Wikipedia:Reference desk/Archives/Mathematics/2006 October 31

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October 31

Exponents of regular singular points

When you say that a differential equation has regular singular points, and those regular singular points have exponents, e.g. with the hypergeometric differential equation, what does the exponent represent? What does it mean? Thanks. enochlau (talk) 03:43, 31 October 2006 (UTC)

The exponent is the r in regular singular point and Frobenius method. If a is an ordinary point (i.e., not a singular point), then the solution can be expanded in a power series:

$c_{0}+c_{1}(x-a)+c_{2}(x-a)^{2}+\cdots ,$

At a regular singular point, this series need to be modified to

$c_{0}(x-a)^{r}+c_{1}(x-a)^{r+1}+c_{2}(x-a)^{r+2}+\cdots .\,$

The number r (which is not necessarily an integer) is called the exponent. For instance, the Bessel function of the first kind has exponent α. -- Jitse Niesen (talk) 02:10, 1 November 2006 (UTC)

Thanks for that. So, just to check, the two exponents at each regular singular point in the hypergeometric DE come from the fact that the indicial equation is a quadratic? And with these two exponents, we get two different expansions that are linearly independent? Thanks again! enochlau (talk) 03:51, 1 November 2006 (UTC)

There are indeed two exponents because the indicial equation is quadratic, which is because the differential equation is second order. The answer to your second question is a bit more complicated. If the difference between two exponents is not an integer, then we get two different expansions that are linearly independent. If the difference is an integer (which includes the case that the exponents are equal), then the method may break down; usually, this means that one of the solutions has a logarithmic factor. I think that this is the reason why the article hypergeometric differential equation mentions that there is a special case when one of the angular parameters (which presumably correspond to differences between exponents) is an integer. However, I don't really know this theory. It should be in most old-fashioned not-too-elementary books on ordinary differential equations; I know it is in Ince, Ordinary differential equations. -- Jitse Niesen (talk) 05:02, 1 November 2006 (UTC)

Liberal paradox

I'm having problem understanding the article. Please pay the talk page at Talk:Liberal paradox a visit for the problem. Any help is fully appreciated. I'll give you an imaginary ice cream if you help me! __earth ^(Talk) 11:10, 31 October 2006 (UTC)

fuzzy logic

Can the process of making a decision on the basis of obtaining more than fifty percent of the votes be considered a form of fuzzy logic? Adaptron 17:57, 31 October 2006 (UTC)

Having a threshold on the voting (50% or else) makes the result determined and not fuzzy at all. On the other hand, imagine having a fuzzy president consisting of both Bush (50.1%) and Gore (49.9%). (Igny 20:03, 31 October 2006 (UTC))

In terms of fuzzy how would this work? How would fuzzy presidents be able to render a determinate decision? Adaptron 20:30, 31 October 2006 (UTC)

Let me rephrase the question. In standard logic the state of a binary variable is either true or false. If the state of a variable is indeterminate then a conclusion can not be reached. Does fuzzy logic provide an answer to this dilemma by allowing indeterminate binary states to be based on statistics? In other words the probability that the state of a variable is either true or false. Adaptron 22:08, 31 October 2006 (UTC)

I would say yes, that's what fuzzy logic does. A good example might be an innocent or guilty verdict in a court, which is rarely based on absolute proof either way (since even DNA evidence can be faked, messed up, etc.). Instead, it must be based on "preponderance of the evidence". StuRat 01:25, 1 November 2006 (UTC)

In the United States legal system, a "guilty" verdict implies a criminal case, for which the standard is "beyond a reasonable doubt". In a civil case, the weaker "preponderance" criterion is used. The rationale is that a criminal sentence can entail imprisonment or death, whereas these are not at risk in a civil case. (See “burden of proof” for details.) A verdict of "innocent" does not exist in the U.S. legal system, only "not guilty". Some systems include the option of "not proven", which has the whiff of fuzzy logic.

The difference between the two burdens received international attention in the case of O. J. Simpson, who a California criminal court found "not guilty" of murder, but who a civil court later found "liable" for wrongful death. However, the facts in evidence were not the same in the two cases, for a variety of reasons, so it is not possible to explain the different verdicts as strictly a function of burden. (The U.S. legal system prohibits trying a person a second time for the same offense, called “double jeopardy”, but criminal/civil or state/federal distinctions have been used get around that, as in the cases of O.J. Simpson and the police assault on Rodney King.) --KSmrq^T 21:27, 1 November 2006 (UTC)

Most texts don't seem to present fuzzy logic in terms of a probability function but its the only thing that makes sense as far as how to deal with a variable who's state is uncertain. After all that is what we learned the function of probability was for, i.e. giving us a way to deal mathematically (if by no other means) with uncertainty. Adaptron 03:15, 1 November 2006 (UTC)

It is not possible to interpret fuzzy logic as representing probabilities, since it is not possible in fuzzy logic to distinguish between the case that variables with identical distributions are independent, and that they are dependent. It is more that a glass is not always full or empty, but can be half-way, which is not a matter of an uncertain state, but possibly of our uncertainty in dealing with an in-between state, say in rules like: "If it rains, the concert will move to an indoor location." Two drops fall; now what?  --Lambiam^Talk 06:04, 1 November 2006 (UTC)