mm-460 Subject: Holograms & StringsAs Above, So Below?Susskind's world hologram suggests something likeLp* = Lp^2/3(c/H)^1/3H = R^-1dR/dt in FRW metric with Einstein cosmological constant/ ~ (H/c)^2 = Einstein's Cosmological Constantalpha' ~ Lp*^2 in Witten's string theoryi.e. (h = c =1 convention)alpha' ~ Lp^4/3//^2/3In terms of ADS the world Hubble horizon is ~ /^-1 in area where the sign is for dark energy repulsionEach Bekenstein Bit has quantum of area Lp*^2The Entropy of the expanding accelerating universe is then~ (Lp*^2/)^-1So is this plausible? It solves the Arrow of Time problem trivially by hooking it to the area of the Hubble horizon. The Second Law of Thermodynamics is explained as well as the origin of gravity and inertia and dark energy/matter in terms of Sakharov's Vision and P.W. Anderson's Battle Tested tried and true More is different.We don't need no damn G-string vibrators in hyperspace, loops and weaves Of course those ideas may play a useful role anyway.More with less.No excess metaphysical baggage and no excess mathematical baggage.The Question is: What is The Question? ;-)I got news for in my book Super Cosmos http://qedcorp.com/destiny/ under constructionBattle of the BooksI am Ralph Nader to 's beating about the Bush. :-)The Higgs ocean is my Vacuum Coherence Field calming the turbulent ZPF. It explains;1. Einstein's GR2. Dark energy and dark matter as virtual exotic vacuum stuff. Dark matter detectors will not click with right stuff.3. Stabililty of spatially extended leptons and quarks and why they look increasingly point like in high energy scattering (partons).4. Universal slope of Regge trajectories of hadronic resonances. alpha' ~ (1 Gev)^-25. Hierarchy problem - Quantum Gravity Planck area is actually ~ (1 fermi)^2 The weave is, perhaps, not as fine as previously thought, at least in the current cosmic epoch.6. Not to mention the unmentionable you know what.For I never use the Big Big U ...Chorus: What never?Well hardly ever ..http://math.boisestate.edu/gas/pinafore/captain.mp3http:// math.boisestate.edu/gas/pinafore/html/index.htmlBought Greene's book this morning - nice chapter on the Higgs fields called vaporizing the vacuum... Greene: If the Higgs ocean is not found, it will require a major rethinking of a theoretical framework that has been in place for more than thirty years === Subject: Re: Some Help is Needed> I need a little help in proving this statement. Thank you very much in> advance for helping me.> Let G be a group and let g E G. If g^2 /= e and g^6 = e, then prove> that g^4 /= e and g^5 /= e. What can you say about the order of g?> I couldn't really figure out how to write not equal to so the symbol I> used is /=.> I understand that since g^2 /= e then g^4 /= e.Ahh, no. Suppose G =Z/4Z. Then 1+1+1+1 = 0, but 1+1 /= 0.> Also the order of g is 6> So if anyone wants to guide me in the right direction it would greatly> be appreciated.Looking at g^4 again ... e = g^6 = (g^4)(g^2). Now supposeg^4 = e, and what can you conclude about g^2? === Subject: Re: Some Help is NeededI need a little help in proving this statement. Thank you very much inadvance for helping me. Let G be a group and let g E G. If g^2 /= e and g^6 = e, then provethat g^4 /= e and g^5 /= e. What can you say about the order of g?I couldn't really figure out how to write not equal to so the symbol Iused is /=.I understand that since g^2 /= e then g^4 /= e.> Ahh, no. Suppose G =Z/4Z. Then 1+1+1+1 = 0, but 1+1 /= 0.He meant that if g^6 = e and g^2 /= e, then g^4 /= e. This is in fact true.For if g^4 = e, then e = g^6 = (g^4)(g^2) = g^2. But g^2 /= e.Contradictcion, so g^4 /= e.Also the order of g is 6So if anyone wants to guide me in the right direction it would greatlybe appreciated.> Looking at g^4 again ... e = g^6 = (g^4)(g^2). Now suppose> g^4 = e, and what can you conclude about g^2? === Subject: Typo Greene Strings & Hologramalpha' ~ Lp^4/3//^2/3should bealpha' ~ Lp^4/3//^1/3 === Subject: Re: transfinite sequences>,>given a transfinite sequence {S_a} of reals, for auncountable ordinal), suppose the sequence is Cauchy (for any epsilon>in R there's countable ordinal c s.t. a,b > c implies |S_a + S_b| <>epsilon), then does {S_a} converge to a real number ? The question is>the generalisation of the usual construction of R from Q : could R be>analogously completed with respect to omega_1 ?Assuming you mean |S_a - S_b|, the answer is that S_a isconstant from some countable point on.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue Universityhrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: 1st order Diff Eqns> Hi there,> I've been studying some kinetics and have come up with this> differential eqn from my experiment.> dy/dx = 2y-xy^2> I can't for the life of me see how you can solve it. As far as I see> it using the substitution y=vx doesn't help. Can anyone lend me a> hand?but the substitution v = 1/y willv' = -y'/(yy)dy/dx [-1/(yy)] = 2y [-1/(yy)] - xy^2 [-1/(yy)]-y'/(yy) = -2/y + xv' = -2v + xv' + 2v = x> Thank you for your help, it is really really appreciated.your equation belongs to a family of differential equations of calledBernoulli equations:http://www.sosmath.com/diffeq/first/bernouilli/ bernouilli.html> Sarah === Subject: Re: 1st order Diff Eqns> http://www.sosmath.com/diffeq/first/bernouilli/ bernouilli.htmlUsually spelled Bernoulli, not in a pseudo-French manner Bernouilli === Subject: Re: JSH: Reply, reply, reply>Come on, reply to my posts!!! Come why don't you say something else,>huh?>Try to point out some mistake Nora Baron you stupid !>Hey, you lapdog David Ullrich, yeah, I called you a LAPDOG Ullrich!!!>Aren't you mad now? Hey, you know, you HAVE to reply right, or I'll>call your school and try to get you fired, right?>Ullrich the university professor cursing in the muck, with no shame.>REPLY TO MY POSTS YOU CUR!>Hey Dik Winter, you ing , reply again! Why don't you throw>more of my old arguments on a webpage, you stupid !!!>Yeah, you'll all be back want you? Along with C. Bond and all the>rest.>lapdogs>You are curs, my , my little traveling circus of obsessive>repliers.>I spit on you.>But you'll be back, now won't you?> Totally bizarre. When people reply to your posts your curse them> and tell them not to. When you make a post or two and people> don't reply immediately, possibly because they're at work or > something, you complain about that.> The strangest thing I've ever seen. And considering that I've> been reading _your_ posts for years that's saying something.>James Harris> ************************> *What Harris said was that you'd be back, and sure enough, there you are.earle* === Subject: Re: JSH: Reply, reply, reply>[...]>*>What Harris said was that you'd be back, and sure enough, there you >are.True. So? I've never taken issue with his claims to be in controlof the rest of us (given that he's ordered some of us not toreply and also ordered us to reply it's hard to see how we couldbehave in a way that could not be construed as giving him justwhat we wants. Why would anyone care about that?)>earle>************************* === Subject: Re: advice on graduate studies> I am an undergrad who will be graduating this semester and> will be going to grad school in fall((if they let me))> and i have a few questions.> I have heard that you should not only think about what you like> but also what is still a growing field.> I really enjoy analysis.((classical,functional,Fourier..))> I really dislike (at least so far) algebra,number theory ,combinatorics.> Now my question is : are the above named analysis's still a > good field to get into or should i look more at say applied> analysis.Someone once told me that unless you are a genious> that harmonic analysis is not a good field to get into.> Is this because the field is so old and developed?> Would the same apply to classical and functional analysis?> > I have done some PDE theory and enjoyed working in the> Sobolev spaces with the functional analysis techniques.> If i was to go into PDE's would I expect more of the same> (which i enjoyed) or does it turn into > a grab bag of techniques ?my limited experience tells me that some topics in harmonic analysisare essential in the study of partial differential equations. if youchoose to work in harmonic analysis problems but find yourselfstalled, you can find your way into PDE, which btw, is a ratherpopular field.> I have never done any measure theoretic probability> but considering my interests would this be another one> to think about?imo, it would be better you explore subjects for which you alreadyhave adequate exposure, like PDE. of course, you should take one ortwo courses on measure theory in grad school.> Any replies would be greatly appreciated> > don === Subject: Re: A newbie's question -- about real number> Two real numbers r1 = 0.89, r2 = 0.889999999..... (with infinite 9s), *Yes.earle* === Subject: Re: Mathcad Upgrade QuestionI actually, finally purchased Mathcad 2001 -- NOT 2001i -- andinstalled it. As far as I can tell, checking my processes and myregistry, c-dilla is not on here.Steve O.>Don't use mathcad. I installed version 2001i this weekend. After>installition I noticed that c-dilla had also been introduced to my registry.>Google c-dilla to find out what this program does. Anyhow I have spent 2>days rebuilding my hard drive because of the damage that Mathcad did.> I am thinking of upgrading from an old version of Mathcad (Version 7)> to the most recent. But past experience with upgrades has taught me> to be a bit wary, so 2 questions:> 1. Did Mathsoft do anything to screw up the product, in terms of> usability or features, since I last purchased it (back in 1997)?> 2. Does the latest product come with any kind of registration> feature, like Mathematica has, that locks your software into one> computer, and possibly locks you out of the software if you upgrade> your hardware? (I respect the rights of companies to make money off> their software, but I'm a firm believer that the only fair deal is> one-user/one-app. If I have three computers -- and I do -- I should> not have any worries or complications with loading the same software> package on all three of them.)> Steve O.> Standard Antiflame Disclaimer: Please don't flame me. I may actually>*be* an idiot, but even idiots have feelings. === Subject: Transcendental numbers other than pi and e?Are there other known, established transcendental numbers besides piand e? If so, I'd be curious to know what they are called, and ifthey can be characterized in a way that would be clear to mymathematically simple brain. (For example, I've never found it easyto understand were e comes from, but the idea that pi is the ratiobetween a circle's circumfirence and diameter is clear enough.)Steve O. === Subject: Re: Transcendental numbers other than pi and e?> Are there other known, established transcendental numbers besides pi> and e? If so, I'd be curious to know what they are called, and if> they can be characterized in a way that would be clear to my> mathematically simple brain. (For example, I've never found it easy> to understand were e comes from, but the idea that pi is the ratio> between a circle's circumfirence and diameter is clear enough.) Well, the sine of 1 degree is a transcendental number. The commonlogarithm of 4 is also a transcendental number. Transcendental numbers are, by definition, numbers that can not bederived from integers solely by arithmetic operations and rootextractions. Michael === Subject: Re: Transcendental numbers other than pi and e?Are there other known, established transcendental numbers besides piand e? If so, I'd be curious to know what they are called, and ifthey can be characterized in a way that would be clear to mymathematically simple brain. (For example, I've never found it easyto understand were e comes from, but the idea that pi is the ratiobetween a circle's circumfirence and diameter is clear enough.)> Well, the sine of 1 degree is a transcendental number. The common> logarithm of 4 is also a transcendental number.> Transcendental numbers are, by definition, numbers that can not be> derived from integers solely by arithmetic operations and root> extractions.Surely there are algebraic numbers that cannot be derived from integerssolely by arithmetic operations and root extractions? The unsolvabilityof the general quintic in radicals shows this.-- G.C. === Subject: Re: Transcendental numbers other than pi and e? <6930a3c6.0402230943.9c1e89@posting.google.com>Are there other known, established transcendental numbers besides piand e? If so, I'd be curious to know what they are called, and ifthey can be characterized in a way that would be clear to mymathematically simple brain. (For example, I've never found it easyto understand were e comes from, but the idea that pi is the ratiobetween a circle's circumfirence and diameter is clear enough.)> Well, the sine of 1 degree is a transcendental number.False: it is the imaginary part of the 90th root of i, hence algebraic.> The common> logarithm of 4 is also a transcendental number.> Transcendental numbers are, by definition, numbers that can not be> derived from integers solely by arithmetic operations and root> extractions. Incomplete or vague: some roots of polynomials of degree 5 or more withinteger coefficients cannot be obtained by radicals, if that's what youmean by root extraction. (Galois classified these polynomials.)Algebraic: roots of polynomials with integer coefficients.Transcendental: real or complex, but not algebraic.> MichaelCheers, ZVK(Slavek). === Subject: Re: Transcendental numbers other than pi and e?> Are there other known, established transcendental numbers besides pi> and e? If so, I'd be curious to know what they are called, and if> they can be characterized in a way that would be clear to my> mathematically simple brain. (For example, I've never found it easy> to understand were e comes from, but the idea that pi is the ratio> between a circle's circumfirence and diameter is clear enough.)It seems that those which arise naturally are few, or at least only afew are _known_ to be transcendental.Liouville constructed an infinity of transcendental numbers usingcontinued fractions. Cantor proved that almost all real numbers aretranscendental. [Btw, e arises when solving dy/dx = y, and in other ways.]-- G.C. === Subject: Re: Transcendental numbers other than pi and e?> >[Btw, e arises when solving dy/dx = y, and in other ways.]Yes, I knew that, but that's still not a geometrically tangible idea,the way pi is. I appreciate the other replies on the thread, and I did go toWolfram's site and some of the others. It seems there are dozens ofspecial mathematical functions that generate transcendentals. But piseems to be the only one that pops up from some picture that is bothsimple and of obvious relevance.Anyway, thanks again.Stee O. === Subject: Re: Transcendental numbers other than pi and e?>>[Btw, e arises when solving dy/dx = y, and in other ways.]> Yes, I knew that, but that's still not a geometrically tangible idea,> the way pi is. > I appreciate the other replies on the thread, and I did go to> Wolfram's site and some of the others. It seems there are dozens of> special mathematical functions that generate transcendentals. But pi> seems to be the only one that pops up from some picture that is both> simple and of obvious relevance.Let f(x) be pretty nearly any function from high school math. Graph y = f(x) from x = 0 to x = 1 and compute the arc length of that graph. Chances are you get a transcendental number. Pi is just the case f(x) = sqrt(1 - x^2), up to some rational multiplier. In particular, are lengths of ellipses, parabolas, and hyperbolas tend to be transcendental. As for e, draw the hyperbola xy = 1 and the vertical line x = 1 and ask yourself how far to the right of that vertical line do you have to go before the area between the hyperbola, the x-axis, and x = 1 accumulates to 1. The answer is e.-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Motivation for e; was : Re: Transcendental numbers other than pi and e?> >[Btw, e arises when solving dy/dx = y, and in other ways.]> >Yes, I knew that, but that's still not a geometrically tangible idea,>the way pi is. Here are three ways to motivate the concept of e.1) Compound interest: The effective annual yield for a nominal annual rate r compounded n times annually is (1 + r/n)^n. As n grows arbitrariliy large, the effective yield approaches e^r.2) With some effort, for positive a, one can show that if f(x) = a^x, then f is differentiable everywhere and f'(x) is proportional to f(x). e is the unique value of a which renders as unity the constant of proportionality. Geometircally tangible? Try drawing the graph of such a function.3) Letting f(x) = log_a (x) (logariithm to the base a) for a > 0, f'(x) is proportional to 1/x. e is the unique value of a which renders as unity the constant of proportionality.-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Transcendental numbers other than pi and e?>Are there other known, established transcendental numbers besides pi>and e? If so, I'd be curious to know what they are called, and if>they can be characterized in a way that would be clear to my>mathematically simple brain. (For example, I've never found it easy>to understand were e comes from, but the idea that pi is the ratio>between a circle's circumfirence and diameter is clear enough.)> >It seems that those which arise naturally are few, or at least only a>few are _known_ to be transcendental.Hmm, aren't the degree measures of the acute angles of a right triangle with integer sides transcendental? This is a sincere question - I don't know if that is actually known.-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Transcendental numbers other than pi and e?>>Are there other known, established transcendental numbers besides pi>>and e? If so, I'd be curious to know what they are called, and if>>they can be characterized in a way that would be clear to my>>mathematically simple brain. (For example, I've never found it easy>>to understand were e comes from, but the idea that pi is the ratio>>between a circle's circumfirence and diameter is clear enough.)>It seems that those which arise naturally are few, or at least only a>few are _known_ to be transcendental.> Hmm, aren't the degree measures of the acute angles of a right triangle> with integer sides transcendental? This is a sincere question - I don't> know if that is actually known.There are 2^{aleph_0} transcendental numbers, betcha no more thanaleph_0 are known. To me, aleph_0 out of 2^{aleph_0} is few :-)-- G.C. === Subject: Re: Transcendental numbers other than pi and e?> Are there other known, established transcendental numbers besides> pi and e? If so, I'd be curious to know what they are called, and> if they can be characterized in a way that would be clear to my> mathematically simple brain. (For example, I've never found it> easy to understand were e comes from, but the idea that pi is> the ratio between a circle's circumfirence and diameter is clear> enough.)> It seems that those which arise naturally are few, or at least>> only a few are _known_ to be transcendental.> Hmm, aren't the degree measures of the acute angles of a right> triangle with integer sides transcendental? This is a sincere> question - I don't know if that is actually known.easy : arctan p/q is transcendental if p and q are integers and p/q is not 0or 1...> There are 2^{aleph_0} transcendental numbers, betcha no more than> aleph_0 are known. To me, aleph_0 out of 2^{aleph_0} is few :-)If (a_i) is a sequence of 1 and 2 (and tehre is 2^(aleph_0) of those, x=sum(a_n /10^(n!)) is trancendental === Subject: Re: Transcendental numbers other than pi and e?> >[...]> Hmm, aren't the degree measures of the acute angles of a right>>triangle with integer sides transcendental? This is a sincere>>question - I don't know if that is actually known.>easy : arctan p/q is transcendental if p and q are integers and p/q is not 0>or 1...1) Proof or reference?2) But I made a claim about (arctan p/q) / pi. Is this still known to be transcendental?-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Transcendental numbers other than pi and e?>>Are there other known, established transcendental numbers besides pi>>and e? If so, I'd be curious to know what they are called, and if>>they can be characterized in a way that would be clear to my>>mathematically simple brain. (For example, I've never found it easy>>to understand were e comes from, but the idea that pi is the ratio>>between a circle's circumfirence and diameter is clear enough.)>It seems that those which arise naturally are few, or at least only a>few are _known_ to be transcendental.> Hmm, aren't the degree measures of the acute angles of a right triangle> with integer sides transcendental? This is a sincere question - I don't> know if that is actually known.> --> Stephen J. Herschkorn herschko@rutcor.rutgers.eduLindemann's Theorem produces plenty of transcendental numbers. Here it is,copied (filling in obvious omissions) from100 Great Problems of Elementary Mathematics, Their History and Solutionby Heinrich Doerrie, Dover, ISBN 0-486-61348-8: As usual, exp(z) means e^z.[begin]The [finite, nonmpty] expression A_1*exp(a_1) + A_2*exp(a_2) + ...in which the [algebraic] coefficients A_j differ from zero and in whichthe exponents a_j are algebraic numbers differing from each other, cannotequal zero.[end]The algebraic numbers are allowed to be complex.The transcendency of e follows from 1*exp(1) + A*exp(0) not being zero forany algebraic A.The transcendency of pi follows from 1*exp(i*pi) + 1*exp(0) = 0(if pi were algebraic then...)Now, natural logarithms (use notation log rather than ln) of (complex)algebraic numbers other than 1 are transcendental (set up the argumentimitating the above lines!). In particular, we learn in complex analysisthat arctan(x) = (1/(2*i)) * log ((1+i*x) / (1-i*x))If a right triangle has all sides integer and one of its acute angles istheta then tan(theta) is a rational number, so its arctangent (expressedby logarithms, and in radians!) is transcendental. And who uses angulardegrees in calculus anyway? :-)=Seriously, the degree measure of theta is simply theta*(180/pi); it may beeasy to complete the answer to your degree inquiry, but I am eager to gohome now.Cheers, ZVK(Slavek). === Subject: Re: Transcendental numbers other than pi and e?Are there other known, established transcendental numbers besides piand e? If so, I'd be curious to know what they are called, and ifthey can be characterized in a way that would be clear to mymathematically simple brain. (For example, I've never found it easyto understand were e comes from, but the idea that pi is the ratiobetween a circle's circumfirence and diameter is clear enough.)> It seems that those which arise naturally are few, or at least only a> few are _known_ to be transcendental.> Liouville constructed an infinity of transcendental numbers using> continued fractions. Cantor proved that almost all real numbers are> transcendental.> [Btw, e arises when solving dy/dx = y, and in other ways.]See Courant, R. and Robbins, H. What Is Mathematics?: An ElementaryApproach to Ideas and Methods for Liouville's construction oftranscendental numbers.-- G.C. === Subject: Re: Transcendental numbers other than pi and e? Adjunct Assistant Professor at the University of Montana.>Are there other known, established transcendental numbers besides pi>and e? Yes; of course, you probably want to avoid things like k*pi or k*e,where k is a rational...> If so, I'd be curious to know what they are called, and if>they can be characterized in a way that would be clear to my>mathematically simple brain. (For example, I've never found it easy>to understand were e comes from, but the idea that pi is the ratio>between a circle's circumfirence and diameter is clear enough.)There's Liouville's constant,L = sum_{i=0 to infinity} 10^{n!};The Gelfond-Schneider constant, 2^{sqrt(2)}.Champernowne's constant, 0.12345678910111213... (list all positiveintegers in order)And others. See for example,http://mathworld.wolfram.com/ TranscendentalNumber.htmlSome of them arise naturally; other's arise when trying to findtranscendental numbers (either for the sake of it, or to answerquestions specifically about transcendental numbers, such as Hilbert's7th problem).-- === ==Arturo Magidinmagidin@math.berkeley.edu===Subject: Re: Transcendental numbers other than pi and e?Isn't natural log of 2 transcendental?? Or have I been mistaken all theseyears?>Are there other known, established transcendental numbers besides pi>and e?> Yes; of course, you probably want to avoid things like k*pi or k*e,> where k is a rational...If so, I'd be curious to know what they are called, and if>they can be characterized in a way that would be clear to my>mathematically simple brain. (For example, I've never found it easy>to understand were e comes from, but the idea that pi is the ratio>between a circle's circumfirence and diameter is clear enough.)> There's Liouville's constant,> L = sum_{i=0 to infinity} 10^{n!};> The Gelfond-Schneider constant, 2^{sqrt(2)}.> Champernowne's constant, 0.12345678910111213... (list all positive> integers in order)> And others. See for example,> http://mathworld.wolfram.com/TranscendentalNumber.html> Some of them arise naturally; other's arise when trying to find> transcendental numbers (either for the sake of it, or to answer> questions specifically about transcendental numbers, such as Hilbert's> 7th problem).> --> === =========================================================== === =====It's not denial. I'm just very selective about> what I accept as reality.> --- Calvin (Calvin and Hobbes)> === =========================================================== === =====> Arturo Magidin> magidin@math.berkeley.edu===Subject: Re: Transcendental numbers other than pi and e?> Isn't natural log of 2 transcendental?? Or have I been mistaken all> these years?It is (by Gelfond-Schneider theorem). Lots of other similar numbers are (ln(a) for all algebraic =/= 1, a^b for all and b algebraic, a =/= 0 or1, and b irrational, etc.) Conjectures are that most usual numbers aretranscendental (like e+pi, or ln(2)/(sqrt(pi)+1),...but no proof are yetknown...> Steven O. > and e?> Yes; of course, you probably want to avoid things like k*pi or k*e,> where k is a rational...>> If so, I'd be curious to know what they are called, and if>> they can be characterized in a way that would be clear to my>> mathematically simple brain. (For example, I've never found it easy>> to understand were e comes from, but the idea that pi is the ratio>> between a circle's circumfirence and diameter is clear enough.)> There's Liouville's constant,> L = sum_{i=0 to infinity} 10^{n!};> The Gelfond-Schneider constant, 2^{sqrt(2)}.> Champernowne's constant, 0.12345678910111213... (list all positive> integers in order)> And others. See for example,> http://mathworld.wolfram.com/TranscendentalNumber.html> Some of them arise naturally; other's arise when trying to find> transcendental numbers (either for the sake of it, or to answer> questions specifically about transcendental numbers, such as> Hilbert's 7th problem).> --> === =========================================================== === =====> It's not denial. I'm just very selective about> what I accept as reality.> --- Calvin (Calvin and Hobbes)> === =========================================================== === =====> Arturo Magidin> magidin@math.berkeley.edu===Subject: Re: Transcendental numbers other than pi and e? charset=iso-8859-7 Arturo Magidin > There's Liouville's constant,> L = sum_{i=0 to infinity} 10^{n!};Arturo, perhaps you mean:L=sum_{i=0,+oo} 1/10^{i!}?Also, an easy consequence (should be) that one could derive countably manytranscendentals from the above construction :*)> Arturo Magidin> magidin@math.berkeley.eduIoannis Galidakishttp://users.forthnet.gr/ath/jgal/------------------- -----------------------Eventually, _everything_ is understandable === Subject: Re: Transcendental numbers other than pi and e? Adjunct Assistant Professor at the University of Montana.Ioannis escribio:> Arturo Magidin [NonBreakingSpace].8b.96.87.8c .97.99.95 .93fi.92.9b.93.87> There's Liouville's constant,> L = sum_{i=0 to infinity} 10^{n!};> Arturo, perhaps you mean:> L=sum_{i=0,+oo} 1/10^{i!}> ?Actually, I meant L = sum_{n=0 to infinity} 10^{n!}... (-:Sorry for the typo...-- === ==Arturo Magidinmagidin@math.berkeley.edu === Subject: Re: Transcendental numbers other than pi and e?Arturo, perhaps you mean:L=sum_{i=0,+oo} 1/10^{i!}?> Actually, I meant L = sum_{n=0 to infinity} 10^{n!}... (-:I think his point was your series does not converge. === Subject: Re: Transcendental numbers other than pi and e?> Arturo, perhaps you mean:>> L=sum_{i=0,+oo} 1/10^{i!}>> ?> > Actually, I meant L = sum_{n=0 to infinity} 10^{n!}... (-:>I think his point was your series does not converge.So, two typos...-- === ==Arturo Magidinmagidin@math.berkeley.edu === Subject: Re: Transcendental numbers other than pi and e?>> Arturo, perhaps you mean:> L=sum_{i=0,+oo} 1/10^{i!}> ?>> >> Actually, I meant L = sum_{n=0 to infinity} 10^{n!}... (-:>I think his point was your series does not converge.>So, two typos...True. But as far as I can see _this_ post was absolutelyperfect! Nice work.************************ === Subject: Re: Transcendental numbers other than pi and e?> Arturo, perhaps you mean:> L=sum_{i=0,+oo} 1/10^{i!}> ?Actually, I meant L = sum_{n=0 to infinity} 10^{n!}... (-:> I think his point was your series does not converge.Right.Liouville's constant is sum_{n=1 to infinity} 10^{-n!} .David === Subject: Re: Transcendental numbers other than pi and e? charset=iso-8859-7 Arturo Magidin > Actually, I meant L = sum_{n=0 to infinity} 10^{n!}... (-:But wouldn't L above diverge? Or am I missing something obvious?> Sorry for the typo...> Arturo Magidin> magidin@math.berkeley.eduIoannis Galidakishttp://users.forthnet.gr/ath/jgal/------------------- -----------------------Eventually, _everything_ is understandable === Subject: Re: Transcendental numbers other than pi and e?En el mensaje:1077489992.165251@athnrd02.forthnet.gr,Ioannis escribio:> Arturo Magidin [NonBreakingSpace].8b.96.87.8c .97.99.95 .93fi.92.9b.93.87> There's Liouville's constant,> L = sum_{i=0 to infinity} 10^{n!};> Arturo, perhaps you mean:> L=sum_{i=0,+oo} 1/10^{i!}> ?> Also, an easy consequence (should be) that one could derive countably> many transcendentals from the above construction :*)AlsoN(b) = Sum(1/b^2^n, n, 1, inf) for b integer >= 2 (I'm not sure for b = 2)The prove is independent of Lioville's Th, but relatively easy:An Oceans of Zeros Proof That a Certain Non-Liouville Number is Transcendental M. J. Knight American Mathematical Monthly, Vol. 98, No. 10. (Dec., 1991), pp. 947-949.-- Ignacio Larrosa CanestroA Coruna (Espana)ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: Advice on future with Math> I'm an upperclass math major that was/is planning on attempting a masters or> phd in math. Here lately, I've wondered if this is a great idea. > My first question is this: How much should you study (reading the material,> working problems) for, say, an abstract algebra or advanced calculus> junior/senior level class? I talked to a graduate student that said> he studied for a couple of his first year grad classes at least 4 hours> a day on weekdays and about 8 hours on Saturday and Sunday. Needless to> say, I do nothhing like that. Should I be doing that or is that just> an insane amount of time to be spending on it? It seems that if you're> good at something, it shouldn't require so much work.it depends - some people are dumber than others. for instance i recallworking even days on single analysis exercises. although, i had quitea number of scummy nitwits in charge of supplying me with the basics.> At times I have difficulty understanding things that, after finding out> where I went wrong, seem pretty simple. I assume others don't have> this problemthat is the learning process. i would say it is somewhat normal.> because normally they can respond right away with what some math> concept meant or, less often, help me where I am having a problem> understanding part of a proof. Is it normal to not follow a proof> or just a sign that I'm just weak mathematically (we're again talking> junior/senior level advanced calc/abstract algebra here)?depends. if you seem to have trouble understanding points and detailsmost of the time, that is likely a sign of a mediocre preparation. onthe other hand, if you experience such difficulty ocassionally, thenit is a matter of spending more time and being persistent.> I have a bad tendency to be hot/cold when it comes to math. It seems> like I'll have the highest/near highest score on an exam and then> really screw up on the next exam, getting something in the bottom> half of the grades. I wish that I could attribute it to something like> when I first started school: I would do no work until after I screwed up> on the first exam, and then work harder after that and do quite well.> Now it seems that I'll do well on the first exam and then screw up on> the next one. I really don't think that I will have studied> much less for the second one but I guess it could be a possibility. When I> do bad on the other exam, I begin to wonder if I just got lucky on the> problems that were given on the good exam and would have done lousy if a> few different problems were picked.do you study by yourself or do you have group studies?group studies seem to be ineffective in higher math, except when onehas successfully solved problems independently prior to the groupstudy session.> Other times, I wonder if I am just not good at proofs, thus making me> about the crappiest grad school candidite out there.quality of students and schools are not high these days. worse casescenario you are in the norm.> I sit and stare at the homework problems and often can't get anywhere.bad sign - if you can't get anywhere in routine exercises, you maybe missing something significant. do you spend sufficient timestudying math? or do you like partying a lot? is it possible thatthere is an assortment of incompetent teachers in charge of provideingyou with the basics?> After I see an answer, it doesn't look too bad but it just seems that> they're impossible at times. I had a bad proofs> class that was designed for just about anyone wanting to take it, so it was> really easy. I'm not sure if I can use that excuse at this point though. My> pre-calculus education was really bad since I was from a really poor school (I> often have no clue what certain things that were assumed as 'knowledge' from> high school) but again this doesn't seem to matter too much in the proof-based> classes.the future seems grim for you - i doubt any good graduate school willtake in a student with a weak backround.quit math while you are not so far behind, unless you really love itfor its own sake.> Well, that was kind of long-winded. I hope that I got the point across. Any> advice would be helpful. === Subject: Re: needing help from a kind soulthanks for all your help. You all are indeed kind souls. === Subject: p_(n+1) < 2 p_nLet p_n be the nth prime. Is there a simple proof that p_(n+1) < 2 p_n for all n?-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: p_(n+1) < 2 p_n> Let p_n be the nth prime. Is there a simple proof that p_(n+1) < 2 p_n> for all n?It follows from the fact that for all n >= 2, there is a prime p such thatn < p < 2n. Here's Erdos' proof: http://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate-- --Tim Smith === Subject: Stepwise regressionI'd like to know the advantages of a stepwise linear regression. In what waydoes it differentiate itself from a normal linear regression?any comments are appreciatedNicolas-- === Subject: Re: Stepwise regression> Z.Stunic @eudoramail.com> replied: to Michael Jrgensen> Certainly, my problem is not so simply. Symbols aj and bj denotes> coefficients(or parameters) for one particular subsets from original> set of data marked B. Only some of xi,yi data is related to for> example a1 and b1 coefficients, other member of original set is> connected with to another pair of coefficients a2,b2 and so far.> Obviously, with index j=1,2,...=K we can specify one particular subset> inside of set B. Number of possible subsets inside original ones B is> a priore unknown. There are two special cases:> 1) a1 = a2=..........aj =.......aK; b1 <> b2 <> b3 <>......bj <> bK ,> and> 2) a1 <> a2 <>........aj <>.....aK; b1 = b2 = b3 =......bj = bK.> Thank you for giving my opportunity to define my problem on better way> Z.StunicIf I understand you, and I'm not so sure of that, you have a bunch ofordered pairs, some subsets of which you want to regress linearly.What is keeping you from using ordinary least squares on thosesubsets? === Subject: Re: Stepwise regressionfor one, if the parameters (a&b in y= ax+b) are not constant then thistechnique would stand a good chance of revealing this.may also be quicker if you have to do it in real time, not sure about thatthough.> I'd like to know the advantages of a stepwise linear regression. In whatway> does it differentiate itself from a normal linear regression?> any comments are appreciated> Nicolas> -- === Subject: Re: Stepwise regression[... moved top post to bottom...]I'd like to know the advantages of a stepwise linear regression. In what> waydoes it differentiate itself from a normal linear regression?> for one, if the parameters (a&b in y= ax+b) are not constant then this> technique would stand a good chance of revealing this.> may also be quicker if you have to do it in real time, not sure about that> though.Rob's reply is pretty nonsensical.Stepwise regression is a way to putpredictor variables into a regression model one at a time. This isespecially useful if you have lots of variables and want to screen them,as only significant predictors are added to the model. (In mostcomputer implementations you get to decide what significant means.)The typical model you strive for is parsimonious, that is, it containsonly few predictors, and every predictor contributes significantly toexplaining the variation of the response.The downside to stepwise regression is that you leave the selectionof important variables to the computer, and all it has to go by isthe data. If you do not check that the resulting model is plausiblebased on your own understanding of the problem, you can end upwith pretty wacky models.Consult an introductory stats text for further information --- this isstandard stuff, and every text should cover it. === Subject: Re: differential equations> lost in finding the necessary differential equation. Any help would be> appreciated. Marine Biologists have determined that when a shark detects the> presence of blood in the water, it will swim in the direction in which the> concentration of the blood increases most rapidly. Based on certain tests> in seawater, the concentration of blood (in parts per million) at a point> P(x,y) on the surface is approximated by C(x,y)= e^ {(-x^2+2y^2)/10^4}where> x and y are measured in meters in a rectangular coordinate system with the> blood source at the origin. Suppose a shark is at the point (Xo,Yo) when it> first detects the presence of blood in the water. Find an equation of the> shark's path by setting up and solving a differential equation.The shark should head in the direction of the gradient vector, which at any point (x,y) is a positive multiple of (-2x,4y). This looks pretty bogus. For example, if the shark is on the y-axis he should swim directly away from the blood source? Are you sure it isn't e^{-(x^2+2y^2)/10^4}? === Subject: Re: differential equationsat alost in finding the necessary differential equation. Any help would beappreciated. Marine Biologists have determined that when a shark detectsthepresence of blood in the water, it will swim in the direction in whichtheconcentration of the blood increases most rapidly. Based on certaintestsin seawater, the concentration of blood (in parts per million) at apointP(x,y) on the surface is approximated by C(x,y)= e^{(-x^2+2y^2)/10^4}wherex and y are measured in meters in a rectangular coordinate system withtheblood source at the origin. Suppose a shark is at the point (Xo,Yo) whenitfirst detects the presence of blood in the water. Find an equation oftheshark's path by setting up and solving a differential equation.> The shark should head in the direction of the gradient vector, which atany> point (x,y) is a positive multiple of (-2x,4y). This looks pretty bogus.> For example, if the shark is on the y-axis he should swim directly away> from the blood source? Are you sure it isn't e^{-(x^2+2y^2)/10^4}?Yes,yes, I know all that and I showed my fiend this but the instruction tothe problem requested solving this with a differential equation. This iswhere I am confused or is it brain dead for missing the obvious. Thank forall your help and I would appreciate a reply. === Subject: Re: differential equations>lost in finding the necessary differential equation. Any help would be>appreciated. Marine Biologists have determined that when a shark detects the>presence of blood in the water, it will swim in the direction in which the>concentration of the blood increases most rapidly. Based on certain tests>in seawater, the concentration of blood (in parts per million) at a point>P(x,y) on the surface is approximated by C(x,y)= e^ {(-x^2+2y^2)/10^4}where>x and y are measured in meters in a rectangular coordinate system with the>blood source at the origin. Suppose a shark is at the point (Xo,Yo) when it>first detects the presence of blood in the water. Find an equation of the>shark's path by setting up and solving a differential equation.Hint: Grad(C) is proportional to <-2x, 4y> so you need dy/dx = -2y/xfor the orthogonal trajectories to the level curves of C...--Lynn === Subject: Re: differential equations> lost in finding the necessary differential equation. Any help would be> appreciated. Marine Biologists have determined that when a shark detects the> presence of blood in the water, it will swim in the direction in which the> concentration of the blood increases most rapidly. Based on certain tests> in seawater, the concentration of blood (in parts per million) at a point> P(x,y) on the surface is approximated by C(x,y)= e^ {(-x^2+2y^2)/10^4}where> x and y are measured in meters in a rectangular coordinate system with the> blood source at the origin. Suppose a shark is at the point (Xo,Yo) when it> first detects the presence of blood in the water. Find an equation of the> shark's path by setting up and solving a differential equation.direction of most rapid increase is related to gradient.Look it up in a multivariable calculus text. === Subject: vertex covering, integer programming, and linear optimizationcovering problem, namely, given a graph G = {E, V}, what is the smallest numberof vertices we must choose in V, so that every edge contains at least one ofthe chosen vertices. What I would like is a way to obtain a 2-approximation (using linearoptimization, and then explaining how the rational solutions lead to integralsolutions). Any assistance would be appriciated.--David Jerrisfield === Subject: Question regarding Archimedes' use of infinity to find a volumeI watched the Nova program Infinite Secrets last September, and Ihave been wondering ever since about a part of his Palimpsest thatthey only briefly mentioned. At one point, they found that Archimedeshad developed a way to use infinite slices to calculate the volume ofan odd shape, but the program does not explain what this method is. Ihave been struggling to find an answer to this question, but have hadno success. Does anyone here happen to know what reasoning Archimedesused to develop his method and what that method was? When I saw theprogram, I was taking multivariable calculus and DifferentialEquations, so you have an idea of where I am in my math education.Please, does anyone know the answer?Renee === Subject: Re: Question regarding Archimedes' use of infinity to find a volume> I watched the Nova program Infinite Secrets last> September, and I have been wondering ever since> about a part of his Palimpsest that they only briefly> mentioned. At one point, they found that Archimedes> had developed a way to use infinite slices to calculate> the volume of an odd shape, but the program does not> explain what this method is. I have been struggling> to find an answer to this question, but have had no> success. Does anyone here happen to know what> reasoning Archimedes used to develop his method and> what that method was?Archimedes was a great inventor and very able in geometryand reckoning. Here is a good reference regarding hisvolume computation:http://www.mathpages.com/home/kmath343. htmArchimedes knew about the two faces of invention:(1) How to play with thoughts and analogies and combining known facts.(2) How to rigorously present the results: as rockhard formulas as well as by elegant proofs.In a letter to a friend he revealed his Method, whichis a more informal description of (1). This Method waswidely known to exist in some copies but all of them seemedto have been lost. Until in 1907 a copy was found in Russia.This was nearly unreadable due to the fact that somebodyhad been using the paper again (reused folia = palimpsest).interesting finding:http://www.wired.com/news/school/0,1383,39870,00.html To reconstruct the original document, Walters set up a competition to search for a team of scientists able to decipher the work. We had lots of applications for the job globally and we whittled it down to two, Noel said. The two winners were the Rochester Institute of Technology (RIT) and Johns Hopkins University. Both teams were given five pages of the palimpsest to work from. The five pages had a variety of problems -- faint text, mold -- one had a forgery on it, Noel said. The manuscript was so badly damaged that traditional methods of imaging, using visible or ultraviolet light, proved ineffective.of mathematics at the U. S. Military Academy at West Point:http://www.ihes.fr/~ilan/sawit.html where he describes, howhe got near to that funny thing. Unfortunately the link isbroken. If you like to search any further, I give you theinformation at the end of that HTML-File:MAA Online is edited by Fernando Q. Gouv.90a (fqgouvea@colby.edu).Last modified: Wed Oct 28 17:57:21 -0500 1998Or if you are interested, I can send it to you by e-mail.Rainer Rosenthalr.rosenthal@web.de === Subject: Re: Question regarding Archimedes' use of infinity to find a volume> I watched the Nova program Infinite Secrets last September, and I> have been wondering ever since about a part of his Palimpsest that> they only briefly mentioned. At one point, they found that Archimedes> had developed a way to use infinite slices to calculate the volume of> an odd shape, but the program does not explain what this method is. I> have been struggling to find an answer to this question, but have had> no success. Does anyone here happen to know what reasoning Archimedes> used to develop his method and what that method was?He divided the volume into infinitesmial slices and used his leverprinicples to change those slices into more managable slices bybalancing the original slices with the new slices.> When I saw the> program, I was taking multivariable calculus and Differential> Equations, so you have an idea of where I am in my math education.> Please, does anyone know the answer?> ReneeLook for the Great Books of the Western World. This is a 50 or sovolume of translations into English of classic books of thewestern world.One volume is partially devoted to the works of Archimedes.These volumes may be on the reference shelf of your library.-- Bill Hale === Subject: Re: Question regarding Archimedes' use of infinity to find a volumeRenee Reavis> I watched the Nova program Infinite Secrets last September, and I> have been wondering ever since about a part of his Palimpsest that> they only briefly mentioned. At one point, they found that Archimedes> had developed a way to use infinite slices to calculate the volume of> an odd shape, but the program does not explain what this method is. I> have been struggling to find an answer to this question, but have had> no success. Does anyone here happen to know what reasoning Archimedes> used to develop his method and what that method was? When I saw the> program, I was taking multivariable calculus and Differential> Equations, so you have an idea of where I am in my math education.> Please, does anyone know the answer?The keyword is method of exhaustion. Google found this:http://www.newton.dep.anl.gov/newton/askasci/1995/math/ MATH015.HTMVery official-looking :)LH === Subject: Re: Fundamental Theorems of CalculusI did some investigating with my graphing calculator last night whileconsidering what the two previous responses had said, and I think I'mstarting to understand. Just to make sure, would it be safe to saythat the following is correct:1. F(x) = INT[from a to x] (f(t)dt) = f(x) + C2. d/dx F(x) = f(x)Joh === Subject: Re: Fundamental Theorems of Calculus> I did some investigating with my graphing calculator last night while> considering what the two previous responses had said, and I think I'm> starting to understand. Just to make sure, would it be safe to say> that the following is correct:> 1. F(x) = INT[from a to x] (f(t)dt) = f(x) + Cwhat went wrong? int_a^x f(t)dt is certainly not f(x) + Ce.g., int_0^x cos(t)dt = sin(x), which is not cos(x)i believe you meant to say, int_a^x f(t)dt = F(x) - F(a)> 2. d/dx F(x) = f(x)and this.the two boils down to d/dx [int_a^x f(t)dt] = F(x), making it clearthat the choice of a (and, ahem, F(a)), is arbitrary and doesn'treally concern us.> Johyou're welcome. :) === Subject: Re: JSH: Understand now? Frustration? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1N1UDb24831;> >> Using P(x) = (x+8a)(x+b), where ab=1, and considering when (8a + b) is> an integer can give you a perspective on what I've been saying, I> hope.>> Since b = 1/a, by your requirement, then (8a + b) = (8a + 1/a). Are you> requiring that this expression be an integer?>> To be fair, he said consider those cases where (8a+b) _is_ an integer.> In a previous post he chose (8a+b) = 9, ab=1 which has the two solutions>> [b=8, a=1/8] and [b=1, a=1]>> so both (8a + b) and (8a + 1/a) are indeed integer even though a=1/8 is not.>> KeithK> > I don't believe I was being unfair. I was simply asking for clarification. James> has a history of posting contradictory requirements and leaving the burden of> choosing what to work with to the reader.> >I've been arguing with posters like Arturo Magidin for YEARS, and>throughout that period, I've found my attempts to clarify were>hampered by attempts by other posters like here C. Bond to confuse,>and I thank KeithK for catching one as he did.Didn't Arturo win every single argument?>The problem isn't with C. Bond using 8a + 1/a as that is correct, as>long as you consider the possibility of 'a' being a factor of 1 in>some more inclusive ring than the ring of algebraic integers.>Circular arguments have gone on and on and on for some years here, as>people like Arturo Magidin, have successfully dodged being pinned down>on this issue as they repeatedly relied on certain numbers not being>algebraic integers to support their claims.So far as I know Arturo was right every time.Are you saying he wasn't?>Notice, I'm NOT saying that 'a' is an algebraic integer when it's>irrational!>If you say that 1/a is not an algebraic integer, when irrational, yes,>you are correct.>But my point is that the label algebraic integer does not contain>all numbers such that 1/a is contained and -1 and 1 are the only>integer units. There's a more inclusive ring, where -1 and 1 are the>only integer units, where 1/a is a unit, and 'a' is not an algebraic>integer.>Here it takes effort if some poster comes in to confuse you, relying>on the *assumption* that if 1/a is not an algebraic integer then it's>more like a fraction like 1/2 than 1/1.>But think of x=(1+sqrt(-3))/2, and someone giving you 1/x, if you were>mostly experienced with integers.>Think of how easy it might be for someone to confuse you with>2/(1+sqrt(-3)) if you weren't careful and it was new to you.Well, x = (1+sqrt(-3))/2 is an algebraic integer _and_ a unit.>That's has to do with why at times I've just gotten totally pissed and>called Arturo Magidin evil, as it's just so frustrating to deal with>people so adept at obscuring the truth.I think you must mean, adept at math. What Arturo said was almost invariably true. What youthe truth while you denied it as long as possible.>My hope is that at least some of you will play with>x^2 + (8a + b)x + 8, with ab = 1, and (8a + b) an integer>so that you can see why *logically* a possibility must still be there>when 'a' and 'b' are irrational, even if 'a' is not an algebraic>integer.If ab = 1 and 8a + b is an integer, then8a + 1/a = m for some integer m. This implies8a^2 - m a + 1 = 0,so a is an algebraic _number_, and in factit equals(m +/- sqrt(m^2 - 32))/16.This can be an algebraic integer if m = 9 or m = -9and not otherwise I think. If m = 7, for example,a = (7 +/- sqrt(17))/16, and in that case 8a + b = 7, but ... so what?Similarly, if m = 11,a = (11 +/- sqrt(89))/16, 8a + b = 11. So what?m = 13:a = (13 +/- sqrt(137))/16, 8a + b = 13. So what? etc.Do you see something strange or mysterious aboutthis? Clearly there are infinitely many cases where a and b are irrational, ab=1, and 8a + b isan integer. All such a and b must be algebraic_numbers_. Usually they are both irrational. So?Are you thinking that for some reason there is a need to invent a new ring for numbers like a?Why? Why do you think the existing ring of algebraicnumbers isn't sufficient? Let's take m = 1. Thena = (1 +/- sqrt(-31))/16. Do you want both of these roots to be in yournew ring? Because if you do, then their sum isalso. That means that 1/8 is in your new ring,which means 8 is a unit. Do you want that?If you only want one of the roots to be in thenew ring, how do you pick which one?>If you can get a handle on that possibility, then maybe discussions>can be more fruitful about those numbers lost in the shuffle that>irrational 'a' here represents.>You know, they are *most* numbers given their cardinality.As I noted above a and b are both algebraic numbers.Therefore their cardinality is the same as that of the integers,ie, countable. There are no more of them than thereare integers or rationals. Most makes no sense.The real mystery with this particular thread is,what the heck is your point?- Whipple>James Harris === Subject: Re: 1st order Diff Eqns by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1N1UEu24872;> http://www.sosmath.com/diffeq/first/bernouilli /bernouilli.html>Usually spelled Bernoulli, not in a pseudo-French manner Bernouilliand you tell me this because...? === Subject: Re: I got low score on math test, please advise me and take a look by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1N1ahY25201;...> don't you think that such rule should have a> provision stateing that students need some> performance feedback prior to the drop deadline?> How do you know that he didn't get any performance> feedback before the deadline?i believe he said that somewhere. now, can you answermy question - perhaps in a direct manner this time?> Fine. RULES ARE RULES. do you always follow rules blind-folded, or are there ocassionswhere you wonder if some rules are well-founded?> How's that?does not seem to address the question i originally asked in adirect manner.>Doug === Subject: Re: I got low score on math test, please advise me and take a look> .> .> .> don't you think that such rule should have a> provision stateing that students need some> performance feedback prior to the drop deadline?>> How do you know that he didn't get any performance> feedback before the deadline?> i believe he said that somewhere. now, can you answer> my question - perhaps in a direct manner this time?Fine. RULES ARE RULES.> do you always follow rules blind-folded, or are there ocassions> where you wonder if some rules are well-founded?Non-sequitur, since I agree completely with this rule.How's that?> does not seem to address the question i originally asked in a> direct manner.It certainly did. You didn't like the answer, but it was direct.Doug === Subject: Re: I got low score on math test, please advise me and take a look...> don't you think that such rule should have a> provision stateing that students need some> performance feedback prior to the drop deadline?>> How do you know that he didn't get any performance> feedback before the deadline?>> i believe he said that somewhere. now, can you answer> my question - perhaps in a direct manner this time?> Fine. RULES ARE RULES.do you always follow rules blind-folded, or are there ocassionswhere you wonder if some rules are well-founded?> Non-sequitur, since I agree completely with this rule.i am not particularly questioning whether you agree with that onerule. my specific question to you, is whether you agree with the ideathat a withdrawal deadline should have a provision stating thatstudents need some performance feedback prior to the deadline.a simple 'yes' or 'no' will suffice.> How's that?does not seem to address the question i originally asked in adirect manner.> It certainly did. You didn't like the answer, but it was direct.> Doug === Subject: Re: I got low score on math test, please advise me and take a look>> don't you think that such rule should have a> provision stateing that students need some> performance feedback prior to the drop deadline?>> How do you know that he didn't get any performance> feedback before the deadline?>> i believe he said that somewhere. now, can you answer> my question - perhaps in a direct manner this time?>> Fine. RULES ARE RULES.> do you always follow rules blind-folded, or are there ocassions> where you wonder if some rules are well-founded?Non-sequitur, since I agree completely with this rule.> i am not particularly questioning whether you agree with that one> rule. my specific question to you, is whether you agree with the idea> that a withdrawal deadline should have a provision stating that> students need some performance feedback prior to the deadline.> a simple 'yes' or 'no' will suffice.I'm not playing your lawyer games. I'm not on trial, and you're notcrossxamining me. Good night.Doug === Subject: Re: P vs NP and the analog machine > Another way of saying it is that the algorithm samples about sqrt(N!)> paths. However, since the sample,is a guassian with about the same variance> and average, the algorithm compares about N!( 1 - 1/sqrt(N!)) paths!. Wouldn't it be sufficient to say that the edge weights are uniformly distributed over some interval of values, and then for large N, a selection of N edges (to make a tour) would make such randomly chosen tours have weights approximating a normal distributio?J === Subject: Re: P vs NP and the analog machineyes, but i want to say a couple of things. First of all, the algorithm isntlimited to a_ij distributed uniformly. This is because, we are depending onthe properties of the central limit theorem.Secondly,> Wouldn't it be sufficient to say that the edge weights are uniformly> distributed over some interval of values, and then for large N, a> selection of N edges (to make a tour) would make such randomly chosen> tours have weights approximating a normal distributio?This is true. But i am not exactly talking about that. I will find betterwords to explain later.Anyways, do u get the idea?Another way of saying it is that the algorithm samples about sqrt(N!)paths. However, since the sample,is a guassian with about the samevarianceand average, the algorithm compares about N!( 1 - 1/sqrt(N!)) paths!.> J === Subject: Re: P vs NP and the analog machine > Secondly, Wouldn't it be sufficient to say that the edge weights are uniformlydistributed over some interval of values, and then for large N, aselection of N edges (to make a tour) would make such randomly chosentours have weights approximating a normal distributio?> This is true. But i am not exactly talking about that. I will find better> words to explain later.> Anyways, do u get the idea? Yeah, I think I get the idea, but my point was that (I think) you were assuming the edge weights to be drawn from a gaussian distribution, and my comment was suggesting that you could have the edge weights generated from many other kinds of distributions as well.J === Subject: Re: P vs NP and the analog machineBtw, looking at the central limit theoremthe average approaches n mu and the standard deviation approachessqrt{n} sigmaTaking the pecentage deviation of the tour costs frac{sqrt{n} sigma}{nmu} = frac{1}{sqrt{n}} frac{sigma}{mu}In a way, it even gets pointless to look for a solution to the salesmanproblem as n->infty, since all paths are approximately going to be equal.Going along one path would equal all paths travelled!!!-sureshSecondly,> Wouldn't it be sufficient to say that the edge weights are uniformly> distributed over some interval of values, and then for large N, a> selection of N edges (to make a tour) would make such randomly chosen> tours have weights approximating a normal distributio?This is true. But i am not exactly talking about that. I will findbetterwords to explain later.Anyways, do u get the idea?> Yeah, I think I get the idea, but my point was that (I think) you were> assuming the edge weights to be drawn from a gaussian distribution, and my> comment was suggesting that you could have the edge weights generated from> many other kinds of distributions as well.> J === Subject: Re: P vs NP and the analog machineWhat i m trying to say is that they will have about the same cost!!! Goingalong one random path would have the same effect as going thru all the otherpaths.-sureshThe Lord of Chaos (Suresh Deva)> Btw, looking at the central limit theorem> the average approaches n mu and the standard deviation approaches> sqrt{n} sigma> Taking the pecentage deviation of the tour costs frac{sqrt{n}sigma}{n> mu} = frac{1}{sqrt{n}} frac{sigma}{mu}> In a way, it even gets pointless to look for a solution to the salesman> problem as n->infty, since all paths are approximately going to be equal.> Going along one path would equal all paths travelled!!!> -suresh> Secondly,> Wouldn't it be sufficient to say that the edge weights areuniformly> distributed over some interval of values, and then for large N, a> selection of N edges (to make a tour) would make such randomlychosen> tours have weights approximating a normal distributio?> This is true. But i am not exactly talking about that. I will find> better> words to explain later.> Anyways, do u get the idea? Yeah, I think I get the idea, but my point was that (I think) you wereassuming the edge weights to be drawn from a gaussian distribution, andmycomment was suggesting that you could have the edge weights generatedfrommany other kinds of distributions as well.J === Subject: Re: P vs NP and the analog machineThis is one thing i can prove.Take an element a_ij. Let x = a_ij Assume that x has the pdf p(x)Now, the total cost along certain tour can written as cost = sum a_mnNow, each a_mn also has the distribution p(x), by definition of p(x) .For a tour of length n, The distribution of costs would have the mean (n) mu and the standarddeviation sqrt(n) sigma % deviation from the mean is frac{sqrt(n) sigma}{ n mu} =frac{1}{sqrt{n}} frac{sigma}{mu}As n->infy, % deviation goes to 0. In a way, all paths relatively haveabout the same n muOf course, the proof makes implicit assumption that p(x) is not changingwith n.Either, i have gone completely crazy, or have discovered somethingcompletely interesting. It looks so deviously simple.-suresh === Subject: Re: P vs NP and the analog machineOne more thingalthough the % deviation = (standard deviation)/(average) goes to 0, thedeviation between the path with the least cost and the average cost doesnt.It can be seen from the following analysis. The notation is sloppyLet P(a) be probability of getting a value anything under a, given that a isnormally distributed.P(a) = 1/N!P ( (x - n mu)/( sqrt {2 n} sigma) ) = 1/N!For asymptocially small values P(a) would be about exp(-a^2)/ aexp(-a^2)/ a = 1/N!-a^2 = -log(N!) a^2 = N (log N - 1) a = sqrt(N) sqrt( log N - 1)x - nmu = sqrt(2n) sqrt(N) sqrt( log N -1)x = n mu + n sqrt(2) sqrt( log N -1)x/(n mu) -> 1 + frac{sqrt(2) sqrt( log N - 1)}{mu}It does us no good to find just the average one.The Lord of Chaos (Suresh Deva)> This is one thing i can prove.> Take an element a_ij. Let x = a_ij Assume that x has the pdf p(x)> Now, the total cost along certain tour can written as> cost = sum a_mn> Now, each a_mn also has the distribution p(x), by definition of p(x) .> For a tour of length n,> The distribution of costs would have the mean (n) mu and the standard> deviation sqrt(n) sigma> % deviation from the mean is frac{sqrt(n) sigma}{ n mu} => frac{1}{sqrt{n}} frac{sigma}{mu}> As n->infy, % deviation goes to 0. In a way, all paths relatively have> about the same n mu> Of course, the proof makes implicit assumption that p(x) is not changing> with n.> Either, i have gone completely crazy, or have discovered something> completely interesting. It looks so deviously simple.> -suresh === Subject: Re: P vs NP and the analog machineerr x - nmu = +/- sqrt(2n) sigma sqrt(N) sqrt( log N -1) x = n mu +/- n sqrt(2) sigma sqrt( log N -1) x/(n mu) -> 1 +/- frac{sqrt(2) sigma sqrt( log N - 1)}{mu}I will leave the math here.The Lord of Chaos (Suresh Deva)> One more thing> although the % deviation = (standard deviation)/(average) goes to 0, the> deviation between the path with the least cost and the average costdoesnt.> It can be seen from the following analysis. The notation is sloppy> Let P(a) be probability of getting a value anything under a, given that ais> normally distributed.> P(a) = 1/N!> P ( (x - n mu)/( sqrt {2 n} sigma) ) = 1/N!> For asymptocially small values P(a) would be about exp(-a^2)/ a> exp(-a^2)/ a = 1/N!> -a^2 = -log(N!)> a^2 = N (log N - 1)> a = sqrt(N) sqrt( log N - 1)> x - nmu = sqrt(2n) sqrt(N) sqrt( log N -1)> x = n mu + n sqrt(2) sqrt( log N -1)> x/(n mu) -> 1 + frac{sqrt(2) sqrt( log N - 1)}{mu}> It does us no good to find just the average one.The Lord of Chaos (Suresh Deva)messageThis is one thing i can prove.Take an element a_ij. Let x = a_ij Assume that x has the pdf p(x)Now, the total cost along certain tour can written as cost = sum a_mnNow, each a_mn also has the distribution p(x), by definition of p(x) .For a tour of length n, The distribution of costs would have the mean (n) mu and the standarddeviation sqrt(n) sigma % deviation from the mean is frac{sqrt(n) sigma}{ n mu} =frac{1}{sqrt{n}} frac{sigma}{mu}As n->infy, % deviation goes to 0. In a way, all paths relativelyhaveabout the same n muOf course, the proof makes implicit assumption that p(x) is not changingwith n.Either, i have gone completely crazy, or have discovered somethingcompletely interesting. It looks so deviously simple.-suresh === Subject: Re: Do Prime Algebraic Numbers even exist? > In sci.math, Jpr2718 > > But, there can be algebraic integer primes in the ring of > algebraic integers of a number field. So, 2^(1/2) is a > prime in the ring of integers of Q(sqrt(2)). > Maybe this is what they are thinking of. > > Maybe, but that's not the ring of algebraic integers; that's > the ring of numbers generated by the combination of any > integer and sqrt(2), which is equivalent to all numbers > a + b * sqrt(2), as it turns out.Indeed, but there are *no* primes in the algebraic integers. Now youcould define class 1 primes to be primes in the integers, but I failto see how you could define class 2 primes or anything else. Notall primes in number fields are of the form n-th root of p with pa prime. For instance, one of the primes in Q(sqrt(2)) is 1 + sqrt(2)(I think).Also you have to be careful with what you call the subring of integers.Not in all quadratic fields (i.e. fields of the form Q(sqrt(m)) ) arethey just combinations of the form a + b*sqrt(m). Assume m square-free,then the integers are of the form (a + b*sqrt(m))/2 with a and b botheven or both odd, when m = 1 mod 4. So (1 + sqrt(5))/2 is an integerin Q(sqrt(5)). > (The units for that ring appear to contain 1, -1, > 1 + sqrt(2), -1 + sqrt(2), 1 - sqrt(2), and -1 - sqrt(2). > Since (1 + sqrt(2))^2 = 3 + 2*sqrt(2), there are apparently > a few additional units as well -- in fact, > a + b * sqrt(2) is a unit if 2*b^2 - a^2 = 1 or -1.)The number of units in quadratic fields (Q(sqrt(m)) ) is as follows:m > 0: infinitely manym = -1: 4m = -3: 6m < 0: 2 in all other cases.(again, assuming m to be square free.)However, when m > 0 it is not easy to always find a unit. I have donesome quite extensive calculations and found that a fundamental unitin Q(sqrt(241)) is: 71011068 + 4574225.sqrt(241).However, anything beyond quadratic fields has been barely studied.the algebraic integers.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Do Prime Algebraic Numbers even exist?In sci.math, Dik T. Winter:In sci.math, Jpr2718> But, there can be algebraic integer primes in the ring of> algebraic integers of a number field. So, 2^(1/2) is a> prime in the ring of integers of Q(sqrt(2)). > Maybe this is what they are thinking of.Maybe, but that's not the ring of algebraic integers; that'sthe ring of numbers generated by the combination of anyinteger and sqrt(2), which is equivalent to all numbersa + b * sqrt(2), as it turns out.> Indeed, but there are *no* primes in the algebraic integers. Now you> could define class 1 primes to be primes in the integers, but I fail> to see how you could define class 2 primes or anything else. Not> all primes in number fields are of the form n-th root of p with p> a prime. For instance, one of the primes in Q(sqrt(2)) is 1 + sqrt(2)> (I think).> Also you have to be careful with what you call the subring of integers.> Not in all quadratic fields (i.e. fields of the form Q(sqrt(m)) ) are> they just combinations of the form a + b*sqrt(m). Assume m square-free,> then the integers are of the form (a + b*sqrt(m))/2 with a and b both> even or both odd, when m = 1 mod 4. So (1 + sqrt(5))/2 is an integer> in Q(sqrt(5)).(The units for that ring appear to contain 1, -1,1 + sqrt(2), -1 + sqrt(2), 1 - sqrt(2), and -1 - sqrt(2).Since (1 + sqrt(2))^2 = 3 + 2*sqrt(2), there are apparentlya few additional units as well -- in fact,a + b * sqrt(2) is a unit if 2*b^2 - a^2 = 1 or -1.)> The number of units in quadratic fields (Q(sqrt(m)) ) is as follows:> m > 0: infinitely many> m = -1: 4> m = -3: 6> m < 0: 2 in all other cases.> (again, assuming m to be square free.)> However, when m > 0 it is not easy to always find a unit. I have done> some quite extensive calculations and found that a fundamental unit> in Q(sqrt(241)) is:> 71011068 + 4574225.sqrt(241).> However, anything beyond quadratic fields has been barely studied.> the algebraic integers.Ah, good. However, there's an issue that what James hasis not a quadratic field, but a field Z[(n +/- sqrt(n^2 - 32))/16],for any integer n (n != 6, -6, 9, -9, as these lead to eitherplain old Z, Z[1/2], Z[1/4], or Z[1/8]).I would assume absent further proof that each of these isa superset of some quadratic field, which basically meansit has at least the units of that quadratic field.-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: Do Prime Algebraic Numbers even exist?> However, when m > 0 it is not easy to always find a unit.For small m there are perfectly easy methods using continued fractions. Seemy webpage on this.For large m the solution gets so large that the challenge is to find a fastway to write it down. See either of the papers below for more info. TheJohn RobertsonH. W. Lenstra Jr., Solving the Pell equation, textit{Notices of the AmericanMathematical Society}, {bf 49} No. 2 (February 2002), pp. 182--192. Theequation $x^2-Dy^2=pm 1$ for large $D$.H. C. Williams, Solving the Pell equation, in Bruce Berndt et al.,textit{Surveys in Number Theory: Papers from the Millennial Conference onNumber Theory}, A. K. Peters, 2002. Also included in textit{Number Theory forthe Millennium}, Volumes 1, 2, 3, M. A. Bennett et al. editors, A. K. Peters,2002. Williams' web page gives this last reference as H. C. Williams, Solvingthe Pell equation, textit{Proc. Millennial Conference on Number Theory}, A. K.Peters, Natick MA, 2002, pp. 397-435. Discusses the equation $x^2-Dy^2=pm 1$. Terrific overview, including discussion when $D$ is large. === Subject: Re: Do Prime Algebraic Numbers even exist? Adjunct Assistant Professor at the University of Montana.>This is probably a slightly stupid question, but Eric W. Weisstein's>definition of Prime Algebraic Number>http://mathworld.wolfram.com/PrimeAlgebraicNumber.html> gives no examples of such a number at all, and therefore>leaves a lot to be desired. I am wondering what such a>number might look like, if one exists at all.The definition does indeed leave much to be desired; for starters, itforgets to specify the ring!Taken at face value, assuming it means in the ring of algebraicintegers, then the definition is correct but vacuous, since noalgebraic integer is irreducible.However, the definition becomes interesting when used in the contextwhich usually occurs in algebraic number theory: in a number field.A number field in this case is a finite extension of Q. Call itK. Then one can consider the ring of integers of K, which is thecollection of all algebraic integers which are in K. In this ring, there ->are<- irreducible elements, and there ->are<-prime elements, though in general they need not be equivalent notions(they are equivalent if and only if this ring of integers is a UFD).For example, consider the classical example K=Q(sqrt(-5)), with ringof integers equal to Z[sqrt(-5)].There are some irreducibles that are not primes: for example, it iseasy to verify that 2 is irreducible; and 2 divides(1+sqrt(-5))^2 = -4+2sqrt(-5) = 2(-2+sqrt(-5))but 2 does not divide (1+sqrt(-5)). So 2 is irreducible, but not aprime.On the other hand, sqrt(-5) itself is irreducible, and is also a prime(any prime must be irreducible): for ifsqrt(-5) divides (a+b*sqrt(-5))*(x+y*sqrt(-5)) = (ax-5by) + (ay+bx)sqrt(-5)then there exists an element r+s*sqrt(-5), r,s integers, such thatsqrt(-5)*(r+s*sqrt(-5)) = -5s + r(sqrt(-5) = (ax-5by) +(ay+bx)sqrt(-5).in particular, -5 = ax-5by, so 5 must divide ax in the integers. Thatmeans that either 5|a (in which case, sqrt(-5) divides a+b*sqrt(-5)),or else 5 divides x (in which case, sqrt(-5) divides x+y*sqrt(-5)). Sosqrt(-5) is a prime in this ring.The following result describes the primes of sqrt(-5) (taken fromDedekind's exposition, but a nice exercise to try):1. The positive rational primes which are congruent to 11, 13, 17, or 19 (mod 20) are primes in Z[sqrt(-5)].2. sqrt(-5) is a prime in Z[sqrt(-5)].3. The rational prime 2 behaves like the square of a prime; that is, 2 satisfies the following two conditions: (I) If 2|a^2*b^2, with a,b in Z[sqrt(-5)], then 2|a^2 or 2|b^2; (II) There exist a in Z[sqrt(-5)] such that 2|a^2 but 2 does not divide a. 4. Each positive rational prime congruent to 1 or to 9 (mod 20) factors as the product of two different factors in Z[sqrt(-5)], each of them primes (they will be conjugate, that is, p=(a+b*sqrt(-5))*(a-b*sqrt(-5)) for suitably chosen integers a,b).5. Each positive rational prime congruent to 3 or 7 (mod 20) behaves like the product of two different primes; that is, it satisfies the following three conditions: (I) If p divides a product x*y*z of elements of Z[sqrt(-5)], then p divides either x*z, x*y, or y*z. (II) There exist elements a,b in Z[sqrt(-5)] such that p divides a*b but p does not divide a nor b. (III) If p divides a^2, then p divides a, for all a in Z[sqrt(-5)]. It is not a good definition in mathworld, certainly...-- === ==Arturo Magidinmagidin@math.berkeley.edu === Subject: Re: Do Prime Algebraic Numbers even exist?In sci.math, Arturo Magidin:>This is probably a slightly stupid question, but Eric W. Weisstein's>definition of Prime Algebraic Number>http://mathworld.wolfram.com/PrimeAlgebraicNumber.html> gives no examples of such a number at all, and therefore>leaves a lot to be desired. I am wondering what such a>number might look like, if one exists at all.> The definition does indeed leave much to be desired; for starters, it> forgets to specify the ring!> Taken at face value, assuming it means in the ring of algebraic> integers, then the definition is correct but vacuous, since no> algebraic integer is irreducible.Perhaps you can correspond with Mr. Weisstein on this. (I'mnot sure I have the technical expertise beyond pointing outwhat I've already discovered -- if one can call it a discovery.But it's certainly a problem.)[rest snipped but saved for future analysis]Your stuff is rather interesting, and in a completely differentdirection from my orders proposal; yours makes more sense. :-)To be sure, it's been too long since I've played with all this.-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: I'm guessing this is an easy problem - but I can't solve it!>Working through calculus... Seeing this phrase makes it clear that the use of L'Hopital's rule(suggested by other posters) is inappropriate here.>Stuck on a problem of factorisation: Please help!Yes, you almost certainly need to factor(ise). Others posted theformula for the factorization of u^n - 1 ; you can then cancel afactor of u-1 from numerator and denominator.>question is:>find lim u--> 1 f(u)> where f(u) = (u^4 - 1) /> (u^3 -1)>Obviously - I cannot just replace u with 1 as this will give me a division>by zero.It's actually a bit worse than this, and it is because no one else waspointing this out that I am following up. You question is about computing a LIMIT as u APPROACHES 1. On theface of it, that makes it completely inappropriate to find the VALUEof the the function when u is EQUAL to 1. Those are, in general,two different things to compute about a function. If I saw studentsdoing willy-nilly what you are proposing, I would certainly set anexam question in which setting u=1 does _not_ lead to a division-by-zeroerror (or any other error) but which does not give the limit either.Of course, in practice, people evaluate limits simply by substitution,as you wanted to do, but it is important to know that this only givesthe correct answer when the function is CONTINUOUS (here, at u=1). Whenever you decide to compute a limit by simple substitution, youneed to (be ready to) defend your step by explaining why you know thefunction is contiuous there. This particular example is typical: aftercancelling common factors of u-1, you have a rational function whosedenominator does not vanish at u=1; there's a theorem (surely in yourtextbook) that says such a function is continuous at u=1. (You also need to know that when two functions agree except at onepoint, then their limits at that point agree. That is, you don't get adifferent limit when you cancel the u-1 's .)The typical calculus student wants to learn to do the problems by manipulation as if calculus were simply a kind of algebra. That's reallykind of pointless; I mean, it's expected that you will learn to do the manipulations, but that's not really the most important thing aboutthe calculus. Learning what the manipulations tell you about theunderlying phenomena is much more important. This problem is a goodexample to show the difference between manipulation and understanding.(I would _almost_ rather have the student plug in u=1.00001 and get1.333366667 than go through the algebra and plug in u=1 without commentingabout the limit process.)dave === Subject: Re: I'm guessing this is an easy problem - but I can't solve it! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1MKlMj01590;>Working through calculus... Stuck on a problem of factorisation: Please>help!>question is:>find lim u--> 1 f(u)> where f(u) = (u^4 - 1) /> (u^3 -1)>Obviously - I cannot just replace u with 1 as this will give me a division>by zero. I tried replacing (u^2 - 1) (u^2 +1) on top as this seemed most>logical - but I can't figure out what to do with the bottom now.>I'm probably going to kick myself for not figuring this one out - but I've>spent hours on this and I'm still lost.>Help!If this is a calculus problem, look up L'Hospital's Rulephil === Subject: Re: How to study for math?>Here is what I usually do:>Read the chapter laying in bed (hey, I was originally a history major). Hens lay, people lie. (Some more than others.) === Subject: Re: How to study for math?>Read the chapter laying in bed (hey, I was originally a history major). > Hens lay, people lie. (Some more than others.)Lying in bed refers to being supported by the bed, but laying in bed wouldbe OK for referring to being spread over the surface of the bed. I'd guesshe was probably doing both.-- --Tim Smith === Subject: Re: How to study for math?>>Read the chapter laying in bed (hey, I was originally a history major). > > Hens lay, people lie. (Some more than others.)>Lying in bed refers to being supported by the bed, but laying in bed would>be OK for referring to being spread over the surface of the bed. Nope. Saying you're laying in bed is not right, regardless of whetheryou're spread all over the surface. (There _is_ a sense of the wordlay under which it might be regarded as correct, but that's notwhat we're talking about here.) Lay is transitive, lie isintransitive.>I'd guess>he was probably doing both.************************ === Subject: Re: How to study for math?>definitions, I'll have to go back and look at what it says. I also have a>problem with waiting to look at the answer. If I can't figure out the>problem in about a minute or so, I'll just look at the back of the book.>I'm not sure how good/bad this is considered but it's a habit I've>developed nevertheless.> Well, if you seriously want to learn mathematics you need to break that> habit. It's not that it's considered bad, it's simply impossible to> learn math without doing a lot of work, including solving problems that> take a long time to figure out. If you look in the back of the book after> a minute you're never going to learn to do it yourself.It might be a good idea for the original poster to pick up a one of thevolumes of Knuth's The Art of Computer Programming. They all have a lotof mathematical content, and in particular, a *lot* of the exercises arebasically math problems. The nice thing about Knuth is that all theexercises include an estimate of about how long they should take. Thatmight give the OP something to shoot for before wimping out and looking inthe back.(There's a lot of very interesting math in those exercises, too. TAoCP isan excellent set of books for mathematics students even if they have nointention of ever going near a computer. I'd say volume 2 is the best, froma mathematical point of view, if you are only going to get one volume).-- --Tim Smith === Subject: Re: bounds for x in Incomplete Gamma function> G(a,x) / G(a) > 0.001> where G(a,x) is the Incomplete Gamma function > and G(a) is the usual Gamma function> Any clues?> Thank you very much / Javier === =====For a> 0, x>0 let G(a,x)= Integral_{t=x to t=infty}t^{a-1}e^{-t}dt , G(a)=G(a,0) , g(a,x)=G(a)-G(a,x).Denote P(a,x)= a*x^{-a}e^{-x} , Q(a,x)= x^{1-a}e^x .Try to use inequalities ((a+1)(a+2)-x)/((a+1)*(a+2+x)) < P(a,x)*g(a,x) < (a+1)/(a+1+x) x/(x+1-a) < Q(a,x)*G(a,x) < (x+1)/(x+2-a) .Perhaps help, Alex === =Subject: Re: Perplexing Patterns of Square Numbers...> I tried to follow these messages about perplex patterns but I am a little lost.> Could someone summarize what has already been done about this subject,> that is how many patterns have been found for n from 1 to 20 (or > something like that) and also how many one can expect to discover....See http://groups.google.com/groups?th=73e11dc0c6222c00 for asci.math thread with details of the above. Briefly, throughout last year B.S. Rangaswamy reported several perplex patterns (found without using a computer, if I'm not mistaken) for n=5, 6, 7, and 8; in December posted some programs, while Ralph Furmaniak posted a more general but slightly slower program. (At n=10, his takes 8 minutes on his machine and mine takes 1 minute on a 750 MHz Pentium.) Below are the number of solutions I find for n=1 to 12. I think no one knowshow many or if any exist for n>12.-jiw n sols(n) Examples (unsquared) 1 3 {1} {2} {3} 2 4 {9 4} {8 7} {4 8} {6 8} 3 7 {29 22 12} {21 22 12} {11 17 14} ... 4 1 {46 35 36 81} 5 16 {304 167 221 136 263} ... 6 17 {865 668 932 476 472 738} ... 7 13 {1913 2636 2333 3134 2343 2643 3114} ... 8 12 {7918 5141 8034 9615 7042 8839 5341 6454} ... 9 11 {30746 20596 23361 17818 13924 25616 22873 10596 25719}10 9 {72927 57786 36719 94789 56828 65462 84523 61608 54432 98038}11 8 {290369 218896 198022 105629 212771 121426 234925 261263 135884 248686 128108}12 2 {957511 358784 827807 934038 475632 868708 628471 391023 767854 403042 504407 411636} {683281 798836 829893 903962 855838 531773 996627 509885 664623 992182 833332 411643} === Subject: Re: differential equation>lost in finding the necessary differential equation. Any help would be>appreciated. Marine Biologists have determined that when a shark detects the>presence of blood in the water, it will swim in the direction in which the>concentration of the blood increases most rapidly. Based on certain tests>in seawater, the concentration of blood (in parts per million) at a point>P(x,y) on the surface is approximated by C(x,y)= e^ {(-x^2+2y^2)/10^4}where>x and y are measured in meters in a rectangular coordinate system with the>blood source at the origin. Suppose a shark is at the point (Xo,Yo) when it>first detects the presence of blood in the water. Find an equation of the>shark's path by setting up and solving a differential equation.Hint: the gradient of a function points in the direction of most rapid increase. === Subject: Analysis helpi am trying to prove the following:if f is differentiable on an interval containing zero and iflim(x->0)f'(x) = Lthen L = f'(0).I have tried a number of tricks, contradiction, etc. but nothing seemsto get me anywhere. Can someone give me a hint as to how a proof likethis might go?This is not homework, btw. === Subject: Re: Analysis help> i am trying to prove the following:> if f is differentiable on an interval containing zero and if> lim(x->0)f'(x) = L> then L = f'(0).> I have tried a number of tricks, contradiction, etc. but nothing seems> to get me anywhere. Can someone give me a hint as to how a proof like> this might go?> This is not homework, btw.It is one of my favourite theorems in Real Analysis; the proof uses theMean Value Theorem: for h not zero but still in the interval, (f(0+h)-f(0))/h = f'(0+theta*h) where 0i am trying to prove the following:>if f is differentiable on an interval containing zero and if>lim(x->0)f'(x) = L>then L = f'(0).Hint: What does the Mean Value Theorem tell you about (f(x) - f(0))/x when x is near 0? === Subject: Re: Analysis help>i am trying to prove the following:>if f is differentiable on an interval containing zero and if>lim(x->0)f'(x) = L>then L = f'(0).> Hint: What does the Mean Value Theorem tell you about > (f(x) - f(0))/x when x is near 0?there exists a c in (x,0) or (0,x) such thatf'(c) = (f(x) - f(0))/x so there is a sequence of c_n's in the interval (and not equal to 0)that approach 0, so (f(c_n) - f(0))/c_n -> f'(0)and thus f'(c_n) -> f'(0) so, since lim(x->0) f'(x) = L, f'(c_n) -> L, therefore L = f'(0).is that correct?Based on the exercise's location in the book (understanding analysisby abbott) i don't think he intends you to use the MVT. so do you seeanother way to do it?i'm happy to have A solution, but i'm curious to know what the authorwas thinking, i have spent way too much time on this. === Subject: Re: Analysis help >if f is differentiable on an interval containing zero and if>lim(x->0)f'(x) = L>then L = f'(0).> Hint: What does the Mean Value Theorem tell you about> (f(x) - f(0))/x when x is near 0?That f'(0) exists and equals L.So the premise can be reduced tof differentiable on (a,b)0, for some a < 0 < b === Subject: Re: Analysis help>if f is differentiable on an interval containing zero and if>lim(x->0)f'(x) = L>then L = f'(0).> Hint: What does the Mean Value Theorem tell you about> (f(x) - f(0))/x when x is near 0?>That f'(0) exists and equals L.>So the premise can be reduced to>f differentiable on (a,b)0, for some a < 0 < bReally? Let f(x) = -1 for x < 0, f(x) = 1 for x > 0.************************ === Subject: Re: Analysis help <5kuj305vf1sk1umiqfc012f0rm3qh55ofr@4ax.com>if f is differentiable on an interval containing zero and if>>lim(x->0)f'(x) = L>>then L = f'(0).>Hint: What does the Mean Value Theorem tell you about>(f(x) - f(0))/x when x is near 0?>That f'(0) exists and equals L.>So the premise can be reduced to>f differentiable on (a,b)0, for some a < 0 < b> Really? Let f(x) = -1 for x < 0, f(x) = 1 for x > 0.Yes, the (top line) of the premise can be reduced to ... === Subject: Re: Analysis help>if f is differentiable on an interval containing zero and if>>lim(x->0)f'(x) = L>>then L = f'(0).> Hint: What does the Mean Value Theorem tell you about>> (f(x) - f(0))/x when x is near 0?>That f'(0) exists and equals L.>>So the premise can be reduced to>f differentiable on (a,b)0, for some a < 0 < b> Really? Let f(x) = -1 for x < 0, f(x) = 1 for x > 0.>Yes, the (top line) of the premise can be reduced to ...Meaning thatif f differentiable on (a,b)0, for some a < 0 < b and iflim(x->0)f'(x) = Lthen L = f'(0).is true? No it's not - I _gave_ you a counterexample.Let f(x) = -1 for x < 0, f(x) = 1 for x > 0. Are youclaiming that f is not differentiable on (a,b){0}for some a < 0 < b, that lim(x->0)f'(x) does notequal 0, or that f'(0) = 0?************************ === Subject: Re: Analysis helpif f differentiable on (a,b)0, for some a < 0 < b and if> lim(x->0)f'(x) = L> then L = f'(0).> is true? No it's not - I _gave_ you a counterexample.> Let f(x) = -1 for x < 0, f(x) = 1 for x > 0. Are you> claiming that f is not differentiable on (a,b){0}> for some a < 0 < b, that lim(x->0)f'(x) does not> equal 0, or that f'(0) = 0?What if we changed the function and defined it at 0, say...f(0) = 0...it's still differentiable on (a,b){0), andlim(x->0)f'(x) = 0 right?but since it's not continuous at 0, f'(0) does not exist (and does notequal 0!). === Subject: [TOC] Mathematical Models and Methods in Applied Sciences - Vol 14 No 2Mathematical Models and Methods in Applied SciencesView table-of-contents and abstracts athttp://www.worldscinet.com/m3as.htmlContents:An Explicit Subparametric Spectral Element Method Of Lines Applied ToA Tumour Angiogenesis System Of Partial Differential EquationsJ. Valenciano and M. A. J. ChaplainLimit Behaviour Of A Dense Collection Of Vortex FilamentsH. Bessaih and F. FlandoliStudy Of A Mathematical Model For Stimulated Raman ScatteringT. Boucheres, T. Colin, B. Nkonga, B. Texier and A. BourgeadeLinear Parabolic Equations In Locally Uniform SpacesJose M. ArrietaNumerical Solution Of Two-Factor Models For Valuation Of FinancialDerivativesAna Berm.9cdez and Mar.92a R. NogueirasFor more information, go to http://www.worldscinet.com/m3as.html === Subject: Binary To TernaryGiven a binary number$$(a_{n-1} a_{n-2} ldots a_2 a_1 a_0)_2,$$is there an efficient way to convert it to the ternary number$$(a_{n-1} a_{n-2} ldots a_2 a_1 a_0)_3$$without explicitly using the normal or binary splitting radix conversiontechniques? === Subject: Re: Binary To Ternary> Given a binary number> $$(a_{n-1} a_{n-2} ldots a_2 a_1 a_0)_2,$$> is there an efficient way to convert it to the ternary number> $$(a_{n-1} a_{n-2} ldots a_2 a_1 a_0)_3$$> without explicitly using the normal or binary splitting radix conversion> techniques?I actually donot know the algorithms but gmp (GNU MP) has some algos.I think they can give you some pointers.The home page is http://www.swox.com/gmp/.Best of luck.Rajsekar === Subject: Re: Binary To Ternary> Given a binary number> $$(a_{n-1} a_{n-2} ldots a_2 a_1 a_0)_2,$$> is there an efficient way to convert it to the ternary number> $$(a_{n-1} a_{n-2} ldots a_2 a_1 a_0)_3$$> without explicitly using the normal or binary splitting radix conversion> techniques?No. What could possibly be faster? === Subject: Re: Binary To TernaryGiven a binary number$$(a_{n-1} a_{n-2} ldots a_2 a_1 a_0)_2,$$is there an efficient way to convert it to the ternary number$$(a_{n-1} a_{n-2} ldots a_2 a_1 a_0)_3$$without explicitly using the normal or binary splitting radix conversiontechniques?> No. What could possibly be faster?I believe the answer is yes rather than no, since radix conversiontechniques usually involve division or subtraction, and have nothing to do with the requested conversion, which only requires n multipliesand adds. Perhaps an example will help. Given a value such as 101101_2 (or 45_10) he wants the value of 101101_3, that is, wants 1+3(0+3(1+3(1+3(0+3(1))))) or 280_10.-jiw === Subject: Re: Binary To TernaryIn sci.math, Josh Liu:> Given a binary number> $$(a_{n-1} a_{n-2} ldots a_2 a_1 a_0)_2,$$> is there an efficient way to convert it to the ternary number> $$(a_{n-1} a_{n-2} ldots a_2 a_1 a_0)_3$$> without explicitly using the normal or binary splitting radix conversion> techniques?Probably not, as 2 and 3 aren't related, really, unlike, say,2 and 4, 2 and 8 (octal), or 2 and 16 (hexadecimal).-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: I got low score on math test, please advise me and take a look> Tim,> I had run into a few professors throughout my college years who were> as arrogant and utterly unthoughtful (in the analytical way, not in> the lovey dovey way) as you are; luckily, I do not find quite so many> in graduate school. Allow me to explain: Your students are> purchasing a service from you, if the course title is Calculus and the> description in the handbook is 'Introduction to Calculus. Topics> include convergent sequences, limits, differentiation [... insert> additional Calculus topics here ...]' then your students are> purchasing a course in Calculus. Your comments about marking students> wrong when it takes additional time to read their answer and your > personal vendetta against students squeezing work in the margin when> there is a back page are fine for your own time - in fact, you can> even explain to your students that you, personally, like this and> don't like that. But the second that you take points off of a> student's grade for work that you don't like, you are commiting fraud> and breaking the law in the worst way. Which is a worse crime, taking off points for work that does not meet the requirements that I make clear the first day of class, or passing a student who has made no effort to learn the subject and has, in fact, not learned the material? Personally, I think the fraud lies in the seond option. I am sorry, but I do not allow students to purchase their grades. They must honestly earn whatever grade they receive.expectations): A rough guide to what work merits which letter grade is as follows. Understanding the solution to every assigned homework problem and most of the concepts very well by test day will be enough to earn a 'C.' The ability to do a large majority of problems and understanding the solution to every problem, assigned or not, and understanding the concepts will earn a 'B.' Being able to do nearly every problem, assigned or not, and understanding the relationships between concepts will be enough to earn an 'A.'Also, I don't think I said I take off points for sqeezing your work into the margins. I do not. However, if you are unclear or I cannot follow the work (which might be because it was squeezed into the margins), I could take off a point or two.> You see, not only have you, mid course, decided that Calculus will no> longer be the topic of discussion, but (addressing the 'in the worst> way' comment) your misleading of the students and change of topic may> very well destroy a student's whole future. Imagine the scenario> where a student is genetically predispositioned to write in margines,> but is Godel smart in mathematical logic and wants to study at> Princeton University. A grade of 'F' in the 'Test taking, my way'> class that the student inadvertently signed up for (entitled> 'Calculus') could very well destroy these chances. If one wishes to study math at Princeton, one should be prepared to be published. Incorrectly written proofs, poorly constructed arguments, and messiness will not get someone published. In my classes, I have always been told to write every homework problem, every test answer, even my notes as if they were going to be published by a scholarly journal. I do not expect that of my students. To be honest, the classes I am teaching are not the students who will be going on further in math. Although, if they do, I applaud them and hope I was helpful in letting them know what to expect.> I don't expect you to understand the above argument, but perhaps> you'll understand this and copy the behavior. Whenever I ran into a> professor who thought so highly of themselves that they dared to> dictate the format that a student could write their test answers,> above and beyond the standard 'show your work, write so that I can> read it and circle your final answer' (but never 'look to see if there> is a back page, if so: do not write in margin if not: blah blah> blah' etc.), they were always very untallented theoretically. That> is, they would never be able to understand the 'theoretical' scenario> that I proposed above, they would never be able to understand the fact> that they are 'theoretically' changing the course topic and commiting> fraud (false advertising). This lack of tallent invariably carried> over into mathematics (or computer science, depending upon the course)> as well. I never ran into a very smart professor who did this sort > of thing. Perhaps you'll want to be thought of as smart, too, and get> down from your high horse.I never change the course topic. We spend time learning whatever is listed in the catalog. My requirements are plain and simple, I require clear, well thought out responses. If I cannot follow your work, it has been from my experience, that the student is trying to drown me in knowledge with the hope I will forget what answer I am supposed to find.> Please note, I think that this 'Spockie Hendrick' guy is a pathetic> whiney excuse for a student who refuses to read his handbook which,> almost certainly, states that a student can't drop a corse past the> drop date for any reason whatsoever. This post has nothing to do with> 'Spockie Hendrick' ... This is only addressed to Tim and his arrogant> thought processes that tell him that his way of test taking is right> and must be spread around the world.I never said my method of test taking was right. In fact, I don't have a method of test taking. All I ask is that students take the time, and actually care enough, to make their responses clear. Perhaps this example will help you understand what I mean. I gave a test recently and one of the questions was, State one of the forms of the Law of Cosines. It was worth three points. For this class sides are labelled with lower case letters and angles with upper case, such as side a is opposite angle A, and that was clearly stated at the beginning of the test. How many points should I assign for a response like this:A^2 = a^2 + b^2 - 2*A*C*cos(a)B^2 = a^2 + c^2 - 2*A*C*cos(b)c^2 = a^2 + b^2 - 2*a*b*cos(C)Yes, the correct answer is one of those. But did the student actually know the answer? Or, was he just guessing hoping that I would see the correct answer and consider it 100% correct?Finally, let me tell you a little about what my students think of me. Where I teach, students are not afraid to give porr evaluations at the end of the semester. They are also quick to remind you that it is their tuition paying my salary. However, I have never received a poor evaluation. I do often receive comments like, high expectations, but they are made clear from day one,A tough, but fair instructor,I learned a lot, whether I wanted to or not. Also, I am a vibrant instructor. I often try to get the students to learn to like math. When I am teaching stats, we go to a fair and collect data on things like what number comes up on a Big-6 wheel, or which horse wins the watergun race most often. Also, we had a Vegas night, where we looked at expected values and how even though the roulette wheel might come up red 4 times in a row, that does not change the probability on the next spin. And what the sucker bets are in craps. And, even though it has little to do with stats, they learn how to count cards in blackjack.Perhaps one of my greatest compliments is that I have had a few students ask me what class I was teaching the next semester so they could take it. In fact, I have taught one student in three different class, Calculus I, Intro. to Statistics, and Discrete Math. Appearently he doesn't think I am draconian in my requirements. If you think I am just making these things up, I can give you email addresses of some of my past students to get their comments. Many times I have been told I make math fun for them, or I have been the best instructor they have ever had (that last one kind of makes me sad because I know the professors they have had and I know how good they are).Maybe, just maybe, I actually expect my students to learn, to be able to present clear thoughts. Maybe I won't just give points for a correct answer when why the answer is correct and how you got there matters even more. Hell, I can use Maple or MatLab or Mathematica or Derive or any other CAS to get answers, if that is all I wanted. If that makes me arrogant, then color me so. - Tim-- Timothy M. BrauchGraduate StudentDepartment of MathematicsWake Forest Universityemail is:news (dot) post (at) tbrauch (dot) com === Subject: Re: I got low score on math test, please advise me and take a look>> assuming that your report about progress feedback is accurate, a> rejection of the request above is a good glimpse into anincompetent> institution. what kind of justification were you offered? some> bureaucratic non-sense?>> My guess is that justification offered was THE RULE THAT YOU CAN'T DROP A> COURSE AFTER A CERTAIN DATE.> in other words, bureaucratic non-sense. don't you think that suchrule> should have a provision stateing that students need some performance> feedback prior to the drop deadline?How do you know that he didn't get any performance feedback before thedeadline?> i believe he said that somewhere. now, can you answer my question -> perhaps in a direct manner this time?Fine. RULES ARE RULES. How's that?Doug === Subject: Re: I got low score on math test, please advise me and take a look> Tim,> I had run into a few professors throughout my college years who were as> arrogant and utterly unthoughtful (in the analytical way, not in the lovey> dovey way) as you are; luckily, I do not find quite so many in graduate> school. Allow me to explain: Your students are purchasing a service from> you, if the course title is Calculus and the description in the handbook is> 'Introduction to Calculus. Topics include convergent sequences, limits,> differentiation [... insert additional Calculus topics here ...]' then your> students are purchasing a course in Calculus. Your comments about marking> students wrong when it takes additional time to read their answer and your> personal vendetta against students squeezing work in the margin when there> is a back page are fine for your own time - in fact, you can even explain to> your students that you, personally, like this and don't like that. But the> second that you take points off of a student's grade for work that you don't> like, you are commiting fraud and breaking the law in the worst way.> You see, not only have you, mid course, decided that Calculus will no longer> be the topic of discussion, but (addressing the 'in the worst way' comment)> your misleading of the students and change of topic may very well destroy a> student's whole future. Imagine the scenario where a student is genetically> predispositioned to write in margines, but is Godel smart in mathematical> logic and wants to study at Princeton University. A grade of 'F' in the> 'Test taking, my way' class that the student inadvertently signed up for> (entitled 'Calculus') could very well destroy these chances.> I don't expect you to understand the above argument, but perhaps you'll> understand this and copy the behavior. Whenever I ran into a professor who> thought so highly of themselves that they dared to dictate the format that a> student could write their test answers, above and beyond the standard 'show> your work, write so that I can read it and circle your final answer' (but> never 'look to see if there is a back page, if so: do not write in margin> if not: blah blah blah' etc.), they were always very untallented> theoretically. That is, they would never be able to understand the> 'theoretical' scenario that I proposed above, they would never be able to> understand the fact that they are 'theoretically' changing the course topic> and commiting fraud (false advertising). This lack of tallent invariably> carried over into mathematics (or computer science, depending upon the> course) as well. I never ran into a very smart professor who did this sort> of thing. Perhaps you'll want to be thought of as smart, too, and get down> from your high horse.> Please note, I think that this 'Spockie Hendrick' guy is a pathetic whiney> excuse for a student who refuses to read his handbook which, almost> certainly, states that a student can't drop a corse past the drop date for> any reason whatsoever. This post has nothing to do with 'Spockie Hendrick'> ... This is only addressed to Tim and his arrogant thought processes that> tell him that his way of test taking is right and must be spread around the> world.> my website states my case and has jpg files of the four pages of the> test please take a look and advise me or give me opinions> http://www.johncho.usAside from the all the politics of whether the grade was fair and if thetest should be returned in time, I have a few comments.First, for all the instructors out there, how many of you take of 1/4 ofa point? To me, if you make the problem worth 8 points, anything lessthan perfection is 7. And, when I am assigning point values toproblems, I usually look over the problem and pick out the two or threeconcepts being tested in that question and give each of those concepts avalue of 1 or 2 or even 3 points. I just can't igaine trying to gradewith fractional points. But, that is just my grading style and I'm notreally saying anything is wrong with giving fractional points,I justfind it uncommon.Secondly, if I am grading and I cannot follow the work (perhaps becauseit is not neatly written), I have a hard time giving full credit. Thatmight sound harsh, but I have often found that students write two orthree possible answers for the problem in their mess and never clearlymark which one they want me to grade. Also, judging from your scans ofthe page, it looks like you had the back of each page left blank (whichis pure speculation on my part). I despise it when students try tosqueeze in something in the margins, leaving the whole back side of apage blank. You aren't going to use that page for anything else, whysqueeze things together? Now, if you really want to get your instructormad, you could ask him (or her) to please write the comments on yourtest neater. But that would serve no purpose otehr than to point youout as a snot-nosed brat.Okay, about the grading. Personally, I don't know what topics werestressed in class so I don't feel like I can make a judgement call. Forexample, I gave a test in Trigonometry II on Tuesday (which I am in theprocess of grading and won't be given back until Monday). For this testthe material I stressed was changing degrees to radians, law of sines,and law of cosines. On one question I asked about the area of atriangle. If a student accidentally used the formula area = base *height, I will not take off many points probably one out of 10 points,because they were not being tested on geometry (and had they asked methe correct formula, I would have told them, since that wasn't the topicthey were being tested on). Or, maybe another example, if someonecomputed 2+3=6 in part of a larger problem, multiplying instead ofadding, and they correctly carried that mistake through the problem(that is if they had used 5 they would have gotten the correct answer) Iwould probably only take off one point out of 10.Those are my thoughts. Your grade, suck it up. If you ask theinstructor to look over the test again, he or she might find some placeswhen you should hae lost more points that you did. At least, if someonequibles over two or three points I say I will look it over, and I do andmake sure I *took off* all the points I should have. - Tim-- Timothy M. BrauchGraduate StudentDepartment of MathematicsWake Forest Universityemail is:news (dot) post (at) tbrauch (dot) comAnonymous,It is my opinion that your comments about Tim being arrogant andunthoughtful and every other rude thing you said about him were wayoutof line. For example, I see no evidence in his post that he assignsgrades based on anything other than the subject matter at hand. Forexample,Tim said:Secondly, if I am grading and I cannot follow the work (perhapsbecauseit is not neatly written), I have a hard time giving full credit. Thatmight sound harsh, but I have often found that students write two orthree possible answers for the problem in their mess and never clearlymark which one they want me to grade.Note that he finds a hard time giving full credit when he cannotfollow thework or when students write more than one answer. How could he givesuch answers full credit? A large percentage of students bluff onexamsby writing any irrelevant thought that comes into their heads to thepoint where their test papers are completely incoherent. Instructors usually do their best to find something good in the answers and go outof their way to give partial credit. I see no evidence that Tim isn'toneof these caring teachers.Also, when he said things like he despised when students squeezed workinto the margins, notice that he did not say such a thing affected thestudents' scores.Tim gave further evidence that he is a fair and caring teacher when he said he doesn't take off very many points for errors that aren'trelated to the material presently being taught.Tim's last comment may seem harsh--that he might even deduct points if a student quibbles over his or her score and he finds a legitimatemathematical reason to deduct those points. But I believe he may bejustified in this behavior. A significant percentage of students (especially at large universities) are immoral and do not care about learning the material. They only want to buy their grades. Suchstudentsoften waste the time of the instructors by asking for regrades (evenwhen they haven't read their exam over closely)! I have seen studentsinsist to instructors that they did a problem correctly and deservedfullcredit, when in fact they had no idea what the question meant, muchlesshow to solve it. When students purchase the services of theinstructor,how much services should they actually get? Unfortunately no contractis made and nothing is written down. I believe it is reasonable for a professor to set certain rules concerning re-grades and coherenttestsolutions. Without these rules, some students may use up too much ofthe professor's time and he won't be able to give deserved time toothermore respectful students.Anyway, your comments about Tim commiting fraud and breaking the lawin the worst way are completely ridiculous. You don't have enoughinformationto judge in this particular matter for one thing. For another, Tim'spostin no way supports such a judgement. Anyway, do you know exactly whatthe law states in such matters? For example, is every student in aclassof 50 entitled to write so illegibly and incoherently on tests that ittakesthe professor three hours to grade it? If they were entitled to suchthings,the price of their education would go way up. -Leoanrd Blackburn === Subject: as usuall , help is needed!!!hi any one can solve this problem please say me.if f(x)=y , f'(x)=y', f(x)=y [(1+x^2)^2]*y+2x*(1+x^2)*y'+y=0now change x=tan(t) now write [(1+x^2)^2]*y+2x*(1+x^2)*y'+y=0 with t and f(t)or y thank you hupo === Subject: Re: as usuall , help is needed!!!Use the chain rule to get dy/dt in terms of dy/dx. === Subject: Re: as usuall , help is needed!!!> Use the chain rule to get dy/dt in terms of dy/dx.> -Michael.dear Michaelif your means is [dy/dx]=[dy/dt]*[dt/dx]i can not solve with it if you can then so0lve === Subject: Bayesian Class and Math/Stat Teaching TechniquesA few weeks ago I posted a message asking about books on BayesianUnfortunately I have since dropped the class and am wondering about whetherI should continue the degree (Masters in Applied Statistics) and would likesome thoughts from the thoughtful people here.I have a BS in Electrical Engineering and an MBA, both from the Universityof Michigan. What I really liked about the MBA program is that it was almost100% applied. Probably 50-70%+ of the classes was case study classes, atrend mostly propagated and refined by Harvard Business School wheresupposedly 100% of the classes are case study classes. Apparently more andmore law programs have more and more case study classes as well. Case studyclasses are really as applied as you get because concepts and theories arelearned in the context of real world situations and circumstances. I am alsothe type of person who is a very intuitive learner and has a much easiertime learning when I see how what I am learning relates to challenges inreal life (eg: business, which is what I do).So given that the degree I am pursuing is called Masters in *APPLIED*Statistics, I thought the the courses would be heavily applied and taught inthe context of solving real world problems. No dice. Both courses I took inthe first semester (part-time evening program) had heavy theory. TheBayesian class was not even as bad as the other one (MathematicalStatistics). There was essentially no attempt on the part of the professorto relate the theory to real world programs or to even give real worldexamples to illustrate the concepts. It was formula, theory, formula,theory, theory, formula, etc. I asked him about that and he said there's noway around the theory. I'm not trying to get around the theory buttheories and formulas mean nothing to me without real world context. I'm notstupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GREand 98+ percentile overall.My thinking right now is that my expectations were just off and disciplineslike Math/Statistics are just not as, ummm, progressive as Business/Law whenit comes to teaching (please -- no hate mail). Those teaching Math/Stat mayalso be too smart and are not interested in mundane day-to-daybusiness/industry problems (hopefully that will stop the hate mail!). Sowhat's up with that? Why is a degree called Masters in Applied Statisticsso heavy in theory? I'm not interested in theory in the absence ofapplication. I enrolled in Masters in Applied Statistics to learn how touse statistical techniques to solve real world problems, how to usestatistical software to solve real world problems, etc. and not to learnesoteric statistical theory in the absence of application that I will surelyforget an hour after the final exam.I am not trying to slam the field. I am interested in some opinions fromthose in the field, especially those teaching it, to help me determine if I === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques> So given that the degree I am pursuing is called Masters in *APPLIED*> Statistics, I thought the the courses would be heavily applied and taught in> the context of solving real world problems. No dice. Both courses I took in> the first semester (part-time evening program) had heavy theory. The> Bayesian class was not even as bad as the other one (Mathematical> Statistics). There was essentially no attempt on the part of the professor> to relate the theory to real world programs or to even give real world> examples to illustrate the concepts. It was formula, theory, formula,> theory, theory, formula, etc. I asked him about that and he said there's no> way around the theory. I'm not trying to get around the theory but> theories and formulas mean nothing to me without real world context. I'm not> stupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GRE> and 98+ percentile overall.There's just a certain amount of theory you'll have to get through.It is building the basis for the practical applications. And, unlessyou got into the measure theory foundations of probability, the theorythey were teaching you was likely a lot of simple calculus appliedto statistics.One small example of learning theory versus practical applications:There was an undergrad operations research course slanted towardsengineering students. The weekly assignment had a problem that had tobe solved with Newton's method -- fine, no problem, there's a formula.But this particular problem was mistated by the prof, and Newton'smethod basically hiked over an asymptote and headed off to infinity,for just about any guessable starting position.It suddenly became a more interesting problem.So, one solution was to slightly modify Newton's method, and basicallygive it a bit of drag. By adjusting the drag, and with a bit of luck,the asymptotes can be avoided, and a solution pops out.But most of the students didn't do that -- they didn't know enough aboutthe basic workings of Newton's method to modify it.Theory is important. Real world problems often don't have standard solutions. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques> A few weeks ago I posted a message asking about books on Bayesian> Unfortunately I have since dropped the class and am wondering about whether> I should continue the degree (Masters in Applied Statistics) and would like> some thoughts from the thoughtful people here.> I have a BS in Electrical Engineering and an MBA, both from the University> of Michigan. What I really liked about the MBA program is that it was almost> 100% applied. Probably 50-70%+ of the classes was case study classes, a> trend mostly propagated and refined by Harvard Business School where> supposedly 100% of the classes are case study classes. Apparently more and> more law programs have more and more case study classes as well. Case study> classes are really as applied as you get because concepts and theories are> learned in the context of real world situations and circumstances. I am also> the type of person who is a very intuitive learner and has a much easier> time learning when I see how what I am learning relates to challenges in> real life (eg: business, which is what I do).> So given that the degree I am pursuing is called Masters in *APPLIED*> Statistics, I thought the the courses would be heavily applied and taught in> the context of solving real world problems. No dice. Both courses I took in> the first semester (part-time evening program) had heavy theory. The> Bayesian class was not even as bad as the other one (Mathematical> Statistics). There was essentially no attempt on the part of the professor> to relate the theory to real world programs or to even give real world> examples to illustrate the concepts. It was formula, theory, formula,> theory, theory, formula, etc. I asked him about that and he said there's no> way around the theory. I'm not trying to get around the theory but> theories and formulas mean nothing to me without real world context. I'm not> stupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GRE> and 98+ percentile overall.That makes your comments even less acceptable.> My thinking right now is that my expectations were just off and disciplines> like Math/Statistics are just not as, ummm, progressive as Business/Law when> it comes to teaching (please -- no hate mail). Those teaching Math/Stat may> also be too smart and are not interested in mundane day-to-day> business/industry problems (hopefully that will stop the hate mail!). So> what's up with that? Why is a degree called Masters in Applied Statistics> so heavy in theory? I'm not interested in theory in the absence of> application. I enrolled in Masters in Applied Statistics to learn how to> use statistical techniques to solve real world problems, how to use> statistical software to solve real world problems, etc. and not to learn> esoteric statistical theory in the absence of application that I will surely> forget an hour after the final exam.Can you think back to your EE course? Differentiation, integration,Coulomb's law, Newton's law, ... -- what would have happened had yourhigh-school teachers decided to skip all that 'theory'?Have you ever attempted to apply theory without having covered thebasis of that theory?Why are statistics books full of mathematics? So that professors canshow off? Or because they like to keep sales down(Assuming that this is not a troll.)Could you elaborate on what you think you mean by: ... how to usestatistical software to solve real world problems, etc...?Loading a set numbers into a calculating program and getting back somenumbers? Doing experiments with dice and balls and urns? And maybe wecould hear something of the law and business studies 'case studies',so that we could attempt to suggest why that approach is notapplicable here?> I am not trying to slam the field. I am interested in some opinions from> those in the field, especially those teaching it, to help me determine if IIf I was hiring an MSc in Applied Statistics (or doing such a degreemyself), I'd be most disappointed if there wasn't a strong theoreticalcontent. At least, I'd expect graduates to be prepared for writing(new) programs (or devising algorithms) to solve (unsolved) problems.What did the rest of the class think?Jon C. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques> Why is a degree called Masters in Applied Statistics so heavy in> theory? I'm not interested in theory in the absence of> application. I enrolled in Masters in Applied Statistics to learn> how to use statistical techniques to solve real world problems, how> to use statistical software to solve real world problems, etc. and> not to learn esoteric statistical theory in the absence of> application that I will surely forget an hour after the final exam.I my understanding, Applied Math is solid, well-funded math whichhas interesting applications and whose development is mostly driven byreal-world needs (as opposed to inner mathematical needs). AppliedMath does not mean that there are less theorems, or that it's justknow how to apply a formula. The problem with just teaching how toapply concept X is that in some situations X cannot be applied asusual. If you don't understand the exact limitations of your methodsthe bridge you build will collapse or the insurance company you advisewill go bankrupt. One of the best ways to make sure you understand atheorem is to prove it. In my experience, proof by trying (doing acase study) is, unfortunately, not an option.If you are lucky, you might find a teacher who gives you both, atranslucent introduction into the theory as well as fascinating casestudies. But these people are rare. And you must be able to adopt topeople will expect that she be able to do both, understandapplications _and_ develop new theory where necessary. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques> There was essentially no attempt on the part of the professor> to relate the theory to real world programs or to even give real world> examples to illustrate the concepts. It was formula, theory, formula,> theory, theory, formula, etc. I asked him about that and he said there'sno> way around the theory.To me, math is like a Swiss army knife: It's a tool. A neat and handy tool,and it can do all sorts of smart things. Some parts I use everyday, otherparts I hardly know what do.You could also compare math to a car. People have different opinions about acar. To some, it is merely a means of transportation. They don't care how itworks, they just want to be able to drive to work every day. For others, acar is a toy. They can spend hours every day fiddling with it, turning aknob, drilling a hole, adjusting a valve, just to see how it works.Tools for me include: Swiss army knife, a car, and math. On the other hand,a computer is like a toy to me, but that is O.T.BUT, that is not all. When using a tool, whether it is math or a car, youneed to learn how to use it. You also need to know the basics of how itworks. For a car, you need to know the traffic laws, basic maintenance, andyou need to know not to drive too fast in adverse weather and roadconditions.In short, you need some theory, before you can start to drive a car. Thesame applies to math.I've taken lots of so-called applied math courses, without seeing a singleapplication. But then again, there are different opinions. I believe someparts of math are more deep than others. For instance, the proof ofFermats Last Theorem involves concepts that I have never heard (e.g. modularforms), even though I have taken several courses in discrete math. So, Iconsider such concepts deep. On the other hand, stuff like calculus andlinear algebra are more likely to find everyday use.One of my prof's expressed the opinion that all undergraduate classes areapplied in the above sense. I agree with that sentiment. Despite that,there were no real-worls applications in any of them. That's too bad.I've been a T.A. in undergraduate math for several years, and it was verycommon to be asked about potential every-day applications. I found it verydifficult to give such answers, but I attribute that to lack oftraining/experience on my part. I've certainly learned that many students -like the O.P. - find it much easier to grasp the concepts when they getrelated to something less abstract.It is the teachers job to explain both the theory *and* give some reasonableapplications. The latter will be the most challenging for me, should I everget a job teaching math.P.S. The O.P. has noticed a difference between learning Math and Business.Perhaps there is - generally - a different teaching culture between the two. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques> To me, math is like a Swiss army knife: It's a tool. A neat and handytool,> and it can do all sorts of smart things. Some parts I use everyday, other> parts I hardly know what do.That is *exactly* why I enrolled in Masters of Applied Statistics -- Iwant to use it like a Swiss Army Knife. I just happen to be an outdoorsmanso let's take your analogy and run with it. If I showed a person who hasnever seen a Swiss Army Knife (or Gerber, Leatherman, etc.) a Swiss ArmyKnife and I cannot explain how/when to use the various tools on the SwissArmy Knife without hours and hours and hours of theory without actuallyreferring *to* the knife or demonstrating with the knife, what's the point?I should just throw the knife away and talk about metallurgy, the history ofman and tools, how to manufacture knives, or something equally theoretical.An *applied* class in Swiss Army Knives should involve lots of actual usageof Swiss Army Knives in real world situations with very short blurbs aboutthe genesis of Swiss Army Knives, metallurgy, etc.> You could also compare math to a car. People have different opinions abouta> car. To some, it is merely a means of transportation. They don't care howit> works, they just want to be able to drive to work every day. For others, a> car is a toy. They can spend hours every day fiddling with it, turning a> knob, drilling a hole, adjusting a valve, just to see how it works.Sure. I also happen to be an amateur mechanic. If I enroll in a classcalled, say, Applied Automotive Repair, the theory necessary for therepairs should be taught in the context of the actual repairs. If theclasses of Applied Automotive Repair have almost not hands on work on carsand are instead lots and lots of formulas and theory (again, without actualyhands on work), it's not really applied -- it's Theory of AutomotiveDesign and Repairs.> One of my prof's expressed the opinion that all undergraduate classes areapplied in the above sense. I agree with that sentiment. Despite that,> there were no real-worls applications in any of them. That's too bad.That is unfortunate but it's not nearly as unfortunate as a part-timegraduate program called Masters of Applied Statistics that's got lots oftheory without applications. Why? Almost all undergrads have nocareer-oriented work experience and wouldn't even know how/where to applywhat they learn. Grads in part-time graduate programs called Masters ofApplied Statistics have years and years of professional work experience andcan all conceptualize where and how they can use what they just learned.It's a pity that those people go back to school and get essentially treatedprograms do not fail their students.> I've been a T.A. in undergraduate math for several years, and it was very> common to be asked about potential every-day applications. I found it very> difficult to give such answers, but I attribute that to lack of> training/experience on my part. I've certainly learned that manystudents -> like the O.P. - find it much easier to grasp the concepts when they get> related to something less abstract.Don't take this as an insult -- because it's not meant as one -- but Isuspect you have been in academia most/all of your life because people whohave been in industry would probably have a much easier job of relating whatthey are teaching to the undergrads and how it might be used in the realworld.> It is the teachers job to explain both the theory *and* give somereasonable> applications. The latter will be the most challenging for me, should Iever> get a job teaching math.> -Michael.Hopefully you succeed. If you have not been in industry you might find itvery useful to be in industry for a few years before you teach math.> P.S. The O.P. has noticed a difference between learning Math and Business.> Perhaps there is - generally - a different teaching culture between thetwo.Almost certainly. Business is focused on making money, period. Everythingelse is a means to that end; It's the ultimate in applied discipline. Allthe professors in the business school I went to still worked/consulted inindustry so they are in touch with the real world and still very much havetheir applied caps on (and almost none had PhDs). Everyone is focused onhow to *solve problems* such that profits are maximized.department was just different. They liked theory and they liked formulas.They liked elegant solutions and proofs, even if they were irrelevant toapplication. I sensed a certain disdain for word problems and real worldanalogies and explanations to help the students conceptualize the theorybecause real math students don't need those crutches. They're very smartpeople who would probably look contemptuously at the description I gaveabove for business schools (No PhDs?!?! Only interested in profit?!?!Theory only useful if taught in conjuction with application?!?! Howgrotesque, how low-brow, how coarse!!!). === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques>department was just different. They liked theory and they liked formulas.>They liked elegant solutions and proofs, even if they were irrelevant to>application. I sensed a certain disdain for word problems and real world>analogies and explanations to help the students conceptualize the theory>because real math students don't need those crutches.My experience, and that of other math/stat instructors whom I'vetalked to, is quite the opposite. It's the STUDENTS who don't likeword problems, and resist applications (eg, to physics), because to dothem they have to actually understand the mathematical material (andeven some physics!), rather than just applying formulas without reallyknowing what they're doing. This may not be true of real mathstudents, however, who ought to be able to do the word problems (butwho may find the standard ones to be too easy to be interesting). Radford Neal---------------------------------------------------------- ------------------Radford M. Neal radford@cs.utoronto.caDept. of Statistics and Dept. of Computer Science radford@utstat.utoronto.caUniversity of Toronto http://www.cs.utoronto.ca/~radford---------------------------- ------------------------------------------------ === Subject: Re: Bayesian Class and Math/Stat Teaching TechniquesI once took a two semester class on Foundations of Applied Mathematics.The textbooks were: and an earlier edition of These are great books. The approach is more case study thantheorem-proof-corollary. We would see examples of how to modelproblems. I seem to remember typically enclosing a flux fieldwith some boundary and doing some accounting for stuff flowingacross the boundary, being created, and destroyed.Typically, we ended up with systems of non-linear differentialequations. We learned about scaling, and how to find solutions toapproximations of the original problems. We found out to check thatthe solutions were small where we assumed that they were, but learnedthat this didn't guarantee that our exact solution to the approximateproblem was an approximate solution to the exact problem.We saw perturbation theory, and such issues as boundary layerseparation.I have doubtless forgot much that I learned. But I thought this wasan extremely cool course. But I know some of my engineer friendsstill thought this course too theoretical.-- Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/ Bukharin.html To solve Linear Programs: .../LPSolver.htmlr c A game: .../Keynes.html v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question fit perhaps to be discussed by n e . slaves in the hearing of their masters, but highly @ r c m unbecoming to reasonable and free men in search of d o the truth. -- Rousseau === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques> One of my prof's expressed the opinion that all undergraduate classes are> applied in the above sense. I agree with that sentiment. Despite that,> there were no real-worls applications in any of them. That's too bad.>That is unfortunate but it's not nearly as unfortunate as a part-time>graduate program called Masters of Applied Statistics that's got lots of>theory without applications. Why?Because any real-life applications would almost certainly be toocomplex and time-consuming to be covered? Besides, you dropped theclass, how do you know there weren't any practical applicationsintroduced later on after the theory was covered?>Almost all undergrads have no>career-oriented work experience and wouldn't even know how/where to apply>what they learn. Grads in part-time graduate programs called Masters of>Applied Statistics have years and years of professional work experience and>can all conceptualize where and how they can use what they just learned.>It's a pity that those people go back to school and get essentially treated>programs do not fail their students.That's because in business and law, how you handle real-lifesituations is what it's all about. In math, while you can certainlycram yourself with formulas and examples for three years, what happensthe first time you run into a problem which was not covered in theApplied Applied Statistics course?>Don't take this as an insult -- because it's not meant as one -- but I>suspect you have been in academia most/all of your life because people who>have been in industry would probably have a much easier job of relating what>they are teaching to the undergrads and how it might be used in the real>world.That's a stretch. Most applied mathematicians will do their masters ordoctorate work on problems directly relating to some real-lifeproblem. I doubt a statistician working in the industry would beable to give any more practical applications for abstract probabilitytheory than someone with years of experience in the academia.Odds are the industry statistician would just blather on about hisown work, never mind if it has any relevance to the topic beingcovered.>Almost certainly. Business is focused on making money, period. Everything>else is a means to that end; It's the ultimate in applied discipline. All>the professors in the business school I went to still worked/consulted in>industry so they are in touch with the real world and still very much have>their applied caps on (and almost none had PhDs). Everyone is focused on>how to *solve problems* such that profits are maximized.If they were really interested in maximizing profits, they would havejust given up with the math and outsourced it to some outfit in China.>department was just different. They liked theory and they liked formulas.>They liked elegant solutions and proofs, even if they were irrelevant to>application.That's why it's called the math department! Would you appreciate abusiness school that dwelled on non-business related topics likephilosophy, physics and women studies?>I sensed a certain disdain for word problems and real world>analogies and explanations to help the students conceptualize the theory>because real math students don't need those crutches.Any math department that discourages against methods that help thestudents learn doesn't sound very promising, I agree, but then againI'm personally frustrated with the eternal what practicalapplications does theory X have? inquiries.Abstraction is an important method for reducing the solution of oneproblem to such a general form that it can be applied to a multitudeof different problems. By teaching abstract theories instead of rigidone-problem-solving methods, you actually enable the students totackle a larger problem set (provided they have the ability to makethe leap from abstract definitions to real-life parameters).>They're very smart>people who would probably look contemptuously at the description I gave>above for business schools (No PhDs?!?! Only interested in profit?!?!>Theory only useful if taught in conjuction with application?!?! How>grotesque, how low-brow, how coarse!!!).This would be an interesting argument if it was an actual quote fromsomeone instead of just a strawman erected to bolster your point ofview.-- I'm not interested in mathematics that might have anythingto do with reality. -- Russell Easterly, in sci.math === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques>One of my prof's expressed the opinion that all undergraduate classesare>applied in the above sense. I agree with that sentiment. Despitethat,>there were no real-worls applications in any of them. That's too bad.>That is unfortunate but it's not nearly as unfortunate as a part-time>graduate program called Masters of Applied Statistics that's got lotsof>theory without applications. Why?> Because any real-life applications would almost certainly be too> complex and time-consuming to be covered?How then are the students going to learn how to use the theory? Only a veryfew of them will be able to work that out by themselves.>Almost all undergrads have no>career-oriented work experience and wouldn't even know how/where to apply>what they learn. Grads in part-time graduate programs called Masters of>Applied Statistics have years and years of professional work experienceand>can all conceptualize where and how they can use what they just learned.>It's a pity that those people go back to school and get essentiallytreated>programs do not fail their students.> That's because in business and law, how you handle real-life> situations is what it's all about. In math, while you can certainly> cram yourself with formulas and examples for three years, what happens> the first time you run into a problem which was not covered in theApplied Applied Statistics course?That's going to the other extreme, completely neglecting the theory andinstead just memorizing a bunch of formulas without knowing how or why theywork.I'm advocating for combining theory and practical applications, teaching thestudents why the tool theory works *and* how to use it.>Don't take this as an insult -- because it's not meant as one -- but I>suspect you have been in academia most/all of your life because peoplewho>have been in industry would probably have a much easier job of relatingwhat>they are teaching to the undergrads and how it might be used in the real>world.> That's a stretch. Most applied mathematicians will do their masters or> doctorate work on problems directly relating to some real-life> problem. I doubt a statistician working in the industry would be> able to give any more practical applications for abstract probability> theory than someone with years of experience in the academia.I disagree, but I have no data to back me up. I'm quite curious and I wouldlike it if someone could give a few data points.Perhaps reality is mixed: You could probably find both good and badapplicationers in both academia and the industry.> Odds are the industry statistician would just blather on about his> own work, never mind if it has any relevance to the topic being> covered.I would think that risc to be equally applicable to a math prof :-)>I sensed a certain disdain for word problems and real world>analogies and explanations to help the students conceptualize the theory>because real math students don't need those crutches.> Any math department that discourages against methods that help the> students learn doesn't sound very promising, I agree, but then again> I'm personally frustrated with the eternal what practical> applications does theory X have? inquiries.> Abstraction is an important method for reducing the solution of one> problem to such a general form that it can be applied to a multitude> of different problems. By teaching abstract theories instead of rigid> one-problem-solving methods, you actually enable the students to> tackle a larger problem set (provided they have the ability to make> the leap from abstract definitions to real-life parameters).Actually, the real difficulty lies in the reverse direction, going fromreal-life descriptions to an abstract model, at least when you trying tosolve a concrete problem. This modelling step is something one really onlylearns through experience and lots of practice. Now, how do the students getthat experience?This is also the reason behind the question what practical applicationsdoes theory X have? The question really boils down to hands-on trainingwith modelling. i.e. teaching the students how to *use* the theory.Some people are able to work out the practical applications without needingany help. They would probably end up doing a PhD a few years down the road.For all the rest of the students, they need training and training and moretraining. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques> Don't take this as an insult -- because it's not meant as one -- but I> suspect you have been in academia most/all of your life because people who> have been in industry would probably have a much easier job of relatingwhat> they are teaching to the undergrads and how it might be used in the real> world.Yep. I had a career in academia, but then I switched to industry. The changewas coincidental at the time, more or less forced upon me by the jobsituation at the time, but I've come to like it out here in the realworld!Nevertheless, I appreciate the theory and its importance. It's not a matterof either/or. I believe one should attempt to combine both theory andapplication, possibly 50% time and effort spent on each. Even in a coursespecifically labelled applied.It is the teachers job to explain both the theory *and* give some> reasonableapplications. The latter will be the most challenging for me, should I> everget a job teaching math.-Michael.> Hopefully you succeed. If you have not been in industry you might find it> very useful to be in industry for a few years before you teach math.Absolutely. For the very same reason, I actually would prefer teachingphysics, simply because it is easier to relate to real-world situations.Anyway, most applied mathematics rests on some kind of model of the realworld, so teaching applied math actually requires the students are wellversed in creating mathematical models describing a specific problem.Converting an every-day problem into a mathematical description is really anart, something that takes years of training. I suppose such training shouldbe a part of the applied courses.P.S. The O.P. has noticed a difference between learning Math andBusiness.Perhaps there is - generally - a different teaching culture between the> two.> Almost certainly. Business is focused on making money, period. Everything> else is a means to that end; It's the ultimate in applied discipline.All> the professors in the business school I went to still worked/consulted in> industry so they are in touch with the real world and still very muchhave> their applied caps on (and almost none had PhDs). Everyone is focused on> how to *solve problems* such that profits are maximized.> department was just different. They liked theory and they liked formulas.> They liked elegant solutions and proofs, even if they were irrelevant to> application. I sensed a certain disdain for word problems and real world> analogies and explanations to help the students conceptualize the theory> because real math students don't need those crutches. They're very smart> people who would probably look contemptuously at the description I gave> above for business schools (No PhDs?!?! Only interested in profit?!?!> Theory only useful if taught in conjuction with application?!?! How> grotesque, how low-brow, how coarse!!!).I'm detecting some contempt on your part towards mathematicians in general,or perhaps I am mistaken.Anyway, mathematics (and mathematicians) have a justification on its own. We(read: civilization) need people who are able to develop and expand on themathematical tools available.But there is a gap between mathematics and business, in what you havedescribed. I suppose it's up to people like you and I to close that gap... === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques> My thinking right now is that my expectations were just off and> disciplines like Math/Statistics are just not as, ummm, progressive as> Business/Law when it comes to teaching (please -- no hate mail). Those> teaching Math/Stat may also be too smart and are not interested in> mundane day-to-day business/industry problems (hopefully that will> stop the hate mail!). So what's up with that? Why is a degree calledMasters in Applied Statistics so heavy in theory? I'm not interested> in theory in the absence of application. I enrolled in Masters in> Applied Statistics to learn how to use statistical techniques to> solve real world problems, how to use statistical software to solve> real world problems, etc. and not to learn esoteric statistical theory> in the absence of application that I will surely forget an hour after> the final exam. Personally, for me, and I am an applied mathematician, I think the applied people actually have to learn more theory than the theoretical mathematicians. The reason being that the theoretical mathematicians are all happy to be doing math just for the sake of math. Me, however, I need to why someone would ever do it, then once I know why someone would, I need to why it actually works. (or sometimes why something works, and then what its purpose is) For example, why does linear regression give the best straight line fit to model data? I can do it no problem, but if it's all for naught, why bother learning it.Those are just my thoughts. - Tim-- Timothy M. BrauchGraduate StudentDepartment of MathematicsWake Forest Universityemail is:news (dot) post (at) tbrauch (dot) com === Subject: Re: Bayesian Class and Math/Stat Teaching TechniquesMy thinking right now is that my expectations were just off anddisciplines like Math/Statistics are just not as, ummm, progressive asBusiness/Law when it comes to teaching (please -- no hate mail). Thoseteaching Math/Stat may also be too smart and are not interested inmundane day-to-day business/industry problems (hopefully that willstop the hate mail!). So what's up with that? Why is a degree calledMasters in Applied Statistics so heavy in theory? I'm not interestedin theory in the absence of application. I enrolled in Masters inApplied Statistics to learn how to use statistical techniques tosolve real world problems, how to use statistical software to solvereal world problems, etc. and not to learn esoteric statistical theoryin the absence of application that I will surely forget an hour afterthe final exam.> Personally, for me, and I am an applied mathematician, I think the applied> people actually have to learn more theory than the theoretical> mathematicians. The reason being that the theoretical mathematicians are> all happy to be doing math just for the sake of math. Me, however, I need> to why someone would ever do it, then once I know why someone would, Ineed> to why it actually works. (or sometimes why something works, and thenwhat> its purpose is) For example, why does linear regression give the best> straight line fit to model data? I can do it no problem, but if it's all> for naught, why bother learning it.> Those are just my thoughts.The way the business schools and law schools would teach that concept (whydoes linear regression give the best straight line fit to model data) wouldbe to have a case where, say, someone using another method got less thanoptimal results that ended in disaster and then show/teach what thecalculations might be with linear regression. The way the AppliedStatistics classes I was in would have taught that would have been toproduce a bunch of incomprehensible formulas without any real world examplesillustrating the concept and the pluses/minuses of each method.In fact, I specifically remember a class in business school illustrating astatistical problem and how/why the scientists at Morton Thiokol did notcatch the potential problem with the o-ring on the space shuttle Challengerthat caused the tragedy -- they didn't do tests nor have data points for theo-ring at low temperatures, and the temperature at launch was very cold forcentral Florida.I am certainly not saying don't teach theory. I am just questioning thepoint of incessantly teaching theory in the complete *absence* ofapplication for a degree called Masters of Applied Statistics. === Subject: Re: Bayesian Class and Math/Stat Teaching TechniquesJust a comment on a detail here -[ delete, some other stuff]> In fact, I specifically remember a class in business school illustrating a> statistical problem and how/why the scientists at Morton Thiokol did not> catch the potential problem with the o-ring on the space shuttle Challenger> that caused the tragedy -- they didn't do tests nor have data points for the> o-ring at low temperatures, and the temperature at launch was very cold for> central Florida.[ ... ]As I recall reading about it, it eventually came out that there were scientists/ engineers around who had wanted to scrubthe cold-weather launch. *Since* they did not have data, they were wise enough to have serious doubts -- but they were unable to convince the administrators, who were not'technical people.'I wonder what the point was, in a business school class? - Keep channels open to your technical people? - or, S**t happens? -- Rich Ulrich, wpilib@pitt.eduhttp://www.pitt.edu/~wpilib/index.htmlTaxes are the price we pay for civilization. === Subject: Re: Explanation with math, error in math worldIn sci.logic, Paul Murray:> (sci.math added)> It turns out that it might actually be easier to explain the problem> that mathematicians have by using mathematics, and rather simple math> at that as consider> Hmm, so you've been proved wrong so many times in sci.math that you are> just doing to post elsewhere, and hope noone notices?Well, look at it this way: he has such a momentous discoverythat he just has to share it.Erm...well, the word starts with an 'm', anyway. But he's astubborn one, I'll grant him that. (Not that it helps hismath any.)-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: Adjacency matrix algorithm re graph theoryJim,Perhaps I am not understanding your reply.I am trying to prove that an algorithm can be devised to guarantee thatabsolute value of the difference between the sum of incoming edges andoutgoing edges for each vertex is less than or equal to 1. The in and outedges for each vertex have to be played with.The idea is to keep the absolute value of the difference less than two. Oneis OK.I have been playing with adjacency matrices, but have not yet come up withan algorithm.DianaI am trying to prove that for every graph G a balanced orientation canbefound. In other words, for a digraph, the absolute value of thedifferencebetween the sum of incoming edges and outgoing edges for every vertex isless than or equal to 1.Any ideas?> Note that every edge contributes 1 to the total in-degree (ID) and 1 to> the total out-degree (OD), so ID = OD. If the graph is not balanced, then> some vertex has more in-degree and some vertex has more out-degree. Is> there an edge between then that will decrease these? If there isn't an> edge, is there a sequence of two edges (going through another vertex) for> this to work?> One might think that if the two vertices are in the same connected> component there will be path of alternations... I think you might need a> little stronger condition than connectedness.> J === Subject: Re: Adjacency matrix algorithm re graph theory > Perhaps I am not understanding your reply.> I am trying to prove that an algorithm can be devised to guarantee that> absolute value of the difference between the sum of incoming edges and> outgoing edges for each vertex is less than or equal to 1. The in and out> edges for each vertex have to be played with. Yes... as I said, the total indegree over all vertices must equal the total outdegree over all vertices. Let D(v) = difference of indegree-outdegree. So, say that there is vertex u with D(u) = 2 and then some vertex w with D(w) = -2. If they have an edge between them such that flipping its orientation would modify D(u) to 1 and D(w) to -1, then flip it. If such an edge does not exist, then maybe there is a path of edges through an intermediate vertex z such that flipping uz and zw would improve those D() values... Again, if such edges don't exist, maybe there are two intermediate vertices, and so on... Is there always a path between u and w? Note that total indegree must equal total outdegree for any connected component, and not just the whole graph. When I said before that 'connectedness' might not be enough, what I meant specifically is that perhaps you need to look at connectedness in the sense of directed edges. I hope this helps,J === Subject: Re: JSH: Understand now? Frustration?> That's has to do with why at times I've just gotten totally pissed and> called Arturo Magidin evil, as it's just so frustrating to deal with> people so adept at obscuring the truth.Is this why you accepted Prof. Magidin's offer to stop responding to you,so that you could continue this cowardly sniping at him without fearof getting a response? You're just demonstrating that you aren't manenough to confront him directly -- something that comes as no surpriseto the rest of us, of course.-- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvisefwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper.e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Re: JSH: Understand now? Frustration?That's has to do with why at times I've just gotten totally pissed andcalled Arturo Magidin evil, as it's just so frustrating to deal withpeople so adept at obscuring the truth.> Is this why you accepted Prof. Magidin's offer to stop responding to you,> so that you could continue this cowardly sniping at him without fear> of getting a response? You're just demonstrating that you aren't man> enough to confront him directly -- something that comes as no surprise> to the rest of us, of course.> -- > Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvise> fwbrown@bellsouth.net | if you're good enough. Otherwise you give> | your pelt to the trapper.e^(i*pi) = -1 -- Euler | -- John Myers Myers, SilverlockWayne, James isn't a man, period.David Moran === Subject: Re: JSH: Understand now? Frustration?>> Using P(x) = (x+8a)(x+b), where ab=1, and considering when (8a +b) is> an integer can give you a perspective on what I've been saying, I> hope.>> Since b = 1/a, by your requirement, then (8a + b) = (8a + 1/a). Areyou> requiring that this expression be an integer?> To be fair, he said consider those cases where (8a+b) _is_ aninteger.> In a previous post he chose (8a+b) = 9, ab=1 which has the twosolutions> [b=8, a=1/8] and [b=1, a=1]> so both (8a + b) and (8a + 1/a) are indeed integer even though a=1/8is not.> KeithKI don't believe I was being unfair. I was simply asking forclarification. Jameshas a history of posting contradictory requirements and leaving theburden ofchoosing what to work with to the reader.> I've been arguing with posters like Arturo Magidin for YEARS, and> throughout that period, I've found my attempts to clarify were> hampered by attempts by other posters like here C. Bond to confuse,> and I thank KeithK for catching one as he did.I was not implying that C. Bond was attempting to confuse. I simplyprovided a clarification.KeithK> The problem isn't with C. Bond using 8a + 1/a as that is correct, as> long as you consider the possibility of 'a' being a factor of 1 in> some more inclusive ring than the ring of algebraic integers.> Circular arguments have gone on and on and on for some years here, as> people like Arturo Magidin, have successfully dodged being pinned down> on this issue as they repeatedly relied on certain numbers not being> algebraic integers to support their claims.> Notice, I'm NOT saying that 'a' is an algebraic integer when it's> irrational!> If you say that 1/a is not an algebraic integer, when irrational, yes,> you are correct.> But my point is that the label algebraic integer does not contain> all numbers such that 1/a is contained and -1 and 1 are the only> integer units. There's a more inclusive ring, where -1 and 1 are the> only integer units, where 1/a is a unit, and 'a' is not an algebraic> integer.> Here it takes effort if some poster comes in to confuse you, relying> on the *assumption* that if 1/a is not an algebraic integer then it's> more like a fraction like 1/2 than 1/1.> But think of x=(1+sqrt(-3))/2, and someone giving you 1/x, if you were> mostly experienced with integers.> Think of how easy it might be for someone to confuse you with> 2/(1+sqrt(-3)) if you weren't careful and it was new to you.> That's has to do with why at times I've just gotten totally pissed and> called Arturo Magidin evil, as it's just so frustrating to deal with> people so adept at obscuring the truth.> My hope is that at least some of you will play with> x^2 + (8a + b)x + 8, with ab = 1, and (8a + b) an integer> so that you can see why *logically* a possibility must still be there> when 'a' and 'b' are irrational, even if 'a' is not an algebraic> integer.> If you can get a handle on that possibility, then maybe discussions> can be more fruitful about those numbers lost in the shuffle that> irrational 'a' here represents.> You know, they are *most* numbers given their cardinality.> James Harris === Subject: Re: JSH: Understand now? Frustration? > Notice, I'm NOT saying that 'a' is an algebraic integer when it's > irrational! > > If you say that 1/a is not an algebraic integer, when irrational, yes, > you are correct. > > But my point is that the label algebraic integer does not contain > all numbers such that 1/a is contained and -1 and 1 are the only > integer units. There's a more inclusive ring, where -1 and 1 are the > only integer units, where 1/a is a unit, and 'a' is not an algebraic > integer.Right, the algebraic integers are not *that* ring. But you can adjoin1/a to the ring of algebraic integers and get a more inclusive ring,such that 1/a is in that ring. On the other hand, when you do adjoin1/a to the ring of algebraic integers, it is almost certain that -1and 1 are no longer the only integer units. It is to you to provethat a ring as you intend does exist. > Here it takes effort if some poster comes in to confuse you, relying > on the *assumption* that if 1/a is not an algebraic integer then it's > more like a fraction like 1/2 than 1/1.But it *is* more like 1/2, 3/2, etc. Note that when p is an algebraicnumber that is not an algebraic integers, it can be written as r/s,where r is an algebraic integer and s is a normal integer. That isall pretty standard. > But think of x=(1+sqrt(-3))/2, and someone giving you 1/x, if you were > mostly experienced with integers. > > Think of how easy it might be for someone to confuse you with > 2/(1+sqrt(-3)) if you weren't careful and it was new to you.I do not understand what you mean here, but indeed, x is a divisor of 1.So, the algebraic integers have more divisors of 1 than the normalintegers. So have the Gaussian integers, the Eisenstein integers (whichcontain this x), and what you have. And it is easy to understand whenyou see that [(1 + sqrt(-3))/2][(1 - sqrt(-3))/2] = 1. So I do notsee what the problem is. > That's has to do with why at times I've just gotten totally pissed and > called Arturo Magidin evil, as it's just so frustrating to deal with > people so adept at obscuring the truth.*What* obscuring? > My hope is that at least some of you will play with > > x^2 + (8a + b)x + 8, with ab = 1, and (8a + b) an integer > > so that you can see why *logically* a possibility must still be there > when 'a' and 'b' are irrational, even if 'a' is not an algebraic > integer.Well, obviously the polynomial is: x^2 + (8a + b)x + 8abwhich is reducible. So there is no sense looking at the quadratic.Note: again you fail to distinguish reducible polynomials fromirreducible polynomials. The theory goes about irreducible polynomialsbecause you can say much more about the roots in that case (see Galoistheory). But I can guide you. Let b be an algebraic integer thatdivides 8, in that case 8a = 8/b is also an algebraic integer. Now,can it be an integer? Or can we split 8 in two factors p and qsuch that p + q is an integer? The answer is, yes, we can. Let rbe an arbitrary integer and consider the polynomial x^2 + rx + 8,the roots multiply to give 8 and add to give an integer, so thereyou are. The roots are: (-r + sqrt(r^2 - 32))/2and (-r - sqrt(r^2 - 32))/2Set b equal to the first and a to 1/8 times the second and you are done.Fill in any algebraic integer number r you can think off. > If you can get a handle on that possibility, then maybe discussions > can be more fruitful about those numbers lost in the shuffle that > irrational 'a' here represents.Eh? > You know, they are *most* numbers given their cardinality.Eh?-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: Understand now? Frustration?> I've been arguing with posters like Arturo Magidin for YEARS, and> throughout that period, I've found my attempts to clarify were> hampered by attempts by other posters like here C. Bond to confuse,The greatest impediment to JSH's attempts to clarify have been JSH's refusal to learn anything about mathematics.> and I thank KeithK for catching one as he did.> The problem isn't with C. Bond using 8a + 1/a as that is correct, as> long as you consider the possibility of 'a' being a factor of 1 in> some more inclusive ring than the ring of algebraic integers.In the field of complex numbers, everything except zero is a factor of everything.> Circular arguments have gone on and on and on for some years here, as> people like Arturo Magidin, have successfully dodged being pinned down> on this issue as they repeatedly relied on certain numbers not being> algebraic integers to support their claims.JSH is an expert at circular arguments, he makes so many of them.And JSH only badmouths Arturo as a person because Arturo's mathematics have so often shot down JSH's maunderings.> Notice, I'm NOT saying that 'a' is an algebraic integer when it's> irrational!> If you say that 1/a is not an algebraic integer, when irrational, yes,> you are correct.You, on the other hand, are not correct even when you are rational.> But my point is that the label algebraic integer does not contain> all numbers such that 1/a is contained and -1 and 1 are the only> integer units. Is this supposed to mean something?There's a more inclusive ring, where -1 and 1 are the> only integer units, where 1/a is a unit, and 'a' is not an algebraic> integer.Show us such a ring.> Here it takes effort if some poster comes in to confuse you, relying> on the *assumption* that if 1/a is not an algebraic integer then it's> more like a fraction like 1/2 than 1/1.The poster who is trying to confuse us is James S Harris.> But think of x=(1+sqrt(-3))/2, and someone giving you 1/x, if you were> mostly experienced with integers.> Think of how easy it might be for someone to confuse you with> 2/(1+sqrt(-3)) if you weren't careful and it was new to you.> That's has to do with why at times I've just gotten totally pissed and> called Arturo Magidin evil, as it's just so frustrating to deal with> people so adept at obscuring the truth.It does not take an adept to conceal the truths from someone as reluctant to accept them as JSH. What JSH does not want to accept, however limpidly clear and true it may be, he will find some way to reject. He has done this so often now that it is a habit that he cannot break.And even after he is forced to accept a truth, as he did recently, he may turn around and later reject it again, as he did recently. As this post of his demonstrates. He seems to have errors hardwired into his brain so deeply that truth can only gain a temporary foothold in his thoughts.> My hope is that at least some of you will play with> x^2 + (8a + b)x + 8, with ab = 1, and (8a + b) an integer> so that you can see why *logically* a possibility must still be there> when 'a' and 'b' are irrational, even if 'a' is not an algebraic> integer.What possibility is that?If a*b = 1 and (8*a+b) is an integer, then 8*a and b are roots ofx^2 - (8a + b)x + 8*a*b = x^2 - (8a + b)x + 8 = 0, so 8*a and b are algebraic integers. This does not require that a, or 2*a, or 4*a be algebraic integers. SO?> If you can get a handle on that possibility, then maybe discussions> can be more fruitful about those numbers lost in the shuffle that> irrational 'a' here represents.> You know, they are *most* numbers given their cardinality.Who are they?As both a and b must be algebraic numbers, and the cardinality of the algebraic numbers is countable, and the cardinality of the trascendental(non-algebraic) numbers is uncountable, there are more numbers not solutions to equation x^2 - (8a + b)x + 8*a*b = 0,where a*b = 1 abd (8*a+b) is an integer, than numbers which are solutions, but why should that be a surprise to anyone? That is the case with most conditional equations.> James HarrisOff in his dreamworld again. === Subject: hi....integral problem...hello........every body.......i saw a integral.int [sqrt{1+4(x^2)}] / {c-(x^2)} dx , c is constant.------------------------um......possible??let me advice, please.... === Subject: Re: hi....integral problem...> hello........every body.......> i saw a integral.> int [sqrt{1+4(x^2)}] / {c-(x^2)} dx , c is constant.> ------------------------> um......possible??> let me advice, please....As usual, I frown at trig substitutions where they can be avoided:Classical rationalization:set 2*x + sqrt(1+4*x^2) = u, so that -2*x + sqrt(1+4*x^2) = 1/ux = (u^2-1)/(4*u)sqrt(1+4*x^2) = (u^2+1)/(2*u)dx/du = (u^2+1)/(4*u^2)and you get a rational function to integrate.(Still lot of work, though.)A rationalization that takes advantage of the situation:set x = (v-2)/(4*sqrt(v-1)) , so one of the inverses is v = (2*x + sqrt(4*x^2+1))^2then you get it rational and simpler than by the classical method.For checking purposes, you should get the transformed integrand-1/v - 4*(4*c+1)/(v^2 - 2*(8*c+1)*v + 1)so the form of the second integral depends on the value of c;discuss separately c=0, c=-1/4, -1/40, c<-1/4.Looks like more than you asked for; where on earth does such anintegral arise?Cheers, ZVK(Slavek). === Subject: Re: hi....integral problem...> hello........every body.......> i saw a integral.> int [sqrt{1+4(x^2)}] / {c-(x^2)} dx , c is constant.> ------------------------> um......possible??I think so :-) Have you tried substituting x = 1/2 tan(t). Or perhaps x =u/(1-u^2)? === Subject: Explaining restricted choiceMany people (even some fairly good maths graduate students) struggle abit with the concept of restricted choice. This principle explains(to take the most common illustration) why, in two-child families, ifyou ask a daughter the gender of her sibling, she will answer male2/3 of the time. [I'm making commonsense mathematical simplifyingassumptions here. In reality, she would, of course, be likely toanswer something like Why are you asking me that for? etc. And theassumption that each birth has a 50/50 probability of being female isopen to question, too.]Here is one example which, I think, clarifies the concept. Considertwo experiments, A and B, which both involve 100 coin tosses. In A,the result of the first toss is predetermined to be heads, but the 99other tosses have the usual 50% heads/ 50% tails probabilities. In B,all 100 coins are tossed in the usual way but a restriction is imposedthat the trial is rejected and the experiment repeated if all 100coins land tails. If all 100 are tails, the experiment keeps gettingrepeated until at least 1 of the 100 coins are heads. The finalsample of 100 coin tosses is considered the result of B.In A, the expected number of heads is 1 + 99/2 = 50.5. In B, there isno restriction in any practical sense because the rejections areastronomically unlikely. Therefore, in B, the expected number ofheads is very, very close to 50.In my opinion, this extreme example clarifies the restricted choiceconcept.Paul Epstein === Subject: Re: Explaining restricted choice> Many people (even some fairly good maths graduate students) struggle a> bit with the concept of restricted choice. This principle explains> (to take the most common illustration) why, in two-child families, if> you ask a daughter the gender of her sibling, she will answer male> 2/3 of the time. [I'm making commonsense mathematical simplifying> assumptions here. In reality, she would, of course, be likely to> answer something like Why are you asking me that for? etc. And the> assumption that each birth has a 50/50 probability of being female is> open to question, too.]How did you go about calculating the 2/3? I'm not aware of anything inbiology that says that the genders of subsequent children aredependent, but I don't think that's what's giving you that numberanyway. Assuming that it's 50/50, we have that the probability of aboy being the younger sibling, given that the first sibling is a girl,is just the probability of a boy being the younger sibling, which is1/2, since the two events in our model are independent.Are you saying that reality somehow adjusts itself to keep theexpected value near 50/50? That by restricting the first birth to be agirl, the second one becomes more likely a boy?> Here is one example which, I think, clarifies the concept. Consider> two experiments, A and B, which both involve 100 coin tosses. In A,> the result of the first toss is predetermined to be heads, but the 99> other tosses have the usual 50% heads/ 50% tails probabilities. In B,> all 100 coins are tossed in the usual way but a restriction is imposed> that the trial is rejected and the experiment repeated if all 100> coins land tails. If all 100 are tails, the experiment keeps getting> repeated until at least 1 of the 100 coins are heads. The final> sample of 100 coin tosses is considered the result of B.> In A, the expected number of heads is 1 + 99/2 = 50.5. In B, there is> no restriction in any practical sense because the rejections are> astronomically unlikely. Therefore, in B, the expected number of> heads is very, very close to 50.Unfortunately, by your own calculations it isn't quite 50. What are weto draw from this example? That small restrictions lead to smallchanges in expected value? You use conditional expectation in A and acompletely revised experiment in B, so it's a bit unclear what you'redriving at.> In my opinion, this extreme example clarifies the restricted choice> concept.Not especially, to me anyway. Stating formally what restricted choiceis, without using examples, might be helpful.> Paul Epstein === Subject: Re: Collatz Conjecture : Symmetry question.>Second: The two divmod's can be combined into one.> How? My thinking was to do the divmod 3 first since it will> fail the test two out of three times allowing the divmod 2> test to be skipped.Well, if n==1 mod 3 and n==0 mod 2, that is the same as n==4 mod 6. === Subject: Re: Collatz Conjecture : Symmetry question.>Second: The two divmod's can be combined into one.How? My thinking was to do the divmod 3 first since it willfail the test two out of three times allowing the divmod 2test to be skipped.> Well, if n==1 mod 3 and n==0 mod 2, that is the same as n==4 mod 6.> -Michael.Ok, I thought you were refering to the nested if statements, which ifcombined would still have done two mod functions. The mod 6 thing neveroccurred to me.I changed p3 = divmod(n,3) if (p3[1]==1): p2 = divmod(n,2) if (p2[1]==0): g.write(str((n-1)/3)+'n')to p3 = divmod(n,6) if (p3[1]==4): g.write(str((n-1)/3)+'n')and found that it saves about half a second (1.4%) at Level 64 where theinput file is 22 MB. By Level 69, where the input file is 76 MB, the difference is up to about 3.5 seconds (4%). I didn't try it, but I wouldexpect that by the time you get to level 84, where the input file is3.3 GB and the run time in hours, the savings would be significant. === Subject: Re: there is no such thing as infinity>640K ought to be enough for anybody.>-BILL GATES, 1981 [I was born the following year :o]Well, since I recall a time when Comp Sci students did an entireundergraduate course with 5Mb of diskspace and all their programs had tofit into 120kb, maybe he had a point. Mind you that was on a machinewith 60 bit words, and the compilers all used overlays extensively.> Ah, hell, the kids today are so spoiled! I write software for cell phones> for a living and *have* to be cheap with the code since we have to live inmore> recent desktop machines and wants to throw extra threads and verbose> processes at the simplest little problems that we face...In college, I crashed one of the largest mainframes in Ohio.The university was charging businesses $700 / CPU minute to use thiscomputer.It took hours to reboot it. The OS required nearly a megabyte.I think the mainframe had less than 5 MB of memory.In the early 1980's, IBM was selling 300 MB disk drives for $100,000.dropped to $1 / kilobyte. Imagine being able to buy a megabyte ofmemory for under $1,000.The average desktop computer has more processing power thanall of the computers at Houston Control when Apollo landed onthe moon.Russell- yes, I'm a fossil === Subject: 2 and only 2 geometries where Euclidean is like Newton's absolute time and absolute spaceI am going to have to revamp File 103 on FLT, and File 120 of 3 andonly 3 geometries and File 125 of two proofs of the RiemannHypothesis in my website of www.iw.net/~a_plutonium/I did not do much mathematics after 1997 and recently when I reviewedmy Riemann Hypothesis proof I realized that it is the p-adics that areon the 1/2 Real line which means that lines are curved when out atinfinity. There are no straightlines. I have to change and revise myPoincare Conjecture proof also.But directly, I have to toss out my early 1990s proof of 3 and only 3geometriesbecause it is really 2 and only 2 geometries. A lot of revision and Ican hack it if I go slowly.What does this say about physic? It says alot. I was sort ofuncomfortable in the early 1990s with the statement of 3 and only 3geometries idea. Because physics is dominated not by threesome butThis is basic and fundamental in physics and so why should mathematicsbe cloaked and dressed in triality when physics is duality.That would mean that Riemannian and Lobachevskian are the only 2geometries where the zeroness of Euclidean flat space is not ageometry. Zeroness is contained in Riemannian geometry as well asLobachevskian.If the Riemann Hypothesis has all the Natural Numbers on the 1/2 RealLine and if the NaturalNumbers are really the P-adics, and since thep-adics curve then there exists no straight lines. There exists noEuclidean Geometry and Euclidean is just a human mental construct withno physical substance. The same as Newtonian absolute space andabsolute time is just a human dreamed up mental construct.I see alot of work ahead in revising that website of mine.Archimedes Plutoniumwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are galaxies(www.iw.net/~a_plutonium) website of the science of AP under revisionwhat used to be my old science websitewww.newphys.se/elektromagnum/physics/LudwigPlutonium from years 1993 === Subject: Re: Statistical independence test for continuous variables> I need to know what are the available statistical independence tests> for continuous variables. I know that the correlation is a test for> linear dependence for continuous variables, but I was wondering if> there were others.Both Kendall's tau and Spearman's rho are sensitive to an overallmonotone relation (which may be nonlinear) between two variables.Beyond that, you pretty much need to specify the form of the relationthat you're looking for. The tighter the specification, the morepowerful the test can be.Terminological note: correlations are not tests. === Subject: Re: Statistical independence test for continuous variables> I need to know what are the available statistical independence tests> for continuous variables. I know that the correlation is a test for> linear dependence for continuous variables, but I was wondering if> there were others.>Both Kendall's tau and Spearman's rho are sensitive to an overall>monotone relation (which may be nonlinear) between two variables.>Beyond that, you pretty much need to specify the form of the relation>that you're looking for. The tighter the specification, the more>powerful the test can be.>Terminological note: correlations are not tests.The analog of the Kolmogorov-Smirnov and the Cramer-von Mises tests are universal testsin any number of variables. Under the nullhypothesis, they are distribution free.However, I am not familiar with a tabulationof their significance levels. One could alsoconsider other tests of a similar nature.However, there are ways of using random significancelevels to get a test of the desired size. Take krandom samples from the null hypothesis, and if j = (k+1)*alpha, rejecting if the observed sampleis in the upper j is at level alpha. For the abovetests, if the sample is large, sqrt(n)*T, T the test statistic, has a limiting distribution, so that if the sample size is large, one might wantto scale using smaller samples for null data.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue Universityhrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: wedge productDo Carmo's _Differential Geometry of Curves and Surfaces_ is thetextbook for a course I'm taking on metric differential geometry. Thebook uses a wedge product (symbol something like '/') where I expected tosee a cross product. Looking up both 'wedge product' and 'cross product',in Mathworld and on Google Groups, it seems that the cross product is aspecial case of the wedge product that acts on 3-dimensional vectors andreturns a 3-dimensional vector. My question is, is this generality usefulin differential geometry -- did Do Carmo use a wedge product in order tobe suggestive of applications in higher-dimensional spaces, or is it justa notational thing (wedge product in three dimensions == cross product)?Tobin === Subject: Re: wedge product> Do Carmo's _Differential Geometry of Curves and Surfaces_ is the> textbook for a course I'm taking on metric differential geometry. The> book uses a wedge product (symbol something like '/') where I expected to> see a cross product. Looking up both 'wedge product' and 'cross product',> in Mathworld and on Google Groups, it seems that the cross product is a> special case of the wedge product that acts on 3-dimensional vectors and> returns a 3-dimensional vector. My question is, is this generality useful> in differential geometry -- did Do Carmo use a wedge product in order to> be suggestive of applications in higher-dimensional spaces, or is it just> a notational thing (wedge product in three dimensions == cross product)?The wedge symbol is often used as notation for vector product inEuclidean 3-space.But in more advanced differential geometry it is usually used forexterior products/powers, especially of the cotangent spaces ofa manifold (leading to differential). As I don't have do Carmo'sbook to hand I cannot tell what his convention is. A text at thislevel could have either (or both) usages.-- === Subject: Re: the anticlassicalist }{ ii: the spectre continues> Excuse the delayed reply. I dropped alt.philosophy, sci.lang,> and sci.physics.Good idea.> | > I know many models whose Heyting structure is far more simplistic> | > than the corresponding Boolean embedding.> |> | Can you name them? Heyting algebras are always infinite, afaik.> Note that simplistic means excessively simplified.> Boolean algebras are a special case of Heyting algebras, and there are> plenty of finite Heyting algebras even excluding finite boolean algebras.You're right. Finite topological spaces, right?Cheers,Herman JurjusPS Thank you for the rest of your contribution as well. === Subject: Re: bound on partitions of subsets> Given a set S of n elements, i need to choose a subset S' of> these elements and then choose a partitioning of S'. Can someone tell> me a compact asymptotic upper bound on the number of ways this> can be done? Anything that is better than (2^n)*(B_n). {B_n is the> nth bell number}. Essentially a bound on the series sigma (nC_i B_i)This number is B_{n+1}. Take your set S and form another S u {*}by adding an extra element. Form any partition of your larger set,and the cast out the part containing *; this yields a partition ofa subset of S.-- === Subject: Re: Category Theory's Demphasis of Sets is a Serious MistakeReply to title alone.Set theory is still the dominant mahematical paradigm.Of course the categorical approach to set theory emphasisesfunctions between sets rather than the sets themselves.-- === Subject: Re: Category Theory's Demphasis of Sets is a Serious Mistake by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1MKlPS01696;Here is a list of some of the implicit or explicit errors of theCategory Theory approach regardless of its direct negative effectson the Set-related Theories that I mentioned in the last posting.A. The assumption that an Object is more general than a General-ized Set. The claim is often made that there is a paradox indefining a Set of Sets, which is allegedly absent from an Object. But nothing is common to Objects since they are notdefined in terms of a common characteristic, and if they were theSet Paradox would affect it since a collection of things withcommon characteristics is a Set! B. The assumption that Functions are more useful than Sets. Ihad an argument with a Belgian Computer-Oriented Logician aboutthis some years ago (Professor Smets who to my recollection foundedthe famous IRIDIA in Belgium and France), and we didn't concludethat one was more useful than the other although each of us remained convinced personally that our favorite was probably moreuseful. However, the assumption that Functions are more usefulthan Sets in Real Analysis would involve the assumption for examplethat the Lebesgue Integral is more useful than the Lebesgue Measure(Measures being on sigma-algebras, which are types of Sets), whichis silly.C. Like B above with Morphisms replacing Functions.D. Like B above with Functors replacing functions.E. The Mis-direction of Composition. The Category Theory claimspriority for Composites of Morphisms, or in Functions Compositesof Functions. The composition f o g (x) = f(g(x)) is not thesame as fg(x) = f(x)g(x) or (f + g)(x) = f(x) + g(x) and so on,although one could always claim that the latter two are somehowspecial cases of the former with a slight generalization. InRare Event Theory, it is shown that fg and f + g are far moreimportant than f o g. This is perhaps the most likely to becriticized by the Mainstream researchers since it goes over tothe whole question of matrix representations, but it's true forRare Event Theory. One may not want something to be true, butthat doesn't change its truth.F. The failure of Category Theory to emphasize discovery of newideas and new concepts in its preoccupation with organizing oldones. It does nothing with Anomalies, Paradoxes, Infinities,Causation, Correlation, Motion, and so on in this respect. Thisincludes Category Theory's failure to match Rare Event Theory'sdiscovery of 1 + y - x as a form across geometry, Fuzzy MultivaluedLogics, and Probability-Statistics. You can organize forever, butlooking for common interdisciplinary forms is unbeatable in myopinion.Osher Doctorow === Subject: Re: Category Theory's Demphasis of Sets is a Serious Mistake> Here is a list of some of the implicit or explicit errors of the> Category Theory approach regardless of its direct negative effects> on the Set-related Theories that I mentioned in the last posting.> A. The assumption that an Object is more general than a General-> ized Set. The claim is often made that there is a paradox in> defining a Set of Sets, which is allegedly absent from anObject. Never heard that before.> B. The assumption that Functions are more useful than Sets. IAre you German?> had an argument with a Belgian Computer-Oriented Logician about> this some years ago (Professor Smets who to my recollection founded> the famous IRIDIA in Belgium and France), and we didn't conclude> that one was more useful than the other although each of us> remained convinced personally that our favorite was probably more> useful. However, the assumption that Functions are more useful> than Sets in Real Analysis would involve the assumption for example> that the Lebesgue Integral is more useful than the Lebesgue Measure> (Measures being on sigma-algebras, which are types of Sets), which> is silly.Which is true. The Lebesgue integral subsumes Lebesgue measure ---which can be recovered as the Lebesgue integrals of the indicatorfunctions.> E. The Mis-direction of Composition. The Category Theory claims> priority for Composites of Morphisms, or in Functions Composites> of Functions. The composition f o g (x) = f(g(x)) is not the> same as fg(x) = f(x)g(x) or (f + g)(x) = f(x) + g(x) and so on,> although one could always claim that the latter two are somehow> special cases of the former with a slight generalization.Well done! f o g is not the same as fg or f+g ... you'redoing better than most of my algebra students here :-)Of course fg anf f+g only make sense if the codomain of f and gadmit addition/multiplication operations.> In> Rare Event Theory, it is shown that fg and f + g are far more> important than f o g. What the **** is Rare Event Theory? Some pet non-concept ofyours?> F. The failure of Category Theory to emphasize discovery of new> ideas and new concepts in its preoccupation with organizing old> ones. It does nothing with Anomalies, Paradoxes, Infinities,> Causation, Correlation, Motion, and so on in this respect. This> includes Category Theory's failure to match Rare Event Theory's> discovery of 1 + y - x as a form across geometry, But that (as evidenced by Doctor Osherow's repeated postingsto geometry.remedial) is a non-discovery.-- === Subject: Re: Minimally simple finite groups?>Which of the finite simple groups are minimally simple, i.e.,>have all of their proper subgroups solvable? Obviously the only>alternating group that qualifies is A_5 =~ L(2,4) =~ L(2,5), and>I know the list also includes L(2,7) =~ L(3,2), L(2,8), L(2,13),>... On the other hand, I also know it *doesn't* include L(2,9) =~>A_6 or L(2,11), both of which contain subgroups isomorphic to>A_5.> The classification of minimal simple groups was a consequence of> John Thomspon's classification of nonsolvable groups in which nontrivial> solvable subgroups have solvable normlaizers, which was one of the big> classification theorems that came before CFSG.> This is in Bull. Amer. Math. Soc. 74, 1968, 383-437.> The simple groups coming out of Thompson's Theorem are L_2(q), Sz(q),> L_3(3), M_11, A_7, U_3(3). Of course, these are not all minimal simple.> L_3(3) is, but M_11, A_7, U_3(3) are not.> Sz(2^e) is minimal simple whenever it does not contain a smaller> Sz(2^f), which I guess is equivalent to e being prime.Is it easy to see why Sz(2^e) can't contain (as sections) anynon-abelian simple groups other than the obvious Sz(2^f) for allf dividing e? Or do you need substantial machinery fromThompson's Theorem?> For L_2(p^e) to be minimal simple, its order must not be divisible by 60> (otherwise it contains A_5).Ah. Why must it contain A_5 if its order is divisible by 60?> Also it must not contain any smaller> simple L_2(p^f), so if p>=3, then we must have e=1. But L_2(2^e) and> L_2(3^e) will be minimal simple for e prime provided their order is> no divisible by 60.You must mean so if p>3, of course. Again, is it easy to seewhy other non-obvious simple groups are excluded from beingsections of L_2(p^e)? (Or are they? Can there be larger groups,but containing A_5?)> Something like that, anyway!As always, thanks for another of your wonderfully informativeposts!-- Jim Heckman === Subject: Re: Minimally simple finite groups?>Which of the finite simple groups are minimally simple, i.e.,>have all of their proper subgroups solvable? Obviously the only>alternating group that qualifies is A_5 =~ L(2,4) =~ L(2,5), and>I know the list also includes L(2,7) =~ L(3,2), L(2,8), L(2,13),>... On the other hand, I also know it *doesn't* include L(2,9) =~>A_6 or L(2,11), both of which contain subgroups isomorphic to>A_5.> The classification of minimal simple groups was a consequence of> John Thomspon's classification of nonsolvable groups in which nontrivial> solvable subgroups have solvable normlaizers, which was one of the big> classification theorems that came before CFSG.> This is in Bull. Amer. Math. Soc. 74, 1968, 383-437.> The simple groups coming out of Thompson's Theorem are L_2(q), Sz(q),> L_3(3), M_11, A_7, U_3(3). Of course, these are not all minimal simple.> L_3(3) is, but M_11, A_7, U_3(3) are not.> Sz(2^e) is minimal simple whenever it does not contain a smaller> Sz(2^f), which I guess is equivalent to e being prime.>Is it easy to see why Sz(2^e) can't contain (as sections) any>non-abelian simple groups other than the obvious Sz(2^f) for all>f dividing e? Or do you need substantial machinery from>Thompson's Theorem?It is easy to see it, because Sz(2^e) are the only finite nonabeliansimple groups with order not divisible by 3. That was actually provedbefore the classification, again by Thompson, but it is was a highlynontrivial result, probably harder than the odd order theorem. I amsure there are much easier ways to analyse the subgroups of Sz(2^e)though!> For L_2(p^e) to be minimal simple, its order must not be divisible by 60> (otherwise it contains A_5).>Ah. Why must it contain A_5 if its order is divisible by 60?> Also it must not contain any smaller> simple L_2(p^f), so if p>=3, then we must have e=1. But L_2(2^e) and> L_2(3^e) will be minimal simple for e prime provided their order is> no divisible by 60.>You must mean so if p>3, of course. Again, is it easy to see>why other non-obvious simple groups are excluded from being>sections of L_2(p^e)? (Or are they? Can there be larger groups,>but containing A_5?)The subgroups of PSL(2,q) were all classified by L.E. Dickson in about1900. I am not sure what the best reference for that is. It is inHuppert's book Endliche Gruppen but that is in German of course!Anyway, the subgroups are roughly cyclic groups, dihedral groupsof order dividing q-1 or q+1 (q odd) or 2(q-1) or 2(q+1) (q even),semidirect products PD for a p-group P of order dividing q and cyclic groupD of order dividing q-1, A_4, S_4 whenever 24 divides order, A_5 whenever60 divides order, and PSL(2,r) and sometimes PGL(2,r) where q is a powerof r. In other words, A_5 is the only `sporadic' simple subgroup ofPSL(2,q). The fact that it occurs whenever 60 divides the order probablyfollows from the fact that SL(2,5) has a 2-dimensional complexrepresentation, but don't push me for details there!Derek Holt. === Subject: Re: the anticlassicalist }{ : mitchism for Daleks <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com> <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com> <1039dr5kpjloif1@corp.supernews.com> <6IvurhFpHeNAFwU6@baesystems.com> <103cn32aodr3m3a@corp.supernews.com>In message <103cn32aodr3m3a@corp.supernews.com>, galathaea >: No, it means that at some point you have to hook into what physicists>: *observe*. Will it tell us the energy levels of a hydrogen atom and>: predict the Lamb shift? If you're interested in developing a better>: mathematical formalism, that's fine, go ahead. Just don't tell us it's>: physics.>Actually, all I did was point to the connection between the foundational>objects of the theory and the observational objects it predicts. I called>them the ontology and the epistemology of the model in a rigorous way to>accord somewhat with common usage, but any other names would work. But the>formalism is all about observational propositions (what is the likelihood>that A and B happen?, if C happens, what does that imply for D?, where>the letters stand for quantum events like spontaneous decay). Anywhere you>have time series of quantum events or concurrent systems, you implicitly or>explicitly use the logic I mention, often in an algebraic setting.OK. So if I'm already unknowingly using this logic, what is the point you're trying to make? Believe me, it hasn't yet emerged from all the verbiage.>It _is_ physics. I _am_ interested in developing a better mathematical>formalism, but alas cannot take credit for this one. This one has been>studied by physicists now for 3 quarters of a century.-- Richard Herring === Subject: Re: the anticlassicalist }{ : mitchism for Daleks <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com> <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com> <1039dr5kpjloif1@corp.supernews.com>> No, it means that at some point you have to hook into what physicists> *observe*. Will it tell us the energy levels of a hydrogen atom and> predict the Lamb shift? If you're interested in developing a better> mathematical formalism, that's fine, go ahead. Just don't tell us it's> physics.>It is not about the formalism. It is about whether assumptions in the >formalism>that make it coherent are even being respected.[...]>Your claimI'm not making a claim, just a request.>translates into a definition for physics as nothing more than a>chaotically organized aggregate of partial results.You make that sound like a bad thing ;-)It's based on inductive reasoning from observation, so it can't help but be partial. As for chaotic, I don't know.> If that is the case, you should>not be relying on mathematical formalism for justifications--as in the >many times I>have seen the statement it works. If mathematics is a symbolic tool with no>inherent connection to the material and the natural language descriptions are>nonsensical, then you are just idiots generating random information.But what I'm requesting *is* your inherent connection to the material. If this stuff is important for physics, you need to explain why, and why it's better than what is presently in use. You need to *sell* it, but so far all I have seen is thousands of words of unnecessary detail that apparently say no more than look how smart I am.-- Richard Herring === Subject: Re: the anticlassicalist } <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com> <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com> <1039dr5kpjloif1@corp.supernews.com>> But what I'm requesting *is* your inherent connection to the material.> If this stuff is important for physics, you need to explain why, and why> it's better than what is presently in use. You need to *sell* it, but> so far all I have seen is thousands of words of unnecessary detail that> apparently say no more than look how smart I am.Fair enough.For my part, what you have are academic disciplines making contradictory claims onthe nature of truth. But, the experimental aspect of physics is directly outsideof that debate--thankfully. Indirectly, there is some impact on how decisions aremade.In contrast to experimental physics, the theoretical physics is pushingmathematical methods to the point of inconsistency. While I want to respect thephysical intuition developed in the study of physics, intepreting mathematicswithout regard for its historical development is unsound.I do respect the fact that mathematical physics is not physics in the experimentalsense. And, I have already removed sci.physics from parts of this thread. Also, Ithank you for your responses which have not been entirely hostile and extend anapology for the parts of my posts that were.Now, you ask if anything I talked about could be better than anything in use. Iwould argue that my interpretation of the mathematics might give a correctexplanation for how information-theoretic interpretations of quantum mechanicsshould be understood. Nevertheless, there are no numeric predictions.Unfortunately, I get a little pissed off when logicians don't think they have anyobligation to justify their explanation for mathematics and when physicists don'tperspective I have to read nonsense like http://plato.stanford.edu/entries/mathematics-inconsistent/ because the bureaucratic structure of universities permits people making contraryclaims on mathematical thought to operate independently.corrective.:-)mitch === Subject: Re: JSH: Apology to Ramsay, why I post>Are you sane?Would an insane person know that they weren't sane? Hence isn't that a rathermeaningless question?Unless of course it was merely rhetorical and you simple wish to be insulting. === Subject: Re: Give me that old time ontology: (was: the anticlassicalist }{ i: linguistic negation) <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com> <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com> <1039dr5kpjloif1@corp.supernews.com> <+No$awZ0HMNAFw0n@baesystems.com> <2a0cceff.0402210820.773d478d@posting.google.com>In message <2a0cceff.0402210820.773d478d@posting.google.com>, Edward > In message <1039dr5kpjloif1@corp.supernews.com>, galathaea>: >You should have realised that anything full of -ism and -ist which is> spewed>: >to sci.lang and sci.logic is not going to contain any physics.>: >>: (Nor language; I can't speak for sci.logic ;-)>:>: I'm well aware of that. What I was wondering was whether the _OP_>: realises that there's a difference between these isms and physics, and>: what it consists of.>>Installment vi> >>Before you lies the Void...>details the properties of the closed linear subspace lattice>of a Hilbert space, detailing the relationship with observables and>prediction in quantum systems. This builds off of the mathematical theory>>You do understand the physical content, do you not?> Haven't seen any.> Dirac compressed quantum mechanics into ~300 pages of terse elegance,> and didn't use ontology once.>Well then, he was tersely codifying the mathematical structure as a>given, which is the easier hard direction. Going in the less easy>hard direction requires talking about something very much like>ontology, whether or not we use the word ... like speaking prose, and>all that. It's our old school chums deduction and induction.>Basically, professor, it goes kind of like this (music please ...):>Models as a Confidence Game>In the beginning we are confronted with a blooming buzzing surface of>sensory confusion. Being cerebral beings, we create a partially>ordered heirarchy of model things, purported to be generative of this>surface which we've been presented. The models are essentially,>roughly and enoughly, our working ontology.>There: I've used ontology in a sentence, and it wasn't so bad, was>it? ;-)>(Richard Herring vigorously spits and loudly calls for a pint).Oh, I'd call for a pint whatever you said ;-)I think what I mostly don't like about the word is that it evokes Anselm's argument for the existence of God. One false step and you'll believe your model really exists.>Some of these models; in a model itself received and conceived -- it's>all heirarchical, you see -- are for us evolutionarily preconceived:>in our lizard brains, ancient and tough, a world populated with>persistent three-dimensional objects is enough.>Onto these models we multiply and add; mathematical annexes, now more>than a fad, allow us to predict with numeric decision the results of>some experiments with admirable precision. This predictive concision>increases our confidence that we're on to somethin', as we laboriously>peel back the skins of the onion -- and well it should: a rigorous>predictive structure is quite good.>Confidence, confidence ... did I not add in my vision that each model>is invested with Bayesian strength; and though we sometimes forget we>should add at greater length they are all always subject to emprical>revision?>Hssst! The doggerels out. But I think you may get the idea:>In the great chain of model being we may focus on some precise>mathematical bits, the triumphs of a natural science known to be>correct in a range of observation with a degree confidence only>predicated on a few assumptions; like sanity and the persistence of>order. Once given these bits, reasoning from their structure is>primarily downward -- deductive -- which is something which can>similarly be carried out with great rigor and confidence. But we may>tend to forget that these models were not themselves handed to us on a>platter -- or rather they were, if we adopt the stance that they are>known givens -- but each the product of protracted groping by groups>of the most talented people of their time, there being no mechanistic>crank we can turn to integrate as readily as we differentiate. And in>this groping, like it or not, something with a strong isometry to the>ologies will out; a discussion of models and our confidence in models,>hic nomen rosa.I think you're really talking about epistemology.But congratulations are in order: you didn't say hermeneutics at all.-- Richard Herring === Subject: Re: little problem> With difficulty, 'cos it aint't true.> The tangent bundle of a sphere T(S^2) has the same fibres> as the trivial bundle S^2 x R^2 yet these undles are not> isomorphic (S^2 is not parallelizable).> --You are true.My proposition isn't true in general.One moment, considerthe fiber bundle P^k(xi,eta) whose fiber over x is the spaceP^k(xi_x,eta_x) of homogeneous polynomial maps of degree k of xi_x toeta_x ;and the fiber bundle L^k_s(xi,eta) whose fiber over xis the space L^k_s(xi_x,eta_x) of k-linearmaps of (xi_x) ^k into eta. P^k(xi,eta) and L^k_s(xi,eta) are fiber bundle over the samemanifold,say M.After having shown that L^k_s(xi_x,eta_x) and P^k(xi_x,eta_x) areisomorphic, Palais says that it follows that L^k_s(xi,eta) andP^k(xi,eta) are naturally equivalent.May you help me please?Tern === Subject: An AB' + A'B Kernel in the Matrix and Algebraic Riccati Control-Filter Equations by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1NBLxl05338;The Riccati Differential Equations for scalars is:1) dy/dt = A(t) + B(t)y + C(t)y^2with A(t), B(t), C(t) scalar functions of time. However, in optimalcontrol theory including Kalman filter-predictors we get MatrixRiccati Equations including for Linear Quadratic Regulators (LQR)and Optimal Linear Filtering for the Covariance Matrix P. TheMatrix Riccati Differential Equation has the form:2) dP/dt = PA' + AP +/- PBP -/+ Qwhere this time prime (') denotes transpose of a matrix and P issymmetric so that P' = P and AP' = AP. There is also a steadystate or asymptotic time-independent version of (2) called theAlgebraic Matrix Riccati Equation (ARE), with both continuous anddiscrete versions, in which dP/dt = 0. This would all be of little surprise if the Riccati Equation werenot the fundamental equation of Growthxpansion-Contraction andsimilarly of Rare Event Theory (RET) and Logic-Based Probability(LBP). For those who do not have the time to examine my longseries of postings to geometry-research and math-history-list andresearchmathematics@yahoogroups.com and superstringtheory.com (which at the last time I looked was under a virus and/or beingmoved to a different server), I'll be glad to explain it later,or you could try to figure it out as an exercise.I call PA' + AP (which is PA' + AP' since P is symmetric) thekernel of the Riccati Equations in the sense that it is the keyterm involving P on the right-hand-side of the equation otherthan the symmetric PBP term. It is what B(t)y of the scalarRiccati equation becomes for matrices in these control scenarios.Should it surprise us to find fg' + f'g, AB' U A'B, AB' + A'B to have the same form for functions, sets, and matrices respect-ively? Let's look again at zz* or ww* = (x + iy)(x - iy) =x^2 + y^2, and as I mentioned in a recent posting, z and z* orw and w* are reflections around the real (x) axis. The transposeA' of matrix A is also a reflection of the matrix. The complement A' of a set A is an outward or inward reflectionfrom A to the rest of the Universe. Is the derivative f' areflection of f in any sense?Let's think of that as an exercise, and I'll give a hint. Weknow that Aristotle (and if I recall even Plato) regarded thePotential and Actual as somewhat reflections of each other.Is there any sense in which change of a function can be thoughtof as the kinetic or dynamic part of whatever the functionrepresents as the potential part? Osher Doctorow === Subject: Re: An AB' + A'B Kernel in the Matrix and Algebraic Riccati Control-Filter Equations> Should it surprise us to find fg' + f'g, AB' U A'B, AB' + A'B> to have the same form for functions, sets, and matrices respect-> ively? Let's look again at zz* or ww* = (x + iy)(x - iy) => x^2 + y^2, and as I mentioned in a recent posting, z and z* or> w and w* are reflections around the real (x) axis. The transpose> A' of matrix A is also a reflection of the matrix. The> complement A' of a set A is an outward or inward reflection> from A to the rest of the Universe. Is the derivative f' a> reflection of f in any sense?No. The derivative has properties such as the Leibniz identitynot shared by complementation. The only link you have demonstratedbetween them is that you use the same notation for both.You talk about reflections but mean involutions --- operationswhich when repeated yield the identity. You ask if the (operation of)derivative is one. The simple answer is no. It is not the case that f'' = fin general. The proof of this should be within even your abilities.-- === Subject: Re: How big can a manifold be?> Is there any limit to the cardinality of a connected n-manifold, > that is a connected Hausdorff space every point of which has a> neighborhood homeomorphic to R^n?and later:> I'm mainly interested in 1-dimentional manifolds anyway.For 1-manifolds, the answer to your question appears to be that the cardinality of the continuum is the limit. David Gauld has a paper on his website http://www.math.auckland.ac.nz/~gauld/giving 88 equivalent conditions for a connected n-manifold to be metrizable. In it, he mentions in passing that there are only four (nonempty) connected 1-manifolds: the circle, the line, the long ray (by which he means what is usually called the long line), and the longline (long in both directions). He doesn't give a reference for this result, but some of the references at the end may be of help. === Subject: Re: Simple idea, mathematics and common-sense> ab = 1, > where again 'a' and 'b' are *irrational* the mathematicians have a> label for 'b' which is algebraic integer, but 'a' cannot be an> algebraic integer.> So they don't have a label for it!> You know how important naming is with human beings, and it makes it> that much harder for me to explain these ideas without a label.> I've named numbers like 'a' objectsTake b = sqrt(2). 'b' is *irrational*. Thus a=1/sqrt(2) is an object. - William Hughes === Subject: Re: Simple idea, mathematics and common-senseab = 1, where again 'a' and 'b' are *irrational* the mathematicians have alabel for 'b' which is algebraic integer, but 'a' cannot be analgebraic integer.So they don't have a label for it!You know how important naming is with human beings, and it makes itthat much harder for me to explain these ideas without a label.I've named numbers like 'a' objects> Take b = sqrt(2). 'b' is *irrational*. Thus a=1/sqrt(2) is an object.> - William HughesI meant with with 8a + b odd.Why not try to consider the idea versus just being a smart-aleck?If you were concerned with the truth, you might have noted that with8a+b even, it's clear that 'a' is a fraction like 1/2, like with yourexample where it is 1/sqrt(2), versus being a smart-aleck by tossingout that special case as if it were definitive.Now then, do you understand why your example doesn't apply for 8a+bodd?Do you understand how your post is a smart-alecky one, which invitesthe kind of response that I just gave it? === Subject: Re: Simple idea, mathematics and common-sense> > ab = 1, > > where again 'a' and 'b' are *irrational* the mathematicians have a> label for 'b' which is algebraic integer, but 'a' cannot be an> algebraic integer.> > So they don't have a label for it!> > You know how important naming is with human beings, and it makes it> that much harder for me to explain these ideas without a label.> > I've named numbers like 'a' objectsTake b = sqrt(2). 'b' is *irrational*. Thus a=1/sqrt(2) is an object. - William Hughes> I meant with with 8a + b odd.And here we all thought that it was only JSH who was odd.> Why not try to consider the idea versus just being a smart-aleck?You cannot expect people to understand conditions which you do not state.He considered the idea that you presented subject to the conditions you stated. That is hardly being smart-aleck. > If you were concerned with the truth, you might have noted that with> 8a+b even, it's clear that 'a' is a fraction like 1/2, like with your> example where it is 1/sqrt(2), versus being a smart-aleck by tossing> out that special case as if it were definitive.> Now then, do you understand why your example doesn't apply for 8a+b> odd?It doesn't apply for a^2 + b^2 = 1, either. So what?> Do you understand how your post is a smart-alecky one, which invites> the kind of response that I just gave it?The only one trying to be smart-alecky here is JSH, but he doesn't quite bring it off.> James Harris === Subject: Re: Simple idea, mathematics and common-senseIn sci.logic, James Harris<3c65f87.0402220637.3e8b6ea0@ posting.google.com>:> In sci.logic, James Harris> > <3c65f87.0402200504.46080ddc@posting.google.com>:> I've had a time explaining some *very* simple mathematical ideas that> lead to a few complexities, but it's been fruitful to explain, or try> to explain, as I work to figure out why these simple ideas either> excite derision, anger or confusion, and I think I have it figured> out.>> I'm sure many of you are put off by mathematics, so I assure you up> front that what I'll be talking about will mostly be *very* simple,> and there will only be a few slightly complicated things at the end.>> First of all, I'm going to talk about a case where mathematicians gave> up because they couldn't see something, and assumed that because they> couldn't see something it didn't exist!>> You know how with simple quadratics like x^2 + 3x + 2, it's easy> enough to see factors of 2 in the roots?>> I mean, it's just (x^2 + 3x + 2) = (x+2)(x+1), and there they are.>> However, if it's something like x^2 + 7x + 2, you can use the> quadratic formula and get the roots to find>> x = (-7 +/- sqrt(41))/2>> and who can see factors of 2 in that thing?> > I think part of the problem here -- and I'll admit to not being> entirely certain where the prime failure occurs -- is that,> if one has an integer such as 6, one can uniquely factor it into> primes: 6 = 2 * 3> That's not the issue here.> It's simple enough, but I have seen *posters* like this person come> forward and confuse the issue enough times that I think I'd better> step in and make sure that the REAL issue isn't easily obscured.> P(x) = (x+8a)(x+b), where ab=1, and consider 8a+b an integer> If a=b=1, notice you have (x+8)(x+1), which is one of two basic> possibilities with integers.It and (a,b) = (-1,-1) are about the only ones in the integer realm.(a,b) = (-1,-1) gives you (x-8)(x-1), and of course 8a+b = -9is an integer.> Another possibility is a=1/2, b=4, which is the second basic> possibility.> That is, 'a' here can be a fraction, like 1/2, or it can be more like> an integer than a fraction, like actually being 1 with *integers* in> the other case.Uh, OK. I fail to see your momentous point here, but you're rightso far.> Remember that making an issue between integers> versus irrationals is a major part of the logical> mistake that mathematicians made. I say> that just because human beings LOVE being able to> count something out on their fingers, it's not a> mathematical constraint!Erm, that sentence made no sense at all.> However, current mathematical dogma is that *if* 'a' and 'b' are> irrational then the first type possibility is eliminated so it> must be the second possibility.Well, if 'b' is irrational then it certainly isn't an integer.An algebraic integer, perhaps, but not an integer. (Oneexample thereof: sqrt(2).)> Part of the problem is that if you imagine the first> type possibility with> ab = 1, > where again 'a' and 'b' are *irrational* the mathematicians have a> label for 'b' which is algebraic integer, but 'a' cannot be an> algebraic integer.a can most certainly be an algebraic integer if b is a unit.Take b = 2 - sqrt(3), for example. 1/b = 1 / (2 - sqrt(3))= (2 + sqrt(3)) / ((2 + sqrt(3)) * (2 - sqrt(3)))= (2 + sqrt(3)) / (4 - 3) = 2 + sqrt(3).Presto.It's a little weird, but that's algebraic integers for you. :-)> So they don't have a label for it!Are we supposed to level everything for your convenience?As it so happens, there is a label for a, namely, thereciprocal of an algebraic integer (if b is a non-unit).As it is, I can fully characterize all integers and reciprocalsof integers satisfying your equation. All one need do isplug them in:ab = 18a+b = n, for some integer n.Therefore, 8a + 1/a = n8a^2 - n*a + 1 = 0a = (n +/- sqrt(n^2 - 32))/16Presto. Two sets of numbers with a 1-1 mapping into Z, theset of all integers. Or, if you prefer, you can combine thesets and establish a 1-1 mapping with various tricks.No doubt you've done this already but it's an obvious direction.> You know how important naming is with human beings, and it makes it> that much harder for me to explain these ideas without a label.> I've named numbers like 'a' objects, but while mathematicians> successfully trash my work or dismiss it, you can see that using the> name I've given might not resolve things.OK, so the set of all algebraic objects is the set W = {w:1/w is analgebraic integer}. No problem but I fail to see how that helpsyou any.At best, it's a subset of all algebraic numbers. It's not even a ring.> That's it. That's the issue as what mathematicians teach in this area> doesn't follow from mathematics. It doesn't follow from logic or any> axioms. It's just some human notion that has settled into dogma.> Some mathematicians come to the logically specious conclusion because> they can't stick the label algebraic integer on it for that reason> 'a' is in no way an integer like 1 but is instead more like a fraction> like 1/2.> There is nothing in mathematics to support that conclusion.> It's just a human preference attached to the label algebraic> integer.> James Harris-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: Simple idea, mathematics and common-sense[SNIP]> Decimal notation is a convenience, not a necessity, for representation> of numbers. The fact that some numbers are not so representable is> merely an inconvenience, not a disaster.But that's where my story begins.The idea behind exponential notation is rather more than a convenience, itfacilitates the definition of real numbers. I you don't believe this thentry doing arithmetic using unitary notation.Tony === Subject: Re: Simple idea, mathematics and common-sense> [SNIP]Decimal notation is a convenience, not a necessity, for representationof numbers. The fact that some numbers are not so representable ismerely an inconvenience, not a disaster.> But that's where my story begins.> The idea behind exponential notation is rather more than a convenience, it> facilitates the definition of real numbers. I you don't believe this then> try doing arithmetic using unitary notation.> Tony The *definitions* of real numbers and of arithmetical operations in the field of real numbers are quite independent of the algorithms we use to implement or approximate them.While decimal and related notations are extremely useful in the execution of those arithmetical operations, they are in no way essential to their definition, which is the point I was trying to make. === Subject: Re: Simple idea, mathematics and common-sense> It's simple enough, but I have seen *posters* like this person comeforward and confuse the issue enough times that I think I'd betterstep in and make sure that the REAL issue isn't easily obscured.P(x) = (x+8a)(x+b), where ab=1, and consider 8a+b an integerIf a=b=1, notice you have (x+8)(x+1), which is one of two basicpossibilities with integers.Another possibility is a=1/2, b=4, which is the second basicpossibility.> This must be a typo, since ab isn't 1.Just a dumb mistake. That should be b=2, or a=1/4.Still, beyond my dumb mistakes there's the real issue of *two*possibilities, where one is that 'a' is a unit in a ring where -1 and1 are the only integer units.Sure I know, yeah, I make a dumb mistake and people jump on it as ifthat proves that everything I say is wrong.Losers. Why can't you people just play straight up? Why can't youworry about the real ideas versus the very human tendency to fail,including in ways so dumb as to have b=4 like I did?It amazes me how often over the years posters have jumped on *every*single little mistake as if it's proof that everything I say is wrong.I guess for those people I'd have to be Jesus Christ or something.As if unless I'm a perfect man, nothing I say matters.Such odd people they are, why do they succeed so much in confusing theissue? === Subject: Re: Simple idea, mathematics and common-senseIn sci.math, James Harris<3c65f87.0402221702.7036eff5@ posting.google.com>:> > > > > > It's simple enough, but I have seen *posters* like this person come> forward and confuse the issue enough times that I think I'd better> step in and make sure that the REAL issue isn't easily obscured.> > P(x) = (x+8a)(x+b), where ab=1, and consider 8a+b an integer> > If a=b=1, notice you have (x+8)(x+1), which is one of two basic> possibilities with integers.> > Another possibility is a=1/2, b=4, which is the second basic> possibility.> > > This must be a typo, since ab isn't 1.> > Just a dumb mistake. That should be b=2, or a=1/4.> Still, beyond my dumb mistakes there's the real issue of *two*> possibilities, where one is that 'a' is a unit in a ring where -1 and> 1 are the only integer units.Well, if a is an integer, then a is definitely a unit in a ringwhere -1 and 1 are the only integer units -- mostly becauseb can't be non-integral in the sum 8a+b, and therefore must beeither +1 or -1, so a is equal to b in that case and 8a+b is either+9 or -9.That's not much of a result.Of course, if you want to generate the ring Z[(n +/- sqrt(n^2 - 32))/16],feel free. (b = n - 8a is also in the ring. One also has someissues because a^2, a^3, etc. have to be in the ring, too.) For whatit's worth, you've defined a meta-generator as well: a method bywhich one can spit out rings given an arbitrary integer 'n'. Some ofthe rings, e.g., n = 9, are merely Z. Others are subsetsof Q (e.g., n = 6), R (|n| > 5) or C (|n| <= 5).I suspect that the units may not be what you expect. One case Ican readily identify is with n = 8; one unit in the resultingring will be 3 - 2*sqrt(2).I'd have to work to see if I can find such a ring where 1 and -1are the only integer units, with n != 9 or -9 (since in thosecase we're only dealing with Z). Ultimately, it should reduceto solving the equationn^2 - m^2 = 32n and m integers. I for one would suspect that the number ofsolutions for this equation are limited, since, if r > 0 isselected to be a positive integer,(r+1)^2 - r^2 = 2*r + 1and therefore if n > 15 no solutions are possible. Therefore,I can enumerate the solutions without too much difficulty,taking sign into account.(6,2) : a = 1/2, b = 2, 8a+b = 6, P(x) = x^2 + 6a + 8(6,-2) : a = 1/4, b = 4, 8a+b = 6, P(x) = x^2 + 6a + 8(-6,2) : a =-1/4, b =-4, 8a+b =-6, P(x) = x^2 - 6a + 8(-6,-2): a =-1/2, b =-2, 8a+b =-6, P(x) = x^2 - 6a + 8(9,7) : a = 1, b = 1, 8a+b = 9, P(x) = x^2 + 9x + 8(9,-7) : a = 1/8, b = 8, 8a+b = 9, P(x) = x^2 + 9x + 8(-9,7) : a =-1/8, b =-8, 8a+b =-9, P(x) = x^2 - 9x + 8(-9,-7): a = -1, b =-1, 8a+b =-9, P(x) = x^2 - 9x + 8and that's it; that's all possible equations in the integer realm.For n != 6, -6, 9, -9, things get rather more interesting.We can of course continue with our nomenclature:(n, sqrt(n^2 - 32))and attempt to find units in the generated ringZ[(n +/- sqrt(n^2 - 32))/16]. (I've already noted one examplewhere that's possible, namely n = 8.)For an arbitrary n other than 6, -6, 9, -9, if we can find integersc and d such thatc + d(n +/- sqrt(n^2 - 32))/16is a unit in our new ring, then it's clear that 1 and -1 arenot the only units.The actual analysis of all these groups looks like a bit ofa schlog, as there are at least 8 cases to consider. Theeasy one is when n = 0 (mod 16); the ring is equivalentin that case to Z[sqrt(n^2 - 32)/16] or Z[sqrt(n^2/256 - 1/8)];one unit in this ring is 4 * (n/16 - sqrt(n^2/256 - 1/8)),with reciprocal 2 * (n/16 + sqrt(n^2/256 - 1/8)).For the more general case, if we multiply(c + d*(n + sqrt(n^2 - 32))/16) * (c - d*(n - sqrt(n^2 - 32))/16)we get (c + d/16)^2 - d^2*(n^2 - 32)/256, or ((16c + d)^2 - d^2*n^2 - d^2*32)/256 = (256c^2 + 32cd - d^2*(n^2 + 31))/256.In order to have a pair of units we have to find (c,d) such that(256c^2 + 32cd - d^2*(n^2 + 31)) = +256 or -256.with d != 0. (If d = 0 we just get +1 or -1 bysetting c to +16 or -16.)Of course these aren't the only possibilities for units. Onecould, for instance, try to solve the pair of 4-variableDiophantine equations generated by(c + d*(n + sqrt(n^2 - 32))/16) * (e - f*(n - sqrt(n^2 - 32))/16) = 1or(c + d*(n + sqrt(n^2 - 32))/16) * (e - f*(n - sqrt(n^2 - 32))/16) = -1To me that looks like the complicated way of doing it, butcertainly possible. Perhaps someone else can think ofa simpler method of either;[1] finding an n, such that the generated ring Z[(n +/- sqrt(n^2 - 32))/16], only has the two units -1 and +1, and n != 6, -6, 9, -9 (since I've already accounted for those), or[2] proving that no such n is possible.I suspect it's [2] but can't prove it without a lot of schlogging.[rest snipped]-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: Simple idea, mathematics and common-sense> > It's simple enough, but I have seen *posters* like this person come> forward and confuse the issue enough times that I think I'd better> step in and make sure that the REAL issue isn't easily obscured.The real issue is whether JSH is really confused or only pretending.> > P(x) = (x+8a)(x+b), where ab=1, and consider 8a+b an integer> > If a=b=1, notice you have (x+8)(x+1), which is one of two basic> possibilities with integers.> > Another possibility is a=1/2, b=4, which is the second basic> possibility.Not with a and b integers, and not with a*b = 1.This must be a typo, since ab isn't 1.> Just a dumb mistake. That should be b=2, or a=1/4.> Still, beyond my dumb mistakes there's the real issue of *two*> possibilities, where one is that 'a' is a unit in a ring where -1 and> 1 are the only integer units.What do you mean by a unit in a ring where -1 and 1 are the only integer units? In at least one sense, 1 and -1 are the only *integer* units in any subring of the complex field as they are the only (rational) integers that are units. Is that what you mean?Until you make what you want to say unambiguously clear, nobody can possibly agree with you.> Sure I know, yeah, I make a dumb mistake and people jump on it as if> that proves that everything I say is wrong.When everything that you claim is based on that mistake, then everything you claim must be redone without mistakes in order to be validated.> Losers. Why can't you people just play straight up?You never play straight up. If your rules do not require straight up play, you should not complain that others use your rules? If your rules do require straight up play, then follow them yourself before you complain that others don't.> Why can't you> worry about the real ideasWe do, but then you bitch them up.> versus the very human tendency to fail,In your case, that tendency transcends mere humanity.> including in ways so dumb as to have b=4 like I did?> It amazes me how often over the years posters have jumped on *every*> single little mistake as if it's proof that everything I say is wrong.A chain is only as strong as its weakest link. If one link is flawed, the whole chain is flawed and will fail.> I guess for those people I'd have to be Jesus Christ or something.Unless JC was more of a mathematician than repute has it, that would not be nearly enough.> As if unless I'm a perfect man, nothing I say matters.All you have to be willing to do is to acknowledge and repair your errors, as actual mathematicians do, rather than dissing everyone who refuses to overlook them. That is a long way short of required perfection, but a lot closer to perfection than you have been these last 8 years. > Such odd people they are, why do they succeed so much in confusing the> issue?By being so clear on the issues that only people with crosswired brains, like you, can still manage to mess things up. === Subject: Re: Simple idea, mathematics and common-sense> Still, beyond my dumb mistakes there's the real issue of *two*> possibilities, where one is that 'a' is a unit in a ring where -1 and> 1 are the only integer units.You have set the condition that ab = 1. For the problem you posed, 'b' is an algebraic integerand 'a' is not an algebraic integer. There is nothing whatsoever unusual or noteworthy aboutthis. For *any* non-unit algebraic integer 'b', there *is* a reciprocal 'a' such that ab = 1,but not in the ring of algebraic integers. For all these cases 'a' would still be an algebraicnumber. So what? Is it your position that since ab = 1 and since 'b' is an algebraic integerthat 'a' *should* be an algebraic integer? Come to think of it, that may the root cause of allthe contention in these recent threads!You seem to find some necessity for 'a' and 'b' to be units whenever one of them is analgebraic integer and their product is 1. Is *that* your position? Is that the source of allthe *should be an algebraic integer* assertions you have made?? Is that why you are reachingout for other rings, to make units of all reciprocal pairs???There are two things you must never attempt to prove: the unprovable -- and the obvious.http://www.crbond.com === Subject: Re: Simple idea, mathematics and common-senseIn sci.logic, James Harris> You know how with simple quadratics like x^2 + 3x + 2, it's easy> enough to see factors of 2 in the roots?> I mean, it's just (x^2 + 3x + 2) = (x+2)(x+1), and there they are.> However, if it's something like x^2 + 7x + 2, you can use the> quadratic formula and get the roots to find> x = (-7 +/- sqrt(41))/2> and who can see factors of 2 in that thing?I think part of the problem here -- and I'll admit to not beingentirely certain where the prime failure occurs -- is that,if one has an integer such as 6, one can uniquely factor it intoprimes:6 = 2 * 3> That's not the issue here.So what _is_ the issue here?> It's simple enough, but I have seen *posters* like this person come> forward and confuse the issue enough times that I think I'd better> step in and make sure that the REAL issue isn't easily obscured.By all means, clarify. I've lost track of what you're even claiminganymore.> P(x) = (x+8a)(x+b), where ab=1, and consider 8a+b an integer> If a=b=1, notice you have (x+8)(x+1), which is one of two basic> possibilities with integers.> Another possibility is a=1/2, b=4, which is the second basic> possibility.> That is, 'a' here can be a fraction, like 1/2, or it can be more like> an integer than a fraction, like actually being 1 with *integers* in> the other case.... more like an integer than a fraction...? To be more precise, Iassume you mean that there are two possibilities (whose order youmomentarily switch in the above paragraph): Given that 'b' is a(rational) integer, and a,b obey the above equations, then either(i) 'a' is a rational number which is an integer.(ii) 'a' is a rational number which is not an integer.> Remember that making an issue between integers versus irrationals is a> major part of the logical mistake that mathematicians made. I say> that just because human beings LOVE being able to count something out> on their fingers, it's not a mathematical constraint!> However, current mathematical dogma is that *if* 'a' and 'b' are> irrational then the first type possibility is eliminated so it must be> the second possibility.Here, it seems that you're saying: Given that neither a nor b arerational numbers, then if b is an algebraic integer, then the twopossibilities are (by analogy):(i) a is an algebraic integer.(ii) a is an irrational complex number which is _not_ an algebraicinteger.And _you_ assert that, according to current mathematical dogma, thesecond case must hold.> Part of the problem is that if you imagine the first type possibility> with> ab = 1, > where again 'a' and 'b' are *irrational* the mathematicians have a> label for 'b' which is algebraic integer, but 'a' cannot be an> algebraic integer.> So they don't have a label for it!This part of the problem is not a problem. Since a is a root of thepolynomial:8a^2 - 17a + 1 = 0which has integer coefficients, the standard label is algebraicnumber; see, for example:http://en.wikipedia.org/wiki/Algebraic_numberNote that, just as every integer is also a rational number, everyalgebraic integer is an algebraic number. And just as not everyrational number is an integer, not every algebraic number is analgebraic integer.Using this terminology, we can restate the two possibilities as:(i) 'a' is an algebraic number which is an algebraic integer.(ii) 'a' is an algebraic number which is _not_ an algebraic integer.You assert that some mathematicians claim to have proven that (ii) isthe only possibility if 'b' is an irrational algebraic integer.> You know how important naming is with human beings, and it makes it> that much harder for me to explain these ideas without a label.Hopefully this will make your argument easier.> I've named numbers like 'a' objects, but while mathematicians> successfully trash my work or dismiss it, you can see that using the> name I've given might not resolve things.Since I have no idea what you mean by like in this context, itdoesn't resolve things for me at all: objects may, or may not, be thesame as algebraic numbers. However, clearly 'a' is an algebraicnumber.> That's it. That's the issue as what mathematicians teach in this area> doesn't follow from mathematics. It doesn't follow from logic or any> axioms. It's just some human notion that has settled into dogma.Huh? _What's_ it? There is a standard label for numbers of thistype. More precisely, there's a definition by which we can determinewhether or not a number deserves this label. 'a' meets thecriteria. Where's the problem?> Some mathematicians come to the logically specious conclusion because> they can't stick the label algebraic integer on it for that reason> 'a' is in no way an integer like 1 but is instead more like a fraction> like 1/2.This sentence doesn't parse; did you mean:Some mathematicians come to the false conclusion that, since 'a' isnot an algebraic integer, 'a' is not like a rational integer. Instead,it is more like a rational number.If so, could you be more specific? What does like an integer andmore like a rational number mean here? As it stands, this statementcould mean just about anything; as a result, it means nothing.> There is nothing in mathematics to support that conclusion._What_ conclusion?The closest I can come to is that you mean:There are certain parallels between the algebraic integers andalgebraic numbers on the one hand, and the (rational) integers and therational numbers on the other hand. For example, every integer is analgebraic integer, and every rational is an algebraic number;furthermore, every integer is a rational number, while every algebraicinteger is an algebraic number.Given the example equations above, some mathematicians come to thefalse conclusion that, since 'a' does not meet the definition ofalgebraic integer, 'a' is not an algebraic integer; instead it is analgebraic number._This_ is clearly a false statement (i.e., it is _not_ a falseconclusion). 'a' has the property meeting the definition of algebraicnumber. The since... part of this statement is unnecessary; 'a' hasthis property since it is the root of a polynomial with integercoefficents. For that matter, 'b' has the same property, and so isalso an algebraic number (as well as being an algebraic integer).What's the problem here?> It's just a human preference attached to the label algebraic> integer._What_ is just a human preference...?> James HarrisCheers - Chas