Wikipedia:Reference desk/Archives/Mathematics/2007 December 14

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December 14

Logorithmic equations

This question was on a quiz I failed. Solve for x.

log (3x+7)+log(x-2)=1

My teacher said that some of the answers you get when you solve this with algebra don't work in the original equation.Please explain why this is because I have a test on this tomorrow. Thanks! --Banana (talk) 00:47, 14 December 2007 (UTC)

A natural (and correct, I believe) assumption is that you're working over the reals, meaning something like

$log(-5)$

doesn't make any sense to the analysis.

Then, by a property of logarithms valid here,

$log(3x+7)+log(x-2)=log(3x^{2}+x-14)$.

:It's easy to see that, if $log(\alpha )=1$, it must be the case that $\alpha =0$.

Hence, solutions to your equation are included in the set of solutions to

$3x^{2}+x-14=0$ (Corrected, see below)

This last expression has $-{\frac {7}{3}}$ and 2 as solutions. However, $-{\frac {7}{3}}$ is not a solution to your original equation.

Why is that? Well, simply because $log(-{\frac {7}{3}}-2)$ has no sense.

In informal terms, you would say the following: take the original equation, apply antilogarithms to both sides and solve the quadratic equation. The point at when you applied antilogarithms is precisely where you add a solution. Pallida  Mors 02:06, 14 December 2007 (UTC)

Would you please forgive my stupidity? If $log(\alpha )=1$, it can not be the case that $\alpha =0$, Indeed, $\alpha =e$. Hence, you have two solutions:

• x=2.1999
• x=-2.53321.

This last value for x is not a solution to your original problem, for a reason similar to the one stated above. Sorry for the mess! Pallida  Mors 02:21, 14 December 2007 (UTC)

It is extremely likely that "log" here means the decimal logarithm, as is common in most elementary contexts. You even get an integer solution this way. -- Meni Rosenfeld (talk) 08:44, 14 December 2007 (UTC)

...although the integer root is -ve, so is not a solution to the original equation. Assuming log means log base 10, I find just one solution to the original equation, rational but non-integer. Thinking graphically, as the left hand side is the sum of two monotonically increasing functions, it is clear that there must be a single solution, and a little more thought shows that it must be between 2.5 and 3. Visual thinking is always a good check on algebraic results. Gandalf61 (talk) 10:02, 14 December 2007 (UTC)

Right, I did mean an integer solution to the original problem, but I accidentally added 10 instead of subtracting. If high school algebra was taught in GF(2) we wouldn't have such problems :) -- Meni Rosenfeld (talk) 11:29, 14 December 2007 (UTC)

Oops! Mathematica has made me forget about decimal logarithms. Too Bad. What is worse, it has made me forget of such basic facts as

$\log _{b}a=1\Rightarrow a=b$ :( Pallida  Mors 14:15, 14 December 2007 (UTC)

Advice for integrating disparate similarity measures?

I'm playing around with a way to cluster journal articles. (It's not really journal articles but it makes the explanation simpler.)

I've come up with 3 different ways to measure the similarity or distance between two given articles, however, each of the resulting scores are on completely different scales.

Method 1: Average citation distance. Compute the average distance between citations of article A and B when they are both cited in the same paper. Possible scores range from 0 to the length of the longest paper in the data set with 0 meaning most similar, or no score if they never appear together.

Method 2: Content similarity. Basically, the number of words both articles have in common, weighted by how unusual those words are in all of the papers that I have, using this method. Higher scores mean higher similarity, the score does not appear to have an obvious upper bound but in practice a score of 100 would be an incredibly high score. Note that A measured against B is different from B measured against A for this score.

Method 3: The number of papers that cite both articles. Score from 0 to the maximum number of papers in the data set minus 2 (papers can't cite themselves), with larger numbers meaning more similar. The number of papers in the data set increases over time.

Obviously methods 1 and 3 are somewhat redundant, and I might ultimately decide to go with either one or the other. Method 2 however captures interesting relationships that 1 or 3 might never capture, and vice versa.

Whatever the resulting amalgamated score is, it needs to be useful for doing two things:

1. Telling whether article A is related to B enough that it's worth mentioning the relationship. There's no reason this doesn't have to be some arbitrary threshold
2. For clustering, that A is related to B, but not as closely as B is related to C, etc..

But it doesn't have to reach anything approaching a scientific level of precision.

So, what are some ways that I could reasonably combine these scores into a single score?

--Clairvoyant walrus (talk) 01:48, 14 December 2007 (UTC)

Not answering your question, but how would you score it if the articles cited each other? Or just one cited the other? 86.132.237.214 (talk) 10:32, 14 December 2007 (UTC)

I hadn't bothered with that because in this data set that happens relatively rarely, though it would be worth adding. Though it's not really a score, it's just two booleans (true/false for: A cites B; B cites A) --Clairvoyant walrus2 (talk) 00:05, 16 December 2007 (UTC)

Let F_i denote a smooth, strictly monotonically increasing approximation of the cumulative distribution function of the i-th component score that should go into the mix, estimated from a representative sample of the kind of papers you want to compute the scores for. Then you can normalize each raw i-th component score s_i by defining n_i = F_i(s_i), which gives a value in the range 0 to 1, more or less uniformly distributed. This makes the scores in some sense comparable.

How you combine these to a single score may depend on what that score is meant to signify for you. You can get false positives and false negatives, so to speak; are these equally bad? If you use this in the hope of proving (and publishing!) that the Declaration of Independence was actually written by Shakespeare, then a false positive is extremely embarassing, while a false negative is a missed chance to fame. Which is worse? Or this could be a service to alert readers of one article that they may be interested in another article. There you may want to minimize the number of false negatives among the N highest-scoring hits, where N is some relatively low value like 10 or 20. It may take some experimentation to find out what works best.

One theory is that if two articles are truly related, they should be related on all scores. Under this theory you can take the product of the normalized component scores.

Another theory is that two articles are related if they are significantly similar in some respect. Here you can use 1 − Π_i (1 − n_i). If one component score reaches 1, the amalgamated score is 1.

Finally, you could take some weighted sum of the values Φ⁻¹(n_i), in which Φ⁻¹ stands for the quantile function (the inverse of the cumulative distribution function) of the (standardized) normal distribution.

 --Lambiam 18:10, 14 December 2007 (UTC)

So, let me see if I understand you: I want to compare a reasonably large number of random papers to each other, record the scores for each method I want to use, and then use that data to tell what the probability of getting any given score is, then use that probability as the new normalized score which I can combine more easily. Is that right?

So, how do I do create the probability function from the data? Do I just define a set number of intervals ("buckets", if you will) between the highest and lowest scores, and then count the relative frequencies of each score falling in each bucket?

To answer your specific points: false positives are more embarrassing, to throw a number out there maybe 25% more embarrassing than a false negative. For this data set I've found that if the score is very high for any one comparison method it usually does signify an interesting relationship worth reporting; a low score for one method shouldn't count against the others, so the scores should be additive. Though I have a feeling that I might actually want to multiply scores for methods 1 and 3 and then add the score for method 2.

Method 2 has the unfortunate property that if the paper is about a popular topic, there is likely to be a large amount of content that happens to match it, and as a result 1) the similarity is likely to be foiled by a random coincidence of words between papers that are roughly on that same subject but not particularly more related than any other given papers on that subject 2) if there are a large number of related papers, it's just not worth reporting all of them. If it's an unusual topic than this method is much more reliable.--Clairvoyant walrus2 (talk) 01:04, 16 December 2007 (UTC)

You can take the fraction of elements in your sample with a score less than X as an estimate of the probability that a random element from the population has a score less than X. As a function of X, this is an increasing step function that ranges from 0 to 1. Smoothing the step function (see e.g. Kernel density estimation for a possible method, replacing h⁻¹K((x-x_i)/h) by Φ((x-x_i)/h), since you are not estimating the probability density function but its antiderivative) gives you a continuous function that has a continuous inverse.

I don't see the problem you raise for method 2; more precisely, I don't see how the popularity of the topic makes a match between two given, marginally related papers more likely than if the topic is abstruse. Is the issue that there will be more papers and therefore more pairs to be compared if the topic is popular? But then the effect comes equally into play for the other methods. For the score according to method 2, you should also take into account that in general longer texts will have more words in common, also if they are not about related topics. You might want to use something like the commonality between the 100 most frequent uncommon words in each paper, discarding words you will find in any text, regardless of the topic.

For the rest, you'll have to experiment with various formulas. If you have a collection of pairs of papers, for half of which you know you want a (true) positive, while the other half should be a (true) negative, you can partly automate the process of finding a formula that works for you. Examining and understanding why it is that some pairs refuse to yield can help in developing ideas for further improvements.  --Lambiam 14:55, 16 December 2007 (UTC)

Sorry, but what does the Φ in Φ((x-x_i)/h) represent? --Clairvoyant walrus2 (talk) 21:11, 17 December 2007 (UTC)

Other possible comparison methods:

• Boolean: paper A is cited by paper B
• Boolean: paper A cites paper B
• Correlation of all of the authors of all of the papers that cite A and all of the authors of all of the papers that cite B

--Clairvoyant walrus2 (talk) 01:04, 16 December 2007 (UTC)

Pentagon equations

Is it possible to create a pentagon using functions on a coordinate plane using less than 10 equations and no piecewise functions ? —Preceding unsigned comment added by 76.221.143.186 (talk) 03:27, 14 December 2007 (UTC)

You mean like, z⁵ = 1? —David Eppstein (talk) 03:36, 14 December 2007 (UTC)

I believe that this would only give you five points in the complex plane, meaning that five vertices. You also need five lines connecting the five vertices. I believe that it can be done because I know that for a square it is possible but I can't imagine what it would be for a pentagon. I would have to play around with it a bit. —Preceding unsigned comment added by A Real Kaiser (talk • contribs) 08:30, 14 December 2007 (UTC)

Ok, then, something like $0=\prod _{i=0}^{4}(x\cos {\frac {2\pi i}{5}}+y\sin {\frac {2\pi i}{5}}-1).$ It's still not quite a pentagon, because the lines extend beyond the vertices, but you could fix that with a second inequality $x^{2}+y^{2}\leq 1$.—David Eppstein (talk) 01:42, 15 December 2007 (UTC)

Does the pentagon have to be regular ? Do you want an equation for just the boundary of the pentagon, or can we include its interior as well ? Are inequalities allowed ? Do the functions have to be algebraic, or can they be transcendental ? Gandalf61 (talk) 09:25, 14 December 2007 (UTC)

A polar equation should work for the convex pentagon, though it won't be continuous. mattbuck (talk) 09:34, 14 December 2007 (UTC)

If inequalities are banned, we could replace the second part of David Eppstein's answer with an equation involving a suitable bump function. Algebraist 04:50, 15 December 2007 (UTC)

Following the suggestion of mattbuck: "A polar equation should work". The formula is r = 1/cos(v) for -2·π/10<v<2·π/10, and the function is periodic with period 2·π/5. Set t = 5·v. As the function r(t) is symmetric, and periodic with period 2·π, it can be written a₀ + a₁·cos(t) + a₂·cos(2·t) + a₃·cos(3·t) + a₄·cos(4·t) + a₅·cos(5·t) + a₆·cos(6·t) + ... See Fourier_series#Canonical_form for the formulas for the coefficients a_n. This solves the problem: "a pentagon using functions on a coordinate plane using less than 10 equations and no piecewise functions". Bo Jacoby (talk) 19:12, 15 December 2007 (UTC).

Riemann zeros

So there's all this talk about the "Riemann Hypothesis" I keep hearing. I am not well educated in mathematics so excuse any ignorance in this question.

Since $1+(1/2)^{s}+(1/3)^{s}+(1/4)^{s}+\cdots =[1+1/(2^{s}-1)][1+1/(3^{s}-1)][1+1/(5^{s}-1)][1+1/(7^{s}-1)][1+1/(11^{s}-1)]\cdots$

which is an astounding formula, by the way ... well, if one of the quantities in that infinite product on the right side is $0$, wouldn't that make the whole thing $0$?

If we set the first expression equal to 0, we get $1+1/(2^{s}-1)=0\quad \Rightarrow \quad 2^{s}=0$, for which there is no complex solution.

If we generalise it to any quantity from that product on the right, we get $1+1/(p^{s}-1)=0$ where $p$ is any prime -- we get $p^{s}=0$. There exists no such s that could possibly satisfy this equation, therefore there are no zeros? Or maybe I'm just talking out of my ass here.

Also, what's with the statement that trivial zeros are negative even integer solutions? I'm pretty sure that $1+1/(2^{-2})+1/(3^{-2})+1/(4^{-2})+\cdots =1+4+9+16+\cdots$ which diverges to infinity. I don't think it is zero. Again, maybe I'm talking out of my ass. —Preceding unsigned comment added by 124.191.113.111 (talk) 12:54, 14 December 2007 (UTC)

I'm not entirely up on my zeta functions, but two comments. First, the formula $1+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots$ only holds for $Re(s)\geq 1$; in particular it's not valid for negative integers. Second, an infinite product is defined only so long as it converges to a non-zero value. The zeros of the zeta function arise from values of $s$ where the product formula does not converge. SmaleDuffin (talk) 16:51, 14 December 2007 (UTC)

There is nothing undefined about an infinite product that has a factor equal to 0. Convergence of the partial products to some limit is all that is required for definedness, and in such a case they converge to the limit 0 – and in fact they do so p.d.q. (which is not a requirement).

The formula given above , however, is not quite correct. It should be is more commonly expressed as:

$1+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots ={\frac {1}{1-2^{-s}}}\times {\frac {1}{1-3^{-s}}}\times {\frac {1}{1-5^{-s}}}\times \cdots \,,$

where the 2, 3, and 5 in the right-hand side are the beginning of the infinite sequence of primes. This remarkable formula, known as the Euler product formula, is only valid for Re(s) > 1; for other values of s both sides do not converge. Can this be used to find any zeros of the Riemann zeta function? No, since it is already known that ζ(s) ≠ 0 if Re(s) > 1. Even if we did not know that, any value that can be expressed in the form 1/X is not 0 (just multiply both by X), so none of the factors of the infinite product is equal to 0.  --Lambiam 17:34, 14 December 2007 (UTC)

P.S. I see the original questioner proposed to use the absence of zero factors to show that the whole thing is not zero. But that doesn't work. ¹/₁ × ¹/₂ × ¹/₃ × ... = 0, even though all factors are greater than 0.  --Lambiam 17:42, 14 December 2007 (UTC)

The definition of infinite product in Nevanlinna and Paatero, "Introduction to Complex Analysis", is the following:
Let $a_{1},a_{2},\cdots ,a_{n},\cdots$ be an infinite sequence of non-zero complex numbers, and set $P_{n}=\Pi _{i=1}^{n}a_{i}$. The infinite product $\Pi _{i=1}^{\infty }a_{i}$ is said to converge if the limit $\lim _{n\to \infty }P_{n}=P$ exists and has a non-zero value.
As the original poster pointed out, the infinite product has no terms that are zero. Hence, the zeros of the Riemann zeta function occur where the infinite product formula fails to converge (or so it seems to me). SmaleDuffin (talk) 20:00, 14 December 2007 (UTC)

Actually the criterion for convergence for an infinite product is a bit complicated and not particularly intuitive. It's not enough that the sequence of product of initial segments converge to some limit. If there are no zero factors, but the sequence of partial products converges to zero, the infinite product is said to "diverge to zero". For the product to be convergent in this case, the sequence of partial products must converge to a finite nonzero real number.

On the other hand if there are finitely many factors that are zero, you decide whether the product as a whole converges by deleting the zero factors, and then checking whether that infinite product converges, by the above definition. If there are infinitely many zero factors, if I recall correctly, the product is considered to be divergent no matter what the nonzero factors do.

At first glance this looks like an artificial and unweildy definition, but it really is the one that turns out to give the "right" notion with the best properties. --Trovatore (talk) 20:01, 14 December 2007 (UTC)

But the case at hand is dealing with an infinite product of non-zero terms, as the original poster had stated. In this context, the infinite product does not converge (and note that `not converging' includes `diverging to zero') unless it converges to a non-zero number. All this stuff about non-zero factors seems somewhat irrelevant to the original question. SmaleDuffin (talk) 20:08, 14 December 2007 (UTC)

Convergence is sufficient for definedness, but not necessary. While the infinite product П 0 does not converge, it is still defined and equal to 0.  --Lambiam 20:25, 14 December 2007 (UTC)

(clearing some indents) It seems we might be working at cross purposes here. The original poster asked about an infinite product of non-zero terms; he asked how could an infinite product of non-zero terms result in zero. My intent in answering as I did was to point out that an infinite product of non-zero terms cannot converge to zero, and so finding the zeros of the zeta function by appealing to the infinite product formulation of the zeta function would probably not work. On the broader question of infinite products in general--yes, having terms that are zero is possible, and have to be handled in some way. The product under discussion, however, has only non-zero terms. SmaleDuffin (talk) 20:47, 14 December 2007 (UTC)

P.S. Wikipedia's page on infinite product explicitly states that the limit to which the product converges has to be non-zero. If this is an error, perhaps it should be corrected? SmaleDuffin (talk) 20:51, 14 December 2007 (UTC)

There are two different questions you can ask for an infinite product: (1) "is it defined?"; and (2) "does it converge?" It may be the case that an infinite product of non-zero terms does not converge (i.e., the answer to question (2) is "no"), but it can nevertheless be a well-defined expression (i.e., the answer to question (1) is "yes") whose value is 0. Convergence is not a necessary condition for definedness. I gave several examples of this: ¹/₁ × ¹/₂ × ¹/₃ × ... = 0 and П 0 = 0. It is not (explicitly or implicitly) stated that the limit to which the partial products converge "has to" be non-zero, as far as the defined meaning is concerned.  --Lambiam 01:31, 15 December 2007 (UTC)

Analytic continuation is essential to the definition of the Riemann zeta function. —Keenan Pepper 18:52, 14 December 2007 (UTC)

So what you're saying here is that the right side can never be 0 for any value of s, but the left side can? Also, why are the trivial zeros of the Riemann zeta function said to be all negative even integers? I seriously think the series would diverge for s = -2, -4, -6.. —Preceding unsigned comment added by 124.191.112.218 (talk) 03:57, 16 December 2007 (UTC)

No, since the infinite series in the left-hand side only is defined (and used to define ζ(s)) for Re(s) > 1. The infinite series, for the domain where it is defined, is nowhere zero. To define ζ(s) for Re(s) ≤ 1, you have to use analytic continuation, as mentioned above by Keenan. See the sections titled "Definition" and "The functional equation" in the Riemann zeta function article.  --Lambiam 05:31, 16 December 2007 (UTC)