mm-2869 === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days > Let's be serious for once. Consider an object being accelerated by a idealistic jet of water or a > continuous 'stream of elastic ping pong balls'. What is its subsequent velocity > pattern? accelerated beyond the operating speed of the accelerating fields, ie at 'c'. I > hope it might also produce a relationship that is equivalent to mass > 'appearing' to increase with velocity by gamma.) >Please define what you mean by the operating speed of an >>accelerating field. I've been in physics research for over 20 >>years, and I've never heard that term. >>Do you mean phase velocity, group velocity, how fast the >>operators turn the knobs, what? >I think you forgot to answer the question, Henry. >Why is that? yes, for some strange reason I 'forget' to answer anything EjP asks. >Paul Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days > In the elastic case, the water (or stream of ping pong balls) ends up moving > backwards at Vo-2v, so the change in momentum per second is 2m(Vo-v) >>v = Vo*(1 - exp(-2*m*t/M)) >Note: >According to Newton the momentum >p = m*Vo*(1 - exp(-2*m*t/M)) decreases with time. >Newton predicts that there is a speed limit because the ping pong >ball will cease to transfer momentum to the mass when its speed approaches Vo. Is that your approximte solution? See my other message. >[..] >>There is no reason for making this more complicated than it is, Henry. >>The mass M will in all cases end up moving with the speed Vo. >>Or rather, v will approach Vo asymptotically. >>But you knew this, didn't you? >>And this have absolutely nothing to do with what happens >>Which you also knew, didn't you? >> Not at all Paul. >> I have to compare the curve given by the above solution with that predicted by >> SR for a charge accelerated in a constant field. >> Can you tell me what that might be? >Sure I can. >But first, I will tell you what Newtonian mechanics predicts. >Let the mass m with the charge q be in a static electric field Eo. >Let v be the normalized speed, that is the speed is v*c. >Let the mass accelerate from standstill. >What is v(t)? >Solution according to Newton: >---------------------- >The force on the the mass is constant Fo = q*Eo >Newton predicts that the momentum p = Fo*t >increases linearly with time. >Fo*t = m*v*c >v = t/T where T = m*c/q*Eo >Newton predicts that the speed will increase lineray with time, >and will pass the speed of light at the time T = m*c/q*Eo. >There is no speed limit because the electric field never >ceases to transfer momentum to the mass. Well we know that is certainly wrong. Why it is wrong remains to be explained 'physically'. >Solution according to SR: >-------------------- >The force on the the mass is constant Fo = q*Eo >SR predicts that the momentum p = Fo*t >increases linearly with time. >Fo*t = m*v*c/sqrt(1 - v^2) >v = (t/T)/sqrt(1+(t/T)^2) where T = m*c/q*Eo >SR predicts that the speed will approach the speed of >light asymptotically. v = 2^-0.5 = 0.707 at the time T = m*c/q*Eo. >c is the speed limit despite the fact that the electric field never >ceases to transfer momentum to the mass. >Of course two curves which both are starting at zero and >are approaching a limit asymptotically will show a superficial >resemblance. >But it makes no sense to compare the Newtonian prediction >for a scenario to the SR prediction for a completely different >scenario, and conclude that because the predictions show >a slight resemblance there must be a causal connection >between the predictions. Except that one is a physical explanation and the other a purely mathematical one that sheds no light on anything. >What makes very much sense, though, is to compare >Which I did above. >They are very different. You didn't take into account any of the limiting factors. You must know this is plain silly. >And we both know which of them are in accordance with >experimental evidence. >Don't we? Yes Paul . Read all about the evidence in 'The Real Einstein' . >Paul Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: I NEED HELP BADLY (sorry, maths not psych) Expires: 28 days >> Let's be serious for once. >> In the elastic case, the water (or stream of ping pong balls) ends up moving >> backwards at Vo-2v, so the change in momentum per second is 2m(Vo-v) >Note that this is based on the assumption that the mass of >a ping-pong ball is very much smaller than M. >This is of course an approximation. I don't want an approximation. the relative masses shouldn't make any difference. >v = Vo*(1 - exp(-2*m*t/M)) >> Incidentally, particularly in the case of the elastic 'ping-pong ball drive', >> momentum is balanced but does the (kinetic) energy equation match? >Not quite, due to the approximation mentioned above. Why are you only producing an approximation? The (classical) equation is Md2x/dt2=2m(Vo-dx/dt) , is it not? That is MD^2+2mD-mVo=0, where m is the mass per second leaving the source and M is the mass of the body.. The roots are -K(+/-)sqrt(K^2-4KVo), where K=m/M So a solution is: x=(e^-mt)[e^root()-e^root()] +A Isn't this tanh(t)? I'm a bit rusty on this stuff. >Paul Henri Wilson. See why relativity is wrong: http://www.users.bigpond.com/HeWn/index.htm === Subject: Re: JSH: My victories get lost > :) good one! > LH Confused by the [original] subject line. Plenty of null sets in the universe, no need to lose one. === Subject: prime primitive root Is it true that every odd prime has a prime primitive root? How does the proof go or where can I find one? === Subject: Re: prime primitive root > Is it true that every odd prime has a prime primitive root? > How does the proof go or where can I find one? Every prime, odd or otherwise, has a prime primitive root. Let p be the prime in question. Let g be a primitive root mod p. Then any x of the form x = g + tp, t =0, 1, 2, ..., is a primitive root mod p. By Dirichlet, the arithmetic progression g + tp contains a prime, indeed, contains infinitely many primes. But perhaps you were looking for a prime primitive root less than p. That's harder. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Number theory congruence > Hi group, > I'm a little confused over some homework problems which dont appear to have > any supporting examples/theorems (to me anyway). > For integer n > 0, use congruence theory to establish the divisibility > statement: > 7 | 5^(2n) + 3 * 2^(5n-2) > among other theorems, i have the one that says > if ca congruent cb (mod n) and gcd(c, n) = 1, > then a congruent b (mod n) > ...which implies factoring something out, which i dont see any way to do. > any help is appreciated > Ira If nothing else seems to work, you can do it by direct evaluation for enough cases from n = 1 , n = 2, ..., to prove the pattern. To simplify some of your calculations, note that 7 | (k^7 - k) for all integers k, or, equivalently, 7 | (k^6 - 1) for every integer k coprime to 7. In mod notation, k^7 == k (mod 7), for all integers k, or k^6 == 1 (mod 7) for integers k such that k =/= 0 (mod 7). === Subject: Re: Product of Reals > It took me a couple of minutes to realise why the uncountable sums did > not work. Is this right? If the sum converges then clearly the > number of terms with absolute value greater than 1 must be finite. > Similarly the number of terms with absolute value between 1 and 1/2 is > also finite, ditto 1/2 and 1/3, 1/3 and 1/4 so the non-zero terms are > the union of a countable set of finite sets and hence are countable. > Simple, once you know. Yes, that's right. > If the structure of N (natural numbers) is ignored and this definition > of sum is used then I expect that only the series which are absolutely > convergent in the common sense will be convergent in this sense. Exactly. > Considering the countability limitation, is this generalised notion of > convergence useful and commonly used? Measure theory also depends heavily on countable additivity, and yet this has not proved to be much of a handicap. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Number theory problem, or something > This is a homework problem from a course I'm not taking. > Prove that none of the numbers 1, 11, 111, 1111, 11111 and so on, are > squares of natural numbers greater than 1. > I can't seem to do it completely. Here's my best attempt so far: Hint: The square of an odd number is congruent to 1 mod 4. === Subject: Re: Number theory problem, or something > Um. Naturals can have values 0, 1, 2 or 3 modulus 4. Squares of even > naturals can only have value 0 modulus 4. Squares of odd naturals are > squares of (even naturals + 1), so if n is an even natural, the square > of (n+1) is equal to n^2 + 2n + 1. Both n^2 and 2n have value 0 > modulus 4, so the square has value 1 modulus 4. 2n isn't 0 mod 4. It is 0 or 2 mod 4. /David === Subject: Re: Number theory problem, or something > Um. Naturals can have values 0, 1, 2 or 3 modulus 4. Squares of even > naturals can only have value 0 modulus 4. Squares of odd naturals are > squares of (even naturals + 1), so if n is an even natural, the square > of (n+1) is equal to n^2 + 2n + 1. Both n^2 and 2n have value 0 > modulus 4, so the square has value 1 modulus 4. > 2n isn't 0 mod 4. It is 0 or 2 mod 4. Joona said: ... so if n is an even natural ... -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: Number theory problem, or something > Joona said: > ... so if n is an even natural ... Oh, true. /David === Subject: Re: Algebraic number theory books > > Can anyone suggest a good book(s) on algebraic number theory? >> I'm partial to Daniel Marcus's _Number Fields_; Ireland and Rosen's _A >> Classical Introduction to Modern Number Theory_ is also a good first step. > I warmly recommend Marcus as well. In addition I have found Problems > in Algebraic Number Theory, By Esmonde and Murty, quite entertaining > and useful. And of course, one should always study the masters - > Hecke's classic is just that. > There's one book that has for long intrigued me, but I never got > around to actually buying it (it's expensive!) or spending serious > time with it in a library - this is the fairly recent book by Neukirch > (trans. by Schappacher). Can anyone comment on this one? It is a very beautiful book with well-deserved rave reviews (e.g. see below). -Bill Dubuque ---------------------------------------------------------------------------- -- Neukirch, Jurgen Algebraische Zahlentheorie. (Algebraic number theory). (German) [B] Berlin etc.: Springer-Verlag. xiii, 595 S. (1992). [ISBN 3-540-54273-6/hbk] http://www.emis.de/cgi-bin/zmen/ZMATH?type=html&an=0747.11001 During the recent decades, algebraic number theory has undergone an enormous and far-reaching development. This process is essentially characterized by the fact that the geometric point of view has gained a central significance in the arithmetic theory of fields. The framework of modern algebraic geometry, its general concepts, methods, and results as well as its complex-analytic aspects have penetrated algebraic number theory in a natural way and to a growing extent. The interrelation between classical algebraic number theory and modern algebraic geometry has led to a particular branch of mathematics, which is commonly called ``arithmetic algebraic geometry''. The recent epoch-making results concerning, for example, the Weil conjectures, the Mordell conjecture, the Birch-Swinnerton-Dyer conjecture, and other long-standing problems have been achieved by systematically using methods of (arithmetic) algebraic geometry, in the course of which the algebro-geometric viewpoint has crucially contributed to a deeper systematization, conceptual unification, and methodical clarity in algebraic number theory as a whole. Eventually, the recent methods and results of arithmetic algebraic geometry have also proved to be of increasing significance for constructing models of physical theories, in particular quantum field theories. In regard to this development, it is unquestionable that algebraic number theory has reached a new stage of both theoretical compactness in itself and intertwining with diophantine geometry. Consequently, it became highly desirable that this new quality would be suitable reflected, in a fundamental and updated manner, in a comprehensive textbook on algebraic number fields itself. On the one hand, there are many excellent textbooks on algebraic number theory, and there are now also a few treatises on topics in arithmetic algebraic geometry. On the other hand, however, the established textbooks on algebraic number theory are rather aged, in the meantime, and mainly impart the classical material, nevertheless from different points of view and in varying degree of theoretical uniformity, whereas the available texts on arithmetic algebraic geometry are, throughout, fairly advanced and specific, already assuming that the reader is sufficiently familiar with both the fundamentals of classical algebraic number theory and the basic framework of modern algebraic geometry. The present book has as its aim to resolve this discrepancy in the relevant textbook literature and, apparently for the first time in such a throughgoing way, to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry. In contrast to the existing textbooks, the present work focuses on elaborating thoroughly the higher theoretical completeness of the subject, with special emphasis on the general functorial and algebro-geometric methods. In this sense, the author has given preference, wherever possible, to the powerful general (and generalizable) geometric aspects, instead of the many common -- though fascinating and ingenious -- special artifices and ad-hoc methods in number theory. Despite this exacting program, the book remains an introduction to algebraic number theory for the beginner, who is required to only have the basic knowledge in university algebra, and it develops the modern abstract framework only as far as necessary for a profound understanding of the rich classical theory of algebraic number fields in its modern, unifying geometric interpretation. Chapter I presents the foundations of the global theory of algebraic number fields from the classical ideal-theoretic point of view. In addition, this chapter includes, as a first special feature, a geometric approach to algebraic orders in number fields. This already illustrates the geometric background, namely by the analogy between algebraic orders and the theory of singular algebraic curves; and it motivates, in the sequel, the introduction of one-dimensional algebraic schemes (curves) in the light of their arithmetic examples. Chapter II deals with the local aspects of number fields, that is, with the classical theory of valuations. This encompasses sections on p-adic numbers, p-adic valuations, general valuations and valuation rings, completions, local fields, Henselian fields, unramified and tamely ramified extensions, the Galois theory of valuations, and an introduction to higher ramification groups. The basic material developed so far is completed in the following Chapter III. This chapter represents, in particular, the aforementioned new character of the textbook under review, in that it turns towards the modern algebro-geometric approach to algebraic number fields. The author discusses the classical concepts and results in number field theory from the actual viewpoint of (one-dimensional) Arakelov theory which, for its part, essentially consists in transferring results from algebraic geometry over algebraically closed fields (i.e., from the function field case) to the number field case. This geometric viewpoint is not entirely new; it can be traced back to an earlier paper by A. Weil [Sur l'analogie entre les corps de nombres algebriques et les corps de fonctions algebriques (1939) in: Collected Papers. Vol. 1 (Springer 1979; Zbl 0424.01027)]. However, the decisive breakthrough in understanding this analogy was provided by S. Yu. Arakelov in 1974, who succeeded in constructing an intersection theory for divisors in arithmetic surfaces, which had precisely the crucial properties known from the theory of nonsingular projective complex surfaces in algebraic geometry [cf. Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1179-1192 (1974; Zbl 0355.14002)]. Arakelov's basic idea of completing the set of points (and divisors) of an arithmetic surface by including those at infinity (i.e., the Archimedean places of the arithmetic ground field) and considering Hermitean structures on the extended sheaves, so that complex analysis got naturally involved, gave the possibility to transpose the fundamental geometric invariants (e.g., divisor class groups, Picard groups, Chow groups, Grothendieck groups, cohomology groups) adequately to arithmetic-geometric objects. As for the one-dimensional case of arithmetic curves, i.e., for spectra of rings of integers in number fields, the analogous concepts of Arakelov's theory provide a new, geometrically disposed insight into classical algebraic number theory. This was sketchily pointed out, for the first time, by L. Szpiro in 1983 [Seminaire sur les pinceaux arithmetiques: La conjecture de Mordell, Asterisque 127, 11-28 (1985; Zbl 0588.14028)] and, later on, generally worked out by E. Hubschke} [Arakelovtheorie fur Zahlkorper, Der Regensburger Trichter 20, Diss. (1987; Zbl 0743.14020)]. Chapter III of the present textbook provides a detailed account of this modern viewpoint in the one-dimensional case and, in addition, a rigorous embedding of this arithmetic subject into the general (geometric) Grothendieck-Riemann-Roch theory for algebraic schemes. This careful, comprehensive introduction takes the reader to the forefront of current research in arithmetic geometry, in that it enables him to study the very recent developments in the higher-dimensional case [e.g., G. Faltings, Lectures on the arithmetic Riemann-Roch theorem, Ann. Math. Stud. 127, Princeton Univ. Press (1992; Zbl 0744.14016)] by already knowing the philosophy from the example of arithmetic curves. The remaining Chapters IV-VII, which form the second half of the book, are devoted to two other central areas of algebraic number theory: class field theory and the theory of algebraic zeta functions and L-series. More precisely, Chapters IV-VI consecutively treat, in an introductory yet very thoroughgoing manner, the main contents of general class field theory, local class field theory, and global class field theory. The presentation given here differs from the one offered in the author's earlier standard textbook ``Class Field Theory'' (Springer-Verlag 1986; Zbl 0587.12001), in that the material is not only structurally appended to the foregoing introductory chapters, but also arranged in another way which leads again to a higher level of coherency, theoretical compactness, methodical effectivity, and alluding actual relatedness. Moreover, the presentation is slightly more detailed than in other textbooks on class field theory, including the author's aforementioned standard work, and particularly rich in illustrating complements, hints for further study, and concrete examples. A peculiar feature is certainly given by the early introduction and use of profinite groups and infinite Galois theory (in Chapter IV), as well as by the thorough treatment of formal groups, the Lubin-Tate theory, and higher ramification theory. This is perhaps more than a textbook is expected to offer, but perfectly serves the author's ambitious aim of exhibiting the actual width of modern algebraic number theory, together with its relations to classical problems. The concluding Chapter VII on zeta functions and L-series is another outstanding advantage of the present textbook. This classical topic, which relates complex analysis and number theory in a fascinating and deep-going way, has gained new interest and central significance during the recent decades, too. Since Hecke's pioneering work on L-series [cf. E. Hecke, Mathematische Werke, 2nd. ed. (Gottingen 1970; Zbl 0205.28902)], which is still barely accessible to the non-specialist, this subject had never been updatedly included in a general textbook on number theory. Certainly, there are some special treatises on L-series, for example J. Tate's famous Ph. D. thesis (Princeton 1950) which provides an ingenious approach to Hecke's L-series by methods of harmonic analysis [as for a published version, cf. J. W. S. Cassels and A. Frohlich (ed.), Algebraic Number Theory, Proc. Int. Conf., London, New York, Acad. Press (1967; Zbl 0153.07403)], or the omnibus volume [A. Frohlich (ed.), Algebraic Number Fields (L-functions and Galois properties). Academic Press (1977; Zbl 0339.00010)]; however, here the author has not only included this topic in a general textbook on algebraic number theory but, beyond that, he has also entered upon the very rewarding and creative task of elaborating the first modern version of Hecke's original approach, followed by a likewise comprehensive account on the Artin theory of L-series and their functional equations. This updated representation of Hecke's theory fills a longstanding gap in number theory. Apart from its didactic value, it is an essential contribution of the author to the further theoretical consistency of number theory, in particular, and of pure mathematics in general. Likewise, the detailed treatment of the Artin L-series, in this context, is new and unique in the textbook literature, also in regard to included hints to the recently discovered (infinite-dimensional) geometric generalization of L-series [cf., e.g., M. Rapoport, N. Schappacher, P. Schneider (ed.), Beilinson's Conjectures on Special Values of L-Functions, Perspectives in Mathematics 4, Boston, Academic Press (1988; Zbl 0635.00005)]. Altogether, the present work is an outstanding textbook. It is, without any doubt, the most actual, systematic and theoretically comprehensive textbook on algebraic number field theory available. Every section goes with a set of selected exercises. Many of them are intended as an invitation to further studies, just to complete the material by related (or deeper-going) topics. This provides the reader with additional guidance to the current literature, besides the excellent preparation of the text itself, which is enhanced by numerous motivating explanations, didactic recalls, and indications to related recent developments. For all that, the book is widely self-contained and, despite the vast material, of great clarity and intelligibility. It is perfectly suited as a modern reference book, too, because of its nearly encyclopedic character, and it is likewise a distinguished source for lectures and seminars. In this preface, the author proposes several variants for teaching by using this book. Undoubtedly, Professor Neukirch's recent book will quickly become a standard text on contemporary algebraic number theory, likewise for students, teachers, researchers, and interested non-specialists in mathematics or theoretical physics. [ W.Kleinert (Berlin) ] ---------------------------------------------------------------------------- -- Neukirch, Jurgen Algebraic number theory. Transl. from the German by Norbert Schappacher. [B] Grundlehren der Mathematischen Wissenschaften. 322. Berlin: Springer. xvii, 571 p. $99.00 (1999). [ISBN 3-540-65399-6/hbk; ISSN 0072-7830] http://www.emis.de/cgi-bin/zmen/ZMATH?type=html&an=0956.11021 This is an English translation of a book, whose German original has been reviewed (Zbl 0747.11001). It brings in seven chapters a well-written introduction into modern number theory. The first chapter presents the fundamental results including ideal theory (with Kummer's factorization theorem), Dirichlet's unit theorem, finiteness of the class-number and Hilbert's ramification theory. One finds here also a study of orders, which is welcome, as this topic is usually omitted in most textbooks. Also a link to algebraic geometry is provided: one-dimensional schemes are defined, as well as the Picard and Chow groups and a connection with the theory of function fields is sketched. The second chapter brings valuation theory, including a study of Henselian fields and their extensions and in the next chapter this is applied to algebraic number fields. The theory of the discriminant and different is presented, Arakelov ideals and Arakelov class group are considered and a proof is given of an analogue of the Riemann-Roch theorem, based on A. Weil's definition of the genus for algebraic number fields. Then metrized modules over rings of integers are introduced and a formalism, which was introduced by A. Grothendieck in the case of algebraic varieties, is developed for these modules. After defining compactified Grothendieck groups (on which the tensor product induces a ring structure), the Chern character and Todd classes this construction culminates in the Grothendieck-Riemann-Roch theorem, relating, for a finite extension L/K of algebraic number fields, the Grothendieck groups corresponding to K and L. As the author states, this result ``integrates completely the theory of algebraic integers into a general programme of algebraic geometry as a special case.'' The next two chapters present local and global class field theory, modelled upon a previous work of the author [``Class Field Theory'', Springer Verlag (1986; Zbl 0587.12001)] but including certain modifications and fresh examples. The last chapter deals with zeta functions and L series. Contrary to most treatments of this topic the author does not use the approach based on harmonic analysis, but proceeds with a careful presentation of ideas of E. Hecke. This seems to be the first modern exposition of Hecke's method. This book is a most welcome addition to the literature and will serve as a learning tool for years to come. The translator made a splendid job, preserving the lucid style of the original. [ Wladyslaw Narkiewicz (Wroclaw) ] ---------------------------------------------------------------------------- -- Neukirch, Jurgen Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher. With a foreword by G. Harder. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322. Springer-Verlag, Berlin, 1999. xviii+571 pp. $99.00. ISBN 3-540-65399-6 http://www.ams.org/mathscinet-getitem?mr=2000m:11104 ---------------------------------------------------------------------------- -- This book is the English translation of Neukirch's Algebraische Zahlentheorie, which appeared in 1992 [Springer, Berlin; Zbl 747.11001]. The book, which stands out due to its style and its content, will now gain an even larger readership in the English-speaking community. The book covers all standard topics in local and global algebraic number theory, and much more. It confesses to the policy of building the theory for its own sake, not avoiding abstraction and generality. This it does in a very efficient way. For instance, the first chapter develops the standard concepts (Dedekind rings, class groups, units) practically from scratch, providing all the required commutative algebra as well. The second chapter presents the theory of local fields, again from the very beginning, including a wealth of material, some of which is not frequently found in books, e.g., a general discussion of Henselian fields. The theory of differents and discriminants expounded in the third chapter is placed in a more general context, so-called Riemann-Roch theory. This adds a geometric flavor. True, many of the analogies between number theory and geometry can only be appreciated by readers who are already somewhat conversant with algebraic geometry, but for those who are not, Neukirch's exposition provides an excellent motivation to learn more about the latter. The geometric side is further stressed, and this is a unique feature of the book, by studying metrized modules, replete (or Arakelov) divisors, and replete (or Arakelov) class groups. The basic idea is to deal simultaneously with the finite places and the infinite places of a given number field. Roughly speaking, Neukirch deals with Arakelov theory in dimension 1; Arakelov theory in dimension 2 played a decisive role in Faltings' proof of the Mordell conjecture in 1984. These geometric considerations can be bypassed by readers who want to go on quickly to the rest of the book. Chapters 4-6 give a complete treatment of class field theory, using the new approach Neukirch discovered himself some years before the publication of Algebraische Zahlentheorie [see Class field theory, Springer, Berlin, 1986; homomorphism d: G -> ~Z and a G-module A endowed with a Henselian valuation with respect to d, one constructs in an elementary albeit involved way the reciprocity homomorphisms; if A satisfies the so-called class field axiom (a statement involving zeroth and (-1) st cohomology of A), then the reciprocity local and global class field theory are deduced, and the classical formulation of global class field theory, using ray class groups instead of the idele class group, is presented as well. Neukirch's approach minimizes the cohomological apparatus needed to construct class field theory. The final seventh chapter deals with zeta functions and L-series, and it is another didactic masterpiece. The author deliberately proves and re-proves some results several times over in increasing generality, for example, the functional equation. As stated in his preface, he goes back to the classical proofs of Hecke, using theta transformation formulas (instead of following Tate's thesis), which apparently have not yet received a modern treatment in book form. (The reviewer is only aware of mimeographed notes by Halter-Koch et al., Essen, 1976, which follow this route; however, these notes do not cover L-series with Grossencharaktere, which Neukirch's book does.) The reviewer thinks that this book is ideal for any student of algebraic number theory, no matter whether she is a serious-minded beginner or already advanced. It is remarkably self-contained: perhaps in the proof of Hilbert's Theorem 90 on p. 281 the linear independence of automorphisms (a fundamental fact) could have been re-proved, but everywhere else all ingredients of proofs, within reason of course, are included. The author constantly guides the reader, pointing out what is important, and highlighting connections between topics. The book can be used for independent study, but lends itself very well to classroom use. The reviewer once taught parts of Chapters 4 and 5, a very technical matter, to a small group of students with not too much background in algebraic number theory, and it is clear that Chapter 1 makes an excellent introductory course, to which more material from later chapters can easily be added if time permits. There are numerous exercises, some of them demanding. As far as the reviewer (who is also German) can judge, the translator N. Schappacher has done a fantastic job, rendering Neukirch's precise and sometimes elaborate style very accurately, and retaining the flavor of the original. This accuracy even applies to minute details: for instance, people quoted in the text often have the epithet mathematician attached to their names, just as in the German original; this is obviously intended as a high distinction. The reader is urged to read the Translator's Note, which explains the history and the particular challenge of the translation better than any comments of the reviewer could. The foreword by G. Harder should not be missed either. from this publisher. The only typographical matter worth mentioning is that in the bibliography the strange accents in references [9], [12], [88] should have come out as a small gothic p. As a last finicky comment, note the widespread inconsistent transliteration of Borevicz (should be Borevic) and Safarevic in reference [14] and elsewhere. Even though Neukirch's renowned book probably needs no further recommendation, the reviewer would like to recommend it anyway, hoping that curiosity, the reputation of the book, or even perhaps this review will induce many people to look at it; they will be richly rewarded. In the German preface Neukirch expresses the hope that (quote abridged) dissatisfaction with the exclusion of Iwasawa theory and etale cohomology would be strong enough to bring about a sequel volume. Neukirch did not live to see this project finished, but the sequel volume Cohomology of number fields by Neukirch, A. Schmidt and K. Wingberg has just appeared as Volume 323 in Springer's Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences) series [Springer, Berlin, 2000; MR 2000j:11168]. Reviewed by Cornelius Greither === Subject: Re: Probability of a Run >As far as I can see, if q is the probabilty of a loss and P(n,m) is >the probability of at least 1 run of at least length n in m trials >then >P(n,m) = 0 if m < n, > = q^n.(1 + (1-q).( (m-n) - sum(i=n..m-n-1, P(n,i) ) ). >Having read the material in this thread, I assume I've made a simple >mistake or that a similar formula is given in Feller but not >considered a practical method of calculation in 1968. > I'm afraid I can't evaluate your formula without a detailed > explanation of the logic behind it, but the change in the computing > environment is precisely the reason I am seeking a new solution at > this time. Nobody uses Weddle's rule (for numerical integration) any > more, but it was a Godsend in the days of pencil and paper arithmetic. > A train of logic used by both Burnside and Uspensky, and which might > have originated with DeMoive, leads to the following difference > equation > u(m + 1) = u(m) + (1 - u(m - n)) (1 - p) p^n > where > u(m) = Pr that a run of n or more has happened by the m'th trial. > The initial values are > u(n) = p^n and u(m - n) = 0 if m < 2n > Burnside and Uspensky develop solutions to the difference equation, > find them unwieldy, and go on to derive approximations. Today, any > kid with a TI-83 or a home computer can calculate particular solutions > numerically. Here is the logic: > [skip] Your formula u(m + 1) = u(m) + (1 - u(m - n)) (1 - p) p^n is deducible from mine, for m > n. Just calculate P(n,m+1)-P(n,m) in my notation, swap P(n,m) to u(m) and q to p and you'll get the same. I had just about realized this couldn't be the difficult part of the problem! I'm slightly puzzled as to why you said I want to calculate the probability of this if you know the answer, unless you're not a kid with a TI-83 or a home computer! Here's some VBA code I used to evaluate it, for m not too big. Public Function Probnmp(n As Long, m As Long, p As Double) Dim C() As Double, result As Double, pn As Double, q As Double Dim i As Long, j As Long, next_j As Long If n <= 0 Or p <= 0 Or p >= 1 Then Probnmp = [#VALUE!] ElseIf n > m Then Probnmp = 0 Else ReDim C(n + 1) For i = 1 To n + 1 C(i) = 0 Next i pn = p ^ n q = 1 - p j = n For i = n To m If j > n Then next_j = 1 Else next_j = j + 1 End If result = pn * (1 + q * ((i - n) - C(next_j))) C(next_j) = C(j) + result j = next_j Next i Probnmp = result End If End Function For example, with p = 0.5, the smallest number of trials so that you have at least a 50% chance of 10 consecutive events is 1421, 20 consecutive events is 1453639, 30 consecutive events is 1488522243. I modified the code to stop when result exceeded 0.5. The last calculation was not quick but then 1488522243 would seem to be quite a lot of turns at the game! The P(n,m) formula I gave allows you to repeatedly substitute more and more complicated formulae back in to give P(n,m) = 0, if m < n = q^n(1+(1-q)(m-n)), if m < 2n = q^n(1+(1-q)((m-n)-q^n((m-2n)+(1-q)*(m-2n-1)(m-2n)/2)), if m < 3n = q^n(1+(1-q)((m-n)-q^n((m-2n)+(1-q)*((m-2n-1)(m-2n)/2)-q^n((m-3n-1)(m-3n)/2)+ ( 1-q)*((m-3n-2)(m-3n-1)(m-3n)/2/3)))))), if m < 4n = q^n.(1+(1-q).( (m-n) - sum(i=n..m-n-1, P(n,i) ) ) with values substituted back in for P(n,i) from above 3 lines. = ... The summations are easy enough using (n+1)sum(i=1..m, product(j=1..n, i+j-1)) = product(j=1..n+1, m+j-1) and it's not difficult to see the general pattern of the formula. Again, I expect all this is well known so I'm not sure how promising this is as a method of calculation although it does seem to give more and more accurate approximations. For example, when calculating 20 consecutive events in 65000 trials with q = 0.5 the expression which is exact for m up to 80 has a relative error of just 1.25e-6, the expression which is exact for m up to 100 has a relative error of 7.71e-9, the expression which is exact for m up to 120 has a relative error of 5.15e-11. The error when calculating 20 consecutive events in 1453639 trials with q = 0.5 with the formula which is exact for up to 120 trials has a relative error of 2.8e-4. I haven't programmed up this approximation to arbitrary accuracy. Perhaps you already know all the problems with it! Ian Smith === Subject: Re: I can't stand it anymore What gave you the idea that I was upset about the term Asian? My only point was that IQ tests to determine people's intelligence > are > bs. Ah, of course. It would be far better indeed to administer > intelligence tests rather than IQ tests to determine people's > intelligence, right? Really, you sound like someone who is disappointed by their > own results on such a test. > I knew that someone would come up with that stupid opinion. It just > happens to be you. Sorry about that > Oh, boo hoo. It was an observation, not an opinion. Do you > understand the difference? Should we test you on that? Test, test, test! Which test invented by human can possibly be perfect? You still haven't got my point, have you? > Don't you realize that us Asians don't need the tests invented by > non-Asians to find out whether we are intellgent or not? > So? You have your own tests then? Fine. Get on with it. Still, no clue about my original post? > It was only at the turn of 2oth century that the Western World > started to reliaze that Asians are not inferior to them as they have > been told by the 19th century European racists who believed that they > were superior to all other humans. > As someone was throwing away a bunch of old books, piling up outisde > the lab I was doing my research worl in grad school, I saw just such a > book written by a European scientists. Anyway, > before that time, did the opinion of the West affected the Asians? > Ever read the book soverign Individual. The author stated that > white people hate Asians but that didn't affect .... He was just > trying to say how the Asians knew who they were and didn't act based > on others' opinion. > We certainly don't need these IQ tests to find out about our > intelligent level. > So what IQ tests would you apply? Stop whining and get on with it. Obviously you cannot understand my point. Let me repharse ot for you. A people secure in themselves do not need stupid tests to know that they are intelligent people, i.e no amount of brainwashing from the colonial masters that the natives were inferior to the colonial masters has affected these people because they knew who they were. Now..accuse me that I still have grudge against the colonialists. === Subject: Re: I can't stand it anymore > Obviously you cannot understand my point. Let me repharse ot for you. > A people secure in themselves do not need stupid tests to know that > they are intelligent people, i.e no amount of brainwashing from the > colonial masters that the natives were inferior to the colonial > masters has affected these people because they knew who they were. Whether an individual considers him or herself to be intelligent is mostly irrelevant from the point of view of someone else looking to hire or admit candidates of of a desired standard to their organization. That is why there are intelligence tests. If you want to play in the sandbox with the big kids, you have to play by their rules or head home to mommy. She'll be more than happy to reinforce your ego no matter what your handicap. > Now..accuse me that I still have grudge against the colonialists. Why? Did you have a grudge before? Or are you just trying to pick a fight? === Subject: Re: I can't stand it anymore > amanda replied: > >What gave you the idea that I was upset about the term Asian? My only point was that IQ tests to determine people's intelligence are >>bs. >>Ah, of course. It would be far better indeed to administer >>intelligence tests rather than IQ tests to determine people's >>intelligence, right? >>Really, you sound like someone who is disappointed by their >>own results on such a test. > > I knew that someone would come up with that stupid opinion. It just > happens to be you. Sorry about that > > Don't you realize that us Asians don't need the tests invented by > non-Asians to find out whether we are intellgent or not? > It was only at the turn of 2oth century that the Western World > started to reliaze that Asians are not inferior to them > It's interesting that in the Asian countries, the inferiority of > non-Asians is still widely accepted and believed. Now, where did you get that idea? I didn't say that, did I? > There is no attempt > for or desire for reciprocity. > as they have been told by the 19th century European racists > Is there an Asian country which is not racist? Whya re you talking abput cpuntries. Why not individuals? > Just look at the nice > things high Japanese govt officials have said about Americans. > who believed that they were superior to all other humans. > As do all Asian races currently. Currently is the keyword but you are wrong to generalize all Asians. > But somehow you don't seem to see > anything wrong with this. That's your opinion. My saying that Asian do not need a test invented by someone else to determine their intelligence level, it didn't imply that I think those who invented that test were inferior. If you can't understand that, that's your problem. > This entire post would be ironic were it > not so hypocritical. That also is your opinion. > Why don't you also complain about Asian racism? Why should I complain about it here? I have complained many times about Indian caste system somewhere else. > Probably because you support it. Now, now. Probably is not the same as you support it, does it? As a minority in the country I was born (my parent were born and raised thare too) who suffered discrimination and were not considered as full-citizens, you must be joking accusing me as a racist. Beside, as a person whose blood covers basically almost all races of the world (not through Caucasoid and Negroid per se but through Indo-Aryan which you probably have no clue about), including semite (not jewish), you gotta be kidding to accuse me as a racist. > It's clear that you will profit > from any pro-Asian racism. What are you talking about? When you said Asian, are you thinking of the slant-eyes, flat nose, dark, straight haired people OR are you referring to ALL THOSE FROM ASIAN COUNTRIES which would include Arabs, Persians, people from Afghanistan some of who are Caucasians, the Mongoloid group, i.e Chinese and Japanese and more, the Indo-Aryans of India, etc.? Note that I have excluded the people of Turkey. I was/am going with the second one, WHICH WAS MY WHOLE POINT OF criticizing the use of the term Asian in IQ test, duh... > It's almost certain that you have. The most stupid thing I have ever heard. What I have had was getting pulled over by the red-neck police in Houston for driving a beat-up car in my student days because I look a bit Hispanic or South Americans. > Rich Note: I will not be wasting any more time defending groundless accusations. === Subject: Re: I can't stand it anymore >I have been biting my tongue about the IQ test but I can't any more. >How reliable is a test that use the term Asian to represent the most >diverse of ethnic and cultural groups? IQ tests are not particularly reliable, but what does this have to do > with it? What test is using that term? And how is any of this relevant > to these newsgroups? If you want to know more about the diversity, ask me and I will give >you tons of specific examples. > ... Your examples may be interesting, but what do they have to do with > sci.math? > Sci.math was one of the group listed in the thread where I saw the > discussion of IQ test. I had no idea who were from which group in that > thread and so decided to posts in all 3 groups. The only science group > I posted before was sci.chem group. > I don't browse sci.chem group regularly either. So when I broswe the > posts once in a while and see some supposedy intelligent people from > these 3 groups talking about IQ tests as a ver important tests (with > some seeing it as the ultimate test to determine human intelligence), > I get very disappointed. > So let's see; in your opinion, talking about IQ tests is > an indication of low intelligence or some social backwardness? Unbelievably childish response. === Subject: Re: I can't stand it anymore > I don't browse sci.chem group regularly either. So when I broswe the > posts once in a while and see some supposedy intelligent people from > these 3 groups talking about IQ tests as a ver important tests (with > some seeing it as the ultimate test to determine human intelligence), > I get very disappointed. > So let's see; in your opinion, talking about IQ tests is > an indication of low intelligence or some social backwardness? > Unbelievably childish response. So then, why are you disappointed when you see supposedly intelligent people talking about IQ tests? === Subject: Re: How do you integrate the function f(x) = x/(tanx) [0,pi/2]? > Ray Steiner > I worked on the indefinite integral a bit. > We want to find I= int(x/tan(x) dx). > Let u= tan x, x= arctan(u), dx= du/(1+u^2). > Then we get > int( arctan(u)/u(1+u^2) du). > Now separate 1/ u(1+u^2) into partial fractions. Then we get > I= int(arctan(u)/u) du ) - int( u*arctan(u)/(1+u^2) du). > The first integral is a standard nonelementary integral. One can use > arguments > similar to those in Wiener's posts to prove this. But it can be > written as > 1/2*I*dilog(1+I*u)-1/2*I*dilog(1-I*u). > The second one also seems to be nonelementary, but I haven't a proof > as yet. > Interestingly enough, I managed to reduce it to integrating > (arctan(u))^2 and > Maple was unable to do this last integral. > ... > Ray Steiner > I doubt arctan^2 or x/tan x has an elementary antiderivative. The > dilogarithm > dilog(x) = sum_{n from 1 to oo}(x^n)/(n^2) > looks like as good a way as any to express such antiderivatives. > Can we say something general about the definite integral (over a period or > half-period) of x y'/ y where y is a periodic function of x? If y=sin x, as > before, we have an infinite product for y: > sin pi.x = xprod_{n from 1 to oo}(1-xx/(pi.n)^2) > and its logarithmic derivative y'/y is the partial fraction breakdown of cot > x. Maybe it generalizes a bit. > LH I came up with another way of showing that x/tan x has no elementary antiderivative. We need only one result from Wiener's 1997 paper: arcsin(x)/x does not have an elementary antiderivative. Let I = int(x/tan x dx)= int (x cot x dx) Use integration by parts to get I = x ln(sin x) - int( ln(sin x) dx) Let I2= int( ln(sin x) dx) Let u= sin x, x = arcsin(u), dx = 1/sqrt(1-u^2) du Then I2= int ( ln(u)/sqrt(1-u^2) du) Finally, use parts again to get I2= ln(u) arcsin(u) - int(arcsin(u)/u du). So, by Wiener's result, the original integral is not elementary. More results: By exactly the same method one can show that I3 = int (x tan x dx) is not elementary. Now, let's substitue u= tan x, x = arctan u, dx= 1/ (u^2 + 1) du in I3. Then it reduces to I4 = int( u*arctan(u)/(1+u^2) du). so the second integral of my previous post is non-elementary. Finally, consider I5= int ( (arctan(x))^2 dx). By parts, one can reduce it to integrating I4, so I5 is also non-elementary. Ray Steiner === Subject: Re: JSH: Clearing up confusion > It seems to me that there's some confusion about constant terms with > polynomials and factors of polynomials, so it might help for me to > post in an attempt to clear up the confusion. > Let's say you have the polynomials x+1 and x+7, and you multiply to > get > (x+1)(x+7) = x^2 + 8x + 7 Ok so far. > so the constant term is 7, but of course, the polynomial doesn't > necessarily have 7 as a factor. That's true, the polynomial does not necessarily have 7 as a factor. But if you EVALUATE the polynomial at various values of x, the result may well have 7 as a factor. For example if f(x) = x^2+8x+7, then f(0) does have 7 as a factor. > Also, dividing back through by x+7 gives you x+1, so the constant term > after the division is 1, whereas before it was 7. > Ok, yeah, it's kid stuff, but remember, I'm facing people who seem to > be confused on these issues, which is why I'm posting. We're all in agreement so far. > But now consider having 7 itself as a factor. 7 itself as a factor of what? > Notice that it's different from x+7 as one is a polynomial while 7 is > a constant. x+7 is a 1-degree poly, and 7 is a 0-degree poly. > Can you ever divide 7 off as a variable? > Some of you seem intent on arguing as long as you can that you can. > However, division of constants by constants is a constant operation, > and is NOT variable dependent. > No matter how much you wish it is, it's not, and refusing to accept > that it's not merely makes you a crank who refuses to believe in > mathematical logic. What are you talking about? > So you may wonder about relevancy. > Well I have an expression where I divide off 49, a constant factor of > a polynomial P(x), which changes you from having 7, 7 and 22 as > constant terms of factors of that polynomial to having 1, 1, and 22 as > the constant terms of the factors of P(x)/49, so it should be > straightforward. Now you are babbling. You and I were in total agreement right up until this paragraph, which has no context and makes no sense. Why don't you keep on going with your x+7 example and say what it is you're talking about? === Subject: Re: JSH: Clearing up confusion > It seems to me that there's some confusion about constant terms with > polynomials and factors of polynomials, so it might help for me to > post in an attempt to clear up the confusion. > Let's say you have the polynomials x+1 and x+7, and you multiply to > get > (x+1)(x+7) = x^2 + 8x + 7 > so the constant term is 7, but of course, the polynomial doesn't > necessarily have 7 as a factor. > Also, dividing back through by x+7 gives you x+1, so the constant term > after the division is 1, whereas before it was 7. > Ok, yeah, it's kid stuff, but remember, I'm facing people who seem to > be confused on these issues, which is why I'm posting. > But now consider having 7 itself as a factor. > Notice that it's different from x+7 as one is a polynomial while 7 is > a constant. > Can you ever divide 7 off as a variable? > Some of you seem intent on arguing as long as you can that you can. > However, division of constants by constants is a constant operation, > and is NOT variable dependent. > No matter how much you wish it is, it's not, and refusing to accept > that it's not merely makes you a crank who refuses to believe in > mathematical logic. > So you may wonder about relevancy. > Well I have an expression where I divide off 49, a constant factor of > a polynomial P(x), which changes you from having 7, 7 and 22 as > constant terms of factors of that polynomial to having 1, 1, and 22 as > the constant terms of the factors of P(x)/49, so it should be > straightforward. Unfortunately, you have still not proven that the individual factors distribute themselves this way for values of 'x' other than x = 0. Certainly P(0) suppresses every contribution from the polynomial except the constant term. But P(1) consists of the *sum* of all coefficients of the polynomial and without performing the calculations, you can't just assert that the partition you claim actually occurs. In fact, P(1)/49 = 281412 so your discussion of 1, 1, 22 isn't even relevant. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === === Subject: Re: NOVA strings and branes your point is taken, because we didn't have to use those bombs on Hiroshima and Nagasaki; those were plain terrorism, along with teh British-US fire-bombings of Dresden, Tokyo etc. (as specified in the Strategic Bombing Survey docs, and at Tavistock (British Psy-ops)). now, it may be totally obvious to a student of government (sic; do you know the meaning of the word, republic?... R don't), but it's just an inference based upon decades of speculation/hype. you do not criticize (or even note) Corso's bald premise, that human beings cannot create ideas. Hell, no; RADAR, transistors, graphite, fiber-optics ... it *all* came from that little pile of crap at Roswell. I know, because I was the only guy who was in charge of handing it out to the assmebled brethren of the MIC!... yeah, right; just say zero point energy and wave your hands about. now, do you know the cache that Roswell has with WW2, or are you one of the residents of Denial, NM? > I think the response is TOTALLY obvious to any student of government. > It has absolutely NOTHING to do with the usual arguements of public > panic and the like and EVERYTHING to do with advantage such technology > would give one country over the others. It's a power trip at LEAST > on the order of the Atomic bomb. Was the atomic bomb kept secret > (at least until they blew a few up)? You bet. Well, except from > the countries with spies in the project. The same goes for UFOs > and more! > And it's even BIGGER than that. Let's consider zero point energy. > If there is indeed zero point energy that can be easily tapped, > then this spells doom for oil/gas/coal and all other fuel industries. > Either the technology is real and being kept secret. In which > case you'll never prove it, because even if you had hard proof, > you'd just disappear if you tried to show it. Or there is nothing > there so there is nothing to prove. > were true, and I am not saying it is true, the fact is that none of the > intelligence people who might have that stuff have the slightest > understanding how that hypothetical and/or alleged alien technology > actually works. It's The Sorceror's Apprentice. > Think so? Back-engineering is quite a separate thing from a true > understanding of physics. And I see no way you can be sure that > no human physicist might understand some of this stuff. You > have no idea who or what is involved with black projects. --ils duces d'Enron! http://www.wlym.com/antidummies/part17.html === Subject: Re: NOVA strings and branes > your point is taken, because we didn't have to use those bombs > on Hiroshima and Nagasaki; those were plain terrorism, > along with teh British-US fire-bombings of Dresden, Tokyo etc. > (as specified in the Strategic Bombing Survey docs, and > at Tavistock (British Psy-ops)). Bull. It was war. The Germans and the Japs started it. The Allies finished it. The bombing of Pearl Harbor was terrorism. The bombing of Hiroshima and Nagasaki was just and condign revenge. We were at -war-. In a war you kill your enemies and bust up their . That is what war is all about. Revisionist dunce! Bob Kolker === Subject: Re: is there a general way to approach series? >i feel that everytime i look at a new series problem i am starting >from the beginning and struggle to finish it. is there a general >approach to simple series? >for example, why does the divergence of Sum[an] imply the divergence >of Sum[an/(1+an)]. > It doesn't. (You're also given that an >= 0 in the exercise, right? > _Then_ it does. Hint: either an tends to 0 or an does not tend to 0.) >it seems to me that self study from rudin's book is nearly impossible. > Possibly you need to read it much more slowly and carefullly. > Or possibly you're not ready for a book that makes the sort > of demands on the reader that Rudin does - you could try > something like Ross Elementary Analysis: the Theory of Calculus > instead. > ************************ > David C. Ullrich >i feel that everytime i look at a new series problem i am starting >from the beginning and struggle to finish it. is there a general >approach to simple series? >for example, why does the divergence of Sum[an] imply the divergence >of Sum[an/(1+an)]. > It doesn't. (You're also given that an >= 0 in the exercise, right? > _Then_ it does. Hint: either an tends to 0 or an does not tend to 0.) >it seems to me that self study from rudin's book is nearly impossible. > Possibly you need to read it much more slowly and carefullly. > Or possibly you're not ready for a book that makes the sort > of demands on the reader that Rudin does - you could try > something like Ross Elementary Analysis: the Theory of Calculus > instead. > ************************ > David C. Ullrich Indeed, i fully agree with you. if it weren't for the fact that i dropped 70 bucks on this book i would try something else, but i am going to stick with it. i bought this book at the same time a bought herstein's topics in algebra, but that book has been going much smoother. i am nearly done with the chapter on vector spaces and i have done nearly every excercise in the book. === Subject: Re: How to define a function to be smooth? >> >>Hey all >>When we say a function f(t) is smooth, does this mean that >>f has infinite differentials with respect to t? >> >> That's _often_ what it means - sometimes it means less than that. >> >> David C. Ullrich >I thought a function was smooth at a point a if there was a >neighborhood of a in which f could be represented as a Taylor series >about a. > That's certainly a _type_ of smoothness, and I suppose there could > be a context in which people define smooth to mean exactly that. > But it's not the usual definition - the usual terminology for > functions with this property is analytic, or sometimes > real-analytic. >So, I had a wrong definition. >Artur > ************************ > David C. Ullrich I have seen smooth used for once continuously differentiable, twice continuously differentiable, C^infty and presumably anything in between, so I agree with Gottschalk on this. I had not seen it used for analytic before. In any case, it is not a term that should be used without a precise definition. === Subject: Re: cardinality of monotone set-functions Content-Length: 1407 Originator: rusin@vesuvius >> Let X be a finite set. A set-function on X is a >> function f: 2^X rightarrow 2^X. f is monotone >> if A subset B implies f(A) subset f(B). >> What is the cardinality of the set of monotone >> set-functions on X ? >> (Actually, I'm looking for the cardinality of >> the set functions such that f(A)subset A for all A.) >One clarification about f(A)subset A: I meant the >cardinality of the monotone set-valued f such that >f(A)subset A. OK: for any y in X, G_y = {A in 2^X: y in f(A)} must 1) be an upset, i.e. if A is in G_y and A subset B then B is in G_y 2) contain only sets A that contain y. Conversely, given G_y for each y satisfying (1) and (2), f(A) = {y: A in G_y} defines a monotone set-valued function f such that f(A) subset A. The possible choices for G_y correspond to the upsets of X {y}. So the number of such possible choices is N(|X|-1) in the notation of my previous posts on this subject in sci.math, where N(n) are the Dedekind numbers, sequence A000372 of the On-line Encyclopedia of Integer Sequences. And thus your answer is N(|X|-1)^|X|. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel === Subject: Solution to an inequation Hello everyone, Here is an important equation I am encountering. b <= x * ( (1-x)^a ) Solve for x in terms of a and b. I can find x where x * ( (1-x)^a ) is maximised in terms of a. But what I really need is the range of x in terms of a and b where the inequation holds true. The grapf for x * ( (1-x)^a ) will look something like this - | | | / | _ / | / / | / / | / / ---|-------|------------------------------------ / | 1 / | Any help will be greatly appreciated. Manan. === Subject: Re: Solution to an inequation >Here is an important equation I am encountering. >b <= x * ( (1-x)^a ) Inequality, not equation. >Solve for x in terms of a and b. Unlikely to have a closed-form solution in general. >I can find x where x * ( (1-x)^a ) is maximised in terms of a. That can help tell you whether there is any solution in 0 < x < 1. > But what I >really need is the range of x in terms of a and b where the inequation holds >true. >The grapf for x * ( (1-x)^a ) will look something like this - > | > | > | / > | _ / > | / / > | / / > | / / >---|-------|------------------------------------ > / | 1 >/ | Maybe... I suppose you are taking a > 0? The part for x > 1 may or may not be there at all. If a = 2m/n is a rational number with even numerator in lowest terms, (1-x)^a can be interpreted as a positive real number when x > 1 (although it's not the principal branch). If a = m/n is a rational number with odd numerator and denominator, it can be a negative real. If a = m/(2n) is a rational with even denominator in lowest terms, it's not real at all. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel === Subject: Re: naive geometry questions > | at 08:00 PM, Elaine Jackson said: > | >1) Are there surfaces that can be bent to each other (the way a flat > | >piece of paper is bent to a cylinder) but not to anything flat? > | What do you mean by surface and what do you mean by bent to each > | other? Are you limiting your question to surfaces in Euclidean > | 3-space? > YES > | >2) Besides planes and spheres, is there any other surface S such > | >that a piece of S can be moved around adlibitum while each of its > | >points remains in contact with S? > | If I understand your question, the cylinder and the torus are > | examples. > rotate it as well. Indeed, if we're talking only about embeddings in 3-space, the torus fails to be an example at all. === Subject: Ellipse Area I'd like to calculate the area of an ellipse using polar coordinates for the purpose of finding arc areas about the center. Given semi-axes of length 'a' and 'b', I've obtained an area formula of (PI/2)*(a^2 + b^2) for the area of the entire ellipse, and I've found at least one website supporting my answer. However, most sources calculate the area to be (PI)(ab) using cartesian coordinates. Where is my mistake? === Subject: Re: Ellipse Area > I'd like to calculate the area of an ellipse using polar coordinates > for the purpose of finding arc areas about the center. Given > semi-axes of length 'a' and 'b', I've obtained an area formula of > (PI/2)*(a^2 + b^2) for the area of the entire ellipse, and I've found > at least one website supporting my answer. However, most sources > calculate the area to be (PI)(ab) using cartesian coordinates. As Robert Israel confirmed, the area of the entire ellipse is Pi*a*b. But reading your first sentence makes me guess that what you really want is the area of a central sector of an ellipse, that is, the area of a sector having its vertex at the center (rather than a focus) of the ellipse. If my guess is correct, then the following copy of an old response of mine should answer your question. David -------------------------------------- > How can I work out the area of a segment/wedge of a non-rotated > ellipse? The starting and ending angles of the wedge sides are known > as is the width and height of the ellipse. I'm guessing that by wedge you really mean sector. (Also note that a segment would be something quite different: a region between the ellipse and one of its chords.) So assuming that you want the area of a sector of the ellipse given by (x/a)^2 + (y/b)^2 = 1 and that the radii (the straight sides of the sector) are at angles of S and T (with T > S), the area is ab/2 ( T - Arctan(sin(2T)/(cos(2T)+(b+a)/(b-a))) -S + Arctan(sin(2S)/(cos(2S)+(b+a)/(b-a))) ) given by (working in polar coordinates) 1/2 integral of r^2 dt from t = S to T, where r^2 = 1/((cos(t)/a)^2 + (sin(t)/b)^2). === Subject: Re: Ellipse Area >I'd like to calculate the area of an ellipse using polar coordinates >for the purpose of finding arc areas about the center. Given >semi-axes of length 'a' and 'b', I've obtained an area formula of >(PI/2)*(a^2 + b^2) for the area of the entire ellipse, and I've found >at least one website supporting my answer. You can find websites with all kinds of nonsense. One way to see that your formula is wrong is to imagine the limit as b -> 0 with a = constant. The ellipse degenerates into a line segment with area 0. But your formula goes to (PI/2)*a^2. > However, most sources >calculate the area to be (PI)(ab) using cartesian coordinates. Where >is my mistake? Where was it last time you saw it? Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel === Subject: Re: Do finite strings over a countable alphabet form a countable set? >How could you have a rational discussion with ZZBunker? :-) > But what's wrong with giving someone who posts a homework problem the > wrong answer he deserves? Well it's automatically wrong when Chapman's the teacher. Since the only posters on sci.math who are more Van Newmann Challenged than Harris are the Cantor Wannabees: Chapman, Goedel, and Gauss Inc. Or as Goedel might say: I once say a plus, then I saw a minus, but I've need seen a lower intelligence score than that the Chapman and Feymann Spelling Bee Club who think their Turings. And as Gauss would say: The World is round except where Einstone is involved. Then is it's all Britian, France, and Chapman Real Estate LTD. against the more advanced civilizations on Mars. > John Savard > http://home.ecn.ab.ca/~jsavard/index.html === Subject: Re: A 3rd Grade Word Problem---HELP > complete solution sent to the e-mail address you so courteously >provided. Dude, the flippant response is funny once or twice. Every single time it's not funny. If you don't want to respond, don't. But this isn't Open Mike Nite at the Laff Shack, and if it were, you'd be booed off the stage because it's the same exact joke every single #@!$# time. Doug === Subject: Re: Continuum I am a novice to the redneck notations accepted in the US math world, and intend to stay this way. To cater whims of philistines, card(N):=aleph_0 and card(Z):=2**aleph0 > yours truly can promptly word it into a hypothesis : there exist an _infinite_ (technically wrong term, i know) amount of > non-interequivalent axioms, each of them completing ZF; cardinality of this > _set_ of axioms is strictly greater then card(N) and strictly lower then > card(Z). Not of much use without a proof, which took inventing another level of > abstraction, but who cares ? yours, > amateur > The proof of this one should be quite interesting, considering that > card(N)=card(Z) > If N and mean , as usual, the set of naturals and the set of > integers, respectively, such a proof would indeed be interesting. > Possibly something that Ross Findlayson or James Harris might assent > to. === Subject: Re: Continuum > I am a novice to the redneck notations accepted in the US math world, and > intend to stay this way. To cater whims of philistines, card(N):=aleph_0 and > card(Z):=2**aleph0 Did you ever tell the philistines what set you think the symbol Z stands for? === Subject: Re: Uncle Al is Sadistic . > ...correlation between SAT scores and grades > ...you're comparing apples and oranges That was when education worked. > Classes are only foreplay. > I Ace'd AP calculus and physics...Ace'd all the technical courses. > But the most ironic part was I didn't feel I was being better educated > I always viewed it as an educational playground. > The most ironic thing is now, 20 years later, > I'm now interested in some of the non-objective subjects I > originally hated, such as history and politics. > Bruce Bruce, you described a very general evolution of interests vs. age. Interests do change in time. That's for sure. Here is a list by the tooth of time as a function of the most common complaints, which are from: 15 - 25: for not getting enough laid, paid or loaded 25 - 35: for not having enough $$ but a greedy stupid boss/spouse 35 - 45: for not having enough $$ but the burden of useless underlings/kids 45 - 55: for not having enough $$ but a ed job/boss/underlings/kids/spouse 55 - 65: for not having achieved enough/any of their hoped for < 55 objectives 65 - to the end..... that the world is *still* not running according to them. Give it a +/- xz years each to make it look scientific. ahahahaha.......ahahahahanson === Subject: Re: Uncle Al is Sadistic . SNIP.... > I didn't hit this so called wall until graduate school at Stanford. >Ace'd AP calculus and physics in high school (wasn't allowed into AP >english, more on this later). Ace'd all the technical courses >throughout my undergraduate years. Graduated Summa Cum Laude Decided >to go to graduate school at Stanford and was dropped in with the cream >of the crop. Boom!!! But the most ironic part was I didn't feel I >was being better educated, it was just harder to get a good grade. It >struck me that Stanford was more of a filter than an educator. Aha, finally some insight. Remember the old prof who said there were 3000 students and 600 spots, so he saw his job as failing 2400 students, not trying to educate them. Lots of folk here thought that was a good thing. josh halpern === Subject: Re: Uncle Al is Sadistic . > SNIP.... > I didn't hit this so called wall until graduate school at Stanford. >Ace'd AP calculus and physics in high school (wasn't allowed into AP >english, more on this later). Ace'd all the technical courses >throughout my undergraduate years. Graduated Summa Cum Laude Decided >to go to graduate school at Stanford and was dropped in with the cream >of the crop. Boom!!! But the most ironic part was I didn't feel I >was being better educated, it was just harder to get a good grade. It >struck me that Stanford was more of a filter than an educator. > Aha, finally some insight. Remember the old prof who said there > were 3000 students and 600 spots, so he saw his job as failing > 2400 students, not trying to educate them. Lots of folk here thought The prof failed nobody. He simply skimmed the top of the curve to allocate precious resources to the most qualified candidates. Of the 1200+ students in that first day of majors organic, about 17 managed to graduate with BS/Chem degrees three years later. I think maybe one third of those did medicine or something other than chemistry as such. Moo U didn't graduate all that many BS chemists, but the ones it did were really hot stuff. The ones that didn't make it still had time to do something they could successfully compete in. Moo U's output has increased and the quality has declined. We've degenerated from degree holders who could do chemistry to those who have been educated in doing it - process has replaced product. Why should anybody be surprised that I can fabricate Pyrex glassware? I'm a chemist, I can blow glass. It's a proper and necessary job skill unless you want to be slipping the real glassblower sixpacks or fifths for priority. Besides, how do you pay for books if not by blowing Pyrex shotguns, bongs, and hookas? -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Uncle Al is Sadistic . >> > >>SNIP.... >> > I didn't hit this so called wall until graduate school at Stanford. >Ace'd AP calculus and physics in high school (wasn't allowed into AP >english, more on this later). Ace'd all the technical courses >throughout my undergraduate years. Graduated Summa Cum Laude Decided >to go to graduate school at Stanford and was dropped in with the cream >of the crop. Boom!!! But the most ironic part was I didn't feel I >was being better educated, it was just harder to get a good grade. It >struck me that Stanford was more of a filter than an educator. > >>Aha, finally some insight. Remember the old prof who said there >>were 3000 students and 600 spots, so he saw his job as failing >>2400 students, not trying to educate them. Lots of folk here thought >> >The prof failed nobody. He simply skimmed the top of the curve to >allocate precious resources to the most qualified candidates. Of the >1200+ students in that first day of majors organic, about 17 managed >to graduate with BS/Chem degrees three years later. I think maybe one >third of those did medicine or something other than chemistry as such. Sure, the I don't fail you you fail you bull. He didn't educate anyone either. He didn't go out and get more resources for more students. He stood up their and pontificated, and you little lap dog danced to his music and were SO proud that you earned his approval. This constitutes education in the Alocracy. >Moo U didn't graduate all that many BS chemists, but the ones it did >were really hot stuff. The ones that didn't make it still had time to >do something they could successfully compete in. Moo U's output has >increased and the quality has declined. We've degenerated from degree >holders who could do chemistry to those who have been educated in >doing it - process has replaced product. In the interest of comity I will refrain from commenting on these assertions. >Why should anybody be surprised that I can fabricate Pyrex glassware? >I'm a chemist, I can blow glass. It's a proper and necessary job >skill unless you want to be slipping the real glassblower sixpacks or >fifths for priority. Besides, how do you pay for books if not by >blowing Pyrex shotguns, bongs, and hookas? So, I'm a physicist I can blow glass, run a lathe and a mill, build electronics. Frankly everyone has to have a few hobbies. josh halpern === Subject: Re: Key Core Error Argument > ... > Actually it is, as Dik Winter is trying to find a way that 7, 7 and 22 > become 1, 1 and 22 based on a *varying* x. > > That is, he's trying to make the change in *constants* dependent on a > variable. > > Eh? > > I want readers to understand that his behavior is crank, while I guess > many of you may sympathize with his strong desire for me to be wrong, > remember, it's not about people as the math didn't just decide to > change. > > What you should be sympathetic to, is the truth. > > You claim that having P(x) = g1(x).g2(x).g3(x) with g1(0) = g2(0) = 7 > and g3(0) = 22; the *only* way to divide P(x) by 49 is by dividing g1(x) > and g2(x) by 7 and g3(x) by 1. Because now g1(0)/7 = 1, g2(0)/7 = 1 and > g3(0)/1 = 22. > > I claim there are other ways to do that. Have w1(x), w2(x), w3(x), such > that w1(0) = w2(0) = 7, w3(0) = 1, w1(x).w2(x).w3(x) = 49. Because now > g1(0)/w1(0) = 1, g2(0)/w2(0) = 1 and g3(0)/w3(0) = 22. > > Where do I change constants? > The basic algebra is that if you have 7 and divide it by *something* > and get 1, then you divided by 7. Yup. When x = 0 I divide by 7. When x != 0 I divide by other numbers. I need only get 1 when x = 0, remember? > And no tricks will change that fact, and it's simply crank behavior to > try and act like there's some complicated way you can divide 7 to get > 1, without actually just dividing it by 7. You are plain stupid. You have (5 a(x) + 7) with constant term 7. You want to divide by something such that the constant term becomes 1. The constant term of (5 a(x) + 7)/w(x) is 1 for *every* function w with constant term 7. And so I get 1 by dividing 7 by 7. Or do you claim that 1 is *not* the constant term of (5 a(x) + 7)/w(x) or that the constant term of w is *not* 7 when w(0) = 7? > I noted that the trick of using ratios of functions to try and hide > things just doesn't work, yet the crank Dik Winter is still back > trying to find a way to divide 7 as a *variable* rather than just > divide it by 7 to get 1. Eh? Where do I treat it as a variable? I only write plain 7 when I mean 7. And I need only get 1 when x = 0. > I think it satisfying to *finally* be able to show that Dik Winter and > the posters like him are indeed the ones who are cranks, as here all > you need to be able to realize is that dividing 7 by *something* to > get 1 means you divided it by 7. Yup, exactly what I did. > That's it. If you're still doubting, you're a crank as well. Shall I reverse the compliment? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Key Core Error Argument >> ... >> Actually it is, as Dik Winter is trying to find a way that 7, 7 and 22 >> become 1, 1 and 22 based on a *varying* x. >> >> That is, he's trying to make the change in *constants* dependent on a >> variable. >> Eh? >> I want readers to understand that his behavior is crank, while I guess >> many of you may sympathize with his strong desire for me to be wrong, >> remember, it's not about people as the math didn't just decide to >> change. >> >> What you should be sympathetic to, is the truth. >> You claim that having P(x) = g1(x).g2(x).g3(x) with g1(0) = g2(0) = 7 >> and g3(0) = 22; the *only* way to divide P(x) by 49 is by dividing g1(x) >> and g2(x) by 7 and g3(x) by 1. Because now g1(0)/7 = 1, g2(0)/7 = 1 and >> g3(0)/1 = 22. >> I claim there are other ways to do that. Have w1(x), w2(x), w3(x), such >> that w1(0) = w2(0) = 7, w3(0) = 1, w1(x).w2(x).w3(x) = 49. Because now >> g1(0)/w1(0) = 1, g2(0)/w2(0) = 1 and g3(0)/w3(0) = 22. >> Where do I change constants? >The basic algebra is that if you have 7 and divide it by *something* >and get 1, then you divided by 7. What never fails to amaze me is how you pick up on the one point over which no one has any objection and announce it as an example of how unreasonable mathematicians can be. Alan -- Defendit numerus === Subject: Re: Key Core Error Argument Note: snipping liberally to keep the major points. Feel free to look back in the thread. >> Define three functions w1(x), w2(x), w3(x), such that w1(x).w2(x).w3(x) = 49 >> for all x, and w1(0) = w2(0) = 7, w3(0) = 1. Now >> >> (5 a1(x)/w1(x)+7/w1(x))(5 a2(x)/w2(x)+7/w2(x))(5 b3(x)/w3(x)+22/w3(x)) = >> 300125 x^3 - 10375 x^2 - 360 x + 22 >> So why is that *not* a possibility? If I understand your terminology >> correctly (you still have *not* defined the concept constant term >> as you use it), the constant terms of the three factors are 1, 1 and 22. >> >> Let's say you have f(x)/g(x) in the ring of algebraic integers. That >> is, both f(x), and g(x) give algebraic integers for algebraic integer >> x. Then necessarily, you have *another* algebraic integer function >> I'll call h(x). >> >> Understand? >>So, h(x) = f(x)/g(x)? > Given algebraic integer functions f(x), and g(x), where f(x)/g(x) is > an algebraic integer as well then obviously you have yet another > algebraic integer function, which I've called h(x) for this disussion. > It's not even a big leap to realize that Dik winter, and strange to > question. > Your reply should have been, yeah, that's right. >> Now then, if you have h(x), then you can set x=0 to get the constant >> term. >>Ok. So you define the constant term of a function as the value of the >>function when the argument is 0. True? That is what I expected. > That's the definition! That's why it's constant!!! [snip rest of yes] >>1. f(x) = (5 a1(x) + 7), g(x) = w1(x), h(x) = f(x)/g(x). >> h(0) = f(0)/g(0) = 7/7 = 1 >>2. f(x) = (5 a2(x) + 7), g(x) = w2(x), h(x) = f(x)/g(x). >> h(0) = f(0)/g(0) = 7/7 = 1 >>3. f(x) = (5 b3(x) + 22), g(x) = w3(x), h(x) = f(x)/g(x). >> h(0) = f(0)/g(0) = 22/1 = 22 >>So the constant terms of the functions work out as 1, 1 and 22. >>This is even independent of whether the h's are algebraic integers >>or not. [James' comment snipped] > I said: >> Otherwise you'd have a backdoor to making numbers like 7, that are >> constants, into variables, which would be mathematically inconsistent. >> >> So if you have 7/w_1(x), and it gives algebraic integers for any >> algebraic integer x, then you have some function h(x) = 7/w_1(x), >> which STILL has to have a constant term of 1, found by checking at >> x=0. >>Note that I maybe do not have that 7/w1(x) is an algebraic integer, I >>do have that (5 a1(x)/w1(x) + 7/w1(x)) is an algebraic integer, and as > Here he tries his latest trick which is to NOW claim that 7/w1(x) is > not necessarily an algebraic integer. This isn't new. It's just that you've finally understood the point. Maybe. > However, remember that 49 is being divided off, so w_1 w_2 w_3 = 49. Also not new. > Think about why the poster tried to float such a puzzling assertion > past you. Perhaps because it is true? -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Key Core Error Argument before I retire, also from lurking: can anyone posit a rough estimate of the number of items that James Harris has posted over the years -- that is, after running-off from a more or less simple query, like Snora B. Barons?... I mean, maybe he gets paid for the number of those, instead of per character of his would-be correspondents (we, the few, the proud, les Beauxeaux Synonyme .-) > Note that ultimately the proof relies on 22 NOT having 7 as a factor, > and constant terms like 7 and 22, being constant, and not variables > dependent on x, which may seem like odd things to emphasize, but I've > One of the factors of your polynomial P(x) is > 5*b3 + 22. > Here I think we all agree that b3 is a function of x, and > that b3(0) = 0. > We all agree that 7 is coprime to 22. > We all know that you can have two numbers, r and s for example, > each of which is coprime to 7, but their sum is not coprime to 7: > for example, r + s = 3 + 4. > Now consider > 5*b3(1) + 22. > Again, we do know the constant term is 22, and 22 is coprime > to 7. It is constant, and no one is saying here that it changes > in any way. > Now let r = 5*b3(1) and s = 22. > Thus s is coprime to 7. We don't know about r. > So now tell us why r + s is coprime to 7. --ils duces d'Enron! http://www.wlym.com/covers/7101contents.png === Subject: Re: Key Core Error Argument > before I retire, also from lurking: > can anyone posit a rough estimate of the number of items > that James Harris has posted over the years -- that is, > after running-off from a more or less simple query, > like Snora B. Barons?... I mean, > maybe he gets paid for the number of those, > instead of per character of his would-be correspondents (we, > the few, the proud, les Beauxeaux Synonyme .-) I am now at 2710, but I am only around in this newsgroup since january 1988. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Key Core Error Argument we, the dedicated followers of JSHism, have to belabor the obviously painful: A proof starts with a statement, itself a link, proceeding by a chain of steps, any one of which can be false, but which break in the chain may be annulled by a surreptitious advent of a new, true link, with further true steps leading to a conclusion that is true.... unless you made *another* mistake, before the final link (you missed the green, dood !-) > A proof begins with a truth and proceeds by logical steps to a > conclusion which then MUST be true. > What if you present a proof, but because people don't like it, they > decide to talk about something else and claim it invalidates your > proof? --ils duces d'Enron! http://www.wlym.com/covers/7101contents.png === Subject: Re: Key Core Error Argument whoah, long division; now we're getting some where! > The basic algebra is that if you have 7 and divide it by *something* > and get 1, then you divided by 7. --ils duces d'Enron! http://www.wlym.com/covers/7101contents.png === Subject: Re: some coin tossing > If I toss a fair coin n times, what is the probability that the outcome of > my experiment will contain m consecutive heads (m This is NOT a binomial distribution > If the exact probability is difficult to compute, can we have a good upper > bound on the probability? > A trivial upperbound is (n-m+1)(0.5^m) > Any help will be greatly appreciated. > Manan. I recall a very nice discussion of this and, in fact, much more general questions, in a book called Combinatorial Algorithms on book that deals with those quetions is, I think, by Apostolico himself (though I may be wrong here). The general problem is this: given an alphabet S (in your case: {0,1}) and a word W from that alphabet (in your case: the word 1111...1 of length m), how many words of length n over the alphabet S are such that they avoid the word w, meaning that they do not contain it as a subword? Some people find it surprising that the answer depends on the structure of the word w, not just its length. by structure I mean its self correlations, namely the various ways of shifting the word to the right and getting a complete match in the overlap. For instance, the number of words that don't contain 11 is not the number of words that don't contain 10. It's not really surprising if you think about. Here's the gist of how this is done - I'm trying to recall this from memory and don't have time to look into the details, if you can't figure it out let me know and I'll work on it a bit. Let a_n be the number of words of length n that do not contain w, and let b_n be the number of words of length n that contain w only once as a suffix (namely w appears only at the rightmost portion of the word). Let f(x) and g(x) be the generating functions of a_n and b_n respectively, namely f(x) = sum a_n x^n n=1..infinity g(x) = sum b_n x^n n=1..infinity Lastly, define a polynomial p(x)=sum c_k x^(m-k) where c_k is the kth correlation coefficient of the word w, namely c_k is 1 iff shifting w k places to the right and looking at the ovelap, you get a perfect match. The trick is now to notice that you can write two linear equations in f and g, and solve them. The polynomial p will appear as a constant in these equations. To do this, you need to consider various possibilities of what happens when letters are appended to the end of words, or when w itself is appended as a suffix to a word. This is where I need to work on the details (and I do realize this is where the heart of the matter is!), but I'm sure it's not too complicated and does indeed lead to the desired equations on f and g. I do remember two very nice related facts. One has to do with the expected time of coin tosses until a certain pattern emerges. How main times do you need to toss a coin until 111 comes out on consecutive tosses? How about 101 and 100? The answer is simply 2p(2) where p(x) is the polynomial defined above. For instance, you have to toss a coin 8 times (on average) to get 100 (namely heads, tails, tails), but 14 times to get 111, and 10 times to get 010. The other nice result has to do with a coin tossing contest. I let you choose a pattern of length 3, then I choose a different pattern of length 3, and we toss a coin until one of the patterns emerges as consecutive results - whichever pattern comes up first determines the winner. The cute thing: whatever pattern you choose, I can choose one that gives me a chance of winning *strictly* higher than 50%. And if you choose my pattern next time, I can again choose a better pattern, etc... A non-transitive game! Sorry for the mostly unrelated rant, I hope at least the reference will be useful. Alan === Subject: Re: Aut(Petersengraph) > Hello everyone. > How can one show directly, without using any theorems about automorphisms of > Kn and L(Kn), that Aut (L(K5)) isomorph to S5? Or how can one prove these > theorems? > sasha.mal The Petersen graph may be represented as the Kneser graph K(5,2). This is the graph with vertex set consisting of the 10 two element subsets of {1,2,3,4,5}. An edge is a pair {A,B} such that A intersect B = empty set. automorphism of the graph. === Subject: Re: Which side of 2D line is Point On > Any help with the following would be appreciated. > I have a set of points in the plane and want to sort them into > cyclical order (say anti-clockwise) about another given point P0. > The naive way would to have a function F1(point PN) which takes one > argument - a point PN. The function makes the vector from P0 to PN and > returns the angle this vector makes with the positive x axis. Rather than working with angles, why not use the slope of the line through P0 and PN? You'll also have a concern regarding which half-plane (or quadrant, depending on the starting vector) a point is in. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Which side of 2D line is Point On > I am using this function to determin this: > (P1.x - Po.x)*(P2.y - Po.y) - (P2.x - Po.x)*(P1.y - Po.y) > It returns < 0.0, > 0.0 or 0.0 depending on the side P2 is from the > line defined in the sense Po p1 > Well, this works if P2 is entirely outside of the point set but not if > P2 is interior to the set. > Is this the correct function, will it always return >0.0 if P2 is on > the left of Po P1. If so I should expect it to work if P2 is interior > to the point set. I'm not sure why you think the above fails in certain cases; it always does what you want. Another approach to your problem might be: Take Po to be the origin for convenience. Let g(x,y) = x/sqrt(x^2+y^2). Compute g at all your points. Then list the points with y >= 0 as P1, P2, ..., Pm so that g(P1) > g(P2) > ... > g(Pm). List the points with y < 0 as Q1, ..., Qn so that g(Q1) < ... < g(Qn). Then the points P1, P2, ..., Pm, Q1, ..., Qn will then be in counterclockwise order. === Subject: Re: hypercube/tesseract help === >Subject: hypercube/tesseract help >I am not at all mathematical, please keep in mind when replying. How could I >find hypercube images for hebrew letters? Does anyone know how to create them? >Is it even possible? I'm not quite sure what the hell you are talking about (i.e. how do hypercubes relate to hebrew letters?)... but here's an answer for you. Any images of hypercubes that you might find are not going to be worth much because you can't draw a hypercube on a 2-d computer screen, or blackboard, or whatever. Here's a mental picture for you: Imagine a red cube. now imagine that on each corner of the cube there is a loop of string (eight loops total, one at each corner). Imagine yourself standing right on one of the corners. You walk all the way around the loop, and as you walk the cube changes color from red to blue... Now you are back in the same place on the cube but the color is different. You walked along the color dimension which is independent of the other three. Just like you could have walked along the x direction without changing y or z or the color. How's that for a hypercube image?... the only problems with that image is that really the loop is all at the same point...and the loops aren't just on the corners, they are everywhere on the cube. adam === Subject: torque T = r x F and basic tensors Here are 2 questions which perhaps are related: 1. Why is there a cross product in the definition of torque T = r x F ? In Feynman's Lectures on Physics, he derives torque in the xy plane as T = xF_x - yF_y (where _x means subscript) in Vol 1 Ch 18. But in Ch 20 he generalizes it into 3 dimensions, and I can't quite follow that, especially how we would know, a priori, to permute the x_1, x_2, x_3 symbols in the way for a right handed co-ord system. (Of course I see that r x F in 3 dimensions reduces to what Feynman gives in the xy plane). 2. How is torque a tensor? Following Feynman vol 2 ch 31, a tensor is a linear map A relating 2 vector quantities, for example p = Ae (where I am using p for the polatization vector of a dielectric and e for the electric field vector) and hence if we change the basis of R^3, A must transform as A' = Q^{-1} A Q where V is the new ordered basis, U is the old and V = UQ^{-1} I can see Feynman gets the 3 by 3 anti-symmetric matrix T_{ij} = x_i F_j - x_j F_i i,j = 1,2,3 How would this matrix object ever be used for torque? If it has any relevance, I can derive the formula Q( a x b ) = 1/(detQ) ( Qa x Qb ) where Q is the matrix of the change of basis, but I'm not quite seeing how that matches the A' = Q^{-1} A Q form. Jeff === Subject: Re: JSH: It begins >Ha! The chickens come home to roost. Your answer to every refutation and counter-example, is to pump up >the volume. Is 'volume' the tool of truth? One of the graduate student instructors that I was friends with at Colorado really couldn't teach. His teaching method was essentially this - he'd teach something, and if someone didn't understand, he would repeat himself slower and LOUDER. Then he'd look at them and see if that helped. Perhaps James Harris is this instructor? Doug === Subject: Re: JSH: It begins > As for my posting, it may increase dramatically if I see that pumping > up the volume is effective. > So there may be many MORE posts from me on sci.math, not fewer. > James Harris Spoken like a sullen, petulant juvenile. Pumping up the volume has nothing to do with mathematical correctness. It is behavior more appropriate for an ape than a man. Did you learn that shouting louder was an algebraic method for proving something? -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: A Question for James Harris >There is a searchable on-line database of mathematicians, >_The Mathematics Genealogy Project_, and the homepage is at > http://genealogy.math.ndsu.nodak.edu/ This site really needs to be promoted and used more; I love checking out my lineage from time-to-time. I'm a descendent of a long line of American mathematicians; on the European side, I'm a descendent of Poisson, Lagrange, Euler, and both Bernoullis. I surely won't be as famous as any of them (not in math, anyhow), but it's neat to see. Doug === Subject: Re: A Question for James Harris ... > That's a rather easy falsehood to correct. > Like, see > http://www.megasociety.net/NoesisHighlights.html I also want to publish a paper. How much does it cost to publish a paper there? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: A Question for James Harris >>James: >>Something's been bothering me. The world population >>is now approximately 6.3 billion. In seven years you >>have not convinced even ONE of them about ANYTHING. >>The thing that is a source of puzzlement to me is that >>in view of this obvious fact, is why have you not just >>once sat down and said to yourself, Hmmm. Maybe. >>Just possibly. Something's wrong here. > That's a rather easy falsehood to correct. > Like, see > http://www.megasociety.net/NoesisHighlights.html You've been published in a journal that objects to the peer review system in most scientific/mathematical journals. This proves what? -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Greek Alphebet ... > Thus, alpha comes from aleph, which means Ox. And gamma comes > from gimel... so one can look at a book on the history of the > alphabet. And the 22 letters of the Hebrew alphabet are alleged to > have meaning on other levels too. This is wrong. Both alphabets are descendants from the Phoenician alphabet. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Greek Alphebet > Thus, alpha comes from aleph, which means Ox. And gamma comes > from gimel... so one can look at a book on the history of the > alphabet. And the 22 letters of the Hebrew alphabet are alleged to > have meaning on other levels too. Sounds like a case of putting two and two together to make twenty-two : ) === Subject: Re: Attacking a proof is attacking yourselves I've talked about the basic truths they're attacking, about how 7 and > 22 are NUMBERS and not variables, as they're constant. Yeah, but what about 17? I think I saw 17 vary a little bit one time. > Gotta > go now. Time for my shots. > That's because 17 is a more random number than 7 or 22. > - Randy > Yeah, but really, I swear that one time while a had my back > turned, 17 was sneakily varying a little bit. But when I > turned around really quickly to try and catch it, it had sneakily > changed back to 17 again. Goddam sneaky variable constants. Driving me nuts. === Subject: Re: Attacking a proof is attacking yourselves I've talked about the basic truths they're attacking, about how 7 > and > 22 are NUMBERS and not variables, as they're constant. Yeah, but what about 17? I think I saw 17 vary a little bit one time. > Gotta > go now. Time for my shots. That's because 17 is a more random number than 7 or 22. - Randy > Yeah, but really, I swear that one time while a had my back > turned, 17 was sneakily varying a little bit. But when I > turned around really quickly to try and catch it, it had sneakily > changed back to 17 again. > Goddam sneaky variable constants. Driving me nuts. And those constant variables are just too much. A man has to believe in something. I believe I'll have a little drink. === Subject: Re: JSH: Attacking a proof is attacking yourselves Richard Gere, here, for Gerbil Viagra (tm). > Being called a ing piece of dog is minor in comparison to the > fear of what happens if people know that my mathematical discoveries --les ducs d'Enron! http://www.wlym.com/antidummies/part17.html === Subject: Re: JSH: Attacking a proof is attacking yourselves >Reality check: if people found out your mathematical discoveries were >valid nothing bad would happen to David Ullrich whatsoever. Untrue. He might be suddenly crushed to death by a pig landing at the end of a transatlantic flight.