mm-949 === Subject: Re: Solving inequality Don't think the solution includes x = 0, agree otherwise. x < 0 1/12 ( 11 - sqrt(73) ) < x < 1 === Subject: Re: Solving inequality Ah, yes of course, missed that one, the denominator of the expression is zero when x=0, so is undefined === Subject: Re: Tricky Quadratic Optimization Problem : : I am trying to optimize a function of a single variable which has a : very simple form. : : F(x) = A*[cos(x) - r*cos(c)]^2 + B*[sin(x)-r*sin(c)]^2 : : where A,B,r,c are all known constants with A,B, and r all positive. : (I know from experience with this problem that numerically A and B : have about the same value, r is close to 1, and x should be close to : c). : : I can use various numerical techniques, but I want a closed form soln : to the eqn. It seems like there should be some simple way to do this, : but after quite some time, I haven't been able to come up with : anything except a solution that involves the soln of a quartic that is : messy and I don't think produces a very good numerical solution. What about: F(x) = |Sqrt(A)[cos(x)-rcos(c)] + iSqrt(B)[sin(x)-rsin(c)]|^2 = |[Sqrt(A)cos(x)+ iSqrt(B)sin(x)] - r[Sqrt(A)cos(c)+ iSqrt(B)sin(c)]|^2 = |g(x) - r*g(c)|^2 , where g(z) = Sqrt(A)cos(z)+ iSqrt(B)sin(z) Hope this helps. D. Baruth x86[]iging[]com -- Summary: EMP spam === === Subject: Re: ZFC++ No, you don't. I understand your distinction between a large set and a small set, but I'm not working in your theory when I do my model theory. This distinction is not relevant to me. No, it's not. It's just that I make mistakes from time to time, like everybody else. Anyway, your theory is equi-interpretable with Morse-Kelly set theory. So, what about it? === Subject: Re: ZFC++ At last! Actually Morse-Kelley is equi-interpretable with a subtheory of this theory,since Morse-Kelley doesn't have ur-elements like those in this. This theory is stronger than Morse- Kelley .I am pretty sure that this theory without ur-elements is equi- interpretable with Morse-Kelley,but with the ur-elements I was not sure, till you confirmed it now(although you said equi-interpretable, while actually M.K is equi-interpretable with a subtheory of this theory). anyhow I have another question, I don't know if you know about this, but in another topic the collection of all sets and proper classes', I introduced the concept of a universe on top of this theory(with no ur-elements), I only want to know if it is consistent. This is a link to it: The idea is to preserve the intuitive feeling of the existance of a universe. Also the idea perhaps can solve the matter of intersection y or /y , but I think to solve it for sure I should axiomatize T in such a manner as to be in itself, and restate regularity to be applicable for sets within T other than T. I will try to do it their Zuhair === Subject: Re: ZFC++ They're still equi-interpretable. We can translate sentences from the language of one into the language of the other in a way that preserves theoremhood. Meaningless since they are in two different languages. They are equi- interpretable. They have the same consistency strength. No, what I said was correct. I'm thinking you don't have a very good understanding of what equi-interpretable means. No, it's not. === Subject: Re: ZFC++ You mean the following theory is not consistent: Theory Y is the set of sentences entailed (from first order logic with identity)by these axioms: 2) Universe: E!xAy(yex & ~Ez(~z=x & xez)). 3) Regularity: 4) Schema of Global comprehension: if P is a formula in which x doesn't occure free, then all closures of: are axioms. Accordingly V is the proper class of all sets. xen))). with 'U' and {x} having the usual definitions. 8) limitation of size: surjective)). xen))). 10) Set Existence: Ex(Ez(~z=T & xez)). If you meant this theory is not consistent, then I want to know were is this incosistency? can you point it exactly? Zuhair - Hide quoted text - === Subject: Re: ZFC++ which theory you mean it is not consistent? the one with the universe. I don't think you saw it so how do you know it is not consistent? - Hide quoted text - === Subject: Re: ZFC++ Yes, I did. I followed your link. It's the theory you gave in this thread, plus an axiom asserting the existence of a universe. It's trivial to prove that that's inconsistent. === Subject: Re: ZFC++ I don't think you saw the same theory I am speaking about, since there are many theories in this thread. it is the last of these theories the one with TeT. It might be trivial to you.Since you are a professional mathematician , but I would like to see this proof even if it is trivial, I want to understand my errors. Since it is so trivial then I don't think it will take a lot of time from you to state this proof, and I would appreciate it very much if you state it here or their. Zuhair === Subject: Re: ZFC++ just not to be confused with another theory I will copy and past it here. I am speaking about this theory: Theory Y is the set of sentences entailed (from first order logic with identity)by these axioms: 2) Universe: E!xAy(yex & ~Ez(~z=x & xez)). 3) Regularity: 4) Schema of Global comprehension: if P is a formula in which x doesn't occure free, then all closures of: are axioms. Accordingly V is the proper class of all sets. xen))). with 'U' and {x} having the usual definitions. 8) limitation of size: surjective)). 10)Empty:ExeVAy(~(yex)). Zuhair === Subject: what is reduced rank regression model? Hi all, material about reduced rank regression. What is reduced rank regression? When and why is it useful? Could anybody give some pointers? === Subject: Re: what is reduced rank regression model? There are 22,000 results using Google so it might not be useful to you - but you don't say what your interest is! I have never heard of this before but see http://isi.cbs.nl/sbr/sbrRev1999.htm#75 Title MULTIVARIATE REDUCED-RANK REGRESSION. Theory and Application. AuthorG.C. Reinsel and R.P. Velu. PublisherNew York: Springer-Verlag, 1998, pp. xiii + 258, US$39.95. Contents: 1. Multivariate linear regression 2. Reduced-rank regression model 3. Reduced-rank regression models with two sets of regressors 4. Reduced-rank regression model with autoregressive errors 5. Multiple time series modeling with reduced ranks 6. The growth curve model and reduced-rank regression methods 7. Seemingly unrelated regression models with reduced ranks 8. Applications of reduced-rank regression in financial economics 9. Alternative procedures for analysis of multivariate regression models Readership: Those familiar with basic matrix theory, plus at least limited exposure to multivariate statistics This well-written and well laid-out monograph deals with multivariate (m Y's depending on n X's) linear models which are of reduced rank, that is, use fewer parameters than mn. In fact, for the model Y = CX + e, rank (C) = r min(m,n) is initially assumed. Later, two sets of regressors are considered, one set with reduced rank parameters, one with full rank parameters, as well as the case where both sets have reduced ranks with distinct structures. Applications in the areas of time series, growth curves, economics and finance are subsequently discussed. Several numerical examples are presented to illustrate the analysis of multivariate data sets using reduced-rank methods. A seven-page final chapter sums up other and related approaches. There are two hundred and five references. The cover is soft-back but sturdy. This text is a must for the library and would be excellent for a seminar course. See this job description http://www.qimr.edu.au/employ/archive/1005.txt Conduct a review of literature on the different methods in formulating dietary patterns (eg reduced rank regression, factor analysis) and recommend the most appropriate method for the dietary data available and objectives of the study. And next time you might be a bit more persistent. This took me 15 minutes. Nick === Subject: Re: what is reduced rank regression model? I was looking for study material -- say lecture notes ,tutorial notes, etc. But those are all research papers, at least the first tens of links on === Subject: Re: what is reduced rank regression model? With 22,000 hits, there is plenty of room for narrowing your request. Say, add introduction or tutorial or FAQ . -- Rich Ulrich, wpilib@pitt.edu http://www.pitt.edu/~wpilib/index.html === Subject: Re: what is reduced rank regression model? haha I was more persistent than you -- I dug out that book only to find that it's not available in our local library... thankx === Subject: Re: what is reduced rank regression model? There are loads of books that are not probably in your library. Whatever happened to Inter-Library Loan. As for persistence, you are the one who is interested in the subject and therefore are the one who has to do the hard work. I was at university in the 70's when we didn't have the Internet and would have to look through book by book to discover minor references if we were luclky on a topic. So that you don't know when you are lucky. Nick === Subject: Re: what is reduced rank regression model? It's like when you're demoted from Sergeant to Private. I can't imagine how it would be useful at all. === Subject: Re: Polynomials What do you mean by form? The polynomial is product_{i=1}^k (x - (X_i)^(X_i)) (or a constant multiple thereof). Do you mean you want to express the coefficients of the new polynomial in terms of the coefficients of the old one? That's not going to have a nice solution. Suppose, for example, the original polynomial has rational coefficients (so the X_i are algebraic), but at least one X_i is irrational. Then X_i^(X_i) is transcendental by Gelfond-Schneider, and some of the coefficients of the new polynomial must be transcendental. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Polynomials yes. I agree. {(x_i)^x_i} without solving for the roots. Is this possible? Any references that may help? TIA Karan === Subject: Re: Polynomials So ... your question is: sum_i (x_i)^(x_i) is symmetric in the x_i , so express it in terms of the elementary symmetric polynomials. A simple version would be: express (a^a)*(b^b) in terms of p=a+b and q=a*b. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ Hello sir~ ------------------------------------------------------- is this a unsolved problem ? infinitely many n. See http://www.research.att.com/~njas/sequences/A088306 1, 260515 and 37362253. --- J K Haugland http://home.no.net/zamunda Howdy you isometric devil, you ;-) What do you think of tan(214112296674652)/214112296674652? FatPhil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./. 122925461 534483448 3083975227 214112296674652 118554299812338354516058 1428599129020608582548671 9322105473781932574489648896 1647533557310758242795542778250 19203062276130315764031455655979057 231767240447593988184889934086223330 16132875282857518417293384808417539001172631 378942232820064582240301274646582332037664078 1169809367327212570704813632106852886389036911 22259360648084057667375368818197310731667745986 1531216647248491128766081193927187028939518325390 4719396382295507308889802400540643563335875857787 197186417171362951534971854903816172537236551228132 4368074519151556502674077262299885926704676221372633 100071341106143073658433833323089743969133079989114295 348065656611924979553181377999664160928613441851213716 6909093424425852275983073604700588403624349958919030856 6666170001744260720956920582292251522427910541800763146162 4062825847242558749454381515818934924447734487275988642228022452 146693013073961620081133606483522315887613577021678543650155007535068 329518791919632670629735531264897175374034718291866616652493301652751797778 1782844948372829034939843193909561799611318597922189434029058975603328120112 1 3299163020883839809260213593120379160500181903506251054111105321983252700748 6815 6654708928700802329346979024351284492994249877994741667914778994141691865601 4596762 3185179659740272554075266933340562179164005327567697985710906047984996145501 03963161 3132915670469030233496338154620314249424955394367930882730566922473610921487 32658062900 7074622758673691648691237724290739292711386664428478134862212803354906574934 753345836141 1077769057656908910300822724460822476866596361568483293201363546318642882196 85765732595787283007 5396038929117835939477133661542110341912284802753607061567287219193079283699 744674488173066659647 8163070518603000766779251550430858310745683336483553497774485060241988494505 046246218503375362664014349 2476497847829225243872571430021209370999602492136613883724400982213012678855 7681607262128273967634934841541 6566636288731834650027209350772283122025850238166486346199651079787259538316 33144141594852184603577778946951537005011 2009650737046327989578154604427752358797383047290204131787825291288828044319 4272152276248822107367873121933748229441250962 1570153714759510073736179625727381214761857814223096101904073972970676806568 770074074895299243501413409994503856532696600398 3688856404316325509285233043157131978510967171900630737408679923835446081414 4499604683901407811229531198309873754403649483872 1979696507925506870269037632272185617328500434775806693754320726988968160159 4134326800984272114465488089723542384041344572622477 ... Old ground, well trod. See mathworld. Tanc. Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./. 18.0078... Interesting. I was unaware of that reference, but now will send a comment to Neil Sloane linking that sequence to mine below. For increasingly large values of tan(n)/n, see together with the corresponding MathWorld entry, which shows the results of further computations by Phil Carmody. David W. Cantrell http://www.math.udel.edu/~lazebnik/papers/tan n.pdf But...I can't understand it. Because... Show that there are infinitely many positive integers n pf) Lemma) Every irrational real u has infinitely many Diophantine approximations q=1 (mod 2) with |u - (p/q)| < 1/(2.q^2). If n = (2k + 1).(pi/2) + y, where k is an integers and y is small, say |y| < 1/4, then = [1 + (y^2)/2! + O(y^4)] / [y - (y^3)/3! + O(y^5)] (Strange...maybe, 1 - (y^2)/2! + O(y^4)) = (1/y) + y + O(y^3).(no from my indication.)--(*) Suppose now that |y| = |n - (2k+1)(pi/2)| < 1/{a(2k+1)} for some constant a, (Maybe, if y is very small, I can find a by Archimedes.) then (*) gives (Why? I can't understand it.) and I had a question. If I show that there are infintely many positive integers n is this possible ? If possible, How do you show that there are infintely many positive integers n Table[N[Tan[n]/n], {n, 1, 100}] {1.55741, -1.09252, -0.0475155, 0.289455, -0.676103, -0.048501, 0.124493, -0.849964, -0.0502573, 0.0648361, -20.541, -0.0529883, 0.035617, 0.517472, -0.0570662, 0.0187895, 0.205524, -0.0631841, 0.00797839, 0.111858, -0.072738, 0.000402348, 0.0690501, -0.088954, -0.00534106, 0.0453367, -0.121248, -0.0100511, 0.0305911, -0.213511, -0.0142482, 0.0206564, -2.28221, -0.0183382, 0.0135376, 0.215291, -0.0227235, 0.00816604, 0.0926809, -0.0279304, 0.00391846, 0.0545569, -0.0348462, 0.00040238, 0.035995, -0.0453612, -0.00264952, 0.0250027, -0.0647532, -0.00543801, 0.0177076, -0.116409, -0.00813506, 0.0124778, -0.821511, -0.0109156, 0.0085035, 0.143635, -0.0139962, 0.00533401, 0.0613634, -0.0177018, 0.00269444, 0.0366853, -0.022616, 0.000402432, 0.0246615, -0.0300012, -0.00167462, 0.0174566, -0.0433468, -0.00364469, 0.012593, -0.0775273, -0.00560935, 0.00903615, -0.419072, -0.00768179, 0.0062744, 0.112546, -0.0100123, 0.00402227, 0.0467542, -0.0128355, 0.00210435, 0.0279852, -0.0165795, 0.000402506, 0.0189419, -0.0221689, -0.0011713, 0.0135247, -0.0321224, -0.00269126, 0.00985003, -0.0567848, -0.00423012, 0.00714136, -0.25346, -0.00587214} How do you think about it ? === Subject: Core 2.0 I download the Core 2.0 library but I am not able to compile it under windows VC . Someone compiled it under windows with some compiler? I am very intersted to the Hypergeom functionality integrated into the Core 2.0 into the Expr class. I tried to use the HyperGeom extension of Core 1.7 but when I get trouble when I compare Hypergeom numbers with algebraic numbers. === Subject: Lucjan E Boettcher (1872 - ? ) Hi I'm looking fo r informations about: Lucjan E Boettcher (1872 - ? ) I have only found this: was born in Warsaw in 1872. He took his doctorate in Leipzig in 1898, working in Iteration Theory, and then moved to Lvov. He published in Polish and Russian . L. E. B.9attcher, The principal laws of convergence of iterates and their aplication to analysis ( in Russian), Izv. Kazan. fiz.-Mat. Obshch. 14) (1904), 155-234. Adam Majewski === Subject: nested intervals theorem, why nonempty intervals as condition? This question is about the Nested Intervals Theorem: The union of an infinite sequence of nested nonempty, bounded closed intervals is nonempty. [interval means interval of real numbers - also described below] I know the statement of the theorem above is imprecise, but it should be enough for readers to recognize which theorem I'm talking about. My question is why books and internet pages include the nonempty qualifier. Is it possible for an interval of real numbers to be empty? As far as I know [a,b] means {x element of reals : a <= x <= b} which can't be empty (because the smallest interval is for some a = b, in which case the interval is the singleton {b} ). I haven't found a reference where the nonempty isn't mentioned, but I can't think of why it's really needed. Can anyone shed some light? === Subject: Re: nested intervals theorem, why nonempty intervals as condition? A very uninteresting proposition. Look it up in your book. The usual theorem with that name is for the intersection, not the union. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: nested intervals theorem, why nonempty intervals as condition? Ooopps typed the wrong one. It should be intersection of nested nonempty closed intervals. My question is about closed intervals, how can they be empty? === Subject: Re: nested intervals theorem, why nonempty intervals as condition? format=flowed; reply-type=original (a,a) === Subject: Re: nested intervals theorem, why nonempty intervals as condition? The original post talked about *closed* intervals. They cannot be empty, can they? === Subject: Re: nested intervals theorem, why nonempty intervals as condition? The interval (a,a) is both open and closed, since it is empty. It contains all of its endpoints, since it has no endpoints. An empty interval is indistinguishable from the empty set, which is both open and closed in every topology. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. === Subject: Re: nested intervals theorem, why nonempty intervals as condition? Ohh right. That makes sense. But the two books I have actually state the theorem using the [a_n, b_n] closed bracket interval notation. Nevertheless the (a,a) is probably what they're fighting against with the nonempty since it explicitly says something that the square brackets only implicitly do. === Subject: Re: nested intervals theorem, why nonempty intervals as condition? format=flowed; reply-type=original [1,0] has the same problem. === Subject: Re: nested intervals theorem, why nonempty intervals as condition? That theorem has a whole bunch of hypotheses, and you're complaining about the only one that is relevant. The union of *any* infinite sequence of nonempty sets is nonempty. You don't need them to be nested, or bounded, or closed, or intervals, but you do need them to be nonempty. Well, you need at least one of them to be nonempty. (If we were talking about intersections, now, it would be a different story.) As to why they say nonempty interval instead of just plain interval, I don't know, I can only guess: Maybe the definition of interval varies from one book to another? Maybe some books define it in such a way that the empty set is an interval? === Subject: Re: nested intervals theorem, why nonempty intervals as condition? I should have typed intersection instead of union. === Subject: Triangles in a rectangle What is the problem that fits triangles into a rectangle called ? Could someone state clearly the problem (rectangle's, triangle's sides, etc) Hoang Long === Subject: Re: Triangles in a rectangle There is no problem. I would say that there are an infinite number of triangles that can be fitted in to a given rectangle. Google does give you some results. Nick === Subject: fitting triangles What is the problem of fitting triangles into a rectangle ? What special points does it have ? What are the rectangle sides'sizes and the rtiangles'? === Subject: on the Heisenberg groupe I am interested on the Heisenberg group. I do some work and I send my questions to a french Maths forum : http://les-mathematiques.u-strasbg.fr/phorum5/file.php?8,file=5883 May I ask you for some explanations on the subject, in particular the existence of left invariant metric on the Heisenberg group (on the dual of its Lie algebra), other that the euclidian metric obtained by identification with $mathbb{r}^3$. Amine my email is as follows : bahayou@altern.org === Subject: on the Heisenberg groupe I am interested on the Heisenberg group. I do some work and I send my questions to a french Maths forum : http://les-mathematiques.u-strasbg.fr/phorum5/file.php?8,file=5883 May I ask you for some explanations on the subject, in particular the existence of left invariant metric on the Heisenberg group (on the dual of its Lie algebra), other that the euclidian metric obtained by identification with $mathbb{r}^3$. Amine my email is as follows : bahayou@altern.org === Subject: Re: Undecidability Theorems in Special Relativity We worked that out last year. Spirit (using June's e-mail to communicate to you)! === Subject: would you stop for a moment Excuse me!! Would you stop for a moment?! O...man...Haven't you thought-one day- about yourself ? Who has made it? Have you seen a design which hasn't a designer ?! Have you seen a wonderful,delicate work without a worker ?! It's you and the whole universe!.. Who has made them all ?!! You know who ?.. It's ALLAH,prise be to him. Just think for a moment. How are you going to be after death ?! Can you believe that this exact system of the universe and all of these great creation will end in in nothing...just after death! Have you thought, for a second, How to save your soul from Allah's punishment?! Haven't you thought about what is the right religion?! Read ... and think deeply before you answer.. It is religion of Islam. It is the religion that Mohammad-peace upon him- the last prophet, had been sent by. It is the religion that the right Bible- which is not distorted-has preached. Just have a look at The Bible of (Bernaba). Don't be emstional. Be rational and judge.. Just look..listen...compare..and then judge and say your word. We advise you visiting : http://www.islam-guide.com/ http://www.thetruereligion.org/ http://www.it-is-truth.org/ http://www.beconvinced.com/ http://www.plaintruth.org/ http://english.islamway.com/ http://www.todayislam.com/ http://www.prophetmuhammed.org/ http://www.islamtoday.net/english/ http://www.islamunveiled.org/ http://www.islamic-knowledge.com/ We willingly recive any inquries at the e-mail : muslim5@hotmail.com === Subject: Re: *** CANADIAN ANTI-TERROR LAW HAS BEEN STRUCK DOWN BY ITS HONORABLE SUPREME COURT UNANIMOUSLY *** (REPOST) === Subject: Re: *** CANADIAN ANTI-TERROR LAW HAS BEEN STRUCK DOWN BY ITS HONORABLE SUPREME COURT UNANIMOUSLY *** (REPOST) === Subject: Finding common references in book indexes Suppose you want to find what 20 history books and biographies have in common. You scan the indexes into your computer, run an OCR program, and now what? Ivan === Subject: Re: Finding common references in book indexes Now you load the OCR output into a database and write a query to crosstabulate the index words with the book titles, for example: TAOCP Vol. 2 Concrete Mathematics Pascal, Blaise 1 1 Pascal's Triangle 10 which indicates there is 1 reference to Pascal, Blaise in each book, no references to Pascal's Triangle in TAOCP Vol. 2 and 10 references to it in Concrete Mathematics. === Subject: Download your own website Heres a website that offers a free web page. There are some good music templates. http://download.gofour.co.uk mgb === Subject: Re: Taylor-Lagrange inequality You have at least 2 possibilities here: 1. Consider the completion of the normed space. 2. Consider the real function x* o f, where x* is in the unit ball of the dual. Using Hahn-Banach you reduce the problem to the real case. Mate === Subject: sylow question Let G be a finite group and p and q be two prime numbers which divide the order of G. Let Q be a q-sylow group in G. If the number of p-sylow groups in the normalizer of Q is one, then the normalizer of Q is in the normalizer of the p-sylow in the normalizer of Q. What does this mean for the normalizer of a p sylow in G? === Subject: Re: sylow question Without some assumptions on the size of the Sylow p-subgroup of the normalizer of Q, you cannot say very much. For instance there is no reason to think p divides the order of the normalizer of Q, so of course the Sylow p-subgroup of the normalizer of Q is normal, but this says nothing about the Sylow p-subgroup of G. Did you intend that the Sylow p-subgroup of the normalizer of Q is also a Sylow p-subgroup of G? === Subject: Bounded function Hello I need some help with a problem. If f is a continous function from R^n to R and for all x in R^n f(x+c)=f(x) where c in Z^n. Show f is bounded and there exist an x0 in R^n so f(x0)=sup(f(x)) (x in R^n). === Subject: Re: Bounded function I still have some problem showing it is bounded. === Subject: Re: Bounded function Sometimes considering a special case helps. Can you convince yourself that it is true in R^1? To be even more specific, consider f(x) = sin(2*pi*x). Why are you so certain that *this* f is bounded and attains its max? Can you prove that? If so, can you generalize your argument to higher dimensions? === Subject: Re: Bounded function Nntp-Posting-Host: apps.cwi.nl Is this supposed to be for all c in Z^n or there exists c in Z^n? In the latter case, try n = 2 f(x1, x2) = x2 c = (1, 0) Then f(x1 + 1, x2) = x2 = f(x1, x2) for all x = (x1, x2), f is continuous but not bounded. Even easier, try c = 0. -- 44 months after Japan attacked Pearl Harbor, Japan surrendered. 47 months after US attacked Iraq, it's time for the US to surrender. pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA === Subject: Re: Bounded function Hello! Since f is continuous, f attains a maximum and minimum on every ball in R^n centered at x with radius c in Z. By the Intermediate Value Theorem, f attains such a maximum at the point x_0. Now, take the sup of all such maximums, and call this maximum M. Then, sup{f(x)} = M for some x = x_0 (this follows since, of course, f is a function). Did this help? === Subject: Re: Bounded function Consider a closed sphere containing the origin that is large enough so that all points in R^n can be writen x+v where x lies in the sphere and v in a vector in Z^n. What do you know about f restricted to such a sphere? hi non-linear map), is there an existing practical or theoretic technique to express that as some combination of square matrices. (I'm not sure it's possible in general) the combination doesn't have to be linear. polynomial combination maybe? hi If I have integer invertible matrix, and use that to transform integer n-tuples, further if I look at the n-tuple aa a single integer, then the transform become one-one integer to integer tranform, can I express that (matrix) as an integer function? Sure. But the answer will depend critically on how you map integer n-tuples to single integers; this can be done in all kinds of ways. What mapping did you have in mind? -- Daniel Mayost === Subject: Re: *** CANADIAN ANTI-TERROR LAW HAS BEEN STRUCK DOWN BY ITS HONORABLE SUPREME COURT UNANIMOUSLY *** (REPOST) === Subject: Re: Big Bertha Thing lightcraft Big Bertha Thing dwarf Cosmic Ray Series Possible Real World System Constructs http://web.onetel.com/~tonylance/dwarf.html 42K Web Page Astrophysics net ring Access site Newsgroup Reviews including alt.astronomy White Dwarfs, Neutron Stars, And Black Holes v1.0 01 sep 99 greg goebel public domain Contents List:- 1.THE DISCOVERY OF WHITE DWARFS 2.WHITE DWARFS AND ELECTRON DEGENERACY 3.THE STRUCTURE AND EVOLUTION OF WHITE DWARFS 4.WHITE DWARFS AND THE AGE OF THE GALAXY 5.BEYOND WHITE DWARFS? 6.NEUTRON STARS DISCOVERED 7.CHARACTERISTICS OF NEUTRON STARS 8.MILLISECOND PULSARS AND OTHER UNUSUAL NEUTRON STARS 9.BLACK HOLES DISCOVERED? 10.MINIHOLES 11.COMMENTS, SOURCES, AND REVISION HISTORY Big Bertha Thing checklist Neighbour checklist 1.Your child beaten up? 2.Rat-proof dustbins? 3.Broken windows? 4.You have apologised? 5.Police called out? 6.Security lighting? 7.New fences? 8.No Leylandii? 9.No Russian Vine? 10.No Solicitors letters sent? Our score is 10, one neighbour scores 1, another scores 5. Tony Lance judemarie@bigberthathing.co.uk === Subject: Big Bertha Thing warlord Big Bertha Thing warlord The last time I heard that an apology given under duress was valid, was in the Monty Python' comedy sketch on the Spanish Inquisition. Everytime anyone said Spanish Inquisition, then 3 red cardinals turned up to organise it. What do I have in comon with a Texas cattle baron? He thinks that he is a bigger liar than I am. I think I am. What does a drill sargeant have in comon with a chinese warlord? He says that the sun will not rise tomorrow. His men believe it. What is the difference between a Texas cattle baron and a chinese warlord? The one knows he is lying. The other has never had the problem. Big Bertha Thing adversity Milton (1644) from The Liberty of Unlicensed Printing. First, when a city shall be as it were besieged and blocked about, her navigable river infested, inroads and incursions round, defiance and battle oft rumoured to be marching up even to her walls and suburb trenches; that then the people, or the greater part, more than at other times, wholly taken up with the study of the highest and most important matters to be reformed, should be disputing, reasoning, reading, inventing, discourcing, even to a rarity and admiration, things not before discourced or written of, argues first a singular good will, contentedness and confidence in your prudent forsight, and safe government, Lords and Commons; and from thence derives itself to a gallant bravery and well-grounded contempt of their enemies, as if there were no small number of as great spirits among us, as his was, who when Rome was nigh besieged by Hannibal, being in the city, bought that piece of ground at no cheap rate whereon Hannibal himself encamped his own regiment. Next, it is a lively and cheerful presage of our happy success and victory. For as in a body, when the blod is fresh, the spirits pure and vigorous, not only to vital, but to rational faculties, and those in the acutes and the pertest operations of wit and subtilty, it argues in what good plight and constitution the body is; so when the cheerfulness of the people is so sprightly up, as it has not only wherewith to guard well its own freedom and safety, but to spare, and to bestow upon the solidest and sublimest points of contyroversy, and new invention, it betokens us not degenerated, nor drooping to a fatal decay, by casting off the old and wrinkled skin of corruption to outlive these pangs, and wax young again, entering the glorious ways of truth and prosperous virtue, destined to become great and honourable in these latter ages. Methinks I see in my mind a noble and puissant nation rousing herself like a strong man after sleep, and shaking her inincible locks; methinks I see her as an eagle nursing her mighty youth, and kindling her undazzled eyes at the full mid-day beam; purging and unscaling her long-abused sight at the fountain itself of heavenly radiance; while the whole noise of timorous and flocking birds, with those also that love the twilight, flutter about amazed at what she means, and in their envious gabble would prognosticate a year of sects and schisms. Big Bertha Thing liberty Milton (1644) from The Liberty of Unlicensed Printing What should ye do then, should ye suppress all this flowery crop of knowledge and new light sprung up and yet springing daily in this city? Should ye set an oligarchy of twenty engrossers over it, to bring a famine upon our minds again, when we shall know nothing but what is measured to us by their bushell? Believe it, Lords and Commons! they who counsel you to such a suppression, do as good as bid ye suppress yourselves; and I will soon show how. If it be desired to know the immediate cause of all this free writing and free speaking, there cannot be assigned a truer than your own mild, and free, and humane government: it is the liberty, Lords and Commons, which your own valorous and happy counsels have purchased us; liberty, which is the nurse of all great wits; this is that which hath rarified and enlightened our spirits like the influence of heaven; this is that which hath enfranchised, enlarged, and lifted up our apprehensions degrees above themselves. Ye cannot make us now less capable, less knowing, less eagerly pursuing of the truth, unless ye first make yourselves, that made us so, less the lovers, less the founders of our true liberty. We can grow ignorant again, brutish, formal, slavish, as ye found us; but you then must first become that which ye cannot be, oppressive, arbitrary, and tyrannous, as they were from whom ye have freed us. That our hearts are now more capacious, our thoughts more erected to the search and expectations of greatest and exactest things, is the issue of your own virtue propagated in us; ye cannot suppress that, unless ye reinforce an abrogated and merciless law, that fathers may despatch at will their own children. And who shall then stick closest to ye, and excite others? not he who takes up arms for coat and conduct, and his four nobles of Dangelt. Although I dispraise not the defence of just immunities, yet love my peace better, if that were all. Give me the liberty to know, to utter, and to argue freely, according to conscience, above all liberties. Big Bertha Thing indomitable (1938) about biography of Lord Grey of Falloden Lord Grey of Falloden sprang from a Northumberland family of country squires, who for generations had played a part in public affairs. His own pleasures lay in the country, but his sense of duty drove him into politics. He was happiest fishing for trout, and watching wild birds, but once he was a member of parliament his abilities and character won for him a prominence that gave him little time for such pursuits. It is strange that the man whose heart was never entirely in politics should have risen to such a high office, should have held it so long, and in such crucial years. It is possible to consider Lord Grey's life as a failure. His sense of duty prevented him from living the life he loved. His efforts to preserve the peace of Europe suffered the defeat of August 1914, that darkened the rest of his life. He sacrificed his eyesight in his wartime service in the government. When at last release came, and he returned to his birds and books, he could no longer see them. Domestic griefs beset him. Yet as our extract from his biography shows, from this tragic material his serene and strong nature won a greatness that is an inspiration and splendid example.(Two extracts follow) He was equally cut off from books, of which as life advanced he had grown scarcely less fond. I classify the different parts of my body as being of different ages, as thus: Sense of smell aged 99 years Eyes 95 Stomach 85 Sense of Hearing 56 (My age) Brain 56 Heart and lungs 45 It makes an unequal team to get along with. === Subject: Irreducibility of polynomials over F_p What is the best way of determining if a polynomial p(x) of high degree over the field F_2 (in particular) is irreducible? Is there an obvious polynomial-time algorithm? -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Irreducibility of polynomials over F_p Nntp-Posting-Host: apps.cwi.nl You can multiply two binary polynomials of degree at most d in time O(d^2), and can also do a product modulo p(x) in this time. Set f_j(x) = x^(2^j) mod p(x) for 0 <= j <= d. You can get f_j from f_{j-1} with one squaring modulo p(x), so the time is O(d^3) to get all of these. Let g(x) = product(f_j(x) - x, j = 1..d-1) mod p(x). This product takes time O(d^3). Your p(x) is reducible precisely when g(x) = 0. Don't neglect the case where p has repeated factors in your proof. -- 44 months after Japan attacked Pearl Harbor, Japan surrendered. 47 months after US attacked Iraq, it's time for the US to surrender. pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA === Subject: Re: Irreducibility of polynomials over F_p I thought at first that was testing if p(x) was primitive rather than irreducible. But I see now that p(x) | x^{2^j} - x for j < d implies p(x) is irreducible. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Detecting random selection in 20 books Sorry. My own query earlier was badly formulated. Let me try again. I want to find out if 20 history books have been chosen at random or not. I scan the indexes into my computer, run an OCR program, and now what is the right way to proceed? What sort of program would I need to identify a pattern in the references? I'm looking for general advice - to be pointed in the right direction, or to the right newsgroup. Ivan === Subject: Re: Detecting random selection in 20 books Try sci.stat.math. Nick === Subject: Re: Detecting random selection in 20 books I think your question becomes clearer if I include your earlier version: === : Subject: Finding common references in book indexes : : Suppose you want to find what 20 history books and biographies : have in common. You scan the indexes into your computer, run : an OCR program, and now what? I think what you may want to do is to divide these books into groups that cover similar topics. If I'm right, then it sounds like you want to do factor analysis http://en.wikipedia.org/wiki/Factor_analysis I am not familiar with that at all, myself. I have heard non-mathematical descriptions of it in discussions of IQ, but that's about it. Also, great use seems to be made of it in the social sciences. I'm pretty sure there are widely used statistics packages that do this, but, again, I'm not familiar with them myself. Jim Burns === Subject: Re: Detecting random selection in 20 books Sounds like my previous Kenkyukeigakusho. I handed in two: one about a finance's option problem, the other is about journal/text categorization but both are eliminated by two gay arseholes. It's been months, but I ams till dumb at both. slow and TOO stooopid === Subject: Re: Detecting random selection in 20 books Randomly chosen from what set? By whom? For what purpose? Suppose one book has been through four editions and the other only one; should the first book be four times more likely to be chosen? If not, what's your criterion for distinguishing different editions of the same book from different books? For that matter, what distinguishes a history book from a non-history-book? Are you counting only books published since 1900, or only books published in English? If you include every history book ever written, including those of which no copies survive, it's rather unlikely you'll see a uniform distribution in the selection, regardless of how the selection is happening, which you haven't specified. Furthermore, even given an unambiguous specification of the set you're choosing from, you still can't determine whether a particular subset was randomly chosen, because any choice is equally likely to be made and therefore any observed choice is equally consistent with the hypothesis. All you can do is formulate alternate hypotheses and try to falsify those, and with a sample size of only 20 books you probably won't be able to falsify them with a very high degree of confidence. I don't understand what the indexes have to do with your question. Maybe I've misunderstood completely. Please give more context. This is inherently impossible. You can't find patterns without a model of the patterns you expect to find, and the right model depends on the context. Incidentally, I don't know much about history, and I probably won't be able to help you with the specifics of your problem; I'm just pointing out that without more details, it's statistically impossible for anybody to help you. -- Ben === Subject: Re: Warning to the Children of the 21st Century... How soon they forget Vince Lombardi. (Or, if Wikipedia is to be believed, Henry Red Sanders.) John Savard === Subject: isomorphism Given a field F, prove that F[x]/(x) is isomorphic to F. here im guessing that (x) is actually the ideal generated byt the polynomial x. === Subject: Re: isomorphism ************************************ Hi: H(f(x)):= a0... Tonio === Subject: Re: isomorphism i didnt quite get what you meant, how you defined this mapping? === Subject: Re: isomorphism days. My association with the Department is that of an alumnus. His mapping is evaluate at 0. It maps f(x) to a0, i.e., to f(0). -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: isomorphism i had another quick questions, that if F is isomorphic to S then S would also be isomorphic to F, is it an if and only if statement? === Subject: Re: isomorphism days. My association with the Department is that of an alumnus. If F is isomorphic to S, then S is isomorphic to F. You seem to agree this statement is true. Now replace F by S and S be F. Then you get: If S is isomorphic to F, then F is isomorphic to S If you agree with the first statement, you must agree with this one. All we did was change the names of the objects. But this is exactly the converse of the one you had. The symmetric property of relations says If a is related to b, then b is related to a. If a relation has the symmetric property, then it follows that a is related to b IF AND ONLY IF b is related to a. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: radius of convergence of SUM sin(exp(n)) x^n My professor, in a practice exam, gave us this exercise: finding out the convergence radius of SUM (n = 0 to +infinity) sin(e^n) x^n. Of course it equals the reciprocal of I assumed that it equalled 1, without proving it, above all because I could see no reason for expecting limsup |sin (e^n)|^(1/n) to be <1. The professor, when correcting the exercise, told: It's enough to know that limsup |sin (e^n)|^(1/n) must be <= 1, because In order to know wheter it is indeed 1 or less, we'd need to study the asymptotic behaviour of the n-th root of |sin(exp(n))|, which isn't a kind of thing we do everyday. Is it known wheter that limsup does equal one? __ Army1987 === Subject: radius of convergence of SUM sin(exp(n)) x^n My professor, in a practice exam, gave us this exercise: finding out the convergence radius of SUM (n = 0 to +infinity) sin(e^n) x^n. Of course it equals the reciprocal of I assumed that it equalled 1, without proving it, above all because I could see no reason for expecting limsup |sin (e^n)|^(1/n) to be <1. The professor, when correcting the exercise, told: It's enough to know that limsup |sin (e^n)|^(1/n) must be <= 1, because In order to know wheter it is indeed 1 or less, we'd need to study the asymptotic behaviour of the n-th root of |sin(exp(n))|, which isn't a kind of thing we do everyday. Is it known wheter that limsup does equal one? __ Army1987 === Subject: Re: radius of convergence of SUM sin(exp(n)) x^n Of course the radius of convergence is at least 1, but the real question is might the radius of convergence be greater than 1 if the lim sup is infinitely many n for which |e^n - (k+1/2)pi| < epsilon2, for some integer k? This then reduces to: do there exist infinitely many k for which log(k+1/2) + log(pi) is within distance epsilon3 of some integer? I think the answer to this last question is yes because log(k+1/2) goes to infinity as k goes to infinity, while log(k+3/2)-log(k+1/2) goes to zero. -- Daniel Mayost === Subject: Re: radius of convergence of SUM sin(exp(n)) x^n Acutally, once our problem is |e^n - (k+1/2)pi| < epsilon2, taking logs and using the first order approximation for log((k+1/2)pi+epsilon2) leads to the inequality: |n - log((k+1/2)pi)| < epsilon2 / ((k+1/2)pi) for which it is not as straightforward to show that it is satisfied for infinitely many k. I would still guess that it can be done though. -- Daniel Mayost === Subject: Re: Product(s?) On Sat, 3 Mar 2007 18:44:33 +0100, bluelabel No, as you showed any isomorphic object in the relevant category to the product (replace the product by any particular product you have in mind) will do. The is supposed to mean the following: An object defined by a universal product (e.g. a product), if it exists, is unique up to a remark, who wants more? Best, G. Rodrigues === Subject: Re: *** CANADIAN ANTI-TERROR LAW HAS BEEN STRUCK DOWN BY ITS HONORABLE SUPREME COURT UNANIMOUSLY *** (REPOST) === Subject: Re: *** CANADIAN ANTI-TERROR LAW HAS BEEN STRUCK DOWN BY ITS HONORABLE SUPREME COURT UNANIMOUSLY *** (REPOST) === Subject: Re: irrationality of polynomial zeros? Am 02.03.2007 18:13 schrieb Gottfried Helms: Hm, most of my assumptions about the properties of the polynomials were false when tried with more examples. Please excuse inconvenience when tried to help. I improved some heuristics about the polynomials and prepared a text, where I collected these. It also explains the context. Still I would like to get some ideas how to proceed to finally be able to prove, that there are no positive integer roots for the whole set of polynomials except the first two ones. http://go.helms-net.de/math/divers/ZerosOfGpFunctions.htm Gottfried Helms === Subject: nC3 Triangles Area Sum here's a programming contest problem from ACM ICPC World Finals Warmup 3 on http://acm.uva.es/contest, Problem A, or http://online-judge.uva.es/p/v111/11186.html already archived. n distinct points are given strictly on the circle of radius r with their polar angles phi[1],phi[2],...,phi[n] and sum of all nC3 triangles is needed. The problem is I can't solve this problem faster than O(n^3), however O(n^2) is enough.Some mathematical modifications are needed to find the O(n^2) solution, which I tried to do. Here's my result. nC3 = n*(n-1)*(n-2)/6 n i-1 j-1 ----- ----- ----- Result = S(i,j,k) ----- ----- ----- i=1 j=1 k=1 where S(i,j,k) is the area of triangle with vertexes i, j and k. We have a formula for computing area of triangle with vertexes (x1,y1), (x2, y2) and (x3, y3). A=Abs(x1*y2-x2*y1+x2*y3-x3*y2+x3*y1-x1*y3)/2 If we come to polar coordinates, we'll have A=Abs(r1*r2*sin(phi2-phi1)+r3*r2*sin(phi3-phi2)+r1*r3*sin(phi1-phi3))/ 2, but r1=r2=r3=r so, S(i,j,k)=(r^2)*Abs( sin(phi[j]-phi[i]) + sin(phi[k]-phi[j]) + sin(phi[i]-phi[k]) ) / 2 n i-1 j-1 ----- ----- ----- Result = r^2 * Abs(sin(phi[j]-phi[i]) + sin(phi[k]-phi[j]) + sin(phi[i]-phi[k]) ) / 2 ----- ----- ----- i=1 j=1 k=1 If not Abs, we could divide the sum in three components and compute each one in O(n^2), because each component depends only on two of three parameters. But what to do in this case? Narek Saribekyan === Subject: How to get conditional expected value? We have know the random variable x with E(x)=a, var(x)=b^2 the observation z=x+w is made, where w is a random variable with E(w)=0, var(w)=c^2, cov(x,w)=p*b*c how to get the conditional expected value E(x|z)? actually, this is a estimation problem, I want to gain Minimum Mean Squared Error(MMSE) Estimator === Subject: Re: How to get conditional expected value? Without further information, you cannot determine E[X | Z]. Example: Let X ~ N(0,1). 1) Let I ~ Ber(1/2), W = (-1)^I X. Then W ~ N(0,1), Cov(X, W) = 0, and E[X | Z] = Z/2. 2) Let W ~ N(0,1) be independent of X. Then E[X | Z] = Z. Do you wish to assume that X and W are jointly normal? (Note that they are not in (1) above.) -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: How to get conditional expected value? actually, there is no pdf given in this problem. In the first question of this problem, ask for the linear MMSE for random variable x. And I have solve it by some equation. And asking for MMSE is the second question. If x and z are jointly Gaussian distributed, them LMMSE=MMSE But if it isn't, I only know MMSE=E(x|z), then this seems no solution? === Subject: three identity Consider a separable real Hilbert space H and denote by e_i = (0,...,0,1, 0,....) with 1 in position i, the i-th element of the fixed canonical base. Let H_n = Span{e_1, ..., e_n}. My ask is: is it true that H / H_n =(1)= the closure of Span(e_{n +1}, e_{n+2},...) =(2)= H_n^perp ? In other words, are both (1) and (2) true ? === Subject: Re: three identity Which canonical base are you talking about? I will assume that {e_k | k natural} is *some* base of H. Equality (1) makes no sense, because the closure of the span of {e_{n + 1},e_{n + 2},...} is a subset of H, whereas H/H_n and H are disjoint. Equality (2) is correct. Jose Carlos Santos === Subject: Re: three identity the canonical base I'm considering -the hilbert space is separable- is: e 1=(1,0,...........) e 2=(0,1,0,........) e 3=(0,0,1,0,.....) ......... I think (1) is true: proof: Denote with H n = Span{e 1,...,e n} and by H^n= closure of Span{e {n +1}, e {n+2},....}. delete the first n coordinates. Then pi^n is linear, surjective and continuos. Besides ker pi^n = H n, hence H/H n and H^n are isomorphic as vectorial space. Besides such an isomorphism is indeed an homeomorphism, since pi^n is an open mapping. What do you think about? === Subject: Re: three identity What if H = L^2(R), which is also a real separable space? What is e_1 then? As before, I think that the span of {e_{n +1}, e_{n+2},....} and H_n live in different worlds. H_n is a subset of H and, on the other hand H/H_n and H are disjoint. All that you tried to prove was that H/H_n and the span of {e_{n +1}, e_{n+2},....} are *isomorphic*, not equal. Jose Carlos Santos === Subject: Re: three identity I know that every infinite dimensional separable hilbert space is linearly isometric to l^2. Yes, sure, I would have written H/H n simeq H^n. === Subject: Re: three identity And... ? The vector space V = { (x,y,z) in R^3 : x + y + z = 0 } is isomorphic to R^2, which has a canonical base. You can't deduce from this that V has a canonical base. Jose Carlos Santos === Subject: Re: three identity You are right, my question should have written as follows: Consider a separable real Hilbert space H and denote by (e i) an orthonormal basis of H. (So, in particular, the base {e i} is countable). Let H n := Span{e 1, ..., e n}. My ask is: is it true that H / H n =(1)= the closure of Span{e {n+1}, e {n+2},...} =(2)= H n^perp ? The answer is: =(1)= is false, since H / H n is toplinear isomophic to the closure of Span(e {n+1}, e {n+2},...) , not equal ! =(2)= is true, since H n is closed, so H is the direct sum of H n and H n^perp. On the other hand, the direct sum of the closure of Span{e {n+1}, e {n+2},...} and H n is H itself, so the closure of Span{e {n+1}, e {n+2},...} must be equal to H n^perp. I think now is all ok, isn't it? Phillips === Subject: Re: three identity they are not equal. They are not equal because the are disjoint sets! Wrong again. Consider R^2 with its usual inner product. Then R^2 is equal to the direct sum of {(x,x) : x real} and {(1,0)}^perp. It is also equal to the direct sum of {(x,0) : x real} and {(1,0)}^perp. I hope that you don't deduce from these two equalities that {(x,x) : x real} = {(x,0) : x real}. Jose Carlos Santos === Subject: Re: three identity I shouldn't say that they are disjoint, they don't intersect too. They are as the set of real numbers and the set of measureble function surely not, it doesen't imply such an equality. So, what is the reason for H n^perp = H^n ? === Subject: Re: three identity What you meant to ask was why is it that the closure of the span of {e_{n + 1},e_{n + 2},...} is equal to H_n^perp. Well, it is clear that H_n^perp. Since furthermore H_n^perp is a closed set, it follows that Now, let _v_ be an element of H_n^perp. Then _v_ can be written as sum_k a_k*e_k, where each a_k is real. If a_k is not 0 for some k <= n, H_n^perp. So, each a_k is equal to 0 when k <= n and therefore, since v = a_{n + 1}*e_{n + 1} + a_{n + 2}*e_{n + 2} + ..., _v_ is in the Jose Carlos Santos === Subject: Re: three identity Is it standard to speak about the canonical base of ell^2 ? In other words, if I speak about the canonical base of ell^2, then people think automatically to {e i} {i in N}, where e i is the vector whose i- th component is 1 with all other equal to zero? === Subject: Re: Is continuum completely filled up? The problem isn't your insufficient explanation, but the fact that you don't understand mathematics and don't wish to learn. Until when? And, that is exactly your problem: you don't care what the words you use actually mean in mathematics. -- David Marcus === Subject: Re: Is continuum completely filled up? I'm sure he is. Just as you are. -- David Marcus === Subject: Re: Is continuum completely filled up? I don't know what a part abobe is. Perhaps you meant the word about here? If so, I still don't understand. 3.14159... is a single number. Do you have many rooms assigned to this one number? If so, it is nothing like the Hilbert case. I don't know what this rule is. Are you are assigning a person in Room #X into Room #X/10 for each real number X? I don't expect numbers to vanish, I expect them to remain numbers. I'm not sure what you are expecting here or why. If you mean to say that the sequence { 0.1, 0.01, 0.001, ... } has a limit of 0, then yes you are correct. What number 1? 1 does not appear anywhere in the sequence you describe. perhaps you mean the digit 1? Yes, it may be replaced with any of the other 9 digits throughout the sequnce (they needn't even be the same digit), and the sequence still has a limit of 0. Perhaps you mean to say that every real number has a unique decimal expression? If so, this is almost true. The exceptions have to do with terminating rationals which have two decimal expressions (one with an infinite trail of 0's and the other with an infinite trail of 9's). Are you saying that there is a bijection between a real numbers decimal expanison and a sequence adding a decimal position at each stage. Again, that is true for many sequences, such as pi ~ { 3, 3.1, 3.14, ... }, but as indicated above, it ids falsse for terminating sequences. For example, both the sequences ( 0.9, 0.99, 0.999, ... } and { 1, 1.0, 1.00, 1.000... } have limits at 1. In the case of { 0.9, 0.99, ... }, the limit is not a member of the sequence. This sentence does not make sense to me. What does it mean for a member of a cardinal number to be managed directly? Countability is a property of cardinal numbers (and thus by extension ordinal numbers). When applied to a set, we are speaking of its size, that is, its cardinality. If a given set has an injection into the natural numbers, then we say that the set is countable. Otherwise, we say that the set is uncountable. No, although Aleph_1 is uncountable, none of its members (nor any of their members, etc.) are uncountable. In fact, Aleph_1 is simply the set of all countable ordinals. Aleph_1 cannot itself be countable, since it would then have to contain itself as a member, and no set in ZFC can do that. Aleph_1 cannot contain any uncountable members, otherwise it would not be the set of all countable ordinals. Please define what you mean by a real number which can be managed. Again, you are mistaken here. N contains only countable members. Furthermore, N itself is countable. Good Luck in your endeavors. Jonathan Hoyle Eastman Kodak === Subject: Re: Is continuum completely filled up? Probably, he means above, i.e., before. -- David Marcus === Subject: Re: Is continuum completely filled up? deny I consider countable real numbers. This is good management of the consept of the infinite. But I think that infinite set is not appropriate basis for mathematics except set theory. You know difficulties accomponying to AC. That's right. The discription of the axiom of infinity is acceptable intact, if it doesn't asserts the existence of complete infinite numbers. Your opinion is a bit different from that of other supporter of set theory. I think that yours is rather near mine.I can't judge about the number of Every one not alwawy accept finite view of the universe. For example, Fred hoyle offered the view of infinitely vast universe which create matters and extend consistently. My difinition are made of dinial of AI, and the diagonal argument, and apprication of the axiom of the least upper bound property, and the axiom countability. When we difine at first that all objects is countable, the rest is not so much different from traditional way. The uncountability of reals is guranteed by Cantor's diagonal argument. This is difinition of uncountability of reals to show the meaning of uncountability, but not proof. There are another interpretations of his argument. This is one of interpletations: We make a list of all reals in binary by following way. At first we make one unit decimals 0.0 and 0.1. Then we gradually increse units and fill them with 0 and 1 in turn from lower unit until all units are filled with 1. Next, we add infinite 0s to the tails of each decimals to make them infinite decimals. 0.0000............... 0.1000............... 0.0100............... 0.1100............... 0.0010............... 0.0110............... ........................................................... By this way we can get a list of all possible permutation of 0 and 1. And we get a diagonal number. Next we make correspondence between digits of diagonal number and decimal on the list. 1st digit of diagonal number is correspondent with a set of one unit decimals of the sequence,because 1st digit of one of them is equal to1st digit of diagonal number. 2nd digit of diagonal number is correspondent with a set of two unit decimals of the sequence,because 1st two digits of one of them.81@is equal to 1st two digits of diagonal number. ............................................................. By this way we can make correspondence between digits of diagonal number and sets of decimals of the sequence. We can find correspondent set for any arbitrary digits of diagonal number, and correspondent finite decimal in that set. Therefore the diagonal number is found on the list. The fact is that one to one correspondence never complete, therefore the production process of diagonal number by step by step denial never ends too. If we accept the diagonal number, we happen to accept the contradiction at the same time. This simple proof may have been already offered many times, and denyed by thhe reason that however many digits might be added to finite decimal, it should not be infinite decimal. But actually a bijection exist between them. In the sense that the amount of menbers is endless like rationals. Some other proofs of uncountability of reals only shows the fact that a set of countable number of members doesn't build continuum. Though we assign reals to the gap of continuum, more than countable number of them are out of our sight. Ozaki Toshiaki === Subject: Traveling waves Hi everyone, Does anybody hear about traveling waves? If so. Could you please help me to find good books and notes or even any traveling waves? What do we mean by the continiuity of traveling wave? === Subject: Re: Traveling waves microwaves all the time, === Subject: Re: Traveling waves Google traveling waves. When I do it, I get 1,320,000 references! The first 10 or 20 should give you all you need. R.G. Vickson === Subject: Plouffe Inverter, new version of Feb. 2007 Hello everybody, I made a new version of my program. This new version contains a lot of new tests, especialy in the generalized expansions. The header of the program contains most the of essential informations to run it. Briefly : 1) Get a copy of the program here : http://www.lacim.uqam.ca/~plouffe/plouffeinverter.txt 2) save the program on your computer, preferably in the directory where maple resides. 3) Open a maple session and read the file (any name will do) like read inverter; 4) voil.88, it is ready to use. PS : this improved version can handle a lot of cases. The 'generalized expansion' part contains 320 different tests. The file is about 500K and contains a bunch of 127 precomputed tables of functions at a precision of 110 decimal digits. PS 2 : any comment is welcome at simon.plouffe@gmail.com Simon Plouffe === Subject: Re: Plouffe Inverter, new version of Feb. 2007 http://www.lacim.uqam.ca/~plouffe/inverter.txt works better. John === Subject: Re: Plouffe Inverter, new version of Feb. 2007 I don't think works better is very helpful in itself. Better in what ways? There seems to be an error, however, in the new pismart function. This calls a function pilooque, which does not seem to exist. I think that should be pilook. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Plouffe Inverter, new version of Feb. 2007 Used with Maple 10 Used with Maple 7 or 8 -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Plouffe Inverter, new version of Feb. 2007 robert israel was kind enough to point out the error, I just realized a small glitch in the recent version. I just corrected the problem, Sorry for the inconvenience. Simon Plouffe === Subject: Re: Rational/Irrational Numbers How about that? Ross === Subject: Re: Solutions of the quintic 1+x+x^5 in radicals? One can use a system like Maple to get that (by brute force). restart; # set the system to some reasonable precision Digits:=24: # try to find all the roots through numerical methods: RootFinding[Analytic](1+x+x^5, x, re=-5..5, im=-5..5): [%]: #nops(%); # now use Maple to find a closed form for the decimal approximation of the roots, # simplify the results, to make them more readable: map(identify,%): simplify(%): L:=combine(%,power): for i from 1 to nops(L) do printf(%a,n, L[i]) end do; i:='i': -1/6*((100+12*69^(1/2))^(2/3)+4-2*(100+12*69^(1/2))^(1/3))/(100+12*69^(1/2)) ^(1/3), -1/12*(-(100+12*69^(1/2))^(2/3)-4-4*(100+12*69^(1/2))^(1/3)+I*(100+12*69^(1/ 2))^(2/3)*3^(1/2)-4*I*3^(1/2))/(100+12*69^(1/2))^(1/3), -1/2-1/2*I*3^(1/2), (1/12*(100+12*69^(1/2))^(2/3)+1/3+1/3*(100+12*69^(1/2))^(1/3)+1/12*I*3^(1/2) *(100+12*69^(1/2))^(2/3)-1/3*I*3^(1/2))/(100+12*69^(1/2))^(1/3), -1/2+1/2*I*3^(1/2) # of course one has to check, that this is correct: P:='product( (x-L[i]), i = 1 .. nops(L))': expand(P): printf(%an, %): simplify(%): printf(%an, %); 1+x+x^5 So the above are the roots in terms of radicals. === Subject: Re: Solutions of the quintic 1+x+x^5 in radicals? For the general trinomial eqn. with coefficient -unity-and for any function f(x) over the real variable x, of the following form: f(x) = x^n + x^m + 1 = 0 x = -[1-(1/n)+(2m-n+1)/(2!*n^2) -(3m-2n+1)*(3m-n+1)/(3!*n^3) +(4m-3n+1)*(4m-2n+1)*(4m-n+1)/(4!*n^4) -(5m-n+1)*(5m-2n+1)*(5m-3n+1)*(5m-4n+1)/(5!*n^5)+...] The general term is SO obvious check it please and inform others For example you can find root of this eqn, to any required degree of accuracy x^101 + x^77 + 1 = 0 B.Karzeddin Al Hussein Bin Talal University JORDAN === Subject: Re: Solutions of the quintic 1+x+x^5 in radicals? There are infinitly many symbolic forms of the solvable quintic eqns. by radicals Here is only one form x^5 + (10*(a*b)^2-5*a^4-b^4)*x + 4*a*(a^4-b^4)=0 Where (a,b) belong to C, complex numbers Bassam Karzeddin Al Hussein Bin Talal University JORDAN === Subject: Re: Solutions of the quintic 1+x+x^5 in radicals? Hi Bassam, i checked your formula up to term 8! for the root x of the trinomial equation x^n+x^m+1=0 for n up to degree 13 and all odd values of m. It sure looks like your formula works. Is there a proof? As for the quintic do you know other forms? Gerry === Subject: Re: Solutions of the quintic 1+x+x^5 in radicals? Hello Gerry Yes make sure it works only for all odd integers, and was proved by me many years back, around 1990 I also went through a copy that was submitted by me to the Third World Academy of Sciences (TWAS) prize for 1994, in cooperation with the Royal Scientific Society (RSS), in JORDAN, their reference (7) 253/39/3/19177 date Oct 30, 1994, and (7) 253/39/3/19743 dated 6/11/1994, where my formula was proved and derived with very elementary methods Here are also some of the reputable Journals replies references about this issue: Journal of Algebra, Dept. of Math. Yale University, their replies dated (Jan. 16, 1986, and July 25, 1990) Monash University, Dept. of Math. Australia, their reply dated 25 October 1990 Cambridge University Press, New York, their replies to me dated (7 and 29), May 1990 Bulletin of the Australian Mathematical Society, their reply dated 20th,July, 1990 American Journal of Mathematics, The Johns Hopkins University, Their reply dated, June 8, 1990 New York University, Courant Institute of Mathematical Sciences, their reply dated April 25, 1990 The University of Western Australia, Nedlands, Dept. of Math. their reply dated 12, June 1990 School of Mathematics, University College of North Wales, Bangor, UK, their reply dated 10/4/1990 Washington State University, Dept. of Pure and Applied Mathematics, there reply dated April 13, 1990 The Australian National University, their reply dated 6, June 1990 The American Mathematical Monthly, their reply dated, May 2, 1990 Quarterly Journal of Mathematics, Oxford University Press, Mathematical Institute, their reply dated 5/4/1990 ????? ??? ?????? ?????- ??????, ?? ?????? 28/3/1995, ???? ??? : ? ?/8/471/95 There is also interesting reply from Monash University in the year 2001, I will tell you about it later But, unfortainatly, It seems that non of them could understand it. Yes, and infinetly many, check if I can remember this form x^5 + (3*a^2*b - b^2 -a^4)*x + a*b*(2*b -a^2) = 0 Where, (a & b) belong to C, complex numbers Bassam Karzeddin Al-Hussein Bin Talal University JORDAN === Subject: average distance between points inside or on a sphere I was looking (again) into the calculation of the average distance between two points in a ball of radius R. There is a calculation explained by Dave Rusin available at http://www.math.niu.edu/~rusin/known-math/96/spheric. I can follow that through. However, an alternative calculation is available at http://www.faqs.org/faqs/puzzles/archive/geometry/part1/ under the heading 'geometry/point.in.sphere.s'. This is the beginning of that text. Use spherical polar coordinates, and w.l.o.g. choose the polar axis through one of the points. Now the distance between the two points is sqrt ( r1^2 + r2^2 - 2 r1 r2 cos(theta)) and cos(theta) is (conveniently) uniformly distributed between -1 and +1, while r1 and r2 have densities 3 r1^2 d(r1) and 3 r2^2 d(r2). Split and it all comes down to integrating polynomials. ... I do not understand 1) why cos(theta) is uniformly distributed between -1 and +1; I would think that for given r1 and r2 a certain value for cos(theta) would correspond to a circular slice through the sphere with radius r2; this would of course imply that certain value of cos(theta) would be more probable than others, because the radius of the circular slice would change with cos(theta). 2) why the densities are given by the expressions 3 r1^2 d(r1) and 3 r2^2 d(r2). Since the volume V of a sphere is 4/3 pi R^3, I would be tempted to think that the density would be something like dV/V, which would be 3dr/R. I am afraid that the integrals will be too difficult as well, but I will look into that matter when I understand the first two points. I apologize already for the naive pseudomathematics; I am only an amateur. Frank === Subject: Re: average distance between points inside or on a sphere Perhaps someone else can provide an obvious derivation. In the meantime here is my lengthier attempt... By symmetry, the intersection with the surface of the sphere of a line through the random point and the centre of the sphere will be randomly distributed on the surface, and this transformation leaves theta unchanged, so we may as well consider points uniformly distributed on the surface. Let R be the radius of the sphere. Picture a cross- section through the sphere, equation x^2 + y^2 = R^2, and wlog let one of the random points be at (R,0). By calculus we have dS = 2*pi*R*dx, where dS is the element of surface area at x (i.e. the surface area of a slice through the sphere at distance x from the centre, of thickness dx). The probability that the second point lies on this surface element is the area of the surface element divided by the total surface area of the sphere, which is dS/ (4*pi*R^2), or dx/(2*R). So, if we choose a random second point on the surface of the sphere, then the pdf of its x-value is 1/(2*R). However, by simple trig we also have x = R*cos(theta), so the pdf of R*cos(theta) is also 1/(2*R), so cos(theta) is also uniformly distributed. The density at r is equal to the volume of a spherical shell element(like an onion skin) of radius r, divided by the total volume of the sphere. This is 4*pi*r^2*dr/(4/3*pi*R^3), or 3*r^2*dr/R^3. I guess they are taking R = 1 (the value of R obviously makes no difference to the answer). === Subject: Re: Nineteen months / Goldbach-FLTsemigroups Hisanobu Shinya === Subject: Re: Nineteen months. I am sorry. I no longer have time to read your paper. Hisanobu Shinya === Subject: Change of Variables If I have dF/dx+dG/dy=0 and I want to transform this into generalized curvilinear coordinates (a,b) to get dFbar/da+dGbar/db=0 how could I define Fbar and Gbar? === Subject: sum of tan(n)/sqrt(n) Today i ran across a website (which I shall mercifully leave unnamed) which states that sum_n tan(n)/sqrt(n) diverges---by applying L'Hospital's rule to the expression! Of course, L'Hospital doesn't apply, since tan(n) doesn't converge to anything. The question of convergence will have to be settled by how close pi/2 + k pi can come to an integer n, which is rather delicate. (And complicated by the tan.) It's not even clear to me that lim sup tan(n)/sqrt(n) = infinity, although it looks like it. I've seen similar developments done here. Does anyone have an answer? -- Ron Bruck === Subject: Re: sum of tan(n)/sqrt(n) According to another thread now running in this same newsgroup series do not go to zero. === Subject: Re: sum of tan(n)/sqrt(n) That would certainly do it. So let me ask another question: can anyone prove that tan(n)/n does not converge? (The sequence, not the series.) It looks to me like the limsup is +infinity, the liminf is -infinity, and every real number is the limit of some subsequence. -- Ron Bruck === Subject: Re: sum of tan(n)/sqrt(n) It seems to be difficult to show that liminf and limsup of a_n = tan(n)/n are -oo and +oo. but it is easy to show that a_n does not converge. On the other side, the irrationality measure for pi is p <= 8.0161 (a result of Hata). It is not difficult to obtain from here that It is conjectured that p=2 (the irrationality measure for e is known to be 2); Mate === Subject: Re: sum of tan(n)/sqrt(n) Here's an approach that might get somewhere. 1. Show there are inifnitely many rationals n / q with q odd and | pi / 2 - n / q | < 17 / q^2. Here, 17 and 42 are arbitrary. -- === Subject: Re: sum of tan(n)/sqrt(n) I got stuck on it, but I asked my Calc 3 students, who had no difficulty coming up with the following obvious solution: tan(n)/sqrt(n) = sin(n)/cos(n)sqrt(n). Cancel everything in sight to get: i/cosqrt = sqrt(-1)/cosqrt and cancel again: -1/co. Clearly, in the limit, co = oo which stands for infinity. So the series obviously converges by the divergence test. Hope that helps, Bart -- The man without a .sig === Subject: Re: sum of tan(n)/sqrt(n) ROFL!!!!!! -Michael. === Subject: Re: JSH: Narcissistic Personality Disorder: Reason for posting? Are you still looking for male volunteers to cut their balls off in your stinking witch ceremony ? === Subject: Re: JSH: Narcissistic Personality Disorder: Reason for posting? YES!!! Witches and Warlock Apprentices/Slaves wanted. Practicing Ceremonial magick and Wiccan Ritual Practice Cabal/Jewish Blood offerings (Mensa) and animal sacrifices (Fish/lobster) We Initiate Witch and warlock apprentices into witchcraft/bondage using a blood ritual. You will stand nude in a circle head down. A sharp razor, cut to your scrotum/clit to make it bleed. Mixed in a cauldren for all to share. Black magick and white magick practiced. Serrious appretices only, 3rd level $5,000.00 up front 1 year of your time, all property and benifits signed over to our farm and magic center. Total commitment is required. If you are not willing to allow total control 24/7. Shock collar or belt may be required. You may be asked to masterbate and eat your cum or . You will not as a student have intercourse, but will be asked to perform for eather male of female company. You may have to milk your self each day before chors. A but plug or vagina plug or chasity belt will be required as well. Once you have mastered your body, the process of breaking your mind will begin, drugs and herbs are invovled at this point. Once you can see and attain some power, you will be of service to the coven and may return to where ever you came, but will be expected to repay the coven all time and matereals spent. You must be willing to be circumsised or de-hooded as we see fit. The mark of our lord/goathead will be tattoed to warn others and mark you as ours. Ritual burning of cow and drinking of urine is part of daily prayers. Reading fowl intestines, reading the sky, shape-shifting, hex casting, group sex, candle driping S&M are but some of the joys and pains available to the right person. Must be drug and alcohol clean, no venerial desease not prenant and willing to get an abortion, not in debt married or follower of christ or jesus allowed. Cathies Cathartic Circle...Cult of the Cloven Clit CCC Dankewicz CCC 798 Sugar Tree Rd SW Willis, VA 24380 Collapse message to short view === Subject: Re: [Erd.9as Straus] the 2521 case What do you mean? There's no such thing as a solution that isn't good. If it's a solution, it's good. The conjecture has been checked up to 10^14, according to MathWorld. -- === Subject: Poisson Summation Hello there, http://en.wikipedia.org/wiki/Poisson_summation_formula#Applications_of_the_P SF the Poisson Summation formula can be applied to sum_{n = 1}^infty frac{1}{n^2} (the sum from 1 to infinity of 1/n^2) to obtain the value pi^2/6. Could someone show me the calculation step by step to obtain this? I've tried - but without any success. Claudia === Subject: Re: Poisson Summation I have a question if you don't mind. In this equation what are t+ and t- standing for? Also, could you tell me what's wrong with the following: In the version of the Poisson summation formula I've found it is said that $$ sum_{n in mathbb{Z}} f(n) = sum_{n in mathbb{Z}} hat{f}(n) $$ where $hat{f}(n)$ denotes the Fourier transform of $f$ defined by $$ hat{f}(t) = int_{-infty}^infty f(x)exp(-2pi i t x)dx. $$ My first attempt was to calculate $hat{f}$. But I had difficulities to evaluate the right handside of the last equation. Claudia === Subject: Re: Poisson Summation Claudia Mathy a .8ecrit : sum_{n=-oo}^oo (f(nT + t+)+f(nT + t-))/2= sum_{m=-oo}^oo F(mw)e^(imwt)/T Another notation is F(x+0) and F(x-0) (Hardy & Rogosinski Fourier series) F(x-) x'