). Team A clinches when this number exceeds the number of games Team B has remaining, since at that point Team B cannot make up the deficit even if Team A fails to win any more games. If there are a total of G games in the season, then the number of games remaining for Team B is given by G − (WB + wB + LB + lB). Thus the condition for Team A to clinch is that (WA + wA) − (WB + wB) = 1 + G − (WB + wB + LB + lB). Canceling the common terms, we obtain wA + lB = G + 1 − WA − LB, which establishes the magic number formula.

Subtlety

Sometimes a team can appear to have a mathematical chance to win even though they have actually been eliminated already, due to scheduling. In this Major League Baseball scenario, there are three games remaining in the season. Teams A, B and C are assumed to be eligible only for the division championship; teams with better records in other divisions have already clinched the two available "wild card" spots:

Team Wins Losses
A 87 72
B 87 72
C 85 74

If Team C were to win all three remaining games, it would finish at 88-74, and if both Teams A and B were to lose their three remaining games, they would finish at 87-75, which would make Team C the division winner. However if Teams A and B are playing against each other in the final weekend (in a 3 game series), it would be impossible for both teams to lose the three remaining games. One of them will win at least two games and thereby clinch the division title with a record of either 90-72 or 89-73. The more direct consequence of this situation is that it is also not possible for Teams A and B to finish in a tie with each other, and Team C can't win the division.

One can say definitely whether a team has been eliminated by use of the algorithm for the maximum flow problem.[1]

Alternative Method

Another method can be used to determine the Elimination Number which uses only the Games Remaining (GRL,GRT) and Games Behind Leader (GBL) statistics, as follows: E=GRL+GRT2−GBL+1,
where GRL means Games Remaining for Leader (similarly, GRT means Games Remaining for Trailer).


Refer back to the example presented above. The elimination number for Team B is once again "5": E=8+72−3.5+1.


It is necessary to use this method if the teams play different numbers of games in the full season, for instance due to cancellations or ties that will not be replayed. Note that this algorithm also is limited by the aforementioned subtleties.

See also

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.

External links

  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
Retrieved from "https://en.formulasearchengine.com/index.php?title=Germ_(mathematics)&oldid=4219"