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1858:∫ ∑ ∏ √ − ± ∞ ≈ ∝ ≡ ≠ ≤ ≥ × · ÷ ∂ ′ ″ ∇ ‰ ° ∴ ℵ ø ∈ ∉ ∩ ∪ ⊂ ⊃ ⊆ ⊇ ¬ ∧ ∨ ∃ ∀ ⇒ ⇐ ⇓ ⇑ ⇔ → ↓ ↑ ← ↔
1911:<ref>{{citation|last1=Blyth|first1=Colin R.|last2=Pathak|first2=Pramod K. |year=1986|title=A note on easy proofs of Stirling's theorem|journal=American Mathematical Monthly|volume=93|issue=5|pages=376–379|url=http://www.jstor.org/stable/2323600}}.</ref> <ref>{{citation|last=Gordon|first=Louis|year=1994 |title=A stochastic approach to the gamma function |journal=American Mathematical Monthly |volume=101|issue=9|pages=858–865|url=http://www.jstor.org/stable/2975134}}.</ref>
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829:(called also "ideal thinker", "ideal mathematician", "omnipotent mathematician", "infinite intelligence" etc.) Every mathematical statement is either true or false (even if neither follows from our poor axioms), since the infinite mind can check all special cases at once, no matter how many cases, — finitely many, countably many or even uncountably many. I prefer to say "infinite machine", but it is still the same idea.
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2802:, but their proposed solution relies on a somewhat arcane and arbitrary invocation of two different utilities, followed by a roundabout filtration through two major software packages, which is necessitated by one of them (pstoedit) requiring a costly proprietary plugin to work properly. And the end result is still unusable if your diagram has diagonal lines. Here's the right way:
2264::<math>\frac{n(n + 1)}{2} + (n+1) = \frac{(n+1)((n+1) + 1)}{2}\,,</math><ref group="derivation"> Derivation of induction formula for summing consecutive positive integers: :<math> \begin{align} \frac{n(n + 1)}{2} + (n+1) & = (n+1)\left( \frac n 2 + 1 \right) \\ & = (n+1)\left( \frac n 2 + \frac 2 2 \right) \dots \end{align} </math></ref>
1939:<ref>{{citation|last=Fulman|first=Jason|year=2001|title=A probabilistic proof of the Rogers–Ramanujan identities|journal=Bulletin of the London Mathematical Society|volume=33|issue=4|pages=397–407 |url=http://blms.oxfordjournals.org/cgi/content/abstract/33/4/397|doi=10.1017/S0024609301008207}}. Also .</ref>
697:" and "finitely additive measure" are (generally) not measures. On the other hand, every measure is both a signed measure and a finitely additive measure. That is, "signed" means here "not necessarily unsigned", "vector" means "not necessarily scalar", and "finitely additive" means "not necessarily countably additive". See also
956:. Naturally, I ask myself: but really, why not textbook? Here is my answer. There is only one Knowledge, and a lot of textbooks (on the same topic, I mean). Why not a single optimal textbook? Just because a textbook cannot be universally optimal. Different students need different textbooks. Quite different, indeed! As long as
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I think that the OR rule together with the
Copyright law make coverage of mathematics (or any other subject) impossible. You have to think (commit 'original research') to do mathematics. The only alternative is to blindly copy from 'reliable' sources which violates copyright. Of course, such copying
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11.4) If editors disagree on how to express a problem and/or solution in mathematics, citations to reliable published sources that both are directly related to the topic of the article and directly support the material as presented must be supplied by the editor(s) who wishes to include the material.
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are small, free (if new and somewhat alpha) programs that work properly. I advocate pdflatex since with the alternative, you might be tempted to go the route of latex→dvips→pstopdf before vectorizing, and that runs into a problem with fonts that has to be corrected with one of the arcane
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Now it generates at random an infinite sequence of bits, finds its equivalence class, picks up the representative of this class, and compares the random sequence and the representative via the bit-wise XOR ("exclusive OR") operation. It gets a random element of the zero equivalence class (a sequence
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The reason we have a rule against OR is to try to avoid disputes about what is correct reasoning by appealing to an outside source. Notice that in mathematics, this is usually only necessary when one or more of the disputing parties is a crank or troll. However, refusing to allow an edit on grounds
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We humans are able to write down all subsets of a 10-element set, surely not of a 1000-element set. Nevertheless we human mathematicians are pretty sure that the idea of a finite machine (or mind), able to write down not only 2 but also 2 objects, does not lead to any contradiction, in other words,
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This one is fairly complicated. I don't think it true that, outside of mathematics, OR and copyright makes coverage impossible. The problem is that an allowable rephrasing in most fields becomes OR in mathematics, as even a change in notation does not fall in the "routine arithmetic calculation"
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defined by...satisfies..." (and instead of "function" it could be a longer noun group, like "separable reflexive Banach space" etc). Or do you think I can always rephrase? How? Really, I also feel more comfortable writing "the", but I got unsure, being confused by opposite opinions, like this: but
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Before proving this negative answer, let me comment on it. For me, our idea of the infinite machine is thus questionable. We want to endow the machine with all our basic abilities, extended to the infinite; but we cannot. Either the choice ability, or the random generator, but not both. The set
2844:, "The generally accepted policy is that all facts and major points of view on a certain subject should be treated in one article. As Knowledge does not view article forking as an acceptable solution to disagreements between contributors, such forks may be merged, or nominated for deletion."
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It may sound strange, but I don't mind if slightly different styles (as long as they are "correct") are used in a longer text. It can help avoiding monotonic repetition, of which there is plenty anyway in mathematical texts. But in the present case, the second option gets my vote too.
2051:*{{citation|last1=Barany|first1=Imre|last2=Vu|first2=Van|year=2007|title=Central limit theorems for Gaussian polytopes|journal=The Annals of Probability|publisher=Institute of Mathematical Statistics|volume=35|issue=4|pages=1593–1621|doi=10.1214/009117906000000791}}. Also .
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I think for art like this it would be preferable to use .svg (a vector format) for the graphics instead of .jpg (a bitmap format), if possible. I use Adobe
Illustrator for that but it's kind of expensive; the most popular free alternative seems to be Inkscape. —David Eppstein
655:". Indeed, in physics we do not discover an infinite hierarchy of levels; rather, at some finite step we recognize that the "empty space", skipped so lightheartedly as containing "only" fields, was just the matter we are looking for. By the way, inside atom, a typical
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is disallowed, WP cannot provide textbook(s). No textbook on tensors could satisfy mathematicians, physicists, engineers and biologists. (Can an encyclopedic article satisfy them all? A good question. Maybe not. But for textbooks the problem is much harder.)
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chosen by the machine. This ability is the idea of the famous choice axiom. We cannot instruct the machine how to choose, but still, it can choose. Free will? Not necessarily; maybe the internal representation (of these sets, and whatever) makes it possible.
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Like the alligator pear that is neither an alligator nor a pear and the biologist’s white ant that is neither white nor an ant, the probabilist’s random variable is neither random nor a variable. (Alligator pear = avocado; white ant = termite.)
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Thank you. Having vote 2:0 (or even 3:0, counting myself) I get more sure. No, I do not find it strange... I also like some variations; but I face this case quite often. But wait, do you say that these "a" and "the" are both correct, or not?
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invocations above. (There is a correct route, which is to replace that chain with dvipdfm, that I have never seen anyone suggest. Somehow, the existence of this useful one-step solution to getting PDFs from plain latex is always ignored.)
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Mind it: a random element of a countably infinite set! Distributed uniformly, that is, with equal probabilities for all elements! This is incompatible with any reasonable probability theory for many reasons. Here is my favorite reason. If
1925:<ref name="RY">{{citation|last1=Revuz|first1=Daniel|last2=Yor|first2=Marc|year=1994|title=Continuous martingales and Brownian motion|edition=2nd|publisher=Springer}} (see Exercise (2.17) in Section V.2, page 187).</ref>
1918:<ref name="RY">{{citation|last1=Revuz|first1=Daniel|last2=Yor|first2=Marc|year=1994|title=Continuous martingales and Brownian motion|edition=2nd|publisher=Springer}} (see Exercise (2.17) in Section V.2, page 187).</ref>
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About the infinite machine (or mind) we are less sure. Otherwise
Hilbert would not ask for an arithmetical proof of consistency of the set theory, and Goedel would not discover that arithmetic cannot prove even its own consistency.
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It considers all infinite sequences of bits (not a harder job than all reals...) and groups them into equivalence classes; here two sequences are called equivalent if they differ only in finitely many positions (that is,
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1932:<ref>{{citation|last=Durrett|first=Richard|author-link=Rick Durrett|year=1984|title=Brownian motion and martingales in analysis|place=California|publisher=Wadsworth|isbn=0-534-03065-3}}.</ref>
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Being a probabilist, I wonder, what about a random generator? Can the infinite machine produce an infinite array of random bits? A countably infinite array would satisfy me. Alas, this is impossible!
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2063:* <cite id=CITEREFArBaBaNa2004> S. Artstein, K. Ball, F. Barthe and A. Naor (2004), , ''Journal of the American Mathematical Society'' '''17''', 975–982. Also .</cite>
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A linguist would be shocked to learn that if a set is not closed this does not mean that it is open, or again that "E is dense in E" does not mean the same thing as "E is dense in itself".
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Novel derivations, applications or conclusions that cannot be supported by sources are likely to constitute original research within the definition used by the
English Knowledge.
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Theorems follow from axioms (and definitions); axioms formalize our intuition. Nowadays, axioms of mathematics are axioms of set theory. Which intuitive idea is thus formalized?
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It's "a
Euclidean" because it's pronounced "yoo-", not "oy-" like in German. Same reason for "a European" instead of "an European". But would still be "an Eulerian" though.
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2449:{\displaystyle {\begin{aligned}{\frac {n(n+1)}{2}}+(n+1)&=(n+1)\left({\frac {n}{2}}+1\right)\\&=(n+1)\left({\frac {n}{2}}+{\frac {2}{2}}\right)\dots \end{aligned}}}
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2132::<cite id="equation1" style="float:right;margin-right:2.5em">(1)</cite> <math>a = 0</math> and have a label somewhere else ("equation ]")
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and the verification that the source is indeed reliable also require thought (OR). So the rule against OR is an absurdity which should be repealed.
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The infinite machine is able to form (and store in its infinite memory) not only a list of all subsets of the real line, but also a list of pairs (
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2019:*{{citation|last=Pollard|first=David|title=A user's guide to measure theoretic probability|year=2002|publisher=Cambridge University Press}}
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Thank you; if you are sure, also I am. Yes, I understand it is a language question; but sometimes math jargon differs from usual
English.
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when you introduce me someone, you say "a friend of mine" even though he is uniquely defined, if not by your words then by your gestures.
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that it is OR is ultimately just an excuse for rejecting what we think is false without having to get the agreement of a crank or troll.
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2075:* <cite id=CITEREFKlartag2007>], Bo'az (2007), , ''Inventiones Mathematicae'' '''168''', 91–131. Also .</cite>
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I'm not a native
English speaker (I'm Swedish), but I'd say, as a guess, that both options are correct, but the first option seems
2012:*{{citation|last=Durrett|first=Richard|author-link=Rick Durrett|title=Probability: theory and examples|edition=Second|year=1996}}
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This is not an encyclopedia article or the talk page for an encyclopedia article. If you find this page on any site other than
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be using SVG, and of those who have, few seem to have an easy way of actually accomplishing it. This is addressed at
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It chooses (and stores in memory) one sequence in each equivalence class, — call it the representative of this class.
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satisfies the definition of random variable even though it does not appear random in the everyday sense of the word.
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I observe repeatedly that good faith editors, striving to make
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A set, however, is not a door: it can be neither open or closed, and it can be both open and closed. (Examples?)
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Thank you. Yes, I feel forced to choose, since really I write more complicated texts :-) such as "...then
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may be a better place to post this question. As a native
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pdflatex file.tex pdfcrop --clip file.pdf tmp.pdf pdf2svg tmp.pdf file.svg (rm tmp.pdf at the end)
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large enough). (Only a continuum of equivalence classes, — much less than all sets of reals...)
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It depends a bit on how you "read it out loud inside your head" when reading. An equation like
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is positive". But it forced to pick from one of the two choices I vote for the second one. --
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That is mathematics, of course. But it reminds me some physics. One often says that "
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theory stipulates the choice and sacrifices the randomness. This fact bothers me.
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In mathematics, a “red herring” need not, in general, be either red or a herring.
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147:{\displaystyle {\begin{array}{r}\zeta (s)=0\\\Re (s)={\frac {1}{2}}\end{array}}}
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means "not necessarily bounded operator, not necessarily defined on the whole
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It has been road-tested on, most notably (for the complexity of its images)
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Derivation of induction formula for summing consecutive positive integers:
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but most stochastic differential equations are not differential equations.
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are two independent, uniformly distributed random positive integers, then
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Using the lost grammatical adage "when in doubt, rephrase", I'd use "if
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Wikipedia_talk:WikiProject_Mathematics/Archive_46#Connected_space/Proofs
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1801:{{nowrap begin}}E ( P ( ''A'' | ''X'' ) ) = P ( ''A'' ).{{nowrap end}}
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appears anew, it was not introduced before. On the other hand, given
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S. Goldberg “Probability: an introduction”, Dower 1986, p. 160.
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see {{harv|Durrett|1996|loc=Sect. 7.7(c), Theorem (7.8)}};
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This is why the choice and the randomness cannot coexist.
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Wikipedia_talk:WikiProject_Mathematics/Archive_46#Proofs
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Mathematics/missing mathematicians
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Here is why the choice and the randomness cannot coexist
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Knowledge:When to cite#Challenging another user's edits
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Knowledge talk:WikiProject Mathematics/Proofs/Archive 2
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Knowledge talk:WikiProject Mathematics/Proofs/Archive 1
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can "be pronounced differently" when given a context.
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A hydrogen atom is about 99.9999999999996% empty space
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Lectures on Probability, Statistics and Econometrics
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math and physics that I don't know a good name for.
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1161:Knowledge:Manual of Style (mathematics)
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