1901:
your second statement? Likewise, if you take any function f from A to B and restrict the codomain of the function to the image, then that new function is surjective. Are we still chasing each other in circles here? Rereading you question 3 outdents above, yes, saying that f is a surjective function from A onto C gives you item 2. but does miss item 1. Saying that f is a bijection from A to C gives 1. (because bijection implies injection) but it also gives you 2. (because bijection implies surjection). (Note the comment in my previous reply about the ambiguities of
1260:) has no impact on the characteristics (injective, surjective, bijective, etc.) of a correspondence or function. As Meni Rosenfeld stated above, a function is defined by its domain, its range, and a rule mapping between them; changing the domain or range can change characteristics of the function because it means defining a new function, even it that new function seems a natural extension of the old one. Likewise a correspondence between sets
455:
in each) so can anyone please point me to a good resource on the internet that I can use to write my own source code. Something fast and efficient preferably. If somebody has an m file laying around that already does this or if someone known how to do this on MATLAB for example, it would be appreciated. This is just one of the steps in a larger scheme. Oh and I also have an irreducible polynomial given as the modulus. Thanks!
3354:
2101:(←)So I believe that I do get a sense of where your unease originates, but I think that with precise language no contradiction appears. I have reread this section, and I believe that all your questions, as stated, are properly answered "bijection". So can you precisely restate your problem with a well defined function or relation? I suggest that for clarity we stick with terms such a
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I don't accept your first assumption, because changing the codomain may change the image. For example, when the domain is the set of positive numbers, and F(x) is defined to be the number whose absoloue value is x, then the image of F is the set of positive/negative numbers - when the codomain is the
363:
I thik they probably duplicated the 3 in the second number. Perhaps this is some sort of alien maze test on the reference desk to see how it scurries around when presented with random stuff? I've noticed mopre and more of these type questions where they just enter some random search query with random
454:
Hello, for a computer science project, I am looking for an algorithm that will invert any given element (zero maps to zero) in the Galois Field GF(2^128) and give me its multiplicative inverse. The problem is that my background in abstract algebra/number theory is very limited (one undergrad course
130:
What is your question? Any calculator will give you a value of about 0.8. Are you wondering why there are two 3's in the denominator? Are you wondering what / means? Are you wondering why your computer didn't instamagically speak the answer to you when you randomly typed numbers into some random
4429:
is currently blowing my mind -- although my friends are less impressed having apparently learned it in grade school. At any rate, what is the proof that it works? Also, I see that it can be generalized to nth roots, when n is an integer -- which lets you calculate nth roots when n is rational -- is
2413:
Now when we say a function from X to Y we mean just like what 58.147.60.130 said; it's an ordered triplet (X,Y,G), where G is a subset of XxY that's a function. Such a function is said to be a bijection if it satisfies the condition I mentioned above. If it helps you can think of when the phrase "A
1900:
Do you agree that if f is a bijection (or even just a surjection) from A to C then then C is the image of any function f redefined by simply changing its codomain? (The codomain of a function is always a superset of its image.) If so, then doesn't saying that f is a bijection between A and C give
1861:
You've replied to an old version of my question. I'd changed my question before you answered (but after you saw the old version). Please review my question again. Anyways, If I simply say that F is a bijection between A and C (or is an injection from A to B, or to C), then I miss the second
3349:{\displaystyle {\begin{aligned}e^{x}&=\left^{x}\\&=\lim _{n\to \infty }\left^{x}\\&=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{nx}\\&=\lim _{n\to \infty }\left(1+{\frac {x}{nx}}\right)^{nx}\\&=\lim _{m\to \infty }\left(1+{\frac {x}{m}}\right)^{m}.\end{aligned}}}
2070:) and 2. it is not functional (2 inv_abs 2 and 2 inv_abs -2). If we restrict the domain to the positive reals (as you suggested) it becomes left-total. If we then restrict the codomain to either the positive reals or the negative reals it becomes functional and is in fact a bijection.
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and B have the same cardinality" as a relation between sets, rather than a function. So when we change the codomain Y to Z, we must make sure G is still a subset of XxZ and is still a function. In actually defining G we use a formula like what
Rosenfeld said.
2004:
There is a slippery semantic slope (one that I may have already crossed) to beware of when speaking of redefining a function by changing its domain or codomain, because, as Meni
Rosenfeld pointed out, they are integral to the definition of the function. (See
339:
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Where the last step is substituting m = nx. The first step can be justified by continuity of exponentiation, and the last step is only justified if we recognize that the limit is taken over the entire real numbers, as opposed to just over the integers.
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For every x in X, there's an order pair with x as 1st coordinate, for any two ordered pairs w,t in C, if they have the same first coordinate then they have the same second coordinate (these two conditions guarantee that it's a function, not a
1955:
To sum up, if I simply say that F is a bijection between A and C (or is an injection from A to B, or to C), then I miss the information that when the domain of F is taken to be A, and the codomain of F is taken to be B, then the image of F is
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The square root of 5.23 was just an example. Actually, I question was inspired by how calculator's are able to evaluate roots, which I believe is through a Taylor series. Is there a way to write a taylor series for the expression (a+x)^1/2?
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You could also divide the operand by four first: the square root of 5.23 is equal to the product of the square root of four (2) and that of 5.23/4 = 1.3075. In that case, the x in the Taylor series is 0.3075, so that would converge.
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Just curious, but is there any way to show that a particular function, such as the one given here, has no elementary antiderivative, or is it simply a matter of not being able to come up with one using know methods if integration?
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2009:) When I spoke of redefining the codomain I was implying that both the "before" and "after" codomains were subsets of some larger codomain where we had still defined a function, but as you point out this need not be the case.
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1505:, and it is all the same function. Were you to expand the domain and range onto the reals, then using the first two rules would give you a bijection; using the third rule would not. This does not change the fact that
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belonging to B may indeed be correspondent (by the inverse correspondence) to many elements, of which one single element only - is in A, or: does the phrase "one to one correspondence between A and B" mean that every
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For every w,t in C, if they have the same second coordinate then they have the same first coordinate (this is for injection). For every y in Y, there's an ordered pair in C with y as second coordinate (this is for
3900:{\displaystyle {\sqrt {N^{2}+d}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!d^{n}}{(1-2n)n!^{2}4^{n}N^{2n-1}}}=N+{\frac {d}{2N}}-{\frac {d^{2}}{8N^{3}}}+{\frac {d^{3}}{16N^{5}}}-{\frac {5d^{4}}{128N^{7}}}+\cdots }
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You could just approximate n by a rational r, and then compute the rth root. If you're going digit-by-digit then just make r so close to n that the rth root and the nth root match at the digit you're computing.
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belonging to B is correspondent (by the inverse correspondence) to many elements, of which one single element only - is in A, or: does the phrase "one to one function between A and B" mean that every
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956:. Usually it is understood that the domain is implicitly specified to be the largest domain possible for the expression, but in your question this seems to be a source of confusion. The function
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It's easy to see why this is true, but for a rigorous proof you need to be careful with respect to the decimal point (and perhaps some induction). It boils down to having a current guess
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1589:(i.e. the information that when the domain of F is taken to be A, and the codomain of F is taken to be B, then: for every a,b belonging to the domain of F, if f(a)=f(b) then a=b)
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How would I be able to approximate a square root, like (5.23)^.5, with a taylor series. I know the Taylor series for (1+x)^(1/2), but that only converges for 0<x<1...
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1300:. There may be what seems to you a natural extension of the correspondence to a larger universe, but that is immaterial for considerations of the correspondence itself.
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By "bijection" I meant "one to one correspondence". I'm unnecessarily talking about a function, although every one to one function is also a one to one correspondence.
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3508:| < 1. The convergence will be slow though. Why do you want to do it and why is using the Taylor series important since most any calculator will give the result?
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I am trying, but no, I am not seeing your problem yet. For a function to be bijective is equivalent to it being both injective and surjective. So saying that
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2388:". This looks like a short way to describe it, but any shorter one will depend on a convention for denoting the induced relation, of which I am not aware. --
1587:
Do you see now my problem? If I simply say that F is a surjective from A onto C (thus informing that C is the image of F), then I miss the first information
1146:
A correspondence, as well as a one to one correspondence, must have a domain of discourse. In our case, the domain of discourse is the set of real numbers.
85:
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
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Given two sets X and Y, we say they are one to one correspondent if there's a subset C of XxY (cartesian product) such that the following are satisfied:
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with its original domain and range is bijective if defined by any of the the three rules, because with that original domain the rules are equivalent.
474:. If you want highly optimized code, web search on "finite field inversion". A huge amount of work has been done to optimize that operation, for
1551:
Let's put it this way: F is a function, and A,B,C are sets. Is there a simple phrase expressing the following two pieces of information (at once)?
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belonging to A may indeed be correspondent (by that correspondence) to many elements, of which one single element only - is in B, and that every
37:
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information, i.e. the information that when the domain of F is taken to be A, and the codomain of F is taken to be B, then the image of F is C.
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I don't think it can be generalized to irrational-order roots in any meaningful way. Using exp and log is the way to go for general powers. --
1591:. However, if I simply say that F is a bijection between A and C (or is an injection from A to B, or to C), then I miss the second information
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1. Prove that for all real x the sequence (1+x/n) (n a positive integer) is eventually increasing; precisely, as soon as n: -->
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Also, when you write "correspondence", do you mean a "binary relation that is both left-total and right-total" as defined in
2250:
402:
by Boris A. Kordemsky presents the following four numbers, each of which is an arrangement of the ten digits 0 through 9:
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Ooh okay, didn't see that article. But one last question (to make sure that I understand Taylor series), is the equation
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One sometimes sees it mentioned that 123456789 × 8 = 987654312, just because the pattern in the digits is amusing.
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334:{\displaystyle \lim _{b\to \infty }{\frac {1}{b-2}}{\frac {\sum _{n=1}^{b}nb^{n-1}}{\sum _{n=1}^{b}nb^{b-n}}}=1}
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You don't need to do numerical integration just because there is no elementary antiderivative. You may use a
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because if Ln(k^2) is a number belonging to the domain of discourse then k is unnecessarily a positive number
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Each of these numbers is divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18. —
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The same question may be asked about a "one to one function between A and B": Does it mean that every
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although when Ln(k^2) belongs to the domain of discourse then k is unnecessarily a positive number
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need the extended
Euclidean algorithm, this time for integers. But these two are separate issues.—
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4733:(up to the decimal point). Since you want only one digit at a time, you take the greatest digit
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5. Use 4 to prove exp(x)exp(y)=exp(x+y) for all x and y, which justifes the notation exp(x)=e.
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there any way to generalize again so a similar method can produce roots for any real number?
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My question is about how to interpret the phrase "one to one correspondence between A and B".
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3981:. Anyway, an excellent and simple way to compute square roots is the Babylonian method. --
854:(belonging to the domain of discourse) is correspondent - by the inverse correspondence of
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2006:
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415:
4220:{\displaystyle \int e^{\sin x}dx=\int \sum _{n=0}^{\infty }{\frac {(\sin x)^{n}}{n!}}dx=}
364:
keywords as if querying google. I think the answer is they should enter it into google.
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2437:∞, but from this definition how one arrive at the equation e^x=lim(1+x/n)^n as n--: -->
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does not depend on the larger universe. So when you speak of a correspondence between
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is surjective. (I doubt that this is the problem, but you do know, don't you, that a
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belonging to B is correspondent (by the inverse correspondence) to one single element
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belonging to B is correspondent (by the inverse correspondence) to one single element
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is increasing and bijective, define the inverse to be log(x) ("natural logarithm").
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is not a bijection between the set of positive numbers and the domain of discourse,
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2. Prove that it is bounded (just observe that 0≤(1+x/n)(1-x/n)≤1 as soon as n: -->
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1959:
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672:, not for integers) does. In order to implement it, you have to do computations in
1108:{\displaystyle g:\mathbb {R} \backslash \{0\}\to \mathbb {R} ,x\mapsto \ln(x^{2})}
164:
Perhaps it was a typo and the OP meant to ask about the progress of the sequence 1
4020:
antiderivative. You can still calculate definite integrals of it numerically. --
3458:
3431:
2172:
It seems what HOOTmag has in mind is not a function or a relation, but rather a
834:
is really a bijection between the positive numbers and the domain of discourse,
827:
I have a technical question about either option which will turn out to be true:
132:
1177:
belonging to A is correspondent (by that correspondence) to one single element
4329:...and then evaluate the sums numerically? Seems to me it's still numerical.
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771:
a bijection between the set of positive numbers and the domain of discourse?
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3579:
A calculator almost certainly would not use the Taylor series. That article
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1164:
Does the phrase "one to one correspondence between A and B" mean that every
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74:
2046:
Consider your inverse absolute value binary relation on the reals so that
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is a bijection satisfies both your 1. and your 2. What am I missing? If
952:
by itself is not a function. To define a function you need to specify its
2150:
1952:. Hence, the image is dependent on the codomain (not only on the domain).
1559:
2062:). This is not a function because 1. it is not left-total (there is no
2955:{\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}
2145:
unless we specifically define them here. (I have been guilty of using
622:, and that's what the extended Euclidean algorithm (for polynomials in
4302:{\displaystyle \sum _{n=0}^{\infty }{\frac {\int (\sin x)^{n}dx}{n!}}}
764:
Let's assume that our domain of discourse is the set of real numbers.
493:
Woah, that works for all finite fields? I thought it only worked for
4346:
No, the terms of the sum involve only the elementary antiderivatives
3377:-x (use the inequality between the arithmetic and geometric means).
3500:
Well you can always approximate the square root of the inverse of
1565:
For every a,b belonging to the domain of F, if F(a)=F(b) then a=b.
1029:{\displaystyle f:(0,+\infty )\to \mathbb {R} ,x\mapsto \ln(x^{2})}
79:
Welcome to the
Knowledge (XXG) Mathematics Reference Desk Archives
3523:
You might expand about the point 5.29 = 2.3 rather than about 0.
4034:
You can also calculate antiderivatives of it numerically. E.g.
2973:
2765:{\displaystyle =\lim _{n\to \infty }(1+{\frac {x}{nx}})^{nx}}
4427:
Methods_of_computing_square_roots#Digit_by_digit_calculation
2691:{\displaystyle =\lim _{n\to \infty }(1+{\frac {1}{n}})^{nx}}
1054:
3427:
8. Prove that exp(x) is the sum of the exponential series.
2831:{\displaystyle =\lim _{m\to \infty }(1+{\frac {x}{m}})^{m}}
785:
numbers and the domain of discourse, because for every two
470:
If you just want a straightforward implementation, use the
405:
2,438,195,760; 4,753,869,120; 3,785,942,160; 4,876,391,520.
3388:
exp(x) to be the limit of (1+x/n) as n tends to infinity.
2438:∞. I've googled it, but couldn't find a proof anywhere.
3583:
says calculators would normally do it via log and exp.
861:
On the other hand, if the second option is correct, and
3446:
In 5 it should be exp(x+y)=exp(x)exp(y), of course. --
2436:
I have seen e defined as limit of (1-1/n)^n as n--: -->
816:
is a number belonging to the domain of discourse, then
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then amounts to computing an inverse polynomial modulo
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610:, you add them as usual, and you multiply them modulo
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and then invert again as the series converges for |
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Also, I suggest you the following alternative path.
858:- to a single number, and this number belongs to A?
881:, and whose image is the domain of discourse (i.e.
808:does not seem to be a bijection between the set of
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4391:{\displaystyle \scriptstyle {\int (\sin x)^{n}dx}}
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2007:Binary relation#Is a relation more than its graph?
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830:On one hand, if the first option is correct, and
425:As long as we're on this topic, this one's cute:
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781:does seem to be a bijection between the set of
1381:{\displaystyle f:\mathbb {Z} \to \mathbb {Z} }
8:
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665:{\displaystyle (\mathbb {Z} /p\mathbb {Z} )}
4425:The method for calculating square roots at
2976:display look better by doing it like this:
1115:is not. They are both surjective, and thus
850:and the domain of discourse, so that every
707:, and in particular to compute inverses in
3384:3. Therefore it converges for any real x:
735:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
700:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
591:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
521:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
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645:
637:
633:
632:
627:
584:
583:
575:
571:
570:
568:
555:prime? You fix an irreducible polynomial
514:
513:
505:
501:
500:
498:
310:
297:
286:
268:
255:
244:
237:
219:
207:
201:
2972:Also, you can make "COVIZAPIBETEFOKY"'s
2332:. The question is about the property of
1498:{\displaystyle x\mapsto x\cos(2\pi x)+1}
49:
36:
65:
543:How do you represent the finite field
43:
2232:{\displaystyle \varphi (x,y)=(|y|=x)}
7:
3974:{\displaystyle -N^{2}<d<N^{2}}
1352:Note that you may define a function
4787:{\displaystyle (2p+x)x\leq S-p^{2}}
2877:Introductio in Analysin Infinitorum
2867:Principles of Mathematical Analysis
131:website that you stumbled upon? --
4249:
4167:
3649:
3301:
3233:
3170:
3095:
3024:
2914:
2794:
2720:
2651:
2572:
2496:
981:
214:
196:, ..., specifically the fact that
32:
4556:{\displaystyle p^{2}+2px+x^{2}=S}
3581:Methods of computing square roots
2247:, the formula induces a relation
1948:set of positive/negative numbers
3454:) 16:47, 23 February 2010 (UTC)
2858:Two books you might look at are
1791:with its codomain restricted to
1731:, then saying that the image of
4613:{\displaystyle (2p+x)x=S-p^{2}}
4421:Digit by digit root calculation
1558:of F is taken to be A, and the
877:is a function, whose domain is
846:is a bijection between the set
4759:
4744:
4717:
4702:
4679:
4664:
4585:
4570:
4480:
4467:
4371:
4358:
4273:
4260:
4188:
4175:
3718:
3703:
3685:
3676:
3667:
3657:
3298:
3230:
3167:
3092:
3021:
2911:
2819:
2799:
2791:
2750:
2725:
2717:
2676:
2656:
2648:
2610:
2600:
2580:
2577:
2569:
2531:
2521:
2501:
2493:
2482:
2316:
2304:
2273:
2269:
2257:
2226:
2216:
2208:
2204:
2198:
2186:
1689:
1657:
1486:
1474:
1462:
1430:
1398:
1370:
1102:
1089:
1080:
1066:
1023:
1010:
1001:
987:
984:
969:
939:
926:
659:
653:
650:
629:
211:
18:Knowledge (XXG):Reference desk
1:
4822:14:43, 24 February 2010 (UTC)
4807:07:43, 24 February 2010 (UTC)
4446:23:04, 23 February 2010 (UTC)
4412:06:30, 27 February 2010 (UTC)
4339:18:27, 25 February 2010 (UTC)
4320:13:30, 25 February 2010 (UTC)
4095:12:01, 24 February 2010 (UTC)
4081:20:30, 23 February 2010 (UTC)
4063:19:39, 23 February 2010 (UTC)
4048:17:46, 23 February 2010 (UTC)
4030:16:44, 23 February 2010 (UTC)
4011:16:36, 23 February 2010 (UTC)
3991:07:31, 24 February 2010 (UTC)
3926:02:33, 24 February 2010 (UTC)
3593:23:06, 23 February 2010 (UTC)
3569:19:49, 25 February 2010 (UTC)
3552:13:34, 23 February 2010 (UTC)
3533:13:30, 23 February 2010 (UTC)
3518:13:07, 23 February 2010 (UTC)
3494:12:49, 23 February 2010 (UTC)
3467:20:56, 23 February 2010 (UTC)
3440:16:13, 23 February 2010 (UTC)
3368:13:18, 23 February 2010 (UTC)
2853:13:07, 23 February 2010 (UTC)
2454:12:10, 23 February 2010 (UTC)
2424:15:54, 24 February 2010 (UTC)
2398:08:30, 24 February 2010 (UTC)
2167:02:25, 24 February 2010 (UTC)
2080:02:25, 24 February 2010 (UTC)
1968:21:21, 23 February 2010 (UTC)
1915:18:48, 23 February 2010 (UTC)
1872:18:18, 23 February 2010 (UTC)
1837:18:05, 23 February 2010 (UTC)
1771:is equivalent to saying that
1621:17:35, 23 February 2010 (UTC)
1562:of F is taken to be B, then:
1535:16:01, 23 February 2010 (UTC)
1338:16:01, 23 February 2010 (UTC)
1310:15:23, 23 February 2010 (UTC)
1276:it makes no sense to talk of
1240:12:25, 23 February 2010 (UTC)
1129:09:47, 23 February 2010 (UTC)
909:09:23, 23 February 2010 (UTC)
754:11:43, 23 February 2010 (UTC)
538:07:29, 23 February 2010 (UTC)
488:07:22, 23 February 2010 (UTC)
465:07:10, 23 February 2010 (UTC)
445:23:19, 26 February 2010 (UTC)
435:(Note: 12345679, with NO 8.)
429:12 345 679 × 81 = 999 999 999
420:22:52, 23 February 2010 (UTC)
393:13:26, 23 February 2010 (UTC)
374:12:07, 23 February 2010 (UTC)
351:06:50, 23 February 2010 (UTC)
156:05:17, 23 February 2010 (UTC)
124:05:08, 23 February 2010 (UTC)
33:
1724:{\displaystyle B\subseteq A}
1445:{\displaystyle x\mapsto 1+x}
1413:{\displaystyle x\mapsto x+1}
774:Let me explain my question:
472:extended Euclidean algorithm
4498:{\displaystyle (p+x)^{2}=S}
4001:How to integrate e^sin(x)?
2352:, "The relation induced by
2176:. For example, the formula
1256:? The larger universe (or
476:elliptic curve cryptography
450:Inversion in a Galois Field
4838:
2965:with "+" rather than "−".
2543:{\displaystyle e^{x}=^{x}}
945:{\displaystyle \ln(x^{2})}
614:. Computing an inverse in
4653:is what is denoted there
4071:can be used for this. --
2460:One way of seeing it is:
1821:one-to-one correspondence
1185:is in B, and that every
4726:{\displaystyle (20p+x)x}
2365:{\displaystyle \varphi }
2345:{\displaystyle \varphi }
1698:{\displaystyle f:D\to A}
1666:{\displaystyle f:D\to B}
4688:{\displaystyle (2p+x)x}
4646:{\displaystyle S-p^{2}}
4085:Very cool. Thank you.
3910:exact for all d<1?
1815:is another name for an
873:express the fact that:
842:express the fact that:
108:98765321 / 1233456789
4788:
4727:
4689:
4647:
4614:
4557:
4499:
4392:
4303:
4253:
4221:
4171:
3975:
3901:
3653:
3457:whatth...corrected! --
3419:7. Prove that log(x)=∫
3350:
2956:
2832:
2766:
2692:
2623:
2544:
2366:
2346:
2326:
2233:
1823:is another name for a
1805:
1785:
1765:
1745:
1725:
1699:
1667:
1519:
1499:
1446:
1414:
1382:
1109:
1030:
946:
736:
701:
666:
592:
522:
335:
302:
260:
87:current reference desk
4789:
4728:
4690:
4648:
4615:
4558:
4500:
4393:
4304:
4233:
4222:
4151:
4016:This function has no
3976:
3902:
3633:
3351:
2957:
2833:
2767:
2693:
2624:
2545:
2367:
2347:
2327:
2234:
2129:, avoiding the terms
1806:
1786:
1766:
1746:
1726:
1700:
1668:
1520:
1500:
1447:
1415:
1383:
1296:is not an element of
1288:is not an element of
1110:
1031:
947:
737:
702:
667:
593:
523:
336:
282:
240:
104:98765321 / 1233456789
4741:
4699:
4661:
4624:
4567:
4509:
4464:
4456:and wanting to find
4350:
4230:
4117:
3936:
3607:
2983:
2894:
2777:
2703:
2634:
2555:
2466:
2432:Exponential function
2356:
2336:
2251:
2180:
1795:
1775:
1755:
1735:
1709:
1677:
1645:
1568:The image of F is C.
1509:
1456:
1424:
1392:
1356:
1040:
1036:is injective, while
960:
917:
838:, then how should I
711:
676:
626:
567:
497:
478:and other reasons.
200:
3932:It's exact for all
3423:dt/t for all x: -->
2143:domain of discourse
2066:so that -1 inv_abs
1813:one-to-one function
1258:domain of discourse
889:belonging to A, if
869:, then how sould I
820:is unnecessarily a
804:On the other hand,
4784:
4723:
4685:
4643:
4610:
4553:
4495:
4388:
4387:
4299:
4217:
3971:
3897:
3405:6. Prove that exp:
3391:4. Prove that if x
3346:
3344:
3305:
3237:
3174:
3099:
3028:
2952:
2918:
2828:
2798:
2762:
2724:
2688:
2655:
2619:
2576:
2540:
2500:
2380:is bijective from
2362:
2342:
2322:
2229:
1825:bijective function
1817:injective function
1801:
1781:
1761:
1741:
1721:
1695:
1663:
1515:
1495:
1442:
1410:
1378:
1105:
1026:
942:
732:
697:
662:
588:
518:
400:The Moscow Puzzles
331:
218:
4436:comment added by
4297:
4206:
3916:comment added by
3889:
3852:
3820:
3788:
3764:
3628:
3484:comment added by
3469:
3455:
3395:→x then also (1+x
3326:
3290:
3263:
3222:
3195:
3159:
3125:
3084:
3049:
3013:
2939:
2903:
2816:
2783:
2747:
2709:
2673:
2640:
2597:
2561:
2518:
2485:
2444:comment added by
2239:. Given a domain
2149:as a synonym for
1804:{\displaystyle B}
1784:{\displaystyle f}
1764:{\displaystyle B}
1744:{\displaystyle f}
1594:
1590:
1518:{\displaystyle f}
1280:corresponding to
1218:- only, and that
1193:- only, and that
1119:is bijective. --
868:
837:
323:
235:
203:
114:comment added by
93:
92:
73:
72:
4829:
4793:
4791:
4790:
4785:
4783:
4782:
4732:
4730:
4729:
4724:
4694:
4692:
4691:
4686:
4652:
4650:
4649:
4644:
4642:
4641:
4619:
4617:
4616:
4611:
4609:
4608:
4563:or equivalently
4562:
4560:
4559:
4554:
4546:
4545:
4521:
4520:
4504:
4502:
4501:
4496:
4488:
4487:
4448:
4397:
4395:
4394:
4389:
4386:
4379:
4378:
4308:
4306:
4305:
4300:
4298:
4296:
4288:
4281:
4280:
4255:
4252:
4247:
4226:
4224:
4223:
4218:
4207:
4205:
4197:
4196:
4195:
4173:
4170:
4165:
4138:
4137:
3980:
3978:
3977:
3972:
3970:
3969:
3951:
3950:
3928:
3906:
3904:
3903:
3898:
3890:
3888:
3887:
3886:
3873:
3872:
3871:
3858:
3853:
3851:
3850:
3849:
3836:
3835:
3826:
3821:
3819:
3818:
3817:
3804:
3803:
3794:
3789:
3787:
3776:
3765:
3763:
3762:
3761:
3743:
3742:
3733:
3732:
3701:
3700:
3699:
3675:
3674:
3655:
3652:
3647:
3629:
3621:
3620:
3611:
3496:
3464:
3456:
3445:
3437:
3381:|x| and use 1).
3355:
3353:
3352:
3347:
3345:
3338:
3337:
3332:
3328:
3327:
3319:
3304:
3278:
3277:
3269:
3265:
3264:
3262:
3251:
3236:
3210:
3209:
3201:
3197:
3196:
3188:
3173:
3147:
3146:
3141:
3137:
3136:
3131:
3127:
3126:
3118:
3098:
3072:
3071:
3066:
3062:
3061:
3060:
3055:
3051:
3050:
3042:
3027:
2999:
2998:
2961:
2959:
2958:
2953:
2951:
2950:
2945:
2941:
2940:
2932:
2917:
2845:COVIZAPIBETEFOKY
2837:
2835:
2834:
2829:
2827:
2826:
2817:
2809:
2797:
2771:
2769:
2768:
2763:
2761:
2760:
2748:
2746:
2735:
2723:
2697:
2695:
2694:
2689:
2687:
2686:
2674:
2666:
2654:
2628:
2626:
2625:
2620:
2618:
2617:
2608:
2607:
2598:
2590:
2575:
2549:
2547:
2546:
2541:
2539:
2538:
2529:
2528:
2519:
2511:
2499:
2478:
2477:
2456:
2371:
2369:
2368:
2363:
2351:
2349:
2348:
2343:
2331:
2329:
2328:
2323:
2276:
2238:
2236:
2235:
2230:
2219:
2211:
1810:
1808:
1807:
1802:
1790:
1788:
1787:
1782:
1770:
1768:
1767:
1762:
1750:
1748:
1747:
1742:
1730:
1728:
1727:
1722:
1704:
1702:
1701:
1696:
1672:
1670:
1669:
1664:
1592:
1588:
1524:
1522:
1521:
1516:
1504:
1502:
1501:
1496:
1451:
1449:
1448:
1443:
1419:
1417:
1416:
1411:
1387:
1385:
1384:
1379:
1377:
1369:
1114:
1112:
1111:
1106:
1101:
1100:
1073:
1053:
1035:
1033:
1032:
1027:
1022:
1021:
994:
951:
949:
948:
943:
938:
937:
866:
835:
741:
739:
738:
733:
731:
723:
718:
706:
704:
703:
698:
696:
688:
683:
671:
669:
668:
663:
649:
641:
636:
597:
595:
594:
589:
587:
579:
574:
527:
525:
524:
519:
517:
509:
504:
340:
338:
337:
332:
324:
322:
321:
320:
301:
296:
280:
279:
278:
259:
254:
238:
236:
234:
220:
217:
150:
147:
144:
141:
138:
135:
126:
75:
38:Mathematics desk
34:
4837:
4836:
4832:
4831:
4830:
4828:
4827:
4826:
4774:
4739:
4738:
4697:
4696:
4695:corresponds to
4659:
4658:
4633:
4622:
4621:
4600:
4565:
4564:
4537:
4512:
4507:
4506:
4479:
4462:
4461:
4431:
4423:
4370:
4348:
4347:
4289:
4272:
4256:
4228:
4227:
4198:
4187:
4174:
4123:
4115:
4114:
4069:Risch algorithm
3999:
3961:
3942:
3934:
3933:
3911:
3878:
3874:
3863:
3859:
3841:
3837:
3827:
3809:
3805:
3795:
3780:
3744:
3734:
3724:
3702:
3691:
3666:
3656:
3612:
3605:
3604:
3479:
3476:
3462:
3435:
3422:
3415:
3398:
3394:
3343:
3342:
3311:
3307:
3306:
3283:
3280:
3279:
3255:
3243:
3239:
3238:
3215:
3212:
3211:
3180:
3176:
3175:
3152:
3149:
3148:
3110:
3106:
3105:
3101:
3100:
3077:
3074:
3073:
3034:
3030:
3029:
3012:
3008:
3007:
3000:
2990:
2981:
2980:
2924:
2920:
2919:
2892:
2891:
2887:Note that it's
2818:
2775:
2774:
2749:
2739:
2701:
2700:
2675:
2632:
2631:
2609:
2599:
2553:
2552:
2530:
2520:
2469:
2464:
2463:
2439:
2434:
2354:
2353:
2334:
2333:
2249:
2248:
2178:
2177:
1793:
1792:
1773:
1772:
1753:
1752:
1733:
1732:
1707:
1706:
1675:
1674:
1643:
1642:
1507:
1506:
1454:
1453:
1422:
1421:
1390:
1389:
1354:
1353:
1254:Binary relation
1181:only, and that
1092:
1038:
1037:
1013:
958:
957:
929:
915:
914:
795:Ln(a^2)=Ln(b^2)
762:
709:
708:
674:
673:
624:
623:
565:
564:
495:
494:
452:
398:Problem 323 in
306:
281:
264:
239:
224:
198:
197:
195:
191:
187:
183:
179:
175:
171:
167:
148:
145:
142:
139:
136:
133:
109:
106:
101:
30:
29:
28:
12:
11:
5:
4835:
4833:
4825:
4824:
4809:
4799:Meni Rosenfeld
4795:
4781:
4777:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4640:
4636:
4632:
4629:
4607:
4603:
4599:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4575:
4572:
4552:
4549:
4544:
4540:
4536:
4533:
4530:
4527:
4524:
4519:
4515:
4494:
4491:
4486:
4482:
4478:
4475:
4472:
4469:
4422:
4419:
4418:
4417:
4416:
4415:
4385:
4382:
4377:
4373:
4369:
4366:
4363:
4360:
4357:
4327:
4326:
4295:
4292:
4287:
4284:
4279:
4275:
4271:
4268:
4265:
4262:
4259:
4251:
4246:
4243:
4240:
4236:
4216:
4213:
4210:
4204:
4201:
4194:
4190:
4186:
4183:
4180:
4177:
4169:
4164:
4161:
4158:
4154:
4150:
4147:
4144:
4141:
4136:
4133:
4130:
4126:
4122:
4104:
4103:
4102:
4101:
4100:
4099:
4098:
4097:
4073:Meni Rosenfeld
4067:I believe the
4050:
4022:Meni Rosenfeld
3998:
3995:
3994:
3993:
3983:Meni Rosenfeld
3968:
3964:
3960:
3957:
3954:
3949:
3945:
3941:
3908:
3907:
3896:
3893:
3885:
3881:
3877:
3870:
3866:
3862:
3856:
3848:
3844:
3840:
3834:
3830:
3824:
3816:
3812:
3808:
3802:
3798:
3792:
3786:
3783:
3779:
3774:
3771:
3768:
3760:
3757:
3754:
3751:
3747:
3741:
3737:
3731:
3727:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3698:
3694:
3690:
3687:
3684:
3681:
3678:
3673:
3669:
3665:
3662:
3659:
3651:
3646:
3643:
3640:
3636:
3632:
3627:
3624:
3619:
3615:
3601:
3600:
3596:
3595:
3572:
3571:
3555:
3554:
3544:Meni Rosenfeld
3521:
3520:
3475:
3472:
3471:
3470:
3448:Meni Rosenfeld
3420:
3413:
3396:
3392:
3372:
3357:
3356:
3341:
3336:
3331:
3325:
3322:
3317:
3314:
3310:
3303:
3300:
3297:
3293:
3289:
3286:
3284:
3282:
3281:
3276:
3273:
3268:
3261:
3258:
3254:
3249:
3246:
3242:
3235:
3232:
3229:
3225:
3221:
3218:
3216:
3214:
3213:
3208:
3205:
3200:
3194:
3191:
3186:
3183:
3179:
3172:
3169:
3166:
3162:
3158:
3155:
3153:
3151:
3150:
3145:
3140:
3135:
3130:
3124:
3121:
3116:
3113:
3109:
3104:
3097:
3094:
3091:
3087:
3083:
3080:
3078:
3076:
3075:
3070:
3065:
3059:
3054:
3048:
3045:
3040:
3037:
3033:
3026:
3023:
3020:
3016:
3011:
3006:
3003:
3001:
2997:
2993:
2989:
2988:
2970:
2969:
2963:
2962:
2949:
2944:
2938:
2935:
2930:
2927:
2923:
2916:
2913:
2910:
2906:
2902:
2899:
2885:
2884:
2881:
2880:
2873:Leonhard Euler
2870:
2856:
2855:
2840:
2839:
2838:
2825:
2821:
2815:
2812:
2807:
2804:
2801:
2796:
2793:
2790:
2786:
2782:
2772:
2759:
2756:
2752:
2745:
2742:
2738:
2733:
2730:
2727:
2722:
2719:
2716:
2712:
2708:
2698:
2685:
2682:
2678:
2672:
2669:
2664:
2661:
2658:
2653:
2650:
2647:
2643:
2639:
2629:
2616:
2612:
2606:
2602:
2596:
2593:
2588:
2585:
2582:
2579:
2574:
2571:
2568:
2564:
2560:
2550:
2537:
2533:
2527:
2523:
2517:
2514:
2509:
2506:
2503:
2498:
2495:
2492:
2488:
2484:
2481:
2476:
2472:
2433:
2430:
2429:
2428:
2427:
2426:
2416:Money is tight
2411:
2407:
2403:
2390:Meni Rosenfeld
2361:
2341:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2275:
2271:
2268:
2265:
2262:
2259:
2256:
2228:
2225:
2222:
2218:
2214:
2210:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2131:correspondence
2099:
2098:
2097:
2096:
2095:
2094:
2093:
2092:
2091:
2090:
2089:
2088:
2087:
2086:
2085:
2084:
2083:
2082:
2027:
2026:
2025:
2024:
2023:
2022:
2021:
2020:
2019:
2018:
2017:
2016:
2015:
2014:
2013:
2012:
2011:
2010:
1985:
1984:
1983:
1982:
1981:
1980:
1979:
1978:
1977:
1976:
1975:
1974:
1973:
1972:
1971:
1970:
1957:
1953:
1930:
1929:
1928:
1927:
1926:
1925:
1924:
1923:
1922:
1921:
1920:
1919:
1918:
1917:
1885:
1884:
1883:
1882:
1881:
1880:
1879:
1878:
1877:
1876:
1875:
1874:
1848:
1847:
1846:
1845:
1844:
1843:
1842:
1841:
1840:
1839:
1800:
1780:
1760:
1740:
1720:
1717:
1714:
1694:
1691:
1688:
1685:
1682:
1662:
1659:
1656:
1653:
1650:
1630:
1629:
1628:
1627:
1626:
1625:
1624:
1623:
1603:
1602:
1601:
1600:
1599:
1598:
1597:
1596:
1578:
1577:
1576:
1575:
1574:
1573:
1572:
1571:
1570:
1569:
1566:
1552:
1542:
1541:
1540:
1539:
1538:
1537:
1514:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1441:
1438:
1435:
1432:
1429:
1409:
1406:
1403:
1400:
1397:
1376:
1372:
1368:
1364:
1361:
1345:
1344:
1343:
1342:
1341:
1340:
1317:
1316:
1315:
1314:
1313:
1312:
1252:Have you read
1245:
1244:
1243:
1242:
1226:
1225:
1224:
1223:
1201:
1200:
1199:
1198:
1159:
1158:
1157:
1156:
1150:
1149:
1148:
1147:
1141:
1140:
1139:
1138:
1132:
1131:
1121:Meni Rosenfeld
1104:
1099:
1095:
1091:
1088:
1085:
1082:
1079:
1076:
1072:
1068:
1065:
1062:
1059:
1056:
1052:
1048:
1045:
1025:
1020:
1016:
1012:
1009:
1006:
1003:
1000:
997:
993:
989:
986:
983:
980:
977:
974:
971:
968:
965:
941:
936:
932:
928:
925:
922:
761:
758:
757:
756:
730:
726:
722:
717:
695:
691:
687:
682:
661:
658:
655:
652:
648:
644:
640:
635:
631:
586:
582:
578:
573:
516:
512:
508:
503:
491:
490:
451:
448:
433:
432:
430:
423:
422:
408:
407:
406:
381:
380:
379:
378:
377:
376:
356:
355:
354:
353:
330:
327:
319:
316:
313:
309:
305:
300:
295:
292:
289:
285:
277:
274:
271:
267:
263:
258:
253:
250:
247:
243:
233:
230:
227:
223:
216:
213:
210:
206:
193:
189:
185:
181:
177:
173:
169:
165:
159:
158:
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:
4834:
4823:
4819:
4815:
4810:
4808:
4804:
4800:
4796:
4779:
4775:
4771:
4768:
4765:
4762:
4756:
4753:
4750:
4747:
4736:
4720:
4714:
4711:
4708:
4705:
4682:
4676:
4673:
4670:
4667:
4656:
4638:
4634:
4630:
4627:
4605:
4601:
4597:
4594:
4591:
4588:
4582:
4579:
4576:
4573:
4550:
4547:
4542:
4538:
4534:
4531:
4528:
4525:
4522:
4517:
4513:
4492:
4489:
4484:
4476:
4473:
4470:
4459:
4455:
4451:
4450:
4449:
4447:
4443:
4439:
4438:71.70.143.134
4435:
4428:
4420:
4413:
4409:
4405:
4401:
4383:
4380:
4375:
4367:
4364:
4361:
4355:
4345:
4344:
4343:
4342:
4341:
4340:
4336:
4332:
4331:Michael Hardy
4325:
4324:
4323:
4321:
4317:
4313:
4309:
4293:
4290:
4285:
4282:
4277:
4269:
4266:
4263:
4257:
4244:
4241:
4238:
4234:
4214:
4211:
4208:
4202:
4199:
4192:
4184:
4181:
4178:
4162:
4159:
4156:
4152:
4148:
4145:
4142:
4139:
4134:
4131:
4128:
4124:
4120:
4112:
4111:Taylor series
4109:
4096:
4092:
4088:
4087:58.147.60.130
4084:
4083:
4082:
4078:
4074:
4070:
4066:
4065:
4064:
4060:
4056:
4055:58.147.60.130
4051:
4049:
4045:
4041:
4040:Michael Hardy
4038:or the like.
4037:
4033:
4032:
4031:
4027:
4023:
4019:
4015:
4014:
4013:
4012:
4008:
4004:
3996:
3992:
3988:
3984:
3966:
3962:
3958:
3955:
3952:
3947:
3943:
3939:
3931:
3930:
3929:
3927:
3923:
3919:
3918:173.179.59.66
3915:
3894:
3891:
3883:
3879:
3875:
3868:
3864:
3860:
3854:
3846:
3842:
3838:
3832:
3828:
3822:
3814:
3810:
3806:
3800:
3796:
3790:
3784:
3781:
3777:
3772:
3769:
3766:
3758:
3755:
3752:
3749:
3745:
3739:
3735:
3729:
3725:
3721:
3715:
3712:
3709:
3706:
3696:
3692:
3688:
3682:
3679:
3671:
3663:
3660:
3644:
3641:
3638:
3634:
3630:
3625:
3622:
3617:
3613:
3603:
3602:
3598:
3597:
3594:
3590:
3586:
3582:
3578:
3577:
3576:
3570:
3566:
3562:
3557:
3556:
3553:
3549:
3545:
3541:
3537:
3536:
3535:
3534:
3530:
3526:
3525:Michael Hardy
3519:
3515:
3511:
3507:
3503:
3499:
3498:
3497:
3495:
3491:
3487:
3486:173.179.59.66
3483:
3473:
3468:
3465:
3460:
3453:
3449:
3444:
3443:
3442:
3441:
3438:
3433:
3428:
3425:
3417:
3412:
3408:
3403:
3400:
3389:
3387:
3382:
3378:
3374:
3370:
3369:
3365:
3361:
3360:Michael Hardy
3339:
3334:
3329:
3323:
3320:
3315:
3312:
3308:
3295:
3287:
3285:
3274:
3271:
3266:
3259:
3256:
3252:
3247:
3244:
3240:
3227:
3219:
3217:
3206:
3203:
3198:
3192:
3189:
3184:
3181:
3177:
3164:
3156:
3154:
3143:
3138:
3133:
3128:
3122:
3119:
3114:
3111:
3107:
3102:
3089:
3081:
3079:
3068:
3063:
3057:
3052:
3046:
3043:
3038:
3035:
3031:
3018:
3009:
3004:
3002:
2995:
2991:
2979:
2978:
2977:
2975:
2968:
2967:
2966:
2947:
2942:
2936:
2933:
2928:
2925:
2921:
2908:
2900:
2897:
2890:
2889:
2888:
2883:
2882:
2878:
2874:
2871:
2868:
2864:
2861:
2860:
2859:
2854:
2850:
2846:
2841:
2823:
2813:
2810:
2805:
2802:
2788:
2780:
2773:
2757:
2754:
2743:
2740:
2736:
2731:
2728:
2714:
2706:
2699:
2683:
2680:
2670:
2667:
2662:
2659:
2645:
2637:
2630:
2614:
2604:
2594:
2591:
2586:
2583:
2566:
2558:
2551:
2535:
2525:
2515:
2512:
2507:
2504:
2490:
2479:
2474:
2470:
2462:
2461:
2459:
2458:
2457:
2455:
2451:
2447:
2446:173.179.59.66
2443:
2431:
2425:
2421:
2417:
2412:
2408:
2404:
2401:
2400:
2399:
2395:
2391:
2387:
2383:
2379:
2375:
2359:
2339:
2313:
2310:
2307:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2266:
2263:
2260:
2246:
2243:and codomain
2242:
2223:
2220:
2212:
2201:
2195:
2192:
2189:
2183:
2175:
2171:
2170:
2169:
2168:
2164:
2160:
2159:58.147.60.130
2156:
2152:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2120:
2116:
2112:
2108:
2104:
2081:
2077:
2073:
2072:58.147.60.130
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2044:
2043:
2042:
2041:
2040:
2039:
2038:
2037:
2036:
2035:
2034:
2033:
2032:
2031:
2030:
2029:
2028:
2008:
2003:
2002:
2001:
2000:
1999:
1998:
1997:
1996:
1995:
1994:
1993:
1992:
1991:
1990:
1989:
1988:
1987:
1986:
1969:
1965:
1961:
1958:
1954:
1951:
1946:
1945:
1944:
1943:
1942:
1941:
1940:
1939:
1938:
1937:
1936:
1935:
1934:
1933:
1932:
1931:
1916:
1912:
1908:
1907:58.147.60.130
1904:
1899:
1898:
1897:
1896:
1895:
1894:
1893:
1892:
1891:
1890:
1889:
1888:
1887:
1886:
1873:
1869:
1865:
1860:
1859:
1858:
1857:
1856:
1855:
1854:
1853:
1852:
1851:
1850:
1849:
1838:
1834:
1830:
1829:58.147.60.130
1826:
1822:
1818:
1814:
1798:
1778:
1758:
1738:
1718:
1715:
1712:
1692:
1686:
1683:
1680:
1660:
1654:
1651:
1648:
1640:
1639:
1638:
1637:
1636:
1635:
1634:
1633:
1632:
1631:
1622:
1618:
1614:
1611:
1610:
1609:
1608:
1607:
1606:
1605:
1604:
1586:
1585:
1584:
1583:
1582:
1581:
1580:
1579:
1567:
1564:
1563:
1561:
1557:
1553:
1550:
1549:
1548:
1547:
1546:
1545:
1544:
1543:
1536:
1532:
1528:
1527:58.147.60.130
1512:
1492:
1489:
1483:
1480:
1477:
1471:
1468:
1465:
1459:
1439:
1436:
1433:
1427:
1407:
1404:
1401:
1395:
1362:
1359:
1351:
1350:
1349:
1348:
1347:
1346:
1339:
1335:
1331:
1330:58.147.60.130
1327:
1323:
1322:
1321:
1320:
1319:
1318:
1311:
1307:
1303:
1302:58.147.60.130
1299:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1259:
1255:
1251:
1250:
1249:
1248:
1247:
1246:
1241:
1237:
1233:
1230:
1229:
1228:
1227:
1221:
1217:
1213:
1209:
1205:
1204:
1203:
1202:
1196:
1192:
1188:
1184:
1180:
1176:
1171:
1167:
1163:
1162:
1161:
1160:
1154:
1153:
1152:
1151:
1145:
1144:
1143:
1142:
1136:
1135:
1134:
1133:
1130:
1126:
1122:
1118:
1097:
1093:
1086:
1083:
1077:
1074:
1060:
1046:
1043:
1018:
1014:
1007:
1004:
998:
995:
978:
975:
972:
966:
963:
955:
934:
930:
923:
920:
913:
912:
911:
910:
906:
902:
898:
896:
892:
888:
884:
880:
876:
872:
864:
859:
857:
853:
849:
845:
841:
833:
828:
825:
823:
819:
815:
811:
807:
802:
800:
796:
792:
788:
784:
780:
777:On one hand,
775:
772:
770:
765:
759:
755:
752:
749:
745:
724:
720:
689:
685:
656:
642:
638:
621:
617:
613:
609:
605:
602:, you define
601:
580:
576:
562:
558:
554:
550:
546:
542:
541:
540:
539:
535:
531:
530:174.29.98.151
510:
506:
489:
485:
481:
480:75.62.109.146
477:
473:
469:
468:
467:
466:
462:
458:
457:174.29.98.151
449:
447:
446:
442:
438:
437:Michael Hardy
431:
428:
427:
426:
421:
417:
413:
409:
404:
403:
401:
397:
396:
395:
394:
390:
386:
385:Michael Hardy
375:
371:
367:
362:
361:
360:
359:
358:
357:
352:
348:
344:
343:58.147.60.130
328:
325:
317:
314:
311:
307:
303:
298:
293:
290:
287:
283:
275:
272:
269:
265:
261:
256:
251:
248:
245:
241:
231:
228:
225:
221:
208:
163:
162:
161:
160:
157:
154:
151:
129:
128:
127:
125:
121:
117:
116:71.143.224.27
113:
103:
98:
96:
88:
84:
83:
80:
77:
76:
68:
61:
57:
53:
47:
42:
39:
35:
27:
23:
19:
4734:
4654:
4505:which means
4457:
4453:
4424:
4328:
4310:
4105:
4000:
3909:
3573:
3522:
3505:
3501:
3477:
3474:Square roots
3429:
3426:
3418:
3410:
3406:
3404:
3401:
3399:/n)→exp(x).
3390:
3385:
3383:
3379:
3375:
3371:
3358:
2971:
2964:
2886:
2876:
2866:
2863:Walter Rudin
2857:
2435:
2410:surjection).
2385:
2381:
2377:
2373:
2244:
2240:
2142:
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2110:
2106:
2102:
2100:
2067:
2063:
2059:
2055:
2051:
2047:
1950:respectively
1949:
1902:
1824:
1820:
1816:
1812:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1261:
1219:
1215:
1211:
1207:
1194:
1190:
1186:
1182:
1178:
1174:
1169:
1165:
1116:
899:
894:
890:
886:
882:
878:
874:
870:
862:
860:
855:
851:
847:
843:
839:
831:
829:
826:
821:
817:
813:
809:
805:
803:
798:
794:
790:
786:
782:
778:
776:
773:
768:
766:
763:
743:
619:
615:
611:
607:
603:
599:
560:
556:
552:
548:
544:
492:
453:
434:
424:
399:
382:
107:
94:
78:
4432:—Preceding
4400:Runge–Kutta
4036:Runge–Kutta
3997:Integration
3912:—Preceding
3561:83.81.42.44
3480:—Preceding
2440:—Preceding
2153:instead of
110:—Preceding
99:February 23
67:February 24
46:February 22
26:Mathematics
4737:for which
4460:such that
4108:convergent
4018:elementary
4003:Chirsgayle
2406:relation).
2139:one-to-one
2123:surjection
1903:one-to-one
598:of degree
4404:Bo Jacoby
4312:Bo Jacoby
3538:See also
2127:bijection
2119:injection
1554:When the
891:f(a)=f(b)
789:numbers,
760:Bijection
50:<<
4814:Staecker
4434:unsigned
3914:unsigned
3482:unsigned
3430:(...) --
2442:unsigned
2151:codomain
2115:codomain
2107:relation
2103:function
2050:inv_abs
1819:, but a
1560:codomain
1222:is in A?
1197:is in A?
824:number!
822:positive
810:positive
787:positive
783:positive
563:) over
112:unsigned
56:February
24: |
22:Archives
20: |
2174:formula
1960:HOOTmag
1864:HOOTmag
1613:HOOTmag
1232:HOOTmag
901:HOOTmag
871:briefly
863:Ln(X^2)
840:briefly
832:Ln(X^2)
814:Ln(k^2)
806:Ln(X^2)
779:Ln(X^2)
769:Ln(X^2)
767:So, is
742:you'll
551:) with
89:pages.
4657:, and
3386:define
2141:, and
2125:, and
2111:domain
2058:= abs(
1556:domain
1292:or if
954:domain
188:, 4321
3542:. --
2372:from
2155:image
2147:range
2135:range
1452:, or
1420:, or
893:then
797:then
793:, if
547:= GF(
412:Bkell
192:/1234
180:, 321
69:: -->
63:: -->
62:: -->
44:<
16:<
4818:talk
4803:talk
4442:talk
4408:talk
4335:talk
4316:talk
4091:talk
4077:talk
4059:talk
4044:talk
4026:talk
4007:talk
3987:talk
3959:<
3953:<
3922:talk
3589:talk
3585:Dmcq
3565:talk
3548:talk
3529:talk
3514:talk
3510:Dmcq
3490:talk
3452:talk
3364:talk
2849:talk
2450:talk
2420:talk
2394:talk
2163:talk
2076:talk
2054:iff
1964:talk
1911:talk
1868:talk
1833:talk
1705:and
1617:talk
1531:talk
1334:talk
1306:talk
1272:and
1264:and
1236:talk
1125:talk
905:talk
844:f(X)
748:Emil
744:also
534:talk
484:talk
461:talk
441:talk
416:talk
389:talk
370:talk
366:Dmcq
347:talk
184:/123
172:, 21
120:talk
4362:sin
4264:sin
4179:sin
4129:sin
3876:128
3424:0.
3292:lim
3224:lim
3161:lim
3086:lim
3015:lim
2974:TeX
2905:lim
2875:'s
2865:'s
2785:lim
2711:lim
2642:lim
2563:lim
2487:lim
2384:to
2376:to
1827:.)
1751:is
1469:cos
1388:by
1284:if
895:a=b
887:a,b
799:a=b
791:a,b
528:.
205:lim
176:/12
60:Mar
52:Jan
4820:)
4805:)
4772:−
4766:≤
4706:20
4631:−
4620:.
4598:−
4444:)
4410:)
4402:.
4365:
4356:∫
4337:)
4322:.
4318:)
4267:
4258:∫
4250:∞
4235:∑
4182:
4168:∞
4153:∑
4149:∫
4132:
4121:∫
4113:.
4093:)
4079:)
4061:)
4046:)
4028:)
4009:)
3989:)
3940:−
3924:)
3895:⋯
3855:−
3839:16
3791:−
3756:−
3710:−
3661:−
3650:∞
3635:∑
3591:)
3567:)
3550:)
3531:)
3516:)
3492:)
3459:pm
3432:pm
3366:)
3302:∞
3299:→
3234:∞
3231:→
3171:∞
3168:→
3096:∞
3093:→
3025:∞
3022:→
2915:∞
2912:→
2851:)
2843:--
2795:∞
2792:→
2721:∞
2718:→
2652:∞
2649:→
2573:∞
2570:→
2497:∞
2494:→
2452:)
2422:)
2396:)
2360:φ
2340:φ
2302:φ
2293:∈
2281:∈
2184:φ
2165:)
2137:,
2133:,
2121:,
2117:,
2113:,
2109:,
2105:,
2078:)
1966:)
1956:C.
1913:)
1905:.)
1870:)
1835:)
1716:⊆
1690:→
1658:→
1619:)
1533:)
1481:π
1472:
1463:↦
1431:↦
1399:↦
1371:→
1336:)
1328:?
1308:)
1238:)
1127:)
1087:
1084:ln
1081:↦
1067:→
1055:∖
1008:
1005:ln
1002:↦
988:→
982:∞
924:
921:ln
907:)
897:?
801:.
751:J.
536:)
486:)
463:)
443:)
418:)
391:)
372:)
349:)
341:.
315:−
284:∑
273:−
242:∑
229:−
215:∞
212:→
168:/1
122:)
58:|
54:|
4816:(
4801:(
4794:.
4780:2
4776:p
4769:S
4763:x
4760:)
4757:x
4754:+
4751:p
4748:2
4745:(
4735:x
4721:x
4718:)
4715:x
4712:+
4709:p
4703:(
4683:x
4680:)
4677:x
4674:+
4671:p
4668:2
4665:(
4655:c
4639:2
4635:p
4628:S
4606:2
4602:p
4595:S
4592:=
4589:x
4586:)
4583:x
4580:+
4577:p
4574:2
4571:(
4551:S
4548:=
4543:2
4539:x
4535:+
4532:x
4529:p
4526:2
4523:+
4518:2
4514:p
4493:S
4490:=
4485:2
4481:)
4477:x
4474:+
4471:p
4468:(
4458:x
4454:p
4440:(
4414:.
4406:(
4384:x
4381:d
4376:n
4372:)
4368:x
4359:(
4333:(
4314:(
4294:!
4291:n
4286:x
4283:d
4278:n
4274:)
4270:x
4261:(
4245:0
4242:=
4239:n
4215:=
4212:x
4209:d
4203:!
4200:n
4193:n
4189:)
4185:x
4176:(
4163:0
4160:=
4157:n
4146:=
4143:x
4140:d
4135:x
4125:e
4089:(
4075:(
4057:(
4042:(
4024:(
4005:(
3985:(
3967:2
3963:N
3956:d
3948:2
3944:N
3920:(
3892:+
3884:7
3880:N
3869:4
3865:d
3861:5
3847:5
3843:N
3833:3
3829:d
3823:+
3815:3
3811:N
3807:8
3801:2
3797:d
3785:N
3782:2
3778:d
3773:+
3770:N
3767:=
3759:1
3753:n
3750:2
3746:N
3740:n
3736:4
3730:2
3726:!
3722:n
3719:)
3716:n
3713:2
3707:1
3704:(
3697:n
3693:d
3689:!
3686:)
3683:n
3680:2
3677:(
3672:n
3668:)
3664:1
3658:(
3645:0
3642:=
3639:n
3631:=
3626:d
3623:+
3618:2
3614:N
3587:(
3563:(
3546:(
3527:(
3512:(
3506:x
3502:x
3488:(
3463:a
3450:(
3436:a
3421:1
3414:+
3411:R
3409:→
3407:R
3397:n
3393:n
3362:(
3340:.
3335:m
3330:)
3324:m
3321:x
3316:+
3313:1
3309:(
3296:m
3288:=
3275:x
3272:n
3267:)
3260:x
3257:n
3253:x
3248:+
3245:1
3241:(
3228:n
3220:=
3207:x
3204:n
3199:)
3193:n
3190:1
3185:+
3182:1
3178:(
3165:n
3157:=
3144:x
3139:]
3134:n
3129:)
3123:n
3120:1
3115:+
3112:1
3108:(
3103:[
3090:n
3082:=
3069:x
3064:]
3058:n
3053:)
3047:n
3044:1
3039:+
3036:1
3032:(
3019:n
3010:[
3005:=
2996:x
2992:e
2948:n
2943:)
2937:n
2934:1
2929:+
2926:1
2922:(
2909:n
2901:=
2898:e
2879:.
2869:;
2847:(
2824:m
2820:)
2814:m
2811:x
2806:+
2803:1
2800:(
2789:m
2781:=
2758:x
2755:n
2751:)
2744:x
2741:n
2737:x
2732:+
2729:1
2726:(
2715:n
2707:=
2684:x
2681:n
2677:)
2671:n
2668:1
2663:+
2660:1
2657:(
2646:n
2638:=
2615:x
2611:]
2605:n
2601:)
2595:n
2592:1
2587:+
2584:1
2581:(
2578:[
2567:n
2559:=
2536:x
2532:]
2526:n
2522:)
2516:n
2513:1
2508:+
2505:1
2502:(
2491:n
2483:[
2480:=
2475:x
2471:e
2448:(
2418:(
2392:(
2386:C
2382:A
2378:B
2374:A
2320:}
2317:)
2314:y
2311:,
2308:x
2305:(
2299:,
2296:B
2290:y
2287:,
2284:A
2278:x
2274:|
2270:)
2267:y
2264:,
2261:x
2258:(
2255:{
2245:B
2241:A
2227:)
2224:x
2221:=
2217:|
2213:y
2209:|
2205:(
2202:=
2199:)
2196:y
2193:,
2190:x
2187:(
2161:(
2074:(
2068:y
2064:y
2060:y
2056:x
2052:y
2048:x
1962:(
1909:(
1866:(
1831:(
1799:B
1779:f
1759:B
1739:f
1719:A
1713:B
1693:A
1687:D
1684::
1681:f
1661:B
1655:D
1652::
1649:f
1615:(
1595:.
1529:(
1513:f
1493:1
1490:+
1487:)
1484:x
1478:2
1475:(
1466:x
1460:x
1440:x
1437:+
1434:1
1428:x
1408:1
1405:+
1402:x
1396:x
1375:Z
1367:Z
1363::
1360:f
1332:(
1304:(
1298:B
1294:b
1290:A
1286:a
1282:b
1278:a
1274:B
1270:A
1266:B
1262:A
1234:(
1220:a
1216:a
1212:b
1208:b
1195:a
1191:a
1187:b
1183:b
1179:b
1175:a
1170:b
1166:a
1123:(
1117:f
1103:)
1098:2
1094:x
1090:(
1078:x
1075:,
1071:R
1064:}
1061:0
1058:{
1051:R
1047::
1044:g
1024:)
1019:2
1015:x
1011:(
999:x
996:,
992:R
985:)
979:+
976:,
973:0
970:(
967::
964:f
940:)
935:2
931:x
927:(
903:(
883:f
879:A
875:f
856:f
852:k
848:A
818:k
729:Z
725:p
721:/
716:Z
694:Z
690:p
686:/
681:Z
660:]
657:x
654:[
651:)
647:Z
643:p
639:/
634:Z
630:(
620:f
616:F
612:f
608:d
604:F
600:d
585:Z
581:p
577:/
572:Z
561:x
559:(
557:f
553:p
549:p
545:F
532:(
515:Z
511:n
507:/
502:Z
482:(
459:(
439:(
414:(
387:(
368:(
345:(
329:1
326:=
318:n
312:b
308:b
304:n
299:b
294:1
291:=
288:n
276:1
270:n
266:b
262:n
257:b
252:1
249:=
246:n
232:2
226:b
222:1
209:b
194:5
190:5
186:4
182:4
178:3
174:3
170:2
166:2
153:™
149:w
146:a
143:n
140:i
137:a
134:k
118:(
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