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324:(namely division of the polynomial by a factor corresponding to the root), and that only if the root still persists as a solution of te new problem does one consider it to occur more than once. The text already says this, and the lead section already says that multiplicity in general means being present multiple times, when a precise meaning can be given to that. So what does the sentence really add?
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isn't, I think none of the definitions given applies, so one can only conclude that the zero has no multiplicity at all (not even multiplicity zero). Other problematic points are infinite multiplicities (which are not allowed in the most common definition of multiset; even those who prefer to be more liberal make it a
323:
The problem is that this is just informal language; there is no such thing as "occurring as a solution" in general; a value either is a solution of a problem, or it isn't. The reason that one can define multiplicity in the case of a polynomial equation is that there is a method to "remove" a solution
271:
Hey guys, I appreciate the precise mathematical definition of the multiplicity of the zeros of a function, but how about a simple one line explanation that the multiplicity of a root is the number of times that root is repeated? Followed by a simple example, like the root of f(x) = (x-1)^3 is 1, with
393:
This could be something simple, like "if k is an underestimate, then the limit evaluates to 0, and if k is an overestimate, then the limit evaluates to ∞, which is not a real number." Or it could give a full-fledged derivation for why both of these is the case. Regardless, an explanation is needed,
470:
In its current state, this section is not correctly formulated, and is a very bad illustration of the general notion of multiplicity. Since nothing is required of the function (an indeed the interval is not even required to not be reduced to a point), it need not be continuous at its zero; if it
1071:
Can anyone suggest a better way of resolving this confusion? I do not now have time to look into this further, but perhaps we have somewhere else an article which includes a description of his ideas of "multiplicity", and we could link
404:
I don't really have any real expertise on this topic; I'm just going by what I see on face value (plus a little, I guess), so if there's any mistaken impression here, feel free to correct me, and there will be no hard feelings. :)
769:) = 0. Thus the formulation "a solution of ... is a root of multiplicity ..." may be confusing for some readers, as using two different words for the same concept. For this reason, I suggest the following formulation:
512:?). I insist that while infinite multiplicities can be admitted in some cases (any "root" of the zero polynomial), ordinary use of "multiplicity" does not allow it to be just any real number (and why not
1055:, ed. Jean van Heijenoort, Cambridge, Massachusetts: Harvard University Press, 1967, pp. 113–117. These references both purport to be a letter from Cantor to Dedekind, dated July 28, 1899. However, as
219:
OK. That looks like a better category. But this raises the question of whether "Numerical analysis" should be a subcategory of "Mathematical analysis". It seems like it should be, but it is not now.
1120:, the meanings of "multiplicity" that refer to other concepts than the one of this article do not belong to it. So, Knowledge rules justify my revert. The only thing that you can do is to use
140:
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913:, saying that this is the common-language meaning. It seems to me however that neither a set, nor an ordered set (which he seems to have meant here), is covered by the definitions at
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seems to me to help no-one. I changed that to link to my new section, but since that has been removed, it just links to the entire page, which is even worse. What I added was this:
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is not a root, one may say that it is a root of multiplicity zero. This may be useful for simplifying some statements and proofs. IMO, it is not useful to mention this here.
184:. My point was that multiplicity of roots is used more widely than numerical analysis. Hence, I should have put it in a more general category. I have now added
308:
I added- Multiplicity can be thought of as "How many times does the solution appear in the original equation?". hopefully making it explicit and more clear.
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In my experience, root, zero, & solution are used more or less interchangeably in this context. We need links to Root_of_a_function and I added a few.
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is not just about multisets. It also and mostly talks about the multiplicity of roots of a function. So it should not just be in the
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by Jané; with biography by Adolf
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Just a suggestion, for easier reading and to be a bit more concise, to change the first 2 sentences of the section
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Multiplicity of a zero section needs an added explanation of why it won't overestimate the multiplicity
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Then edit it... However, in analysis, I think that the zeroes of a function is more common than roots.
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The article should be consistent in using root or zeros. Root is the more common in scientific use.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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The system Ω of all numbers is an inconsistent, absolutely infinite multiplicity.
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reverted an edit of mine to explain the usage of "multiplicity" by Cantor at
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It could also be pointed out that k need not necessarily be an integer. Take
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with several elements. This the common meaning of the term as expressed in
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Multiplicity of a zero section should be rewritten for precision or removed
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Other senses: usage by Cantor not (evidently) the common-language meaning.
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if it fulfills the condition that every sub-multiplicity has a first
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has discovered, this is in fact an amalgamation by Cantor's editor,
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From Frege to Gödel: A Source Book in
Mathematical Logic, 1879-1931
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In the standard mathematical terminology, a root of the polynomial
1280:. Readers of this page are welcome to comment on this redirect at
1228:. Readers of this page are welcome to comment on this redirect at
1176:. Readers of this page are welcome to comment on this redirect at
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Gesammelte
Abhandlungen mathematischen und philosophischen Inhalts
170:. If I was wrong, then what other category would you put it in?
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to this page to explain that usage, as linking "multiplicity" to
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Some older or translated texts use this term differently:
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in the previous post, and removed a duplicate signature.
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Now I envisage the system of all numbers and denote it
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If you agree, feel free to insert this in the article.
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to determine whether its use and function meets the
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to determine whether its use and function meets the
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to determine whether its use and function meets the
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Jahresbericht der
Deutschen Mathematiker-Vereinigung
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423:* A little knowledge is a dangerous thing.
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537:Multiplicity of a root of a polynomial
267:Multiplicity of a root of a polynomial
1039:(1974/75), pp. 104–139, at p. 126 ff.
1007:. Berlin: Verlag von Julius Springer.
441:And drinking largely sobers us again.
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570:in one variable and coefficients in
501:{\displaystyle x\mapsto x^{\alpha }}
95:This article is within the scope of
38:It is of interest to the following
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801:) (that is a solution of the
508:for any positive real number
457:07:38, 13 December 2018 (UTC)
433:Drink deep, or taste not the
109:and see a list of open tasks.
1140:13:03, 6 November 2022 (UTC)
1088:12:06, 6 November 2022 (UTC)
892:16:43, 4 November 2017 (UTC)
748:16:09, 4 November 2017 (UTC)
664:in one variable. A solution
401:at 0, for a simple example.
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354:22:52, 5 December 2007 (UTC)
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598:) if there is a polynomial
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231:Category:Numerical analysis
188:. Does that make sense? --
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917:. I had added a section
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816:) has the multiplicity
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274:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
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986:References
974:Therefore:
950:is called
836:) ≠ 0 and
784:polynomial
740:Bob K31416
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568:polynomial
420:"None"?...
158:Categories
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222:JRSpriggs
172:JRSpriggs
1291:1234qwer
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361:mattbuck
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276:unsigned
1099:talkref
956:element
782:) be a
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139:on the
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394:IMHO.
36:scale.
844:) = (
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450:Jerzy
1136:talk
1125:wikt
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788:root
774:Let
744:talk
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652:Let
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