mm-102 a is primitive in GF(4),> then how to nd a^4+a^2, why it is equal to 1 ?> a^4+a^3, why it is equal to a^2 ? Anybody can tell me what is the trick?>also, in GF(4), or any GF(q),is all the -a equal to +a, thats to say, there is no minus sign inGF(q), and once we see - in computation, all,> I have the following question: suppose a is primitive in GF(4),> then how to nd> a^4+a^2, why it is equal to 1 ?> a^4+a^3, why it is equal to a^2 ?GF(4) has 4 elements. One of them is 0, another is 1. Let a be a third. The fourth has to be a + 1, but it also has to be a^2. So a^2 a + 1. Then a^3 a(a + 1) a^2 + a (a + 1) + a 1. Etc. > also, in GF(4), or any GF(q),is all the -a equal to +a, thats to say, there is no minus sign in> GF(q), and once we see - in computation, we just change it to +?am I right?No. Only if q is even. By the way, its spelled Galois.-- all,> I have the following question: suppose a is primitive in GF(4),> then how to nd> a^4+a^2, why it is equal to 1 ?> a^4+a^3, why it is equal to a^2 ? GF(4) has 4 elements. One of them is 0, another is 1. Let a be a third.> The fourth has to be a + 1, but it also has to be a^2.----------------------------------->>why The fourth has to be a + 1, but it also has to be a^2?> So a^2 a + 1.> Then a^3 a(a + 1) a^2 + a (a + 1) + a 1.> Etc. > also, in GF(4), or any GF(q),> is all the -a equal to +a, thats to say, there is no minus signin> GF(q), and once we see - in computation, we just change it to +?> am I right? No. Only if q is even.>when q is even, say 10, then in GF(10), -a^4 a^4, right?when q is odd, say 225, why in GF(225), -a^4 / a^4?thank you very much!-Walala when q is even, say 10, then in GF(10), -a^4 a^4, right?>when q is odd, say 225, why in GF(225), -a^4 / a^4?What GF(10) and GF(225)? A nite elds cardinality has to be a power of a prime.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 [cut]> GF(4) has 4 elements.> One of them is 0, another is 1. Let a be a third.> The fourth has to be a + 1, but it also has to be a^2.-----------------------------------> why The fourth has to be a + 1, but it also has to be a^2?GF(4) has 4 elements.Three have been accounted for: 0, 1, a.Note that they are distinct since a does not equal 0 or 1 etc.Consider a + 1. It is certainly in GF(4).Could it be equal to one of the accounted three elements so far?No, for a + 1 0 says a 1; a + 1 1 says a 0; anda + 1 a says 1 0. Thus, a + 1 must be the fourth unaccounted element.As an aside, you can approach your original question as follows.GF(4) has 4 elements and the element a is primitive.Since the non-zero elements of GF(4) form a cyclic group of order 3,the distinct elements of GF(4) are: 0, a, a^2, a^3 1. This is alsothe way that a primitive element is used for other GF(q), q not 4.You want to nd out what a^4 + a^2 is.First note that a^4 a^3 * a 1 * a a.Thus, a^4 + a^2 a + a^2.You have four choices for a + a^2: a + a^2 0 a + a^2 a a + a^2 a^2 a + a^2 1The second and third are easy to rule outsince they imply the false statements a^2 0 and a 0.The rst is also easy to rule outsince you get 1 + a 0 by dividing both sides by a.But, 1 + a 0 implies the false statement a 1.Thus, you are left with just the last statement,which is what you want.They may be the trick you mentioned: just set the expressionyou want to evaluate to the four possible choicesand eliminate the three that lead to false statementsby simple algebraic manipulations.-- Bill Hale one can make a dystinction between a)omniprescience, b)broad predictions, c)valuable forecasts, d)promoting **** that youre in on. which do you claim?>thank you Jedi Knight Noostradangit;> [blah, blah,blah]8 instances of references from the same source on record in USENET> made to things that hadnt happened yet counter the assertion> that the future cant be known in advance. QED.Whining or ridiculing about it is irrelevant. The original> assertion (by Popper) is proven wrong. Period.--les Dicks dEnron! Ive called the factoring of polynomial into non-polynomial factors> advanced for a reason....> To move beyond factoring polynomials into polynomial factors, you need> to have a *thorough* understanding of coprimeness, ring operations,> and logical argument.LOL.www.crank.net/harris.html Does anyone know how to convert .JMP data to comma delimited format (orbetter yet, MINITAB format)? I really, really need to convert this databecause I dont own JMP--only MINITAB.. any points to conversion utilities(or if someone would be so kind as to convert it), I would be very grateful.Alex Chavezalex@divide0.netData (2 les):http://www-stat.wharton.upenn.edu/~lzhao/stat431/ DataFiles/Call%20Center%20Arrivals.JMPhttp:// www-stat.wharton.upenn.edu/~lzhao/stat431/DataFiles/ Philadelphia%20(Suburban).JMP I found a program that does .JMP data to comma delimited format (or> better yet, MINITAB format)? I really, really need to convert this data> because I dont own JMP--only MINITAB.. any points to conversion utilities> (or if someone would be so kind as to convert it), I would be verygrateful. Alex Chavez> alex@divide0.net Data (2 les):http://www-stat.wharton.upenn.edu/~lzhao/stat431/ DataFiles/Call%20Center%20Arrivals.JMPhttp:// www-stat.wharton.upenn.edu/~lzhao/stat431/DataFiles/ Philadelphia%20(Suburban).JMP what a beautiful evocation of rationality, that old report! a lot more than I can say for Jacks hearsayabout Roswell, which had a couple of interesting associationsaround WW2, which Art Bell (e.g.) never mentions (althoughhe must know about one of them). what Arts promoted is the thesisof the late Lt.Col. Corso, thatthis little pile of **** was parcelled-outto the big companies, and we reverse-engineered ber-optics,the transistor, RADAR ... you name it; I forgot. (fortunately,Corsos co-author exposed him, although I dont knowif its in the book; he did in the promotional tour !-) > The Religious Experience of Philip K. Dick by R. Crumb> http://www.philipkdick.com/weirdo.htm > Yes! There is no religion, no Way of God, no Way of Divine> Realization, no Way of Enlightenment, and no Way of> .... Therefore, Reality (Itself) Is Truth,> and Reality (Itself) Is the Only Truth.> -- Adi Da Samraj, a.k.a. Bubba Free John,> a.k.a. Da Free John, a.k.a. Da Kalki,> a.k.a. the Ruchira Avatar, Adi Da Love-Ananda Samraj,> a.k.a. Franklin Jones> http://adidam.org/> http://www.daplastique.com/home.html > [1968] SCIENTIFIC STUDY OF UNIDENTIFIED FLYING OBJECTS Conducted by the University of Colorado> Under contract No. 44620-67-C-0035 With > the United States Air Force Dr. Edward U. Condon, Scientic Director [1968]> C O N T E N T S> http://ncas.sawco.com/condon/text/contents.htmSection IConclusions and RecommendationsEdward U. Condon> http://ncas.sawco.com/condon/text/sec-i.htm > This formulation carries with it the corollary that we do not > think that at this time the federal government ought to set up > a major new agency, as some have suggested, for the scientic > study of UFOs. This conclusion may not be true for all time. > If, by the progress of research based on new ideas in this eld, > it then appears worthwhile to create such an agency, the decision> to do so may be taken at that time. We nd that there are important areas of atmospheric optics, > including radio wave propagation, and of atmospheric electricity> in which present knowledge is quite incomplete. These topics came> to our attention in connection with the interpretation of some > UFO reports, but they are also of fundamental scientic interest, > and they are relevant to practical problems related to the> improvement of safety of military and civilian ying. > As the reader of this report will readily judge, we have focussed > attention almost entirely on the physical sciences. This was in part > a matter of determining priorities and in part because we found > rather less than some persons may have expected in the way of > psychiatric problems related to belief in the reality of UFOs as > craft from remote galactic or intergalactic civilizations. We believe > that the rigorous study of the beliefs--unsupported by valid > evidence--held by individuals and even by some groups might prove of > scientic value to the social and behavioral sciences. There is no > implication here that individual or group psychopathology is a > principal area of study. Reports of UFOs offer interesting challenges > to the student of cognitive processes as they are affected by > individual and social variables. By this connection, we conclude that > a content-analysis of press and television coverage of UFO reports > might yield data of value both to the social scientist and the > communications specialist. The lack of such a study in the present > report is due to a judgment on our part that other areas of > investigation were of much higher priority. We do not suggest, however, > that the UFO phenomenon is, by its nature, more amenable to study in > these disciplines than in the physical sciences. On the contrary, we > conclude that the same specicity in proposed research in these areas > is as desirable as it is in the physical sciences. > It has been contended that the subject has been shrouded in > ofcial secrecy. We conclude otherwise. We have no evidence > of secrecy concerning UFO reports. What has been miscalled > secrecy has been no more than an intelligent policy of delay in> releasing data so that the public does not become confused by > premature publication of incomplete studies of reports. The subject of UFOs has been widely misrepresented to the public > by a small number of individuals who have given sensationalized > presentations in writings and public lectures. So far as we can > judge, not many people have been misled by such irresponsible > behavior, but whatever effect there has been has been bad. > Therefore we strongly recommend that teachers refrain from > giving students credit for school work based on their reading > Teachers who nd their students strongly motivated in this > direction should attempt to channel their interests in the > direction of serious study of astronomy and meteorology, and > in the direction of critical analysis of arguments for fantastic > propositions that are being supported by appeals to fallacious > reasoning or false data. We hope that the results of our study will prove useful to > scientists and those responsible for the formation of public > policy generally in dealing with this problem which has now > been with us for 21 years. The Condon Report - 1968 CE> http://ncas.sawco.com/condon/index.html> http://www.csicop.org/klassles/posner_klass.html > Parallel Universes [...] http://www.sciam.com/issue.cfm> Not just a staple of science ction, other universes> are a direct implication of cosmological observations> By Max Tegmark--les ducs dEnron! grava ? la saucisse et au marteau: >By the way, what Le Roux calls an application is usually called a >>>Oops, sorry. Thats because there is a difference between fonction and>application in French, and I thought there was the same difference in>English.>>Thats interesting. Whats the difference? I have not seen >>application used this way in English, but I am not a professional >>mathematician.>> >>Could it be the same distinction as between function and map?Marctranslate to the English map. However, I always thought that map and function were synonymous.-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu [...]>>Oops, sorry. Thats because there is a difference between fonction and>>application in French, and I thought there was the same difference in>>English.>>Thats interesting. Whats the difference? I have not seen >application used this way in English, but I am not a professional >mathematician.>Could it be the same distinction as between function and map?>>Marc> translate to the English map. However, I always thought that map > and function were synonymous.It could be just a hard dying habit (cf. Funktion and Abbildung),from Analysis, where maps with the reals as codomain are more equal than the restMarc Of interest perhaps are these formulas:> When U subset A subspace S> cl_A (U) A / cl_S (U)> int_A (U) A / int_S (SA / U)Prove the rst and for the second use. int_A(U) A - cl_A (A-U)As adapted from the theorem S - cl U int (S-U) Of interest perhaps are these formulas:>>When U subset A subspace S>>cl_A (U) A / cl_S (U)>>int_A (U) A / int_S (SA / U)>>Ben Scott>And thank *you* for using my preferred plural of formula!-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu Many of you may be confused by the use by James Harrisof an overly complicated cubic polynomial and unneededvariables. Here is a somewhat simpler version.Lemma 1: Let: P(m) be a polynomial with coefcients inA, the algebraic integers. Let each coefcient bedivisible (in A) by the algebraic integer f.Let P(m) g_1(m) * g_2(m), with g_1(m) and g_2(m) functions from A to A. Then: there exist s and t in Asuch that s*tf; and for all m in A,s divides g_1(m) (in A) and t divides g_2(m) (in A).Core Error Proof: Consider the functionP(m,f,x) mfx^2 +2fx + f^2 (m,f,x in A)[Among other possibilities we can consider this as a polynomial in m, with parameters f and x, or a polynomial in x with parameters m and f. The latter only matters if we want to motivate the following]Consider the quadratict^2 + 2t + mf (*)Let a_1(m,f) 1 - sqrt(1-mf) and a_2(m,f) 1 + sqrt(1-mf)be the negatives of the roots of (*). Some simple algebra givesP(m,f,x) ( a_1(m,f) x + f ) ( a_2(m,f) x + f )or (with a minor abuse of notation)letting P(m) P(m,f,x), g_1(m) a_1(m,f) x + fand g_2(m) a_2(m,f) x + fP(m) g_1(m) * g_2(m)Noting that P(m) can be considered a polynomial in mwith coefcients in A, all coefcients divisible by fwe apply lemma 1 and obtain:There exists s and t, such that s*tf; and for all m, s divides g_1(m) a_1(m,f) x + ft divides g_2(m) a_2(m,f) x + fLet m 0. g_1(0) f and g_2(0) 2x + f.The only choice for s and t (up to units)is sf and t1.thus f divides (a_1(m,f) x + f) thus f divides a_1(m,f)Now specialize by letting m1, f 33 divides a_1(1,3) 1-sqrt(1-(1)(3)) 1 - i sqrt(2)but (1- i sqrt(2))/3 is a root of (3x^2 - 2x + 1)At this point the mathematical universe collapsesOr does it, The problem is with Lemma 1.While this holds for integer polynomials[? does this hold for algebraic integer polynomials]it is not true in general. Counterexamples areeasy to construct if g_1 and g_2 are not continuous.We get a more interesting example by lettingx1 and f 3 above.Let P(m) 3m + 15, this is divisible by 3.By simple algebraP(m) 3m+15 (4-sqrt(1-3m)) (4+sqrt(1-3m))so let g_1(m) (4-sqrt(1-3m)) and g_2(m) (4+sqrt(1-3m))P( -1) 12, g_1( -1) 2, g_2( -1) 6P( -5) 0, g_1( -5) 0. g_2( -5) 8P( -8) -9, g_1( -8) -1. g_2( -8) 9P(-16) -33, g_1(-16) -3. g_2(-16) 11We note that while P(m) is always divisible by3, the factor of 3 switches back and forthbetween g_1 and g_2. In general all we have isLemma 1: Let: P(m) be a polynomial with coefcients inA, the algebraic integers. Let each coefcient bedivisible (in A) by the algebraic integer f.Let P(m) g_1(m) * g_2(m), with g_1(m) and g_2(m) functions from A to A. Then: for all m in A there exist s(m) and t(m) in Asuch that s(m)*t(m)f;,s(m) divides g_1(m) and t(m) divides g_2(m).So all we can conclude above is that s(0)fand t(0)1. We cannot conclude that s(1)f,so we cannot conclude that f divides a_1(1,f),so we cannot conclude that 3 divides a_1(1,3),so mathematics does not collapse in ruins. William Hughes Let me start by saying that I didnt mean to offend or annoy you in> case youre offended or annoyed. Next, let me just say, for the> record, that as a probabilist, in particular, one who works with> stochastic ordinary and partial differential equations, Im very> familiar with white noise and the techniques for working with it in a> mathematically rigorous fashion. I am, however, a student of pure> mathematics, so anything I say will be from that perspective. Ill try> to answer your questions below.I guess I did y off the handle a bit fast there. I will certainlykeep your perspective in mind. Indeed it is what I expected of thisnewsgroup. And I am thankful that a person of your qualications istrying to answer my questions - I appreciate that. I apologize forjumping to the wrong conclusions about your motives.>> [..]>from a mathematical perspective, your question doesnt even make>sense. You state, let X(t) be a zero-mean IID process with variance>of sigma^2. By this, I assume you mean that X(t) is a mean zero>stationary process such that>>(i) X(t) and X(s) are independent whenever t < s, and>>Er, last I checked IID includes dependent, so you have simply>>repeated my conditions.>(ii) E|X(t)|^2 sigma^2 < infty.>>Do you really have to be that anal? Actually, youre not being anal>>enough - if the covariance was not nite, then I wouldve stated that>>it does not exist.> Its true that, here, I am simply repeating your conditions. Im> trying to use the terminology you will be more likely to see in the> pure math literature. For example, when it comes to a process of a> real parameter, t, the term IID is less common. The reason is> twofold: (1) postulating independence can lead to difculties> regarding the existence of the process, as I mention below; and (2)> processes of a real parameter which are identically distributed are> more commonly referred to as stationary.Point taken. I am still getting comfortable with the language andconventions.> The < infty is there to emphasize that this is the important part of> the assumption. If the variance is allowed to be innite, then such a> process will have a measurable version, though it wont be continuous.>Well, there is no reasonable process that satises these conditions.>(In this case, such a process would not even be measurable. If you>drop (ii), such a process would not be continuous.)>>And how did you get to these conclusions?> See the rst page of Chapter III of Stochastic Differential> Equations by Bernt Oksendal and the references therein.> (Incidentally, white noise is *not* the non-continuous process> constructed by dropping assumption (ii). See below.)>On the applied side, there is a lot of intuition about white noise.>Translating that into rigorous mathematical denitions is no small>task. The white noise mathematicians work with is a completely>different kind of object than youre probably used to. (In fact, its>not even a process in the traditional sense.)>>Please, enlighten me, or point me to the subjects and/or books>>that will enlighten me.> An ordinary process is a function X(t,omega) of two variables. For> each omega, we have a function of t, and for each t, we have a random> variable. Even though Papoulis and others state this, I somehow got this wronglong ago and Id appreciate it if you could conrm it: Are the omegasthe elements of the sample space, so that for any specic t, say, t0,X(t0, omega) is the mapping from S (the sample space) to R (the reals)that we normally speak of when considering a random variable?> It is possible to regard white noise as a process in the> sense that for each t we have an object called a generalized random> variable or a generalized Wiener functional. See Stochastic Partial> Differential Equations: A Modeling White Noise Approach by Oksendal,> et al, and the references therein for an introduction to this> approach. More commonly, though, white noise is regarded as a random> variable taking values in the space of distributions: for each omega,> we get a distribution rather than a function of t. To be precise,> white noise would be the distributional derivative of Brownian motion.Do you mean distributions in the sense that the Dirac delta functionis a distribution? So X(t, omega) for each omega gives a distribution(kinda like a pathological function) instead of a normal function?> In the former approach, white noise does not have pointwise values for> each omega, in the latter it does not have pointwise values for each> t.Hmmm.>I know this doesnt answer your question, but I hope it at least>partially explains why you havent received more responses. In some>sense, youre posting to the wrong group. >>I am absolutely not posting to the wrong group. If the answer to this>>this question involves delving into pure math, so be it.>If you post your question in>an engineering group, they may be able to give you an answer based on>their own intuition and jargon, which may be what youre looking for.>>No, thats not what Im looking for. And I HAVE posted it to an engineering>>group (comp.dsp) SEVERAL times now over the last two or three years. It>>is not resolvable by the folks there.>If you want a mathematical answer, then the question must be posed in>a more formal way.>>Hey Jason, bend a little - give me a clue as to what is lacking in formality>>here.> Okay, Ill give it a try. If someone said to me, consider an IID> process whose covariance function is EX(s)X(t) delta(t-s), I would> understand that to mean the white noise I mentioned earlier, which I> would think of as the distributional derivative of Brownian motion.> The connection between the mathematics and the heuristics is this: if> X_r(t) [B(t+r)-B(t)]/r, where B(t) is Brownian motion, then> EX_r(s)X_r(t) f_r(t-s) for some functions f_r which coverge in the> sense of distributions to the delta distribution.Now, I think you want to consider an IID process whose covariance> function is EX(s)X(t) sigma^2 if st, 0 otherwise.Yes, in the one sense thats exactly what I want. Heuristicallyspeaking, it seems like it should be reasonable to constructa process which is stationary and independent at any time differencetau (i.e., X(t) and X(t+tau) are independent random variables when tauis not zero), so that the autocorrelation function is zero when tau is notzero. Yet each random variable in this processs family of randomvariables could have the same distribution and that distribution couldhave a nite variance - for example, a simple N(0,1) distribution - sothat the autocorrelation function is the nite value sigma^2 at tau 0and zero otherwise. Yes, thats precisely the animal I want to construct.> How should I> interpret this in a rigorous sense? Is it the distributional> derivative of some ordinary process? If so, which one?Yeah, yeah!> Okay, once we establish what the process is, we come to next part,> which is that it has a white PSD. As I understand it, the PSD is> simply the Fourier transform of the covariance function. You keep on saying the covariance function, but most engineeringtexts Ive seen say autocorrelation function (the probabilisticautocorrelation, i.e., Rxx(tau) E[X(t)X(t+tau)], NOT the onewhich does time-domain correlation, i.e., the convolution ofthe time-domain function with a time-reversed version of itself).> But two> functions which differ only a set of measure zero have the same> Fourier transform. This process has a covariance function which is 0> almost everywhere. Therefore, its Fourier transform is the zero> function. Right. Thats the problem. > Notice that we dont have this problem when taking the> Fourier transform of the delta function, even though the delta> function is zero everywhere except the origin. This is because the> delta function is not a function at all, but a distribution, so we> must take the Fourier transform in the sense of distributions.So, in summary, the process youve introduced does not exist as an> ordinary process. Perhaps it exists as a random variable taking values> in the space of Schwartz distributions. If so, its not clear exactly> what that random variable should be, so you need to explicitly dene> it in the language of distribution theory. Once done, you need to be> more explicit about the white PSD part, since, as it stands, the> Fourier transform of your covariance function is the zero function,> which is not white. A good place to start reading about distribution> theory (if youre not already familiar with it) is Introduction to> the Theory of Distributions by F. G. Friedlander.Excellent - Ive been looking for a reference to enjoy discussing this stuff because I> enjoy stochastic analysis. But I dont want to annoy or offend you and> I dont want to engage in any kind of emotionally charged discussion> over this stuff. Good luck in your investigations. Please let me know> if you come upon the formal way to describe your process. Id be very> interested in seeing it.help.The key seems to be in understanding why a continuous-time independentprocess isnt a normal process. I have the Oksendal book on order.-- % Randy Yates % ...the answer lies within your soul%% Fuquay-Varina, NC % cause no one knows which side%%% 919-577-9882 % the coin will fall.%%%% % Big Wheels, *Out of the Blue*, ELOhttp://home.earthlink.net/~yatescr An ordinary process is a function X(t,omega) of two variables. For> each omega, we have a function of t, and for each t, we have a random> variable. Even though Papoulis and others state this, I somehow got this wrong> long ago and Id appreciate it if you could conrm it: Are the omegas> the elements of the sample space, so that for any specic t, say, t0,> X(t0, omega) is the mapping from S (the sample space) to R (the reals)> that we normally speak of when considering a random variable?Yes, thats right.It is possible to regard white noise as a process in the> sense that for each t we have an object called a generalized random> variable or a generalized Wiener functional. See Stochastic Partial> Differential Equations: A Modeling White Noise Approach by Oksendal,> et al, and the references therein for an introduction to this> approach. More commonly, though, white noise is regarded as a random> variable taking values in the space of distributions: for each omega,> we get a distribution rather than a function of t. To be precise,> white noise would be the distributional derivative of Brownian motion.Do you mean distributions in the sense that the Dirac delta function> is a distribution? So X(t, omega) for each omega gives a distribution> (kinda like a pathological function) instead of a normal function?Right again.In the former approach, white noise does not have pointwise values for> each omega, in the latter it does not have pointwise values for each> t.Hmmm.> ...> Okay, Ill give it a try. If someone said to me, consider an IID> process whose covariance function is EX(s)X(t) delta(t-s), I would> understand that to mean the white noise I mentioned earlier, which I> would think of as the distributional derivative of Brownian motion.> The connection between the mathematics and the heuristics is this: if> X_r(t) [B(t+r)-B(t)]/r, where B(t) is Brownian motion, then> EX_r(s)X_r(t) f_r(t-s) for some functions f_r which coverge in the> sense of distributions to the delta distribution.Now, I think you want to consider an IID process whose covariance> function is EX(s)X(t) sigma^2 if st, 0 otherwise.Yes, in the one sense thats exactly what I want. Heuristically> speaking, it seems like it should be reasonable to construct> a process which is stationary and independent at any time difference> tau (i.e., X(t) and X(t+tau) are independent random variables when tau> is not zero), so that the autocorrelation function is zero when tau is not> zero. Yet each random variable in this processs family of random> variables could have the same distribution and that distribution could> have a nite variance - for example, a simple N(0,1) distribution - so> that the autocorrelation function is the nite value sigma^2 at tau 0> and zero otherwise. Yes, thats precisely the animal I want to construct.Is this animal that you want to construct something that others on theapplied side are working with? I mean, is it well known in theoutside world, or is it just something you thought of at some pointand were curious about?Ive thought some more about this. Lets let sigma1. For small r, theprocess X_r(t)[B(t+r)-B(t)]/sqrt{r} is pretty close to what yourelooking for. The question is, do these processes, as r-->0, coverge inany reasonable sense? I dont see it, off hand. For xed omega, theyconverge to zero as Schwartz distributions. For xed t, they convergeto zero as generalized random variables. In fact, if we writeEX_r(t+tau)X_r(t)f_r(tau), then f_r converges to 0 almost everywhereand in L^1. Of course, for xed t, the X_r(t)s coverge in law (infact theyre all identical in law) to a normal(0,1) random variable. Idont think this helps though, because you want some kind ofconvergence that deals with all t simultaneously. Theres a lot onteresting behavior in these processes X_r relating to the law of theiterated logarithm and its variants. Chapter 2, section 2.9,subsection E of Brownian Motion and Stochastic Calculus by Karatzasand Shreve contains the law of the iterated logarithm. The paragraphafter the proof of it discusses this process X_r. It containsreferences to two previous remarks and to comments in the notes at theend of the chapter. Part of remark 9.15 is worth posting here:It is quite possible that for each xed t>0, a certain propertyholds almost surely, but then it fails to hold for all t>0simultaneously on an event whose probability is one (or evenpositive!). As an extreme and rather trivial example, consider thatP[B(t)!1]1 holds for every 0 interpret this in a rigorous sense? Is it the distributional> derivative of some ordinary process? If so, which one?Yeah, yeah!Okay, once we establish what the process is, we come to next part,> which is that it has a white PSD. As I understand it, the PSD is> simply the Fourier transform of the covariance function. You keep on saying the covariance function, but most engineering> texts Ive seen say autocorrelation function (the probabilistic> autocorrelation, i.e., Rxx(tau) E[X(t)X(t+tau)], NOT the one> which does time-domain correlation, i.e., the convolution of> the time-domain function with a time-reversed version of itself).For a process X(t), the covariance function is the function of twovariables given by p(s,t)EX(s)X(t). If X(t) is stationary, then pdepends only on the difference t-s, so that we may writeEX(s)X(t)f(t-s) for some function f. This f is the autocorrelationfunction. Ive been being sloppy. Although Ive been saying covariancefunction, Ive meant autocorrelation function.But two> functions which differ only a set of measure zero have the same> Fourier transform. This process has a covariance function which is 0> almost everywhere. Therefore, its Fourier transform is the zero> function. Right. Thats the problem.Notice that we dont have this problem when taking the> Fourier transform of the delta function, even though the delta> function is zero everywhere except the origin. This is because the> delta function is not a function at all, but a distribution, so we> must take the Fourier transform in the sense of distributions.So, in summary, the process youve introduced does not exist as an> ordinary process. Perhaps it exists as a random variable taking values> in the space of Schwartz distributions. If so, its not clear exactly> what that random variable should be, so you need to explicitly dene> it in the language of distribution theory. Once done, you need to be> more explicit about the white PSD part, since, as it stands, the> Fourier transform of your covariance function is the zero function,> which is not white. A good place to start reading about distribution> theory (if youre not already familiar with it) is Introduction to> the Theory of Distributions by F. G. Friedlander.Excellent - Ive been looking for a reference to discussing this stuff because I> enjoy stochastic analysis. But I dont want to annoy or offend you and> I dont want to engage in any kind of emotionally charged discussion> over this stuff. Good luck in your investigations. Please let me know> if you come upon the formal way to describe your process. Id be very> interested in seeing it.help.The key seems to be in understanding why a continuous-time independent> process isnt a normal process. I have the Oksendal book on order. The key seems to be in understanding why a continuous-time independent> process isnt a normal process. The problem with a continuous-time iid process X(t) is not that it doesnt exist---Kolmogorovs theorem guarantees existence---but that as a function X(t,w) of time and sample point it behaves rather pathologically. For instance, it cannot be a measurable function of (t,w), so various things one would like to do are problematic. For example, an integral like int_u^v X(s) ds need not exist, let alone be a random variable.The process you may be looking for can be described as follows. Let H be the class of square-integrable real-valued functions on the real line R. The white noise W {W(h) : h in H} is a collection of random variables dened on some probability space with joint normal distributions; each W(h) is mean zero, and for g,h in HE[W(g)W(h)] int_R g(t)h(t) dt.In particular, if g and h are orthogonal, then W(g) and W(h) are independent. Notice that W is stationary in the sense that the joint distributions of the W(h) are unchanged if, for xed s in R, each h is replaced by the translate h(.+s).-- A. about picking up a higher level book so I can read some more about it. It> really intrigues me that this one is not elementary, because it seems it> should be. Integral of e^x is elementary, e^-(x^2) is elementary, e^x^2 is> not. I need to read up some more about this. I did nd an approximation>> The indenite integral of e^(-x^2) is no more elementary than that of> e^x^2.> I was working this integral out last night (a denite integral) and power> series didnt seem to work. What am I overlooking?Its obvious we are not talking about the same integral. What makes youthink you are overlooking anything?-- Dave SeamanJudge Yohns mistakes revealed in Mumia Abu-Jamal ruling. >thinking> about picking up a higher level book so I can read some more about it.It> really intrigues me that this one is not elementary, because it seems it> should be. Integral of e^x is elementary, e^-(x^2) is elementary, e^x^2is> not. I need to read up some more about this. I did nd anapproximation>> The indenite integral of e^(-x^2) is no more elementary than that of> e^x^2. > I was working this integral out last night (a denite integral) andpower> series didnt seem to work. What am I overlooking? Its obvious we are not talking about the same integral. What makes you> think you are overlooking anything? -- > Dave Seaman> Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling.> I was evaluating the integral of e^(x^2) from 2 to 3 and got somethingaround 12, which I know cannot be right (and I veried it on my TI-83).David Moran > thinking>> about picking up a higher level book so I can read some more about it.> It>> really intrigues me that this one is not elementary, because it seems it>> should be. Integral of e^x is elementary, e^-(x^2) is elementary, e^x^2> is>> not. I need to read up some more about this. I did nd an> approximation>> The indenite integral of e^(-x^2) is no more elementary than that of>> e^x^2.>> I was working this integral out last night (a denite integral) and> power>> series didnt seem to work. What am I overlooking?>> Its obvious we are not talking about the same integral. What makes you>> think you are overlooking anything?> I was evaluating the integral of e^(x^2) from 2 to 3 and got something> around 12, which I know cannot be right (and I veried it on my TI-83).I dont see what this has to do with what I said about the indeniteintegrals, but since you ask, it depends on what series you wereintegrating and how many terms you used. For example, if we askMathematica to integrate the MacLaurin series, we can get something likeIn[1]: Integrate[Normal[Series[Exp[x^2],{x,0,20}]],x] 3 5 7 9 11 13 15 17 19 x x x x x x x x xOut[1] x + -- + -- + -- + --- + ---- + ---- + ----- + ------ + ------- + 3 10 42 216 1320 9360 75600 685440 6894720 21 x> -------- 76204800where, if you substitute x 3, you nd that the x^21 term stillevaluates to around 137.3 and therefore lots more terms are needed forconvergence. I found that going to x^100 seems to work. But youprobably need more signicant digits than you can get on a calculator toadd up all those numbers without losing accuracy.On the other hand, if we use a Taylor series centered at the midpointof the interval of integration, we getIn[2]: Integrate[Normal[Series[Exp[x^2],{x,5/2,10}]],x] 25/4 3 5Out[2] (E (-408240 (5 - 2 x) - 177093 (5 - 2 x) + 5 7 5 8 5 9> 5615280 (-(-) + x) + 4789350 (-(-) + x) + 3752610 (-(-) + x) + 2 2 2 5 11 20393423 (-(-) + x) 5 10 2> 2727745 (-(-) + x) + --------------------- - 8346240 x + 2 11 6 2 4 373275 (-5 + 2 x)> 1814400 x + 292950 (-5 + 2 x) + ------------------)) / 725760 4and this converges much more quickly, since the powers of (-5/2 + x)go quickly to zero when evaluated at x2 or x3.-- Dave SeamanJudge Yohns mistakes revealed in Mumia Abu-Jamal ruling. theres abundant evidence to show tahtNewton is a hoax of the Venetian Party. to say that publishers are biased is a gross understatement; > I was a bit surprised to read this. I looked in a few American> calculus texts on my shelf, and found Newton & Leibniz with equal> billing for this.--les ducs dEnron!http://www.wlym.com/antidummies/part17.html thats a great little book! > Getting back to Newtons mathematics, Huygens & Barrow, Newton & Hooke> by V.I. Arnold (Birkhauser 1990) is denitely worth reading.--les ducs dEnron!http://www.wlym.com/antidummies/part17.html r.e.s. ~ you (and many others out there) are all above my head. However,it is nice to know you are out there making sure the information given iscorrect. I am really in awe. Keep us all on track!!Kavon> I second that. r.e.s. > Kavon>> and is perfectly consistent (and no, Im NOT talking about> Non-Standard>> analysis here).>If d^2 0, and the analysis is *not* non-standard,> ... which has nothing to do with anything that was said that youre> replying to...> http://search.yahoo.com/search?pHow To Search The Web> http://search.yahoo.com/search?pNon-Standard Analysis> http://search.yahoo.com/search?pWikipedia>then how can d differ from 0?> *Sigh.*> http://search.yahoo.com/search?pAssociative Linear Algebras> http://search.yahoo.com/search?pVector Spaces> http://search.yahoo.com/search?pNilpotent Algebras> http://search.yahoo.com/search?pFinitely Presented Algebras> http://search.yahoo.com/search?pIdeals+Innitesimals> http://search.yahoo.com/search?pHypercomplex Numbers+Innitesimals> I got redirected to this newgroup for this problem:5 customers (A thru E) requesting a load of supply: 25, 10, 3, 6 and 7> tons respectively. Delivery-cars (1 thru 6), capacity 8, 16, 24, 32, 32> and 32 tons resp. go each to 1 customer, go back and stay.In matrix are the haulage-costs:Cust: A B C D E> Car:> 1 6 7 8 9 10> 2 2 6 9 11 14> 3 6 7 10 11 12> 4 6 8 10 12 13> 5 8 10 11 13 13> 6 7 8 10 12 13Request:25 10 3 6 7 tonsHow do I solve this using Linear Programming (with objective function > and constraints) so that total haulagecosts are minimal? Im required to > use Maple, tho in that newsgroup they say it can hardly be done that way.> I have difculties with this since its an un-equal matrix.Theres no requirement in Linear Programming that the number of rowsand columns in the constraints be equal. By the way, I learned aboutLPs before interior point methods were shown to be good ways to solvethem. My program (see sig) for solving them is dated, in that sense.Im not sure if I fully understand what your LP should be. But to ndit, heres how I would proceed:1. Dene the decisions variables. (Some of these will be zero in thenal solution.) Must your decision variables be non-negative?2. Express total haulage costs in terms of the decision variables. Ithink the matrix of the haulage-costs will be used here. Thisis the objective function you want to minimize.3. Express constraints. It seems to me that satisfying the request of acustomer will yield a constraint. I think the capacities of thedelivery cars will provide coefcients for the constraint. The sizeof the requested supply will provide the right-hand-side, I think. Andyou should not get an equality for the constraint.What algorithm have you been taught to solve LPs?If this is not sufcient help, say so.-- 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 t 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 Georg Goerg> How do you integrate the function f(x) x/(tanx) ?> please the denite integral.Sint{0 to pi/2} t cos t / sin t int{0 to pi/4} t cos t / sin t + int{pi/4 to pi/2} t cos t / sin t int{0 to pi/4} t cos t / sin t + int{0 to pi/4} (pi/2 - t) sin t / cos t int{0 to pi/4} t [ cos t / sin t - sin t / cos t ] - pi/2 log cos pi/4 int{0 to pi/4} t [ cos 2t / sin 2t ] - pi/2 log 1/sqrt(2) S/2 + pi/4 log 2and thereforeS pi/2 log 2.Another way:pi cot pi.t 1/t + 2t/(tt-1) + 2t/(tt-4) + 2t/(tt-9) + ...Now multiply by t and integrate termwise. Its quite cute. Try it.Larry Georg Goerg>> How do you integrate the function f(x) x/(tanx) ?>> Let S be the denite integral.> Sint{0 to pi/2} t cos t / sin t> int{0 to pi/4} t cos t / sin t + int{pi/4 to pi/2} t cos t / sin t> int{0 to pi/4} t cos t / sin t + int{0 to pi/4} (pi/2 - t) sin t / cos> t int{0 to pi/4} t [ cos t / sin t - sin t / cos t ] - pi/2 log cos> pi/4 int{0 to pi/4} t [ cos 2t / sin 2t ] - pi/2 log 1/sqrt(2)> S/2 + pi/4 log 2> and therefore> S pi/2 log 2.> Another way:> pi cot pi.t 1/t + 2t/(tt-1) + 2t/(tt-4) + 2t/(tt-9) + ...> Now multiply by t and integrate termwise. Its quite cute. Try it.> LarryTHANK YOU very much,georg-- Georg Goerg If a sequence of reals {s_n} converges, then the corresponding Limsup>equals its Liminf. These 2, however, appear to be independent of the>particular ordering of {s_n}. >>This would be true, by looking at their denitions and considering that >>for any epsilon, the s_i that cause a problem can be excluded nitely >>far into the sequence, regardless of the order.>> Is it accurate to state, then, that>LimsupLiminf does not necessarily imply the convergence of {s_n}? >Any help/explanation appreciated.>>{s_n} converges iff LimsupLiminf. Think about epsilon/N proofs.> for a given relationship (n,s_n). For an arbitrary e, there exists an> N such that n>N ----> |s_n - A| (k,s_k)it is entirely possible that the implication does not hold.> Obiously Im confused here. Is there some simple proof that, no> matter what the relationship, the limit of the resulting sequence natural numbers m other news readers or means of posting besides Google Groups because I> use it all the time and I think it has transformed Usenet.Really... how?Besides the archiving part, I mean. > I wouldnt put in a lot of technical things, nor would I talk about>> other news readers or means of posting besides Google Groups because I>> use it all the time and I think it has transformed Usenet.> Really... how?> Besides the archiving part, I mean.Well, its made it easier to get access to USENET for people who eitherdont want, or are too clueless, to learn to operate a proper newsreader.James himself is a prime example of the latter class.Sometime I miss the days when getting to USENET required conguringUUCP, compiling, installing and conguring B News or C News, ndingand negotiating with an admin somewhere for a news feed... or, if youwere a student, getting an account from your university news admin.James would be excluded from both methods: too technically illiterateto do it himself, and too afraid of education to have a student account.-- Wayne Brown (HPCC #1104) | When your tails in a crack, you improvisefwbrown@bellsouth.net | if youre good enough. Otherwise you give | your pelt to the trapper.e^(i*pi) -1 -- Euler | -- John Myers Myers, Silverlock I wouldnt put in a lot of technical things, nor would I talk about> other news readers or means of posting besides Google Groups because I> use it all the time and I think it has transformed Usenet.Really... how?Besides the archiving part, I mean.Google Groups makes Usenet easily accessible over the Internet. Somenewsgroups, like sci.math, are mirrored to the Internet, so that youhave what I now call Usenet-Internet.That vastly increases the potential audience.Usenet is the one sure-re way to say something, and probably get aresponse--if you know what youre doing--as youre read around theworld.Sure, you can put up a webpage, and watch as no one comes. And ittakes more effort to start and *maintain* a webpage, as remember, itsnot just about throwing up a webpage, as you have to keep it up aswell. And that applies to blogs as well.Now posting, youre competing with a lot of other posters, but youhave the possibility of standing out based on your content and style,so theres a greater potential I think to step out and have an impact.My feeling is that Usenet-Internet will explode into an ever moreimportant medium of exchange as people see its important features.If you want to see the power of knowing what youre doing, do a Googlesearch on core error and see what person has number 1 and 2.Or you can do a search on my name and math, yes, its a lot of hatestuff, but my point remains, as notice how many *different* LANGUAGESyoull bump into if you dig, where for some reason, my work is forwardenough that Google highlights it, so you end up directed to it.James Harrishttp://lostincomment.blogspot.com/ If you want to see the power of knowing what youre doing, do a Google> search on core error and see what person has number 1 and 2.Repeating the same phrase over and over despite the fact that it hasno real mathematical meaning at all is proof of knowing what youredoing?> Or you can do a search on my name and math, yes, its a lot of hate> stuff, but my point remains, as notice how many *different* LANGUAGES> youll bump into if you dig, where for some reason, my work is forward> enough that Google highlights it, so you end up directed to it.Google highlights based on how forward your mathematics is?I tell ya, Ive learned all about this Usenet-Internet thing today.-- Jesse F. HughesWhat you call reasonable is suspect since youve proven yourself tobe an enemy of mathematics. -- James S. Harris defends the cause. I wouldnt put in a lot of technical things, nor would I talk about> other news readers or means of posting besides Google Groups because I> use it all the time and I think it has transformed Usenet.Really... how?Besides the archiving part, I mean.Google Groups makes Usenet easily accessible over the Internet. Some> newsgroups, like sci.math, are mirrored to the Internet, so that you> have what I now call Usenet-Internet.That vastly increases the potential audience.*snip*Youre basically singing about how good usenet is. We all know that.anyone to post, but how does that affect usenet itself?> My feeling is that Usenet-Internet will explode into an ever more> important medium of exchange searches always turned up a mush ofusenet-internet then wed have the kind of noise problem that ourweblogs are creating now. I dont want this post to turn up whensomeone types noise ratio on usenet is terrible, and Im expectingweblogs to go that way--useful and intellectual for a while, buteventually degenerating into ames, ames about ames, histories ofames being amed, etc.My point is that just a core error search doesnt mean anyone has totake those results seriously. Plus that mirroring would happen withoutGoogle Groups too. Google groups have not transformed Usenet. Crossposts to writing newsgroups left in because, for once,theyre relevant.> I wouldnt put in a lot of technical things, nor would I talk about> other news readers or means of posting besides Google Groups because I> use it all the time and I think it has transformed Usenet.Really... how?Besides the archiving part, I mean.Google Groups makes Usenet easily accessible over the Internet. Some> newsgroups, like sci.math, are mirrored to the Internet, so that you> have what I now call Usenet-Internet.Quibble: Usenet was available over the internet before therewas widespread web access. I think you mean Google Groupsmakes Usenet accessible over the Web. The Internet isboth much older and much larger than the Web.archive. Google extended the capabilities of the searchengine and made some improvements in the interface.> That vastly increases the potential audience.Usenet is the one sure-re way to say something, and probably get a> response--if you know what youre doing--as youre read around the> world.provide the same heard around the world capability. - Randy I wouldnt put in a lot of technical things, nor would I talk about>> other news readers or means of posting besides Google Groups because I>> use it all the time and I think it has transformed Usenet.Really... how?Besides the archiving part, I mean.Google Groups makes Usenet easily accessible over the Internet. As opposed to the days before Google, when usenet was easilyaccessible over the internet? (I think you mean easily accessibleover the web...)>Some>newsgroups, like sci.math, are mirrored to the Internet, so that you>have what I now call Usenet-Internet.Um, usenet is _part_ of the internet - always has been. Again,you mean the web, not the internet.You should denitely write that guide to posting. Just what weneed, authoritative information from someone who thinks thatthe internet means the web.>That vastly increases the potential audience.Usenet is the one sure-re way to say something, and probably get a>response--if you know what youre doing--as youre read around the>world.Sure, you can put up a webpage, and watch as no one comes. And it>takes more effort to start and *maintain* a webpage, as remember, its>not just about throwing up a webpage, as you have to keep it up as>well. And that applies to blogs as well.Now posting, youre competing with a lot of other posters, but you>have the possibility of standing out based on your content and style,>so theres a greater potential I think to step out and have an impact.My feeling is that Usenet-Internet will explode into an ever more>important medium of exchange as people see its important features.If you want to see the power of knowing what youre doing, do a Google>search on core error and see what person has number 1 and 2.How does this show that usenet is important? All it shows is thatanyone can post any nonsense he wants. Youve derived somesort of power from being able to make a fool of yourself in frontof a global audience? Good for you.>Or you can do a search on my name and math, yes, its a lot of hate>stuff, but my point remains, as notice how many *different* LANGUAGES>youll bump into if you dig, where for some reason, my work is forward>enough that Google highlights it, so you end up directed to it.>James Harris>http://lostincomment.blogspot.com/******************** ****David C. Ullrich ... >Google Groups makes Usenet easily accessible over the Internet. > As opposed to the days before Google, when usenet was easily > accessible over the internet? (I think you mean easily accessible > over the web...)Indeed, James is confused. >Some >newsgroups, like sci.math, are mirrored to the Internet, so that you >have what I now call Usenet-Internet. > Um, usenet is _part_ of the internet - always has been.No, usenet has various ways of transmission. Internet is one of them,but it can be done (and has been done) with UUCP through dial-up lines.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ...>Google Groups makes Usenet easily accessible over the Internet. As opposed to the days before Google, when usenet was easily> accessible over the internet? (I think you mean easily accessible> over the web...)Indeed, James is confused. >Some>newsgroups, like sci.math, are mirrored to the Internet, so that you>have what I now call Usenet-Internet.Um, usenet is _part_ of the internet - always has been.No, usenet has various ways of transmission. Oh. Id always assumed that internet was the same asdescribed in RFCs. I guess not.>Internet is one of them,>but it can be done (and has been done) with UUCP through dial-up lines.************************David C. Ullrich ... >Google Groups makes Usenet easily accessible over the Internet. > As opposed to the days before Google, when usenet was easily > accessible over the internet? (I think you mean easily accessible > over the web...) >Indeed, James is confused. >Some >newsgroups, like sci.math, are mirrored to the Internet, so that you >have what I now call Usenet-Internet. > Um, usenet is _part_ of the internet - always has been. >No, usenet has various ways of transmission. > Oh. Id always assumed that internet was the same as > described in RFCs. I guess not.transport mechanisms than the Internet. Dial-up lines withUUCP connection, FIDO, BITNET, EARN. The RFCs for Usenetwhen transmitted through the Internet.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ Given my *years* of posting on several newsgroups, and my current> funding situation (as in, few funds) Im considering writing up a> guide to posting on newsgroups. The way I gure it, in keeping with> the open source mentality, Id give it away, with people having the> option to contribute, if they found it useful.> Do a search rst, it appears that this has already been done.> I wouldnt put in a lot of technical things, nor would I talk about> other news readers or means