Knowledge:Reference desk/Archives/Mathematics/2007 August 5

328:: how does one, like myself, manually calculate these results? That being said, a year is simply one date to the exact same date a year later: example = June 12, 1965 to June 12, 1966. Sure, whenever you are doing calculations with dates / calendars, the issue of leap years and February 29th comes up. Nonetheless, that does not convolute or obfuscate my original question. The anomaly or ambiguity would lie in the algorithm, which is not my original question. In any event, the February 29th "issue" (if you will) will be negligible at best and will probably hardly ever surface. If it does, I would posit that it works like this: if we start out at February 5, 2004 … a year later will be February 5, 2005 – regardless of the fact that there is an extra day (Feb. 29th) wedged in there. In other words, in that case, the term "year" simply means 366 as opposed to 365 days. 538:(for "fun", I appreciate there are plenty of other ways to do this on a computer). The implementation continues until the value of the n term is below a certain threshold, at which point the algorithm returns; so I can count how many terms I need to evaluate for a given degree of accuracy. There are marked optimisations to be had in exploiting the symmetries of the sine function: identities that move values into the 0..2π range, and further that move those from π..2π into 0..π make for marked improvements - clearly the series converges much faster in the 0..π range than for other values. I've coded a futher (trivial) symmetric optimisation that flips values in π/2..π down into 0..π/2, but that doesn't produce any improvement. Am I correct in thinking that the convergence of the Taylor series for sine between 0 and π/2 is not much faster than between π/2 and π? -- 321:

equally unhelpful to have output that says "Adrien Brody is 29 years, 11 months, and 9 days old." The chart in the above link is formatted with an individual's age reported as Y years old and D days old. Thus, that is the format of the data output that I would like to receive. If the results are given simply in terms of D days old … or even in terms of Y years, M months, and D days old – that will not be useful to me. All of the websites that I have found give me the output in one of those two ways that are not helpful to me. I am looking for a website (or function in Excel) that will output the result in the format I need (age expressed as Y years and D days).

313:. As you see, that list includes the individuals’ ages. I am proofreading that list (and many others) and checking for errors. And I am also involved in several projects of a similar nature. Ultimately, there will be many, many, many ages that I need to compute – with the above link merely being an example. I certainly know how to calculate these figures manually. That is, I know the mathematical / numerical algorithm that will achieve the answer. However, because I have so many dates to calculate, I am trying to make the task easy / manageable / as painless as possible. I don’t want to manually do all this … I want a computer / calculator to do it for me. 6901:

the problem greatly by discarding the higher-order terms. Depending on the size of the perturbation and on the accuracy required and on the problem itself, we may wish to keep second-order terms as well (or more); but a great deal of applied mathematics works with a first-order approximation. We then apply the powerful machinery of linear algebra to the "linearized problem". This is such a habit that folks may forget that nature itself often

6897:, we can create a model of the reals extended with infinitesimals. In this approach, δ is taken to be such a nonstandard quantity, a genuine infinitesimal, and the result of the subtraction and division is coerced back to a standard quantity. Another approach is to extend the reals with a quantity ε defined to satisfy ε = 0, but ε ≠ 0. If this is used for δ, the higher-order terms automatically disappear. 118:

result is in the desired format)? (Question #3) In Excel 2007, I cannot seem to work with dates prior to 1/1/1900. That is, the date of January 1, 1900 -- and all subsequent dates -- are valid. Any date prior to that (December 31, 1899 and earlier) seem to be invalid dates that trigger an error message. Does any one know a way around this limitation in Excel 2007? Thanks. (

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Well, not mistaken, but perhaps not interpreting terminology correctly. I understood "small" to mean "infinitesimal", but not that "produces no change" meant "produces change of second order or higher". If this is standard language in calculus, could you give some other examples of how "X happens" is

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To avoid the vagaries of excel just put the years, months and dates in separate columns, say A1,B1,C1,D1,E1,F1 for (year1,month1,day1,year2,month2,day2), to find the number of years difference use G1=IF(B1<E1,D1-A1,IF(C1<F1,D1-A1,D1-A1-1)), the birthday will then be H1=DATE(A1+G1,B1,C1) if the

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Before you rush off to finding a program that calculates what you want, you need to know what you want. It astonishes me how you have avoided answering a question that was posed twice. I have no problem with "a year is simply one date to the exact same date a year later" whenever this exists. This is

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in the format of 18 years and 8 months and 14 days. I have found web sites that will give me the latter two formats (which I don't need), but I was not able to find the former format (which I do need). (Question #2) Does any one know how to perform the above calculation in Excel 2007 (such that the

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as "the exact same date a year later", as there is no such thing as February 29 2005. So what will the answer be? 1 year, or 1 year and a day? 1 year and a day will be weird, as there is no date defined as 1 year after 2.29.05. Another peculiarity is that both 2.29.04 to 3.1.05 and 2.28.04 to 3.1.05

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Also – back to Excel for a moment. It seems hard to believe that a behemoth like Microsoft would not allow us to account for dates pre-1900. Dates pre-1900 would be relatively common for various uses. (How many days was George Washington in office? How many days did the American Revolution or the

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That being said, I would like to be able to merely enter the two dates (start date and end date) and have the results calculated for me automatically. Thus, as my original question states … I would like to do this with either (a) a web site that can do so or (b) Excel if possible. I do not want to

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I wasn't sure what Help Desk to use for this question ... so I am going with Math (here). Given two dates, I want to be able to compute the amount of time elapsed between the two dates. So, for example, if I have May 3, 1962 ... and January 17, 1981 ... I want to know that the time elapsed between

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The general idea of discarding higher-order terms has many practical applications. For example, we may have a physical problem where the full description is quite complicated. However, if we can write the effect of a small perturbation as a series of terms of increasing order, we can often simplify

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Isn't this ambiguous? What is the amount of time, in your format, elapsed between March 1, 2003 and March 1, 2004? And what is it for February 28, 2003 and February 28, 2004? What about February 28, 2003 and February 29, 2004? And March 1, 2003 and February 29, 2004? And, finally, February 28, 2003

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To quantify the effect: if you stop when your term drops below 2 in absolute value, then for x = π you will stop at x/29!, while for x = π/2 you can stop "already" at x/23!. That is, indeed, not a big deal. However, for x = π−3 you only need to go to x/11!, which is a considerable gain compared to

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So, to address Lambian’s point - it is outside of the scope of the question. I am asking if there is a computer / calculator out there that will allow me to input two dates and receive as output the age I need. How this computer / calculator does this calculation is of no interest to me. So can

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If you had bothered to try to answer Lambiam's questions, you would have found out the ambiguity\anomaly. The ambiguity is "what is the difference between February 29 2004 to March 1 2005?". The anomaly is that however you answer this question, you will get something weird. We can help you more if

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I want to know whether LHS = RHS or not. As a background, R(p,q) is the Euclidean distance between points P and Q in a meshed unit cube (in all there are N = n^3 regularly spaced points), and x(p,q) is just x-component of the distance. Also, N can be arbitrarily increased, so that the points are

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To be more specific: Let's use the first person on the list as an example. Adrien Brody is 29 years and 343 days old. So, if I entered the beginning and ending dates, I want that as my result. It is not helpful to me to have output that merely says "Adrien Brody is 10,935 days old." It is

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amount to simply finding where the derivative is 0. The minimum can either be at such a point, or at the endpoints of the domain of the function (the problem specification should also describe what the domain is). Sometimes a minimum does not at all exist. In your case, the domain is probably

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to get slightly simpler expressions. Have you tried it numerically with some values? That way you could prove that there are cases where the equality does not hold, but perhaps that is not good enough for you. Anyway, I find myself trying to figure out more about what your problem really is

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There be dragons! As I mentioned in response to a question about associativity, floating point addition rounds to a fixed precision. If you add the terms from largest to smallest, the contributions of the small terms can disappear. Either add from small to large, or use something like the

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you explain what do you need this for. As it is, I think your best bet will be to find the number of days (easy, there should be a function for this in excel) and divide it by 365.2425 - the quotient will be the number of years, and the remainder (rounded) will be the number of days. --

765:. Once again, it is up to you and your desired accuracy to decide if the difference is great. If what you're asking is about the largest term in the series (which bounds the greatest precision you can achieve with given internal precision), it is roughly 1.5 and 5, respectively. -- 211:

Then to find the numbers of days difference, you can use (forcing valid excel years) something like "=date(2007,B1,C1)-date(2007,B2,C2)" possibly +365 or +366 depending upon whether years are leap or not. Note that this isn't easy because, say your dates are from leapyear-feb-14 to

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n ‖Re(LHS-RHS)‖/N² ‖Im(LHS-RHS)‖/N² 2 0 0 3 0.2672 0.1263 4 0.3554 0.1951 5 0.4033 0.2542 6 0.4329 0.3001 7 0.4529 0.3355 8 0.4672 0.3631

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Is this right and if yes, as I'm sure it isn't conincidence that it is my previous answer times minus one, could I have arrived at it without doing any of this working, i.e. just using my first answer and the fact that the square of an imaginary number is minus one?

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Yep, that's right. And to be honest, I have no idea if you can simply do that every time without doing it out... the fact that it's equal to x^4 might be why such a thing is the case, but to be honest I'm not entirely sure -- someone else will have to answer that.

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inf (i.e., we consider all points in R^3), then shouldn't this "error" should keep decreasing with increasing n? The result for n = 2 makes sense because in that case, we have 8 points, which all have three neighbouring points "filled" and other three "empty."

1510: 1317: 6866:δ, but we also have a second-order change, δ. One way to look at the derivative is to subtract, divide, and take the limit as δ goes to zero. Instead of taking limits, we may ignore the second-order change; the result is the same. If we try again with 385:

Last but not least, do you know any programming languages? It shouldn't be hard to write a program which can do as much as take the wiki or html code of a page and automatically calculate the date differences and check the consistency of the data. --

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these two dates is 18 years and 259 days. So, here are my questions. (Question #1) Does any one know of any web sites that perform this calculation? Important: I need the output to be in the format of 18 years and 259 days ... and

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The HP48 series of calculaors can calculate the number of days between any 2 dates. the newer 49 series and the current HP50G also does it. if one does not have a actual HP calc, there are free HP48-49-50 emulators available online.

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What is the ambiguity? How much more precise can I be? I want the format in Y years and D days ... as opposed to simply D days ... or as opposed to Y years, M months, and D days. What is unclear / ambiguous about my question?

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Thanks but I'm an amateur mathematician so what you said about limits means little to me. I have heard about minimums, and maximums, existing at 'end points' before. Could you show me where to find out more about them? Thanks

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birthday after 1900 you can then subtract I1=date(D1,E1,F1)-H1. For Feb 29 you can do it by hand, taking either Feb 28 or March 1 for non leap years. If you need to work pre 1901 then you'll need to correct for leap years.

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As to why, its probably historical. Excel was initially for financial applications and they didn't think it would be needed, and it saved space. They haven't changed since then to ensure compatibility. Indeed it might be

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Thanks to everyone for the input. As was suggested by Meni Rosenfeld (above), I will provide some context and clarification for my questions. As it may be helpful, please take a look at the following Knowledge page:

4971: 6191:| has a minimum at zero, even though the derivative does not exist.) We have no general method to decide if a local minimum is also a global minimum. And as others have noted, a minimum can also occur at a boundary. 5557: 7065: 175:

Yes, I know that. Have you read my question? I want a program to do this for me. I have many, many, many to do ... and I would like to simplify the matter. Either with a website / program ... or with Excel.

6020:= 0 is not even a local minimum.) Remember that the derivative is always 0 at a maximum or minimum (excepting boundary points), but not every point where the derivative is 0 is necessarily a maximum or minimum. 5491: 2432:{\displaystyle {\frac {(ikR_{pq}+1)\left|x_{pq}\right|e^{-ik(R_{pq}+R_{qr})}}{R_{pq}^{3}R_{qr}}}={\frac {(ikR_{qr}+1)\left|x_{qr}\right|e^{-ik(R_{pq}+R_{qr})}}{R_{pq}R_{qr}^{3}}}\ \forall q\in \{0,1,\ldots ,N\}} 4906: 2543: 371:

In case there are places where you have no say in the chosen format, the question is asked if you are willing for the results to be a day off sometimes. If so, my suggestion of dividing days by 365.2425 is

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really close to each other, and the summation can be replaced by an integration (if required). Hopefully there is a way to prove that these two are (not) equal. Any help will be highly appreciated.

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Just break it up into two parts. First calculate the number of full years between May 1962 and January 1981. Then calculate the distance from May 3, 1980 to January 17, 1981. You can just use the

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nonleapyear-march-5, then the year difference is easy, but are the days those at the start of the period (ie a leap number of days) or at the end of the period (ie a nonleap number of days). --

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Find out which day of the year the first date was on, and the day of the year the second was on, and subtract modulo 365. This ought to do the trick? Finding the difference in years is trivial.

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To address another concern: There really is no anomaly or ambiguity in the question I have posed. The question is: Is there a program out there that does what I want? The question is

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preserves symmetry. I see no immediate reason why that should be the case. Did you try disproving the claim by computing the two for some small randomly chosen symmetric input matrix

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do these manually. So, I will indeed go through the time / expense of typing in a lot of dates for the data input. But I am not willing to manually calculate each and every result.

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To Salix alba: What exactly is the difference between an "age" and a "duration" ...? Aren't they both simply an elapse of time from Date 1 to Date 2? I am confused. Thanks. (

3139: 3027: 763: 715: 3508: 2903: 7500: 2653: 6783:). Small here is obviously taken in the sense of infinitesimal, and zero in the sense of neglecting effects beyond first order. This is standard calculus-speak. Certainly any 1643: 1609: 4577: 3185: 3073: 1560: 5705: 5117:

and the other solution(s) could have resulted in a smaller answer. (and that the imaginary solutions give you a smaller minimum, so if you count those your answer is wrong)

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goes to 0 and infinity. Since the value of the function is positive infinity in both limits, which is greater than the value you have found, this is indeed the minimum. --

3740: 3289: 3239: 3786: 3696: 3652: 7680:). It's often easy to forget the stuff you learned in junior high when looking at things from the perspective of calculus and beyond. You want to avoid that temptation. 3846: 2961: 1670: 6082: 3568: 3538: 5776: 7678: 7629: 7600: 5900: 5822: 2731: 7649: 5842: 5796: 3806: 3608: 3588: 3309: 2923: 2839: 2093: 1752: 336:

Civil War last?) Is there any way in Excel to get around the limitation that December 31, 1899 (and earlier dates) are considered "errors" or invalid? Thanks. (

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That's correct. k is the wave number. This comes from an acoustics problem. Taking this further, I guess even the exponential can be eliminated, so that we have

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Otherwise, you should follow JayHenry's suggestion - he didn't say you need to do this manually. Have your spreadsheet software do the calculation in two parts.

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when the calenders got changed about. If you need dates before that you may need to look at something like Java, perl seems to have some advanced packages.

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The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the

856:(compensated summation), or use double precision to get an accurate single precision result. Otherwise, your "theoretical" accuracy will be meaningless. 5354: 4782: 6826:

In somewhat old-fashioned parlance of analysis, a finite quantity is larger (in absolute magnitude) than any infinitesimal, and thus excludes zero.

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Actually not so minor, really. This sort of substitution makes life easier at many places, not just in this problem, but in many related problems.

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This is why I often hesitate to question you--it seems to always turn out either that I'm wrong, or that there was a miscommunication. Oh well. =)

7189:{\displaystyle \Rightarrow 3x^{2}+{\frac {1}{7x^{2}}}=\left({\sqrt {3}}x-{\frac {1}{{\sqrt {7}}x}}\right)^{2}+{\frac {2{\sqrt {3}}}{\sqrt {7}}}} 2761:

Ahrgg, you are right! I was going to object to deeptrivia's simplification for that very reason, but apparently I did the same mistake myself. —

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Using excel, if you can get the year numbers, month numbers and day numbers into 6 separate cells then the calculations can be done as follows:

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the case x = 3 (which is like x = π). The reduction also gives you better accuracy, and the effect is stronger when you get close to π. --

37: 6187:), then we are surely not at a minimum, even locally. This is the source of the "derivative equals zero" condition. (Caution: note that | 5073:

You are correct. Keep in mind for future reference, though, that there are actually two (real, four if you count imaginary) solutions to

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Yes, I see there's a problem with the simplification. Any other ideas? To me it appears that it is more likely to be false (LHS <: -->

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That depends mostly on your definition of "much". When evaluating the sine by putting the variable in the Taylor series and calculating

429:'s fault as I think the functions were chosen to be compatible with that. Many other computer systems start counting quite recently see 130:

and March 1, 2004? Somehow you cannot avoid anomalies if you reckon this way, and you have to be precise how you want them resolved. --

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OH! Yes I'm with you 100%. Thanks. So it's only when you have selected a specific range - in which the derivatove never equals zero?

7054:{\displaystyle \left({\sqrt {3}}x-{\frac {1}{{\sqrt {7}}x}}\right)^{2}=3x^{2}-{\frac {2{\sqrt {3}}}{\sqrt {7}}}+{\frac {1}{7x^{2}}}} 21: 2448: 980:{\displaystyle \sin(\theta )=\sin \left(-{\tfrac {\theta }{3}}\right)\left(4\sin ^{2}\left(-{\tfrac {\theta }{3}}\right)-3\right).} 378:

As for the 1900 problem, it shouldn't be too difficult to do this with the right formulae in excel, but why should you - there are

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Now that we have seen why the year & day format is problematic, we need to ask - why do we even need days for pages like

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you need to specify a convention when the staring date is February 29, and be prepared for anomalies whatever choice you make

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is 1y1d. If it is only 1 year, you also get things such as both 2.29.04 to 3.1.05 and 3.1.04 to 3.1.05 is 1y. So, I repeat -

7684: 7510: 7454: 7396: 6909: 6837: 6821: 6795: 6750: 6238: 6160: 6024: 5935: 5906: 5730: 5720: 5630: 5612: 5121: 5067: 3924: 3906: 3428: 3390: 3322: 2796: 2765: 2744: 2548: 2060: 1762: 1684: 1112: 1010: 844: 826: 805: 787: 774: 546: 514: 493: 483: 468: 455: 442: 395: 340: 294: 282: 257: 235: 216: 180: 166: 148: 134: 122: 531: 6295:). I believe the real reason for the "derivative equals zero" condition is more like the following (you should have the 4178: 7260: 6230:, which may be positive, negative, or zero. For a minimum, we require it to be positive, so that as we move away from 559: 5182: 410:. In words find the closest birthday less than you target date and then find the number of days between then and now. 6087: 5563: 5133: 1505:{\displaystyle RHS=\sum _{q=0}^{N}{\frac {(ikR_{qr}+1)\left|x_{qr}\right|e^{-ik(R_{pq}+R_{qr})}}{R_{pq}R_{qr}^{3}}}} 1312:{\displaystyle LHS=\sum _{q=0}^{N}{\frac {(ikR_{pq}+1)\left|x_{pq}\right|e^{-ik(R_{pq}+R_{qr})}}{R_{pq}^{3}R_{qr}}}} 5021: 4977: 2657: 7353: 6813:

used for "X is a first-order approximation"? (I'm not doubting you, I'm actually interested in this phenomenon.)

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RHS). Any way to demonstrate this? I know plugging in a few numbers should do, but any better ideas? Thanks,

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to lie in the range as Bromskloss did above. Let's try to find the minimum in this domain; differentiating

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This is, indeed, not exactly a coincidence, but isn't very general either. It results from the facts that:

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I may be mistaken, but it seems to me both of those removals are only valid if we break the two sums over

511: 465: 452: 430: 337: 177: 145: 119: 7527: 6890:δ+δ, the higher-order terms retain a factor of delta. This explains why they disappear in the derivative. 3937:

Hi. I'm back again with a new problem I think I've solved, which I again would like you to check for me.

3441: 641: 479:. I've not tried it to see how it deals with leap years, or, say, the fact that 1900 wasn't a leap year. 4498: 4002: 859:

Since you're doing this for fun, here's an amusing observation on range reduction. First, observe that

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One interesting observation is that any difference will have to be entirely due to the restriction of

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Then recall that as θ approaches zero, sin(θ) approaches θ. Thus we can use repeated range reductions

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Are you asking about my justification for the above estimates? This is just a simple application of

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Fun. I haven't come up with anything great, just noticed that you can multipy both sides with

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A1=later year, B1=later month, C1=later day, A2=earlier year, B2=earlier month, C2=earlier day

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and free yourself the MS behemoth, it has a better selection of date functions than excel.

274: 249: 7654: 7605: 7576: 5343:{\displaystyle 3({1 \over i{\sqrt{21}}})^{2}+{1 \over 7({1 \over i{\sqrt{21}}})^{2}}\,\!} 5879: 5801: 2710: 2034:{\displaystyle RHS=\sum _{q=0}^{N}{\frac {(ikR_{qr}+1)\left|x_{qr}\right|}{R_{qr}^{2}}}} 1901:{\displaystyle LHS=\sum _{q=0}^{N}{\frac {(ikR_{pq}+1)\left|x_{pq}\right|}{R_{pq}^{2}}}} 7451: 7393: 6817:

Also, why do you emphasize "finite" in your last sentence? I would expect "nonzero"...

5627: 5118: 4771:{\displaystyle 3({1 \over {\sqrt{21}}})^{2}+{1 \over 7({1 \over {\sqrt{21}}})^{2}}\,\!} 3413:

I divided the norms by N^2 because both LHS and RHS matrices have N^2 elements. If, as

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what I am looking for. Thanks. I guess I didn't have to look too far, huh? Thanks! (

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As to dates before 1900 (1904 on a mac) Excels too dense to be able to work with them.

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I have to disagree: it is entirely possible to have a minimum where a small change in

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linear, and there can be a great deal of interesting behavior in the nonlinearity. --

6882:δ+δ. Notice that in the ratio of output change to input change (which is δ), namely 3 6818: 6791:

will produce a change in a non-constant function, but that's why we have calculus. --

6747: 6021: 5847: 5662:(which itself comes from the fact that they are all fourth roots of the same number). 4394: 2441:

which is a stronger claim than simply the equality of the sums. While both sides of

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where the rational function is undefined which may rule out some of the zeroes from

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As ever I am only looking to be told if I am correct or not. Thank you in advance.

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Okay, here are my results. Consider a unit cube, each of whose side is meshed with

5430:{\displaystyle 3({1 \over -{\sqrt {21}}})+{1 \over ({7 \over -{\sqrt {21}}})}\,\!} 6651:

is differentiable, then these two limits must be equal, and so we conclude that

4848:{\displaystyle 3({1 \over {\sqrt {21}}})+{1 \over ({7 \over {\sqrt {21}}})}\,\!} 1755: 426: 418: 6906: 6792: 6296: 6235: 1007: 263: 159: 3915:

Oh yes, I get what you're saying now. Thanks a ton, it was really helpful.

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There is a simple way of doing it, here on Knowledge. There is a template;

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It's also worth noting that in finding the zeroes of a rational function,

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A minor remark. The problem is equivalent to finding the minimum value of

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An alternative approach to the original problem by completing the square:

4966:{\displaystyle {3{\sqrt {21}} \over {21}}+{3{\sqrt {21}} \over 21}\,\!} 999: 6767:

example should make that clear. The very meaning of the derivative of

5552:{\displaystyle -{3{\sqrt {21}} \over 21}-{3{\sqrt {21}} \over 21}\,\!} 6830: 6420:

0; it follows from properties of limits that the "limit from above",

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Let's throw in a dash of geometric intuition. If a small change in

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into their component terms and equate those simultaneously, i.e.:

2733:. So I don't think this approach looks particularly helpful. — 267: 3610:

must lie within the unit cube. For example, consider the case

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is equivalent to the statement that the transformation turning

6324:. (This is true if you restrict to a small enough area around 2538:{\displaystyle {\frac {e^{-ik(R_{pq}+R_{qr})}}{R_{pq}R_{qr}}}} 7402:

A complex valued functions does not have a minimum, because

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in years and days that is more well defined problem, than a

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if your not too fussed about leap years this may do you. --

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6226:) is maximizing for its negative. We must also look at the 5707:, so you should have checked the limits of the function as 7506:

be non-negative. This results in simpler computations. --

6640:{\displaystyle \lim _{h\to 0^{-}}{\frac {f(t+h)-f(t)}{h}}} 6495:{\displaystyle \lim _{h\to 0^{+}}{\frac {f(t+h)-f(t)}{h}}} 1002:. Hardware implementations take a different approach; the 352:(2004, say) to March 1 2005. The problem is that there is 3403:

nodes (so there are in all N = n^3 points.) Then, we get:

244:

There is a YEARFRAC function in excel which requires the

7343:{\displaystyle {\sqrt {3}}x-{\frac {1}{{\sqrt {7}}x}}=0} 5844:

to be any real number, we wanted it to lie in the range

2600:{\displaystyle \left|x_{pq}\right|=\left|x_{qr}\right|} 638:(assuming your internal precision is high enough). For 556:

terms, the number of correct digits you get is roughly

6356:. Going to the definition of derivative, we have that 5902:

even though the derivative isn't zero there. Get it? —

3808:

in the unit cube that could give such a high value of

947: 901: 652: 489:

I've just checked, and it seems fine with leap years.

7657: 7637: 7608: 7579: 7530: 7467: 7410:

is undefined for complex numbers. For real values of

7356: 7301: 7263: 7208: 7068: 6936: 6657: 6571: 6508: 6426: 6362: 6090: 6053: 5882: 5850: 5830: 5804: 5784: 5757: 5678: 5566: 5500: 5444: 5357: 5243: 5185: 5136: 5079: 5024: 4980: 4915: 4862: 4785: 4681: 4631: 4586: 4545: 4501: 4451: 4397: 4333: 4247: 4181: 4112: 4054: 4005: 3946: 3854: 3814: 3794: 3748: 3704: 3660: 3616: 3596: 3576: 3546: 3516: 3487: 3444: 3297: 3247: 3197: 3147: 3081: 3035: 2969: 2931: 2911: 2882: 2847: 2827: 2713: 2660: 2613: 2551: 2451: 2103: 2081: 1921: 1788: 1740: 1696: 1652: 1618: 1568: 1522: 1326: 1133: 1025: 867: 727: 679: 644: 562: 368:? Is it not better to simply delete the days column? 7573:, we really are only concerned about the zeroes of 4233:{\displaystyle {2 \over 7}*{21x^{4}-1 \over x^{3}}} 7672: 7643: 7623: 7594: 7565: 7494: 7384: 7342: 7287: 7249: 7188: 7053: 6738: 6639: 6553: 6494: 6407: 6316:(for example) is a minimum, then on both sides of 6194:But we have another problem. If the derivative of 6148: 6076: 5894: 5868: 5836: 5816: 5790: 5770: 5699: 5655:i raised to a power which is 2 modulo 4 is -1, and 5597: 5551: 5485: 5429: 5342: 5225: 5171: 5109: 5052: 5008: 4965: 4900: 4847: 4770: 4663: 4616: 4571: 4530: 4486: 4432: 4381: 4318: 4232: 4164: 4097: 4039: 3987: 3887: 3840: 3800: 3780: 3734: 3690: 3646: 3602: 3582: 3562: 3532: 3502: 3473: 3303: 3283: 3233: 3187:. (In particular, this implies that the equality 3179: 3133: 3067: 3021: 2955: 2917: 2897: 2868: 2841:into the unit cube: if the sum was taken over all 2833: 2725: 2699: 2647: 2599: 2537: 2431: 2087: 2033: 1900: 1746: 1725: 1664: 1637: 1603: 1554: 1504: 1311: 1100: 979: 757: 709: 665: 630: 7288:{\displaystyle {\frac {2{\sqrt {3}}}{\sqrt {7}}}} 3417:said, LHS should become equal to RHS as N -: --> 631:{\displaystyle 2n(\log _{10}(2n)-\log _{10}(xe))} 6679: 6573: 6428: 5226:{\displaystyle x=\pm {1 \over i{\sqrt{21}}}\,\!} 6565:< 0, we conclude that the limit from below, 6149:{\displaystyle 1/(i{\sqrt{21}})=-i/{\sqrt{21}}} 5598:{\displaystyle -{6{\sqrt {21}} \over {21}}\,\!} 5593: 5547: 5481: 5425: 5338: 5221: 5172:{\displaystyle x=\pm {1 \over {\sqrt{21}}}\,\!} 5167: 5105: 5048: 5004: 4961: 4896: 4843: 4766: 4659: 4612: 4567: 5798:is a real number. The derivative is zero when 5053:{\displaystyle {6{\sqrt {21}} \over {21}}\,\!} 5009:{\displaystyle {6{\sqrt {21}} \over {21}}\,\!} 2700:{\displaystyle \forall q\in \{0,1,\ldots ,N\}} 7446:= − 1 and − 1 < 0, so even when the value 7385:{\displaystyle x^{2}={\frac {1}{\sqrt {21}}}} 8: 3438:of points (indeed, in the continuous limit, 2694: 2670: 2426: 2402: 1101:{\displaystyle (((+))))=(((-((((=-(=)\neq 0} 998:What do professionals use? In software, try 104:Calculating the Difference Between Two Dates 5751:Oh, that's easy! Let's find the minimum of 4664:{\displaystyle x={1 \over {\sqrt{21}}}\,\!} 4165:{\displaystyle {2(21x^{4}-1) \over 7x^{3}}} 2545:, I think what's left reduces down to just 783:Interesting. Any source pointer for that? — 208:Where A3 is the number of years difference. 7250:{\displaystyle 3x^{2}+{\frac {1}{7x^{2}}}} 162:calculators that you've found for that. -- 7656: 7636: 7607: 7578: 7531: 7529: 7477: 7466: 7370: 7361: 7355: 7321: 7315: 7302: 7300: 7270: 7264: 7262: 7238: 7225: 7216: 7207: 7171: 7165: 7156: 7138: 7132: 7119: 7101: 7088: 7079: 7067: 7042: 7029: 7011: 7005: 6996: 6980: 6962: 6956: 6943: 6935: 6829:This ought to be noted in our article on 6694: 6682: 6656: 6595: 6587: 6576: 6570: 6509: 6507: 6450: 6442: 6431: 6425: 6363: 6361: 6139: 6134: 6129: 6110: 6105: 6094: 6089: 6057: 6052: 5881: 5849: 5829: 5803: 5783: 5762: 5756: 5677: 5667:Anyway, note that finding a minimum does 5584: 5576: 5570: 5565: 5532: 5526: 5510: 5504: 5499: 5467: 5465: 5453: 5448: 5443: 5407: 5398: 5389: 5373: 5364: 5356: 5327: 5313: 5308: 5299: 5287: 5278: 5264: 5259: 5250: 5242: 5209: 5204: 5195: 5184: 5156: 5151: 5146: 5135: 5093: 5084: 5078: 5039: 5031: 5025: 5023: 4995: 4987: 4981: 4979: 4946: 4940: 4930: 4922: 4916: 4914: 4882: 4880: 4868: 4863: 4861: 4826: 4821: 4812: 4797: 4792: 4784: 4755: 4742: 4737: 4732: 4720: 4711: 4698: 4693: 4688: 4680: 4648: 4643: 4638: 4630: 4600: 4591: 4585: 4553: 4544: 4520: 4511: 4500: 4470: 4461: 4450: 4419: 4410: 4396: 4368: 4359: 4334: 4332: 4305: 4296: 4285: 4274: 4264: 4248: 4246: 4222: 4205: 4195: 4182: 4180: 4153: 4129: 4113: 4111: 4086: 4065: 4055: 4053: 4028: 4015: 4004: 3976: 3963: 3954: 3945: 3853: 3833: 3824: 3815: 3813: 3793: 3767: 3758: 3749: 3747: 3703: 3659: 3615: 3595: 3575: 3551: 3545: 3521: 3515: 3494: 3490: 3489: 3486: 3465: 3443: 3296: 3246: 3196: 3168: 3152: 3146: 3126: 3117: 3108: 3100: 3091: 3082: 3080: 3056: 3040: 3034: 3014: 3005: 2996: 2988: 2979: 2970: 2968: 2930: 2910: 2889: 2885: 2884: 2881: 2860: 2856: 2855: 2846: 2826: 2712: 2659: 2634: 2618: 2612: 2584: 2560: 2550: 2523: 2510: 2491: 2475: 2458: 2452: 2450: 2382: 2374: 2361: 2341: 2325: 2308: 2291: 2265: 2249: 2234: 2224: 2216: 2196: 2180: 2163: 2146: 2120: 2104: 2102: 2080: 2023: 2015: 1997: 1971: 1955: 1949: 1938: 1920: 1890: 1882: 1864: 1838: 1822: 1816: 1805: 1787: 1739: 1714: 1701: 1695: 1651: 1625: 1617: 1592: 1573: 1567: 1543: 1527: 1521: 1493: 1485: 1472: 1452: 1436: 1419: 1402: 1376: 1360: 1354: 1343: 1325: 1297: 1287: 1279: 1259: 1243: 1226: 1209: 1183: 1167: 1161: 1150: 1132: 1024: 946: 929: 900: 866: 726: 678: 651: 643: 604: 576: 561: 402:(After edit conflict) If your wanting an 6554:{\displaystyle {\frac {f(t+h)-f(t)}{h}}} 6502:, is zero or positive. Similarly, since 6408:{\displaystyle {\frac {f(t+h)-f(t)}{h}}} 6000:= 0. But this is not the minimum, since 5235:Taking the imaginary roots would lead to 4098:{\displaystyle {42x^{4}-2 \over 7x^{3}}} 3510:), but about the restrictions placed on 311:List of Best Actor winners by age at win 49: 36: 5591: 5545: 5479: 5423: 5336: 5219: 5165: 5103: 5046: 5002: 4959: 4894: 4841: 4764: 4657: 4610: 4565: 3988:{\displaystyle 3x^{2}+{1 \over 7x^{2}}} 3291:will stay within the unit cube for any 2643: 1006:algorithm is a traditional favorite. -- 65: 5648:Your function involved only powers of 5110:{\displaystyle x^{4}={1 \over 21}\,\!} 4617:{\displaystyle x^{4}={1 \over 21}\,\!} 3997:First I differentiated it, leading to 43: 6771:being zero is that a small change in 4487:{\displaystyle 21x-{1 \over x^{3}}=0} 3434:My previous comment wasn't about the 2869:{\displaystyle q\in \mathbb {R} ^{3}} 2445:equation can certainly be divided by 1125:Consider the following LHS and RHS. 7: 7566:{\displaystyle {\frac {f(x)}{g(x)}}} 6862:δ+δ. The first-order change is the 2 3474:{\displaystyle \left]0,1\right[^{3}} 666:{\displaystyle x={\tfrac {\pi }{2}}} 113:in the format of 6,834 days ... and 6846:example. If we add a quantity δ to 4531:{\displaystyle 21x={1 \over x^{3}}} 4040:{\displaystyle 6x-{2 \over 7x^{3}}} 7430:does not have a minimum value for 5691: 2661: 2393: 32: 7392:. Same answer, different method. 3134:{\displaystyle |x_{ps}|=|x_{qr}|} 3022:{\displaystyle |x_{pq}|=|x_{sr}|} 758:{\displaystyle n(0.87\ln n-1.26)} 710:{\displaystyle n(0.87\ln n-0.66)} 534:) code to calculate sine using a 348:why we have asked about February 6763:Sorry, you're mistaken, and the 6320:the function is greater than at 3503:{\displaystyle \mathbb {R} ^{3}} 2898:{\displaystyle \mathbb {Z} ^{3}} 7502:subject to the constraint that 7495:{\displaystyle 3z+{1 \over 7z}} 5658:The ratio between the roots is 5130:OK. The real solutions must be 2648:{\displaystyle R_{pq}=R_{qr}\,} 464:Thanks to all for the input. ( 7667: 7661: 7618: 7612: 7589: 7583: 7557: 7551: 7543: 7537: 7069: 6727: 6721: 6712: 6700: 6686: 6672: 6666: 6647:, is zero or negative. But if 6628: 6622: 6613: 6601: 6580: 6542: 6536: 6527: 6515: 6483: 6477: 6468: 6456: 6435: 6396: 6390: 6381: 6369: 6117: 6099: 5863: 5851: 5694: 5679: 5417: 5395: 5383: 5361: 5324: 5296: 5275: 5247: 4835: 4818: 4806: 4789: 4752: 4729: 4708: 4685: 4427: 4398: 4376: 4347: 4313: 4261: 4141: 4119: 3882: 3861: 3834: 3816: 3768: 3750: 3729: 3711: 3685: 3667: 3641: 3623: 3272: 3254: 3222: 3204: 3127: 3109: 3101: 3083: 3015: 2997: 2989: 2971: 2500: 2468: 2350: 2318: 2280: 2252: 2205: 2173: 2135: 2107: 1986: 1958: 1853: 1825: 1638:{\displaystyle i={\sqrt {-1}}} 1604:{\displaystyle x_{pq}=-x_{qp}} 1461: 1429: 1391: 1363: 1268: 1236: 1198: 1170: 1089: 1083: 1074: 1071: 1068: 1065: 1059: 1056: 1053: 1047: 1044: 1041: 1038: 1032: 1029: 1026: 880: 874: 752: 731: 704: 683: 625: 622: 613: 594: 585: 569: 526:Convergence of a Taylor series 477:Template:Age in years and days 266:is a bit better going back to 1: 4572:{\displaystyle 21x^{4}=1\,\!} 3570:by the requirement that both 3180:{\displaystyle R_{ps}=R_{qr}} 3068:{\displaystyle R_{pq}=R_{sr}} 2925:there would be another point 1555:{\displaystyle R_{pq}=R_{qp}} 33: 7450:is real, it is not minimum. 6267:). For example, considering 5876:. Now our minimum occurs at 5700:{\displaystyle (0,+\infty )} 5179:and the imaginary solutions 4382:{\displaystyle {2 \over 7}*} 4319:{\displaystyle {2 \over 7}*} 3888:{\displaystyle s=(-0.5,0,0)} 1726:{\displaystyle R_{pq}R_{qr}} 380:arguably better alternatives 304:Clarification of My Question 7422:= 0. For complex values of 6008:(-1) = -1. (In fact, since 3735:{\displaystyle r=(0.5,0,0)} 3481:has just as many points as 3284:{\displaystyle s=(1,1,1)-q} 3234:{\displaystyle r=(1,1,1)-p} 7703: 7685:18:58, 6 August 2007 (UTC) 7511:18:07, 6 August 2007 (UTC) 7455:12:41, 6 August 2007 (UTC) 7397:09:38, 6 August 2007 (UTC) 6910:06:28, 7 August 2007 (UTC) 6850:, the output changes from 6838:03:34, 7 August 2007 (UTC) 6822:02:05, 7 August 2007 (UTC) 6796:06:29, 6 August 2007 (UTC) 6751:05:40, 6 August 2007 (UTC) 6239:23:36, 5 August 2007 (UTC) 6161:21:37, 5 August 2007 (UTC) 6025:22:09, 5 August 2007 (UTC) 5936:21:57, 5 August 2007 (UTC) 5907:21:51, 5 August 2007 (UTC) 5731:21:35, 5 August 2007 (UTC) 5721:21:30, 5 August 2007 (UTC) 5631:21:17, 5 August 2007 (UTC) 5613:21:02, 5 August 2007 (UTC) 5122:20:51, 5 August 2007 (UTC) 5068:20:42, 5 August 2007 (UTC) 3940:Find the minimum value of 3925:18:13, 7 August 2007 (UTC) 3907:21:10, 6 August 2007 (UTC) 3781:{\displaystyle |x_{pq}|=1} 3429:19:11, 6 August 2007 (UTC) 3391:17:42, 6 August 2007 (UTC) 3323:10:29, 6 August 2007 (UTC) 2797:22:40, 5 August 2007 (UTC) 2766:21:22, 5 August 2007 (UTC) 2745:20:37, 5 August 2007 (UTC) 2707:, which, I think, implies 2061:19:45, 5 August 2007 (UTC) 1763:19:10, 5 August 2007 (UTC) 1685:18:13, 5 August 2007 (UTC) 1113:22:07, 5 August 2007 (UTC) 1011:18:08, 5 August 2007 (UTC) 845:17:53, 5 August 2007 (UTC) 827:07:53, 6 August 2007 (UTC) 806:07:26, 6 August 2007 (UTC) 788:22:04, 5 August 2007 (UTC) 775:14:29, 5 August 2007 (UTC) 547:14:02, 5 August 2007 (UTC) 515:17:37, 7 August 2007 (UTC) 494:09:42, 7 August 2007 (UTC) 484:09:37, 7 August 2007 (UTC) 469:22:22, 6 August 2007 (UTC) 456:17:06, 9 August 2007 (UTC) 443:22:10, 5 August 2007 (UTC) 396:21:19, 5 August 2007 (UTC) 341:20:42, 5 August 2007 (UTC) 295:19:32, 8 August 2007 (UTC) 283:15:04, 5 August 2007 (UTC) 258:14:43, 5 August 2007 (UTC) 236:12:44, 5 August 2007 (UTC) 217:07:14, 5 August 2007 (UTC) 181:05:31, 5 August 2007 (UTC) 167:05:12, 5 August 2007 (UTC) 149:05:31, 5 August 2007 (UTC) 135:04:10, 5 August 2007 (UTC) 123:03:21, 5 August 2007 (UTC) 7257:takes a minimum value of 3691:{\displaystyle q=(1,0,0)} 3647:{\displaystyle p=(0,0,0)} 854:Kahan summation algorithm 7418:has a minimum value for 6297:definition of derivative 6206:, the same is true for − 6175:can produce a change in 4673:Subbing this in, we get 3895:, would do the trick. — 3841:{\displaystyle |x_{sr}|} 794:Stirling's approximation 201:B3="=IF((B2*100+C2): --> 18:Knowledge:Reference desk 7426:, the complex function 6348:) is positive whenever 6234:the slope increases. -- 4440:must be equal to zero. 2956:{\displaystyle s=r+p-q} 2905:), then for each point 1665:{\displaystyle k\geq 0} 7674: 7645: 7625: 7596: 7567: 7496: 7386: 7344: 7289: 7251: 7190: 7055: 6775:produces no change in 6740: 6641: 6555: 6496: 6409: 6150: 6078: 6077:{\displaystyle 1/i=-i} 5896: 5870: 5838: 5818: 5792: 5772: 5701: 5599: 5553: 5487: 5431: 5344: 5227: 5173: 5111: 5060:is the minimum value. 5054: 5010: 4967: 4902: 4849: 4772: 4665: 4618: 4573: 4532: 4488: 4434: 4383: 4320: 4234: 4166: 4099: 4041: 3989: 3889: 3842: 3802: 3782: 3736: 3692: 3648: 3604: 3584: 3564: 3563:{\displaystyle x_{pq}} 3534: 3533:{\displaystyle R_{pq}} 3504: 3475: 3347:are free variables in 3305: 3285: 3235: 3181: 3135: 3069: 3023: 2957: 2919: 2899: 2870: 2835: 2727: 2701: 2649: 2601: 2539: 2433: 2089: 2035: 1954: 1902: 1821: 1748: 1727: 1666: 1639: 1605: 1556: 1506: 1359: 1313: 1166: 1102: 981: 759: 711: 667: 632: 431:Epoch (reference date) 332:anyone help? Thanks. 87:current reference desk 7675: 7646: 7626: 7597: 7568: 7497: 7387: 7345: 7290: 7252: 7191: 7056: 6870:, the output becomes 6741: 6642: 6561:is negative whenever 6556: 6497: 6415:is positive whenever 6410: 6328:.) Thus we have that 6151: 6079: 5897: 5871: 5839: 5819: 5793: 5773: 5771:{\displaystyle x^{2}} 5702: 5652:which are 2 modulo 4, 5600: 5554: 5488: 5432: 5345: 5228: 5174: 5112: 5055: 5011: 4968: 4903: 4850: 4773: 4666: 4619: 4574: 4533: 4489: 4435: 4384: 4321: 4235: 4167: 4100: 4042: 3990: 3890: 3843: 3803: 3783: 3737: 3693: 3649: 3605: 3585: 3565: 3535: 3505: 3476: 3306: 3286: 3236: 3182: 3136: 3070: 3024: 2958: 2920: 2900: 2871: 2836: 2728: 2702: 2650: 2602: 2540: 2434: 2090: 2036: 1934: 1903: 1801: 1749: 1728: 1667: 1640: 1606: 1557: 1507: 1339: 1314: 1146: 1103: 982: 760: 712: 668: 633: 530:I'm implementing (in 7673:{\displaystyle f(x)} 7655: 7635: 7624:{\displaystyle g(x)} 7606: 7595:{\displaystyle f(x)} 7577: 7528: 7465: 7414:, the real function 7354: 7299: 7261: 7206: 7066: 6934: 6655: 6569: 6506: 6424: 6360: 6214:); yet a minimizing 6088: 6051: 5968:, where we restrict 5964:Not quite: consider 5880: 5848: 5828: 5802: 5782: 5755: 5676: 5564: 5498: 5442: 5355: 5241: 5183: 5134: 5077: 5022: 4978: 4913: 4860: 4783: 4679: 4629: 4584: 4543: 4499: 4449: 4395: 4331: 4245: 4179: 4172:(eqn fixed 41 -: --> 4110: 4052: 4003: 3944: 3852: 3812: 3792: 3746: 3702: 3658: 3614: 3594: 3574: 3544: 3514: 3485: 3442: 3295: 3245: 3195: 3145: 3079: 3033: 2967: 2929: 2909: 2880: 2845: 2825: 2711: 2658: 2611: 2549: 2449: 2101: 2079: 1919: 1786: 1738: 1694: 1650: 1616: 1566: 1520: 1324: 1131: 1023: 865: 725: 677: 642: 560: 6287:yields a change in 6283:= 0, any change in 6016:< 0, this point 5996:, which is zero at 5895:{\displaystyle x=1} 5817:{\displaystyle x=0} 2726:{\displaystyle p=r} 2387: 2229: 2028: 1895: 1498: 1292: 7670: 7641: 7631:to find values of 7621: 7592: 7563: 7492: 7382: 7340: 7285: 7247: 7186: 7051: 6842:Let's look at the 6736: 6693: 6637: 6594: 6551: 6492: 6449: 6405: 6146: 6074: 5892: 5866: 5834: 5814: 5788: 5768: 5697: 5595: 5594: 5592: 5549: 5548: 5546: 5483: 5482: 5480: 5427: 5426: 5424: 5340: 5339: 5337: 5223: 5222: 5220: 5169: 5168: 5166: 5107: 5106: 5104: 5050: 5049: 5047: 5006: 5005: 5003: 4963: 4962: 4960: 4898: 4897: 4895: 4845: 4844: 4842: 4768: 4767: 4765: 4661: 4660: 4658: 4614: 4613: 4611: 4569: 4568: 4566: 4528: 4484: 4430: 4379: 4316: 4230: 4162: 4095: 4037: 3985: 3885: 3838: 3798: 3778: 3732: 3688: 3644: 3600: 3580: 3560: 3530: 3500: 3471: 3351:. Defining matrix 3301: 3281: 3231: 3177: 3131: 3065: 3019: 2953: 2915: 2895: 2866: 2831: 2723: 2697: 2645: 2644: 2597: 2535: 2429: 2370: 2212: 2085: 2031: 2011: 1898: 1878: 1744: 1723: 1662: 1635: 1601: 1552: 1502: 1481: 1309: 1275: 1098: 977: 956: 910: 755: 707: 663: 661: 628: 417:Alternatively get 7644:{\displaystyle x} 7561: 7490: 7380: 7379: 7332: 7326: 7307: 7283: 7282: 7275: 7245: 7184: 7183: 7176: 7149: 7143: 7124: 7108: 7049: 7024: 7023: 7016: 6973: 6967: 6948: 6816: 6734: 6678: 6635: 6572: 6549: 6490: 6427: 6403: 6355: 6228:second derivative 6144: 6115: 5837:{\displaystyle x} 5791:{\displaystyle x} 5589: 5581: 5543: 5537: 5521: 5515: 5477: 5472: 5460: 5458: 5421: 5415: 5412: 5381: 5378: 5334: 5321: 5318: 5272: 5269: 5217: 5214: 5163: 5161: 5101: 5044: 5036: 5000: 4992: 4957: 4951: 4935: 4927: 4892: 4887: 4875: 4873: 4839: 4833: 4831: 4804: 4802: 4762: 4749: 4747: 4705: 4703: 4655: 4653: 4608: 4526: 4476: 4425: 4374: 4342: 4311: 4291: 4256: 4228: 4190: 4160: 4093: 4035: 3983: 3933:Problem Solving 3 3905: 3801:{\displaystyle s} 3788:, but there's no 3603:{\displaystyle q} 3583:{\displaystyle p} 3321: 3304:{\displaystyle q} 2918:{\displaystyle q} 2834:{\displaystyle q} 2743: 2533: 2392: 2389: 2244: 2088:{\displaystyle q} 2051:Is that correct? 2029: 1896: 1747:{\displaystyle k} 1633: 1500: 1307: 1121:Prove or disprove 955: 909: 660: 512:Joseph A. Spadaro 466:Joseph A. Spadaro 453:Joseph A. Spadaro 338:Joseph A. Spadaro 202:(B1*100+C1),1,0)" 178:Joseph A. Spadaro 146:Joseph A. Spadaro 120:Joseph A. Spadaro 93: 92: 73: 72: 7694: 7679: 7677: 7676: 7671: 7650: 7648: 7647: 7642: 7630: 7628: 7627: 7622: 7601: 7599: 7598: 7593: 7572: 7570: 7569: 7564: 7562: 7560: 7546: 7532: 7501: 7499: 7498: 7493: 7491: 7489: 7478: 7391: 7389: 7388: 7383: 7381: 7375: 7371: 7366: 7365: 7349: 7347: 7346: 7341: 7333: 7331: 7327: 7322: 7316: 7308: 7303: 7294: 7292: 7291: 7286: 7284: 7278: 7277: 7276: 7271: 7265: 7256: 7254: 7253: 7248: 7246: 7244: 7243: 7242: 7226: 7221: 7220: 7195: 7193: 7192: 7187: 7185: 7179: 7178: 7177: 7172: 7166: 7161: 7160: 7155: 7151: 7150: 7148: 7144: 7139: 7133: 7125: 7120: 7109: 7107: 7106: 7105: 7089: 7084: 7083: 7060: 7058: 7057: 7052: 7050: 7048: 7047: 7046: 7030: 7025: 7019: 7018: 7017: 7012: 7006: 7001: 7000: 6985: 6984: 6979: 6975: 6974: 6972: 6968: 6963: 6957: 6949: 6944: 6895:Abraham Robinson 6814: 6745: 6743: 6742: 6737: 6735: 6730: 6695: 6692: 6665: 6646: 6644: 6643: 6638: 6636: 6631: 6596: 6593: 6592: 6591: 6560: 6558: 6557: 6552: 6550: 6545: 6510: 6501: 6499: 6498: 6493: 6491: 6486: 6451: 6448: 6447: 6446: 6414: 6412: 6411: 6406: 6404: 6399: 6364: 6353: 6202:) has a zero at 6155: 6153: 6152: 6147: 6145: 6143: 6135: 6133: 6116: 6114: 6106: 6098: 6083: 6081: 6080: 6075: 6061: 5901: 5899: 5898: 5893: 5875: 5873: 5872: 5869:{\displaystyle } 5867: 5843: 5841: 5840: 5835: 5823: 5821: 5820: 5815: 5797: 5795: 5794: 5789: 5777: 5775: 5774: 5769: 5767: 5766: 5706: 5704: 5703: 5698: 5604: 5602: 5601: 5596: 5590: 5588: 5583: 5582: 5577: 5571: 5558: 5556: 5555: 5550: 5544: 5539: 5538: 5533: 5527: 5522: 5517: 5516: 5511: 5505: 5492: 5490: 5489: 5484: 5478: 5473: 5468: 5466: 5461: 5459: 5454: 5449: 5436: 5434: 5433: 5428: 5422: 5420: 5416: 5414: 5413: 5408: 5399: 5390: 5382: 5380: 5379: 5374: 5365: 5349: 5347: 5346: 5341: 5335: 5333: 5332: 5331: 5322: 5320: 5319: 5317: 5309: 5300: 5288: 5283: 5282: 5273: 5271: 5270: 5268: 5260: 5251: 5232: 5230: 5229: 5224: 5218: 5216: 5215: 5213: 5205: 5196: 5178: 5176: 5175: 5170: 5164: 5162: 5160: 5152: 5147: 5116: 5114: 5113: 5108: 5102: 5094: 5089: 5088: 5059: 5057: 5056: 5051: 5045: 5043: 5038: 5037: 5032: 5026: 5015: 5013: 5012: 5007: 5001: 4999: 4994: 4993: 4988: 4982: 4972: 4970: 4969: 4964: 4958: 4953: 4952: 4947: 4941: 4936: 4934: 4929: 4928: 4923: 4917: 4907: 4905: 4904: 4899: 4893: 4888: 4883: 4881: 4876: 4874: 4869: 4864: 4854: 4852: 4851: 4846: 4840: 4838: 4834: 4832: 4827: 4822: 4813: 4805: 4803: 4798: 4793: 4777: 4775: 4774: 4769: 4763: 4761: 4760: 4759: 4750: 4748: 4746: 4738: 4733: 4721: 4716: 4715: 4706: 4704: 4702: 4694: 4689: 4670: 4668: 4667: 4662: 4656: 4654: 4652: 4644: 4639: 4623: 4621: 4620: 4615: 4609: 4601: 4596: 4595: 4578: 4576: 4575: 4570: 4558: 4557: 4537: 4535: 4534: 4529: 4527: 4525: 4524: 4512: 4493: 4491: 4490: 4485: 4477: 4475: 4474: 4462: 4439: 4437: 4436: 4433:{\displaystyle } 4431: 4426: 4424: 4423: 4411: 4388: 4386: 4385: 4380: 4375: 4373: 4372: 4360: 4343: 4335: 4325: 4323: 4322: 4317: 4312: 4310: 4309: 4297: 4292: 4290: 4289: 4280: 4279: 4278: 4265: 4257: 4249: 4239: 4237: 4236: 4231: 4229: 4227: 4226: 4217: 4210: 4209: 4196: 4191: 4183: 4171: 4169: 4168: 4163: 4161: 4159: 4158: 4157: 4144: 4134: 4133: 4114: 4104: 4102: 4101: 4096: 4094: 4092: 4091: 4090: 4077: 4070: 4069: 4056: 4046: 4044: 4043: 4038: 4036: 4034: 4033: 4032: 4016: 3994: 3992: 3991: 3986: 3984: 3982: 3981: 3980: 3964: 3959: 3958: 3899: 3894: 3892: 3891: 3886: 3847: 3845: 3844: 3839: 3837: 3832: 3831: 3819: 3807: 3805: 3804: 3799: 3787: 3785: 3784: 3779: 3771: 3766: 3765: 3753: 3741: 3739: 3738: 3733: 3697: 3695: 3694: 3689: 3653: 3651: 3650: 3645: 3609: 3607: 3606: 3601: 3589: 3587: 3586: 3581: 3569: 3567: 3566: 3561: 3559: 3558: 3539: 3537: 3536: 3531: 3529: 3528: 3509: 3507: 3506: 3501: 3499: 3498: 3493: 3480: 3478: 3477: 3472: 3470: 3469: 3464: 3460: 3315: 3310: 3308: 3307: 3302: 3290: 3288: 3287: 3282: 3240: 3238: 3237: 3232: 3186: 3184: 3183: 3178: 3176: 3175: 3160: 3159: 3140: 3138: 3137: 3132: 3130: 3125: 3124: 3112: 3104: 3099: 3098: 3086: 3074: 3072: 3071: 3066: 3064: 3063: 3048: 3047: 3028: 3026: 3025: 3020: 3018: 3013: 3012: 3000: 2992: 2987: 2986: 2974: 2962: 2960: 2959: 2954: 2924: 2922: 2921: 2916: 2904: 2902: 2901: 2896: 2894: 2893: 2888: 2875: 2873: 2872: 2867: 2865: 2864: 2859: 2840: 2838: 2837: 2832: 2737: 2732: 2730: 2729: 2724: 2706: 2704: 2703: 2698: 2654: 2652: 2651: 2646: 2642: 2641: 2626: 2625: 2606: 2604: 2603: 2598: 2596: 2592: 2591: 2572: 2568: 2567: 2544: 2542: 2541: 2536: 2534: 2532: 2531: 2530: 2518: 2517: 2504: 2503: 2499: 2498: 2483: 2482: 2453: 2438: 2436: 2435: 2430: 2391: 2390: 2388: 2386: 2381: 2369: 2368: 2355: 2354: 2353: 2349: 2348: 2333: 2332: 2303: 2299: 2298: 2273: 2272: 2250: 2245: 2243: 2242: 2241: 2228: 2223: 2210: 2209: 2208: 2204: 2203: 2188: 2187: 2158: 2154: 2153: 2128: 2127: 2105: 2094: 2092: 2091: 2086: 2040: 2038: 2037: 2032: 2030: 2027: 2022: 2010: 2009: 2005: 2004: 1979: 1978: 1956: 1953: 1948: 1907: 1905: 1904: 1899: 1897: 1894: 1889: 1877: 1876: 1872: 1871: 1846: 1845: 1823: 1820: 1815: 1753: 1751: 1750: 1745: 1734:about. ;-) That 1732: 1730: 1729: 1724: 1722: 1721: 1709: 1708: 1671: 1669: 1668: 1663: 1644: 1642: 1641: 1636: 1634: 1626: 1610: 1608: 1607: 1602: 1600: 1599: 1581: 1580: 1561: 1559: 1558: 1553: 1551: 1550: 1535: 1534: 1511: 1509: 1508: 1503: 1501: 1499: 1497: 1492: 1480: 1479: 1466: 1465: 1464: 1460: 1459: 1444: 1443: 1414: 1410: 1409: 1384: 1383: 1361: 1358: 1353: 1318: 1316: 1315: 1310: 1308: 1306: 1305: 1304: 1291: 1286: 1273: 1272: 1271: 1267: 1266: 1251: 1250: 1221: 1217: 1216: 1191: 1190: 1168: 1165: 1160: 1107: 1105: 1104: 1099: 986: 984: 983: 978: 973: 969: 962: 958: 957: 948: 934: 933: 916: 912: 911: 902: 764: 762: 761: 756: 716: 714: 713: 708: 672: 670: 669: 664: 662: 653: 637: 635: 634: 629: 609: 608: 581: 580: 246:Analysis ToolPak 75: 38:Mathematics desk 34: 7702: 7701: 7697: 7696: 7695: 7693: 7692: 7691: 7653: 7652: 7633: 7632: 7604: 7603: 7575: 7574: 7547: 7533: 7526: 7525: 7482: 7463: 7462: 7434:= 0. Note that 7357: 7352: 7351: 7320: 7297: 7296: 7266: 7259: 7258: 7234: 7230: 7212: 7204: 7203: 7167: 7137: 7118: 7114: 7113: 7097: 7093: 7075: 7064: 7063: 7038: 7034: 7007: 6992: 6961: 6942: 6938: 6937: 6932: 6931: 6696: 6658: 6653: 6652: 6597: 6583: 6567: 6566: 6511: 6504: 6503: 6452: 6438: 6422: 6421: 6365: 6358: 6357: 6086: 6085: 6049: 6048: 6012:< 0 implies 5933:172.188.191.126 5878: 5877: 5846: 5845: 5826: 5825: 5800: 5799: 5780: 5779: 5758: 5753: 5752: 5728:172.188.191.126 5674: 5673: 5610:172.188.191.126 5572: 5562: 5561: 5528: 5506: 5496: 5495: 5440: 5439: 5403: 5394: 5369: 5353: 5352: 5323: 5304: 5292: 5274: 5255: 5239: 5238: 5200: 5181: 5180: 5132: 5131: 5080: 5075: 5074: 5065:172.188.191.126 5027: 5020: 5019: 4983: 4976: 4975: 4942: 4918: 4911: 4910: 4858: 4857: 4817: 4781: 4780: 4751: 4725: 4707: 4677: 4676: 4627: 4626: 4587: 4582: 4581: 4549: 4541: 4540: 4516: 4497: 4496: 4466: 4447: 4446: 4415: 4393: 4392: 4364: 4329: 4328: 4301: 4281: 4270: 4266: 4243: 4242: 4218: 4201: 4197: 4177: 4176: 4173:21 2007-08-06) 4149: 4145: 4125: 4115: 4108: 4107: 4082: 4078: 4061: 4057: 4050: 4049: 4024: 4020: 4001: 4000: 3972: 3968: 3950: 3942: 3941: 3935: 3850: 3849: 3820: 3810: 3809: 3790: 3789: 3754: 3744: 3743: 3700: 3699: 3656: 3655: 3612: 3611: 3592: 3591: 3572: 3571: 3547: 3542: 3541: 3517: 3512: 3511: 3488: 3483: 3482: 3450: 3446: 3445: 3440: 3439: 3411: 3360: 3293: 3292: 3243: 3242: 3193: 3192: 3164: 3148: 3143: 3142: 3113: 3087: 3077: 3076: 3052: 3036: 3031: 3030: 3001: 2975: 2965: 2964: 2927: 2926: 2907: 2906: 2883: 2878: 2877: 2854: 2843: 2842: 2823: 2822: 2709: 2708: 2656: 2655: 2630: 2614: 2609: 2608: 2580: 2576: 2556: 2552: 2547: 2546: 2519: 2506: 2505: 2487: 2471: 2454: 2447: 2446: 2357: 2356: 2337: 2321: 2304: 2287: 2283: 2261: 2251: 2230: 2211: 2192: 2176: 2159: 2142: 2138: 2116: 2106: 2099: 2098: 2077: 2076: 1993: 1989: 1967: 1957: 1917: 1916: 1860: 1856: 1834: 1824: 1784: 1783: 1736: 1735: 1710: 1697: 1692: 1691: 1648: 1647: 1614: 1613: 1588: 1569: 1564: 1563: 1539: 1523: 1518: 1517: 1468: 1467: 1448: 1432: 1415: 1398: 1394: 1372: 1362: 1322: 1321: 1293: 1274: 1255: 1239: 1222: 1205: 1201: 1179: 1169: 1129: 1128: 1123: 1021: 1020: 942: 938: 925: 921: 917: 896: 892: 863: 862: 723: 722: 675: 674: 640: 639: 600: 572: 558: 557: 540:Finlay McWalter 528: 306: 106: 101: 30: 29: 28: 12: 11: 5: 7700: 7698: 7690: 7689: 7688: 7687: 7669: 7666: 7663: 7660: 7640: 7620: 7617: 7614: 7611: 7591: 7588: 7585: 7582: 7559: 7556: 7553: 7550: 7545: 7542: 7539: 7536: 7519: 7518: 7517: 7516: 7488: 7485: 7481: 7476: 7473: 7470: 7400: 7399: 7378: 7374: 7369: 7364: 7360: 7339: 7336: 7330: 7325: 7319: 7314: 7311: 7306: 7281: 7274: 7269: 7241: 7237: 7233: 7229: 7224: 7219: 7215: 7211: 7199: 7198: 7197: 7196: 7182: 7175: 7170: 7164: 7159: 7154: 7147: 7142: 7136: 7131: 7128: 7123: 7117: 7112: 7104: 7100: 7096: 7092: 7087: 7082: 7078: 7074: 7071: 7061: 7045: 7041: 7037: 7033: 7028: 7022: 7015: 7010: 7004: 6999: 6995: 6991: 6988: 6983: 6978: 6971: 6966: 6960: 6955: 6952: 6947: 6941: 6926: 6925: 6921: 6920: 6919: 6918: 6917: 6916: 6915: 6914: 6913: 6912: 6898: 6891: 6840: 6803: 6802: 6801: 6800: 6799: 6798: 6756: 6755: 6754: 6753: 6733: 6729: 6726: 6723: 6720: 6717: 6714: 6711: 6708: 6705: 6702: 6699: 6691: 6688: 6685: 6681: 6677: 6674: 6671: 6668: 6664: 6661: 6634: 6630: 6627: 6624: 6621: 6618: 6615: 6612: 6609: 6606: 6603: 6600: 6590: 6586: 6582: 6579: 6575: 6548: 6544: 6541: 6538: 6535: 6532: 6529: 6526: 6523: 6520: 6517: 6514: 6489: 6485: 6482: 6479: 6476: 6473: 6470: 6467: 6464: 6461: 6458: 6455: 6445: 6441: 6437: 6434: 6430: 6402: 6398: 6395: 6392: 6389: 6386: 6383: 6380: 6377: 6374: 6371: 6368: 6303: 6302: 6301: 6300: 6251:can produce a 6242: 6241: 6192: 6168: 6167: 6166: 6165: 6164: 6163: 6142: 6138: 6132: 6128: 6125: 6122: 6119: 6113: 6109: 6104: 6101: 6097: 6093: 6073: 6070: 6067: 6064: 6060: 6056: 6047:Remember that 6040: 6039: 6038: 6037: 6036: 6035: 6034: 6033: 6032: 6031: 6030: 6029: 6028: 6027: 6004:(0) = 0 while 5949: 5948: 5947: 5946: 5945: 5944: 5943: 5942: 5941: 5940: 5939: 5938: 5918: 5917: 5916: 5915: 5914: 5913: 5912: 5911: 5910: 5909: 5891: 5888: 5885: 5865: 5862: 5859: 5856: 5853: 5833: 5813: 5810: 5807: 5787: 5765: 5761: 5740: 5739: 5738: 5737: 5736: 5735: 5734: 5733: 5713:Meni Rosenfeld 5696: 5693: 5690: 5687: 5684: 5681: 5665: 5664: 5663: 5656: 5653: 5638: 5637: 5636: 5635: 5634: 5633: 5618: 5617: 5616: 5615: 5605: 5587: 5580: 5575: 5569: 5559: 5542: 5536: 5531: 5525: 5520: 5514: 5509: 5503: 5493: 5476: 5471: 5464: 5457: 5452: 5447: 5437: 5419: 5411: 5406: 5402: 5397: 5393: 5388: 5385: 5377: 5372: 5368: 5363: 5360: 5350: 5330: 5326: 5316: 5312: 5307: 5303: 5298: 5295: 5291: 5286: 5281: 5277: 5267: 5263: 5258: 5254: 5249: 5246: 5236: 5233: 5212: 5208: 5203: 5199: 5194: 5191: 5188: 5159: 5155: 5150: 5145: 5142: 5139: 5125: 5124: 5100: 5097: 5092: 5087: 5083: 5042: 5035: 5030: 4998: 4991: 4986: 4956: 4950: 4945: 4939: 4933: 4926: 4921: 4891: 4886: 4879: 4872: 4867: 4837: 4830: 4825: 4820: 4816: 4811: 4808: 4801: 4796: 4791: 4788: 4758: 4754: 4745: 4741: 4736: 4731: 4728: 4724: 4719: 4714: 4710: 4701: 4697: 4692: 4687: 4684: 4651: 4647: 4642: 4637: 4634: 4607: 4604: 4599: 4594: 4590: 4564: 4561: 4556: 4552: 4548: 4523: 4519: 4515: 4510: 4507: 4504: 4483: 4480: 4473: 4469: 4465: 4460: 4457: 4454: 4429: 4422: 4418: 4414: 4409: 4406: 4403: 4400: 4378: 4371: 4367: 4363: 4358: 4355: 4352: 4349: 4346: 4341: 4338: 4315: 4308: 4304: 4300: 4295: 4288: 4284: 4277: 4273: 4269: 4263: 4260: 4255: 4252: 4225: 4221: 4216: 4213: 4208: 4204: 4200: 4194: 4189: 4186: 4156: 4152: 4148: 4143: 4140: 4137: 4132: 4128: 4124: 4121: 4118: 4089: 4085: 4081: 4076: 4073: 4068: 4064: 4060: 4031: 4027: 4023: 4019: 4014: 4011: 4008: 3979: 3975: 3971: 3967: 3962: 3957: 3953: 3949: 3934: 3931: 3930: 3929: 3928: 3927: 3910: 3909: 3897:Ilmari Karonen 3884: 3881: 3878: 3875: 3872: 3869: 3866: 3863: 3860: 3857: 3836: 3830: 3827: 3823: 3818: 3797: 3777: 3774: 3770: 3764: 3761: 3757: 3752: 3731: 3728: 3725: 3722: 3719: 3716: 3713: 3710: 3707: 3687: 3684: 3681: 3678: 3675: 3672: 3669: 3666: 3663: 3643: 3640: 3637: 3634: 3631: 3628: 3625: 3622: 3619: 3599: 3579: 3557: 3554: 3550: 3527: 3524: 3520: 3497: 3492: 3468: 3463: 3459: 3456: 3453: 3449: 3415:Ilmari Karonen 3408: 3407: 3406: 3405: 3404: 3394: 3393: 3366:, equality of 3358: 3336: 3335: 3334: 3333: 3332: 3331: 3330: 3329: 3328: 3327: 3326: 3325: 3313:Ilmari Karonen 3300: 3280: 3277: 3274: 3271: 3268: 3265: 3262: 3259: 3256: 3253: 3250: 3230: 3227: 3224: 3221: 3218: 3215: 3212: 3209: 3206: 3203: 3200: 3174: 3171: 3167: 3163: 3158: 3155: 3151: 3129: 3123: 3120: 3116: 3111: 3107: 3103: 3097: 3094: 3090: 3085: 3062: 3059: 3055: 3051: 3046: 3043: 3039: 3017: 3011: 3008: 3004: 2999: 2995: 2991: 2985: 2982: 2978: 2973: 2952: 2949: 2946: 2943: 2940: 2937: 2934: 2914: 2892: 2887: 2863: 2858: 2853: 2850: 2830: 2808: 2807: 2806: 2805: 2804: 2803: 2802: 2801: 2800: 2799: 2775: 2774: 2773: 2772: 2771: 2770: 2769: 2768: 2752: 2751: 2750: 2749: 2748: 2747: 2735:Ilmari Karonen 2722: 2719: 2716: 2696: 2693: 2690: 2687: 2684: 2681: 2678: 2675: 2672: 2669: 2666: 2663: 2640: 2637: 2633: 2629: 2624: 2621: 2617: 2595: 2590: 2587: 2583: 2579: 2575: 2571: 2566: 2563: 2559: 2555: 2529: 2526: 2522: 2516: 2513: 2509: 2502: 2497: 2494: 2490: 2486: 2481: 2478: 2474: 2470: 2467: 2464: 2461: 2457: 2439: 2428: 2425: 2422: 2419: 2416: 2413: 2410: 2407: 2404: 2401: 2398: 2395: 2385: 2380: 2377: 2373: 2367: 2364: 2360: 2352: 2347: 2344: 2340: 2336: 2331: 2328: 2324: 2320: 2317: 2314: 2311: 2307: 2302: 2297: 2294: 2290: 2286: 2282: 2279: 2276: 2271: 2268: 2264: 2260: 2257: 2254: 2248: 2240: 2237: 2233: 2227: 2222: 2219: 2215: 2207: 2202: 2199: 2195: 2191: 2186: 2183: 2179: 2175: 2172: 2169: 2166: 2162: 2157: 2152: 2149: 2145: 2141: 2137: 2134: 2131: 2126: 2123: 2119: 2115: 2112: 2109: 2096: 2084: 2068: 2067: 2066: 2065: 2064: 2063: 2044: 2043: 2042: 2041: 2026: 2021: 2018: 2014: 2008: 2003: 2000: 1996: 1992: 1988: 1985: 1982: 1977: 1974: 1970: 1966: 1963: 1960: 1952: 1947: 1944: 1941: 1937: 1933: 1930: 1927: 1924: 1911: 1910: 1909: 1908: 1893: 1888: 1885: 1881: 1875: 1870: 1867: 1863: 1859: 1855: 1852: 1849: 1844: 1841: 1837: 1833: 1830: 1827: 1819: 1814: 1811: 1808: 1804: 1800: 1797: 1794: 1791: 1778: 1777: 1776: 1775: 1768: 1766: 1765: 1743: 1720: 1717: 1713: 1707: 1704: 1700: 1661: 1658: 1655: 1632: 1629: 1624: 1621: 1598: 1595: 1591: 1587: 1584: 1579: 1576: 1572: 1562: 1549: 1546: 1542: 1538: 1533: 1530: 1526: 1496: 1491: 1488: 1484: 1478: 1475: 1471: 1463: 1458: 1455: 1451: 1447: 1442: 1439: 1435: 1431: 1428: 1425: 1422: 1418: 1413: 1408: 1405: 1401: 1397: 1393: 1390: 1387: 1382: 1379: 1375: 1371: 1368: 1365: 1357: 1352: 1349: 1346: 1342: 1338: 1335: 1332: 1329: 1303: 1300: 1296: 1290: 1285: 1282: 1278: 1270: 1265: 1262: 1258: 1254: 1249: 1246: 1242: 1238: 1235: 1232: 1229: 1225: 1220: 1215: 1212: 1208: 1204: 1200: 1197: 1194: 1189: 1186: 1182: 1178: 1175: 1172: 1164: 1159: 1156: 1153: 1149: 1145: 1142: 1139: 1136: 1122: 1119: 1118: 1117: 1116: 1115: 1097: 1094: 1091: 1088: 1085: 1082: 1079: 1076: 1073: 1070: 1067: 1064: 1061: 1058: 1055: 1052: 1049: 1046: 1043: 1040: 1037: 1034: 1031: 1028: 1014: 1013: 996: 989: 988: 987: 976: 972: 968: 965: 961: 954: 951: 945: 941: 937: 932: 928: 924: 920: 915: 908: 905: 899: 895: 891: 888: 885: 882: 879: 876: 873: 870: 857: 848: 847: 836: 835: 834: 833: 832: 831: 830: 829: 813: 812: 811: 810: 809: 808: 798:Meni Rosenfeld 778: 777: 767:Meni Rosenfeld 754: 751: 748: 745: 742: 739: 736: 733: 730: 706: 703: 700: 697: 694: 691: 688: 685: 682: 659: 656: 650: 647: 627: 624: 621: 618: 615: 612: 607: 603: 599: 596: 593: 590: 587: 584: 579: 575: 571: 568: 565: 527: 524: 523: 522: 521: 520: 519: 518: 499: 498: 497: 496: 462: 461: 460: 459: 446: 445: 422: 415: 411: 399: 398: 388:Meni Rosenfeld 383: 376: 373: 369: 362: 305: 302: 300: 298: 297: 286: 285: 260: 241: 240: 239: 238: 228:Meni Rosenfeld 220: 219: 209: 206: 205:A3="=A1-A2-B3" 203: 199: 196: 192: 191: 187: 186: 185: 184: 170: 169: 155: 154: 153: 152: 138: 137: 105: 102: 100: 97: 95: 91: 90: 82: 81: 71: 70: 64: 48: 41: 40: 31: 15: 14: 13: 10: 9: 6: 4: 3: 2: 7699: 7686: 7683: 7664: 7658: 7638: 7615: 7609: 7586: 7580: 7554: 7548: 7540: 7534: 7523: 7522: 7521: 7520: 7514: 7513: 7512: 7509: 7505: 7486: 7483: 7479: 7474: 7471: 7468: 7460: 7459: 7458: 7456: 7453: 7449: 7445: 7441: 7437: 7433: 7429: 7425: 7421: 7417: 7413: 7409: 7405: 7398: 7395: 7376: 7372: 7367: 7362: 7358: 7337: 7334: 7328: 7323: 7317: 7312: 7309: 7304: 7279: 7272: 7267: 7239: 7235: 7231: 7227: 7222: 7217: 7213: 7209: 7201: 7200: 7180: 7173: 7168: 7162: 7157: 7152: 7145: 7140: 7134: 7129: 7126: 7121: 7115: 7110: 7102: 7098: 7094: 7090: 7085: 7080: 7076: 7072: 7062: 7043: 7039: 7035: 7031: 7026: 7020: 7013: 7008: 7002: 6997: 6993: 6989: 6986: 6981: 6976: 6969: 6964: 6958: 6953: 6950: 6945: 6939: 6930: 6929: 6928: 6927: 6923: 6922: 6911: 6908: 6904: 6899: 6896: 6892: 6889: 6885: 6881: 6877: 6873: 6869: 6865: 6861: 6857: 6853: 6849: 6845: 6841: 6839: 6836: 6832: 6828: 6825: 6824: 6823: 6820: 6811: 6810: 6809: 6808: 6807: 6806: 6805: 6804: 6797: 6794: 6790: 6786: 6782: 6778: 6774: 6770: 6766: 6762: 6761: 6760: 6759: 6758: 6757: 6752: 6749: 6731: 6724: 6718: 6715: 6709: 6706: 6703: 6697: 6689: 6683: 6675: 6669: 6662: 6659: 6650: 6632: 6625: 6619: 6616: 6610: 6607: 6604: 6598: 6588: 6584: 6577: 6564: 6546: 6539: 6533: 6530: 6524: 6521: 6518: 6512: 6487: 6480: 6474: 6471: 6465: 6462: 6459: 6453: 6443: 6439: 6432: 6418: 6400: 6393: 6387: 6384: 6378: 6375: 6372: 6366: 6351: 6347: 6343: 6339: 6335: 6331: 6327: 6323: 6319: 6315: 6311: 6307: 6306: 6305: 6304: 6298: 6294: 6290: 6286: 6282: 6278: 6274: 6270: 6266: 6262: 6258: 6254: 6250: 6246: 6245: 6244: 6243: 6240: 6237: 6233: 6229: 6225: 6221: 6217: 6213: 6209: 6205: 6201: 6197: 6193: 6190: 6186: 6182: 6178: 6174: 6170: 6169: 6162: 6159: 6140: 6136: 6130: 6126: 6123: 6120: 6111: 6107: 6102: 6095: 6091: 6071: 6068: 6065: 6062: 6058: 6054: 6046: 6045: 6044: 6043: 6042: 6041: 6026: 6023: 6019: 6015: 6011: 6007: 6003: 5999: 5995: 5991: 5987: 5983: 5979: 5975: 5971: 5967: 5963: 5962: 5961: 5960: 5959: 5958: 5957: 5956: 5955: 5954: 5953: 5952: 5951: 5950: 5937: 5934: 5930: 5929: 5928: 5927: 5926: 5925: 5924: 5923: 5922: 5921: 5920: 5919: 5908: 5905: 5889: 5886: 5883: 5860: 5857: 5854: 5831: 5811: 5808: 5805: 5785: 5763: 5759: 5750: 5749: 5748: 5747: 5746: 5745: 5744: 5743: 5742: 5741: 5732: 5729: 5724: 5723: 5722: 5718: 5714: 5710: 5688: 5685: 5682: 5670: 5666: 5661: 5657: 5654: 5651: 5647: 5646: 5644: 5643: 5642: 5641: 5640: 5639: 5632: 5629: 5624: 5623: 5622: 5621: 5620: 5619: 5614: 5611: 5606: 5585: 5578: 5573: 5567: 5560: 5540: 5534: 5529: 5523: 5518: 5512: 5507: 5501: 5494: 5474: 5469: 5462: 5455: 5450: 5445: 5438: 5409: 5404: 5400: 5391: 5386: 5375: 5370: 5366: 5358: 5351: 5328: 5314: 5310: 5305: 5301: 5293: 5289: 5284: 5279: 5265: 5261: 5256: 5252: 5244: 5237: 5234: 5210: 5206: 5201: 5197: 5192: 5189: 5186: 5157: 5153: 5148: 5143: 5140: 5137: 5129: 5128: 5127: 5126: 5123: 5120: 5098: 5095: 5090: 5085: 5081: 5072: 5071: 5070: 5069: 5066: 5061: 5040: 5033: 5028: 5016: 4996: 4989: 4984: 4973: 4954: 4948: 4943: 4937: 4931: 4924: 4919: 4908: 4889: 4884: 4877: 4870: 4865: 4855: 4828: 4823: 4814: 4809: 4799: 4794: 4786: 4778: 4756: 4743: 4739: 4734: 4726: 4722: 4717: 4712: 4699: 4695: 4690: 4682: 4674: 4671: 4649: 4645: 4640: 4635: 4632: 4624: 4605: 4602: 4597: 4592: 4588: 4579: 4562: 4559: 4554: 4550: 4546: 4538: 4521: 4517: 4513: 4508: 4505: 4502: 4494: 4481: 4478: 4471: 4467: 4463: 4458: 4455: 4452: 4444: 4441: 4420: 4416: 4412: 4407: 4404: 4401: 4389: 4369: 4365: 4361: 4356: 4353: 4350: 4344: 4339: 4336: 4326: 4306: 4302: 4298: 4293: 4286: 4282: 4275: 4271: 4267: 4258: 4253: 4250: 4240: 4223: 4219: 4214: 4211: 4206: 4202: 4198: 4192: 4187: 4184: 4174: 4154: 4150: 4146: 4138: 4135: 4130: 4126: 4122: 4116: 4105: 4087: 4083: 4079: 4074: 4071: 4066: 4062: 4058: 4047: 4029: 4025: 4021: 4017: 4012: 4009: 4006: 3998: 3995: 3977: 3973: 3969: 3965: 3960: 3955: 3951: 3947: 3938: 3932: 3926: 3922: 3918: 3914: 3913: 3912: 3911: 3908: 3903: 3898: 3879: 3876: 3873: 3870: 3867: 3864: 3858: 3855: 3828: 3825: 3821: 3795: 3775: 3772: 3762: 3759: 3755: 3726: 3723: 3720: 3717: 3714: 3708: 3705: 3682: 3679: 3676: 3673: 3670: 3664: 3661: 3638: 3635: 3632: 3629: 3626: 3620: 3617: 3597: 3577: 3555: 3552: 3548: 3525: 3522: 3518: 3495: 3466: 3461: 3457: 3454: 3451: 3447: 3437: 3433: 3432: 3431: 3430: 3426: 3422: 3416: 3402: 3398: 3397: 3396: 3395: 3392: 3389: 3385: 3381: 3377: 3373: 3369: 3365: 3361: 3354: 3350: 3346: 3342: 3338: 3337: 3324: 3319: 3314: 3298: 3278: 3275: 3269: 3266: 3263: 3260: 3257: 3251: 3248: 3241:, since then 3228: 3225: 3219: 3216: 3213: 3210: 3207: 3201: 3198: 3190: 3172: 3169: 3165: 3161: 3156: 3153: 3149: 3121: 3118: 3114: 3105: 3095: 3092: 3088: 3060: 3057: 3053: 3049: 3044: 3041: 3037: 3009: 3006: 3002: 2993: 2983: 2980: 2976: 2950: 2947: 2944: 2941: 2938: 2935: 2932: 2912: 2890: 2861: 2851: 2848: 2828: 2820: 2819: 2818: 2817: 2816: 2815: 2814: 2813: 2812: 2811: 2810: 2809: 2798: 2794: 2790: 2785: 2784: 2783: 2782: 2781: 2780: 2779: 2778: 2777: 2776: 2767: 2764: 2760: 2759: 2758: 2757: 2756: 2755: 2754: 2753: 2746: 2741: 2736: 2720: 2717: 2714: 2691: 2688: 2685: 2682: 2679: 2676: 2673: 2667: 2664: 2638: 2635: 2631: 2627: 2622: 2619: 2615: 2593: 2588: 2585: 2581: 2577: 2573: 2569: 2564: 2561: 2557: 2553: 2527: 2524: 2520: 2514: 2511: 2507: 2495: 2492: 2488: 2484: 2479: 2476: 2472: 2465: 2462: 2459: 2455: 2444: 2440: 2423: 2420: 2417: 2414: 2411: 2408: 2405: 2399: 2396: 2383: 2378: 2375: 2371: 2365: 2362: 2358: 2345: 2342: 2338: 2334: 2329: 2326: 2322: 2315: 2312: 2309: 2305: 2300: 2295: 2292: 2288: 2284: 2277: 2274: 2269: 2266: 2262: 2258: 2255: 2246: 2238: 2235: 2231: 2225: 2220: 2217: 2213: 2200: 2197: 2193: 2189: 2184: 2181: 2177: 2170: 2167: 2164: 2160: 2155: 2150: 2147: 2143: 2139: 2132: 2129: 2124: 2121: 2117: 2113: 2110: 2097: 2082: 2074: 2073: 2072: 2071: 2070: 2069: 2062: 2058: 2054: 2050: 2049: 2048: 2047: 2046: 2045: 2024: 2019: 2016: 2012: 2006: 2001: 1998: 1994: 1990: 1983: 1980: 1975: 1972: 1968: 1964: 1961: 1950: 1945: 1942: 1939: 1935: 1931: 1928: 1925: 1922: 1915: 1914: 1913: 1912: 1891: 1886: 1883: 1879: 1873: 1868: 1865: 1861: 1857: 1850: 1847: 1842: 1839: 1835: 1831: 1828: 1817: 1812: 1809: 1806: 1802: 1798: 1795: 1792: 1789: 1782: 1781: 1780: 1779: 1773: 1772: 1771: 1770: 1769: 1764: 1761: 1757: 1754:looks like a 1741: 1718: 1715: 1711: 1705: 1702: 1698: 1689: 1688: 1687: 1686: 1682: 1678: 1672: 1659: 1656: 1653: 1645: 1630: 1627: 1622: 1619: 1611: 1596: 1593: 1589: 1585: 1582: 1577: 1574: 1570: 1547: 1544: 1540: 1536: 1531: 1528: 1524: 1515: 1512: 1494: 1489: 1486: 1482: 1476: 1473: 1469: 1456: 1453: 1449: 1445: 1440: 1437: 1433: 1426: 1423: 1420: 1416: 1411: 1406: 1403: 1399: 1395: 1388: 1385: 1380: 1377: 1373: 1369: 1366: 1355: 1350: 1347: 1344: 1340: 1336: 1333: 1330: 1327: 1319: 1301: 1298: 1294: 1288: 1283: 1280: 1276: 1263: 1260: 1256: 1252: 1247: 1244: 1240: 1233: 1230: 1227: 1223: 1218: 1213: 1210: 1206: 1202: 1195: 1192: 1187: 1184: 1180: 1176: 1173: 1162: 1157: 1154: 1151: 1147: 1143: 1140: 1137: 1134: 1126: 1120: 1114: 1111: 1095: 1092: 1086: 1080: 1077: 1062: 1050: 1035: 1018: 1017: 1016: 1015: 1012: 1009: 1005: 1001: 997: 994: 990: 974: 970: 966: 963: 959: 952: 949: 943: 939: 935: 930: 926: 922: 918: 913: 906: 903: 897: 893: 889: 886: 883: 877: 871: 868: 861: 860: 858: 855: 850: 849: 846: 843: 838: 837: 828: 825: 821: 820: 819: 818: 817: 816: 815: 814: 807: 803: 799: 795: 791: 790: 789: 786: 782: 781: 780: 779: 776: 772: 768: 749: 746: 743: 740: 737: 734: 728: 720: 701: 698: 695: 692: 689: 686: 680: 657: 654: 648: 645: 619: 616: 610: 605: 601: 597: 591: 588: 582: 577: 573: 566: 563: 555: 551: 550: 549: 548: 545: 541: 537: 536:Taylor series 533: 525: 516: 513: 509: 506:Yes, that is 505: 504: 503: 502: 501: 500: 495: 492: 488:<edit: --> 487: 486: 485: 482: 478: 474: 473: 472: 470: 467: 457: 454: 450: 449: 448: 447: 444: 440: 436: 432: 428: 423: 420: 416: 412: 409: 405: 401: 400: 397: 393: 389: 384: 381: 377: 374: 370: 367: 363: 360: 355: 354:no such thing 351: 346: 345: 344: 342: 339: 333: 329: 327: 322: 318: 314: 312: 303: 301: 296: 293: 288: 287: 284: 280: 276: 272: 269: 265: 261: 259: 255: 251: 247: 243: 242: 237: 233: 229: 224: 223: 222: 221: 218: 215: 210: 207: 204: 200: 197: 194: 193: 189: 188: 182: 179: 174: 173: 172: 171: 168: 165: 161: 157: 156: 150: 147: 142: 141: 140: 139: 136: 133: 128: 127: 126: 124: 121: 116: 112: 103: 98: 96: 88: 84: 83: 80: 77: 76: 68: 61: 57: 53: 47: 42: 39: 35: 27: 23: 19: 7682:Donald Hosek 7503: 7447: 7443: 7439: 7435: 7431: 7427: 7423: 7419: 7415: 7411: 7407: 7403: 7401: 6902: 6887: 6883: 6879: 6875: 6871: 6867: 6863: 6859: 6855: 6851: 6847: 6843: 6788: 6784: 6780: 6776: 6772: 6768: 6764: 6648: 6562: 6416: 6349: 6345: 6341: 6337: 6333: 6329: 6325: 6321: 6317: 6313: 6309: 6292: 6288: 6284: 6280: 6276: 6272: 6268: 6264: 6260: 6256: 6252: 6248: 6231: 6227: 6223: 6219: 6215: 6211: 6207: 6203: 6199: 6195: 6188: 6184: 6180: 6176: 6172: 6017: 6013: 6009: 6005: 6001: 5997: 5993: 5989: 5985: 5981: 5977: 5973: 5969: 5965: 5708: 5668: 5659: 5649: 5062: 5017: 4974: 4909: 4856: 4779: 4675: 4672: 4625: 4580: 4539: 4495: 4445: 4442: 4390: 4327: 4241: 4175: 4106: 4048: 3999: 3996: 3939: 3936: 3435: 3412: 3400: 3383: 3379: 3375: 3371: 3367: 3363: 3356: 3352: 3348: 3344: 3340: 3188: 2442: 1767: 1673: 1646: 1612: 1516: 1514:Given that 1513: 1320: 1127: 1124: 992: 822:Thank you. — 721:= π this is 718: 553: 529: 507: 463: 407: 403: 358: 353: 349: 334: 330: 325: 323: 319: 315: 307: 299: 245: 114: 110: 107: 94: 78: 6352:is nonzero 1756:wave number 427:Lotus 1-2-3 419:Open Office 26:Mathematics 7350:i.e. when 6893:Following 6787:change in 6158:Bromskloss 5904:Bromskloss 5018:Therefore 4391:Therefore 3917:deeptrivia 3742:: clearly 3421:deeptrivia 3339:Note that 2963:such that 2789:deeptrivia 2763:Bromskloss 2053:deeptrivia 1760:Bromskloss 1677:deeptrivia 1110:Bromskloss 993:instead of 824:Bromskloss 785:Bromskloss 717:, and for 673:, this is 435:Salix alba 275:Salix alba 264:OpenOffice 250:Salix alba 160:Julian day 7452:Bo Jacoby 7394:Gandalf61 6746:is zero. 6354:and small 6299:in mind): 5984:, we get 5628:Gscshoyru 5119:Gscshoyru 995:a series. 491:Richard B 481:Richard B 50:<< 7602:(we use 6819:Tesseran 6748:Tesseran 6022:Tesseran 5778:, where 3191:hold if 408:duration 214:SGBailey 164:JayHenry 99:August 5 67:August 6 46:August 4 24:‎ | 22:Archives 20:‎ | 7508:Lambiam 6835:Lambiam 3388:Lambiam 1108:. ;-) — 842:Lambiam 508:exactly 292:Zeno333 132:Lambiam 89:pages. 7442:gives 6903:is not 6831:Finite 6785:finite 6253:change 3436:number 1019:Hmmm, 1004:CORDIC 1000:fdlibm 532:python 56:August 7406:< 7295:when 6907:KSmrq 6833:. -- 6793:KSmrq 6419:: --> 6236:KSmrq 6084:, so 3386:? -- 3378:into 3311:.) — 1008:KSmrq 796:. -- 372:good. 69:: --> 63:: --> 62:: --> 44:< 16:< 6275:) = 6218:for 5980:) = 5717:talk 3921:talk 3902:talk 3698:and 3590:and 3540:and 3425:talk 3370:and 3343:and 3318:talk 3189:will 3141:and 2876:(or 2793:talk 2740:talk 2607:and 2443:this 2057:talk 1681:talk 802:talk 771:talk 750:1.26 735:0.87 702:0.66 687:0.87 544:Talk 439:talk 433:. -- 392:talk 366:this 279:talk 268:1583 254:talk 232:talk 7202:so 6878:δ+3 6854:to 6680:lim 6574:lim 6429:lim 6308:If 6279:at 6255:in 6156:. — 5992:)=3 5669:not 4443:So 3868:0.5 3715:0.5 3372:RHS 3368:LHS 3364:LHS 3362::= 3355:by 3349:RHS 1758:. — 927:sin 887:sin 869:sin 602:log 574:log 404:age 326:not 115:not 111:not 60:Sep 52:Jul 7457:. 7438:= 7377:21 7313:− 7130:− 7070:⇒ 7003:− 6954:− 6886:+3 6874:+3 6858:+2 6716:− 6687:→ 6617:− 6589:− 6581:→ 6531:− 6472:− 6436:→ 6385:− 6340:)- 6312:= 6259:= 6179:= 6137:21 6124:− 6108:21 6069:− 5986:f' 5719:) 5692:∞ 5586:21 5579:21 5568:− 5541:21 5535:21 5524:− 5519:21 5513:21 5502:− 5470:21 5463:− 5456:21 5446:− 5410:21 5405:− 5376:21 5371:− 5311:21 5262:21 5207:21 5193:± 5154:21 5144:± 5099:21 5041:21 5034:21 4997:21 4990:21 4955:21 4949:21 4932:21 4925:21 4885:21 4871:21 4829:21 4800:21 4740:21 4696:21 4646:21 4606:21 4547:21 4503:21 4459:− 4453:21 4408:− 4402:21 4357:− 4351:21 4345:∗ 4294:− 4268:21 4259:∗ 4212:− 4199:21 4193:∗ 4136:− 4123:21 4072:− 4059:42 4013:− 3923:) 3865:− 3654:, 3427:) 3359:pr 3276:− 3226:− 3075:, 3029:, 2948:− 2852:∈ 2795:) 2686:… 2668:∈ 2662:∀ 2460:− 2418:… 2400:∈ 2394:∀ 2310:− 2165:− 2059:) 1936:∑ 1803:∑ 1683:) 1657:≥ 1628:− 1586:− 1421:− 1341:∑ 1228:− 1148:∑ 1093:≠ 1081:− 1063:− 964:− 950:θ 944:− 936:⁡ 904:θ 898:− 890:⁡ 878:θ 872:⁡ 804:) 773:) 747:− 741:⁡ 738:ln 699:− 693:⁡ 690:ln 655:π 611:⁡ 606:10 598:− 583:⁡ 578:10 542:| 471:) 441:) 394:) 350:29 343:) 281:) 273:-- 256:) 234:) 125:) 58:| 54:| 7668:) 7665:x 7662:( 7659:f 7639:x 7619:) 7616:x 7613:( 7610:g 7590:) 7587:x 7584:( 7581:f 7558:) 7555:x 7552:( 7549:g 7544:) 7541:x 7538:( 7535:f 7504:z 7487:z 7484:7 7480:1 7475:+ 7472:z 7469:3 7448:x 7444:x 7440:i 7436:x 7432:x 7428:x 7424:x 7420:x 7416:x 7412:x 7408:b 7404:a 7373:1 7368:= 7363:2 7359:x 7338:0 7335:= 7329:x 7324:7 7318:1 7310:x 7305:3 7280:7 7273:3 7268:2 7240:2 7236:x 7232:7 7228:1 7223:+ 7218:2 7214:x 7210:3 7181:7 7174:3 7169:2 7163:+ 7158:2 7153:) 7146:x 7141:7 7135:1 7127:x 7122:3 7116:( 7111:= 7103:2 7099:x 7095:7 7091:1 7086:+ 7081:2 7077:x 7073:3 7044:2 7040:x 7036:7 7032:1 7027:+ 7021:7 7014:3 7009:2 6998:2 6994:x 6990:3 6987:= 6982:2 6977:) 6970:x 6965:7 6959:1 6951:x 6946:3 6940:( 6888:x 6884:x 6880:x 6876:x 6872:x 6868:x 6864:x 6860:x 6856:x 6852:x 6848:x 6844:x 6789:x 6781:x 6779:( 6777:f 6773:x 6769:x 6765:x 6732:h 6728:) 6725:t 6722:( 6719:f 6713:) 6710:h 6707:+ 6704:t 6701:( 6698:f 6690:0 6684:h 6676:= 6673:) 6670:t 6667:( 6663:′ 6660:f 6649:f 6633:h 6629:) 6626:t 6623:( 6620:f 6614:) 6611:h 6608:+ 6605:t 6602:( 6599:f 6585:0 6578:h 6563:h 6547:h 6543:) 6540:t 6537:( 6534:f 6528:) 6525:h 6522:+ 6519:t 6516:( 6513:f 6488:h 6484:) 6481:t 6478:( 6475:f 6469:) 6466:h 6463:+ 6460:t 6457:( 6454:f 6444:+ 6440:0 6433:h 6417:h 6401:h 6397:) 6394:t 6391:( 6388:f 6382:) 6379:h 6376:+ 6373:t 6370:( 6367:f 6350:h 6346:t 6344:( 6342:f 6338:h 6336:+ 6334:t 6332:( 6330:f 6326:t 6322:t 6318:t 6314:t 6310:x 6293:x 6291:( 6289:f 6285:x 6281:x 6277:x 6273:x 6271:( 6269:f 6265:x 6263:( 6261:f 6257:y 6249:x 6232:x 6224:x 6222:( 6220:f 6216:x 6212:x 6210:( 6208:f 6204:x 6200:x 6198:( 6196:f 6189:x 6185:x 6183:( 6181:f 6177:y 6173:x 6141:4 6131:/ 6127:i 6121:= 6118:) 6112:4 6103:i 6100:( 6096:/ 6092:1 6072:i 6066:= 6063:i 6059:/ 6055:1 6018:x 6014:x 6010:x 6006:f 6002:f 5998:x 5994:x 5990:x 5988:( 5982:x 5978:x 5976:( 5974:f 5970:x 5966:x 5890:1 5887:= 5884:x 5864:] 5861:2 5858:, 5855:1 5852:[ 5832:x 5812:0 5809:= 5806:x 5786:x 5764:2 5760:x 5715:( 5709:x 5695:) 5689:+ 5686:, 5683:0 5680:( 5660:i 5650:x 5574:6 5530:3 5508:3 5475:7 5451:3 5418:) 5401:7 5396:( 5392:1 5387:+ 5384:) 5367:1 5362:( 5359:3 5329:2 5325:) 5315:4 5306:i 5302:1 5297:( 5294:7 5290:1 5285:+ 5280:2 5276:) 5266:4 5257:i 5253:1 5248:( 5245:3 5211:4 5202:i 5198:1 5190:= 5187:x 5158:4 5149:1 5141:= 5138:x 5096:1 5091:= 5086:4 5082:x 5029:6 4985:6 4944:3 4938:+ 4920:3 4890:7 4878:+ 4866:3 4836:) 4824:7 4819:( 4815:1 4810:+ 4807:) 4795:1 4790:( 4787:3 4757:2 4753:) 4744:4 4735:1 4730:( 4727:7 4723:1 4718:+ 4713:2 4709:) 4700:4 4691:1 4686:( 4683:3 4650:4 4641:1 4636:= 4633:x 4603:1 4598:= 4593:4 4589:x 4563:1 4560:= 4555:4 4551:x 4522:3 4518:x 4514:1 4509:= 4506:x 4482:0 4479:= 4472:3 4468:x 4464:1 4456:x 4428:] 4421:3 4417:x 4413:1 4405:x 4399:[ 4377:] 4370:3 4366:x 4362:1 4354:x 4348:[ 4340:7 4337:2 4314:] 4307:3 4303:x 4299:1 4287:3 4283:x 4276:4 4272:x 4262:[ 4254:7 4251:2 4224:3 4220:x 4215:1 4207:4 4203:x 4188:7 4185:2 4155:3 4151:x 4147:7 4142:) 4139:1 4131:4 4127:x 4120:( 4117:2 4088:3 4084:x 4080:7 4075:2 4067:4 4063:x 4030:3 4026:x 4022:7 4018:2 4010:x 4007:6 3978:2 3974:x 3970:7 3966:1 3961:+ 3956:2 3952:x 3948:3 3919:( 3904:) 3900:( 3883:) 3880:0 3877:, 3874:0 3871:, 3862:( 3859:= 3856:s 3835:| 3829:r 3826:s 3822:x 3817:| 3796:s 3776:1 3773:= 3769:| 3763:q 3760:p 3756:x 3751:| 3730:) 3727:0 3724:, 3721:0 3718:, 3712:( 3709:= 3706:r 3686:) 3683:0 3680:, 3677:0 3674:, 3671:1 3668:( 3665:= 3662:q 3642:) 3639:0 3636:, 3633:0 3630:, 3627:0 3624:( 3621:= 3618:p 3598:q 3578:p 3556:q 3553:p 3549:x 3526:q 3523:p 3519:R 3496:3 3491:R 3467:3 3462:[ 3458:1 3455:, 3452:0 3448:] 3423:( 3401:n 3384:R 3380:T 3376:R 3357:T 3353:T 3345:r 3341:p 3320:) 3316:( 3299:q 3279:q 3273:) 3270:1 3267:, 3264:1 3261:, 3258:1 3255:( 3252:= 3249:s 3229:p 3223:) 3220:1 3217:, 3214:1 3211:, 3208:1 3205:( 3202:= 3199:r 3173:r 3170:q 3166:R 3162:= 3157:s 3154:p 3150:R 3128:| 3122:r 3119:q 3115:x 3110:| 3106:= 3102:| 3096:s 3093:p 3089:x 3084:| 3061:r 3058:s 3054:R 3050:= 3045:q 3042:p 3038:R 3016:| 3010:r 3007:s 3003:x 2998:| 2994:= 2990:| 2984:q 2981:p 2977:x 2972:| 2951:q 2945:p 2942:+ 2939:r 2936:= 2933:s 2913:q 2891:3 2886:Z 2862:3 2857:R 2849:q 2829:q 2791:( 2742:) 2738:( 2721:r 2718:= 2715:p 2695:} 2692:N 2689:, 2683:, 2680:1 2677:, 2674:0 2671:{ 2665:q 2639:r 2636:q 2632:R 2628:= 2623:q 2620:p 2616:R 2594:| 2589:r 2586:q 2582:x 2578:| 2574:= 2570:| 2565:q 2562:p 2558:x 2554:| 2528:r 2525:q 2521:R 2515:q 2512:p 2508:R 2501:) 2496:r 2493:q 2489:R 2485:+ 2480:q 2477:p 2473:R 2469:( 2466:k 2463:i 2456:e 2427:} 2424:N 2421:, 2415:, 2412:1 2409:, 2406:0 2403:{ 2397:q 2384:3 2379:r 2376:q 2372:R 2366:q 2363:p 2359:R 2351:) 2346:r 2343:q 2339:R 2335:+ 2330:q 2327:p 2323:R 2319:( 2316:k 2313:i 2306:e 2301:| 2296:r 2293:q 2289:x 2285:| 2281:) 2278:1 2275:+ 2270:r 2267:q 2263:R 2259:k 2256:i 2253:( 2247:= 2239:r 2236:q 2232:R 2226:3 2221:q 2218:p 2214:R 2206:) 2201:r 2198:q 2194:R 2190:+ 2185:q 2182:p 2178:R 2174:( 2171:k 2168:i 2161:e 2156:| 2151:q 2148:p 2144:x 2140:| 2136:) 2133:1 2130:+ 2125:q 2122:p 2118:R 2114:k 2111:i 2108:( 2083:q 2055:( 2025:2 2020:r 2017:q 2013:R 2007:| 2002:r 1999:q 1995:x 1991:| 1987:) 1984:1 1981:+ 1976:r 1973:q 1969:R 1965:k 1962:i 1959:( 1951:N 1946:0 1943:= 1940:q 1932:= 1929:S 1926:H 1923:R 1892:2 1887:q 1884:p 1880:R 1874:| 1869:q 1866:p 1862:x 1858:| 1854:) 1851:1 1848:+ 1843:q 1840:p 1836:R 1832:k 1829:i 1826:( 1818:N 1813:0 1810:= 1807:q 1799:= 1796:S 1793:H 1790:L 1742:k 1719:r 1716:q 1712:R 1706:q 1703:p 1699:R 1679:( 1660:0 1654:k 1631:1 1623:= 1620:i 1597:p 1594:q 1590:x 1583:= 1578:q 1575:p 1571:x 1548:p 1545:q 1541:R 1537:= 1532:q 1529:p 1525:R 1495:3 1490:r 1487:q 1483:R 1477:q 1474:p 1470:R 1462:) 1457:r 1454:q 1450:R 1446:+ 1441:q 1438:p 1434:R 1430:( 1427:k 1424:i 1417:e 1412:| 1407:r 1404:q 1400:x 1396:| 1392:) 1389:1 1386:+ 1381:r 1378:q 1374:R 1370:k 1367:i 1364:( 1356:N 1351:0 1348:= 1345:q 1337:= 1334:S 1331:H 1328:R 1302:r 1299:q 1295:R 1289:3 1284:q 1281:p 1277:R 1269:) 1264:r 1261:q 1257:R 1253:+ 1248:q 1245:p 1241:R 1237:( 1234:k 1231:i 1224:e 1219:| 1214:q 1211:p 1207:x 1203:| 1199:) 1196:1 1193:+ 1188:q 1185:p 1181:R 1177:k 1174:i 1171:( 1163:N 1158:0 1155:= 1152:q 1144:= 1141:S 1138:H 1135:L 1096:0 1090:) 1087:= 1084:( 1078:= 1075:( 1072:( 1069:( 1066:( 1060:( 1057:( 1054:( 1051:= 1048:) 1045:) 1042:) 1039:) 1036:+ 1033:( 1030:( 1027:( 975:. 971:) 967:3 960:) 953:3 940:( 931:2 923:4 919:( 914:) 907:3 894:( 884:= 881:) 875:( 800:( 769:( 753:) 744:n 732:( 729:n 719:x 705:) 696:n 684:( 681:n 658:2 649:= 646:x 626:) 623:) 620:e 617:x 614:( 595:) 592:n 589:2 586:( 570:( 567:n 564:2 554:n 517:) 458:) 437:( 390:( 382:. 361:. 277:( 252:( 230:( 183:) 176:( 151:) 144:(

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Knowledge

:Reference desk/Archives/Mathematics/2007 August 5 - Knowledge

Index