algebraic numbers
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From: Daniel Lichtblau on 8 Jan 2010 04:15 DrMajorBob wrote: >> But Mathematica does or if you prefer "simulates" a lot of mathematics >> that only makes sense under the assumption of continuity. > > Continuity of a function does NOT depend on completeness in the domain, > and I suspect that > > Resolve[Exists[x, x^2 == 2], Reals] > > True > > succeeds based on algebra, not topology. It's based on cylindrical algebraic decomposition (CAD). As the name indicates, that has an algebraic basis. But it is also intimately connected to what is called real algebraic geometry, and that has a bit of a topological flavor to it. > You or I might (MIGHT) treat it as a topological problem, but I doubt > Resolve can do so. > > A better example might be > > Reduce[Exists[x, Exp[x^7 + 3 x - 11] + x - 6/10 == 0], Reals] > > True That might almost be called real analytic (as opposed to algebraic) geometry. [It has something of a local flavor. In the sense of geography, not math; it is relatively recent work at WRI. Serious development is found in Adam W. Strzebonski: Real root isolation for exp-log functions. ISSAC 2008: 303-314 This received the conference Best Paper award, I might add. It would be a breach of the MathGroup rules to mention the prize...] > A human might have great difficulty solving the equation, but he might > easily establish that the LHS is negative for some value and positive for > another, hence a solution exists in between. > > Yet, since this works: As of version 7 of Mathematica... > FindInstance[ Exp[x^7 + 3 x - 11] + x - 6/10 == 0, x] > > {{x -> Root[{-3 + 5 E^(-11 + 3 #1 + #1^7) + 5 #1 &, > 0.599896128076431511686789719766}]}} > > I suspect that algebra and root-search was used in Resolve. > > Unless a developer can confirm that Resolve didn't find a solution, merely > proved that one could be bracketed? I do not know but I'd imagine it could be done either way. Probably the latter if at all possible, since that could be faster. > To do that, Resolve would have to know the LHS is continuous on the real > line, and haven't we found, frequently, that Mathematica CAN'T identify > continuous functions? Reduce, FindInstance, and probably Resolve have a reasonable understanding of exp-log-quasipolynomial functions. To answer a possible question from the philosophically-minded, yes, the software really does understand these things. > And what does THIS mean? > > 0.5998961280764315116867897197655402817356291002252018609367`30. // \ > RootApproximant > > Root[1 - #1 + #1^3 - 7 #1^4 + 10 #1^5 - 16 #1^6 + 15 #1^7 - 15 #1^8 + > 15 #1^9 - 16 #1^10 + 12 #1^11 - 10 #1^12 + 11 #1^13 - > 2 #1^14 + #1^15 + 8 #1^16 &, 3] > > (Note the constant included in the output from FindInstance.) > > Did FindInstance (and Resolve) generate and solve a series approximation > to -3 + 5 E^(-11 + 3 #1 + #1^7) + 5 #1 & ? > > Or is the RootApproximant result a pure accident? > > Bobby An accident, of sorts. It is designed behavior, but it is not obvious whether the design is without flaw. I say something about this (not much, but something) in a response to another post from this thread-of-all-threads. Daniel Lichtblau Wolfram Research
From: DrMajorBob on 8 Jan 2010 04:16 > And in > particular, 1.2==12/10 in Mathematica should trouble you if you believe > Mathematica speaks meaningfully on these issues. That's it, in a nutshell. Bobby On Thu, 07 Jan 2010 01:33:33 -0600, Richard Fateman <fateman(a)cs.berkeley.edu> wrote: > Andrzej Kozlowski wrote: >> Just one more comment, I hope my last one on this subject. Obviously >> RandomReal make it choices out of a countable set of entities. One would >> have to be insane to claim otherwise and I am not that yet. >> >> But, Mathematica does not regard these entities as rational numbers and >> so they are not that. If you call them rationals the it does not make >> *mathematical* sense (because rationals have measure 0). So, if >> Mathemaitca does not regard them as rationals they are not rationals. > > You can refer to Rationals as whatever Mathematica > calls Rationals. But the rational numbers include all numbers that are > represented by finite explicit binary strings in a floating-point > format. They also include other numbers whose binary expansions are > infinite, but repeat. > > Can Mathematica represent Reals that are NOT RATIONAL? Sure. Here are > examples: Sqrt[2], 3*Pi, 4*E. 3*E +4*E^E + 5*E^E^E. > Incidentally, it is not known if E+Pi is rational. > >> How could they be that ? Until they are interpreted by Mathematica, they >> are not numbers at all but just some data stored in computer memory - >> which are not numbers of any kind. > true, but other programs can also interpret them. As numbers, as ASCII > character strings, as pointers into memory. > Mathematica interprets them as >> non-computable irrationals > > No, that's not the way computer programs work. Mathematica allows some > set of operations like +, *, printing. That's all. They are obviously > computable and finitely representable as well, but Mathematica doesn't > need to have an opinion on this, and neither do we have to attribute > opinions to Mathematica. > > If you think that the operations that Mathematica performs are > consistent with YOUR view that these numbers are non-computable > irrationals, I suppose that is your view, but it is certainly > unnecessary for others to hold this view. > > in order to make mathematical sense when >> returning them while simulation a real distribution, because all other >> numbers have measure 0. > > There is a literature on pseudo-random numbers that makes mathematical > sense without any such interpretation. >> >> This is all about "simulating mathematics" - numbers do not live in any >> sense inside computers. To say that "all computer numbers are rational" >> is weird - there is no such things as "computer numbers". Numbers exist >> only and (probably) exclusively in the human mind. > > Actually, you just said that Mathematica interprets --blah blah. Maybe > you think that Mathematica has a human mind? >> >> To say that 1.2 is rational in Mathematica even if Mathematica says >> >> Element[1.2, Rationals] >> >> False >> >> does not make any sense at all. > > It makes perfect sense to say (in Mathematica) that 1.2 is a rational > number because it is equal to a rational number. Huh?? > 1.2==12/10 > True. > > (A better example would be 1.25, since 1.2 is not representable exactly > in binary. This example of 1.2 actually reveals a "misfeature of > mathematica. > > 1.2==5404319552844595/4503599627370496 > True. > > So 1.2 is actually Mathematica-equal to another rational number. Many, > in fact. > ) > > > If you capitalize the term and wish to say that 1.2 is not a Rational > in Mathematica, that is just a convention based on the "type" of data > that is input to Mathematica with a decimal point and is therefore > stored in a memory format that is labeled "Real" which (in Mathematica) > is a superclass of "Rational". That is, "1/2" is a Rational but is also > a Real and is incidentally also a Complex. But 0.5 is not a Real. > > This categorization of types in Mathematica does not determine the > membership (or not) of a particular numeric VALUE in a mathematical > category such as "rational". From a mathematical perspective, any > legal number in an ordinary floating-point format can represent only a > rational value. > >> >> Andrzej Kozlowski >> >> >> >> >> On 6 Jan 2010, at 07:04, Andrzej Kozlowski wrote: >> >>> Well, you are obviously misunderstanding what I am trying to explain >> but I have no desire to spend any more time on it. I give up. > > OK. > >>> Perhaps you should try to explain yourself why Mathematica gives >>> >>> In[1]:= Element[1.2, Rationals] >>> >>> Out[1]= False >>> >>> In[2]:= Element[1.2, Reals] >>> >>> Out[2]= True > > the explanation is that Mathematica takes numbers written with a decimal > point and labels them "Real". This has nothing to do with their values, > which are, most assuredly, equal to rational numbers. And in > particular, 1.2==12/10 in Mathematica should trouble you if you believe > Mathematica speaks meaningfully on these issues. > >>> >>> and you might also read >>> >>> http://en.wikipedia.org/wiki/Computable_number >>> > But this would be irrelevant. > > The Mathematica documentation says, > "When domain membership cannot be decided the Element statement remain > [sic] unevaluated". > > "cannot be decided" > is not a statement about decidability in the technical "computability" > sense. It is a statement about this version of Mathematica not being > programmed to make a decision. Thus the fact that Mathematica 6.0 > cannot decide if e+pi is rational is not a deep result, and it is > referring to the mathematical literature about conjectures on the > matter. It just happens that the program fails to decide. The program > seems to be a jumble of some sort, since it knows that > > > > Mathematica 6.0 does not know e^e is definitely NOT Rational. (It is > known not to be rational). > > It just hasn't been programmed. Yet. It would be nice if the > documentation were clearer on this. > > Regards > RJF > > ..snip.. > -- DrMajorBob(a)yahoo.com
From: DrMajorBob on 8 Jan 2010 04:17 > A consequence is that all randoms can be made to fit some algebraic > number of whatever degree, simply by allowing siufficiently large > coefficients. And sufficiently large degree? A combination of the two? Bobby On Thu, 07 Jan 2010 10:19:39 -0600, Daniel Lichtblau <danl(a)wolfram.com> wrote: > DrMajorBob wrote: >>> Well, I think when you are using Mathematica it is the designers of >>> Mathematica who decide what is rational and what is not. >> Not to repeat myself, but RootApproximant said 100 out of 100 >> randomly chosen machine-precision reals ARE algebraic. >> If your interpretation is correct and consistent with Mathematica, and >> if Mathematica is internally consistent on the topic, virtually all of >> those reals should NOT have been algebraic. >> Mathematica designers wrote RootApproximant, I assume? >> Hence, I'd have to say your interpretation is no better than mine. >> Bobby > > Regarding RootApproximant design, the missing functionality is this. > There is no limiting of coefficient size (or if there is, it's not > obvious to me how it might be done). Rationalize has such limiting > capability, more or less (though it is really built into the algorithm; > the optional second argument does not impose it). > > A consequence is that all randoms can be made to fit some algebraic > number of whatever degree, simply by allowing siufficiently large > coefficients. > > I am not sure whether this is a design flaw. It might alternatively have > been intentional, due to possible implementational difficulties in doing > otherwise. In retrospect, it kinda surprises me that I am not familiar > with the history of this particular design issue, but there you have it. > > Daniel Lichtblau > Wolfram Research > > -- DrMajorBob(a)yahoo.com
From: Noqsi on 8 Jan 2010 04:13 On Jan 7, 12:29 am, DrMajorBob <btre...(a)austin.rr.com> wrote: > > Bobby, you're looking at how machine reals are *encoded*, and a > > *subset* of them is encoded as if they were precise rationals, as you > > say. But what matters is how they *behave*. > > If RootApproximant(a)RandomReal[] succeeds, that random number is algebraic, > according to Mathematica... yes or no? Since the result of RandomReal[] represents a neighborhood, success of RootApproximant here indicates the presence of an algebraic number in that neighborhood. Seems unremarkable to me.
From: DrMajorBob on 8 Jan 2010 04:16
Agreed. In the US at least, the safest reason an employer can use for hiring or firing is "no reason". They can't be sued unless they tell you (or record somewhere, discuss in front of witnesses, etc.) the reason they didn't hire (or admit) you. That flexibility is lost with a rule like Texas has, where the top 10% of high school students MUST be accepted to state schools. If I'm not mistaken, though, it seems I heard students are only guaranteed admission to SOME state school... not necessarily the one they want. Bobby On Wed, 06 Jan 2010 18:00:45 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > Well, I can't tell much about SATs and SAT II as I never had to take > them, but from my perspective I see quite a different problem > > My daughter, who was educated in Japan and the UK, took the SAT and SAT > II's while attending an international school in Britain. She got the > maximum possible scores on all of them. (Later she also got a maximum > possible score on GRE). > > Although we did not really want her to go to a US university she wanted > to see if she could get in, so she applied to 3 of the top Ivy League > schools and was turned down by all three, in spite of her perfect scores > (and a record high score in the European International Baccalaureate > examinations). > > So she went to Cambridge University in the UK (which accepted her on the > basis of her IB results) and then did a doctorate in Germany and > published a paper in the top ranked journal in her field. She then > applied for post-doc position to one of the places that turned her down > as an undergraduate, and they accepted her this time giving her at least > the satisfaction of being able to turn down an offer from a famous Ivy > League university. > > We have a pretty good idea why she was turned down as an undergraduate > in spite of having as good an academic record as one could possibly > have - but this it not the right place to write about that. But, at > least from my perspective, the problem with these SATs, SAT IIs, GREs > etc., is that the supposedly best institution ignore them and make > selectons on the basis of their own highly dubious (and secret) > criteria. However bad these exams are, they are better than that. > > Andrzej Kozlowski > > On 6 Jan 2010, at 20:02, DrMajorBob wrote: > >> I've always told people, "I test smarter than I really am," and now I >> see... I was right! >> >> But not because I worked hard or my parents got involved in my >> schoolwork, >> as the New Yorker article suggests. >> >> At least, I didn't think so, until I thought about it some more and came >> up with some factoids: >> >> a) My grandmother bought me comic books... and I READ them. >> >> b) I participated in summer reading programs at the local library >> (voluntarily). >> >> c) My mother coached me for spelling bees, twice. >> >> d) She took dictation for my history notebook one summer when I >> (voluntarily) went to summer school. >> >> e) Nobody told me math was hard, that I can remember. >> >> f) Comics led me to science fiction, which I read like a house on fire. >> >> So the article makes more sense than I originally thought. >> >> Highly recommended. Thanks for the link! >> >> Bobby >> >> On Tue, 05 Jan 2010 00:44:27 -0600, Noqsi <jpd(a)noqsi.com> wrote: >> >>> On Jan 4, 4:00 am, DrMajorBob <btre...(a)austin.rr.com> wrote: >>>>> The issue here is >>>>> whether the student has enough common culture with the test writer to >>>>> find the same answer. And that's *always* an issue. >>>> >>>> So those are cultural conformity questions?!? >>> >>> One might not need to conform, but one must at least understand the >>> culture. Mathematics is a human cultural artifact, and students are >>> going to need to understand some things about that artifact and its >>> expression to be successful in college. >>> >>> Specifically in this case series are often presented as specific terms >>> and ellipsis, judged to be easier to comprehend in some ways than a >>> formula, so the student should be able to comprehend that form. >>> >>> And this continues into professional life. Today I'm looking over the >>> specs of a megapixel image sensor. The drawings that document its >>> structure contain "..." in a number of places: it's not practical to >>> show every pixel! I can, of course, think of all kinds of perverse and >>> stupid ways to misunderstand what's omitted, but that wouldn't be >>> helpful in any way. >>> >>>> >>>> That's even worse than I thought! >>> >>> It's still worse. The intentions behind the widespread adoption of the >>> SAT didn't really address the need to establish that the student could >>> comprehend the academic cultural context: instead, they were >>> consciously bigoted. >>> >>> http://www.newyorker.com/archive/2001/12/17/011217crat_atlarge >>> >>>> >>>> Bobby >>>> >>>> >>>> >>>> On Sun, 03 Jan 2010 02:40:36 -0600, Noqsi <j...(a)noqsi.com> wrote: >>>>> On Jan 2, 3:05 am, DrMajorBob <btre...(a)austin.rr.com> wrote: >>>>>> When I clicked on the link below, the search field was already >>>> filled = >>> >>>>>> with >>>>>> the sequence >>>> >>>>>> target = {1, 2, 3, 6, 11, 23, 47, 106, 235}; >>>> >>>>>> Searching yielded "A000055 Number of trees with n unla= >>> beled >>>>>> nodes." >>>> >>>>>> I tried a few Mathematica functions on it: >>>> >>>>>> FindLinearRecurrence(a)target >>>> >>>>>> FindLinearRecurrence[{1, 2, 3, 6, 11, 23, 47, 106, 235}] >>>> >>>>>> (fail) >>>> >>>>>> FindSequenceFunction(a)target >>>> >>>>>> FindSequenceFunction[{1, 2, 3, 6, 11, 23, 47, 106, 235}] >>>> >>>>>> (fail) >>>> >>>>>> f[x_] = InterpolatingPolynomial[target, x] >>>> >>>>>> 1 + (1 + (1/ >>>>>> 3 + (-(1/ >>>>>> 12) + (7/ >>>>>> 120 + (-(1/ >>>>>> 60) + (1/144 - (41 (-8 + x))/20160= >>> ) (-7 + x)) (-6 + >>>>>> x)) (-5 + x)) (-4 + x)) (-3 + x) (-= >>> 2 + x)) (-1 + x) >>>> >>>>>> and now the next term: >>>> >>>>>> Array[f, 1 + Length(a)target] >>>> >>>>>> {1, 2, 3, 6, 11, 23, 47, 106, 235, 322} >>>> >>>>>> But, unsurprisingly, the next term in A000055 is 551, not 322. >>>> >>>>>> A000055 actually starts with another three 1s, but that doesn't >>>> change >>>>>> things much: >>>> >>>>>> target = {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}; >>>> >>>>>> FindLinearRecurrence(a)target >>>> >>>>>> FindLinearRecurrence[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}] >>>> >>>>>> (fail) >>>> >>>>>> FindSequenceFunction(a)target >>>> >>>>>> FindSequenceFunction[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}] >>>> >>>>>> (fail) >>>> >>>>>> f[x_] = InterpolatingPolynomial[target, x] >>>> >>>>>> 1 + (1/24 + (-(1/ >>>>>> 40) + (1/ >>>>>> 90 + (-(1/ >>>>>> 280) + (1/ >>>>>> 1008 + (-(43/ >>>>>> 181440) + (191/3628800 - (4= >>> 37 (-11 + x))/ >>>>>> 39916800) (-10 + x)) (-9 + = >>> x)) (-8 + x)) (-7 + >>>>>> x)) (-6 + x)) (-5 + x)) (-4 + x) (-3 + x) = >>> (-2 + x) (-1 + >>>>>> x) >>>> >>>>>> Array[f, 1 + Length(a)target] >>>> >>>>>> {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, -502} >>>> >>>>>> So I ask you, from the data alone: what's the next term? >>>> >>>>> It's the sort of question where one might expect a specialist to >>>>> recognize a familiar sequence. It's all context. >>>> >>>>> Consider that in a narrow mathematical sense, spectroscopy is an >>>>> utterly ambiguous, "ill conditioned" problem. But show me a gigagauss >>>>> cyclotron spectrum, and I'll recognize it as such (see the >>>>> acknowledgment at the end of arxiv.org/pdf/astro-ph/0306189: the >>>>> authors were struggling to contrive an interpretation from atomic >>>>> physics before one of them showed the spectrum to me). But I expect >>>>> very few could do this, since few have the background. >>>> >>>>>> If one had the Encyclopedia of Integer Sequences handy, those SAT >>>>>> questions could be interesting. But they'd still be nonsense. >>>> >>>>> No they are not. Remember that the SAT isn't about the ability of a >>>>> student to function in some ideal abstract world of infinite >>>>> possibility. In the real world of academia, every single question >>>>> they >>>>> will encounter will be ambiguous in some sense. The issue here is >>>>> whether the student has enough common culture with the test writer to >>>>> find the same answer. And that's *always* an issue. >>>> >>>>>> Bobby >>>> >>>>>> On Fri, 01 Jan 2010 04:32:58 -0600, Noqsi <j...(a)noqsi.com> wrote: >>>>>>> On Dec 31, 1:16 am, DrMajorBob <btre...(a)austin.rr.com> wrote: >>>> >>>>>>>> This is a little like those idiotic SAT and GRE questions that ask >>>>>>>> "What's >>>>>>>> the next number in the following series?"... where any number >>>> will = >>> >>>>>> do. >>>>>>>> Test writers don't seem to know there's an interpolating >>>> polynomial= >>> >>>>>> (for >>>>>>>> instance) to fit the given series with ANY next element. >>>> >>>>>>> Explanations in terms of epicycles may be mathematically adequate >>>> in= >>> a >>>>>>> narrow sense, but an explanation in terms of a single principle >>>>>>> applied repeatedly is to be preferred in science. The ability to >>>>>>> recognize such a principle is important. >>>> >>>>>>> And my mathematical logician son (who's looking over my shoulder) >>>>>>> directed me tohttp://www.research.att.com/~njas/sequences/for >>>>>>> research on this topic. When he encounters such a sequence in his >>>>>>> research, he finds that knowledge of a simple genesis for the >>>> sequen= >>> ce >>>>>>> can lead to further insight. >>>> >>>>>> -- >>>>>> DrMajor...(a)yahoo.com >>>> >>>> -- >>>> DrMajor...(a)yahoo.com >>> >>> >> >> >> -- >> DrMajorBob(a)yahoo.com >> > -- DrMajorBob(a)yahoo.com |
