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Talk:Martingale (probability theory)

gamble's fortune
Why is the gambler's fortune called out specifically? Isn't it just a type of random walk?

tau
I don't think saying that [τ=t] is independent of Xt+1 etc. is quite what we want, but the issue is a little messy since I've been trying to avoid talking about filtrations. The definition of stopping time I'm used to is with respect to a filtration; the natural filtration to use (given that we aren't talking about filtrations) would seem to be the one generated by the Xi. In this case [τ=t] is a function of X1 through Xt, which is what I was aiming for in the definition given here.

I'm not sure how to make the "independent of the future" style definition work---for example, we could have a martingale where Xt+1 encodes in some subtle way (e.g. by minute perturbations of the lower-order bits) all of the preceding values. So for the moment I am pulling this paragraph out, since unless I am missing something it looks like the last claim is just plain false:

 * To say that "the occurrence or non-occurrence of the event τ=t depends only on the values of X1, X2, ..., Xt" need not mean that the occurrence or non-occurrence of that event is completely determined by those values. It can mean rather that, although the occurrence or non-occurrence of the event τ=t may be correllated with the values of X1, X2, ..., Xt, that occurrence or non-occurrence is  probabilistically independent of the values of Xt+1, Xt+2, Xt+3, ....

Populus 03:06, 13 Nov 2003 (UTC)

Just added an article on filtrations, putting the link here in case anybody wants to link to it ... Bryan Barnard 22:40, 14 Jun 2004 (UTC)

"1 = E[Y1] = E[Yτ] = m² - E[τ]. We immediately get E[τ] = m²+1" Seems to me we get $$E[\tau] = m^2 - 1$$. Typo? - Gauge 00:49, 21 Aug 2004 (UTC)

Why on earth is a Martingale called a Martingale? One sentence on the historical reason would be interesting Torfason 18:52, 22 October 2005 (UTC)

Recent addition
I added a general mathematical definition for martingales that take value in general topological vector spaces. They have quite a lot a applications in mathematical finance and in stochastic partial differential equations, so I think it is an important definition.

But I am not sure that the remark is placed in the right position (quite at the beginning). On one hand, it is a definition, and should be written in the beginning. On the other hand, it uses some abstract mathematical tools, and should be put at the end of the article in order to avoid to make it too hostile for non-mathematical readers. gala.martin
 * Good points. For now I moved it down. See the article martingale and its history for my explanation. Oleg Alexandrov (talk) 01:51, 4 December 2005 (UTC)

I'm a wikipedia neophyte, so I won't try to sign this or anything like that. Nor am I an expert on sailing vessels, so I won't try to edit this page, either. However-- I do believe there is a type of sail called a martingale, which could benefit from its own page disambiguated from this one.
 * I added it to the disambig at the top --14:18, 8 March 2006 (UTC)

Gambler's Fallacy
"Of course in reality the exponential growth of the bets would quickly bankrupt those foolish enough to use the martingale after even a moderately long run of bad luck."

I think this sentence qualifies as gambler's fallacy as it implies a long run of bad luck would affect future bets badly. --Kurulananfok 21:00, 20 May 2006 (UTC)

 * Maybe yhe sentence should be changed (do we need to say that somebody is foolish?). Anyway, since nobody has infinite worth, after a long run of bad luck there are not future bets: the gambler bankrupts and cannot afford further betting. --gala.martin ( what? ) 13:38, 21 May 2006 (UTC)

Compared to others
I think that we had better to put supmartingale, submartingale, semimartingale together in the article. Jackzhp 16:29, 9 September 2006 (UTC)

suggestions
In optional sampling theorem condition (a) is redundant since you require (b). Second, the link of the word constant to mathematical_constant is irrelevant - it's not the same meaning/intention

In Optional Sampling Theorem there is a reference to undefined condition (c): "a gambler with a finite lifetime (which gives conditions (a) and (b)) and a house limit on bets (condition (c))"

Martingale systems in investment, and the value of stating the obvious
It seems that more snake oil is being sold, and more ignorance spread, in the form of martingale systems promoted as a sound investment strategy, for stocks or foreign exchange. Shouldn't it perhaps be mentioned that martingale systems have been used, not just by casino gamblers, but by investors who perceive some kind of scientifically proven advantage to the use of such systems?

On the subject of stating the obvious, perhaps a plain language explanation of the fallacy of martingale systems would be suitable. Simply put, if you you're betting on a flip of a fair coin, then your expectation is zero (that is, your current status +/- 0), and by the very definition of random outcomes there is no need to refer to previous results to draw this conclusion. When betting in a "supermartingale" situation, such as a bet on black on a roulette wheel with two green spots, your expectation is determined by the odds of one trial, again without reference to previous trials, and this is largely self-evident without lengthy mathematical proof (I'm not saying we shouldn't show the proof! Only that it's fair to make a plain language statement of the obvious fact as well).

A similar plain language explanation might be to point out that, "if it makes sense, after six consecutive losses, to bet 2^6 times your original bet, and if results of each bet are independent, then it makes just as much sense to bet that amount right now. If it ever makes sense to bet $64.00 (or $64,000.00), then it always makes sense to do so.  If it ever doesn't make sense to bet that amount, then it never makes sense to, unless somehow the conditions of the game itself are changed... or unless you take the rational approach and constrain your bets, not by tying them to a chain of random past events according to an irrational system, but rather by the limits of one's own tolerance for risk.

Maybe this doesn't sound scientific enough, but sometimes common sense should be expressed in a common way.zadignose 04:56, 22 August 2007 (UTC)

 * Zadi, the idea of doubling your bet until you win can work, but you'd have to wait until the right point in time. If there was already 10 heads in a row and THEN you started the double bets, your are ARE INDEED much better off. The reason for that is that 10 in a row is very rare and 15 in a row is even more rare. If you withheld the start of your betting until a major statistical anomoly was already present, then what are are actually doing is beting AGAINST the continuation of the MAJOR anomoly. The key would be to know how often 10, 15 and 20, etc. in a row actually occur. Just by the seat of my pants, I'd venture to say you almost never see 20 in a row - even with millions of flips. 216.153.214.89 (talk) 00:09, 13 September 2008 (UTC)

 * Eek! Gambler's fallacy! C RETOG 8(t/c) 00:20, 13 September 2008 (UTC)

No - you are wrong. The odds are added differently than you think. If you flip a coin, it's 50/50. If you've flipped and gotten 10 heads in a row, is it still 50/50 on the 11th? No. Why? Because the more you get of the same in a row, the less likely the streak is to continue. If not, then you'd see 5,000 in a row all the time - but you never do. The way to count the odds is the difference between the frequency of each count in a row. For example, 5 in a row would take X number of flips to see one occurance of 5 in a row. Well it takes a LOT more flips to see a 10 in row than a 5 in a row. The longer the streak, the less often it occurs. Let's just say for example 10 in a row happens once every 100,000 times and and 11 in a row happens once every 110,000 times. The odds are .909 lower (100/110) that 10 in a row will continue to 11 in a row. However, if you took the results of the 11th flip after 1,000 ten-in-a-row runs, it would be 50/50. But we ARE NOT talking about doing 10 in a row many times and then going for the 11 in a row afterwards. We are only talking about 1 series and betting against that 1 series until it reverses. By waiting until there is a higher number of flips in a row (starting at 10 in a row, say), before starting the opposite betting, what you are doing is betting ONLY during a small slice of time, where a larger slice of time is required to get to 11 in a row. You are in effect exploiting the known temporary anomaly in the odds. I've discussed this at length in the past with a oddsmaking math genius - it is correct and could be proven with a qbasic program. The odds of 50/50 flips runs in small peaks and valleys. Imagine a timeline fluctuating up and down as it goes - these are the flips recorded as a fluctuating line. Up is heads, down is tails, left to right are the flips. Up one notch for heads, back to one below the line for tails and vice versa. As long as you get the same in row, the peak or valley gets larger. As soon as the series breaks, you go to position #1 on the other side of the line. The line would look something like seismograph printout. But on our paper, no matter how long our sheet was, you'd almost NEVER find a 10 valley or a 10 peak (10 tails or 10 heads in a row). And it's that much more difficult to get to 11, 12. It's not that you are beating the odds, it's that you are only playing when the odds are in your favor. The longer you let the steak get before you bet, the better the odds it will reverse while you still have money. If you don't believe me, challenge yourself to prove it. Let's write the qbasic code for this proof and post it here. BTW, the reason why no one benefits from this odds anomaly is that you'd have to stand by the roulette wheel all day just to see a goodly enough number in a row that your starting point would be affordable. You might have to wait all week to see one occurance of 10 in a row (must count 0 and 00 as streak continue - so your odd are less than 50/50 to start). So, if you went against it at 10, it would likely reverse by 13. So you bet 100(11-lose), bet 200(12-lose), bet 400(13-win). You lost $300, won $400, net $100 and you worked all week to get it. The anomaly is there, but the risk and time required to exploit it is too great. No matter how many times you double bet before you finally win, in the end, your net winnings equals the amount of your 1st bet. 216.153.214.89 (talk) 05:50, 18 September 2008 (UTC)

 * Yep, that's exactly the gambler's fallacy. You wrote, "If you flip a coin, it's 50/50. If you've flipped and gotten 10 heads in a row, is it still 50/50 on the 11th?" To which my answer is, yes, it is. C RETOG 8(t/c) 06:04, 18 September 2008 (UTC)
 * P.S. The talk page of an article isn't really the place to go into this in depth, but if you want to discuss it more, feel free to hit me at my talk page. C RETOG 8(t/c) 06:05, 18 September 2008 (UTC)

No, you are wrong. In any lot of odds-smoothing series, there will be outlier events. The longer a continous run is from the mean, the greater the odds that it must revert to the mean. The only fallacy about the "gamblers fallacy" is that some idiots think that all people who look at this and think "not so", would actually try to make $$ betting on it. Let me ask you a question: If you had just witenessed 100 heads in a row and had $900 billion dollars reserve, are you going to tell me you couldn't win the even money bet that the next would be tails - even for say $10>20>40>80>160>320>640>1,280>2,560>5,120>10,240>$20,480? Are you going to go from seeing 100 in a row (which NEVER happens) to 110 in a row (which NEVER NEVER happens). It's not a "fallacy" it's an odds-smoothing anomaly and it could be exploited, but the amount you could bet is too tightly controlled by the demands for a HUGE capital reserve, so even if you waited for a string of many in a row before you start, it wouldn't be worth the effort. To understand this, you need to step back from the absolute of the total odds being spread over all flipping occurrences and see that localized odds-smoothing anomalies do rarely occur and when they do, they are detectable. 216.153.214.89 (talk) 11:28, 19 September 2008 (UTC)

 * What are you talking about? To illustrate the point: I just flipped my quarter and it landed heads three times in a row.  Is it going to magically be more likely to land on tails when I toss it again?  Of course not.  If you suggest that it is so, you are contradicting practically all of probability theory and the theory of martingales.  When you say "given that we tossed a coin 10 times, what's the probability that it will land on heads?"  You calculate the conditional probability of an 11th heads given 10 heads, which by Bayes Rule would amount to the equation: P[11 heads]/P[10 heads] = (1/2)^11/(1/2)^10 = 1/2.  There you have it, 1/2 probability of getting a heads on the next toss.  I'm not trying to discredit you, but I find it highly unlikely that you spoke at length with a "math genius" on this topic.  Furthermore, you cannot "prove" such a thing with a computer simulation (lol @ qbasic).  Also, given that you are discussing in the mathematics section, please try to use mathematically precise language.  —Preceding unsigned comment added by 207.237.81.59 (talk) 18:07, 11 February 2009 (UTC)

207: Thanks for the feedback. What I am referring to is reversion to the mean. Please read this and after you have done so, please re-consider my comments in light of that information. 216.153.214.89 (talk) 01:09, 23 February 2009 (UTC)

 * 216.153.214.89, you're still wrong, and committing the gambler's fallacy by misunderstanding reversion to the mean. Reversion to the mean states that, e.g., a run of 9 heads out of 10 is likely to followed by the next 10 flips having fewer than 9 heads, closer to 5.  Reversion to the mean only comes into play when you're comparing two of the same observations-- a run of 100 heads is unlikely to be followed by another identical run of 100 heads, but reversion to the mean says nothing about observing the next 1 or 10 flips, as that's a different variate.  In practice, here is what reversion to the mean means:
 * 1) Getting eight heads out of ten flips can mean that you're flipping a very biased coin. But it's much more likely that you're flipping a fair coin but got an unusual number of heads.  Therefore, we expect that the next ten flips will have a number of heads closer to average, five.
 * 2) This applies in say, IQ tests. There are many more people who have an IQ score of 130 than 140.  So if you take an IQ test and get a score of 135, it's more likely that you were a person with an IQ score of 130 scoring 5 points high out of luck than a person with a score of 140 scoring 5 points low out of bad luck.  So your next IQ test is more likely to be closer to the mean of 100.  It's the same thing for height; a 6'10" person is more likely to have 6'6" genes and a great environment than 7'2" genes and a poor environment, simply because many more people have 6'6" genes.  —Preceding unsigned comment added by 132.228.195.207 (talk) 21:31, 31 March 2009 (UTC)

Origin of the name
There is no citation of the etymology of Martingale. In fact, I am reasonably confident that the stochastic process called a Martingale had its origin from the horse collar, not the gambling system. It is to give the imagery that there is a constraint on where the horse's head can move in the next time instant. It was a coincidence that this also happened to be a name of a gambling system. —Preceding unsigned comment added by 124.171.59.106 (talk) 11:37, 24 April 2008 (UTC)

 * A coincidence? The gambling system has an obvious connection to the mathematical concept.  I find that hard to believe that it would be coincidence.  The entire point of the gambling system is a claim that the optional sampling theorem can be violated.  —Preceding unsigned comment added by 132.228.195.207 (talk) 21:35, 31 March 2009 (UTC)
 * You may wish to see [] Melcombe (talk) 13:31, 2 April 2009 (UTC)

Demonstration of R software
I think that the R-programmed simulation of a Brownian motion is just a sequence of independent variables, whereas it should be the sum of independent random variables. I haven't changed it, because I'd like someone to provide a second opinion. Thanks. —Preceding unsigned comment added by Philip Maton (talk • contribs) 16:46, 27 May 2009 (UTC)
 * I agree. As it stands, the Excel formula and the R command/graphic are inconsistent. -- Chadhoward (talk) 22:18, 14 March 2010 (UTC)

Polya's urn redirect
Polya's urn redirects here, Wouldn't it be better if redirected to Urn problem? --Squidonius (talk) 23:20, 23 June 2010 (UTC)

Why this section was hidden in the article instead of deleted?
{xx{Hidden begin}xx}

A more general definition
One can define a martingale which is an uncountable family of random variables. Also, those random variables may take values in a more general space than just the real numbers.

Let $$\scriptstyle \mathcal{I} $$ be a directed set, $$\scriptstyle V$$ be a real topological vector space, and $$ \scriptstyle V^\star $$ its topological dual (denote by $$ \scriptstyle\left( \cdot, \cdot\right) $$ this duality). Moreover, let $$\scriptstyle \left( \Omega, \mathbb{P}, \mathcal{F}, \mathcal{F}_i\right) $$ be a filtered probability space, that is a probability space $$\scriptstyle \left( \Omega, \mathbb{P}, \mathcal{F}\right) $$ equipped with a family of sigma-algebras $$\scriptstyle \{\mathcal{F}_i\}_{i \in \mathcal{I}} $$ with the following property: for each $$\scriptstyle i, j \in \mathcal{I} $$ with $$\scriptstyle i \leq j $$, one has $$\scriptstyle \mathcal{F}_i \subset \mathcal{F}_j \subset \mathcal{F} $$.

A family of random variables $$\scriptstyle \{X_i\}_{i \in \mathcal{I}} $$:

 * $$ X_i : \Omega \to V, $$

is called a martingale if for each $$ \scriptstyle u \in V^\star $$ and $$\scriptstyle i, j \in \mathcal{I} $$ with $$\scriptstyle i \leq j $$, the three following properties are satisfied:

 * $$\scriptstyle \left(X_i, u\right) $$ is $$\scriptstyle \mathcal{F}_i$$-measurable.

 * $$ \mathbb{E}[|(X_i, u)|] < \infty. $$

 * $$ \mathbb{E} [\left(X_j,u\right) | \mathcal{F}_i] = \left(X_i, u\right). $$

If the directed set $$\scriptstyle \mathcal{I} $$ is a real interval (or the whole real axis, or a semiaxis) then a martingale is called a continuous time martingale. If $$\scriptstyle \mathcal{I} $$ is the set of natural numbers it is called a discrete time martingale. {xx{Hidden end}xx}

Coined by Ville
I think the term martingale was coined by Ville, somewhat before Lévy used them. About the Girsanov theorem, it only permits to construct a measure that makes SOME Ito processes a martingale (not any Ito process!). And Balsara and Kleinert seem out of place in the references. Why not link to Bachelier or some other financial applicant instead of referencing Balsara's book "for futures traders". Likewise, treatises of Doob, Meyer or Neveu are actually about martingales, whereas Kleinert's book seems to be about "a lot of other stuff".90.27.21.180 (talk) 13:50, 28 April 2010 (UTC)

Relation to Markov chains ???
Surely, there must be some theorems relating martingales to Markov chains? The definitions are quite similar; one speaks of probabilities, the other of expectations. Surely there must be overlap! Yet the current article breaths narry a word of this. linas (talk) 03:40, 20 November 2010 (UTC)
 * They are only related in that they both describe stochastic processes. The Markov property (roughly) states that after observing the prior state of the system, there is no additional information that can be gained by observing earlier states. In a Markov chain, once we enter one link of the chain, the next transitions out of the chain only depend on that link; they do not depend on how we got to that link. However, in a Martingale (roughly), the conditional expectation of the next observation given this and all prior observations is equal to this observation. So Martingales need not have a Markov property as the next transition can depend on more than the present state of the system. —TedPavlic (talk/contrib/@) 02:15, 15 August 2011 (UTC)

 * I had the same confusion reading the article for the first time. It says that martingale is "a model of a fair game where knowledge of past events never helps predict the mean of the future winnings and only the current event matters." This is just misleading, and sounds like a special case of Markov property. It should be changed to "a model in which the expected value of the future state is always equal to the current state."Sprlzrd (talk) 18:56, 20 January 2017 (UTC)

Lead is gibberish
Please explain in English, someone! -- cheers, Michael C. Price talk 12:56, 14 August 2011 (UTC)
 * The lede gives the standard rough definition of a martingale. In a prototypical martingale stochastic process, a realization (i.e., a "draw") of the current value of the stochastic process is exactly the mean of the next value of the process. For example, imagine a game where you win a dollar every time a fair coin comes up heads and you lose a dollar every time that fair coin turns up tails. Your winnings are a martingale. That is, your winnings now are the expected value (the "mean" in probability) of your winnings after the next coin flip. In other words, the next coin flip will increase your winnings by a dollar with 50% probability and will decrease your winnings by a dollar with a 50% probability, and so your expended gain on the next flip is $0, and your expected total winnings on the next flip is identical to your current winnings. Because the expected winnings on the next flip are equal to your known winnings on this flip, the process is a martingale. Does that help? Do you think there is something that could be added to the lede that could make things more clear? At the moment, the wikilinks should help clarify some of the muddiness. Where that doesn't help, some of the rest of the page should. However, the lede is a little short, and so maybe something (but what?) could be added. —TedPavlic (talk/contrib/@) 02:07, 15 August 2011 (UTC)
 * To clarify, I'm describing a "prototypical" Martingale by focusing on only the immediately prior observation. The broader definition of Martingale states that the conditional expectation of the next observation given all prior observations (including this one) is equal to this observation. So it's easier to talk about Martingales when they are Markov chains, but the generic Martingale need not have this property. —TedPavlic (talk/contrib/@) 02:33, 15 August 2011 (UTC)
 * Thanks, that helps a great deal. I was struggling with stuff like "value of an observation" which I think means "value of an observable" or "value of the observables".  The term "equal to the observation" needs clarification as well, IMO. -- cheers, Michael C. Price talk 06:05, 15 August 2011 (UTC)
 * Hm. I think "observation" is probably a better word than observable. This topic doesn't really relate to observability. That is, we aren't measuring an output of a system and trying to estimate the internal states of that system. Here, "observation" is a "realization" of a random variable. A stochastic process is a sequence (i.e., an ordered list) of random variables that are typically indexed by something related to time.
 * For example, a continuous-time stochastic process $$C(t)$$ maps each continuous time $$t \in \mathbb{R}$$ to a random variable $$C(t)$$. So, at time 0, there is a random variable $$C(0)$$, and at time 3.4, there is a random variable $$C(3.4)$$. To "observe" these random variables means to "draw" from them according to their probability distributions just as you would draw from a card deck or flip a coin or pull the arm of a one-armed bandit. Making an observation of $$C(0)$$ may tell you nothing about what you would draw from $$C(3.4)$$. On the other hand, if $$C(3.4)$$ is strongly correlated with $$C(0)$$, then drawing from $$C(0)$$ might allow you to predict a future draw from $$C(3.4)$$ with absolute certainty.
 * A martingale (at least the conventional definition of one) is a discrete-time stochastic process (because it requires a notion of "immediately before"). A discrete-time stochastic process $$D(k)$$ maps each discrete time $$k \in \mathbb{N}$$ (i.e., a natural number (or an integer, if you like)) to a random variable $$D(k)$$. In a continuous-time stochastic process, there is no way to describe the "next" random variable at any given time. However, in a discrete-time stochastic process, at time $$k$$ the "next" random variable is at time $$k+1$$. It is not necessary for a discrete-time stochastic process to be a martingale. However, there is a discrete-time stochastic process $$M(k)$$ with the property that observing ("drawing") $$M(k)$$ tells you the (conditional) expected value of the probability distribution at $$M(k+1)$$, then that process is a martingale.
 * So martingales are stochastic processes (i.e., ordered lists of random variables) where the conditional expected value of the probability distribution at time $$k$$ given all draws of variables before time $$k$$ is equal to the value drawn at time $$k-1$$. —TedPavlic (talk/contrib/@) 14:28, 15 August 2011 (UTC)
 * These comments still hold for observables in physics too. "Observation" really is a better term ("realization" probably is even better). —TedPavlic (talk/contrib/@) 17:49, 17 August 2011 (UTC)
 * I've edited the lede to try to make a distinction between observed values and random variables. —TedPavlic (talk/contrib/@) 17:53, 17 August 2011 (UTC)
 * Thanks for your efforts. It all helps! -- cheers, Michael C. Price talk 20:35, 17 August 2011 (UTC)

I've taken a stab at it myself, philosophy being that anybody should be able to at least understand what the article is about (and those who want details will read further for the rigorous definition). Hence I've put the opening sentence into plain English, avoided defining any symbols, and incorporated mention of a prototypical well-known example, but fundamentally I've tried to preserve all that was being said. Cesiumfrog (talk) 01:12, 30 August 2011 (UTC)
 * Hm. I think those edits might go too far. Terms like "likely" and "rise" and "fall" have too much loaded meaning and may communicate the wrong idea to the reader. For example, the conditional probability distribution of the next sample may have significant skew asymmetry. In that case, saying that the mean of that distribution is equivalent to the previous sample does not imply that the process is "equally likely" to "rise" and "fall" unless you put strict qualifications on what it means to rise and fall. You can imagine a probability distribution where 99% of the time it rises but the 1% of the time it falls it falls tremendously. In that case, you still have a Martingale, but the process would spend most of its time rising. So I prefer the old definition perhaps with a little rounding off of edges. This "lay definition" goes too far (IMHO). —TedPavlic (talk/contrib/@) 19:05, 30 August 2011 (UTC)

Well, for future reference of anybody wanting simpler, here was my proposal for the lead:
 * In probability theory, a martingale is a stochastic process (i.e., a sequence of random variables) which is likely to rise just as much as it is likely to fall. That is, given all of the previous observed values (i.e., the realizations), the expectation for the next value is always equal to the current value.
 * An unbiased random walk is an example of a martingale. Martingales are models of fair games.

and I disagree with the current version. Much of this pertains directly to the MOS: Lastly, somewhere in the article I think an example should explain more clearly why the definition is not simply E(X_{n+1}|X_n)=X_n since the obvious examples are actually not conditional on X_1,..X_{n-1}. Similarly, a more tangible example (or at least less abstractly explained) of a martingale with respect to another sequence could be beneficial. Cesiumfrog (talk) 02:46, 31 August 2011 (UTC)
 * Since it is the norm for all mathematical articles to have a separate section for the precise definition, I don't think this article has an exceptional necessity for its lead to be cluttered with a self-referential explanation of that fact. Besides, there's already the TOC. (If there are still concerns, perhaps adding "rigorous" to the section title would alleviate them?)
 * The lead is always supposed to be informal and "without rigor". (At the very least it should be less rigorous than those "mathematically rigorous definitions" it says are "given below".) Now, the lower section contains the mathematical definition (expressed E(X_{n+1}|X_1,...,X_n)=X_n) and also restates the definition into words (rigorously), saying it is a "stochastic process (i.e., a sequence of random variables) [..for which..] the conditional expected value of the next observation, given all the past observations, is equal to the last observation." Alas, the current version of the lead not only preserves just as much rigor but also defines more terms and introduces more jargon: "..is a stochastic process (i.e., a sequence of random variables) [for which,] given all of the previous observed values (i.e., the realizations), the conditional expected value (i.e., the expectation or mean of all cases that share the same previous observed values) for the next value is equal to the current value."
 * An article should never open with the words "Very roughly speaking,". Furthermore, in cases (such as this) where the lead is really not speaking that innacurately (ideally just isn't using strict jargon), a disclaimer isn't warranted at all. (Those seeking a completely rigorous definition will read down to that section, still.)
 * Yes, it is true that the chance of a martingale rising is not always equal to the chance of it falling. But the expecation for how much it will rise does equal the amount it is expected to fall. Yes, this isn't perfectly unambiguous, which is why my proposal immediately follows up with a clarification of exactly this point. (The approach is to start simple and move toward more technical as the article proceeds. And not to proceed too long before getting to an example and motivation. The concept of martingales really shouldn't be too difficult for someone unfamiliar with expectation values to understand.)
 * The use of the (common speech) clause "given .." makes redundant the term conditional expectation. Better to stick with what terminology is more likely to be familiar, and save more specialised jargon for outside of the lead.
 * But I'm happy we're no longer introducing symbols in the lead. (I used the word "always" to replace the use of "t".)
 * Your qualification of what "always" means suggests a purpose for symbols. In any case, technical jargon can be resolved with Wikilinks as it is done on many other technical pages. I think it's important not to relegate the lead to be an auxiliary example (that is better suited for an "Examples" section). Regarding the conditional expectation and the other realizations, you need more than just the previous realization if the process does not have the Markov property (also see Markov chain). Imagine you had a deck of cards and every time you drew a card off the top of the deck, you did not replace the card back into the deck. In that case, the conditional probability distribution for the next card draw depends not only on the previous card drawn but on all of the previous cards drawn. Moreover, you could implement a "coin" based on such a card deck and thus the conditional probability of "heads" and "tails" would differ if you knew only the previous realization versus knowing all prior realizations. —TedPavlic (talk/contrib/@) 18:29, 31 August 2011 (UTC)

 * I would suggest the following:

 * In probability theory, a martingale is a sequence of random variables (i.e. a stochastic process) where the expectation for the next value is always equal to the most recent value, notwithstanding all of the earlier observed values (i.e., the realizations).
 * An unbiased random walk is an example of a martingale. Martingales are models of fair games because the expectation of winning on the next round always equals the expectation of losing.
 * --agr (talk) 12:13, 31 August 2011 (UTC)
 * "Notwithstanding" is not the right word. The conditional expected value is really the expectation conditioned on having knowledge of all prior realizations. If one of those realizations change, the expectation may change. Moreover, the phrase about martingales being models of fair games is so strong that it seems to suggest that martingales are only applied in the analysis of games. A wikilink from games to a generic definition may solve this problem. —TedPavlic (talk/contrib/@) 18:29, 31 August 2011 (UTC)

 * I don't understand what you mean here by "If one of those realizations change, the expectation may change." Isn't that what the definition disallows?--agr (talk) 21:19, 31 August 2011 (UTC)

 * The martingale definition restricts the conditional probability distribution of the next sample. Each possible prefix (i.e., each sequence of prior realizations) maps to a conditional probability distribution of the next sample. The "notwithstanding" sounds like the conditional probability distribution is somehow invariant of the history of the system. In fact it's the opposite. The history (i.e., all of the prior realizations) tell you which conditional probability distribution (or at least conditional expected value) to use to describe the next sample. If it is a martingale, then the conditional expected value (which is parameterized by the entire history of the process at that point) will always be equal to the most recent realization. In other words, if I only knew the current observation, my expected value for the next observation might be different if I knew the entire history. Imagine drawing cards out of a deck without replacement. If all you know is the previous card drawn, you have a lot of uncertainty about the next card drawn. However, if you know the entire history (i.e., every card drawn so far), then your expected value of the next card changes substantially. It is that latter expected value that is used in the definition of the martingale. This is why the conditional modifier is so important (see conditional expected value). —TedPavlic (talk/contrib/@) 03:08, 1 September 2011 (UTC)
 * agr: If you'd like to use the "notwithstanding" language, it may be a good idea to take a hint from the definition in Probability and statistics: the science of uncertainty by Evans and Rosenthal (note that they explicitly assume that the Martingale is a Markov chain "for simplicity", but they also say that "this is not really necessary"). In particular, you can rephrase the definition in terms of the conditional expectation of $$X_{n+1}-X_{n}$$ (given all prior observations). In that case, the conditional expectation is 0, notwithstanding the current observation. As they summarize, "on average", the chain's value "does not change, regardless of" what the current value actually is. Again, this definition assumes a Markov chain (explicitly, for simplicity), but a similar form could be used here in the general definition. If a Markov chain is needed to help explain, then it should be stated explicitly as a device for simplicity. —TedPavlic (talk/contrib/@) 20:07, 1 September 2011 (UTC)
 * No, you're misimagining it or else it isn't a martingale. If all you know is the current observation, then that is your expectation value for the next observation, and if it's a martingale then you're correct. If knowing more would have changed your expectation to anything other than that, then it isn't a martingale. Yes, the probability distribution shape may change depending on earlier realisations, but not its center of gravity. For example, a drunkard may lurch with greater amplitudes if it has stumbled into the bar more recently than the coffee shop, but provided the walk stays unbiased it stays a martingale. Likewise, the random idling of a unicyclist can be a martingale despite being likely to go back and forth more than side to side (hence the probability distribution depends on the last two observations, but the expectation value depends only on the last one).
 * agr, this suggests a great simplification of the lead. For example, is the definition "a martingale is a sequence of random observations for which the latest realised value is always the expectation value for the following observation" anything short of completely rigorous? Can you fully elucidate a counterexample TedPavlic? Cesiumfrog (talk) 03:44, 1 September 2011 (UTC)
 * Cesiumfrog: You need the whole history because that's the definition of a Martingale. For confirmation, see Randomized algorithms by Motwani and Raghavan. Now scroll down to page 87. It says that the knowledge of the past bets does not help to predict the future. That's the essence of a martingale, and perhaps this is where agr's "notwithstanding" came from. In a Martingale, if you know all of your past history, you still are clueless about the future outcome. The point of a martingale is not simply that knowing what you have now fixes your expectation of what you have later, it's that even if you did know the history of how you got here, it wouldn't help you to predict where you would go next. Thus, that's conditioned on the entire history and does not require a Markov property. As mentioned in my previous edit, some authors add the Markov property for simplicity, but it is not the general definition of a martingale. A martingale is a fair game because there's no way to predict the future based on any knowledge of past events. —TedPavlic (talk/contrib/@) 20:21, 1 September 2011 (UTC)

Please review new lede. —TedPavlic (talk/contrib/@) 21:01, 1 September 2011 (UTC)

Biased Coin?
Something about the biased coin as an example of a sub- and supermartingale seems misleading. The expectation value of a biased coin doesn't depend on its history – it's always p. Even in the case where you have a coin and are told that it's biased (but not by how much), you wouldn't know whether it was biased up or down, so still it's not a sub- or supermartingale. And if you were told that it was biased in a particular direction, the initial expectation value probably isn't 0.5.

206.124.141.187 (talk) 19:56, 29 June 2013 (UTC)

 * Never mind – I realized that the bankroll is the random variable, not the outcome of the coin flips. 206.124.141.187 (talk) 03:44, 2 July 2013 (UTC)

Applications
I would like to suggest adding a section on applications for martingales. The abstract of The least variable phase type distribution is Erlang concludes stating that their "proof is simple and elegant and is a nice example of the power of martingales; it seems intractible without them." Could we collect together some examples of other insights given by the theory of martingales? Gareth Jones (talk) 16:50, 12 November 2013 (UTC)

Chance, Not Skill
"To contrast, in a process that is not a martingale, it may still be the case that the expected value of the process at one time is equal to the expected value of the process at the next time. However, knowledge of the prior outcomes (e.g., all prior cards drawn from a card deck) may be able to reduce the uncertainty of future outcomes. Thus, the expected value of the next outcome given knowledge of the present and all prior outcomes may be higher than the current outcome if a winning strategy is used. Martingales exclude the possibility of winning strategies based on game history, and thus they are a model of fair games." [emphasis mine]

A martingale illustrates a fair game of chance, not skill. A fair game of skill simply requires that all parties have equal access to the same body of information, if they have the presence of mind to seek it out. That means that parties that are willfully ignorant (that is, "stupid" ["Boko Haram!"]) are not at a disadvantage in a game of chance, but are at a disadvantage in a game of skill. Life is arguably just as much a game of skill as it is of chance, or none of us would bother reading and writing an online encyclopedia. This may be a small difference in the nature of this article, but it's a ginormous difference in world-view. I'm not sure how to proceed from here, beyond pointing this out on a talk page, but the anomaly struck me rather hard at the time, sorry. TheLastWordSword (talk) 13:37, 20 June 2014 (UTC)

Reason for removal of paragraph
The post of 20 June 2014 has been largely ignored. Currently the second paragraph in the lead says

 * To contrast, in a process that is not a martingale, it may still be the case that the expected value of the process at one time is equal to the expected value of the process at the next time. However, knowledge of the prior outcomes (e.g., all prior cards drawn from a card deck) may be able to reduce the uncertainty of future outcomes. Thus, the expected value of the next outcome given knowledge of the present and all prior outcomes may be higher than the current outcome if a winning strategy is used. Martingales exclude the possibility of winning strategies based on game history, and thus they are a model of fair games.

This makes no sense. First, it refers to the expected rather than actual value of “the process at one time”. Second, it conflates the outcome of the stochastic process with the outcome of a bet chosen based on skill. And it talks about reducing the “uncertainty” of future outcomes, which is irrelevant to the issues at hand. So I’m removing this paragraph. Loraof (talk) 01:01, 17 November 2017 (UTC)

False statement about word origin
English has been using the word "martingale" since 1580 at the latest. It originally meant a strap that could be added to a horse's harness to limit how far the horse could lift its head; this was used during training as a check against out-of-control behavior. TooManyFingers (talk) 22:45, 6 May 2024 (UTC)

 * Please find a good RS reference and we can add it. Limit-theorem (talk) 06:06, 8 May 2024 (UTC)
 * Actually, for the word origin we should start with the gambling use, as this is not about etymology but mathematics and probability. Limit-theorem (talk) 12:27, 8 May 2024 (UTC)
 * The word does not come from the strap for horses, to my knowledge. If I remember correctly, there were other things that were also called a martingale in the past. The origin of the word in mathematics is to my knowledge unkown. There is also a French article about it: "Histoire de martingales" by Roger Mansuy (Roger Mansuy, “Histoire de martingales”, Mathématiques et sciences humaines, 169, Printemps 2005, DOI: 10.4000/msh.2945 ) Tensorproduct --talk 20:59, 8 May 2024 (UTC)

title: Talk:Martingale (probability theory)


text:
Talk:Accumulation point

Changes
Hi User:Dcoetzee

You seem to be in the middle of a list of changes to this page. I won't interrupt, other than to point out:
 * the version of the page without the TeX code formats better. The <math> stuff should be as a last resort, from what I've read.
 * your last paragraph is incorrect, in that open sets usually do contain limit points. For instance, the set of limit points of the open interval (0,1) is the closed interval [0,1], not just the endpoints.  Some qualification is necessary - e.g. limit points other those in the set.

Cheers, AndrewKepert 03:14, 13 Nov 2003 (UTC)

 * Okay, since Dcoetzee's changes have finished, I think almost all the changes should be reverted back to the 7 Jun 2003 version. The last paragraph is factually incorrect.  Okay if you say "boundary point" instead of "limit point" but otherwise no.  So I am removing it - sorry.  The use of math code instead of html character entities is nicer in some ways but really makes the document less readable, mainly  due to it pinning the font size to a particular pixel size, and hence resolution dependent.  The rule of thumb is that if it can be done without <math> then this is preferable.
 * The two improvements are the speling corection (containg -> containing) and the change of "infinitely many" to "at least one other".
 * So my action is to revert to the 7 Jun version and incorporate these changes. Someone else may want to put in a corrected version of the last paragraph.
 * AndrewKepert 06:58, 13 Nov 2003 (UTC)

 * I'm sorry, you're right, I wasn't thinking straight. The part about the closure being the union of a set and its limit points is true, but the rest wasn't. Sorry for the bad changes.

 * I'm still a little new to Wikipedia, I just figured out talk pages. The HTML codes you use appear in my browser (IE) as plaintext, so I can't tell what the article is supposed to be saying. I think the font problems associated with the TeX markup are better than this. This is a subject of some debate on the page

 * http://meta.wikipedia.org/wiki/Math_Markup

 * Derrick Coetzee 03:23, 15 Nov 2003 (UTC)

Proposition
I added a comment to the definition, I hope this is okay. Also, Proposition 1 is not sharp; it holds even if the space is not T 0. It would probably be better to substitute a proof which does not use it (I know of at least one simple such proof).

Derrick Coetzee

HTML codes versus math tags
Because most viewers use IE, and it does not support the HTML tags originally used, I believe it's preferable to use the math tags. Although inline images can be unpleasant, some of the symbols translate into characters in the Symbol font, which IE will render, and these are just as good. Also, the user can choose how to render math tag symbols in their preferences, and in the future when browsers support MathML and Wikipedia translates these tags to MathML, it will be able to render even the more complex symbols without sending inline images. I hope this is sufficient justification for you. View the web page I linked above for more discussion of this.

Derrick Coetzee

 * Oh okay - fair enough. I use Mozilla, and the symbols were listed on How to edit a page and looked ok to me.  So I assumed they were widely implemented in common browsers.  Maybe the wiki engine should re-encode these on the fly if (a) user prefs dictate or (b) it detects IE?  (via <math> or symbol fonts)  AndrewKepert 01:36, 20 Nov 2003 (UTC)

I had written about an hour's worth of explanation and comments, but since I forgot to cut and paste, and lost contact with wikipedia (not the net), it's gone forever, so I'm not even going to try to rewrite it, I don't have time. Be warned, though: IF YOU USE INTERNET EXPLORER, ALWAYS CUT AND PASTE TO A NOTEPAD OR WORD SOMETHING YOU'RE GOING TO SEND TO WIKIPEDIA BEFORE YOU FINALLY SEND IT. IF YOU DON'T, YOU COULD LOSE IT FOREVER. So, you might think the following comments are terse and/or rude, I'm sorry, blame Bill Gates or the idiots who wrote Explorer or the wiki servers or something.

First, there are 3 distinct definitions, whether you count the point, look for a distinct point, or look for infinitely many points. By "limit point", most people mean a distinct point. The first is really "point in the closure". The second is NOT the same! So, I suggest that limit point be used for what it is currently, and "accumulation point" for infinitely many (which is not the same as simply distinct -- consider Sierpinski space I believe). "Cluster point" is another matter, depending on what opinion people who use filters have for terminology.

[edit out]

My bad! I must really be fuming at these computers. The reformulation is correct...but the Sierpinski space is a counterexample for LP vs. AP, I think. Revolver

I looked up in some old textbooks and found:

 * "limit point" is hardly ever used for "point of closure", sometimes this is called "point of adhesion" or something similar, so I'll take that part out.

 * T1 is equivalent to every limit point being accumulation point -- proof to come. (It's an interesting problem)

I'll clean up a bit, add proof later today... Revolver

 * I like the discussion of the three different definitions and terms, but there's enough consistent usage of all three for definition (2) in particular that I think it's a mistake to define accumulation point to refer to (3). I changed this.

 * Derrick Coetzee 14:10, 3 Dec 2003 (UTC)

I still think accumulation point should be defined to mean (3), and I'll explain. The problem is that by defining limit point and accumulation point to mean the same thing, then all the articles and examples that involve non-T1 spaces immediately lose a piece of terminology to lose. Put another way, Wikipedia terminology (at least in mathematics) has to appease even the slightest minorities. Yes, for 80-90% (or more) of the people (or articles) there's consistent usage, but for 10-20% of the people or articles, it's not consistent enough, in the sense that the distinction must be made. Put another way, "consistent usage" is a very subjective judgement, esp. when made by people in mathematics who may work in differing areas of research or work on different problems. So, I think this outweighs the confusion that comes from defining them both to mean what we mean by limit point. To give an example, I could write articles on Sierpinski space or the upper- or lower-limit topologies on the reals, or other articles, and in all of these, I might at some time want to talk about "limit points" and "accumulation points" as separate definitions. To define them to be the same takes this possibility away. At the same time, if an article only deals with T1 spaces, then the authors will be able to use both "limit point" and "accumulation point" freely, because in those cases, it won't make a difference. In other words, if articles have been using "accumulation point" when they really meant "limit point", there's no need to go back and change everything, or even to mention the distinction, so long as the articles in question are only discussing T1 spaces. Now, if the articles ARE discussing non-T1 spaces, then there IS a need to go back and possibly cause confusion, because the article is attempting to tackle examples where the two definitions aren't the same. So, even by defining them differently, it shouldn't be confusing for those articles where they're equivalent. Revolver

Sorry...I keep changing things back, to explain the alternate definition..."neighbourhood" def is not the same as "open set" def, not every neighbourhood is open (see wikipedia definition of neighbourhood, contains an open set), so there is technically a slight distinction between the two definitions. Revolver

 * In reference to this particular point, I have seen neighbourhood used in this sense. I would use the term open neighbourhood.
 * Derrick Coetzee 23:54, 4 Dec 2003 (UTC)

 * Oops, I see what you mean now. You're actually making a stronger statement.
 * Derrick Coetzee 23:57, 4 Dec 2003 (UTC)

Okay, I've been thinking about this more. I went back and looked at several standard textbooks, and most (not all) do define accumulation point the same as we've defined limit point. So it might be confusing not to do this. At the same time, I think there's a need for some terminology to refer to what I want to call "accumulation point". Perhaps a compromise (or the solution that would be most comfortable to the most articles) would be to let accumulation point be the same as limit point, but have another term for "accumulation point", a standard one seems to be "omega-accumulation point" (I assume omega refers to the ordinal number, which is strange since it's a matter of cardinality, not ordinality). Then, maybe this whole discussion of the distinction could be moved to a separate article on the new definition. How does this sound? Revolver

Sorry if I seem a bit touchy above. General (point-set, descriptive) topology is a sort of guilty pleasure hobby for me, so I've explored more than a bit "life without T2" or some of the bizarre spaces one can get from considering various cardinalities or even different axiom systems. But I'm also quite rusty on some of the terminology. Let me check out some stuff at the library and see if there's a consistent terminology for some of this stuff. I know there is some terminology that distinguishes say nbhds of a point that meet a set in a distinct, infinitely many, uncountably many points, e.g. the "uncountably many" concept is sometimes called a "condensation point", although presumably one could define the concept for any cardinal at all. Revolver

Trivial topology statement
If a space has more than one point, then its topology is trivial if and only if every point is a limit point of every nonempty subset.

I'm afraid this simply isn't true; we already pondered this for a while in the Trivial topology article, and our conclusion was instead this:

If S is any subset of X with more than one element, then all elements of X are limit points of S. If S is a singleton, then every point of X \ S is still a limit point of S.

This is easy to see if you think about the closure-based definition of limit point. I'm replacing the statement and proof.

Derrick Coetzee 13:33, 5 Dec 2003 (UTC)

Ambiguity
"infinitely many points $$x_n$$ such that $$d(x,x_n)<\epsilon$$" is ambiguous when there may be infinitely many n for which $$x_n$$ are equal. Probably this should be "infinitely many values of n such that $$d(x,x_n)<\epsilon$$.--Patrick 23:22, 2 June 2007 (UTC)

 * I fixed this. Since, if for infinitely many n the values of $$x_n$$ are equal, this point is a cluster point of the sequence but not necessarily a limit point of the set of points in the sequence, care should be taken with terminology so that the two are not confused. I edited the article accordingly. Any comments?--Patrick 12:30, 6 June 2007 (UTC)

Differences between limit point, accumulation point, and cluster point?
According to PlanetMath and MathWorld they are the same. Others sources including this one disagree. What is correct? Ht686rg90 (talk) 16:30, 27 June 2008 (UTC)
 * I think these notions are the same. Oleg Alexandrov (talk) 02:54, 28 June 2008 (UTC)

 * Agreed. By introducing sequences, the article section in question manages to confuse the concept of "limit point of a set" with the separate concept of "limit of a sequence". So the set {0.1, 0.9, 0.01, 0.99, 0.001, 0.999 ...} has two limit points/cluster points/accumulation points at 0 and 1, but the sequence 0.1, 0.9, 0.01, 0.99, 0.001, 0.999 ... has no limit.
 * I think the confusion was introduced in a set of edits in June 2007 by User:Patrick, which he mentions in the talk page section above. I suggest we remove all mention of sequence limits from this article and go back to something close to this May 2007 version of the article. Gandalf61 (talk) 08:59, 28 June 2008 (UTC)
 * Done. This is not the first time I see Patrick doing odd edits in math articles. Oleg Alexandrov (talk) 16:00, 28 June 2008 (UTC)

Characterization of limit point in sequential space
"Alternatively, if the space X is sequential, we may say that x ∈ X is a limit point of S if and only if there is an ω-sequence of points in S whose limit is x; hence, x is called a limit point."

I think the "only if" part is wrong. It is true for "point of closure" (also called "adherent point"), but not for isolated points. E.g., in R, let S be the union of [0,1] and {5}. Then {5} is an isolated point; it is not a limit point (in the definition of "every nhbd meets a point in S different from it"), but is a point of closure. But obviously the constant sequence equal to 5 is a sequence in S, which converges to 5.

What is true (and useful) is replacing S by S-{x}, i.e. requiring the sequence to consist of elements different from x itself:

"Alternatively, if the space X is sequential, we may say that x ∈ X is a limit point of S if and only if there is an ω-sequence of points in S-{x} whose limit is x; hence, x is called a limit point."

Urysohn (talk) 20:03, 17 June 2010 (UTC)
 * Yes, that's a silly omission. Fixed.—Emil J. 11:10, 18 June 2010 (UTC)

 * The statement is wrong for general sequential spaces. It works however for Fréchet-Urysohn spaces, see here.—yadaddy_ag 08:21, 10 August 2016 (UTC)

Point Of Accumulation KEY CONCEPT
"Every finite interval or bounded interval that contains an infinite number of points must have at least one point of accumulation."

 * I added this to the bottom of the opening paragraph and it was quickly deleted. This is the key concept.  Please see Richard Courant.

This article is actually very poorly written. It does not include this key concept nor does it expand on it. This concept is fundamental to calculus. 01001 (talk) 18:42, 10 March 2011 (UTC)

Limit point of sets with cardinality 1
"[...] every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself."

Having S:={x} consisting of one element only and, for instance, the discrete topology on X, x is the limit of any sequence on S, respectively ALL elements of S are apparently in a neighbourhood of x, thus x should be "intuitively" a limit point. However, it fails the condition above. I think the definition is not well chosen and should include that case. --85.0.115.52 (talk) 16:46, 1 January 2014 (UTC)

 * Ok, I see, the definition is chosen intentionally, as "No isolated point is a limit point of any set." --85.0.115.52 (talk) 17:01, 1 January 2014 (UTC)

Derived Set
I noticed that there is no mention of the derived set, the set of all limit points of a given set. Since these concepts are closely related, it seems logical to cross-reference. — Preceding unsigned comment added by Fonsf (talk • contribs) 15:44, 15 June 2016 (UTC)

Possible simplification of introductory sentence
The first sentence of the article currently reads"In mathematics, a limit point of a set S in a topological space X is a point x (which is in X, but not necessarily in S) that can be 'approximated' by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself."Could the phrase "with respect to the topology on X" be removed, to read as follows -- ? "In mathematics, a limit point of a set S in a topological space X is a point x (which is in X, but not necessarily in S) that can be 'approximated' by points of S in the sense that every neighbourhood of x also contains a point of S other than x itself." In other words, could it be left to the reader to assume that only one topology at a time is under discussion? I think the change would make the first sentence easier to read and understand. Dratman (talk) 20:42, 28 February 2017 (UTC)

Renaming article Limit point of a set
I suggest that this article (currently named "Limit point") be renamed "Limit point of a set" and that this page (i.e. "Limit point") become a disambiguation page. The reason being that the term "Limit point" is commonly used for notions that are not equivalent to "Limit point of a set" such as: "Limit point of a filter" (Bourbaki Topology 7.1), "Limit point of a net/sequence", "Limit point of a function with respect to a filter" (Bourbaki Topology 7.3). Mgkrupa 15:51, 21 June 2022 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Accumulation point


text: User:Dcoetzee

title: User:Dcoetzee


text:
Sciatica

Sciatica is pain going down the leg from the lower back. This pain may go down the back, outside, or front of the leg. Onset is often sudden following activities like heavy lifting, though gradual onset may also occur. The pain is often described as shooting. Typically, symptoms are only on one side of the body. Certain causes, however, may result in pain on both sides. Lower back pain is sometimes present. Weakness or numbness may occur in various parts of the affected leg and foot.

About 90% of sciatica is due to a spinal disc herniation pressing on one of the lumbar or sacral nerve roots. Spondylolisthesis, spinal stenosis, piriformis syndrome, pelvic tumors, and pregnancy are other possible causes of sciatica. The straight-leg-raising test is often helpful in diagnosis. The test is positive if, when the leg is raised while a person is lying on their back, pain shoots below the knee. In most cases medical imaging is not needed. However, imaging may be obtained if bowel or bladder function is affected, there is significant loss of feeling or weakness, symptoms are long standing, or there is a concern for tumor or infection. Conditions that may present similarly are diseases of the hip and infections such as early shingles (prior to rash formation).

Initial treatment typically involves pain medications. However, evidence for pain medication and muscle relaxants is lacking. It is generally recommended that people continue with normal activity to the best of their abilities. Often all that is required for sciatica resolution is time; in about 90% of people symptoms resolve in less than six weeks. If the pain is severe and lasts for more than six weeks, surgery may be an option. While surgery often speeds pain improvement, its long term benefits are unclear. Surgery may be required if complications occur, such as loss of normal bowel or bladder function. Many treatments, including corticosteroids, gabapentin, pregabalin, acupuncture, heat or ice, and spinal manipulation, have limited or poor evidence for their use.

Depending on how it is defined, less than 1% to 40% of people have sciatica at some point in time. Sciatica is most common between the ages of 40 and 59, and men are more frequently affected than women. The condition has been known since ancient times. The first known use of the word sciatica dates from 1451.

Definition
The term "sciatica" usually describes a symptom—pain along the sciatic nerve pathway—rather than a specific condition, illness, or disease. Some use it to mean any pain starting in the lower back and going down the leg. The pain is characteristically described as shooting or shock-like, quickly traveling along the course of the affected nerves. Others use the term as a diagnosis (i.e. an indication of cause and effect) for nerve dysfunction caused by compression of one or more lumbar or sacral nerve roots from a spinal disc herniation. Pain typically occurs in the distribution of a dermatome and goes below the knee to the foot. It may be associated with neurological dysfunction, such as weakness and numbness.

Risk factors
Modifiable risk factors for sciatica include smoking, obesity, occupation, and physical sports where back muscles and heavy weights are involved. Non-modifiable risk factors include increasing age, being male, and having a personal history of low back pain.

Spinal disc herniation
Spinal disc herniation pressing on one of the lumbar or sacral nerve roots is the most frequent cause of sciatica, being present in about 90% of cases. This is particularly true in those under age 50. Disc herniation most often occurs during heavy lifting. Pain typically increases when bending forward or sitting, and reduces when lying down or walking.

Spinal stenosis
Other compressive spinal causes include lumbar spinal stenosis, a condition in which the spinal canal, the space the spinal cord runs through, narrows and compresses the spinal cord, cauda equina, or sciatic nerve roots. This narrowing can be caused by bone spurs, spondylolisthesis, inflammation, or a herniated disc, which decreases available space for the spinal cord, thus pinching and irritating nerves from the spinal cord that become the sciatic nerve. This is the most frequent cause after age 50. Sciatic pain due to spinal stenosis is most commonly brought on by standing, walking, or sitting for extended periods of time, and reduces when bending forward. However, pain can arise with any position or activity in severe cases. The pain is most commonly relieved by rest.

Piriformis syndrome
Piriformis syndrome is a condition that, depending on the analysis, varies from a "very rare" cause to contributing up to 8% of low back or buttock pain. In 17% of people, the sciatic nerve runs through the piriformis muscle rather than beneath it. When the piriformis shortens or spasms due to trauma or overuse, it is posited that this causes compression of the sciatic nerve. Piriformis syndrome has colloquially been referred to as "wallet sciatica" since a wallet carried in a rear hip pocket compresses the buttock muscles and sciatic nerve when the bearer sits down. Piriformis syndrome may be suspected as a cause of sciatica when the spinal nerve roots contributing to the sciatic nerve are normal and no herniation of a spinal disc is apparent.

Deep gluteal syndrome
Deep gluteal syndrome is non-discogenic, extrapelvic sciatic nerve entrapment in the deep gluteal space. Piriformis syndrome was once the traditional model of sciatic nerve entrapment in this anatomic region. The understanding of non-discogenic sciatic nerve entrapment has changed significantly with improved knowledge of posterior hip anatomy, nerve kinematics, and advances in endoscopic techniques to explore the sciatic nerve. There are now many known causes of sciatic nerve entrapment, such as fibrous bands restricting nerve mobility, that are unrelated to the piriformis in the deep gluteal space. Deep gluteal syndrome was created as an improved classification for the many distinct causes of sciatic nerve entrapment in this anatomic region. Piriformis syndrome is now considered one of many causes of deep gluteal syndrome.

Endometriosis
Sciatic endometriosis, also called catamenial or cyclical sciatica, is a sciatica whose cause is endometriosis. Its incidence is unknown. Diagnosis is usually made by an MRI or CT-myelography.

Pregnancy
Sciatica may also occur during pregnancy, especially during later stages, as a result of the weight of the fetus pressing on the sciatic nerve during sitting or during leg spasms. While most cases do not directly harm the woman or the fetus, indirect harm may come from the numbing effect on the legs, which can cause loss of balance and falls. There is no standard treatment for pregnancy-induced sciatica.

Other
Pain that does not improve when lying down suggests a nonmechanical cause, such as cancer, inflammation, or infection. Sciatica can be caused by tumors impinging on the spinal cord or the nerve roots. Severe back pain extending to the hips and feet, loss of bladder or bowel control, or muscle weakness may result from spinal tumors or cauda equina syndrome. Trauma to the spine, such as from a car accident or hard fall onto the heel or buttocks, may also lead to sciatica. A relationship has been proposed with a latent Cutibacterium acnes infection in the intervertebral discs, but the role it plays is not yet clear.

Pathophysiology
The sciatic nerve comprises nerve roots L4, L5, S1, S2, and S3 in the spine. These nerve roots merge in the pelvic cavity to form the sacral plexus and the sciatic nerve branches from that. Sciatica symptoms can occur when there is pathology anywhere along the course of these nerves.

Intraspinal sciatica
Intraspinal, or discogenic sciatica refers to sciatica whose pathology involves the spine. In 90% of sciatica cases, this can occur as a result of a spinal disc bulge or herniation. Sciatica is generally caused by the compression of lumbar nerves L4 or L5 or sacral nerve S1. Less commonly, sacral nerves S2 or S3 may cause sciatica.

Intervertebral spinal discs consist of an outer anulus fibrosus and an inner nucleus pulposus. The anulus fibrosus forms a rigid ring around the nucleus pulposus early in human development, and the gelatinous contents of the nucleus pulposus are thus contained within the disc. Discs separate the spinal vertebrae, thereby increasing spinal stability and allowing nerve roots to properly exit through the spaces between the vertebrae from the spinal cord. As an individual ages, the anulus fibrosus weakens and becomes less rigid, making it at greater risk for tear. When there is a tear in the anulus fibrosus, the nucleus pulposus may extrude through the tear and press against spinal nerves within the spinal cord, cauda equina, or exiting nerve roots, causing inflammation, numbness, or excruciating pain. Inflammation of spinal tissue can then spread to adjacent facet joints and cause facet syndrome, which is characterized by lower back pain and referred pain in the posterior thigh.

Other causes of sciatica secondary to spinal nerve entrapment include the roughening, enlarging, or misalignment (spondylolisthesis) of vertebrae, or disc degeneration that reduces the diameter of the lateral foramen through which nerve roots exit the spine. When sciatica is caused by compression of a dorsal nerve root, it is considered a lumbar radiculopathy or radiculitis when accompanied by an inflammatory response.

Extraspinal sciatica
The sciatic nerve is highly mobile during hip and leg movements. Any pathology which restricts normal movement of the sciatic nerve can put abnormal pressure, strain, or tension on the nerve in certain positions or during normal movements. For example, the presence of scar tissue around a nerve can cause traction neuropathy.

A well known muscular cause of extraspinal sciatica is piriformis syndrome. The piriformis muscle is directly adjacent to the course of the sciatic nerve as it traverses through the intrapelvic space. Pathologies of the piriformis muscle such as injury (e.g. swelling and scarring), inflammation (release of cytokines affecting the local cellular environment), or space occupying lesions (e.g. tumor, cyst, hypertrophy) can affect the sciatic nerve. Anatomic variations in nerve branching can also predispose the sciatic nerve to further compression by the piriformis muscle, such as if the sciatic nerve pierces the piriformis muscle.

The sciatic nerve can also be entrapped outside of the pelvic space and this is called deep gluteal syndrome. Surgical research has identified new causes of entrapment such as fibrovascular scar bands, vascular abnormalities, heterotropic ossification, gluteal muscles, hamstring muscles, and the gemelli-obturator internus complex. In almost half of the endoscopic surgery cases, fibrovascular scar bands were found to be the cause of entrapment, impeding the movement of the sciatic nerve.

Diagnosis
Sciatica is typically diagnosed by physical examination, and the history of the symptoms.

Physical tests
Generally, if a person reports the typical radiating pain in one leg, as well as one or more neurological indications of nerve root tension or neurological deficit, sciatica can be diagnosed.

The most frequently used diagnostic test is the straight leg raise to produce Lasègue's sign, which is considered positive if pain in the distribution of the sciatic nerve is reproduced with passive flexion of the straight leg between 30 and 70 degrees. While this test is positive in about 90% of people with sciatica, approximately 75% of people with a positive test do not have sciatica. Straight leg raising of the leg unaffected by sciatica may produce sciatica in the leg on the affected side; this is known as the Fajersztajn sign. The presence of the Fajersztajn sign is a more specific finding for a herniated disc than Lasègue's sign. Maneuvers that increase intraspinal pressure, such as coughing, flexion of the neck, and bilateral compression of the jugular veins, may transiently worsen sciatica pain.

Medical imaging
Imaging modalities such as computerised tomography or magnetic resonance imaging can help with the diagnosis of lumbar disc herniation. The utility of MR neurography in the diagnosis of piriformis syndrome is controversial.

Discography could be considered to determine a specific disc's role in an individual's pain. Discography involves the insertion of a needle into a disc to determine the pressure of disc space. Radiocontrast is then injected into the disc space to assess for visual changes that may indicate an anatomic abnormality of the disc. The reproduction of an individual's pain during discography is also diagnostic.

Differential diagnosis
Cancer should be suspected if there is previous history of it, unexplained weight loss, or unremitting pain. Spinal epidural abscess is more common among those who have diabetes mellitus or immunodeficiency, or who have had spinal surgery, injection or catheter; it typically causes fever, leukocytosis and increased erythrocyte sedimentation rate. If cancer or spinal epidural abscess is suspected, urgent magnetic resonance imaging is recommended for confirmation. Proximal diabetic neuropathy typically affects middle aged and older people with well-controlled type-2 diabetes mellitus; onset is sudden, causing pain, usually in multiple dermatomes, quickly followed by weakness. Diagnosis typically involves electromyography and lumbar puncture. Shingles is more common among the elderly and immunocompromised; typically, pain is followed by the appearance of a rash with small blisters along a single dermatome. Acute Lyme radiculopathy may follow a history of outdoor activities during warmer months in likely tick habitats in the previous 1–12 weeks. In the U.S., Lyme is most common in New England and Mid-Atlantic states and parts of Wisconsin and Minnesota, but it is expanding to other areas. The first manifestation is usually an expanding rash possibly accompanied by flu-like symptoms. Lyme can also cause a milder, chronic radiculopathy an average of 8 months after the acute illness.

Management
Sciatica can be managed with a number of different treatments with the goal of restoring a person's normal functional status and quality of life. When the cause of sciatica is lumbar disc herniation (90% of cases), most cases resolve spontaneously over weeks to months. Initially treatment in the first 6–8 weeks should be conservative. More than 75% of sciatica cases are managed without surgery. In persons that smoke who also have sciatica, smoking cessation should be strongly considered, in order to promote healing. Treatment of the underlying cause of nerve compression is needed in cases of epidural abscess, epidural tumors, and cauda equina syndrome.

Physical activity
Physical activity is often recommended for the conservative management of sciatica for persons who are physically able. However, the difference in outcomes between physical activity compared to bed rest have not been fully elucidated. The evidence for physical therapy in sciatica is unclear though such programs appear safe. Physical therapy is commonly used. Nerve mobilization techniques for sciatic nerve is supported by tentative evidence.

Medication
There is no one medication regimen used to treat sciatica. Evidence supporting the use of opioids and muscle relaxants is poor. Low-quality evidence indicates that NSAIDs do not appear to improve immediate pain, and all NSAIDs appear to be nearly equivalent in their ability to relieve sciatica. Nevertheless, NSAIDs are commonly recommended as a first-line treatment for sciatica. In those with sciatica due to piriformis syndrome, botulinum toxin injections may improve pain and function. While there is little evidence supporting the use of epidural or systemic steroids, systemic steroids may be offered to individuals with confirmed disc herniation if there is a contraindication to NSAID use. Low-quality evidence supports the use of gabapentin for acute pain relief in those with chronic sciatica. Anticonvulsants and biologics have not been shown to improve acute or chronic sciatica. Antidepressants have demonstrated some efficacy in treating chronic sciatica, and may be offered to individuals who are not amenable to NSAIDs or who have failed NSAID therapy.

Surgery
If sciatica is caused by a herniated disc, the disc's partial or complete removal, known as a discectomy, has tentative evidence of benefit in the short term. If the cause is spondylolisthesis or spinal stenosis, surgery appears to provide pain relief for up to two years.

For non-discogenic sciatica, the surgical treatment is typically a nerve decompression. A decompression seeks to remove tissue around the nerve that may be compressing it or restricting movement of the nerve.

Alternative medicine
Low to moderate-quality evidence suggests that spinal manipulation is an effective treatment for acute sciatica. For chronic sciatica, the evidence supporting spinal manipulation as treatment is poor. Spinal manipulation has been found generally safe for the treatment of disc-related pain; however, case reports have found an association with cauda equina syndrome, and it is contraindicated when there are progressive neurological deficits.

Prognosis
About 39% to 50% of people with sciatica still have symptoms after one to four years. In one study, around 20% were unable to work at their one-year followup, and 10% had surgery for the condition.

Epidemiology
Depending on how it is defined, less than 1% to 40% of people have sciatica at some point in time. Sciatica is most common between the ages of 40 and 59, and men are more frequently affected than women.

title: Sciatica


text:
Constitution of France

The current Constitution of France was adopted on 4 October 1958. It is typically called the Constitution of the Fifth Republic (French: la Constitution de la Cinquième République), and it replaced the Constitution of the Fourth Republic of 1946 with the exception of the preamble per a 1971 decision of the Constitutional Council. The current Constitution regards the separation of church and state, democracy, social welfare, and indivisibility as core principles of the French state.

Charles de Gaulle was the main driving force in introducing the new constitution and inaugurating the Fifth Republic, while the text was drafted by Michel Debré. Since then, the constitution has been amended twenty-five times, through 2024.

Preamble
The preamble of the constitution recalls the Declaration of the Rights of Man and of the Citizen from 1789 and establishes France as a secular and democratic country, deriving its sovereignty from the people.

Since 2005 it includes the ten articles of the Charter for the Environment.

Government institutions and practices
The French Constitution established a semi-presidential system of government, with two competing readings. On one hand, the executive branch has both a president of the republic and a prime minister, which is commonly seen in parliamentary systems with a symbolic president and a prime minister who directs the government. This reading is supported by Articles 5 and 21 of the Constitution, which respectively states that the president is a Guardian of the State and of the Constitution, while the prime minister has the power to decide on Government's actions and policies.

On the other hand, the Parliament is very weak for a parliamentary system. Parliament has a limited legislative competence: article 34 of the Constitution lists domains exclusive to parliamentary legislation, but the remaining domains are left to the executive's regulations. The president also has the crucial powers to call a referendum and to dissolve the National Assembly. While Parliament may make a vote of no confidence on the government, since 1962 a majority in the National Assembly has supported the Government.

Charles de Gaulle, the first president of the Fifth Republic, was instrumental in the adoption of the new constitution, as he was called back from retirement and narrowly avoided a coup resulting from the Algerian War. De Gaulle always supported the second interpretation of the constitution, in favor of a powerful president. The first socialist president, François Mitterrand, elected in 1981, also supported this interpretation.

Beginning in 1986, elections have from time to time resulted in Parliaments with a majority that did not support the president. Such periods as known in France as cohabitation, where a president appoints a prime minister from the new parliamentary majority. During cohabitation, besides powers reserved to the president by the Constitution, all other government powers would be exercised by the prime minister. In 2000, the Constitution was amended by shortening the president's term of office from seven years to five, coinciding with the term of Parliament. The amendment means the presidential election would take place around the parliamentary election, making it more likely to have winners who agree with one another and make cohabitation less likely.

The Constitution provides for the election of the president and the Parliament, the selection of the Government, the powers of each and the relations between them. It ensures judicial authority and creates a High Court (a never-as-yet-convened court for trying the Government), a Constitutional Council (an innovation of the Fifth Republic), and an Economic and Social Council.

Shared lawmaking power
A unique feature of the Constitution of the Fifth Republic is that it establishes a shared law-making power between two branches of government, the legislative branch, where such powers resided in previous constitutions, and the executive branch headed by the president and his appointed prime minister.

Parliament has the fundamental responsibility for passing legislation in the Fifth Republic. There are two Houses of Parliament, the National Assembly and the Senate. The Assembly is directly elected, and the more important, and has primary power in passing legislation; the Senate can delay, but ultimately not block it.

Traditionally, the prime minister is the executive branch's liaison with Parliament; Article 49 says they must pledge this role. This is weaker than the constitutions of the Third or Fourth republics, where the government could not be installed until Parliament had received the pledge from the prime minister.

The unique aspect in the Fifth Republic is in Article 21 of the French Constitution, where the prime minister has the power to make legislation. In another unique feature in Article 38 of the French Constitution, the Parliament can Legislative delegation in France a portion of its constitutional law-making power to the government to enable rapid consummation of urgent legislation, by passing an enabling law of legislative delegation called a loi d'habilitation ("enabling law").

According to Article 21, a loi d'habilitation may be issued by Parliament upon request of the government to temporarily delegate Parliament's constitutional law-making power to the government in a specifically defined subject area and for a specific length of time. While in effect, Parliament is blocked from issuing statutes in that area, and the government is permitted to draw up ordonnances that normally would be beyond their remit. The ordonnance comes into effect immediately, but must be ratified by Parliament before the end of the period or it expires. Until ratification, the ordonnance has the same status as a réglement (regulation), and can therefor be challenged by the Council of State; but after ratification, it takes on the same status as a statute (loi), and can no longer be challenged. In practice, there have been 23 such lois d'habilitation from 1960 to 1990, with effective periods from one month to three and a half years, resulting in 150 ordonnances. About a third of them were subsequently ratified by Parliament. The loi d'habilitation is a new constitutional feature, not present in earlier constitutions.

Power sharing was unique in being part of the constitution in the Fifth Republic, but the practice was not recent. The determination that Parliament has responsibility for the law goes back to article 6 of the Declaration of the Rights of Man of 1789, and the role of the executive branch was only to execute it. In theory, Parliament would specify general laws, and the executive could only make regulations about how to apply the laws to day-to-day situations. In practice, this turned out differently, as Parliament on its own initiative sometimes passed acts delegating to the executive the right to alter or void acts of Parliament, called décrets-lois. This practice slowly found its way into the Constitution.

Although Article 1 of the Third Republic's explicitly forbade the Parliament to delegate its responsibility, within five years this was ignored and had occurred several times. In 1939, in the runup to the Second World War, Parliament gave the government the power to enact decrees to protect the country. This practice became entrenched after the war, despite the fact that Article 13 of the 1946 Constitution of the newly founded Fourth Republic expressly forbade it. Part of the reason for this, was a lot of squabbling among numerous small political parties in Parliament, who were unable to agree on anything and were ineffective in passing legislation. This became especially problematic in the 1950s, as the crisis in Algeria began to heat up, and Parliament was unable to deal with it. Charles de Gaulle, a private citizen at the time, realized that the way out was to have a more powerful executive and a weaker Parliament; when he was finally invited to form a new government in 1958 and write a constitution, his ideas were incorporated into the Constitution of the Fifth Republic, including the legislative power sharing defined in Articles 21 and 38.

In the original version of Article 37, everything that was not reserved to the legislative domain in article was of a regulatory character (i.e., under control of the executive branch), although that clause was removed later. Since 1982, the legislative domain expanded, and since the landmark 1971 decision of the Constitutional Council, additional sources were defined as part of the constitutional block, such as the 1789 Declaration of the Rights of Man and the 1946 constitutional preamble, which were henceforth part of the legislative domain. and after further reforms in 1996, the legislative domain has more power than was originally thought in 1958.

Treaties and the EU
It enables the ratification of international treaties and those associated with the European Union. It is unclear whether the wording, especially the reserves of reciprocity, is compatible with European Union law.

Amendment
The Constitution also sets out methods for its own amendment: a referendum (article 11) or a parliamentary process with presidential consent. The normal procedure of constitutional amendment is that the amendment must be adopted in identical terms by both houses of Parliament and then must be adopted by a simple majority in a referendum or by a three-fifths supermajority of the French Congress, a joint session of both houses of Parliament (article 89).

Judicial review
Prior to 1971, though executive, administrative and judicial decisions had to comply with the general principles of law (jurisprudence derived from law and the practice of law in general), there were no such restrictions on legislation. It was assumed that unelected judges and other appointees should not be able to overrule laws voted for by the directly elected French parliament.

Constitutional block
One of the cornerstones of the Constitution of the French Fifth Republic was the establishment of the Constitutional Council, composing of nine justices, who oversaw the constitutionality of legislation (treaties, statutes, regulations), ensured election and referendum oversight, and arbitrated legislative disputes between the President and National Assembly. This followed a broader trend during post-war Europe to establish specialized judiciary tribunals to serve as a bulwark against unconstitutional legislative activities. However, the Council was quite limited in its power under de Gaulle’s presidency and was only decisive when it erroneously upheld a popular referendum to streamline the popular presidential election via the Constitution. Following de Gaulle’s resignation in 1969, the Council entertained greater judicial power and discretion upon adjudicating in the consequential political crisis. The staunch Gaullist Georges Pompidou was elected as de Gaulle’s replacement. He faced a political crisis when his Prime Minister Jacques Chaban-Delmas pressured the National Assembly into banning the radical Proletarian Left (La gauche prolétarienne) twice, which he deemed a threat to the public order and national security. This led to opposition from the French Senate, whose president appealed to the Constitutional Council.

Consequently, in 1971, the Constitutional Council ruled its landmark decision no. 71-44 DC, better known as the 1971 Freedom of Association Decision. In such, the Council broke precedent by striking down legislation that allegedly violated the right to freedom of association, thereby fostering the “Constitutional Block.” The Block consisted of the 1958 Constitution, explicit standards (Declaration of the Rights of Man and of the Citizen of 1789, the Preamble to the Constitution of the Fourth Republic or 1946 Constitution), and implicit standards (the fundamental principles of the Republic—indivisibility, secularism, democracy, equal opportunity). Thus, according to the Council, the actions taken violated the collective principles of the Constitutional Block. Prior to the 1971 Freedom of Association Decision, the Council could only verify laws under the explicit textual stipulation of the 1958 Constitution. Since the 1971 decision, the Constitutional Court obtained an enhanced role in judicial review by having a broader constitutional basis to review alleged legislative breaches, curbing the goal of Gaullists from 1958 of maintaining a strong executive. Since the ruling, the Constitutional Council has added the 2004 Charter of the Environment to France’s Constitutional Block, demonstrating France’s newfound tenacity in judicial review.

Principles of the Republic
In the Constitution are written the principles of the French Republic:
 * Social welfare, which means that everybody must be able to access free public services and be helped when needed.
 * Laïcité, which means that the churches are separated from the State and the freedom from religion is protected.
 * Democracy, which means that the Parliament and the Government are elected by the people.
 * Indivisibility, which means that the French people are united in a single sovereign country with one language, the French language, and all people are equal.

Amendments
The Constitution, in Article 89, has an amending formula. First, a constitutional bill must be approved by both houses of Parliament. Then, the bill must either be approved by the Congress, a special joint session of both houses, or submitted to a referendum.

In 1962, Charles de Gaulle proposed that the president be elected by direct suffrage. He bypassed the amendment procedure by directly sending a constitutional amendment to referendum (article 11). The Art. 11 procedure was envisioned as a procedure for proposing legislation, including changing the organization of constitutional institutions. The 1962 referendum was approved by 62% of the vote but only 46% of registered voters. The amendment permitted the establishment of a popularly-elected presidency, which would otherwise have been vetoed by the Parliament.

The referendum was highly controversial at the time, but the Constitutional Council ruled that it can only review legislative acts for unconstitutionality, not executive acts; since the referendum was proposed by the executive, it was unreviewable. Since a referendum expressed the will of the sovereign people, the Council ruled that the amendment had been adopted. Some scholars had regarded the amendment as a post hoc manifestation of the constituent power, which is the inherent power of the people to bypass an existing constitution to adopt a new constitution.

Article 11 was used for constitutional changes for the second and final time in 1969, but the "No" prevailed, causing Charles de Gaulle to resign from the presidency.

On 21 July 2008, Parliament passed constitutional reforms championed by President Nicolas Sarkozy by a margin of two votes. The changes, when finalized, introduced a consecutive two-term limit for the presidency, gave Parliament a veto over some presidential appointments, ended government control over Parliament's committee system, allowed Parliament to set its own agenda, allowed the president to address Parliament in-session and ended the president's right of collective pardon. (See French constitutional law of 23 July 2008).

On 4 March 2024, Parliament amended Article 34 in a 780 to 72 vote. This amendment made France, as of passage, the only nation to guarantee the right to an abortion. The amendment describes abortion as a "guaranteed freedom"; while Yugoslavia included similar measures in 1974 guaranteeing the right to "decide on having children", the French amendment is the first to explicitly guarantee abortion.

Notes and references

 * Notes

 * Citations

 * Works cited

title: Constitution of France


text:
List of national parks of Argentina

The National Parks of Argentina make up a network of 35 national parks in Argentina. The parks cover a very varied set of terrains and biotopes, from Baritú National Park on the northern border with Bolivia to Tierra del Fuego National Park in the far south of the continent. The Administración de Parques Nacionales (National Parks Administration) is the agency that preserves and manages these national parks along with Natural monuments and National Reserves within the country.

The headquarters of the National Parks Service are in downtown Buenos Aires, on Santa Fe Avenue. A library and information centre are open to the public. The administration also covers the national monuments, such as the Jaramillo Petrified Forest, and natural and educational reserves.

History
The creation of the National Parks dates back to the 1903 donation of 75 km2 of land in the Lake District in the Andes foothills by Francisco Moreno, the renowned explorer and academic in Argentina. This formed the nucleus of a larger protected area in Patagonia around San Carlos de Bariloche. At the beginning of the 1920s there was an amazing revival of the national park idea at Lake Nahuel Huapí, which led to the official establishment of the Parque Nacional del Sud on 8 April 1922 by decree of President Yrigoyen. In 1934, a law was passed creating the National Parks system, formalising the protected area as the Nahuel Huapi National Park and creating the Iguazú National Park. Thus, Argentina was the third country in the Americas, after United States and Canada to establish a national parks system. The National Park Police Force was born, enforcing the new laws preventing tree-felling and hunting. Under the direction of the founding director of the National Direction of National Parks Exequiel Bustillo, a conception of national parks focused on economic development and colonization of the region was developed. Beyond touristic development the aim of the parks was to establish national sovereignty over peripherical areas and to protect national borders. Five further national parks were declared in 1937 in Patagonia and the service planned new towns and facilities to promote tourism and education. Six more were declared by 1970.

In 1970 a new law established new categories of protection: National Parks, National Monuments, Educational Reserves, and Natural Reserves. Three national parks were designated in the 1970s. In 1980, another new law affirmed the status of national parks – this law is still in place. The 1980s saw the service reaching out to local communities and local government to help in the running and development of the national parks. Ten more national parks were created with local co-operation, sometimes at local instigation. In 2000, Mburucuyá and Copo National Parks were declared, and El Leoncito natural reserve was upgraded to a national park. Currently, there are 41 protected areas in Argentina, which cover an area of 37000 km2 or about 1.5% of the total land area in Argentina.

National Parks

 * {{Legend|#cfecd2|‡ World Heritage Site|outline=silver}}

title: List of national parks of Argentina


text:
Konoe family

Konoe (近衛) is a Japanese aristocratic family. The family is a branch of Hokke and, by extension, a main branch of the Fujiwara clan.

History
The Konoe claim descent from Konoe Iezane (1179–1242). The origin of the family name was the residence of Iezane's grandfather Konoe Motozane, which was located on a road in Kyoto named "Konoe-Ōji" (近衛大道). Despite Konoe at first being the senior line of the Fujiwara clan, the clan was eventually split up into Five regent houses during the Kamakura period, with each of the five families having the right to assume the regency. During the following Nanboku-chō period, a succession dispute of Konoe emerged, between Tsunetada and his cousin Mototsugu – they served in rival courts, the Southern and the Northern Court respectively; later, when the Southern court lost its political influence in 1392, records of Tsunetada's descendants became lost and have stayed lost ever since.

There were at least five Imperial Consorts who came from the Konoe family,  including Konoe Sakiko, who was adopted by Toyotomi Hideyoshi in 1586. The most recent consort from the Konoe was Konoe Koreko (近衛維子, 1760–1783), who married the 118th Emperor of Japan, Emperor Go-Momozono; they had one daughter Princess Yoshiko.

In the early 17th century the Konoe family was imperiled, for Konoe Nobutada was the last of his line and had no male heir. He decided to adopt one of his nephews, who was also the fourth son of Emperor Go-Yozei. The child was renamed Konoe Nobuhiro (1599–1649), who later married Nobutada's daughter. From there the Konoe lineage was renewed and continued until 1956, when the eldest son of Prime Minister Fumimaro Konoe, Fumitaka, died in the Soviet Union without a legitimate male heir. Fumitaka's wife subsequently adopted his nephew Tadateru Konoe, second son of Fumitaka's sister, as their heir. Tadateru's patrilineal descent, through his biological father Morisada Hosokawa, comes from the Hosokawa clan, a cadet branch of Seiwa Genji and descended from Emperor Seiwa; he also has an elder brother Morihiro Hosokawa, the Prime Minister of Japan from 1993 to 1994. Tadateru married, in 1966, a granddaughter of Emperor Taishō, Yasuko (formerly Princess Yasuko of Mikasa).

Tokiwai family
The Tokiwai family (常磐井家) was founded by a son of Konoe Tadahiro, Gyōki (尭熈), who was the lead Buddhist monk of Senju-ji, and he took the family name "Tokiwai" since 1872.

title: Konoe family