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| In [[probability theory]], '''coupling''' is a [[mathematical proof|proof]] technique that allows one to compare two unrelated variables by "forcing" them to be related in some way.
| | Oral herpes is an infection due to the herpes simplex virus, characterized by an eruption of small and normally painful blisters on the skin of the lips, oral cavity, gums or the skin around the mouth. Newborns contaminated during or soon after birth may create the characteristic blisters, redness, discomfort, or swelling of the eyes, or general severe sickness symptoms within several times to six months. <br><br>There are two other styles of supplement B-3: niacinamide, which is possibly helpful in relieving outward indications of osteoarthritis and delaying the progression of type 1 diabetes; and inositol hexanicotinate, which is always under study instead of niacin and niacinamide. <br><br>Once the virus gets to be reactivated, it travels across the nerves to your skin, where it multiplies at first glance at or near the site of the initial herpes sores, causing new sores to erupt. However, people with genital herpes can shed the virus from the genital location and infect others actually without a blister being found. Cold sores on the mouth are a potential source of genital infections during mouth-to-genital contact (oral intercourse).<br><br>Lists the feeling of strain below the abdomen being an additional female herpes symptom, and estimates that ten to 15% of women of all ages will experience difficulties urinating throughout a herpes outbreak. Because these signs mimic those of different common female conditions, various mistake herpes symptoms for urinary system or yeast infections. That is why if vaginal discharge or pain is persistent, it might be wise to consider ruling out herpes because the source. The virus remains in the body for life and will produce symptoms that could reappear intermittently.<br><br>Some other outward indications of the initial outbreak range from muscle aches, headache, fever, painful urination, swollen glands in the groin area, and vaginal discharge. Lesions surrounding the mouth and on lips will be one of the most recognizable symptoms of herpes labialis.<br><br>If you have any kind of questions regarding where and ways to use [http://www.garagebesson.net/?option=com_k2&view=itemlist&task=user&id=8847 herpes symptoms mouth ulcers], you could contact us at the web site. |
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| ==Definition==
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| Using the [[probability theory#A_somewhat_more_abstract_view_of_probability|standard formalism]] of probability, let <math>X_1</math> and <math>X_2</math> be two random variables defined on probability spaces <math>(\Omega_1,F_1,P_1)</math> and <math>(\Omega_2,F_2,P_2)</math>. Then a coupling of <math>X_1</math> and <math>X_2</math> is a ''new'' probability space <math>(\Omega,F,P)</math> over which there are two random variables <math>Y_1</math> and <math>Y_2</math> such that <math>Y_1</math> has the same distribution as <math>X_1</math> while <math>Y_2</math> has the same distribution as <math>X_2</math>.
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| The interesting case is when <math>Y_1</math> and <math>Y_2</math> are ''not'' independent.
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| ==Examples==
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| ===Random walk===
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| Assume two particles ''A'' and ''B'' perform a simple [[random walk]] in two dimensions, but they start from different points. The simplest way to couple them is simply to force them to walk together. On every step, if ''A'' walks up, so does ''B'', if ''A'' moves to the left, so does ''B'', etc. Thus, the difference between the two particles stays fixed. As far as ''A'' is concerned, it is doing a perfect random walk, while ''B'' is the copycat. ''B'' holds the opposite view, i.e. that he is, in effect, the original and that ''A'' is the copy. And in a sense they both are right. In other words, any mathematical theorem, or result that holds for a regular random walk, will also hold for both ''A'' and ''B''.
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| Consider now a more elaborate example. Assume that ''A'' starts from the point (0,0) and ''B'' from (10,10). First couple them so that they walk together in the vertical direction, i.e. if ''A'' goes up, so does ''B'', etc., but are mirror images in the horizontal direction i.e. if ''A'' goes left, ''B'' goes right and vice versa. We continue this coupling until ''A'' and ''B'' have the same horizontal coordinate, or in other words are on the vertical line (5,''y''). If they never meet, we continue this process forever (the probability for that is zero, though). After this event, we change the coupling rule. We let them walk together in the horizontal direction, but in a mirror image rule in the vertical direction. We continue this rule until they meet in the vertical direction too (if they do), and from that point on, we just let them walk together.
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| This is a coupling in the sense that neither particle, taken on its own, can feel anything we did. Nor that fact that the other particle follows him in one way or the other, nor the fact that we changed the coupling rule or when we did it. Each particle performs a simple random walk. And yet, our coupling rule forces them to meet [[almost surely]] and to continue from that point on together permanently. This allows one to prove many interesting results that say that "in the long run", it is not important where you started.
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| ===Biased coins===
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| Assume two biased coins, the first with probability ''p'' of turning up heads and the second with probability ''q'' > ''p'' of turning up heads. Intuitively, if both coins are tossed the same number of times, the first coin should turn up fewer heads than the second one. More specifically, for any fixed ''k'', the probability that the first coin produces at least ''k'' heads should be less than the probability that the second coin produces at least ''k'' heads. However proving such a fact can be difficult with a standard counting argument.<ref>{{cite book |last1=Dubhashi |first1=Devdatt |last2=Panconesi |first2=Alessandro |title=Concentration of Measure for the Analysis of Randomized Algorithms |publisher=Cambridge University Press |date=June 15, 2009|edition=1st |isbn=978-0-521-88427-3| page=91}}</ref> Coupling easily circumvents this problem.
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| Let ''X<sub>1</sub>'', ''X<sub>2</sub>'', ..., ''X<sub>n</sub>'' be indicator variables for heads in a sequence of flips of the first coin. For the second coin, define a new sequence ''Y<sub>1</sub>'', ''Y<sub>2</sub>'', ..., ''Y<sub>n</sub>'' such that
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| * if ''X<sub>i</sub>'' = 1, then ''Y<sub>i</sub>'' = 1,
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| * if ''X<sub>i</sub>'' = 0, then ''Y<sub>i</sub>'' = 1 with probability (''q''-''p'')/(1-''p'').
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| Then the sequence of ''Y<sub>i</sub>'' has exactly the probability distribution of tosses made with the second coin. However, because ''Y<sub>i</sub>'' depends on ''X<sub>i</sub>'', a toss by toss comparison of the two coins is now possible. That is, for any ''k'' ≤ ''n''
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| :<math> \Pr(X_1 + \cdots + X_n > k) \leq \Pr(Y_1 + \cdots + Y_n > k).</math>
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| ==See also== | |
| [[Copula (probability theory)]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * T. Lindvall, ''Lectures on the coupling method''. Wiley, New York, 1992.
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| * H. Thorisson, ''Coupling, Stationarity, and Regeneration''. Springer, New York, 2000.
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| [[Category:Probability theory]]
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Oral herpes is an infection due to the herpes simplex virus, characterized by an eruption of small and normally painful blisters on the skin of the lips, oral cavity, gums or the skin around the mouth. Newborns contaminated during or soon after birth may create the characteristic blisters, redness, discomfort, or swelling of the eyes, or general severe sickness symptoms within several times to six months.
There are two other styles of supplement B-3: niacinamide, which is possibly helpful in relieving outward indications of osteoarthritis and delaying the progression of type 1 diabetes; and inositol hexanicotinate, which is always under study instead of niacin and niacinamide.
Once the virus gets to be reactivated, it travels across the nerves to your skin, where it multiplies at first glance at or near the site of the initial herpes sores, causing new sores to erupt. However, people with genital herpes can shed the virus from the genital location and infect others actually without a blister being found. Cold sores on the mouth are a potential source of genital infections during mouth-to-genital contact (oral intercourse).
Lists the feeling of strain below the abdomen being an additional female herpes symptom, and estimates that ten to 15% of women of all ages will experience difficulties urinating throughout a herpes outbreak. Because these signs mimic those of different common female conditions, various mistake herpes symptoms for urinary system or yeast infections. That is why if vaginal discharge or pain is persistent, it might be wise to consider ruling out herpes because the source. The virus remains in the body for life and will produce symptoms that could reappear intermittently.
Some other outward indications of the initial outbreak range from muscle aches, headache, fever, painful urination, swollen glands in the groin area, and vaginal discharge. Lesions surrounding the mouth and on lips will be one of the most recognizable symptoms of herpes labialis.
If you have any kind of questions regarding where and ways to use herpes symptoms mouth ulcers, you could contact us at the web site.